Introduction to Modern Economic Growth

Introduction to Modern Economic Growth INTRODUCTION TO MODERN ECONOMIC GROWTH INTRODUCTION TO MODERN ECONOMIC GROWTH DARON ACEMOGLU PRINCETON UNIVERSITY PRESS Princeton and Oxford Copyright 2009 by Princeton University Press Published by Princeton University Press 41 William Street Princeton New Jersey 08540 In the United Kingdom Princeton University Press 6 Oxford Street Woodstock Oxfordshire OX20 1TW All Rights Reserved Library of Congress CataloginginPublication Data Acemoglu Daron Introduction to modern economic growth Daron Acemoglu p cm Includes bibliographical references and index ISBN 9780691132921 hardcover alk paper 1 Economic development 2 Macroeconomics I Title HD75A24 2009 3389dc22 2008038853 British Library CataloginginPublication Data is available This book has been composed in Times Roman and Myriad using ZzTEX by Princeton Editorial Associates Inc Scottsdale Arizona Printed on acidfree paper pressprincetonedu Printed in the United States of America 1 3 5 7 9 10 8 6 4 2 To Asu for her unending love and support Contents Preface xv Part I Introduction Chapter 1 Economic Growth and Economic Development The Questions 3 11 CrossCountry Income Differences 3 12 Income and Welfare 7 13 Economic Growth and Income Differences 9 14 Origins of Todays Income Differences and World Economic Growth 11 15 Conditional Convergence 15 16 Correlates of Economic Growth 18 17 From Correlates to Fundamental Causes 19 18 The Agenda 21 19 References and Literature 23 Chapter 2 The Solow Growth Model 26 21 The Economic Environment of the Basic Solow Model 27 22 The Solow Model in Discrete Time 34 23 Transitional Dynamics in the DiscreteTime Solow Model 43 24 The Solow Model in Continuous Time 47 25 Transitional Dynamics in the ContinuousTime Solow Model 51 26 A First Look at Sustained Growth 55 27 Solow Model with Technological Progress 56 28 Comparative Dynamics 67 29 Taking Stock 68 210 References and Literature 69 211 Exercises 71 Chapter 3 The Solow Model and the Data 77 31 Growth Accounting 77 32 The Solow Model and Regression Analyses 80 33 The Solow Model with Human Capital 85 vii viii Contents 34 Solow Model and CrossCountry Income Differences Regression Analyses 90 35 Calibrating Productivity Differences 96 36 Estimating Productivity Differences 100 37 Taking Stock 105 38 References and Literature 106 39 Exercises 107 Chapter 4 Fundamental Determinants of Differences in Economic Performance 109 41 Proximate versus Fundamental Causes 109 42 Economies of Scale Population Technology and World Growth 112 43 The Four Fundamental Causes 114 44 The Effect of Institutions on Economic Growth 123 45 What Types of Institutions 136 46 Disease and Development 137 47 Political Economy of Institutions First Thoughts 140 48 Taking Stock 140 49 References and Literature 141 410 Exercises 143 Part II Toward Neoclassical Growth Chapter 5 Foundations of Neoclassical Growth 147 51 Preliminaries 147 52 The Representative Household 149 53 Infinite Planning Horizon 156 54 The Representative Firm 158 55 Problem Formulation 160 56 Welfare Theorems 161 57 Proof of the Second Welfare Theorem Theorem 57 168 58 Sequential Trading 171 59 Optimal Growth 174 510 Taking Stock 176 511 References and Literature 176 512 Exercises 178 Chapter 6 InfiniteHorizon Optimization and Dynamic Programming 182 61 DiscreteTime InfiniteHorizon Optimization 182 62 Stationary Dynamic Programming 185 63 Stationary Dynamic Programming Theorems 187 64 The Contraction Mapping Theorem and Applications 190 65 Proofs of the Main Dynamic Programming Theorems 194 66 Applications of Stationary Dynamic Programming 201 67 Nonstationary InfiniteHorizon Optimization 211 68 Optimal Growth in Discrete Time 215 69 Competitive Equilibrium Growth 219 Contents ix 610 Computation 221 611 Taking Stock 221 612 References and Literature 222 613 Exercises 223 Chapter 7 An Introduction to the Theory of Optimal Control 227 71 Variational Arguments 228 72 The Maximum Principle A First Look 235 73 InfiniteHorizon Optimal Control 240 74 More on Transversality Conditions 250 75 Discounted InfiniteHorizon Optimal Control 253 76 Existence of Solutions Concavity and Differentiability 259 77 A First Look at Optimal Growth in Continuous Time 268 78 The qTheory of Investment and SaddlePath Stability 269 79 Taking Stock 274 710 References and Literature 275 711 Exercises 278 Part III Neoclassical Growth Chapter 8 The Neoclassical Growth Model 287 81 Preferences Technology and Demographics 287 82 Characterization of Equilibrium 293 83 Optimal Growth 298 84 SteadyState Equilibrium 300 85 Transitional Dynamics and Uniqueness of Equilibrium 302 86 Neoclassical Growth in Discrete Time 305 87 Technological Change and the Canonical Neoclassical Model 306 88 The Role of Policy 312 89 Comparative Dynamics 313 810 A Quantitative Evaluation 315 811 Extensions 317 812 Taking Stock 317 813 References and Literature 318 814 Exercises 319 Chapter 9 Growth with Overlapping Generations 327 91 Problems of Infinity 328 92 The Baseline Overlapping Generations Model 329 93 The Canonical Overlapping Generations Model 335 94 Overaccumulation and Pareto Optimality of Competitive Equilibrium in the Overlapping Generations Model 336 95 Role of Social Security in Capital Accumulation 339 96 Overlapping Generations with Impure Altruism 342 97 Overlapping Generations with Perpetual Youth 345 98 Overlapping Generations in Continuous Time 348 99 Taking Stock 353 x Contents 910 References and Literature 354 911 Exercises 355 Chapter 10 Human Capital and Economic Growth 359 101 A Simple Separation Theorem 359 102 Schooling Investments and Returns to Education 361 103 The BenPorath Model 363 104 Neoclassical Growth with Physical and Human Capital 367 105 CapitalSkill Complementarity in an Overlapping Generations Model 371 106 Physical and Human Capital with Imperfect Labor Markets 374 107 Human Capital Externalities 379 108 The NelsonPhelps Model of Human Capital 380 109 Taking Stock 382 1010 References and Literature 384 1011 Exercises 384 Chapter 11 FirstGeneration Models of Endogenous Growth 387 111 The AK Model Revisited 388 112 The AK Model with Physical and Human Capital 393 113 The TwoSector AK Model 395 114 Growth with Externalities 398 115 Taking Stock 402 116 References and Literature 404 117 Exercises 404 Part IV Endogenous Technological Change Chapter 12 Modeling Technological Change 411 121 Different Conceptions of Technology 411 122 Science and Profits 414 123 The Value of Innovation in Partial Equilibrium 416 124 The DixitStiglitz Model and Aggregate Demand Externalities 422 125 Individual RD Uncertainty and the Stock Market 428 126 Taking Stock 429 127 References and Literature 430 128 Exercises 431 Chapter 13 Expanding Variety Models 433 131 The LabEquipment Model of Growth with Input Varieties 433 132 Growth with Knowledge Spillovers 444 133 Growth without Scale Effects 446 134 Growth with Expanding Product Varieties 448 135 Taking Stock 452 136 References and Literature 453 137 Exercises 453 Contents xi Chapter 14 Models of Schumpeterian Growth 458 141 A Baseline Model of Schumpeterian Growth 459 142 A OneSector Schumpeterian Growth Model 468 143 Innovation by Incumbents and Entrants 472 144 StepbyStep Innovations 479 145 Taking Stock 489 146 References and Literature 490 147 Exercises 491 Chapter 15 Directed Technological Change 497 151 Importance of Biased Technological Change 498 152 Basics and Definitions 500 153 Baseline Model of Directed Technological Change 503 154 Directed Technological Change with Knowledge Spillovers 514 155 Directed Technological Change without Scale Effects 518 156 Endogenous LaborAugmenting Technological Change 519 157 Generalizations and Other Applications 522 158 An Alternative Approach to LaborAugmenting Technological Change 523 159 Taking Stock 526 1510 References and Literature 527 1511 Exercises 529 Part V Stochastic Growth Chapter 16 Stochastic Dynamic Programming 537 161 Dynamic Programming with Expectations 537 162 Proofs of the Stochastic Dynamic Programming Theorems 544 163 Stochastic Euler Equations 549 164 Generalization to Markov Processes 552 165 Applications of Stochastic Dynamic Programming 554 166 Taking Stock 561 167 References and Literature 561 168 Exercises 562 Chapter 17 Stochastic Growth Models 566 171 The BrockMirman Model 567 172 Equilibrium Growth under Uncertainty 571 173 Application Real Business Cycle Models 579 174 Growth with Incomplete Markets The Bewley Model 583 175 The Overlapping Generations Model with Uncertainty 586 176 Risk Diversification and Growth 588 177 Taking Stock 603 178 References and Literature 604 179 Exercises 605 xii Contents Part VI Technology Diffusion Trade and Interdependences Chapter 18 Diffusion of Technology 611 181 Productivity Differences and Technology 611 182 A Benchmark Model of Technology Diffusion 613 183 Technology Diffusion and Endogenous Growth 619 184 Appropriate and Inappropriate Technologies and Productivity Differences 623 185 Contracting Institutions and Technology Adoption 630 186 Taking Stock 642 187 References and Literature 643 188 Exercises 644 Chapter 19 Trade and Growth 648 191 Growth and Financial Capital Flows 648 192 Why Does Capital Not Flow from Rich to Poor Countries 653 193 Economic Growth in a HeckscherOhlin World 655 194 Trade Specialization and the World Income Distribution 663 195 Trade Technology Diffusion and the Product Cycle 674 196 Trade and Endogenous Technological Change 678 197 LearningbyDoing Trade and Growth 680 198 Taking Stock 684 199 References and Literature 685 1910 Exercises 687 Part VII Economic Development and Economic Growth Chapter 20 Structural Change and Economic Growth 697 201 Nonbalanced Growth The Demand Side 697 202 Nonbalanced Growth The Supply Side 703 203 Agricultural Productivity and Industrialization 715 204 Taking Stock 719 205 References and Literature 720 206 Exercises 721 Chapter 21 Structural Transformations and Market Failures in Development 725 211 Financial Development 726 212 Fertility Mortality and the Demographic Transition 729 213 Migration Urbanization and the Dual Economy 736 214 Distance to the Frontier and Changes in the Organization of Production 744 215 Multiple Equilibria from Aggregate Demand Externalities and the Big Push 752 216 Inequality Credit Market Imperfections and Human Capital 758 217 Toward a Unified Theory of Development and Growth 764 218 Taking Stock 768 219 References and Literature 769 2110 Exercises 771 Contents xiii Part VIII The Political Economy of Growth Chapter 22 Institutions Political Economy and Growth 781 221 The Impact of Institutions on LongRun Development 781 222 Distributional Conflict and Economic Growth in a Simple Society 784 223 The Canonical CobbDouglas Model of Distributional Conflict 792 224 Distributional Conflict and Competition 793 225 Subgame Perfect versus Markov Perfect Equilibria 799 226 Inefficient Economic Institutions A First Pass 802 227 Heterogeneous Preferences Social Choice and the Median Voter 805 228 Distributional Conflict and Economic Growth Heterogeneity and the Median Voter 814 229 The Provision of Public Goods Weak versus Strong States 817 2210 Taking Stock 822 2211 References and Literature 823 2212 Exercises 825 Chapter 23 Political Institutions and Economic Growth 831 231 Political Regimes and Economic Growth 832 232 Political Institutions and GrowthEnhancing Policies 834 233 Dynamic Tradeoffs 837 234 Understanding Endogenous Political Change 850 235 Taking Stock 856 236 References and Literature 857 237 Exercises 858 Epilogue Mechanics and Causes of Economic Growth 861 What Have We Learned 861 A Possible Perspective on Growth and Stagnation over the Past 200 Years 864 Many Remaining Questions 872 Part IX Mathematical Appendixes Appendix A Odds and Ends in Real Analysis and Applications to Optimization 877 A1 Distances and Metric Spaces 878 A2 Mappings Functions Sequences Nets and Continuity 880 A3 A Minimal Amount of Topology Continuity and Compactness 885 A4 The Product Topology 889 A5 Absolute Continuity and Equicontinuity 891 A6 Correspondences and Berges Maximum Theorem 894 A7 Convexity Concavity QuasiConcavity and Fixed Points 898 A8 Differentiation Taylor Series and the Mean Value Theorem 900 A9 Functions of Several Variables and the Inverse and Implicit Function Theorems 904 A10 Separation Theorems 907 xiv Contents A11 Constrained Optimization 910 A12 Exercises 915 Appendix B Review of Ordinary Differential Equations 917 B1 Eigenvalues and Eigenvectors 917 B2 Some Basic Results on Integrals 918 B3 Linear Differential Equations 920 B4 Solutions to Linear FirstOrder Differential Equations 921 B5 Systems of Linear Differential Equations 924 B6 Local Analysis and Stability of Nonlinear Differential Equations 926 B7 Separable and Exact Differential Equations 927 B8 Existence and Uniqueness of Solutions 929 B9 Continuity and Differentiability of Solutions 930 B10 Difference Equations 930 B11 Exercises 932 Appendix C Brief Review of Dynamic Games 934 C1 Basic Definitions 934 C2 Some Basic Results 937 C3 Application Repeated Games with Perfect Observability 941 C4 Exercises 942 Appendix D List of Theorems 944 Chapter 2 944 Chapter 5 944 Chapter 6 944 Chapter 7 945 Chapter 10 945 Chapter 16 945 Chapter 22 946 Appendix A 946 Appendix B 947 Appendix C 947 References 949 Name Index 971 Subject Index 977 Preface As long as a branch of science offers an abundance of problems so long is it alive David Hilbert Paris 1900 T his book is intended to serve two purposes First and foremost this is a book about economic growth and longrun economic development The process of economic growth and the sources of differences in economic performance across nations are some of the most interesting important and challenging areas in modern social science The primary purpose of this book is to introduce graduate students to these major questions and to the theoretical tools necessary for studying them The book therefore strives to provide students with a strong background in dynamic economic analysis since only such a background will enable a serious study of economic growth and economic development I also try to provide a clear discussion of the broad empirical patterns and historical processes underlying the current state of the world economy This narrative is motivated by my belief that to understand why some countries grow and others fail to do so economists have to move beyond the mechanics of models and pose questions about the fundamental causes of economic growth Second in a somewhat different capacity this book is also a graduatelevel introduction to modern macroeconomics and dynamic economic analysis It is sometimes commented that unlike basic microeconomic theory there is no core of current macroeconomic theory that is shared by all economists This is not entirely true While there is disagreement among macroeconomists about how to approach shortrun macroeconomic phenomena and what the boundaries of macroeconomics should be there is broad agreement about the workhorse models of dynamic macroeconomic analysis These include the Solow growth model the neoclassical growth model the overlapping generations model and models of technological change and technology adoption Since these are all models of economic growth a thorough treatment of modern economic growth can also provide and perhaps should provide an introduction to this core material of modern macroeconomics Although there are several good graduatelevel macroeconomic textbooks they typically spend relatively little time on the basic core material and do not develop the links between modern macroeconomic analysis and economic dynamics on the one hand and general equilibrium theory on the other In contrast the current book does not cover any of the shortrun topics in macroeconomics but provides a thorough and rigorous introduction to what I view to be the core of macroeconomics The selection of topics is designed to strike a balance between the two purposes of the book Chapters 1 3 and 4 introduce many of the salient features of the process of economic growth and the sources of crosscountry differences in economic performance Even though these xv xvi Preface chapters cannot do justice to the large literature on economic growth empirics they provide a sufficient background for students to appreciate the issues that are central to the study of economic growth and a platform for further study of this large literature Chapters 57 cover the conceptual and mathematical foundations of modern macro economic analysis Chapter 5 provides the microfoundations for much of the rest of the book and for much of modern macroeconomics while Chapters 6 and 7 supply a quick but rel atively rigorous introduction to dynamic optimization Most books on macroeconomics or economic growth use either continuous time or discrete time exclusively I believe that a seri ous study of both economic growth and modern macroeconomics requires the student and the researcher to be able to move between formulations using discrete and continuous time choos ing the more convenient or appropriate approach for the set of questions at hand Therefore I have deviated from standard practice and included both continuoustime and discretetime material throughout the book Chapters 2 8 9 and 10 introduce the basic workhorse models of modern macroeconomics and traditional economic growth while Chapter 11 presents the firstgeneration models of sustained endogenous economic growth Chapters 1215 cover models of technological progress which are an essential part of any modern economic growth course Chapter 16 generalizes the tools introduced in Chapter 6 to stochastic environments Using these tools Chapter 17 presents a number of models of stochastic growthmost notably the neoclassical growth model under uncertainty which is the foundation of much of modern macroeconomics though it is often left out of courses on economic growth The canonical Real Business Cycle model is presented as an application This chapter also covers another major workhorse model of modern macroeconomics the incomplete markets model of Bewley Finally Chapter 17 also presents a number of other approaches to modeling the interaction between incomplete markets and economic growth and shows how models of stochastic growth can be useful in understanding how economies transition from stagnation or slow growth to an equilibrium with sustained growth Chapters 1821 cover topics that are sometimes left out of economic growth textbooks These include models of technology adoption technology diffusion the interaction between international trade and technology the process of structural change the demographic transi tion the possibility of poverty traps the effects of inequality on economic growth and the interaction between financial and economic development These topics are important for cre ating a bridge between the empirical patterns we observe in practice and the theory Most traditional growth models consider a single economy in isolation often after it has already embarked on a process of steady economic growth A study of models that incorporate cross country interdependences structural change and the possibility of takeoffs makes it possible to link core topics of development economics such as structural change poverty traps or the demographic transition to the theory of economic growth Finally Chapters 22 and 23 consider another topic often omitted from macroeconomics and economic growth textbooks political economy Inclusion of this material is motivated by my belief that the study of economic growth would be seriously hampered if we failed to ask questions about the fundamental causes of differences among countries in their economic performances These questions inexorably bring us to differences in economic policies and institutions across nations Political economy enables us to develop models to understand why economic policies and institutions differ across countries and must therefore be an integral part of the study of economic growth A few words on the philosophy and organization of the book might also be useful for students and teachers The underlying philosophy of the book is that all the results that are stated should be proved or at least explained in detail This implies a somewhat different organization than found in other books Most textbooks in economics do not provide proofs for many of the Preface xvii results that are stated or invoked and mathematical tools that are essential for the analysis are often taken for granted or developed in appendixes In contrast I have strived to provide simple proofs of almost all results stated in this book It turns out that once unnecessary generality is removed most results can be stated and proved in a way that is easily accessible to graduate students In fact I believe that even somewhat long proofs are much easier to understand than general statements made without proof which leave the reader wondering why these statements are true I hope that the style I have chosen not only makes the book selfcontained but also gives students an opportunity to develop a thorough understanding of the material In line with this philosophy I present the basic mathematical tools necessary for the development of the main material in the body of the text My own experience suggests that a linear progression where the necessary mathematical tools are introduced when needed makes it easier for students to follow and appreciate the material Consequently analysis of the stability of dynamical systems dynamic programming in discrete time and optimal control in continuous time are all introduced in the main body of the text This should both help students appreciate the foundations of the theory of economic growth and provide them with an introduction to the main tools of dynamic economic analysis which are increasingly used in every subdiscipline of economics Throughout when some material is technically more difficult and can be skipped without loss of continuity it is marked with an asterisk Material that is only tangentially related to the main results in the text or that should be familiar to most graduate students is left for the appendixes I have also included a large number of exercises Students can gain a thorough understanding of the material only by working through the exercises Exercises that are somewhat more difficult are also marked with an asterisk This book can be used in a number of different ways First it can be used in a onequarter or onesemester course on economic growth Such a course might start with Chapters 14 then depending on the nature of the course use Chapters 57 either for a thorough study of the general equilibrium and dynamic optimization foundations of growth theory or only for reference Chapters 811 cover traditional growth theory and Chapters 1215 provide the basics of endogenous growth theory Depending on time and interest any selection of Chapters 1623 can be used for the last part of such a course Second the book can be used for a onequarter firstyear graduatelevel course in macro economics In this case Chapter 1 would be optional Chapters 2 57 811 1617 and a selection from 1215 would be the core of such a course The same material could also be covered in a onesemester course but in this case it could be supplemented either with some of the later chapters or with material from one of the leading graduatelevel macroeconomic textbooks on shortrun macroeconomics fiscal policy asset pricing or other topics in dynamic macroeconomics Third the book can be used for an advanced secondyear course in economic growth or economic development An advanced course on growth or development could use Chapters 111 as background and then focus on selected chapters from among Chapters 1223 Finally since the book is selfcontained I also hope that it can be used for selfstudy Acknowledgments This book grew out of the first graduatelevel introduction to macroeconomics course I taught at MIT Parts of the book have also been taught as part of secondyear graduate courses on macroeconomics and economic growth I thank the students who attended these lectures and made comments that have improved the manuscript I owe special thanks to Nathan xviii Preface Hendren Derek Li Monica MartinezBravo Plamen Nemov Samuel Pienknagura Anna Zabai and especially to Georgy Egorov Michael Peters and Alp Simsek for outstanding research assistance Alp deserves more than a special mention He has been involved with almost every aspect of the book for more than two years Without Alps help the book would have taken me much longer to complete and would have contained many more errors I am deeply indebted to him I also thank Pol Antras Gabriel Carroll Francesco Caselli Melissa Dell Jesus Fernandez Villaverde Kiminori Matsuyama James Robinson and Pierre Yared for very valuable sug gestions on multiple chapters and GeorgeMarios Angeletos Binyamin Berdugo Truman Bewley Olivier Blanchard Leopoldo Fergusson Peter Funk Oded Galor Hugo Hopenhayn Simon Johnson Chad Jones Christos Koulovatianos Omer Moav Eduardo Morales Ismail Saglam Ekkehart Schlicht Patricia Waeger Luis Zermeno and Jesse Zinn for useful sugges tions and corrections on individual chapters Last but not least I thank Lauren Fahey for editorial suggestions on multiple chapters and help with the references Cyd Westmoreland for truly exceptional copyediting and editorial suggestions and Seth Ditchik and his colleagues at Princeton University Press for support and help throughout the process PART I INTRODUCTION 1 Economic Growth and Economic Development The Questions 11 CrossCountry Income Differences T here are very large differences in income per capita and output per worker across countries today Countries at the top of the world income distribution are more than 30 times as rich as those at the bottom For example in 2000 gross domestic product GDP or income per capita in the United States was more than 34000 In contrast income per capita is much lower in many other countries about 8000 in Mexico about 4000 in China just over 2500 in India only about 1000 in Nigeria and much much lower in some other subSaharan African countries such as Chad Ethiopia and Mali These numbers are all in 2000 US dollars and are adjusted for purchasing power parity PPP to allow for differences in relative prices of different goods across countries1 The crosscountry income gap is considerably larger when there is no PPP adjustment For example without the PPP adjustment GDP per capita in India and China relative to the United States in 2000 would be lower by a factor of four or so Figure 11 provides a first look at these differences It plots estimates of the distribution of PPPadjusted GDP per capita across the available set of countries in 1960 1980 and 2000 A number of features are worth noting First the 1960 density shows that 15 years after the end of World War II most countries had income per capita less than 1500 in 2000 US dollars the mode of the distribution is around 1250 The rightward shift of the distributions for 1980 and 2000 shows the growth of average income per capita for the next 40 years In 2000 the mode is slightly above 3000 but now there is another concentration of countries between 20000 and 30000 The density estimate for the year 2000 shows the considerable inequality in income per capita today The spreading out of the distribution in Figure 11 is partly because of the increase in average incomes It may therefore be more informative to look at the logarithm log of 1 All data are from the Penn World tables compiled by Heston Summers and Aten 2002 Details of data sources and more on PPP adjustment can be found in the References and Literature section at the end of this chapter 3 4 Chapter 1 Economic Growth and Economic Development The Questions 1960 1980 2000 Density of countries 0 20000 40000 60000 GDP per capita FIGURE 11 Estimates of the distribution of countries according to PPPadjusted GDP per capita in 1960 1980 and 2000 income per capita It is more natural to look at the log of variables such as income per capita that grow over time especially when growth is approximately proportional as suggested by Figure 18 below This is for the simple reason that when x t grows at a proportional rate log x t grows linearly and if x1 t and x2 t both grow by the same proportional amount log x1 t log x2 t remains constant while x1 t x2 t increases Figure 12 shows a similar pattern but now the spreading is more limited because the absolute gap between rich and poor countries has increased considerably between 1960 and 2000 while the proportional gap has increased much less Nevertheless it can be seen that the 2000 density for log GDP per capita is still more spread out than the 1960 density In particular both figures show that there has been a considerable increase in the density of relatively rich countries while many countries still remain quite poor This last pattern is sometimes referred to as the stratification phenomenon corresponding to the fact that some of the middleincome countries of the 1960s have joined the ranks of relatively highincome countries while others have maintained their middleincome status or even experienced relative impoverishment Figures 11 and 12 demonstrate that there is somewhat greater inequality among nations today than in 1960 An equally relevant concept might be inequality among individuals in the world economy Figures 11 and 12 are not directly informative on this since they treat each country identically regardless of the size of its population An alternative is presented in Figure 13 which shows the populationweighted distribution In this case countries such as China India the United States and Russia receive greater weight because they have larger populations The picture that emerges in this case is quite different In fact the 2000 distribution looks less spread out with a thinner left tail than the 1960 distribution This reflects the fact that 1960 1980 2000 Density of countries 6 8 10 12 Log GDP per capita FIGURE 12 Estimates of the distribution of countries according to log GDP per capita PPP adjusted in 1960 1980 and 2000 1960 1980 2000 Density of countries weighted by population 6 8 10 12 Log GDP per capita FIGURE 13 Estimates of the populationweighted distribution of countries according to log GDP per capita PPP adjusted in 1960 1980 and 2000 6 Chapter 1 Economic Growth and Economic Development The Questions 1960 1980 2000 Density of countries 6 8 10 12 Log GDP per worker FIGURE 14 Estimates of the distribution of countries according to log GDP per worker PPP adjusted in 1960 1980 and 2000 in 1960 China and India were among the poorest nations in the world whereas their relatively rapid growth in the 1990s puts them into the middlepoor category by 2000 Chinese and Indian growth has therefore created a powerful force for relative equalization of income per capita among the inhabitants of the globe Figures 11 12 and 13 look at the distribution of GDP per capita While this measure is relevant for the welfare of the population much of growth theory focuses on the produc tive capacity of countries Theory is therefore easier to map to data when we look at output GDP per worker Moreover key sources of difference in economic performance across coun tries are national policies and institutions So for the purpose of understanding the sources of differences in income and growth across countries as opposed to assessing welfare ques tions the unweighted distribution is more relevant than the populationweighted distribution Consequently Figure 14 looks at the unweighted distribution of countries according to PPP adjusted GDP per worker Workers here refers to the total economically active population according to the definition of the International Labour Organization Figure 14 is very simi lar to Figure 12 and if anything it shows a greater concentration of countries in the relatively rich tail by 2000 with the poor tail remaining more or less the same as in Figure 12 Overall Figures 1114 document two important facts first there is great inequality in income per capita and income per worker across countries as shown by the highly dispersed distributions Second there is a slight but noticeable increase in inequality across nations though not necessarily across individuals in the world economy 12 Income and Welfare 7 12 Income and Welfare Should we care about crosscountry income differences The answer is definitely yes High income levels reflect high standards of living Economic growth sometimes increases pollution or may raise individual aspirations so that the same bundle of consumption may no longer satisfy an individual But at the end of the day when one compares an advanced rich country with a lessdeveloped one there are striking differences in the quality of life standards of living and health Figures 15 and 16 give a glimpse of these differences and depict the relationship between income per capita in 2000 and consumption per capita and life expectancy at birth in the same year Consumption data also come from the Penn World tables while data on life expectancy at birth are available from the World Bank Development Indicators These figures document that income per capita differences are strongly associated with differences in consumption and in health as measured by life expectancy Recall also that these numbers refer to PPPadjusted quantities thus differences in consumption do not at least in principle reflect the differences in costs for the same bundle of consumption goods in different countries The PPP adjustment corrects for these differences and attempts to measure the variation in real consumption Thus the richest countries are not only producing more than 30 times as much as the poorest countries but are also consuming 30 times as much Similarly crosscountry differences in health are quite remarkable while life expectancy at birth is as AFG ALB DZA AGO ATG ARG ARM AUS AUT AZE BHS BHR BGD BRB BLR BEL BLZ BEN BMU BTN BOL BIH BWA BRA BRN BGR BFA BDI KHM CMR CAN CPV CAF TCD CHL CHN COL COM ZAR COG CRI CIV HRV CUB CYP CZE DNK DJI DMA DOM ECU EGY SLV GNQ ERI EST ETH FJI FIN FRA GAB GMB GEO GER GHA GRC GRD GTM GIN GNB GUY HTI HND HKG HUN ISL IND IDN IRN IRQ IRL ISR ITA JAM JPN JOR KAZ KEN KIR PRK KOR KWT KGZ LAO LVA LBN LSO LBR LBY LTU LUX MAC MKD MDG MWI MYS MDV MLI MLT MRT MUS MEX FSM MDA MNG MAR MOZ NAM NPL NLD ANT NZL NIC NER NGA NOR OMN PAK PLW PAN PNG PRY PER PHL POL PRT PRI QAT ROM RUS RWA WSM STP SAU SEN SCG SYC SLE SGP SVK SVN SLB SOM ZAF ESP LKA KNA LCA VCT SDN SUR SWZ SWE CHE SYR TWN TJK TZA THA TGO TON TTO TUN TUR TKM UGA UKR ARE GBR USA URY UZB VUT VEN VNM YEM ZMB ZWE 10 11 12 13 14 15 Log consumption per capita 2000 6 7 8 9 10 11 Log GDP per capita 2000 FIGURE 15 The association between income per capita and consumption per capita in 2000 For a definition of the abbreviations used in this and similar figures in the book see httpunstatsunorgunsd methodsm49m49alphahtm 8 Chapter 1 Economic Growth and Economic Development The Questions AFG AGO ALB ANT ARE ARG ARM AUS AUT AZE BDI BEL BEN BFA BGD BGR BHR BHS BIH BLR BLZ BOL BRA BRB BRN BTN BWA CAF CAN CHE CHL CHN CIVCMR COG COL COM CPV CRI CUB CYP CZE DJI DNK DOM DZA ECU EGY ERI ESP EST FIN FJI FRA FSM GAB GBR GEO GHA GIN GMB GNB GNQ GRC GTM GUY HKG HND HRV HTI HUN IDN IND IRL IRN IRQ ISRISL ITA JAM JOR JPN KAZ KEN KGZ KHM KOR KWT LAO LBN LBR LBY LCA LKA LSO LTU LUX LVA MAC MAR MDA MDG MDV MEX MKD MLI MLT MNG MOZ MRT MUS MWI MYS NAM NER NGA NIC NLD NOR NPL NZL OMN PAK PAN PER PHL PNG POL PRI PRK PRT PRY QAT ROM RUS RWA SAU SCG SDN SEN SGP SLB SLE SLV SOM STP SUR SVK SVN SWE SWZ SYR TCD TGO THA TJK TKM TON TTO TUN TUR TZA UGA UKR URY USA UZB VCT VEN VNM VUT WSM YEM ZAF ZMB ZWE ETH GER 30 40 50 60 70 80 Life expectancy 2000 years 6 7 8 9 10 11 Log GDP per capita 2000 FIGURE 16 The association between income per capita and life expectancy at birth in 2000 high as 80 in the richest countries it is only between 40 and 50 in many subSaharan African nations These gaps represent huge welfare differences Understanding why some countries are so rich while some others are so poor is one of the most important perhaps the most important challenges facing social science It is important both because these income differences have major welfare consequences and because a study of these striking differences will shed light on how the economies of different nations function and how they sometimes fail to function The emphasis on income differences across countries implies neither that income per capita can be used as a sufficient statistic for the welfare of the average citizen nor that it is the only feature that we should care about As discussed in detail later the efficiency properties of the market economy such as the celebrated First Welfare Theorem or Adam Smiths invisible hand do not imply that there is no conflict among individuals or groups in society Economic growth is generally good for welfare but it often creates winners and losers Joseph Schumpeters famous notion of creative destruction emphasizes precisely this aspect of economic growth productive relationships firms and sometimes individual livelihoods will be destroyed by the process of economic growth because growth is brought about by the introduction of new technologies and creation of new firms replacing existing firms and technologies This process creates a natural social tension even in a growing society Another source of social tension related to growth and development is that as emphasized by Simon Kuznets and discussed in detail in Part VII growth and development are often accompanied by sweeping structural transformations which can also destroy certain established relationships and create yet other winners and losers in the process One of the important questions of 13 Economic Growth and Income Differences 9 political economy which is discussed in the last part of the book concerns how institutions and policies can be arranged so that those who lose out from the process of economic growth can be compensated or prevented from blocking economic progress via other means A stark illustration of the fact that growth does not always mean an improvement in the living standards of all or even most citizens in a society comes from South Africa under apartheid Available data from gold mining wages suggest that from the beginning of the twentieth century until the fall of the apartheid regime GDP per capita grew considerably but the real wages of black South Africans who make up the majority of the population likely fell during this period This of course does not imply that economic growth in South Africa was not beneficial South Africa is still one of the richest countries in subSaharan Africa Nevertheless this observation alerts us to other aspects of the economy and also underlines the potential conflicts inherent in the growth process Similarly most existing evidence suggests that during the early phases of the British industrial revolution which started the process of modern economic growth the living standards of the majority of the workers may have fallen or at best remained stagnant This pattern of potential divergence between GDP per capita and the economic fortunes of large numbers of individuals and society is not only interesting in and of itself but it may also inform us about why certain segments of the society may be in favor of policies and institutions that do not encourage growth 13 Economic Growth and Income Differences How can one country be more than 30 times richer than another The answer lies in differences in growth rates Take two countries A and B with the same initial level of income at some date Imagine that country A has 0 growth per capita so its income per capita remains constant while country B grows at 2 per capita In 200 years time country B will be more than 52 times richer than country A This calculation suggests that the United States might be considerably richer than Nigeria because it has grown steadily over an extended period of time while Nigeria has not We will see that there is a lot of truth to this simple calculation In fact even in the historically brief postwar era there are tremendous differences in growth rates across countries These differences are shown in Figure 17 for the postwar era which plots the density of growth rates across countries in 1960 1980 and 2000 The growth rate in 1960 refers to the geometric average of the growth rate between 1950 and 1969 the growth rate in 1980 refers to the average growth rate between 1970 and 1989 and 2000 refers to the average between 1990 and 2000 in all cases subject to data availability Figure 17 shows that in each time interval there is considerable variability in growth rates the crosscountry distribution stretches from negative rates to average rates as high as 10 per year It also shows that average growth in the world was more rapid in the 1950s and 1960s than in the subsequent decades Figure 18 provides another look at these patterns by plotting log GDP per capita for a number of countries between 1960 and 2000 in this case I plot GDP per capita instead of GDP per worker because of the availability of data and to make the figures more comparable to the historical figures below At the top of the figure US and UK GDP per capita increase at a steady pace with a slightly faster growth in the United States so that the log or proportional gap between the two countries is larger in 2000 than it is in 1960 Spain starts much poorer than the United States and the United Kingdom in 1960 but grows very rapidly between 1960 and the mid1970s thus closing the gap between itself and the latter two countries The three countries that show the most rapid growth in this figure are Singapore South Korea and Botswana Singapore starts much poorer than the United Kingdom and Spain in 1960 but 10 Chapter 1 Economic Growth and Economic Development The Questions 1960 1980 2000 Density of countries 01 00 01 02 Average growth rate of GDP per worker FIGURE 17 Estimates of the distribution of countries according to the growth rate of GDP per worker PPP adjusted in 1960 1980 and 2000 grows rapidly and by the mid1990s it has become richer than both South Korea has a similar trajectory though it starts out poorer than Singapore and grows slightly less rapidly so that by the end of the sample it is still a little poorer than Spain The other country that has grown very rapidly is the African success story Botswana which was extremely poor at the beginning of the sample Its rapid growth especially after 1970 has taken Botswana to the ranks of the middleincome countries by 2000 The two Latin American countries in this picture Brazil and Guatemala illustrate the often discussed Latin American economic malaise of the postwar era Brazil starts out richer than South Korea and Botswana and has a relatively rapid growth rate between 1960 and 1980 But it experiences stagnation from 1980 on so that by the end of the sample South Korea and Botswana have become richer than Brazil Guatemalas experience is similar but even more bleak Contrary to Brazil there is little growth in Guatemala between 1960 and 1980 and no growth between 1980 and 2000 Finally Nigeria and India start out at similar levels of income per capita as Botswana but experience little growth until the 1980s Starting in 1980 the Indian economy experiences relatively rapid growth though this has not been sufficient for its income per capita to catch up with the other nations in the figure Finally Nigeria in a pattern that is unfortunately all too familiar in subSaharan Africa experiences a contraction of its GDP per capita so that in 2000 it is in fact poorer than it was in 1960 The patterns shown in Figure 18 are what we would like to understand and explain Why is the United States richer in 1960 than other nations and able to grow at a steady pace thereafter How did Singapore South Korea and Botswana manage to grow at a relatively rapid pace for 14 Todays Income Differences and World Economic Growth 11 United States United Kingdom Spain Singapore Brazil South Korea Botswana Guatemala Nigeria India 7 8 9 10 11 Log GDP per capita 1960 1970 1980 1990 2000 FIGURE 18 The evolution of income per capita in the United States the United Kingdom Spain Singapore Brazil Guatemala South Korea Botswana Nigeria and India 19602000 40 years Why did Spain grow relatively rapidly for about 20 years but then slow down Why did Brazil and Guatemala stagnate during the 1980s What is responsible for the disastrous growth performance of Nigeria 14 Origins of Todays Income Differences and World Economic Growth The growth rate differences shown in Figures 17 and 18 are interesting in their own right and could also be in principle responsible for the large differences in income per capita we observe today But are they The answer is largely no Figure 18 shows that in 1960 there was already a very large gap between the United States on the one hand and India and Nigeria on the other This pattern can be seen more easily in Figure 19 which plots log GDP per worker in 2000 versus log GDP per capita in 1960 in both cases relative to the US value superimposed over the 45 line Most observations are around the 45 line indicating that the relative ranking of countries has changed little between 1960 and 2000 Thus the origins of the very large income differences across nations are not to be found in the postwar era There are striking growth differences during the postwar era but the evidence presented so far suggests that world income distribution has been more or less stable with a slight tendency toward becoming more unequal 12 Chapter 1 Economic Growth and Economic Development The Questions DZA ARG AUS AUT BRB BEL BEN BOL BRA BFA BDI CMR CAN CPV TCD CHL CHN COL COM COG CRI CIV DNK DOM ECU EGY SLV GNQ ETH FIN FRA GAB GMB GHA GRC GTM GIN GNB HND HKG ISL IND IDN IRN IRL ISR ITA JAM JPN JOR KEN KOR LSO LUX MDG MWI MYS MLI MUS MEX MAR MOZ NPL NLD NZL NIC NER NGA NOR PAK PAN PRY PER PHL PRT ROM RWA SEN SGP ZAF ESP LKA SWE CHE SYR TZA THA TGO TTO TUR UGA GBR USA URY VEN ZMB ZWE 06 07 08 09 10 11 Log GDP per worker relative to the United States 2000 06 07 08 09 10 11 Log GDP per worker relative to the United States 1960 FIGURE 19 Log GDP per worker in 2000 versus log GDP per worker in 1960 together with the 45 line If not in the postwar era when did this growth gap emerge The answer is that much of the divergence took place during the nineteenth and early twentieth centuries Figures 110 112 give a glimpse of these developments by using the data compiled by Angus Maddison for GDP per capita differences across nations going back to 1820 or sometimes earlier These data are less reliable than SummersHestons Penn World tables since they do not come from standardized national accounts Moreover the sample is more limited and does not include observations for all countries going back to 1820 Finally while these data include a correction for PPP this is less complete than the price comparisons used to construct the price indices in the Penn World tables Nevertheless these are the best available estimates for differences in prosperity across a large number of nations beginning in the nineteenth century Figure 110 illustrates the divergence It depicts the evolution of average income among five groups of countries Africa Asia Latin America Western Europe and Western offshoots of Europe Australia Canada New Zealand the United States It shows the relatively rapid growth of the Western offshoots and West European countries during the nineteenth century while Asia and Africa remained stagnant and Latin America showed little growth The rela tively small proportional income gap in 1820 had become much larger by 1960 Another major macroeconomic fact is visible in Figure 110 Western offshoots and West European nations experience a noticeable dip in GDP per capita around 1929 because of the famous Great Depression Western offshoots in particular the United States only recovered fully from this large recession in the wake of World War II How an economy can experience a sharp decline in output and how it recovers from such a shock are among the major questions of macroeconomics 14 Todays Income Differences and World Economic Growth 13 Western offshoots Western Europe Africa Asia Latin America 6 7 8 9 10 Log GDP per capita 1820 1850 1900 1950 2000 FIGURE 110 The evolution of average GDP per capita in Western offshoots Western Europe Latin America Asia and Africa 18202000 A variety of evidence suggests that differences in income per capita were even smaller before 1820 Maddison also has estimates for average income for the same groups of countries going back to 1000 AD or even earlier Figure 110 can be extended back in time using these data the results are shown in Figure 111 Although these numbers are based on scattered evidence and informed guesses the general pattern is consistent with qualitative historical evidence and the fact that income per capita in any country cannot have been much less than 500 in terms of 2000 US dollars since individuals could not survive with real incomes much less than this level Figure 111 shows that as we go further back in time the gap among countries becomes much smaller This further emphasizes that the big divergence among countries has taken place over the past 200 years or so Another noteworthy feature that becomes apparent from this figure is the remarkable nature of world economic growth Much evidence suggests that there was only limited economic growth before the eighteenth century and certainly before the fifteenth century While certain civilizations including ancient Greece Rome China and Venice managed to grow their growth was either not sustained thus ending with collapses and crises or progressed only at a slow pace No society before nineteenthcentury Western Europe and the United States achieved steady growth at comparable rates Notice also that Maddisons estimates show a slow but steady increase in West European GDP per capita even earlier starting in 1000 This assessment is not shared by all economic historians many of whom estimate that there was little increase in income per capita before 1500 or even before 1800 For our purposes this disagreement is not central however What is important is that using Walter Rostows terminology Figure 111 shows a pattern of takeoff into sustained growth the economic growth experience of Western Europe and Western offshoots appears to have changed dramatically about 200 years or so ago Economic historians also debate whether there was a discontinuous change in economic activity that deserves the 14 Chapter 1 Economic Growth and Economic Development The Questions Western offshoots Western Europe Africa Asia Latin America 6 7 8 9 10 Log GDP per capita 1000 1200 1400 1600 1800 2000 FIGURE 111 The evolution of average GDP per capita in Western offshoots Western Europe Latin America Asia and Africa 10002000 terms takeoff or industrial revolution This debate is again secondary to our purposes Whether or not the change was discontinuous it was present and transformed the functioning of many economies As a result of this transformation the stagnant or slowly growing economies of Europe embarked upon a path of sustained growth The origins of todays riches and also of todays differences in prosperity are to be found in this pattern of takeoff during the nineteenth century In the same time that Western Europe and its offshoots grew rapidly much of the rest of the world did not experience a comparable takeoff or did so much later Therefore an understanding of modern economic growth and current crosscountry income differences ultimately necessitates an inquiry into the causes of why the takeoff occurred why it did so about 200 years ago and why it took place only in some areas and not in others Figure 112 shows the evolution of income per capita for the United States the United Kingdom Spain Brazil China India and Ghana This figure confirms the patterns shown in Figure 110 for averages with the United States the United Kingdom and Spain growing much faster than India and Ghana throughout and also much faster than Brazil and China except during the growth spurts experienced by these two countries Overall on the basis of the available information we can conclude that the origins of the current crosscountry differences in income per capita are in the nineteenth and early twentieth centuries or perhaps even during the late eighteenth century This crosscountry divergence took place at the same time as a number of countries in the world took off and achieved sustained economic growth Therefore understanding the origins of modern economic growth are not only interesting and important in their own right but also holds the key to understanding the causes of crosscountry differences in income per capita today 15 Conditional Convergence 15 United Kingdom United States Spain China Brazil India Ghana 6 7 8 9 10 Log GDP per capita 1820 1850 1900 1950 2000 FIGURE 112 The evolution of income per capita in the United States the United Kindgom Spain Brazil China India and Ghana 18202000 15 Conditional Convergence I have so far documented the large differences in income per capita across nations the slight divergence in economic fortunes over the postwar era and the much larger divergence since the early 1800s The analysis focused on the unconditional distribution of income per capita or per worker In particular we looked at whether the income gap between two countries increases or decreases regardless of these countries characteristics eg institutions policies technology or even investments Barro and SalaiMartin 1991 1992 2004 argue that it is instead more informative to look at the conditional distribution Here the question is whether the income gap between two countries that are similar in observable characteristics is becoming narrower or wider over time In this case the picture is one of conditional convergence in the postwar period the income gap between countries that share the same characteristics typically closes over time though it does so quite slowly This is important both for understanding the statistical properties of the world income distribution and also as an input into the types of theories that we would like to develop How do we capture conditional convergence Consider a typical Barro growth regression gitt1 α log yit1 XT it1β εit 11 where gitt1 is the annual growth rate between dates t 1 and t in country i yit1 is output per worker or income per capita at date t 1 X is a vector of other variables included in the regression with coefficient vector β XT denotes the transpose of this vector and εit 16 Chapter 1 Economic Growth and Economic Development The Questions DZA ARG AUS AUT BRB BEL BEN BOL BRA BFA BDI CMR CAN CPV TCD CHL CHN COL COM COG CRI CIV DNK DOM ECU EGY SLV GNQ ETH FIN FRA GAB GMB GHA GRC GTM GIN GNB HND HKG ISL IND IDN IRN IRL ISR ITA JAM JPN JOR KEN KOR LSO LUX MDG MWI MYS MLI MUS MEX MAR MOZ NPL NLD NZL NIC NER NGA NOR PAK PAN PRY PER PHL PRT ROM RWA SEN SGP ZAF ESP LKA SWE CHE SYR TWN TZA THA TGO TTO TUR UGA GBR USA URY VEN ZMB ZWE 002 000 002 004 006 Average growth rate of GDP 19602000 7 8 9 10 11 Log GDP per worker 1960 FIGURE 113 Annual growth rate of GDP per worker between 1960 and 2000 versus log GDP per worker in 1960 for the entire world is an error term capturing all other omitted factors The variables in X are included because they are potential determinants of steadystate income andor growth First note that without covariates 11 is quite similar to the relationship shown in Figure 19 In particular since gitt1 log yit log yit1 11 can be written as log yit 1 α log yit1 εit Figure 19 showed that the relationship between log GDP per worker in 2000 and log GDP per worker in 1960 can be approximated by the 45 line so that in terms of this equation α should be approximately equal to 0 This observation is confirmed by Figure 113 which depicts the relationship between the geometric average growth rate between 1960 and 2000 and log GDP per worker in 1960 This figure reiterates that there is no unconditional convergence for the entire worldno tendency for poorer nations to become relatively more prosperousover the postwar period While there is no convergence for the entire world when we look among the member nations of the Organisation for Economic Cooperation and Development OECD2 we see a different pattern Figure 114 shows that there is a strong negative relationship between log GDP per worker in 1960 and the annual growth rate between 1960 and 2000 What distinguishes this sample from the entire world sample is the relative homogeneity of the OECD countries which 2 OECD here refers to the members that joined the OECD in the 1960s this excludes Australia New Zealand Mexico and Korea The figure also excludes Germany because of lack of comparable data after reunification 15 Conditional Convergence 17 AUS BEL CAN DNK FIN FRA GRC ISL IRL ITA JPN LUX NLD NOR PRT ESP SWE CHE TUR GBR USA 001 002 003 004 Average growth rate of GDP 19602000 85 90 95 100 105 Log GDP per worker 1960 FIGURE 114 Annual growth rate of GDP per worker between 1960 and 2000 versus log GDP per worker in 1960 for core OECD countries have much more similar institutions policies and initial conditions than for the entire world Thus there might be a type of conditional convergence when we control for certain country characteristics potentially affecting economic growth This is what the vector X captures in 11 In particular when this vector includes such variables as years of schooling or life expectancy using crosssectional regressions Barro and SalaiMartin estimate α to be approximately 002 indicating that the income gap between countries that have the same human capital endowment has been narrowing over the postwar period on average at about 2 percent per year When this equation is estimated using panel data and the vector X includes a full set of country fixed effects the estimates of α become more negative indicating faster convergence In summary there is no evidence of unconditional convergence in the world income distribution over the postwar era in fact the evidence suggests some amount of divergence in incomes across nations But there is some evidence for conditional convergence meaning that the income gap between countries that are similar in observable characteristics appears to narrow over time This last observation is relevant both for recognizing among which countries the economic divergence has occurred and for determining what types of models we should consider for understanding the process of economic growth and the differences in economic performance across nations For example we will see that many growth models including the basic Solow and the neoclassical growth models suggest that there should be transitional dynamics as economies below their steadystate target level of income per capita grow toward that level Conditional convergence is consistent with this type of transitional dynamics 18 Chapter 1 Economic Growth and Economic Development The Questions 16 Correlates of Economic Growth The previous section emphasized the importance of certain country characteristics that might be related to the process of economic growth What types of countries grow more rapidly Ideally this question should be answered at a causal level In other words we would like to know which specific characteristics of countries including their policies and institutions have a causal effect on growth Causal effect refers to the answer to the following counterfactual thought experiment if all else being equal a particular characteristic of the country were changed exogenously ie not as part of equilibrium dynamics or in response to a change in other observable or unobservable variables what would be the effect on equilibrium growth Answering such causal questions is quite challenging precisely because it is difficult to isolate changes in endogenous variables that are not driven by equilibrium dynamics or by omitted factors For this reason let us start with the more modest question of what factors correlate with postwar economic growth With an eye to the theories to come in the next two chapters the two obvious candidates to look at are investments in physical and human capital education Figure 115 shows a positive association between the average investment to GDP ratio and economic growth between 1960 and 2000 Figure 116 shows a positive correlation between average years of schooling and economic growth These figures therefore suggest that the countries that have grown faster are typically those that have invested more in physical and human capital It has to be stressed that these figures do not imply that physical or human capital investment are the causes of economic growth even though we expect from basic economic theory that they should contribute to growth So far these are simply correlations and they ARG AUS AUT BEL BEN BOL BRA BFA CAN CHL CHN COL CRI DNK DOM ECU EGY SLV ETH FIN FRA GHA GRC GTM GIN HND ISL IND IRN IRL ISR ITA JAM JPN JOR KEN KOR LUX MWI MYS MUS MEX MAR NLD NZL NIC NGA NOR PAK PAN PRY PER PHL PRT ZAF ESP LKA SWE CHE TWN THA TTO TUR UGA GBR USA URY VEN ZMB ZWE 000 002 004 006 008 Average growth rate of GDP per capita 19602000 00 01 02 03 04 Average investment rate 19602000 FIGURE 115 The relationship between average growth of GDP per capita and average growth of investments to GDP ratio 19602000 17 From Correlates to Fundamental Causes 19 ARG AUS AUT BDI BEL BEN BOL BRA BRB CAN CHE CHL CHN CMR COG COL CRI DNK DOM DZA ECU EGY ESP FIN FRA GBR GHA GMB GRC GTM HKG HND IDN IND IRL IRN ISL ISR ITA JAM JOR JPN KEN KOR LKA LSO MEX MLI MOZ MUS MWI MYS NER NIC NLD NOR NPL NZL PAK PAN PER PHL PRT PRY RWA SEN SGP SLV SWE SYR TGO THA TTO TUN TUR TWN UGA URY USA VEN ZAF ZMB ZWE 002 000 002 004 006 Average growth rate of GDP per capita 19602000 0 2 4 6 8 10 12 Average years of schooling 19602000 FIGURE 116 The relationship between average growth of GDP per capita and average years of schooling 19602000 are likely driven at least in part by omitted factors affecting both investment and schooling on the one hand and economic growth on the other We investigate the role of physical and human capital in economic growth further in Chapter 3 One of the major points that emerges from the analysis in Chapter 3 is that focusing only on physical and human capital is not sufficient Both to understand the process of sustained economic growth and to account for large crosscountry differences in income we also need to understand why societies differ in the efficiency with which they use their physical and human capital Economists normally use the shorthand expression technology to capture factors other than physical and human capital that affect economic growth and performance It is therefore important to remember that variations in technology across countries include not only differences in production techniques and in the quality of machines used in production but also disparities in productive efficiency see in particular Chapter 21 on differences in productive efficiency resulting from the organization of markets and from market failures A detailed study of technology broadly construed is necessary for understanding both the worldwide process of economic growth and crosscountry differences The role of technology in economic growth is investigated in Chapter 3 and later chapters 17 From Correlates to Fundamental Causes The correlates of economic growth such as physical capital human capital and technology is our first topic of study But these are only proximate causes of economic growth and economic success even if we convince ourselves that there is an element of causality in the correlations 20 Chapter 1 Economic Growth and Economic Development The Questions shown above It would not be entirely satisfactory to explain the process of economic growth and crosscountry differences with technology physical capital and human capital since presumably there are reasons technology physical capital and human capital differ across countries If these factors are so important in generating crosscountry income differences and causing the takeoff into modern economic growth why do certain societies fail to improve their technologies invest more in physical capital and accumulate more human capital Let us return to Figure 18 to illustrate this point further This figure shows that South Korea and Singapore have grown rapidly over the past 50 years while Nigeria has failed to do so We can try to explain the successful performances of South Korea and Singapore by looking at the proximate causes of economic growth We can conclude as many have done that rapid capital accumulation has been a major cause of these growth miracles and debate the relative roles of human capital and technology We can simply blame the failure of Nigeria to grow on its inability to accumulate capital and to improve its technology These perspectives are undoubtedly informative for understanding the mechanics of economic successes and failures of the postwar era But at some level they do not provide answers to the central questions How did South Korea and Singapore manage to grow while Nigeria failed to take advantage of its growth opportunities If physical capital accumulation is so important why did Nigeria fail to invest more in physical capital If education is so important why are education levels in Nigeria still so low and why is existing human capital not being used more effectively The answer to these questions is related to the fundamental causes of economic growththe factors potentially affecting why societies make different technology and accumulation choices At some level fundamental causes are the factors that enable us to link the questions of economic growth to the concerns of the rest of the social sciences and ask questions about the roles of policies institutions culture and exogenous environmental factors At the risk of oversimplifying complex phenomena we can think of the following list of potential fundamen tal causes 1 luck or multiple equilibria that lead to divergent paths among societies with identical opportunities preferences and market structures 2 geographic differences that af fect the environment in which individuals live and influence the productivity of agriculture the availability of natural resources certain constraints on individual behavior or even individual attitudes 3 institutional differences that affect the laws and regulations under which individ uals and firms function and shape the incentives they have for accumulation investment and trade and 4 cultural differences that determine individuals values preferences and beliefs Chapter 4 presents a detailed discussion of the distinction between proximate and fundamental causes and what types of fundamental causes are more promising in explaining the process of economic growth and crosscountry income differences For now it is useful to briefly return to the contrast between South Korea and Singapore versus Nigeria and ask the questions even if we are not in a position to fully answer them yet Can we say that South Korea and Singapore owe their rapid growth to luck while Nigeria was unlucky Can we relate the rapid growth of South Korea and Singapore to geographic factors Can we relate them to institutions and policies Can we find a major role for culture Most detailed accounts of postwar economics and politics in these countries emphasize the role of growthpromoting policies in South Korea and Singaporeincluding the relative security of property rights and investment incentives provided to firms In contrast Nigerias postwar history is one of civil war military coups endemic corruption and overall an environment that failed to provide incentives to businesses to invest and upgrade their technologies It therefore seems necessary to look for fundamental causes of economic growth that make contact with these facts Jumping ahead a little it already appears implausible that luck can be the major explanation for the differences in postwar economic performance there were already significant economic differences between South Korea Singapore and Nigeria at the beginning of the postwar era It is also equally implausible to link the divergent fortunes of these countries 18 The Agenda 21 to geographic factors After all their geographies did not change but the growth spurts of South Korea and Singapore started in the postwar era Moreover even if Singapore benefited from being an island without hindsight one might have concluded that Nigeria had the best environment for growth because of its rich oil reserves3 Cultural differences across countries are likely to be important in many respects and the rapid growth of many Asian countries is often linked to certain Asian values Nevertheless cultural explanations are also unlikely to adequately explain fundamental causes since South Korean or Singaporean culture did not change much after the end of World War II while their rapid growth is a distinctly postwar phenomenon Moreover while South Korea grew rapidly North Korea whose inhabitants share the same culture and Asian values has endured one of the most disastrous economic performances of the past 50 years This admittedly quick and partial account suggests that to develop a better understanding of the fundamental causes of economic growth we need to look at institutions and policies that affect the incentives to accumulate physical and human capital and improve technology Institutions and policies were favorable to economic growth in South Korea and Singapore but not in Nigeria Understanding the fundamental causes of economic growth is largely about understanding the impact of these institutions and policies on economic incentives and why for example they have enhanced growth in South Korea and Singapore but not in Nigeria The intimate link between fundamental causes and institutions highlighted by this discussion motivates Part VIII which is devoted to the political economy of growth that is to the study of how institutions affect growth and why they differ across countries An important caveat should be noted at this point Discussions of geography institutions and culture can sometimes be carried out without explicit reference to growth models or even to growth empirics After all this is what many social scientists do outside the field of economics However fundamental causes can only have a big impact on economic growth if they affect parameters and policies that have a firstorder influence on physical and human capital and technology Therefore an understanding of the mechanics of economic growth is essential for evaluating whether candidate fundamental causes of economic growth could indeed play the role that is sometimes ascribed to them Growth empirics plays an equally important role in distinguishing among competing fundamental causes of crosscountry income differences It is only by formulating parsimonious models of economic growth and confronting them with data that we can gain a better understanding of both the proximate and the fundamental causes of economic growth 18 The Agenda The three major questions that have emerged from the brief discussion are 1 Why are there such large differences in income per capita and worker productivity across countries 2 Why do some countries grow rapidly while other countries stagnate 3 What sustains economic growth over long periods of time and why did sustained growth start 200 years or so ago 3 One can turn this reasoning around and argue that Nigeria is poor because of a natural resource curse that is precisely because it has abundant natural resources But this argument is not entirely compelling since there are other countries such as Botswana with abundant natural resources that have grown rapidly over the past 50 years More important the only plausible channel through which abundance of natural resources may lead to worse economic outcomes is related to institutional and political economy factors Such factors take us to the realm of institutional fundamental causes 22 Chapter 1 Economic Growth and Economic Development The Questions For each question a satisfactory answer requires a set of wellformulated models that illustrate the mechanics of economic growth and crosscountry income differences together with an investigation of the fundamental causes of the different trajectories which these nations have embarked upon In other words we need a combination of theoretical models and empirical work The traditional growth modelsin particular the basic Solow and the neoclassical models provide a good starting point and the emphasis they place on investment and human capital seems consistent with the patterns shown in Figures 115 and 116 However we will also see that technological differences across countries either because of their differential access to technological opportunities or because of differences in the efficiency of production are equally important Traditional models treat technology and market structure as given or at best as evolving exogenously rather like a black box But if technology is so important we ought to understand why and how it progresses and why it differs across countries This motivates our detailed study of endogenous technological progress and technology adoption Specifically we will try to understand how differences in technology may arise persist and contribute to differences in income per capita Models of technological change are also useful in thinking about the sources of sustained growth of the world economy over the past 200 years and the reasons behind the growth process that took off 200 years or so ago and has proceeded relatively steadily ever since Some of the other patterns encountered in this chapter will inform us about the types of models that have the greatest promise in explaining economic growth and crosscountry differences in income For example we have seen that crosscountry income differences can be accounted for only by understanding why some countries have grown rapidly over the past 200 years while others have not Therefore we need models that can explain how some countries can go through periods of sustained growth while others stagnate Nevertheless we have also seen that the postwar world income distribution is relatively stable at most spreading out slightly from 1960 to 2000 This pattern has suggested to many economists that we should focus on models that generate large permanent crosscountry differences in income per capita but not necessarily large permanent differences in growth rates at least not in the recent decades This argument is based on the following reasoning with substantially different longrun growth rates as in models of endogenous growth where countries that invest at different rates grow at permanently different rates we should expect significant divergence We saw above that despite some widening between the top and the bottom the crosscountry distribution of income across the world is relatively stable over the postwar era Combining the postwar patterns with the origins of income differences over the past several centuries suggests that we should look for models that can simultaneously account for long periods of significant growth differences and for a distribution of world income that ultimately becomes stationary though with large differences across countries The latter is particularly challenging in view of the nature of the global economy today which allows for the free flow of technologies and large flows of money and commodities across borders We therefore need to understand how the poor countries fell behind and what prevents them today from adopting and imitating the technologies and the organizations and importing the capital of richer nations And as the discussion in the previous section suggests all of these questions can be and perhaps should be answered at two distinct but related levels and in two corresponding steps The first step is to use theoretical models and data to understand the mechanics of economic growth This step sheds light on the proximate causes of growth and explains differences in income per capita in terms of differences in physical capital human capital and technology 19 References and Literature 23 and these in turn will be related to other variables such as preferences technology market structure openness to international trade and economic policies The second step is to look at the fundamental causes underlying these proximate factors and investigate why some societies are organized differently than others Why do societies have different market structures Why do some societies adopt policies that encourage economic growth while others put up barriers to technological change These questions are central to a study of economic growth and can only be answered by developing systematic models of the political economy of development and looking at the historical process of economic growth to generate data that can shed light on these fundamental causes Our next task is to systematically develop a series of models to understand the mechanics of economic growth I present a detailed exposition of the mathematical structure of a number of dynamic general equilibrium models that are useful for thinking about economic growth and related macroeconomic phenomena and I emphasize the implications of these models for the sources of differences in economic performance across societies Only by understanding these mechanics can we develop a useful framework for thinking about the causes of economic growth and income disparities 19 References and Literature The empirical material presented in this chapter is largely standard and parts of it can be found in many books though interpretations and emphases differ Excellent introductions with slightly different emphases are provided in Joness 1998 Chapter 1 and Weils 2005 Chapter 1 undergraduate economic growth textbooks Barro and SalaiMartin 2004 also present a brief discussion of the stylized facts of economic growth though their focus is on postwar growth and conditional convergence rather than the very large crosscountry income differences and the longrun perspective stressed here Excellent and very readable accounts of the key questions of economic growth with a similar perspective to the one here are provided in Helpman 2005 and in Aghion and Howitts new book 2008 Aghion and Howitt also provide a very useful introduction to many of the same topics discussed in this book Much of the data used in this chapter come from SummersHestons Penn World dataset latest version Summers Heston and Aten 2006 These tables are the result of a careful study by Robert Summers and Alan Heston to construct internationally comparable price indices and estimates of income per capita and consumption PPP adjustment is made possible by these data Summers and Heston 1991 give a lucid discussion of the methodology for PPP adjustment and its use in the Penn World tables PPP adjustment enables the construction of measures of income per capita that are comparable across countries Without PPP adjustment differences in income per capita across countries can be computed using the current exchange rate or some fundamental exchange rate There are many problems with such exchange rate based measures however The most important one is that they do not allow for the marked differences in relative prices and even overall price levels across countries PPP adjustment brings us much closer to differences in real income and real consumption GDP consumption and investment data from the Penn World tables are expressed in 1996 constant US dollars Information on workers economically active population consumption and investment are also from this dataset Life expectancy data are from the World Banks World Development Indicators CDROM and refer to the average life expectancy of males and females at birth This dataset also contains a range of other useful information Schooling data are from Barro and Lees 2001 dataset which contains internationally comparable information on years of schooling Throughout crosscountry figures use the World Bank labels to denote the identity The geometric average growth rate is more appropriate to use in the context of income per capita than the arithmetic average since the growth rate refers to proportional growth It can be easily verified from this formula that if yt1 1 g yt for all t then g tT g Historical data are from various works by Angus Maddison in particular Maddison 2001 2003 While these data are not as reliable as the estimates from the Penn World tables the general patterns they show are typically consistent with evidence from a variety of different sources Nevertheless there are points of contention For example in Figure 111 and Rodney 2005 estimates show a slow but relatively steady growth of income per capita in Western Europe starting in 1000 This growth pattern is disputed by some historians and economic historians Pomeranz 2000 who argues that income per capita in Western Europe and the Yangtze Valley in China were broadly comparable as late as 1800 is less widely quoted However this also receives support from recent research by Allen 2004 which documents that the levels of agricultural productivity in 1800 were comparable in Western Europe and China Acemoglu Johnson and Robinson 2002 also confirm that there were very limited income differences across countries as late as the 1500s and that the process of rapid economic growth started in the nineteenth century or perhaps in the late eighteenth century Recent research by Broadberry and Gupta 2006 also disputes Pomeranzs arguments and gives more support to a pattern in which there was already an income gap between Western Europe and China by the end of the eighteenth century 19 References and Literature 25 relationship between the level of schooling and economic growth than between the change in schooling and economic growth see Chapter 10 Finally the relationship between the level of schooling and economic growth may be partly spurious in the sense that it may be capturing the influence of some other omitted factors also correlated with the level of schooling if this is the case these omitted factors may be removed when we look at changes While we cannot reach a firm conclusion on these alternative explanations the strong correlation between average schooling and economic growth documented in Figure 116 is interesting in itself The narrowing of differences in income per capita in the world economy when countries are weighted by population is explored in SalaiMartin 2005 Deaton 2005 contains a critique of SalaiMartins approach The point that incomes must have been relatively equal around 1800 or before because there is a lower bound on real incomes necessary for the survival of an individual was first made by Maddison 1991 and was later popularized by Pritchett 1997 Maddisons estimates of GDP per capita and Acemoglu Johnson and Robinsons 2002 estimates based on urbanization confirm this conclusion The estimates of the density of income per capita reported in this chapter are similar to those used by Quah 1993 1997 and Jones 1997 These estimates use a nonparametric Gaussian kernel The specific details of the kernel estimation do not change the general shape of the densities Quah was also the first to emphasize the stratification in the world income distribution and the possible shift toward a bimodal distribution which is visible in Figure 13 He dubbed this the Twin Peaks phenomenon see also Durlauf and Quah 1999 Barro 1991 and Barro and SalaiMartin 1992 2004 emphasize the presence and importance of conditional convergence and argue against the relevance of the stratification pattern emphasized by Quah and others The estimate of conditional convergence of about 2 per year is from Barro and SalaiMartin 1992 Caselli Esquivel and Lefort 1996 show that panel data regressions lead to considerably higher rates of conditional convergence Marris 1982 and Baumol 1986 were the first economists to conduct crosscountry studies of convergence However the data at the time were of lower quality than the SummersHeston data and also were available for only a selected sample of countries Barros 1991 and Barro and SalaiMartins 1992 work using the SummersHeston dataset has been instrumental in generating renewed interest in crosscountry growth regressions The data on GDP growth and black real wages in South Africa are from Wilson 1972 Wages refer to real wages in gold mines Feinstein 2005 provides an excellent economic history of South Africa The implications of the British industrial revolution for real wages and living standards of workers are discussed in Mokyr 1993 Another example of rapid economic growth with falling real wages is provided by the experience of the Mexican economy in the early twentieth century see GomezGalvarriato 1998 There is also evidence that during this period the average height of the population might have been declining which is often associated with falling living standards see LopezAlonso and Porras Condey 2004 There is a major debate on the role of technology and capital accumulation in the growth experiences of East Asian nations particularly South Korea and Singapore See Young 1991 1995 for the argument that increases in physical capital and labor inputs explain almost all of the rapid growth in these two countries See Klenow and Rodriguez 1997 and Hsieh 2002 for the opposite point of view The difference between proximate and fundamental causes is discussed further in later chapters This distinction is emphasized in a different context by Diamond 1997 though it is also implicitly present in North and Thomass 1973 classic book It is discussed in detail in the context of longrun economic development and economic growth in Acemoglu Johnson and Robinson 2005a I revisit these issues in greater detail in Chapter 4 2 The Solow Growth Model T he previous chapter introduced a number of basic facts and posed the main questions concerning the sources of economic growth over time and the causes of differences in economic performance across countries These questions are central not only to growth theory but also to macroeconomics and the social sciences more generally Our next task is to develop a simple framework that can help us think about the proximate causes and the mechanics of the process of economic growth and crosscountry income differences We will use this framework both to study potential sources of economic growth and also to perform simple comparative statics to gain an understanding of which country characteristics are conducive to higher levels of income per capita and more rapid economic growth Our starting point is the socalled SolowSwan model named after Robert Bob Solow and Trevor Swan or simply the Solow model named after the more famous of the two economists These economists published two pathbreaking articles in the same year 1956 Solow 1956 Swan 1956 introducing the Solow model Bob Solow later developed many implications and applications of this model and was awarded the Nobel prize in economics for his contributions This model has shaped the way we approach not only economic growth but also the entire field of macroeconomics Consequently a byproduct of our analysis of this chapter is a detailed exposition of a workhorse model of macroeconomics The Solow model is remarkable in its simplicity Looking at it today one may fail to appreciate how much of an intellectual breakthrough it was Before the advent of the Solow growth model the most common approach to economic growth built on the model developed by Roy Harrod and Evsey Domar Harrod 1939 Domar 1946 The HarrodDomar model emphasized potential dysfunctional aspects of economic growth for example how economic growth could go handinhand with increasing unemployment see Exercise 223 on this model The Solow model demonstrated why the HarrodDomar model was not an attractive place to start At the center of the Solow growth model distinguishing it from the Harrod Domar model is the neoclassical aggregate production function This function not only enables the Solow model to make contact with microeconomics but as we will see in the next chapter it also serves as a bridge between the model and the data An important feature of the Solow model which is shared by many models presented in this book is that it is a simple and abstract representation of a complex economy At first it may appear too simple or too abstract After all to do justice to the process of growth or macroeconomic equilibrium we have to consider households and individuals with different tastes abilities incomes and roles in society various sectors and multiple social interactions The Solow model cuts through these complications by constructing a simple one 26 21 The Economic Environment of the Basic Solow Model 27 good economy with little reference to individual decisions Therefore the Solow model should be thought of as a starting point and a springboard for richer models In this chapter I present the basic Solow model The closely related neoclassical growth model is presented in Chapter 8 21 The Economic Environment of the Basic Solow Model Economic growth and development are dynamic processes and thus necessitate dynamic models Despite its simplicity the Solow growth model is a dynamic general equilibrium model though importantly many key features of dynamic general equilibrium models emphasized in Chapter 5 such as preferences and dynamic optimization are missing in this model The Solow model can be formulated in either discrete or continuous time I start with the discretetime version because it is conceptually simpler and more commonly used in macroeconomic applications However many growth models are formulated in continuous time and I then provide a detailed exposition of the continuoustime version of the Solow model and show that it is often more convenient to work with 211 Households and Production Consider a closed economy with a unique final good The economy is in discrete time running to an infinite horizon so that time is indexed by t 0 1 2 Time periods here may correspond to days weeks or years For now we do not need to specify the time scale The economy is inhabited by a large number of households Throughout the book I use the terms households individuals and agents interchangeably The Solow model makes rela tively few assumptions about households because their optimization problem is not explicitly modeled This lack of optimization on the household side is the main difference between the Solow and the neoclassical growth models The latter is the Solow model plus dynamic con sumer household optimization To fix ideas you may want to assume that all households are identical so that the economy trivially admits a representative householdmeaning that the demand and labor supply side of the economy can be represented as if it resulted from the behavior of a single household The representative household assumption is discussed in detail in Chapter 5 What do we need to know about households in this economy The answer is not much We have not yet endowed households with preferences utility functions Instead for now households are assumed to save a constant exogenous fraction s 0 1 of their disposable incomeregardless of what else is happening in the economy This assumption is the same as that used in basic Keynesian models and the HarrodDomar model mentioned above It is also at odds with reality Individuals do not save a constant fraction of their incomes if they did then an announcement by the government that there will be a large tax increase next year should have no effect on their savings decisions which seems both unreasonable and empirically incorrect Nevertheless the exogenous constant saving rate is a convenient starting point and we will spend a lot of time in the rest of the book analyzing how consumers behave and make intertemporal choices The other key agents in the economy are firms Firms like consumers are highly hetero geneous in practice Even within a narrowly defined sector of an economy no two firms are identical But again for simplicity let us start with an assumption similar to the representa tive household assumption but now applied to firms suppose that all firms in this economy have access to the same production function for the final good or that the economy admits a 28 Chapter 2 The Solow Growth Model representative firm with a representative or aggregate production function The conditions under which this representive firm assumption is reasonable are also discussed in Chapter 5 The aggregate production function for the unique final good is written as Yt FKt Lt At 21 where Yt is the total amount of production of the final good at time t Kt is the capital stock Lt is total employment and At is technology at time t Employment can be measured in different ways For example we may want to think of Lt as corresponding to hours of employment or to number of employees The capital stock Kt corresponds to the quantity of machines or more specifically equipment and structures used in production and it is typically measured in terms of the value of the machines There are also multiple ways of thinking of capital and equally many ways of specifying how capital comes into existence Since the objective here is to start with a simple workable model I make the rather sharp simplifying assumption that capital is the same as the final good of the economy However instead of being consumed capital is used in the production process of more goods To take a concrete example think of the final good as corn Corn can be used both for consumption and as an input as seed for the production of more corn tomorrow Capital then corresponds to the amount of corn used as seed for further production Technology on the other hand has no natural unit and At is simply a shifter of the production function 21 For mathematical convenience I often represent At in terms of a number but it is useful to bear in mind that at the end of the day it is a representation of a more abstract concept As noted in Chapter 1 we may often want to think of a broad notion of technology incorporating the effects of the organization of production and of markets on the efficiency with which the factors of production are utilized In the current model At represents all these effects A major assumption of the Solow growth model and of the neoclassical growth model we will study in Chapter 8 is that technology is free it is publicly available as a nonexcludable nonrival good Recall that a good is nonrival if its consumption or use by others does not pre clude an individuals consumption or use It is nonexcludable if it is impossible to prevent another person from using or consuming it Technology is a good candidate for a nonexclud able nonrival good once the society has some knowledge useful for increasing the efficiency of production this knowledge can be used by any firm without impinging on the use of it by others Moreover it is typically difficult to prevent firms from using this knowledge at least once it is in the public domain and is not protected by patents For example once the society knows how to make wheels everybody can use that knowledge to make wheels without di minishing the ability of others to do the same thus making the knowledge to produce wheels nonrival Moreover unless somebody has a wellenforced patent on wheels anybody can de cide to produce wheels thus making the knowhow to produce wheels nonexcludable The implication of the assumptions that technology is nonrival and nonexcludable is that At is freely available to all potential firms in the economy and firms do not have to pay for making use of this technology Departing from models in which technology is freely available is a major step toward understanding technological progress and will be our focus in Part IV As an aside note that some authors use xt or Kt when working with discrete time and reserve the notation xt or Kt for continuous time Since I go back and forth between continuous and discrete time I use the latter notation throughout When there is no risk of confusion I drop the time arguments but whenever there is the slightest risk of confusion I err on the side of caution and include the time arguments Let us next impose the following standard assumptions on the aggregate production function 21 The Economic Environment of the Basic Solow Model 29 Assumption 1 Continuity Differentiability Positive and Diminishing Marginal Products and Constant Returns to Scale The production function F R3 R is twice differentiable in K and L and satisfies FKK L A FK L A K 0 FLK L A FK L A L 0 FKKK L A 2FK L A K2 0 FLLK L A 2FK L A L2 0 Moreover F exhibits constant returns to scale in K and L All of the components of Assumption 1 are important First the notation F R3 R implies that the production function takes nonnegative arguments ie K L R and maps to nonnegative levels of output Y R It is natural that the level of capital and the level of employment should be positive Since A has no natural units it could have been negative But there is no loss of generality in restricting it to be positive The second important aspect of Assumption 1 is that F is a continuous function in its arguments and is also differentiable There are many interesting production functions that are not differentiable and some interesting ones that are not even continuous But working with differentiable functions makes it possible to use differential calculus and the loss of some generality is a small price to pay for this convenience Assumption 1 also specifies that marginal products are positive so that the level of production increases with the amount of inputs this restriction also rules out some potential production functions and can be relaxed without much complication see Exercise 28 More importantly Assumption 1 requires that the marginal products of both capital and labor are diminishing that is FKK 0 and FLL 0 so that more capital holding everything else constant increases output by less and less And the same applies to labor This property is sometimes also referred to as diminishing returns to capital and labor The degree of diminishing returns to capital plays a very important role in many results of the basic growth model In fact the presence of diminishing returns to capital distinguishes the Solow growth model from its antecedent the HarrodDomar model see Exercise 223 The other important assumption is that of constant returns to scale Recall that F exhibits constant returns to scale in K and L if it is linearly homogeneous homogeneous of degree 1 in these two variables More specifically Definition 21 Let K N The function g RK2 R is homogeneous of degree m in x R and y R if gλx λy z λmgx y z for all λ R and z RK It can be easily verified that linear homogeneity implies that the production function F is concave though not strictly so see Exercise 22 Linearly homogeneous constant returns to scale production functions are particularly useful because of the following theorem Theorem 21 Eulers Theorem Suppose that g RK2 R is differentiable in x R and y R with partial derivatives denoted by gx and gy and is homogeneous of degree m in x and y Then mgx y z gxx y zx gyx y zy for all x R y R and z RK Moreover gxx y z and gyx y z are themselves homogeneous of degree m 1in x and y Proof We have that g is differentiable and λm gx y z gλx λy z 22 Differentiate both sides of 22 with respect to λ which gives mλm1gx y z g xλx λy zλx g yλx λy zλy for any λ Setting λ 1 yields the first result To obtain the second result differentiate both sides of 22 with respect to x λg xλx λy z λm g xx y z Dividing both sides by λ establishes the desired result 21 The Economic Environment of the Basic Solow Model 31 The complementary slackness formulation ensures that labor market clearing does not happen at a negative wageor that if labor demand happens to be low enough employment could be below Lt at zero wage However this will not be an issue in most of the models studied in this book because Assumption 1 and competitive labor markets ensure that wages are strictly positive see Exercise 21 In view of this result I use the simpler condition 23 throughout and denote both labor supply and employment at time t by Lt The households also own the capital stock of the economy and rent it to firms Let us denote the rental price of capital at time t by Rt The capital market clearing condition is similar to 23 and requires the demand for capital by firms to be equal to the supply of capital by households Kt Kt where Kt is the supply of capital by households and Kt is the demand by firms Capital market clearing is straightforward to ensure in the class of models analyzed in this book In particular it is sufficient that the amount of capital Kt used in production at time t from firms optimization behavior be consistent with households endowments and saving behavior Let us take households initial holdings of capital K0 0 as given as part of the description of the environment For now how this initial capital stock is distributed among the households is not important since households optimization decisions are not modeled explicitly and the economy is simply assumed to save a fraction s of its income When we turn to models with household optimization below an important part of the description of the environment will be to specify the preferences and the budget constraints of households At this point I could also introduce the price of the final good at time t say Pt But there is no need since there is a choice of a numeraire commodity in this economy whose price will be normalized to 1 In particular as discussed in greater detail in Chapter 5 Walrass Law implies that the price of one of the commodities the numeraire should be normalized to 1 In fact throughout I do something stronger and normalize the price of the final good to 1 in all periods Ordinarily one cannot choose more than one numeraireotherwise one would be fixing the relative price between the numeraires But as explained in Chapter 5 we can build on an insight by Kenneth Arrow Arrow 1964 that it is sufficient to price securities assets that transfer one unit of consumption from one date or state of the world to another In the context of dynamic economies this implies that we need to keep track of an interest rate across periods denoted by rt which determines intertemporal prices and enables us to normalize the price of the final good to 1 within each period Naturally we also need to keep track of the wage rate wt which determines the price of labor relative to the final good at any date t This discussion highlights a central fact all of the models in this book should be thought of as general equilibrium economies in which different commodities correspond to the same good at different dates Recall from basic general equilibrium theory that the same good at different dates or in different states or localities is a different commodity Therefore in almost all of the models in this book there will be an infinite number of commodities since time runs to infinity This raises a number of special issues which are discussed in Chapter 5 and later Returning to the basic Solow model the next assumption is that capital depreciates meaning that machines that are used in production lose some of their value because of wear and tear In terms of the corn example above some of the corn that is used as seed is no longer available for consumption or for use as seed in the following period Let us assume that this depreciation takes an exponential form which is mathematically very tractable Thus capital depreciates exponentially at the rate δ 0 1 so that out of 1 unit of capital this period only 1 δ is left for next period Though depreciation here stands for the wear and tear of the machinery it can also represent the replacement of old machines by new ones in more realistic models see Chapter 14 32 Chapter 2 The Solow Growth Model The loss of part of the capital stock affects the interest rate rate of return on savings faced by households Given the assumption of exponential depreciation at the rate δ and the normalization of the price of the final good to 1 the interest rate faced by the households is rt Rt δ where recall that Rt is the rental price of capital at time t A unit of final good can be consumed now or used as capital and rented to firms In the latter case a household receives Rt units of good in the next period as the rental price for its savings but loses δ units of its capital holdings since δ fraction of capital depreciates over time Thus the household has given up one unit of commodity dated t 1and receives 1 rt Rt 1 δ units of commodity dated t so that rt Rt δ The relationship between rt and Rt explains the similarity between the symbols for the interest rate and the rental rate of capital The interest rate faced by households plays a central role in the dynamic optimization decisions of households below In the Solow model this interest rate does not directly affect the allocation of resources 213 Firm Optimization and Equilibrium We are now in a position to look at the optimization problem of firms and the competitive equilibrium of this economy Throughout the book I assume that the objective of firms is to maximize profits Given the assumption that there is an aggregate production function it is sufficient to consider the problem of a representative firm Throughout unless otherwise stated I also assume that capital markets are functioning so firms can rent capital in spot markets For a given technology level At and given factor prices Rt and wt the profit maximization problem of the representative firm at time t can be represented by the following static problem max K0L0 FK L At RtK wtL 25 When there are irreversible investments or costs of adjustments as discussed for example in Section 78 the maximization problem of firms becomes dynamic But in the absence of these features maximizing profits separately at each date t is equivalent to maximizing the net present discounted value of profits This feature simplifies the analysis considerably A couple of additional features are worth noting 1 The maximization problem is set up in terms of aggregate variables which given the representative firm is without any loss of generality 2 There is nothing multiplying the F term since the price of the final good has been normalized to 1 Thus the first term in 25 is the revenues of the representative firm or the revenues of all of the firms in the economy 3 This way of writing the problem already imposes competitive factor markets since the firm is taking as given the rental prices of labor and capital wt and Rt which are in terms of the numeraire the final good 4 This problem is concave since F is concave see Exercise 22 An important aspect is that because F exhibits constant returns to scale Assumption 1 the maximization problem 25 does not have a welldefined solution see Exercise 23 either there does not exist any K L that achieves the maximum value of this program which is infinity or K L 0 or multiple values of K L will achieve the maximum value of this program when this value happens to be 0 This problem is related to the fact that in a world with constant returns to scale the size of each individual firm is not determinate only aggregates are determined The same problem arises here because 25 is written without imposing the condition that factor markets should clear A competitive equilibrium 21 The Economic Environment of the Basic Solow Model 33 requires that all firms and thus the representative firm maximize profits and factor markets clear In particular the demands for labor and capital must be equal to the supplies of these factors at all times unless the prices of these factors are equal to zero which is ruled out by Assumption 1 This observation implies that the representative firm should make zero profits since otherwise it would wish to hire arbitrarily large amounts of capital and labor exceeding the supplies which are fixed It also implies that total demand for labor L must be equal to the available supply of labor Lt Similarly the total demand for capital K should equal the total supply Kt If this were not the case and L Lt then there would be an excess supply of labor and the wage would be equal to zero But this is not consistent with firm maximization since given Assumption 1 the representative firm would then wish to hire an arbitrarily large amount of labor exceeding the supply This argument combined with the fact that F is differentiable Assumption 1 implies that given the supplies of capital and labor at time t Kt and Lt factor prices must satisfy the following familiar conditions equating factor prices to marginal products1 wt FLKt Lt At 26 and Rt FKKt Lt At 27 Eulers Theorem Theorem 21 then verifies that at the prices 26 and 27 firms or the representative firm make zero profits Proposition 21 Suppose Assumption 1 holds Then in the equilibrium of the Solow growth model firms make no profits and in particular Yt wtLt RtKt Proof This result follows immediately from Theorem 21 for the case of constant returns to scale m 1 Since firms make no profits in equilibrium the ownership of firms does not need to be specified All we need to know is that firms are profitmaximizing entities In addition to these standard assumptions on the production function the following bound ary conditions the Inada conditionsare often imposed in the analysis of economic growth and macroeconomic equilibria Assumption 2 Inada Conditions F satisfies the Inada conditions lim K0 FKK L A and lim K FKK L A 0 for all L 0 and all A lim L0 FLK L A and lim L FLK L A 0 for all K 0 and all A Moreover F0 L A 0 for all L and A The role of these conditionsespecially in ensuring the existence of interior equilibria will become clear later in this chapter They imply that the first units of capital and labor 1 An alternative way to derive 26 and 27 is to consider the cost minimization problem of the representative firm which takes the form of minimizing rK wL with respect to K and L subject to the constraint that FK L A Y for some level of output Y This problem has a unique solution for any given level of Y Then imposing market clearing that is Y FK L A with K and L corresponding to the supplies of capital and labor yields 26 and 27 34 Chapter 2 The Solow Growth Model 0 K A FK L A FK L A 0 K B FIGURE 21 Production functions A satisfies the Inada conditions in Assumption 2 while B does not are highly productive and that when capital or labor are sufficiently abundant their marginal products are close to zero The condition that F0 L A 0 for all L and A makes capital an essential input This aspect of the assumption can be relaxed without any major implications for the results in this book Figure 21 shows the production function FK L A as a function of K for given L and A in two different cases in panel A the Inada conditions are satisfied while in panel B they are not I refer to Assumptions 1 and 2 which can be thought of as the neoclassical technology assumptions throughout much of the book For this reason they are numbered independently from the equations theorems and proposition in this chapter 22 The Solow Model in Discrete Time I next present the dynamics of economic growth in the discretetime Solow model 221 Fundamental Law of Motion of the Solow Model Recall that K depreciates exponentially at the rate δ so that the law of motion of the capital stock is given by Kt 1 1 δ Kt It 28 where It is investment at time t From national income accounting for a closed economy the total amount of final good in the economy must be either consumed or invested thus Yt Ct It 29 where Ct is consumption2 Using 21 28 and 29 any feasible dynamic allocation in this economy must satisfy Kt 1 FKt Lt At 1 δKt Ct 2 In addition we can introduce government spending Gt on the righthand side of 29 Government spending does not play a major role in the Solow growth model thus its introduction is relegated to Exercise 27 22 The Solow Model in Discrete Time 35 for t 0 1 The question is to determine the equilibrium dynamic allocation among the set of feasible dynamic allocations Here the behavioral rule that households save a constant fraction of their income simplifies the structure of equilibrium considerably this is a behavioral rule since it is not derived from the maximization of a welldefined utility function One implication of this assumption is that any welfare comparisons based on the Solow model have to be taken with a grain of salt since we do not know what the preferences of the households are Since the economy is closed and there is no government spending aggregate investment is equal to savings St It Yt Ct The assumption that households save a constant fraction s 0 1 of their income can be expressed as St sYt 210 which in turn implies that they consume the remaining 1 s fraction of their income and thus Ct 1 s Yt 211 In terms of capital market clearing 210 implies that the supply of capital for time t 1 resulting from households behavior can be expressed as Kt 1 1 δKt St 1 δKt sYt Setting supply and demand equal to each other and using 21 and 28 yields the fundamental law of motion of the Solow growth model Kt 1 sFKt Lt At 1 δKt 212 This is a nonlinear difference equation The equilibrium of the Solow growth model is described by 212 together with laws of motion for Lt and At 222 Definition of Equilibrium The Solow model is a mixture of an oldstyle Keynesian model and a modern dynamic macro economic model Households do not optimize when it comes to their savings or consumption decisions Instead their behavior is captured by 210 and 211 Nevertheless firms still maximize profits and factor markets clear Thus it is useful to start defining equilibria in the way that is customary in modern dynamic macro models Definition 22 In the basic Solow model for a given sequence of Lt At t0 and an initial capital stock K0 an equilibrium path is a sequence of capital stocks output levels consumption levels wages and rental rates Kt Yt Ct wt Rt t0 such that Kt satisfies 212 Yt is given by 21 Ct is given by 211 and wt and Rt are given by 26 and 27 respectively The most important point to note about Definition 22 is that an equilibrium is defined as an entire path of allocations and prices An economic equilibrium does not refer to a static object it specifies the entire path of behavior of the economy Note also that Definition 22 incorporates the market clearing conditions 26 and 27 into the definition of equilibrium This practice 223 Equilibrium without Population Growth and Technological Progress It is useful to start with the following assumptions which are relaxed later in this chapter 1 There is no population growth total population is constant at some level L 0 Moreover since households supply labor inelastically this implies Lt L 2 There is no technological progress so that At A Let us define the capitallabor ratio of the economy as kt Kt L which is a key object for the analysis Now using the assumption of constant returns to scale output income per capita yt YtL can be expressed as yt FKtL 1 A fkt In other words with constant returns to scale output per capita is simply a function of the capitallabor ratio Note that fk here depends on A so I could have written fk A I do not do this to simplify the notation and also because until Section 27 there will be no technological progress Thus for now A is constant and can be normalized to A 1 The marginal product and the rental price of capital are then given by the derivative of F with respect to its first argument which is fk The marginal product of labor and the wage rate are then obtained from Theorem 21 so that Rt fkt 0 and wt fkt ktfkt 0 22 The Solow Model in Discrete Time 37 It can easily be verified that this production function satisfies Assumptions 1 and 2 including the constant returns to scale feature imposed in Assumption 1 Dividing both sides by Lt the per capita production function in 214 becomes yt Aktα where yt again denotes output per worker and kt is capitallabor ratio as defined in 213 The representation of factor prices as in 215 can also be verified From the per capita production function representation in particular 215 the rental price of capital can be expressed as Rt Aktα kt αAkt1α Alternatively in terms of the original production function 216 the rental price of capital in 27 is given by Rt αAKtα1Lt1α αAkt1α which is equal to the previous expression and thus verifies the form of the marginal product given in 215 Similarly from 215 wt Aktα αAkt1α kt 1 αAKtαLtα which verifies the alternative expression for the wage rate in 26 Returning to the analysis with the general production function the per capita representation of the aggregate production function enables us to divide both sides of 212 by L to obtain the following simple difference equation for the evolution of the capitallabor ratio kt 1 sf kt 1 δkt 217 Since this difference equation is derived from 212 it also can be referred to as the equilibrium difference equation of the Solow model it describes the equilibrium behavior of the key object of the model the capitallabor ratio The other equilibrium quantities can all be obtained from the capitallabor ratio kt At this point let us also define a steadystate equilibrium for this model Definition 23 A steadystate equilibrium without technological progress and population growth is an equilibrium path in which kt k for all t In a steadystate equilibrium the capitallabor ratio remains constant Since there is no population growth this implies that the level of the capital stock will also remain constant Mathematically a steadystate equilibrium corresponds to a stationary point of the equilibrium difference equation 217 Most of the models in this book admit a steadystate equilibrium This is also the case for this simple model The existence of a steady state can be seen by plotting the difference equation that governs the equilibrium behavior of this economy 217 which is done in Figure 22 The thick curve represents the righthand side of 217 and the dashed line corresponds to the 45 line Their positive intersection gives the steadystate value of the capitallabor ratio k which satisfies fk k δ s investment per capita at the steadystate equilibrium equal to δk while the vertical distance between the function fk and the δk line at k gives the level of consumption per capita Clearly the sum of these two terms make up fk 40 Chapter 2 The Solow Growth Model 0 k kt Output Consumption Investment fkt kt sfkt fk sfk FIGURE 24 Investment and consumption in the steadystate equilibrium where the last equality in 221 uses 215 Since f kk is everywhere strictly decreasing there can only exist a unique value k that satisfies 218 Equations 219 and 220 then follow by definition Through a series of examples Figure 25 shows why Assumptions 1 and 2 cannot be dispensed with for establishing the existence and uniqueness results in Proposition 22 In the first two panels the failure of Assumption 2 leads to a situation in which there is no steady state equilibrium with positive activity while in the third panel the failure of Assumption 1 leads to nonuniqueness of steady states So far the model is very parsimonious it does not have many parameters and abstracts from many features of the real world An understanding of how crosscountry differences in certain parameters translate into differences in growth rates or output levels is essential for our focus This connection will be made in the next proposition But before doing so let us generalize the production function in one simple way and assume that f k A f k where A 0 so that A is a shift parameter with greater values corresponding to greater productivity of factors This type of productivity is referred to as Hicksneutral see below For now it is simply a convenient way of parameterizing productivity differences across countries Since f k satisfies the regularity conditions imposed above so does f k Proposition 23 Suppose Assumptions 1 and 2 hold and f k A f k Denote the steady state level of the capitallabor ratio by kA s δ and the steadystate level of output by yA s δ when the underlying parameters are A s and δ Then kA s δ A 0 kA s δ s 0 and kA s δ δ 0 yA s δ A 0 yA s δ s 0 and yA s δ δ 0 42 Chapter 2 The Solow Growth Model Therefore countries with higher saving rates and better technologies will have higher capital labor ratios and will be richer Those with greater technological depreciation will tend to have lower capitallabor ratios and will be poorer All of the results in Proposition 23 are intuitive and they provide us with a first glimpse of the potential determinants of the capitallabor ratios and output levels across countries The same comparative statics with respect to A and δ also apply to c However it is straightforward to see that c is not monotone in the saving rate eg think of the extreme case where s 1 In fact there exists a unique saving rate sgold referred to as the golden rule saving rate which maximizes the steadystate level of consumption Since we are treating the saving rate as an exogenous parameter and have not specified the objective function of households yet we cannot say whether the golden rule saving rate is better than some other saving rate It is nevertheless interesting to characterize what this golden rule saving rate corresponds to To do this let us first write the steadystate relationship between c and s and suppress the other parameters cs 1 sf ks f ks δks where the second equality exploits the fact that in steady state sf k δk Now differentiating this second line with respect to s again using the Implicit Function Theorem we obtain cs s f ks δk s 222 Let us define the golden rule saving rate sgold to be such that csgolds 0 The corre sponding steadystate golden rule capital stock is defined as k gold These quantities and the relationship between consumption and the saving rate are plotted in Figure 26 The next prop osition shows that sgold and k gold are uniquely defined Proposition 24 In the basic Solow growth model the highest level of steadystate con sumption is reached for sgold with the corresponding steadystate capital level k gold such that f k gold δ 223 Proof By definition csgolds 0 From Proposition 23 ks 0 thus 222 can be equal to zero only when f ksgold δ Moreover when f ksgold δ it can be verified that 2csgolds2 0 so f ksgold δ indeed corresponds to a local maximum That f ksgold δ also yields the global maximum is a consequence of the following observations for all s 0 1 we have ks 0 and moreover when s sgold f ks δ 0 by the concavity of f so css 0 for all s sgold By the converse argument css 0 for all s sgold Therefore only sgold satisfies f ks δ and gives the unique global maximum of consumption per capita In other words there exists a unique saving rate sgold and also a unique corresponding capitallabor ratio k gold given by 223 that maximize the level of steadystate consump tion When the economy is below k gold a higher saving rate will increase consumption whereas when the economy is above k gold steadystate consumption can be raised by saving less In the latter case lower savings translate into higher consumption because the capitallabor ratio of the economy is too high households are investing too much and not consuming enough This is the essence of the phenomenon of dynamic inefficiencydiscussed in greater detail in Chapter 9 For now recall that there is no explicit utility function here so statements about inefficiency must be considered with caution In fact the reason this type of dynamic inefficiency does not generally apply when consumptionsaving decisions are endogenized may be apparent to many of you 44 Chapter 2 The Solow Growth Model difference equation 217 Thus the question is whether 217 will take us to the unique steady state starting from an arbitrary initial capitallabor ratio Before answering this question recall some definitions and key results from the theory of dynamical systems Appendix B provides more details and a number of further results Consider the nonlinear system of autonomous difference equations xt 1 Gxt 224 where xt Rn and G Rn Rn where n R Let x be a fixed point of the mapping G that is x Gx I refer to x as a steady state of the difference equation 2245 The relevant notion of stability is introduced in the next definition Definition 24 A steady state x is locally asymptotically stable if there exists an open set Bx containing x such that for any solution xt t0 to 224 with x0 Bx xt x Moreover x is globally asymptotically stable if for all x0 Rn for any solution xt t0 xt x The next theorem provides the main results on the stability properties of systems of linear difference equations The following theorems are special cases of the results presented in Appendix B Theorem 22 Stability for Systems of Linear Difference Equations Consider the following linear difference equation system xt 1 Axt b 225 with initial value x0 where xt Rn for all t A is an n n matrix and b is a n 1column vector Let x be the steady state of the difference equation given by Ax b x Suppose that all of the eigenvalues of A are strictly inside the unit circle in the complex plane Then the steady state of the difference equation 225 x is globally asymptotically stable in the sense that starting from any x0 Rn the unique solution xt t0 satisfies xt x Unfortunately much less can be said about nonlinear systems but the following is a standard local stability result Theorem23LocalStabilityforSystemsofNonlinearDifferenceEquations Con sider the following nonlinear autonomous system xt 1 Gxt 226 with initial value x0 where G Rn Rn Let x be a steady state of this system that is Gx x and suppose that G is differentiable at x Define A DGx 5 Various other terms are used to describe x for example equilibrium point or critical point Since these other terms have different meanings in economics I refer to x as a steady state throughout xt 1 x gαt gx xt x fk f0 kf k kf k Recall that when the economy starts with too little capital relative to its labor supply the capitallabor ratio will increase Thus the marginal product of capital will fall due to diminishing returns to capital and the wage rate will increase Conversely if it starts with too much capital it will decumulate capital and in the process the wage rate will decline and the rate of return to capital will increase Recall that the time periods t 0 1 can refer to days weeks months or years In some sense the time unit is not important This arbitrariness suggests that perhaps it is more convenient to look at dynamics by making the time unit as small as possible that is by going to continuous time While much of modern macroeconomics outside of growth theory uses discretetime models many growth models are formulated in continuous time 24 The Solow Model in Continuous Time 49 Recall that kt Kt Lt which implies that kt kt Kt Kt Lt Lt Kt Kt n where I used the fact that from 232 LtLt n From the limiting argument leading to equation 231 in the previous subsection the law of motion of the capital stock is given by Kt sFKt Lt At δKt Using the definition of kt as the capitallabor ratio and the constant returns to scale properties of the production function the fundamental law of motion of the Solow model in continuous time is obtained as kt kt s f kt kt n δ 233 where following usual practice I have transformed the lefthand side to the proportional change in the capitallabor ratio by dividing both sides by kt6 Definition 25 In the basic Solow model in continuous time with population growth at the rate n no technological progress and an initial capital stock K0 an equilibrium path is given by paths sequences of capital stocks labor output levels consumption levels wages and rental rates Kt Lt Yt Ct wt Rt t0 such that Lt satisfies 232 kt KtLt satisfies 233 Yt is given by 21 Ct is given by 211 and wt and Rt are given by 26 and 27 respectively As before a steadystate equilibrium involves kt remaining constant at some level k It is easy to verify that the equilibrium differential equation 233 has a unique steady state at k which is given by a slight modification of 218 to incorporate population growth f k k n δ s 234 In other words going from discrete to continuous time has not changed any of the basic economic features of the model Thus the steady state can again be plotted in a diagram similar to Figure 21 except that it now also incorporates population growth This is done in Figure 28 which also highlights that the logic of the steady state is the same with population growth as it was without population growth The amount of investment sf k is used to replenish the capitallabor ratio but now there are two reasons for replenishments The capital stock depreciates exponentially at the flow rate δ In addition the capital stock must also increase as 6 Throughout I adopt the notation xt t0 to denote the continuoustime path of variable xt An alternative notation often used in the literature is xt t 0 I prefer the former both because it is slightly more compact and also because it is more similar to the discretetime notation for the time path of a variable xt t0 When referring to xt t0 I use the terms path sequence and function of time t interchangeably Consider the basic Solow growth model in continuous time and suppose that Assumptions 1 and 2 hold Then there exists a unique steadystate equilibrium where the capitallabor ratio is equal to k 0 and satisfies 234 Per capita output is given by y fk and per capita consumption is given by c 1 sfk 25 Transitional Dynamics in the ContinuousTime Solow Model 51 capita The reason for this is simple a higher population growth rate means there is more labor to use the existing amount of capital which only accumulates slowly and consequently the equilibrium capitallabor ratio ends up lower This result implies that countries with higher population growth rates will have lower incomes per person or per worker 25 Transitional Dynamics in the ContinuousTime Solow Model The analysis of transitional dynamics and stability with continuous time yields similar results to those in Section 23 but the analysis is slightly simpler Let us first recall the basic results on the stability of systems of differential equations Once again further details are contained in Appendix B Theorem 24 Stability of Linear Differential Equations Consider the following lin ear differential equation system xt Axt b 235 with initial value x0 where xt Rn for all t A is an n n matrix and b is a n 1 column vector Let x be the steady state of the system given by Ax b 0 Suppose that all eigenvalues of A have negative real parts Then the steady state of the differential equation 235 x is globally asymptotically stable in the sense that starting from any x0 Rn xt x Theorem 25 Local Stability of Nonlinear Differential Equations Consider the following nonlinear autonomous differential equation xt Gxt 236 with initial value x0 where G Rn Rn Let x be a steady state of this system that is Gx 0 and suppose that G is differentiable at x Define A DGx and suppose that all eigenvalues of A have negative real parts Then the steady state of the differential equation 236 x is locally asymptotically stable in the sense that there exists an open neighborhood of x Bx Rn such that starting from any x0 Bx xt x Once again an immediate corollary is as follows Corollary 22 1 Let xt R Then the steady state of the linear differential equation xt axt is globally asymptotically stable in the sense that xt 0 if a 0 2 Let g R R be differentiable in the neighborhood of the steady state x defined by gx 0 and suppose that gx 0 Then the steady state of the nonlinear differential equation xt gxt x is locally asymptotically stable 3 Let g R R be continuously differentiable Suppose that gx 0 and that gx 0 for all x xand gx 0 for all x x Then the steady state of the nonlinear dif ferential equation xt gxt x is globally asymptotically stable that is starting with any x0 xt x Proof See Exercise 210 Example 23 The Constant Elasticity of Substitution Production Function The CobbDouglas production function which features an elasticity of substitution equal to 1 is a special case of the Constant Elasticity of Substitution CES production function first introduced by Arrow et al 1961 This production function imposes a constant elasticity σ not necessarily equal to 1 Consider a vectorvalued index of technology At AHt AKt ALt Then the CES production function can be written as Yt FKt Lt At AHt YAKt Ktσσ 1 1 γALtLtσσ 1 1σ where AHt 0 AKt 0 and ALt 0 are three different types of technological change that are discussed further in Section 27 γ 0 1 is a distribution parameter that determines how important labor and capital services are in determining the production of the final good and α 0 is the elasticity of substitution To verify that α is the constant elasticity of substitution let us use 237 The ratio of the marginal product of capital to the marginal productive labor FKFL is then given by FKFL γ AKt σσ 1Ktσσ 1 1 γ ALt σσ 1Lt1 σ so that the elasticity of substitution is indeed given by σ that is σ logFKFL logKL1 As σ the CES production function becomes linear that is Yt γ AHt AKt Kt 1 γ AHt ALtLt Finally as σ 0 the CES production function converges to the Leontief production function with no substitution between factors Yt AHt min γ AKt Kt 1 γ ALt Lt This solution illustrates that starting from any k0 the equilibrium kt k sAn δ11 α and in fact the rate of adjustment is related to 1 αn δ More specifically the gap between k0 and the steadystate value k narrows at the exponential rate 1 αn δ This result is intuitive a higher α implies less diminishing returns to capital which slows down the rate at which the marginal and average products of capital decline as capital accumulates and this reduces the rate of adjustment to the steady state Similarly a smaller b means less depreciation and a smaller n means slower population growth both of which slow down the adjustment of capital per worker and thus the rate of transitional dynamics 26 A First Look at Sustained Growth 55 The special feature of the Leontief production function is that if γ AKtKt 1 γ ALtLt either capital or labor will be partially idle in the sense that a small reduction in capital or labor will have no effect on output or factor prices Exercise 223 illus trates a number of the properties of the CES production function while Exercise 216 provides an alternative derivation of this production function along the lines of the original article by Arrow et al 1961 Notice also that the CES production function with σ 1violates Assump tion 1 see Exercise 224 so in the context of aggregate production functions with capital and labor we may take σ 1 as the benchmark 26 A First Look at Sustained Growth Can the Solow model generate sustained growth without technological progress The answer is yes but only if some of the assumptions imposed so far are relaxed The CobbDouglas example Example 22 already showed that when α is close to 1 the adjustment of the capital labor ratio back to its steadystate level can be very slow A very slow adjustment toward a steady state has the flavor of sustained growth rather than the economy quickly settling down to a steady state In fact the simplest model of sustained growth essentially takes α 1in terms of the CobbDouglas production function To construct such a model let us relax Assumptions 1 and 2 which do not allow α 1 and consider the socalled AK model where FKt Lt At AKt 239 and A 0 is a constant The results here apply with more general constant returns to scale production functions that relax Assumption 2 for example with FKt Lt At AKt BLt 240 Nevertheless it is simpler to illustrate the main insights with 239 leaving the analysis of the case when the production function is given by 240 to Exercise 222 Let us continue to assume that population grows at a constant rate n as before see 232 Then combining 232 with the production function 239 the fundamental law of motion of the capital stock becomes kt kt sA δ n This equation shows that when the parameters satisfy the inequality sA δ n 0 there will be sustained growth in the capitallabor ratio and thus in output per capita This result is summarized in the next proposition Proposition 210 Consider the Solow growth model with the production function 239 and suppose that sA δ n 0 Then in equilibrium there is sustained growth of output per capita at the rate sA δ n In particular starting with a capitallabor ratio k0 0 the economy has kt exp sA δ n t k0 and yt exp sA δ n t Ak0 This proposition not only establishes the possibility of sustained growth but also shows that when the aggregate production function is given by 239 sustained growth is achieved without transitional dynamics 27 Solow Model with Technological Progress 57 Capital Labor 00 01 02 03 04 05 06 07 08 09 10 Labor and capital share in total value added 1929 1934 1939 1944 1949 1954 1959 1964 1969 1974 1979 1984 1989 1994 1999 FIGURE 211 Capital and labor share in the US GDP progressed tremendously over the past 200 years and even more so over the past 1000 or 10000 years This suggests that an attractive way of introducing economic growth in the framework developed so far is to allow technological progress in the form of changes in At The key question is how to model the effects of changes in At on the aggregate production function The standard approach is to impose discipline on the form of technological progress and its impact on the aggregate production function by requiring that the resulting allocations be consistent with balanced growth as defined by the socalled Kaldor facts Kaldor 1963 Kaldor noted that while output per capita increases the capitaloutput ratio the interest rate and the distribution of income between capital and labor remain roughly constant Figure 211 for example shows the evolution of the shares of capital and labor in the US national income Throughout the book balanced growth refers to an allocation where output grows at a constant rate and capitaloutput ratio the interest rate and factor shares remain constant Clearly three of these four features imply the fourth Figure 211 shows that despite fairly large fluctuations there is no trend in factor shares Moreover a range of evidence suggests that there is no apparent trend in interest rates over long time horizons see eg Homer and Sylla 1991 These facts and the relative constancy of capitaloutput ratios until the 1970s make many economists prefer models with balanced growth to those without The share of capital in national income and the capitaloutput ratio are not exactly constant For example since the 1970s both the share of capital in national income and the capitaloutput ratio may have increased depending on how one measures them Nevertheless constant factor shares and a constant capitaloutput ratio provide a good approximation to reality and a very useful starting point for our models Also for future reference note that in Figure 211 the capital share in national income is about 13 while the labor share is about 23 This estimate ignores the share of land land is not a major factor of production in modern economies though this has not been true 58 Chapter 2 The Solow Growth Model historically and is not true for the lessdeveloped economies of today Exercise 211 illustrates how incorporating land into this framework changes the analysis Note also that this pattern of the factor distribution of income combined with economists desire to work with simple models often makes them choose a CobbDouglas aggregate production function of the form AK13L23 as an approximation to reality especially since it ensures that factor shares are constant by construction However Theorem 26 below shows that CobbDouglas technology is not necessary for balanced growth and as noted in Example 22 the CobbDouglas form is both special and restrictive Another major advantage of models with balanced growth is that they are much easier to analyze than those with nonbalanced growth Analysis is facilitated because with balanced growth the equations describing the law of motion of the economy can be represented by difference or differential equations with welldefined steady states in transformed variables thus balanced growth will imply k 0 except that now the definition of k is different This enables us to apply the tools used in the analysis of stationary models to study economies with sustained growth It is nonetheless important to bear in mind that in reality growth has many nonbalanced features For example the share of different sectors changes systematically over the growth process with agriculture shrinking and manufacturing first increasing and then shrinking Ultimately we would like to develop models that combine certain balanced features with these types of structural transformations I return to these issues in Part VII of the book 272 Types of Neutral Technological Progress What types of restrictions does balanced growth place on our models It turns out that the answer to this question is quite a few The production function FKt Lt At is too general to achieve balanced growth and only some very special types of production functions are consistent with balanced growth To develop this point consider an aggregate production function F and let us define different types of neutral technological progress A first possibility is FKt Lt At AtFKt Lt for some constant returns to scale function F This functional form implies that the technology term At is simply a multiplicative constant in front of another quasi production function F This type of technological progress is referred to as Hicksneutral after the famous British economist John Hicks Figure 212 illustrates this form of technological progress by plotting the isoquants of the function FKt Lt At which correspond to combinations of labor and capital for a given technology At such that the level of production is constant Hicks neutral technological progress in the first panel corresponds to a relabeling of the isoquants without any change in their shape Another alternative is to have capitalaugmenting or Solowneutral technological progress in the form FKt Lt At FAtKt Lt which is also referred to as capitalaugmenting progress because a higher At is equivalent to the economy having more capital This type of technological progress corresponds to the isoquants shifting inward as if the capital axis were being shrunk since a higher A now corresponds to a greater level of effective capital This type of progress is shown in panel B of Figure 212 for a doubling of At FKt Lt At FKt AtLt Yt FKt Lt At YT 0 Therefore i must apply and gY gK gC as claimed in the first part of the theorem Part 2 For any t T the aggregate production function for time T can be written as exp gYt T Yt F expgKt T Kt expnt T Lt AT yields Yt Fexpt TgY gK Kt expt TgY n Lt AT 242 62 Chapter 2 The Solow Growth Model The latter case creates an asymmetry between capital and labor in the sense that capital is accumulating faster than labor Constancy of growth then requires technological change to make up for this asymmetrythat is technology should take a laboraugmenting form This intuition does not provide a reason for why technology should take this labor augmenting Harrodneutral form however The theorem and its corollary simply state that if technology did not take this form an asymptotic allocation with constant growth rates of output capital and consumption and thus balanced growth would not be possible At some level this result is distressing since it implies that balanced growth in fact something weaker than balanced growth is only possible under a very stringent assumption Chapter 15 shows that when technology is endogenous the same intuition also implies that technology should be endogenously more laboraugmenting than capitalaugmenting Notice also that Theorem 26 and its corollary do not state that technological change has to be laboraugmenting all the time Instead technological change ought to be laboraugmenting after time T along the balanced growth path This is the pattern that certain classes of endogenous technology models will generate see again Chapter 15 for a discussion More importantly contrary to common claims in textbooks and the literature Theorem 26 does not even state that capitalaugmenting Solowneutral technological change is impossible as t It states that such technological progress is not possible if there is balanced growth after some date T Exercise 217 provides a simple example where asymptotic balanced growth with the conditions in Theorem 26 being satisfied as t is possible in the presence of asymptotic capitalaugmenting technological progress It should also be emphasized that Theorem 26 does not require that Yt FKt AtLt but only that it has a representation of the form Yt FKt AtLt For example if the aggregate production function is CobbDouglas that is Yt AKtKtαALtLt1α then both AKt and ALt could grow at constant rates while maintaining balanced growth However in this CobbDouglas example we can define At AKtα1αALt and the production function can be represented as Yt Ktα AtLt1α so that technological change is represented as purely laboraugmenting which is what Theorem 26 requires Intuitively the differences between laboraugmenting and capitalaugmenting and Hicksneutral forms of technological progress matter when the elasticity of substitution between capital and labor is not equal to 1 In the CobbDouglas case as we have seen above this elasticity of substitution is equal to 1 thus Harrodneutral Solowneutral and Hicks neutral forms of technological progress are simple transforms of one another Theorem 26 does not specify how factor prices behave As noted at the beginning of this section the Kaldor facts also require constant factor shares Since capital and output are growing at the same rate the rental rate of capital must be constant Does Theorem 26 combined with competitive factor markets imply constant factor shares Unfortunately the answer is not necessarily This is related to an implicit limitation in Theorem 26 The theorem states that the original production function FKt Lt At has a representation of the form FKt AtLt along an asymptotic path with constant growth rates This does not guarantee that the derivatives of F and F with respect to K and L agree Exercise 219 provides an example of production function F that satisfies all of the conditions of Theorem 26 and thus admits a representation of the form FKt AtLt as t but has derivatives that do not agree with those of F In fact the exercise shows that with competitive markets this F Theorem 27 Uzawas Theorem II Suppose that all hypotheses in Theorem 26 are satisfied so that F R² A R has a representation of the form FKt AtLt and FK0 L0 A0 FK0 A0L0 and FLK0 L0 A0 FK0 A0L0 243 Moreover if 243 holds and factor markets are competitive then Rt R and αKt αK for all t T 64 Chapter 2 The Solow Growth Model with AtAt g Define ˆF R2 R as ˆFK AL ˆF1K ALK ˆF2K ALAL 245 From Theorem 21 ˆFKt AtLt FKt Lt At and thus ˆF is homogeneous of degree 1 in its two arguments and provides a representation of F along the path Kt Lt t0 Since ˆF is homogeneous of degree 1 245 implies that its partial derivatives are given by ˆF1 and ˆF2 and thus agree with those of F establishing 243 To prove the second part of the theorem simply note that with competitive factor markets we have that for t T αKt RtKt Yt Kt Yt FKt Lt At Kt α K where the second line uses the definition of the rental rate of capital in a competitive market and the third line uses 243 together with the fact that F is homogeneous of degree 1 Theorem 27 implies that any allocation with constant growth rates for output capital and consumption must be a balanced growth path where factor shares in national income are also constant It also implies that balanced growth can only be generated by an aggregate production function that features Harrodneutral technological change A further intuition for Theorem 26 comes from Theorem 27 Suppose the production function takes the special form FAKtKt ALtLt Theorem 27 implies that factor shares must be constant as t Thus given constant returns to scale balanced growth after some time T is possible only when total capital inputs AKtKt and total labor inputs ALtLt grow at the same rate otherwise the share of either capital or labor will not be constant But if total labor and capital inputs grow at the same rate then output Yt must also grow at this rate again because of constant returns to scale The fact that the capitaloutput ratio is constant in steady state then implies that Kt must grow at the same rate as output and thus at the same rate as ALtLt Therefore balanced growth is only possible if AKt is constant after date T 274 The Solow Growth Model with Technological Progress Continuous Time I now present an analysis of the Solow growth model with technological progress in continuous time The discretetime case can be analyzed analogously and I omit the details to avoid repetition Theorem 26 implies that when the economy is experiencing balanced growth the production function must have a representation of the form Yt FKt AtLt with purely laboraugmenting technological progress Most macroeconomic and growth analy ses then assume that it takes this form throughout for all t and that there is technological progress at the rate g 0 that is At At g 0 246 Let us also start with this assumption Suppose also that population grows at the rate n as in 232 Again using the constant saving rate capital accumulates according to the differential equation Kt sFKt AtLt δKt 247 66 Chapter 2 The Solow Growth Model Next substituting for Kt from 247 into 249 kt kt sFKt AtLt Kt δ g n Using 248 kt kt sf kt kt δ g n 251 which is very similar to the law of motion of the capitallabor ratio in the model without technological progress 233 The only difference is the presence of g which reflects the fact that now k is no longer the capitallabor ratio but the effective capitallabor ratio Thus for k to remain constant in the BGP the capitallabor ratio needs to increase at the rate g An equilibrium in this model is defined similarly to before A steady state or a BGP is in turn defined as an equilibrium in which the effective capitallabor ratio kt is constant Consequently the following proposition holds proof omitted Proposition 211 Consider the basic Solow growth model in continuous time with Harrod neutral technological progress at the rate g and population growth at the rate n Suppose that Assumptions 1 and 2 hold and define the effective capitallabor ratio as in 248 Then there exists a unique BGP where the effective capitallabor ratio is equal to k 0 given by f k k δ g n s 252 Per capita output and consumption grow at the rate g Equation 252 which determines the BGP steadystate effective capitallabor ratio emphasizes that now total savings sf k are used for replenishing the capital stock for three distinct reasons The first is again depreciation at the rate δ The second is population growth at the rate n which reduces capital per worker The third is Harrodneutral technological progress which reduces effective capitallabor ratio at the rate g when the capitallabor ratio is constant Thus the replenishment of the effective capitallabor ratio requires total investment to be equal to δ g n k which is the intuitive explanation for 252 The comparative static results are also similar to before with the additional comparative static with respect to the initial level of the laboraugmenting technology A0 the level of technology at all points in time At is completely determined by A0 given the assumption in 246 Proposition 212 Suppose Assumptions 1 and 2 hold and let A0 be the initial level of technology Denote the BGP level of effective capitallabor ratio by kA0 s δ n g and the level of output per capita by yA0 s δ n g t the latter is a function of time since it is growing over time Then kA0 s δ n g A0 0 kA0 s δ n g s 0 kA0 s δ n g n 0 and kA0 s δ n g δ 0 28 Comparative Dynamics 67 and also yA0 s δ n g t A0 0 yA0 s δ n g t s 0 yA0 s δ n g t n 0 and yA0 s δ n g t δ 0 for each t Proof See Exercise 225 Finally the transitional dynamics of the economy with technological progress are similar to the dynamics without technological change Proposition 213 Suppose that Assumptions 1 and 2 hold Then the BGP of the Solow growth model with Harrodneutral technological progress and population growth in continu ous time is asymptotically stable that is starting from any k0 0 the effective capitallabor ratio converges to the BGP value k kt k Proof See Exercise 226 Therefore with Harrodneutral technological change the dynamics of the equilibrium path and the comparative statics are very similar to those in the model without technological progress The major difference is that now the model generates growth in output per capita so it can be mapped to the data more successfully However the disadvantage is that growth is driven entirely exogenously The growth rate of the economy is exactly the same as the exogenous growth rate of the technology stock The model specifies neither where this technology stock comes from nor how fast it grows 28 Comparative Dynamics This section briefly undertakes some simple comparative dynamics exercises Comparative dynamics are different from the comparative static results in Propositions 23 28 or 212 in that the focus is now on the entire path of adjustment of the economy following a shock or a change in parameters The basic Solow model is particularly well suited to such an analysis because of its simplicity These exercises are also useful because the basic Solow model and its neoclassical cousin are often used for analysis of policy changes mediumrun shocks and business cycle dynamics so an understanding of how the basic model responds to various shocks is useful in a range of applications Recall that the law of motion of the effective capitallabor ratio in the continuoustime Solow model is given by 251 that is ktkt sf ktkt δ g n The right hand side of this equation is plotted in Figure 213 The intersection with the horizontal axis gives the unique BGP with effective capitallabor ratio k This figure is sufficient for the analysis of comparative dynamics Consider for example a onetime unanticipated perma nent increase in the saving rate from s to s This shifts the curve to the right as shown by the dashed line with a new intersection with the horizontal axis at k The dashed arrows under the horizontal axis show how the effective capitallabor ratio adjusts gradually to the new BGP effective capitallabor ratio k Immediately after the increase in the saving rate is realized the capital stock and the effective capitallabor ratio remain unchanged since they are state vari ables After this point k follows the dashed arrows and converges monotonically to k The Dynamics following an increase in the saving rate from s to s The solid arrows show the dynamics for the initial steady state while the dashed arrows show the dynamics for the new steady state 210 References and Literature 69 progress and as long as we are not in the AK world ruled out by Assumption 2 there will be no sustained growth In this case we can talk about crosscountry output differences but not about growth of countries or of the world economy The Solow model generates per capita output growth only by introducing exogenous tech nological progress But in this case everything is driven by technological progress and tech nological progress itself is exogenous just a black box outside the model and outside the influence of economic incentives If technological progress is where its at then we have to study and understand which factors generate technological progress what makes some firms and societies invent better technologies and what induces firms and societies to adopt and use these superior technologies Even on the question of capital accumulation the Solow model is not entirely satisfactory The rate of capital accumulation is determined by the saving rate the depreciation rate and the rate of population growth All these rates are taken as exogenous In this light the Solow growth model is most useful as a framework for laying out the general issues and questions It emphasizes that to understand growth we have to understand physical capital accumulation and human capital accumulation which is discussed in the next chapter and perhaps most importantly technological progress All of these are black boxes in the Solow growth model Therefore much of the rest of the book will be about digging deeper trying to uncover what lies in these black boxes I start by introducing consumer optimization in Chapter 8 which enables a more systematic study of capital accumulation Then I turn to models in which human capital accumulation and technological progress are endogenous A model in which the rate of accumulation of factors of production and technology are endogenous gives us a framework for posing and answering questions related to the fundamental causes of economic growth Nevertheless even in its barebones form the Solow model is useful in helping us think about the world and bringing useful perspectives especially related to the proximate causes of economic growth This is the topic of the next chapter 210 References and Literature The model analyzed in this chapter was first developed in Solow 1956 and Swan 1956 Solow 1970 gives a nice and accessible treatment with historical references Barro and Sala iMartins 2004 Chapter 1 textbook presents a more uptodate treatment of the basic Solow model at the graduate level while Jones 1998 Chapter 2 presents an excellent undergraduate treatment The treatment in the chapter made frequent references to basic consumer and general equilibrium theory These are prerequisites for an adequate understanding of the theory of economic growth Some of the important results from dynamic general equilibrium theory are discussed in Chapter 5 MasColell Whinston and Greens 1995 graduate microeconomics textbook contains an excellent treatment of most of the necessary material including producer theory and an accessible presentation of the basic notions of general equilibrium theory including a discussion of Arrow securities and the definition of ArrowDebreu commodities Properties of homogeneous functions and Eulers Theorem can be found for example in Simon and Blume 1994 Chapter 20 The reader should be familiar with the Implicit Function Theorem and properties of concave and convex functions which are used throughout the book A review is given in Appendix A 70 Chapter 2 The Solow Growth Model Appendix B provides an overview of solutions to differential and difference equations and a discussion of stability Theorems 22 23 24 and 25 follow from the results presented there In addition the reader may want to consult Boyce and DiPrima 1977 Luenberger 1979 or Simon and Blume 1994 for various results on difference and differential equations Knowledge of solutions to simple differential equations and stability properties of difference and differential equations at the level of Appendix B is assumed in the text In addition the material in Luenberger 1979 is particularly useful since it contains a unified treatment of difference and differential equations Galor 2005 gives an introduction to difference equations and discretetime dynamical systems for economists The golden rule saving rate was introduced by Edmund Phelps 1966 It is called the golden rule rate with reference to the biblical golden rule do unto others as you would have them do unto you applied in an intergenerational settingthat is presuming that those living and consuming at each different date form a different generation While the golden rule saving rate is of historical interest and is useful for discussions of dynamic efficiency it has no intrinsic optimality property since it is not derived from welldefined preferences Optimal savings are discussed in greater detail in Chapter 8 The balanced growth facts were first noted by Kaldor 1963 Figure 211 uses data from Piketty and Saez 2003 Homer and Sylla 1991 discuss the history of interest rates over many centuries and across different societies they show that there is no notable upward or downward trend in interest rate Nevertheless not all aspects of the economic growth process are balanced and the nonbalanced nature of growth is discussed in detail in Part VII of the book which also contains references to changes in the sectoral composition of output in the course of the growth process A simpler version of Theorem 26 was first proved by Uzawa 1961 There are various different proofs in the literature though many are not fully rigorous The proof given here is adapted from Schlicht 2006 which is also discussed in Jones and Scrimgeour 2006 A similar proof also appears in Wan 1971 Barro and SalaiMartin 2004 Chapter 1 also suggest a proof Nevertheless their argument is incomplete since it assumes that technological change must be a combination of Harrod and Solowneutral technological change which is rather restrictive and is not necessary for the proof The theorem and the proof provided here are therefore more general and complete There are also a variety of misconceptions about the implications of Theorem 26 Many textbooks claim that this theorem rules out asymptotic capitalaugmenting technological progress unless the production function is CobbDouglas Exercise 217 shows that this claim is not true and balanced growth is possible even with asymptotic capitalaugmenting technological progress with nonCobbDouglas production functions Theorem 26 holds when balanced growth applies after some finite time T or under additional conditions as discussed in Exercise 214 Moreover it is also important to emphasize as I did in the text that Theorem 26 only provides a representation for a particular path of capital and labor Consequently this representation cannot always be used for equilibrium analysis or for pricing capital and labor as shown by Exercise 219 Theorem 27 was provided as a way of overcoming this difficulty I am not aware of other results analogous to Theorem 27 in the literature As noted in the text the CES production function was first introduced by Arrow et al 1961 This production function plays an important role in many applied macroeconomic and economic growth models The Inada conditions introduced in Assumption 2 are from Inada 1963 Finally the interested reader should look at the paper by Hakenes and Irmen 2006 for why Inada conditions can introduce an additional equilibrium path other than the noactivity equilibrium at k 0 in continuous time even when f 0 0 Here it suffices to say that 211 Exercises 71 whether this steady state exists is a matter of the order in which limits are taken In any case as noted in the text the steady state at k 0 has no economic content and is ignored throughout the book 211 Exercises 21 Show that competitive labor markets and Assumption 1 imply that the wage rate must be strictly positive and thus 24 implies 23 22 Prove that Assumption 1 implies that FA K L is concave in K and L but not strictly so 23 Show that when F exhibits constant returns to scale and factor markets are competitive the maximization problem in 25 either has no solution the firm can make infinite profits a unique solution K L 0 or a continuum of solutions ie any K L with KL κ for some κ 0 is a solution 24 Consider the Solow growth model in continuous time with the following per capita production function f k k4 6k3 11k2 6k a Which parts of Assumptions 1 and 2 does the underlying production function FK L violate b Show that with this production function there exist three steadystate equilibria c Prove that two of these steadystate equilibria are locally stable while one of them is locally unstable Can any of these steadystate equilibria be globally stable 25 Prove Proposition 27 26 Prove Proposition 28 27 Let us introduce government spending in the basic Solow model Consider the basic model without technological change and suppose that 29 takes the form Yt Ct It Gt with Gt denoting government spending at time t Imagine that government spending is given by Gt σYt a Discuss how the relationship between income and consumption should be changed Is it reasonable to assume that Ct sYt b Suppose that government spending partly comes out of private consumption so that Ct s λσYt where λ 0 1 What is the effect of higher government spending in the form of higher σ on the equilibrium of the Solow model c Now suppose that a fraction φ of Gt is invested in the capital stock so that total investment at time t is given by It 1 s 1 λ σ φσ Yt Show that if φ is sufficiently high the steadystate level of capitallabor ratio will increase as a result of higher government spending corresponding to higher σ Is this reasonable How would you alternatively introduce public investments in this model 28 Suppose that FK L A is concave in K and L though not necessarily strictly so and satisfies Assumption 2 Prove Propositions 22 and 25 How do we need to modify Proposition 26 What have we learned from the Solow model At some level a lot We now have a simple and tractable framework that allows us to study capital accumulation and the implications of technological progress As we will see in the next chapter this framework is already quite useful in helping us think about the data Define and characterize the steadystate equilibrium of this economy and study its stability What is the relationship between the steadystate capitallabor ratio k and the golden rule capital stock kgold defined in Section 23 Consider the environment in Exercise 217 Suppose that F takes a CES form as in 238 with the elasticity of substitution between capital and labor 1 σ sL and there is constant saving rate s Show that as τ the economy converges to a BGP where the share of labor in national income is equal to 1 and capital output and consumption all grow at the rate gL In light of this result discuss the claim in the literature that capitalaugmenting technological change is inconsistent with balanced growth Why is the claim in the literature incorrect Relate your answer to Exercise 214 Show that when such k1 k2 exist there may also exist a stable steady state Show that such cycles are not possible in the continuoustime Solow growth model for any possibly nonneoclassical continuous production function fk and continuous sk 76 Chapter 2 The Solow Growth Model Therefore we can think of qt as the inverse of the relative price of machinery to output When qt is high machinery is relatively cheaper Gordon 1990 documented that the relative prices of durable machinery have been declining relative to output throughout the postwar era This decline is quite plausible especially given recent experience with the decline in the relative price of computer hardware and software Thus we may want to suppose that qt 0 This exercise asks you to work through a model with this feature based on Greenwood Hercowitz and Krusell 1997 a Suppose that qtqt γK 0 Show that for a general production function FK L there exists no BGP b Now suppose that the production function is CobbDouglas FK L KαL1α and char acterize the unique BGP c Show that this steadystate equilibrium does not satisfy the Kaldor fact of constant KY Is this discrepancy a problem Hint how is K measured in practice How is it measured in this model 3 The Solow Model and the Data I n this chapter we see how the Solow model or its simple extensions can be used to interpret both economic growth over time and crosscountry output differences The focus is on proximate causes of economic growth that is on such factors as investment or capital accumulation highlighted by the basic Solow model as well as technology and human capital differences What lies underneath these proximate causes is the topic of the next chapter There are multiple ways of using the basic Solow model to look at the data I start with the growth accounting framework which is most commonly applied for decomposing the sources of growth over time After briefly discussing the theory of growth accounting and some of its uses I discuss applications of the Solow model to crosscountry output and growth differences In this context I also introduce the augmented Solow model with human capital and show how various different regressionbased approaches can be motivated from this framework Finally I illustrate how the growth accounting framework can be modified to a development accounting framework to form another bridge between the Solow model and the data A constant theme that emerges from many of these approaches concerns the importance of productivity differences over time and across countries The chapter ends with a brief discussion of various other approaches to estimating crosscountry productivity differences 31 Growth Accounting As discussed in the previous chapter at the center of the Solow model is the aggregate production function 21 which I rewrite here in its general form Yt FKt Lt At 31 Another major contribution of Bob Solow to the study of economic growth was the observation that this production function combined with competitive factor markets also gives us a framework for accounting for the sources of economic growth In particular Solow 1957 developed what has become one of the most common tools in macroeconomics the growth accounting framework For our purposes it is sufficient to expose the simplest version of this framework Con sider a continuoustime economy and suppose that the production function 31 satisfies 77 78 Chapter 3 The Solow Model and the Data Assumptions 1 and 2 from Chapter 2 Differentiating with respect to time dropping time de pendence and denoting the partial derivatives of F with respect to its arguments by FA FK and FL we obtain Y Y FAA Y A A FKK Y K K FLL Y L L 32 Now denote the growth rates of output capital stock and labor by g YY gK KK and gL LL respectively and also define x FAA Y A A as the contribution of technology to growth Defining εk FKKY and εl FLLY as the elasticities of output with respect to capital and labor respectively see also equation 39 32 implies x g εkgK εlgL This equation is no more than an identity However with competitive factor markets it becomes useful for estimating the role of technological progress and economic growth In particular factor prices in competitive markets are given by w FL and R FK equations 26 and 27 from the previous chapter so that the elasticities εk and εl correspond to the factor shares αK RKY and αL wLY Putting all these together we have x g αKgK αLgL 33 Equation 33 is the fundamental growth accounting equation which can be used to estimate the contribution of technological progress to economic growth using data on factor shares output growth labor force growth and capital stock growth The contribution from technolog ical progress x is typically referred to as total factor productivity TFP or sometimes as multifactor productivity In particular denoting an estimate by a hat the estimate of TFP growth at time t is ˆxt gt αKtgKt αLtgLt 34 I put the hat only on x but one may want to take into account that all terms on the righthand side are also estimates obtained with a range of assumptions from national accounts and other data sources In continuous time 34 is exact because it is defined in terms of instantaneous changes derivatives In practice we look at changes over discrete time intervals for example over a year or sometimes with better data over a quarter or a month With discrete time intervals there is a potential problem in using 34 over the time horizon in question factor shares can change should we use beginningofperiod or endofperiod values of αK and αL It can be shown that the use of either beginningofperiod or endofperiod values might lead to biased estimates of the contribution of TFP to output growth ˆx Such a bias is particularly likely when the distance between the two time periods is large see Exercise 31 The best way of avoiding such biases is to use data that are as highfrequency as possible For now taking the available data as given let us look at how one could use the growth accounting framework with data over discrete intervals The most common way of dealing with the problems pointed out above is to use factor shares calculated as the average of the 31 Growth Accounting 79 beginningofperiod and endofperiod values Therefore in discrete time for a change between times t and t 1 the analogue of 34 becomes ˆxt1t gt1t αKt1tgKt1t αLt1tgLt1t 35 where gtt1 is the growth rate of output between t and t 1 other growth rates are defined analogously and αKt1t αKt αKt 1 2 and αLt1t αLt αLt 1 2 are average factor shares between t and t 1 Equation 35 would be a fairly good approx imation to 34 when the difference between t and t 1 is small and the capitallabor ratio does not change much during this time interval Solows 1957 article not only developed this growth accounting framework but also applied it to US data for a preliminary assessment of the sources of growth during the early twentieth century The question Bob Solow asked was how much of the growth of the US economy can be attributed to increased labor and capital inputs and how much of it is due to the residual technological progress Solows conclusion was quite striking a large part of the growth was due to technological progress This has been a landmark finding emphasizing the importance of technological progress as the driver of economic growth not only in theory but also in practice It focused the attention of economists on sources of technology differences over time and across nations industries and firms From early days however it was recognized that calculating the contribution of technolog ical progress to economic growth in this manner has a number of pitfalls Moses Abramovitz 1957 famously dubbed the ˆx term the measure of our ignoranceafter all it was the resid ual we could not explain and decided to call technology In its extreme form this criticism is unfair since ˆx does correspond to technology according to 34 thus the growth accounting framework is an example of using theory to inform measurement Yet at another level the criticism has validity If we underestimate the growth rates of labor and capital inputs gL and gK we will arrive at inflated estimates of ˆx And in fact there are good reasons for suspect ing that Solows estimates and even the higher quality estimates that came later may be mis measuring the growth of inputs The most obvious reason for this error is that what matters is not labor hours but effective labor hours so it is importantthough difficultto make adjust ments for changes in the human capital of workers I discuss issues related to human capital in Section 33 and in greater detail in Chapter 10 Similarly measurement of capital inputs is not straightforward In the theoretical model capital corresponds to the final good used as input to produce more goods But in practice capital comprises equipment machinery as well as structures buildings In measuring the amount of capital used in production one has to make assumptions about how relative prices of different types of equipment change over time The typical approach adopted for a long time in national accounts and also naturally in applications of the growth accounting framework is to use capital expenditures However if the same machines are cheaper today than they were in the past eg as has been the case for computers then this methodology would underestimate gK recall Exercise 227 in the previous chapter Therefore when applying 34 underestimates of gL and gK will nat urally inflate the estimates of the role of technology as a source of economic growth Finally changes in relative prices and the quality of products may also lead to the mismeasurement of the growth rate of output g If g is underestimated then there will be a countervailing force toward underestimating ˆx There is still an active debate on how to adjust for the changes in the quality of labor and capital inputs to arrive at the best estimate of technology Dale Jorgensen for example has shown that the residual technology estimates can be reduced very substantially perhaps almost to zero by making adjustments for changes in the quality of labor and capital see eg Jorgensen Gollop and Fraumeni 1987 Jorgensen 2005 These issues also become relevant when we attempt to decompose the sources of crosscountry output differences Before doing this let us look at applications of the Solow model to data using regression analysis The use of the symbol here is to emphasize that this is an approximation ignoring secondorder terms In particular the first line follows simply by differentiating ktkτ with respect to log kt and evaluating the derivatives at k and ignoring secondorder terms The second line uses the fact that the first term in the first line is equal to zero by the definition of the steadystate value k recall that from 252 in the previous chapter sfkk δ g n and the definition of the elasticity of the f function εkkt Now substituting this approximation into 38 we have ytyt g εkkδ g nlog yt log k Let us define yt Atfk as the steadystate level of output per capita that is the level of per capita output that would apply if the effective capitallabor ratio were at its steadystate value and technology were at its time t level A firstorder Taylor expansion of log yt with respect to log kt around log k then gives log yt log yt εkklog kt log k Using a discretetime approximation 310 yields the regression equation gitt1 b0 b1 log yit1 εit where gitt1 is the growth rate of country i between dates t 1 and t log yit1 is the initial time t 1 log output per capita of this country and εit is a stochastic term capturing all omitted influences Regressions of this form have been estimated by among others Baumol 1986 Barro 1991 and Barro and SalaiMartin 1992 If such an equation is estimated in the sample of core OECD countries b1 is indeed estimated to be negative countries like Ireland Greece Spain and Portugal that were relatively poor at the end of World War II have grown faster than the rest as shown in Figure I14 in Chapter 1 Yet Figure I13 in Chapter 1 shows that when we look at the world there is no evidence of worldwide convergence However as discussed in that chapter this notion of unconditional convergence may be too demanding It requires that there should be a tendency for the income gap between any two countries to decline regardless of the technological opportunities investment behavior policies and institutions of these countries If they do differ with respect to these factors the Solow model would not predict that they should converge in income level With this motivation Barro 1991 and Barro and SalaiMartin 1992 2004 force by the notion of conditional convergence which means that the convergence effects emphasized by the Solow model should lead to negative estimates of b1 once b0p is allowed to vary across countries To implement this idea of conditional convergence empirically they estimate models where b0p is assumed to be a function of among other things the male schooling rate the female schooling rate the fertility rate the investment rate the governmentconsumption ratio the inflation rate changes in terms of trades openness and such institutional variables as rule of law and democracy The corresponding regression equation then takes the form gitt1 Xitβ b1 log yit1 εit where Xit is a column vector including the variables mentioned above as well as a constant with a vector of coefficients β recall that X denotes the transpose of X In other words this specification supposes that b0p in 313 can be approximated by Xitβ Consistent with the emphasis on conditional convergence regressions of 314 tend to show a negative estimate of b1 but the magnitude of this estimate is much smaller than that suggested by the computations in Example 31 84 Chapter 3 The Solow Model and the Data and uit is a random and serially uncorrelated error term When the variable log yit is used in the regression the error term uit1 appears both on the left and righthand sides of 314 In particular note that log yit log yit1 log yit log yit1 uit uit1 Since the measured growth is gitt1 log yit log yit1 log yit log yit1 uit uit1 the measurement error uit1 will be part of both the error term εit and the righthand side variable log yit1 log yit1 uit1 in the regression equation gitt1 XT itβ b1 log yit1 εit This will naturally lead to a negative bias in the estimation of b1 Therefore we can end up with a negative estimate of b1 even when there is no conditional convergence 2 The economic interpretation of regression equations like 314 is not always straight forward Many of the regressions used in the literature include the investment rate as part of the vector Xit and all of them include the schooling rate However in the Solow model differences in investment rates and in the extended Solow model differ ences in schooling rates are the primary channel by which the potential determinants included in the vector Xit eg institutions openness will influence economic growth Therefore once we condition on the investment and schooling rates the coefficients on the other variables in Xit no longer measure their full impact on economic growth Consequently estimates of 314 with investmentlike variables on the righthand side are difficult to link to theory 3 Finally the motivating equation for the growth regression 310 is derived for a closed Solow economy When we look at crosscountry income differences or growth experiences the use of this equation imposes the assumption that each country is an island In other words the world is being interpreted as a collection of noninteracting closed economies In practice countries trade goods exchange ideas and borrow and lend in international financial markets These interactions imply that the behavior of different countries will not be given by 310 but by a system of equations characterizing the entire world equilibrium Interpreting crosscountry growth experiences by 310 in a world with interacting economies can often lead to misleading results see the discussion in Chapters 18 and 19 This discussion does not imply that growth regressions are uninformative At some basic level these regressions at least leaving aside the difficulties associated with the estimation of b1 can be interpreted as providing information on salient correlations in the data Knowing what these correlations are is an important input into the process of formulating empirically plausible theories In this context a complementary or perhaps a more natural regression framework for investigating the conditional correlations in the data is log yit α log yit1 XT itβ δi μt εit 315 where δi denotes a full set of country fixed effects and μt denotes a full set of year effects This regression framework differs from the growth regressions in a number of respects First the 33 The Solow Model with Human Capital 85 regression equation is specified in levels rather than with the growth rate on the lefthand side But this difference is mainly a rearrangement of 314since gitt1 log yit log yit1 More importantly by including the country fixed effects this regression equation takes out fixed country characteristics that might be simultaneously affecting economic growth or the level of income per capita and the righthandside variables of interest Therefore panel data regressions as in 315 may be more informative about the statistical relationship between a range of factors and income per capita However it is important to emphasize that including country fixed effects is not a panacea against all omitted variable biases and econometric endogeneity problems Simultaneity bias often results from timevarying influences which cannot be removed by including fixed effects Moreover to the extent that some of the variables in the vector Xit are slowly varying themselves the inclusion of country fixed effects will make it difficult to uncover the statistical relationship between these variables and income per capita and may increase potential biases due to measurement error in the righthandside variables In the remainder of this chapter I discuss how the structure of the Solow model can be further exploited to look at the data But first I present an augmented version of the Solow model incorporating human capital which is useful in these empirical exercises 33 The Solow Model with Human Capital Human capital is a term we use to represent the stock of skills education competencies and other productivityenhancing characteristics embedded in labor Put differently human capital represents the efficiency units of labor embedded in raw labor hours The term human capital originates from the observation that individuals will invest in their skills competencies and earning capacities in the same way that firms invest in their physical capitalto increase their productivity The seminal work by Ted Schultz Jacob Mincer and Gary Becker brought the notion of human capital to the forefront of economics For now all we need to know is that labor hours supplied by different individuals do not contain the same efficiency units a highly trained carpenter can produce a chair in a few hours while an amateur would spend many more hours to perform the same task Economists capture this notion by thinking that the trained carpenter has greater human capital that is he has more efficiency units of labor embedded in the labor hours he supplies The theory of human capital is vast and some of the important notions of this theory are discussed in Chapter 10 For now our objective is more modest to investigate how including human capital makes the Solow model a better fit to the data The inclusion of human capital enables us to embed all three of the main proximate sources of income differences physical capital human capital and technology For the purposes of this section let us focus on continuoustime models and suppose that the aggregate production function of the economy is given by a variant of 21 Y FK H AL 316 where H denotes human capital Notice that this production function is somewhat un usual since it separates human capital H from labor L as potential factors of production I start with this form because it is commonly used in the growth literature The more micro founded models in Chapter 10 assume that human capital is embedded in workers How human capital is measured in the data is discussed below Let us also modify Assumptions 1 and 2 as follows Assumption 1 The production function F R3 R in 316 is twice differentiable in K H and L and satisfies FKHAL K 0 FKHAL H 0 FKHAL L 0 2FKHAL K2 0 2FKHAL H2 0 2FKHAL L2 0 Moreover F exhibits constant returns to scale in its three arguments Assumption 2 F satisfies the India conditions lim K0 FKHAL K and lim K FKHAL K 0 for all H 0 and AL 0 lim H0 FKHAL H and lim H FKHAL H 0 for all K 0 and AL 0 lim L0 FKHAL L and lim L FKHAL L 0 for all K H A 0 In addition let us assume that investments in human capital take a similar form to investments in physical capital households save a fraction sH of their income to invest in physical capital and a fraction sI to invest in human capital Human capital also depreciates in the same way as physical capital and we denote the depreciation rates of physical and human capital by δk and δh respectively There is an exogenous constant population growth and a constant rate of laboraugmenting technological progress that is LtL0 n and AtA0 g Defining effective human and physical capital ratios as kt Kt AtLt and ht Ht AtLt and using the constant returns to scale feature in Assumption 1 output per effective unit of labor can be written as ŷt Yt AtLt F Kt AtLt Ht AtLt 1 fkt ht Using the same steps as in Chapter 2 the laws of motion of kt and ht are kt sKfkt ht δk g nkt ht sHfkt ht δh g nht A steadystate equilibrium is now defined by effective human and physical capital ratios k h satisfying the following two equations sKfk h δk g nk 0 sHfk h δh g nh 0 FIGURE 31 Dynamics of physical capitallabor and human capitallabor ratios in the Solow model with human capital As in the basic Solow model the focus is on steadystate equilibria with k 0 and h 0 If f0 0 0 then there exists a trivial steady state with k h 0 which I ignore for the same reasons as in the previous chapter Let us first prove that this steadystate equilibrium is unique To see this heuristically consider Figure 31 which is drawn in k h space The two curves represent 317 and 318 corresponding to k 0 and h 0 Both curves are upward sloping so that higher human capital is associated with higher physical capital in equilibrium Moreover the proof of the next proposition shows that 318 is always shallower in k h space so the upwardsloping curves can only intersect once Proposition 31 Suppose Assumptions 1 and 2 are satisfied Then in the augmented Solow model with human capital there exists a unique steadystate equilibrium k h Proposition 32 Suppose Assumptions I and II are satisfied Then the unique steadystate equilibrium of the augmented Solow model with human capital k h is globally stable in the sense that starting with any k0 0 and h0 0 we have kt ht k h Example 32 Augmented Solow Model with CobbDouglas Production Functions Let us now work through a special case of the above model with a CobbDouglas production function In particular suppose that the aggregate production function is Yt Ktα Htβ AtLt1αβ Common technology advances Ajt Aj exp gt 92 Chapter 3 The Solow Model and the Data TABLE 31 Estimates of the basic Solow model MRW Updated data 1985 1985 2000 logsk 142 101 122 14 11 13 logn g δ 197 112 131 56 55 36 Adjusted R2 59 49 49 Implied α 59 50 55 Number of observations 98 98 107 Note Standard errors are in parentheses this equation also includes εj as an error term capturing all omitted factors and influences on income per capita Their results from this estimation exercise are replicated in column 1 of Table 31 using the original MRW data standard errors in parentheses Their estimates suggest a coefficient of about 14 for α 1 α which implies a value of α about 23 Since α is also the share of capital in national income it should be about 13 recall Figure 211 Thus the regression estimates without human capital appear to lead to overestimates of α Columns 2 and 3 report the same results with updated data The fit of the model is slightly less good than was the case with the MRW data but the general pattern is similar The implied values of α are also a little smaller than the original estimates but still substantially higher than the value of 13 one would expect on the basis of the underlying model The most natural reason for the high implied values of the parameter α in Table 31 is that εj is correlated with logskj either because the orthogonal technology assumption is not a good approximation to reality or because there are also human capital differences correlated with logskj MRW favor the second interpretation and estimate the augmented model log y j constant α 1 α β logskj α 1 α β lognj g δk 326 β 1 α β logshj β 1 α β lognj g δh εj The original MRW estimates are given in column 1 of Table 32 Now the estimation is more successful Not only is the adjusted R2 quite high about 78 the implied value for α is about 13 On the basis of this estimation result MRW and others have interpreted the fit of the augmented Solow model to the data as a success with common technology human and physical capital investments appear to explain about threequarters of the differences in cross country income per capita and the implied parameter values are reasonable Columns 2 and 3 of the table show the results with updated data The implied values of α are similar though the adjusted R2 is somewhat lower To the extent that these regression results are reliable they give a big boost to the augmented Solow model In particular the estimate of adjusted R2 suggests that a significant fraction of 34 Regression Analyses 93 TABLE 32 Estimates of the augmented Solow model MRW Updated data 1985 1985 2000 logsk 69 65 96 13 11 13 logn g δ 173 102 106 41 45 33 logsh 66 47 70 07 07 13 Adjusted R2 78 65 60 Implied α 30 31 36 Implied β 28 22 26 Number of observations 98 98 107 Note Standard errors are in parentheses the differences in income per capita across countries can be explained by differences in their physical and human capital investment behavior The immediate implication is that technology TFP differences have a somewhat limited role If this conclusion were appropriate it would imply that as far as the proximate causes of prosperity are concerned we could confine our attention to physical and human capital and also assume that countries have access to more or less the same world technology The implications for the modeling of economic growth are of course quite major 343 Challenges to the Regression Analyses of Growth Models There are two major and related problems with the regression approach and the conclusion that the importance of technology differences is limited The first relates to the assumption that technology differences across countries are orthogo nal to all other variables While the constant technology advances assumption may be defended the orthogonality assumption is too strong almost untenable When Aj varies across countries it should also be correlated with measures of sh j and sk j countries that are more productive also invest more in physical and human capital This correlation is for two reasons The first is a version of the omitted variable bias problem technology differences are also outcomes of in vestment decisions Thus societies with high levels of Aj are those that have invested more in technology for various reasons it is then natural to expect the same reasons to induce greater investment in physical and human capital as well Second even if we ignore omitted variable bias there is a reverse causality problem complementarity between technology and physical or human capital implies that countries with high Aj find it more beneficial to increase their stock of human and physical capital In terms of the regression 326 omitted variable bias and reverse causality problems imply that the key righthandside variables are correlated with the error term εj Consequently ordinary least squares regressions of 326 lead to upwardly biased estimates of α and β In addition the estimate of R2 which is a measure of how much 94 Chapter 3 The Solow Model and the Data of the crosscountry variability in income per capita can be explained by physical and human capital will also be biased upward The second problem with the regression analyses relates to the magnitudes of the estimates of α and β in 326 The regression framework above is attractive in part because we can gauge whether the estimate of α is plausible We should do the same for the estimate of β However such an exercise reveals that the coefficient on the investment rate in human capital sh j appears too large relative to microeconometric evidence Recall first that MRW use the fraction of the workingage population enrolled in secondary school This variable ranges from 04 to more than 12 in the sample of countries used for this regression Their estimates therefore imply that holding all other variables constant a country with approximately 12 school enrollment should have income per capita of about 9 times that of a country with sh j 04 More explicitly the predicted log difference in incomes between these two countries is β 1 α β log 12 log 04 070 log 12 log 04 238 Thus holding all other factors constant a country with school enrollment of more than 12 should be about exp238 108 times richer than a country with a level of schooling investment of about 04 In practice the difference in average years of schooling between any two countries in the MRW sample is less than 12 Chapter 10 shows that there are good economic reasons to expect additional years of schooling to increase earnings proportionally for example as in Mincer regressions of the form log wi XT i γ φSi ui 327 where wi denotes the wage earnings of individual i Xi is a set of demographic controls Si is years of schooling and ui is an error term The estimate of the coefficient φ is the rate of returns to education measuring the proportional increase in earnings resulting from one more year of schooling The microeconometrics literature suggests that 327 provides a good approximation to the data and estimates φ to be between 006 and 010 implying that a worker with one more year of schooling earns about 610 more than a comparable worker with one less year of schooling If labor markets are competitive or at the very least if wages are on average proportional to productivity 327 also implies that one more year of schooling increases worker productivity by about 610 Can we deduce from this information how much richer a country with 12 more years of average schooling should be The answer is yes but with two caveats First we need to assume that the microlevel relationship as captured by 327 applies identically to all countries Let us for now ignore other potential determinants of wages and write the earnings of individual i as wi φSi where Si denotes the individuals level of schooling The first key assumption is that this φ function is identical across countries and can be approximated by an exponential function of the form φSi expφSi so that we obtain 327 Why this assumption may be reasonable is further discussed in Chapter 10 Second we need to assume that there are no human capital externalitiesmeaning that the human capital of a worker does not directly increase the productivity of other workers There are reasons why human capital externalities may exist and some economists believe that they are important The evidence discussed in Chapter 10 however suggests that human capital externalitiesexcept those working through innovationare unlikely to be large Thus it is reasonable to start without them The key result which will enable us to go from the microeconometric wage regressions to crosscountry differences that is with constant returns to scale perfectly competitive markets and no human capital externalities differences in worker productivity directly translate into differences in income per capita To see this suppose that each firm f in country j has access to the production function yfj Kfj Aj Hfj1α where Aj is the productivity of all firms in the country Kf is the capital stock and Hf denotes the efficiency units of human capital employed by firm f thus this production function takes the more usual form in which human capital is embedded in workers rather than the form in 316 Here the CobbDouglas production function is chosen for simplicity and does not affect the argument Suppose also that firms in this country face a cost of capital equal to Rj With perfectly competitive factor markets profit maximization implies that the cost of capital must equal its marginal product Rj α Kf Aj Hfj1α 328 Therefore all firms ought to function at the same physical to human capital ratio and consequently all workers regardless of their level of schooling ought to work at the same physical to human capital ratio Another direct implication of competitive labor markets is that in country j wages per unit of human capital are equal to wj 1 α α1 α Aj Rjα1 α Consequently a worker with human capital hj receives a wage income of wj hj Once again this is a more general result with aggregate constant returns to scale production technology wage earnings are linear in the effective human capital of the worker so that a worker with twice as much effective human capital as another should earn twice as much see Exercise 39 96 Chapter 3 The Solow Model and the Data cause of this overestimation is in turn most likely related to the possible correlation between the error term εj and the key righthand side regressors in 326 Consequently regression analyses based on 326 appear unlikely to provide us with an accurate picture of the extent of crosscountry productivity differences and of the proximate causes of income differences 35 Calibrating Productivity Differences What other approach can we use to gauge the importance of physical and human capital and technology differences An alternative is to calibrate the total factor productivity TFP differences across countries rather than estimating them using a regression framework These TFP estimates are then interpreted as a measure of the contribution of technology to cross country income differences The calibration approach was proposed and used by Klenow and Rodriguez 1997 and Hall and Jones 1999 Here I follow Hall and Joness approach which is slightly simpler The advantage of the calibration approach is that the omitted variable bias underlying the estimates of MRW will be less important since microlevel evidence is used to anchor the contribution of human capital to economic growth The disadvantage is that certain assumptions on functional forms have to be taken much more seriously and we must explicitly assume that there are no human capital externalities 351 Basics Suppose that each country j has access to the CobbDouglas aggregate production function Yj Kα j AjHj1α 329 where Hj is the stock of human capital of country j capturing the amount of efficiency units of labor available to this country Kj is its stock of physical capital and Aj is laboraugmenting technology Since our focus is on crosscountry comparisons time arguments are omitted Suppose that each worker in country j has Sj years of schooling Then using the Mincer equation 327 from the previous section ignoring the other covariates and taking exponents Hj can be estimated as Hj expφSjLj where Lj is employment in country j and φ is the rate on returns to schooling estimated from 327 This approach may not lead to accurate estimates of the stock of human capital of a country however First it does not take into account differences in other human capital factors such as training or experience which are discussed in greater detail in Chapter 10 Second countries may differ not only in the years of schooling of their labor forces but in the quality of schooling and the amount of postschooling human capital Third the rate of return to schooling may vary systematically across countries eg it may be lower in countries with a greater abundance of human capital It is possible to deal with each of these problems to some extent by constructing better estimates of the stocks of human capital Following Hall and Jones let us make a partial correction for the last factor Assume that the rate of return to schooling does not vary across countries but is potentially different for different years of schooling For example one year of primary schooling may be more valuable than one year of graduate school eg because learning how to read might increase productivity more than a solid understanding of growth theory In particular let the rate of return to acquiring the Sth year of schooling be φS The above equation would be the special case where φS φ for all S Given this assumption the estimate of the stock of human capital can be constructed as Hj S expφS S LjS where LjS now refers to the total employment of workers with S years of schooling in country j A series for Kjt can be constructed from the SummersHeston dataset using investment data and the perpetual inventory method In particular recall that with exponential depreciation the stock of physical capital evolves according to Kjt 1 1 δ Kjt Ijt where Ijt is the level of investment in country j at time t The perpetual inventory method involves using information on the depreciation rate δ and investments Ijt to estimate Kjt Let us assume following Hall and Jones that δ 006 With a complete series for Ijt this equation can be used to calculate the stock of physical capital at any point in time However the SummersHeston dataset does not contain investment information before the 1960s The equation can still be used by assuming that each countrys investment was growing at the same rate before the sample to compute the initial capital stock Using this assumption Hall and Jones calculate the physical capital stock for each country in the year 1985 I do the same here for 1980 and 2000 Finally with the same arguments as before I choose a value of 13 for α and given series for Hj and Kj and a value for α we can construct predicted incomes at a point in time using Ȳj Kj13 AU S Hj23 for each country j where AU S is the laboraugmenting technology level of the United States computed so that this equation fits the United States perfectly YU S KU S AU S HU S23 Throughout time indices are dropped Once a series for Ȳj has been constructed it can be compared to the actual output series The gap between the two series represents the contribution of technology Alternatively we could explicitly back out countryspecific technology terms relative to the United States as Aj AU S Yj YU S32 KU S Kj12 HU S Hj 98 Chapter 3 The Solow Model and the Data ARG AUS AUT BEL BEN BGD BOL BRA BRB BWA CAF CAN CHE CHL CHN CMR COG COL CRI CYP DNK DOM ECU EGY ESP FIN FJI FRA GBR GHA GMB GRC GTM GUY HKG HND HUN IDN IND IRL IRN ISL ISR ITA JAM JOR JPN KEN KOR LKA LSO MEX MLI MOZ MUS MWI MYS NER NIC NLD NOR NPL NZL PAK PAN PER PHL PNG PRT PRY RWA SEN SGP SLE SLV SWE SYR TGO THA TTO TUN TUR UGA URY USA VEN ZAF ZMB ZWE 45 7 8 9 10 11 Predicted log GDP per worker 1980 7 8 9 10 11 Log GDP per worker 1980 ARG AUS AUT BEL BEN BGD BOL BRA BRB CAN CHE CHL CHN CMR COG COL CRI DNK DOM DZA ECU EGY ESP FIN FRA GBR GER GHA GMB GRC GTM HKG HND HUN IDN IND IRL IRN ISL ISR ITA JAM JOR JPN KEN KOR LKA LSO MEX MLI MOZ MUS MWI MYS NER NIC NLD NOR NPL NZL PAK PAN PER PHL PRT PRY RWA SEN SLV SWE SYR TGO THA TTO TUN TUR UGA URY USA VEN ZAF ZMB ZWE 45 7 8 9 10 11 Predicted log GDP per worker 2000 7 8 9 10 11 12 Log GDP per worker 2000 FIGURE 32 Predicted and actual log GDP per worker across countries 1980 and 2000 3 The same conclusion follows from Figure 33 which plots the estimates of the tech nology differences AjAUS against log GDP per capita These differences are often substantial 4 Also interesting is the pattern indicating that the empirical fit of the Solow growth model seems to deteriorate over time In Figure 32 the observations are further above the 45 line in 2000 than in 1980 and in Figure 33 the relative technology differences become larger over time Why the fit of the simple Solow growth model is better in 1980 than in 2000 is an interesting and largely unanswered question 352 Challenges In the same way as the regression analysis was based on a number of stringent assumptions in particular the assumption that technology differences across countries were orthogonal to other factors the calibration approach also relies on certain important assumptions The above exposition highlighted several of them In addition to the standard assumption that factor markets are competitive the calibration exercise had to assume no human capital externalities impose a CobbDouglas production function and make a range of approximations to measure crosscountry differences in the stocks of physical and human capital 35 Calibrating Productivity Differences 99 ARG AUS AUT BEL BEN BGD BOL BRA BRB BWA CAF CAN CHE CHL CHN CMR COG COL CRI CYP DNK DOM ECU EGY ESP FIN FJI FRA GBR GHA GMB GRC GTM GUY HKG HND HUN IDN IND IRL IRN ISL ISR ITA JAM JOR JPN KEN KOR LKA LSO MEX MLI MOZ MUS MWI MYS NER NIC NLD NOR NPL NZL PAK PAN PER PHL PNG PRT PRY RWA SEN SGP SLE SLV SWE SYR TGO THA TTO TUN TUR UGA URY USA VEN ZAF ZMB ZWE 00 05 10 15 Predicted relative technology level 7 8 9 10 11 Log GDP per worker 1980 ARG AUS AUT BEL BEN BGD BOL BRA BRB CAN CHE CHL CHN CMR COG COL CRI DNK DOM DZA ECU EGY ESP FIN FRA GBR GER GHA GMB GRC GTM HKG HND HUN IDN IND IRL IRN ISL ISR ITA JAM JOR JPN KEN KOR LKA LSO MEX MLI MOZ MUS MWI MYS NER NIC NLD NOR NPL NZL PAK PAN PER PHL PRT PRY RWA SEN SLV SWE SYR TGO THA TTO TUN TUR UGA URY USA VEN ZAF ZMB ZWE 00 05 10 15 Predicted relative technology level 7 8 9 10 11 12 Log GDP per worker 2000 FIGURE 33 Calibrated technology levels relative to US technology from the Solow growth model with human capital versus log GDP per worker 1980 and 2000 Let us focus on the assumptions about functional form Could we relax the assumption that the production function is CobbDouglas The answer is partly yes The exercise here is similar to growth accounting which does not need to make strong functional form assumptions and this similarity to growth accounting is the reason this exercise is sometimes referred to as development accounting or levels accounting In particular recall 35 which showed how TFP estimates can be obtained from a general constant returns to scale production function under competitive labor markets by using average factor shares Now instead imagine that the production function of all countries is given by FKj Hj Aj and that countries differ according to their physical and human capital as well as technologybut not according to F Suppose also that we have data on Kj and Hj and on the share of capital in national income for each country Then a natural adaptation of 35 can be used across countries rather than over time In particular let us rank countries in descending order according to their physical capital to human capital ratios KjHj use Exercise 31 to see why this is the right way to rank countries rather than doing so randomly Then we have ˆxjj1 gjj1 αKjj1gKjj1 αLjj1gHjj1 330 where gjj1 is the proportional difference in output between countries j and j 1 gKjj1 is the proportional difference in capital stock between these countries and gHjj1 is the 100 Chapter 3 The Solow Model and the Data proportional difference in human capital stocks In addition αKjj1 and αLjj1 are the average capital and labor shares between the two countries ˆxjj1 in 330 is then the estimate of the proportional TFP difference between the two countries Using this method and taking one of the countries eg the United States as the base we can calculate relative technology differences across countries This levels accounting exercise faces two major challenges however One is data related and the other is theoretical First data on capital and labor shares across countries are not available for most countries This paucity of data makes the use of equation 330 far from straightforward Consequently almost all calibration or levels accounting exercises that estimate technology productivity differences use the CobbDouglas approach of the previous subsection ie a constant value of αK equal to 13 Second even if data on capital and labor shares were available the differences in factor proportions eg differences in KjHj across countries are large An equation like 330 is a good approximation for small changes As illustrated in Exercise 31 when factor proportion differences between observations are large significant biases are possible To sum up the approach of calibrating productivity differences across countries is a useful alternative to crosscountry regression analysis but has to rely on a range of stringent assump tions on the form of the production function and can lead to biased estimates of technology differences The biases come about both because these functional form assumptions may not be a good approximation to the data and because of mismeasurement of differences in the quality and quantity of physical and human capital across countries 36 Estimating Productivity Differences In the previous section productivity technology differences are obtained as residuals from a calibration exercise so we have to trust the functional form assumptions used in this strategy But if we are willing to trust the functional forms we can also estimate these differences econometrically rather than rely on calibration The great advantage of econometrics relative to calibration is that not only do we obtain estimates of the objects of interest but we also have standard errors which show how much these estimates can be trusted In this section I briefly discuss two different approaches to estimating productivity differences 361 A Naıve Approach The first possibility is to take a production function of the form 329 as given and try to estimate it using crosscountry data In particular taking logs log Yj α log Kj 1 α log Hj α log Aj 331 Given series for Yj Kj and Hj 331 can be estimated with ordinary least squares with the restriction that the coefficients on log Kj and log Hj sum to 1 and the residuals can be interpreted as estimates of technology differences Unfortunately this approach is not particularly attractive since the potential correlation between log Aj and log Kj or log Hj implies that the estimates of α need not be unbiased even when constant returns to scale is imposed Moreover when constant returns is not imposed the restriction that these coefficients sum to 1 will be rejected Thus this regression approach runs into the same difficulties as the MRW approach discussed in Section 34 36 Estimating Productivity Differences 101 Thus even if we are willing to presume that we know the functional form of the aggregate production function it is difficult to directly estimate productivity differences So how can we do better than this naıve approach The answer involves making more use of economic theory Estimating an equation of the form 331 does not make use of the fact that we are looking at the equilibrium of an economic system A more sophisticated approach would use more of the restrictions imposed by equilibrium behavior and would bring in additional relevant data I next illustrate this approach using a specific attempt based on international trade theory The reader who is not familiar with trade theory may want to skip this subsection 362 Learning from International Trade Models of growth and international trade are studied in Chapter 19 Even without a detailed discussion of international trade theory we can use data from international trade flows and some simple principles of international trade theory to obtain an alternate way of estimating productivity differences across countries Let us follow an important paper by Trefler 1993 which uses an augmented version of the standard HeckscherOhlin approach to international trade The standard Heckscher Ohlin approach assumes that countries differ according to their factor proportions eg some countries have much more physical capital relative to their labor supply than others In a closed economy this disparity leads to differences in relative factor costs and in the relative prices of products using these factors in different intensities International trade provides a way of taking advantage of these relative price differences The most stylized form of the theory assumes no costs of shipping goods and no policy impediments to trade so that international trade takes place costlessly between countries Trefler starts from the standard HeckscherOhlin model of international trade but allows for factorspecific productivity differences so that capital in country j has productivity Ak j thus a stock of capital Kj in this country is equivalent to an effective supply of capital Ak jKj Similarly for labor human capital country j has productivity Ah j In addition Trefler assumes that all countries have the same homothetic preferences and there are sufficient differences in factor intensity across goods to ensure international trade between countries to arbitrage relative differences in factor cost or in the jargon of international trade countries are said to be in the cone of diversification The latter assumption is important when all countries have the same productivities both in physical and human capital it leads to the celebrated factor price equalization resultall factor prices would be equal in all countries because the world economy is sufficiently integrated When there are productivity differences across countries this assumption instead leads to conditional factor price equalization meaning that factor prices are equalized once their different effective productivities are taken into consideration Under these assumptions a standard equation in international trade links the net factor exports of each country to the abundance of that factor in the country relative to the world as a whole The term net factor exports needs some explanation It does not refer to actual trade in factors eg migration of people capital flows Instead trading goods is a way of trading the factors that are embodied in that particular good For example a country that exports cars made with capital and imports corn made with labor is implicitly exporting capital and importing labor More specifically the net export of capital by country j XK j is calculated by considering the total exports of country j and computing how much capital is necessary to produce these exports and then subtracting the amount of capital necessary to produce its total imports For our purposes how factor contents are calculated is not important it suffices to say that as with all things empirical the devil is in the details and these calculations are far from straightforward and require a range of assumptions Then the absence of trading frictions across countries and identical homothetic preferences imply that XjK AjK Kj γj Cj i1J AiK Ki and XjH AjH Hj γj Cj i1J AiH Hi 332 where γj is the share of country j in world consumption the value of this countrys consumption divided by world consumption and J is the total number of countries in the world These equations simply restate the conclusion in the previous paragraph that a country will be a net exporter of capital if its effective supply of capital AjK Kj exceeds a fraction here γj of the worlds effective supply of capital j1 AjK Kj Consumption shares are easy to calculate Then given estimates for Kj and XjH the above system of 2 J equations can be solved for the same number of unknowns the AjK and AjH for J countries This solution gives estimates for factorspecific productivity differences across countries that are generated from an entirely different source of variation than those exploited before In fact this exercise provides us with a separate laboraugmenting or human capitalaugmenting term and a capitalaugmenting productivity term for each country How do we know that these numbers provide a good approximation to crosscountry factor productivity differences This problem is the same one we encountered in the previous section in judging whether the calibrated productivity technology differences were reliable As noted above under the assumption that the world economy is sufficiently integrated there is conditional factor price equalization Thus for any two countries j and j we have Rj AjK Rj AjK and wj AjH wj AjH 333 where Rj is the rental rate of capital in country j and wj is the observed wage rate which includes the compensation to human capital in country j The second equation in 333 for example states that if workers in a particular country have on average half the efficiency units of those in the United States their earnings should be roughly half those of American workers Using data on factor prices we can therefore construct an alternative series for AjK and AjH AUT BGD BEL CAN COL DNK FIN FRA GRC HKG IDN IRL ISR ITA JPN NLD NZL NOR PAK PAN PRT SGP ESP LKA SWE CHE THA TTO GBR USA URY DEU YUG 02 04 06 08 10 Capital productivity 02 04 06 08 10 Labor productivity FIGURE 34 Comparison of laborproductivity and capitalproductivity differences across countries AUT BGD BEL CAN COL DNK FIN FRA GRC HKG IDN IRL ISR ITA JPN NLD NZL NOR PAK PAN PRT SGP ESP LKA SWE CHE THA URY 00 05 10 15 Calibrated productivity differences 1988 00 02 04 06 08 10 Estimated labor productivity differences FIGURE 35 Comparison of the labor productivity estimates from the Trefler approach with the calibrated productivity differences from the HallJones approach 104 Chapter 3 The Solow Model and the Data AUT BGD BEL CAN COL DNK FIN FRA GRC HKG IDN IRL ISR ITA JPN NLD NZL NOR PAK PAN PRT SGP ESP LKA SWE CHE THA URY 00 05 10 15 Calibrated productivity differences 1988 00 02 04 06 08 10 Estimated capital productivity differences FIGURE 36 Comparison of the capital productivity estimates from the Trefler approach with the calibrated productivity differences from the HallJones approach section The similarity between the two series is remarkable suggesting that both approaches are capturing some features of reality and that in fact there are significant productivity technol ogy differences across countries Interestingly however Figure 36 shows that the relationship between the calibrated productivity differences and the capital productivity differences is con siderably weaker than for labor productivity Despite its apparent success it is important to emphasize that Treflers approach also relies on stringent assumptions The four major assumptions are 1 No international trading costs 2 Identical production functions except for factoraugmenting technology differences 3 Identical homothetic preferences and 4 Sufficiently integrated world economy leading to conditional factor price equalization All four of these assumptions are rejected in the data in one form or another There are clearly international trading costs including freight costs tariff costs and other trading restrictions Productivity differences in practice are more complex than the simple factoraugmenting form assumed by Trefler There is a very welldocumented home bias in consumption violating the assumption of identical homothetic preferences Finally most trade economists believe that conditional factor price equalization is not a good description of factor price differences across countries In view of these concerns the results from the Trefler exercise have to be interpreted with caution Nevertheless this approach is important both in showing how different sources 37 Taking Stock 105 of data and additional theory can be used to estimate crosscountry technology differences and in providing a crossvalidation for the calibration and estimation results discussed in Section 35 37 Taking Stock What have we learned The major point of this chapter has not been the development of new theory Instead it has been to see whether we could use the Solow model to obtain a more informed interpretation of crosscountry differences and to use data to gauge the strengths and shortcomings of the Solow growth model At the end of this brief journey the message is somewhat mixed On the positive side despite its simplicity the Solow model has enough substance that we can take it to data in various different forms including TFP accounting regression analysis and calibration Moreover each of these different methods gives us some idea about the sources of economic growth over time and of income differences across countries On the negative side however no single approach is entirely convincing Each relies on a range of stringent auxiliary assumptions Consequently no firm conclusions can be drawn The simplest applications of the Solow accounting framework suggest that technology is the main source of economic growth over time However this conclusion is disputed by those who point out that adjustments to the quality of physical and human capital substantially reduce or perhaps even totally eliminate residual TFP growth The same debate recurs in the context of cross country income differences while some believe that accounting for differences in physical and human capital across countries leaves little need for technology differences others show that with reasonable models most of the crosscountry differences are due to technology While complete agreement is not possible it is safe to say that the consensus in the literature today favors the interpretation that crosscountry differences in income per capita cannot be understood solely on the basis of differences in physical and human capital in other words there are technology differences across countries and these technology differences are likely to be at the heart of crosscountry income and growth differences Hence an important potential lesson from this data detour is that technological progress is not only important in generating economic growth in the basic Solow model but also likely to be a major factor in crosscountry differences in prosperity A detailed study of technological progress and technology adoption decisions of households and firms is therefore necessary as part of the study of economic growth This conclusion motivates the detailed analysis of technological progress and technology adoption later in the book It is also useful to emphasize once again that differences in TFP are not necessarily due to technology in the narrow sense If two countries have access to the same technology but make use of the available techniques in different ways with different degrees of efficiency or if they are subject to different degrees of market or organizational failures these differences will show up as TFP differences One indication that TFP differences arising from market or organizational failures are important comes from episodes of severe crises When countries have large drops in their income due to civil wars political instability financial crises or other reasons these drops are almost always associated with corresponding declines in TFP along with little change in capital stocks and much smaller changes in labor inputs Naturally these drops in TFP are not caused by technological regress but result from the breakdown of the market or increases in other sources of inefficiency Therefore technology differences should always be construed rather broadly and we should pay special attention to crosscountry differences in the efficiency of production By implication to understand TFP differences across countries we must study not 106 Chapter 3 The Solow Model and the Data only differences in the techniques that they use but also the way they organize markets and firms and how they provide incentives to different agents in the economy This insight again shapes our agenda for the rest of the book especially paving the way for investigating endogenous technological change in Part IV and differences in technology and productive efficiency across countries in Parts VI and VII There is one more sense in which what we have learned in this chapter is limited What the Solow model makes us focus onphysical capital human capital and technologyare proximate causes of economic growth in crosscountry differences It is important to know which of these proximate causes are important and how they affect economic performance both to have a better understanding of the mechanics of economic growth and also to know which classes of models to focus on But at some level and exaggerating somewhat to say that a country is poor because it has insufficient physical capital human capital and inefficient technology is like saying that a person is poor because he or she does not have money There are in turn other reasons some countries are more abundant in physical capital human capital and technology in the same way as there are factors that cause one person to have more money than another In Chapter 1 I referred to these as the fundamental causes of differences in prosperity contrasting them with the proximate causes A satisfactory understanding of economic growth and differences in prosperity across countries requires both an analysis of proximate causes and of fundamental causes of economic growth The former is essential for the study of the mechanics of economic growth and to develop the appropriate formal models incorporating these insights The latter is important for understanding why some societies make choices that lead them to low physical capital low human capital and inefficient technology and thus to relative poverty This is the issue I turn to in the next chapter 38 References and Literature The growth accounting framework is introduced and applied in Solow 1957 Jorgensen Gol lop and Fraumeni 1987 give a comprehensive development of this framework emphasizing that competitive markets are necessary and essentially sufficient for this approach to work They also highlight the measurement difficulties and emphasize that underestimates of improve ments in the quality of physical and human capital lead to overestimates of the contribution of technology to economic growth Jorgensen 2005 contains a more recent survey Regression analysis based on the Solow model has a long history More recent contributions include Baumol 1986 Barro 1991 and Barro and SalaiMartin 1992 Barro 1991 has done more than anybody else to popularize growth regressions which have become a very commonly used technique over the past two decades See Durlauf 1996 Durlauf Johnson and Temple 2005 and Quah 1993 for various critiques of growth regressions especially focusing on issues of convergence Wooldridge 2002 contains an excellent discussion of issues of omitted variable bias and the different approaches that can be used see eg Chapters 4 5 and 811 in Wooldridges book You should read more about the economic limitations of growth regressions and the econometric problems facing such regressions before embarking upon your own empirical analyses The augmented Solow model with human capital is a generalization of the model presented in Mankiw Romer and Weil 1992 As noted in the text treating human capital as a separate factor of production is somewhat unusual and difficult to microfound Different ways of introducing human capital in the basic growth model are discussed in Chapter 10 Mankiw Romer and Weil 1992 also provide the first regression estimates of the Solow and the augmented Solow models A detailed critique of Mankiw Romer and Weil is provided in Klenow and Rodriguez 1997 Hall and Jones 1999 and Klenow and Rodriguez 1997 provide the first calibrated estimates of productivity technology differences across countries Caselli 2005 gives an excellent overview of this literature with a detailed discussion of how one might correct for differences in the quality of physical and human capital across countries He reaches the conclusion that such corrections will not change the basic conclusions of Klenow and Rodriguez and Hall and Jones that crosscountry technology differences are important Subsection 362 draws on Trefer 1993 Trefer does not emphasize the productivity estimates implied by this approach focusing more on the method as a way of testing the HeckscherOhlin model Nevertheless these productivity estimates are an important input for growth economists Trefers approach has been criticized for various reasons which are secondary for our focus here The interested reader should look at Gabaix 2000 and Davis and Weinstein 2001 Consider the basic Solow model with no population growth and no technological progress and a production function of the form FK H where H denotes the efficiency units of labor human capital given by H iNhi where N is the set of all individuals in the population and hi is the human capital of individual i Assume that H is fixed Suppose there are no human capital externalities and factor markets are competitive a Calculate the steadystate equilibrium of this economy b Prove that if 10 higher h at the individual level is associated with increase in earnings then a 10 increase in the countrys stock of human capital H will lead to increase in state output Compare this result to the immediate impact of an unanticipated 10 increase in H ie consider the impact of a 10 increase in H with the stock of capital unchanged c Consider a collection of Solow economies each with different levels of b s and n Show that an equivalent of the conditional convergence regression equation 313 can be derived from an analogue of 310 in this case d Prove Proposition 32 e In the augmented Solow model see Propositions 31 and 32 determine the impact of increases in sk sy and n on h and k f Consider a world economy consisting of countries represented by the augmented Solow growth model with the production functions given by 316 Derive the equivalent of the fundamental production accounting equation in this case and explain how one might use available data to estimate TFP growth using this equation 4 Fundamental Determinants of Differences in Economic Performance 41 Proximate versus Fundamental Causes The factors we have listed innovation economies of scale education capital accu mulation etc are not causes of growth they are growth North and Thomas 1973 p 2 italics in original The previous chapter illustrated how the Solow growth model can be used to understand crosscountry income differences and the process of economic growth In the context of the Solow growth model the process of economic growth is driven by technological progress Crosscountry income differences on the other hand are due to a combination of technology differences and differences in physical capital per worker and in human capital per worker While this approach provides us with a good starting point and delineates potential sources of economic growth and crosscountry income differences these sources are only proximate causes of economic growth and economic success Let us focus on crosscountry income dif ferences for example As soon as we attempt to explain these differences with technology physical capital and human capital differences an obvious question presents itself if technol ogy physical capital and human capital are so important in understanding differences in the wealth of nations and if they can account for 5fold 10fold 20fold or even 30fold differ ences in income per capita across countries then why is it that some societies do not improve their technologies invest in physical capital and accumulate human capital as much as others It appears therefore that any explanation that simply relies on technology physical capital and human capital differences across countries is at some level incomplete There must be other deeper reasons that we will refer to as fundamental causes of economic growth It is these reasons that are preventing many countries from investing enough in technology physical capital and human capital 109 110 Chapter 4 Fundamental Determinants of Differences in Economic Performance An investigation of fundamental causes of economic growth is important for at least two reasons First any theory that focuses on the intervening variables proximate causes alone without understanding the underlying driving forces would be incomplete Thus growth theory will not fulfill its full promise until it comes to grips with these fundamental causes Second if part of our study of economic growth is motivated by improving the growth performance of certain nations and the living standards of their citizens understanding fundamental causes is central to this objective since attempting to increase growth merely by focusing on proximate causes would be tantamount to dealing with symptoms of diseases without understanding what the diseases themselves are While such attacks on symptoms can sometimes be useful they are no substitute for a fuller understanding of the causes of the disease which may allow a more satisfactory treatment In the same way we may hope that an understanding of the fundamental causes of economic growth could one day offer more satisfactory solutions to the major questions of social sciences concerning why some countries are poor and some are rich and how we can ensure that more nations grow faster What could these fundamental causes be Can we make progress in understanding them And perhaps most relevant for this book is growth theory useful in such an endeavor In this chapter I develop some answers to these questions Let us start with the last two questions The argument in this book is that a good understanding of the mechanics of economic growth and thus the construction of detailed models of the growth process are essential for a successful investigation of the fundamental causes of economic growth This understanding is crucial for at least two reasons first we can only pose useful questions about the fundamental causes of economic growth by understanding what the major proximate causes are and how they impact economic outcomes Second only models that provide a good approximation to reality and are successful in qualitatively and quantitatively matching the major features of the growth process can inform us about whether the potential fundamental causes that are proposed could indeed play a significant role in generating the huge differences observed in income per capita across countries Our analysis of the mechanics of economic growth will often enable us to discard or refine certain proposed fundamental causes As to the question of whether we can make progress the vast economic growth literature is evidence that progress is being made and more progress is certainly achievable In some sense it is part of the objective of this book to convince you that the answer to this question is yes Returning to the first question there are innumerable fundamental causes of economic growth that various economists historians and social scientists have proposed over the ages Clearly listing and cataloging them is neither informative nor useful Instead I classify the major candidate fundamental causes of economic growth into four categories of hypotheses While such a classification undoubtedly fails to do justice to some of the nuances of the literature it is satisfactory for our purposes of highlighting the main factors affecting cross country income differences and economic growth These are 1 The luck hypothesis 2 The geography hypothesis 3 The culture hypothesis and 4 The institutions hypothesis By luck I refer to the set of fundamental causes that explain divergent paths of economic performance among countries that are otherwise identical either because some small uncer tainty or heterogeneity between them has led to different choices with farranging consequences or because of different selection among multiple equilibria Multiple equilibria correspond to different equilibrium configurations arising for the same underlying economic environment 41 Proximate versus Fundamental Causes 111 When models exhibit multiple equilibria we are often unable to make specific predictions as to which of these equilibria will be selected by different countries and it is possible for two otherwise identical countries to end up in different equilibria with quite distinct implications for economic growth and living standards Luck and multiple equilibria can manifest themselves through any of the proximate causes discussed so far and through some additional mechanisms discussed later in the book For example multiple equilibria can exist in technology adoption or in models that focus on investments in human and physical capital Therefore explanations based on luck or multiple equilibria are often theoretically well grounded Whether they are empirically plausible is another matter By geography I refer to all factors that are imposed on individuals as part of the physical geographic and ecological environment in which they live Geography can affect economic growth through a variety of proximate causes Geographic factors that can influence the growth process include soil quality which can affect agricultural productivity natural resources which directly contribute to the wealth of a nation and may facilitate industrialization by providing certain key resources such as coal and iron ore during critical times climate which may affect productivity and attitudes directly topography which can affect the costs of transportation and communication and disease environment which can affect individual health productivity and incentives to accumulate physical and human capital For example in terms of the aggregate production function of the Solow model poor soil quality lack of natural resources or an inhospitable climate may correspond to a low level of A that is to a type of inefficient technology Many philosophers and social scientists have suggested that climate also affects preferences in a fundamental way so perhaps individuals living in certain climates have a preference for earlier rather than later consumption thus reducing their saving rates of both physical and human capital Finally differences in the disease burden across areas may affect the productivity of individuals and their willingness to accumulate human capital Thus geographybased explanations can easily be incorporated into both the simple Solow model and the more sophisticated models discussed later in the book By culture I refer to beliefs values and preferences that influence individual economic behavior Differences in religious beliefs across societies are among the clearest examples of cultural differences that may affect economic behavior Differences in preferences for exam ple regarding how important wealth is relative to other statusgenerating activities and how patient individuals should be might be as important asor even more important thanluck geography and institutions in affecting economic performance Broadly speaking culture can affect economic outcomes through two major channels First it can influence the willingness of individuals to engage in different activities or to tradeoff consumption today versus consump tion tomorrow Via this channel culture influences societies occupational choices market structure saving rates and individuals willingness to accumulate physical and human capital Second culture may also affect the degree of cooperation and of trust in society which are important foundations for productivityenhancing activities By institutions I refer to rules regulations laws and policies that affect economic incentives and thus the incentives to invest in technology physical capital and human capital It is a truism of economic analysis that individuals only take actions that are rewarded Institutions which shape these rewards must therefore be important in affecting all three of the proximate causes of economic growth What distinguishes institutions from geography luck and culture is that they are social choices Although laws and regulations are not directly chosen by individuals and some institutional arrangements may be historically persistent in the end the laws policies and regulations under which a society lives are the choices of the members of that society If the members of the society collectively decide to change them they can do so This possibility implies that if institutions are a major fundamental cause of 112 Chapter 4 Fundamental Determinants of Differences in Economic Performance economic growth and crosscountry differences in economic performance they can potentially be reformed to achieve better outcomes Such reforms may not be easy they may encounter stiff opposition and often we may not exactly know which reforms will work But they are still within the realm of the possible and further research might clarify how such reforms will affect economic incentives and how they can be implemented There is a clear connection between institutions and culture Both affect individual behavior and both are important determinants of incentives Nevertheless a crucial difference between the theories in these two categories justifies their separation Institutions are directly under the control of the members of the society in the sense that by changing the distribution of resources constitutions laws and policies individuals can collectively influence the institutions under which they live In contrast culture refers to a set of beliefs that have evolved over time and are outside the direct control of individuals1 Even though institutions might be hard to change in practice culture is much harder to influence and any advice to a society that it should change its culture is almost vacuous It is also important to emphasize that institutions themselves even if they are a fundamental cause of differences in economic growth and income across countries are endogenous They are equilibrium choices made either by the society at large or by some powerful groups in society One can then argue that luck geography or culture should be more important because they may be more exogenous in the sense that they are not equilibrium choices in the same way as institutions are and institutions vary across societies largely because of geographic cultural or random factors While at some philosophical level this argument is correct it is not a particularly useful observation It neither obviates the need to understand the direct effects of luck geography culture and institutions and these direct effects have been the focus of much of the debate in this area nor does it imply that understanding the specific role of institutions and economic development is secondary in any sense After all if we can understand what the effects of institutions are and which specific types of institutions matter institutional reform can lead to major changes in economic behavior even if part of the original variation in institutions was due to geography luck or culture In the rest of this chapter I explain the reasoning motivating these different hypotheses and provide a brief overview of the empirical evidence pertaining to various fundamental causes of economic growth The theoretical underpinnings and implications of the institutions view are further developed in Part VIII of the book At this point the reader should be warned that I am not an objective outside observer in this debate but a strong proponent of the institutions hypothesis Therefore not surprisingly this chapter concludes that the institutional differences are at the root of the important proximate causes that I have listed Nevertheless the same evidence can be interpreted in different ways and the reader should feel free to draw his or her own conclusions Before delving into a discussion of the fundamental causes one other topic deserves a brief discussion This is where I start in the next section 42 Economies of Scale Population Technology and World Growth As emphasized in Chapter 1 crosscountry income differences result from the differential growth experiences of countries over the past two centuries This makes it important for us to understand the process of economic growth Equally remarkable is the fact that world economic growth is by and large a phenomenon of the past 200 years or so Thus other major questions 1 A major and important exception to this lack of control is the effect of education on the beliefs and values of individuals concern why economic growth started so recently and why there was little economic growth before The growth literature has provided a variety of interesting answers to these questions Much of the literature focuses on the role of economies of scale and population The argument goes as follows in the presence of economies of scale or increasing returns to scale the population needs to have reached a certain critical level so that technological progress can gather speed Alternatively some natural steady progress of technology that may have been going on in the background needs to reach a critical threshold for the process of growth to begin These scenarios are quite plausible World population has indeed increased tremendously over the past million years and the worlds inhabitants today have access to a pool of knowledge and technology unimaginable to our ancestors Could these longrun developments of the world economy also account for crosscountry differences Is the increase in world population a good explanation for the takeover of the world economy Let us focus on population to give a preliminary answer to these questions The simplest way of thinking of the relationship between population and technological change is the SimonKremer model named after the demographer Julian Simon and the economist Michael Kremer This model is implicitly one of the entire world economy since there are no crosscountry differences Imagine that there is a small probability that each individual will discover a new idea that will contribute to the knowledge pool of the society Crucially these random discoveries are independent across individuals so that a larger pool of individuals implies the discovery of more new ideas increasing aggregate productivity Let output be determined simply by technology this condition can be generalized so that technology and capital determine output as in the Solow model but this does not affect the point I make here Yt Ltα AtZt1α where α 0 1 Yt is world output At is the world stock of technology Lt is world population and Z is some other fixed factor of production eg land I normalize Z 1 without loss of any generality Time is continuous and ideas are discovered at the rate λ so that the knowledge pool of the society evolves according to the differential equation At λLt A0 0 taken as given Population in turn is a function of output for example because of the Malthusian channels discussed in Chapter 21 For instance suppose that population increases linearly in output Lt φYt Combining these three equations we obtain see Exercise 41 At λφYtAt The solution to this differential equation involves At expλαφ1αtA0 Equation 44 shows how a model of economies of scale increasing returns in population can generate a steady increase in technology It is also straightforward to verify that Yt φ1αAt 114 Chapter 4 Fundamental Determinants of Differences in Economic Performance so that aggregate income also grows at the constant level λφ11α Such a model would generate steady growth but no acceleration Simon and Kremer instead assume that there are stronger externalities to population than in 41 They impose the following equation governing the accumulation of ideas At At λLt This implies that the law of motion of technology is given by see Exercise 42 At 1 A01 λφ11αt 45 In contrast to 44 this equation implies an accelerating output level Starting from a low level of A0 or L0 this model would generate a long period of low output followed by an acceleration or takeoff reminiescent to the modern economic growth experience discussed in Chapter 1 Therefore a model with significant economies of scale is capable of generating the pattern of takeoff we see in the data While such a story which has been proposed by many economists may have some appeal for accounting for world growth it is important to emphasize that it has little to say about crosscountry income differences or why modern economic growth started in some countries Western Europe and not others Asia South America Africa In fact if we take Western Europe and Asia as the relevant economic units the European population has consistently been less than that of Asia over the past 2000 years see eg Figure 211 thus it is unlikely that simple economies of scale in population are responsible for the economic takeoff in Western Europe while Asia stagnated This discussion therefore suggests that models based on economies of scale of one sort or another do not provide us with fundamental causes of crosscountry income differences At best they are theories of growth of the world taken as a whole Moreover once we recognize that the modern economic growth process has been uneven meaning that it took place in some parts of the world and not others the appeal of such theories diminishes further If economies of scale were responsible for modern economic growth this phenomenon should also be able to explain when and where this process of economic growth started Existing models based on economies of scale do not In this sense they are unlikely to provide the fundamental causes of modern economic growth Then are these types of economies of scale and increasing returns to population unimportant Certainly not They may well be part of the proximate causes of the growth process eg the part lying in the black box of technology But this discussion suggests that these models need to be augmented by other fundamental causes to explain why when and where the takeoff occurred This further motivates the investigation of the fundamental causes 43 The Four Fundamental Causes 431 Luck and Multiple Equilibria Chapter 21 presents a number of models in which multiple equilibria or multiple steady states can arise because of coordination failures in the product market or imperfections in credit markets These models suggest that an economy with given parameter values can exhibit significantly different types of equilibrium behavior some with higher levels of income or perhaps sustained growth while other equilibria involve poverty and stagnation To give a 43 The Four Fundamental Causes 115 flavor of these models consider the following simple game of investment played by a large number of agents in the society Everybody else High Low investment investment High investment yH yH yL ε yL Individual Low investment yL yL ε yL yL Let us focus on symmetric equilibria The first column indicates that all agents except the individual in question have chosen high investment while the second corresponds to low investment by all agents The first row on the other hand corresponds to high investment by the individual in question and the second row is for low investment In each cell the first number refers to the income of the individual in question while the second number is the payoff to each of the other agents in the economy Suppose that yH yL and ε ε 0 This payoff matrix then implies that high investment is more profitable when others are also undertaking high investment For example this may be because of technological complementarities or aggregate demand externalities see Chapter 21 It is then clear that there are two purestrategy symmetric equilibria in this game In one equilibrium the individual expects all other agents to choose high investment and he does so himself Since the same calculus applies to each agent each agent will also ex pect high investment by all others and will choose high investment himself This establishes that high investment by all agents is an equilibrium Similarly when the individual expects all others to choose low investment it is a best response for him to choose low investment so that there also exists an equilibrium with low investment Thus this simple game exhibits two symmetric purestrategy equilibria Two features are worth noting First depending on the extent of complementarities and other economic interactions yH can be quite large relative to yL so there may be significant income differences in the allocations implied by the two different equilibria Thus if we believe that such a game is a good approximation to reality and different countries can end up in different equilibria the economic interactions here could help explain large differences in income per capita Second the two equilibria in this game are also Paretorankedall individuals are better off in the equilibrium in which everybody chooses high investment see Chapter 5 on the Pareto criterion Both of these features are shared by the Big Push models discussed in Chapter 21 In addition to models of multiple equilibria stochastic models in which the realization of certain random variables determines when a particular economy transitions from low to high productivity technologies and starts the process of takeoff might also be relevant in this context see Section 176 Both models of multiple equilibria and those in which stochastic variables determine the longrun growth properties of the economy are attractive as descriptions of certain aspects of the development process They are also informative about the mechanics of economic development in an interesting class of models But do they inform us about the fundamental causes of economic growth Can we say that the United States is rich today while Nigeria is poor because the former has been lucky in its equilibrium selection while the latter has been unlucky Can we pinpoint their divergent development paths to some small stochastic events 200 300 or 400 years ago The answer seems to be no 116 Chapter 4 Fundamental Determinants of Differences in Economic Performance US economic growth is the cumulative result of a variety of processes ranging from innovations and free entrepreneurial activity to significant investments in human capital and rapid capital accumulation It is difficult to reduce these processes to a simple lucky break or the selection of the right equilibrium Even 400 years ago conditions were significantly different in the United States and in Nigeria and this led to different opportunities institutional paths and incentives It is the combination of the historical experiences of countries and different economic incentives that underlies their different processes of economic growth Equally important models based on luck or multiple equilibria can explain why there might be a 20year or perhaps a 50year divergence between two otherwise identical economies But how are we to explain a 500year divergence It certainly does not seem plausible to imagine that Nigeria today can suddenly switch equilibria and quickly achieve the level of income per capita in the United States2 Most models of multiple equilibria are unsatisfactory in another sense As in the simple example discussed above most models of multiple equilibria involve the presence of Paretoranked equilibria This implies that one equilibrium gives higher utility or welfare to all agents than another While such Paretoranked equilibria are a feature of parsimonious models which do not specify many relevant dimensions of heterogeneity that are important in practice it is not clear whether they are useful in thinking about why some countries are rich and others are poor If indeed it were possible for Nigerians to change their behavior and for all individuals in the nation to become better off say by switching from low to high investment in terms of the game above it is very difficult to believe that for 200 years they have not been able to coordinate on such a better action Most readers are aware that Nigerian history is shaped by religious and ethnic conflict and by a civil war that ravaged the nation and that the country is still adversely affected by the extreme corruption of politicians bureaucrats and soldiers who have enriched themselves at the expense of the population at large That an easy Paretoimproving change exists against this historical and social background seems improbable to say the least To be fair not all models of multiple equilibria allow easy transitions from a Paretoinferior equilibrium to a superior one In the literature a useful distinction can be made between models of multiple equilibria in which different equilibria can be reached if individuals change their beliefs and behaviors simultaneously versus models of multiple steady states with history dependence in which once a particular path of equilibrium is embarked upon it becomes much harderperhaps impossibleto transition to the other steadystate equilibrium see Chapter 21 Models with multiple steady states are more attractive for understanding persistent differences in economic performance across countries than models with multiple equilibria Nevertheless unless some other significant source of conflict of interest or distortions are incorporated it seems unlikely that the difference between the United States and Nigeria can be explained by using models in which the two countries have identical parameters but have made different choices and stuck with them The mechanics of how a particular steady state equilibrium can be maintained would be the most important element of such a theory and other fundamental causes of economic growth including institutions policies or perhaps culture must play a role in explaining this type of persistence Put differently in todays world of free information technology and capital flows if Nigeria had the same parameters the same opportunities and the same institutions as the United States there should exist some 2 Naturally one can argue that reforms or major changes in the growth trajectory are always outcomes of a switch from one equilibrium to another But such an explanation would not have much empirical content unless it is based on a wellformulated model of equilibrium selection and can make predictions about when we might expect such switches 43 The Four Fundamental Causes 117 arrangement such that these new technologies could be imported and everybody could be made better off Another challenge to models of multiple steady states concerns the ubiquity of growth miracles such as South Korea and Singapore which we discussed in Chapter 1 If cross country income differences are due to multiple steady states from which escape is totally or nearly impossible then how can we explain countries that embark upon a very rapid growth process The example of China may be even most telling here While China stagnated under communism until Maos death the changes in economic institutions and policies that took place thereafter have led to very rapid economic growth If China were in a lowgrowth steady state before Maos death then we need to explain how it escaped from this steady state after 1978 and why it did not do so before Inevitably this line of reasoning brings us to the role of other fundamental causes such as institutions policies and culture A different and perhaps more promising argument about the importance of luck can be made by emphasizing the role of leaders Perhaps it was Mao who held back China and his death and the identity beliefs and policies of his successors were at the root of its subsequent growth Perhaps the identity of the leader of a country can thus be viewed as a stochastic event shaping economic performance This point of view probably has a lot of merit Recent empirical work by Jones and Olken 2005 shows that leaders seem to influence the economic performance of nations Thus luck could play a major role in crosscountry income and growth differences by determining whether growthenhancing or growthretarding leaders are selected Nevertheless such an explanation is closer to the institutional approaches than the pure luck category First leaders often influence the economic performance of their societies by the policies they set and the institutions they develop Second the selection and behavior of leaders and the policies that they pursue are part of the institutional explanations Third Jones and Olkens research points to an important interaction between the effect of leaders and a societys institutions Leaders seem to matter for economic growth only in countries where institutions are nondemocratic or weak in the sense of not placing constraints on politicians or elites In democracies and in societies where other institutions appear to place checks on the behavior of politicians and leaders the identity of the leaders seems to play almost no role in economic performance Given these considerations I tentatively conclude that models emphasizing luck and mul tiple equilibria are useful for our study of the mechanics of economic development but they are unlikely to provide us with the fundamental causes of why world economic growth started 200 years ago and why some countries are rich while others are poor today 432 Geography While the approaches in the last subsection emphasize the importance of luck and multiple equilibria among otherwise identical societies an alternative is to emphasize the deep hetero geneity across societies The geography hypothesis is first and foremost about the fact that not all areas of the world are created equal Nature that is the physical ecological and geographical environment of nations plays a major role in their economic experiences As pointed out above geographic factors can play this role by determining both the preferences and the opportunity set of individual economic agents in different societies There are at least three main versions of the geography hypothesis each emphasizing a different mechanism for how geography affects prosperity The first and earliest version of the geography hypothesis goes back to Montesquieu 1748 1989 Montesquieu who was a brilliant French philosopher and an avid supporter 118 Chapter 4 Fundamental Determinants of Differences in Economic Performance of republican forms of government was also convinced that climate was among the main determinants of the fate of nations He believed that climate in particular heat shaped human attitudes and effort and through this channel affected both economic and social outcomes He wrote in his classic book The Spirit of the Laws 1989 p 234 The heat of the climate can be so excessive that the body there will be absolutely without strength So prostration will pass even to the spirit no curiosity no noble enterprise no generous sentiment inclinations will all be passive there laziness there will be happiness People are more vigorous in cold climates The inhabitants of warm countries are like old men timorous the people in cold countries are like young men brave Today some of the pronouncements in these passages appear somewhat naıve and perhaps bordering on political incorrectness They still have many proponents however Even though Montesquieus eloquence makes him stand out among those who formulated this perspective he was neither the first nor the last to emphasize such geographic fundamental causes of economic growth Among economists a more revered figure is one of the founders of our discipline Alfred Marshall Almost a century and a half after Montesquieu Marshall 1890 p 195 wrote Vigor depends partly on race qualities but these so far as they can be explained at all seem to be chiefly due to climate While the first version of the geography hypothesis appears naıve and raw to many of us its second version which emphasizes the impact of geography on the technologies available to a society especially in agriculture is more palatable and has many more supporters This view is developed by an early Nobel Prize winner in economics Gunnar Myrdal 1968 vol 3 p 2121 who wrote Serious study of the problems of underdevelopment should take into account the climate and its impacts on soil vegetation animals humans and physical assets in short on living conditions in economic development More recently Jared Diamond in his widely popular Guns Germs and Steelespouses this view and argues that geographical differences between the Americas and Europe or more appropriately Eurasia have determined the timing and nature of settled agriculture and by means of this channel shaped whether societies have been able to develop complex organi zations and advanced civilian and military technologies 1997 eg p 358 The economist Jeffrey Sachs 2001 p 2 has been a recent and forceful proponent of the importance of geography in agricultural productivity stating that By the start of the era of modern economic growth if not much earlier temperatezone technologies were more productive than tropicalzone technologies There are also reasons for questioning this second and more widelyheld view of geo graphic determinism Most of the technological differences emphasized by these authors refer to agriculture But as Chapter 1 emphasized the origins of differential economic growth across countries goes back to the age of industrialization Modern economic growth came with indus try and it is the countries that have failed to industrialize that are poor today Low agricultural productivity if anything should create a comparative advantage in industry and encourage those countries with unfavorable geography to start investing in industry before others did One might argue that reaching a certain level of agricultural productivity is a prerequisite for industrialization While this suggestion is plausible or at least possible many of the societies that later failed to industrialize had already achieved a certain level of agricultural productivity 43 The Four Fundamental Causes 119 and in fact were often ahead of those who later industrialized very rapidly see Section 44 Thus a simple link between unfavorable agricultural conditions and the failure to take off seems to be absent3 The third variant of the geography hypothesis which has become particularly popular over the past decade links poverty in many areas of the world to their disease burden emphasizing that the burden of infectious disease is higher in the tropics than in the temperate zones Sachs 2000 p 32 Bloom and Sachs 1998 and Gallup and Sachs 2001 p 91 claim that the prevalence of malaria alone reduces the annual growth rate of subSaharan African economies by as much as 26 a year Such a magnitude implies that had malaria been eradicated in 1950 income per capita in subSaharan Africa would have been double what it is today If we add to this the effect of other diseases we would obtain even larger effects This third version of the geography hypothesis may be much more plausible than the first two especially since the microeconomics literature shows that unhealthy individuals are less productive and perhaps less able to learn and thus accumulate human capital I discuss both the general geography hypothesis and this specific version of it in greater detail in the next two sections But an important caveat needs to be mentioned The fact that the burden of disease is heavier in poor nations today is as much a consequence as a cause of poverty European nations in the eighteenth and even nineteenth centuries were plagued by many diseases It was the process of economic development that enabled them to eradicate these diseases and create healthier living environments The fact that many poor countries have unhealthy environments is at least in part a consequence of their failure to develop economically 433 Institutions An alternative fundamental cause of differences in economic growth and income per capita is institutions One problem with the institutions hypothesis is that it is somewhat difficult to define what institutions are In daily usage the word institutions refers to many different things and the academic literature is sometimes not clear about its definition The economic historian Douglass North was awarded the Nobel Prize in economics largely because of his work emphasizing the importance of institutions in the historical development process North 1990 p 3 offers the following definition Institutions are the rules of the game in a society or more formally are the humanly devised constraints that shape human interaction He goes on to emphasize the key implications of institutions In consequence institutions structure incentives in human exchange whether polit ical social or economic This definition encapsulates the three important elements that make up institutions First they are humanly devised that is in contrast to geography which is outside human control institutions refer to manmade factors Institutions are about the effect of societies own choices on their own economic fates Second institutions place constraints on individual behavior These constraints do not need to be unassailable any law can be broken any regulation can be ignored Nevertheless policies regulations and laws that punish certain types of behavior 3 Ex post one can in fact tell the opposite story perhaps the poor nations of today had agriculturally superior land and this created a comparative advantage against industry This is not an entirely convincing explanation either since as discussed in Chapter 20 most lessdeveloped economies today have lower agricultural as well as lower industrial productivity than the relatively advanced nations 120 Chapter 4 Fundamental Determinants of Differences in Economic Performance while rewarding others will naturally have an effect on behavior And this brings us to the third important element in the definition The constraints placed on individuals by institutions shape human interaction and affect incentives In some deep sense institutions much more than the other candidate fundamental causes are about the importance of incentives The reader may have already noted that the above definition makes the concept of institutions rather broad In fact this is precisely the sense in which I use the concept throughout this book institutions refer to a broad cluster of arrangements that influence various economic interactions among individuals These include economic political and social relations among households individuals and firms The importance of political institutions which determine the process of collective decision making in society cannot be overstated and is the topic of analysis in Part VIII of this book A more natural starting point for the study of the fundamental causes of income differences across countries is in economic institutions which comprise such things as the structure of property rights the presence and well or ill functioning of markets and the contractual op portunities available to individuals and firms Economic institutions are important because they influence the structure of economic incentives in society Without property rights individuals do not have the incentive to invest in physical or human capital or adopt more efficient technolo gies Economic institutions are also important because they ensure the allocation of resources to their most efficient uses and determine who obtains profits revenues and residual rights of control When markets are missing or ignored as was the case in many former socialist soci eties for example gains from trade go unexploited and resources are misallocated Economic theory therefore suggests that societies with economic institutions that facilitate and encour age factor accumulation innovation and the efficient allocation of resources should prosper relative to societies that do not have such institutions The hypothesis that differences in economic institutions are a fundamental cause of differ ent patterns of economic growth is intimately linked to the models I develop in this book All economic models start with a specification of economic institutions for example the struc ture of markets the set of feasible contracts and transactions and allocations of endowments and ownership rights to individuals Moreover in all of these models individuals respond to incentives It is the economic institutions determined broadly by the way in which individ uals organize their societies that shape these incentives Some ways of organizing societies encourage people to innovate take risks save for the future find better ways of doing things learn and educate themselves solve problems of collective action and provide public goods Others do not Our theoretical models pinpoint what specific policy and institutional variables are important in retarding or encouraging economic growth Part VIII of the book develops theoretical approaches to the analysis of what constitutes good economic institutions that encourage physical and human capital accumulation and the development and adoption of better technologies though good economic institutions do change with environment and time It should already be intuitive to the reader that economic institu tions that tax productivityenhancing activities will not encourage economic growth Economic institutions that ban innovation will not lead to technological improvements Therefore en forcement of some basic property rights and some amount of free enterprise are indispens able But other aspects of economic institutions matter as well Human capital for example is important both for increasing productivity and for technology adoption However for a broad cross section of society to be able to accumulate human capital some degree of equality of opportunity is necessary Economic institutions that only protect the rights of a rich elite or the privileged will not achieve such equality of opportunity and will often create other distortions potentially retarding economic growth Chapter 14 emphasizes that the process of Schum peterian creative destruction in which new firms improve over and destroy incumbents is 43 The Four Fundamental Causes 121 an essential element of economic growth Schumpeterian creative destruction requires a level playing field so that incumbents are unable to block technological progress Economic growth based on creative destruction therefore also requires economic institutions that guarantee some degree of equality of opportunity in the society Another question may have already occurred to the reader why should any society have economic and political institutions that retard economic growth Would it not be better for all parties to maximize the size of the national pie level of GDP consumption or economic growth There are two possible answers to this question The first takes us back to multiple equilibria It may be that the members of the society cannot coordinate on the right eg growthenhancing institutions This answer is not satisfactory for the same reasons as other broad explanations based on multiple equilibria are unsatisfactory if there exists an equilibrium institutional improvement that will make all members of a society richer and better off it seems unlikely that the society will be unable to coordinate on this improvement for extended periods of time The second answer recognizes that there are inherent conflicts of interest within the society There are no reforms changes or advances that would make everybody better off as in the Schumpeterian creative destruction stories each reform change or advance creates winners and losers Part VIII shows that institutional explanations are intimately linked with conflicts of interest in society Put simply the distribution of resources cannot be separated from the aggregate economic performance of the economyor perhaps in a more familiar form efficiency and distribution cannot be decoupled Institutions that fail to maximize the growth potential of an economy may nonetheless create benefits for some segments of the society who then form a constituency in favor of these institutions Thus to understand the sources of institutional variation we have to study the winners and losers of different institutional reforms and why even when the institutional change in question may increase the size of the national pie winners are unable to buy off or compensate losers and why they are not powerful enough to overwhelm the potential losers Such a study will not only help explain why some societies choose or end up with institutions that do not encourage economic growth but it will also enable us to make predictions about institutional change After all the fact that institutions can and do change is a major difference between the institutions hypothesis and the geography and culture hypotheses Questions about equilibrium institutions and endogenous institutional change are central for the institutions hypothesis but must be postponed until Part VIII Here note that the endogeneity of institutions has another important implication the endogeneity of institutions makes empirical work on assessing the role of institutions more challenging because it implies that the standard simultaneity biases in econometrics will be present when we look at the effect of institutions on economic outcomes4 In this chapter I focus on the empirical evidence in favor of and against the various hypotheses I argue that this evidence by and large suggests that institutional differences that societies choose and end up with are a primary determinant of their economic fortunes The discussion below provides a summary of recent empirical work to bolster this case Nevertheless it is important to emphasize that luck geography and culture are also potentially important and the four fundamental causes are complementary The evidence suggests that institutions are the most important one among these four causes but it does not deny the potential role of other factors such as cultural influences 4 Note also that although geography is exogenous in the sense that with some notable exceptions eg climate change global warming it is not much influenced by economic decisions this does not make it econometrically exogenous Geographic characteristics may still be and in fact likely are correlated with other factors that influence economic growth 122 Chapter 4 Fundamental Determinants of Differences in Economic Performance 434 Culture The final fundamental explanation for economic growth emphasizes the idea that different societies or perhaps different races or ethnic groups have distinct cultures because of different shared experiences or different religions Culture is viewed by some social scientists as a key determinant of the values preferences and beliefs of individuals and societies and the argument goes these differences play a key role in shaping economic performance At some level culture can be thought of as influencing equilibrium outcomes for a given set of institutions Recall that in the presence of multiple equilibria there is a central question of equilibrium selection For example in the simple game discussed in Section 431 culture may be one of the factors determining whether individuals coordinate on the high or the low investment equilibrium Good cultures can be thought of as ways of coordinating on better Paretosuperior equilibria Naturally the arguments discussed abovethat an entire society being stuck in an equilibrium in which all individuals are worse off than in an alternative equilibrium is implausiblewould militate against the importance of this particular role of culture Alternatively different cultures generate different sets of beliefs about how people behave and these distinctions can alter the set of equilibria for a given specification of institutions eg some beliefs allow punishment strategies to be used whereas others do not The most famous link between culture and economic development is that proposed by Max Weber 1930 p 11 who argued that the origins of industrialization in Western Europe could be traced to a cultural factorthe Protestant reformation and particularly the rise of Calvinism Interestingly Weber provided a clear summary of his views as a comment on Montesquieus arguments Montesquieu says of the English that they had progressed the farthest of all peoples of the world in three important things in piety in commerce and in freedom Is it not possible that their commercial superiority and their adaptation to free political institutions are connected in some way with that record of piety which Montesquieu ascribes to them Weber argued that English piety in particular Protestantism was an important driver of capitalist development Protestantism led to a set of beliefs that emphasized hard work thrift and saving It also interpreted economic success as consistent with even as signaling being chosen by God Weber contrasted these characteristics of Protestantism with those of other religions such as Catholicism which Weber argued did not promote capitalism More recently similar ideas have been applied to emphasize different implications of other religions Many historians and scholars have argued that the rise of capitalism the process of economic growth and industrialization are intimately linked to cultural and religious beliefs Similar ideas have been proposed as explanations for why Latin American countries are relatively poor because of their Iberian culture while their North American neighbors are more prosperous because of their AngloSaxon culture A related argument originating in anthropology argues that societies may become dys functional because their cultural values and their system of beliefs do not encourage co operation An original and insightful version of this argument is developed in Banfields 1958 analysis of poverty in southern Italy His ideas were later popularized by Putnam 1993 who suggested the notion of social capital as a standin for cultural attitudes that lead to cooper ation and other good outcomes Many versions of these ideas are presented in one form or another in the economics literature as well Two challenges confront theories of economic growth based on culture The first is the difficulty of measuring culture While there has been some progress in measuring certain cultural characteristics with selfreported beliefs and attitudes in social surveys simply stating 44 The Effect of Institutions on Economic Growth 123 that the north of Italy is rich because it has good social capital while the south is poor because it has poor social capital runs the risk of circularity The second difficulty confronting cultural explanations is accounting for growth miracles such as those of South Korea and Singapore As mentioned above if some Asian cultural values are responsible for the successful growth experiences of these countries it becomes difficult to explain why these Asian values did not lead to growth before Why do these values not spur economic growth in North Korea If Asian values are important for Chinese growth today why did they not lead to a better economic performance under Maos dictatorship Both of these challenges are in principle surmountable One may be able to develop models of culture with better mapping to data and also with an associated theory of how culture may change rapidly under certain circumstances While possible in principle such theories have not been developed Moreover the evidence presented in the next section suggests that cultural effects are not the major force behind the large differences in economic growth experienced by many countries over the past few centuries In this light culture may be best viewed as a complement to institutional factors for example acting as one of the forces responsible for institutional persistence 44 The Effect of Institutions on Economic Growth I now argue that there is convincing empirical support for the hypothesis that differences in economic institutions more than luck geography or culture cause differences in incomes per capita Let us start by looking at the simplest correlation between a measure of economic institutions and income per capita Figure 41 shows the crosscountry correlation between the log of GDP per capita in 1995 and a broad measure of property rights protection against expropriation risk averaged over the period 1985 to 1995 The data on this measure of economic institutions come from Political Risk Services a private company that assesses the expropriation risk that foreign investments face in different countries These data are not perfect They reflect the subjective assessments of some analysts about how secure property rights are Nevertheless they are useful for our purposes First they emphasize the security of property rights which is an essential aspect of economic institutions especially in regard to their effect on economic incentives Second these measures are purchased by businessmen contemplating investment in these countries thus they reflect the market assessment of security of property rights Figure 41 shows that countries with more secure property rightsthus better economic institutionshave higher average incomes One should not interpret the correlation in this figure as depicting a causal relationshipthat is as establishing that secure property rights cause prosperity First the correlation might reflect reverse causation it may be that only countries that are sufficiently wealthy can afford to enforce property rights Second and more importantly there might be a problem of omitted variable bias It could be something else for example geography or culture that explains both why countries are poor and why they have insecure property rights Thus if omitted factors determine institutions and incomes we would spuriously infer the existence of a causal relationship between economic institutions and incomes when in fact no such relationship exists This is the standard identification prob lem in econometrics resulting from simultaneity or omitted variable biases Finally security of property rightsor other proxy measures of economic institutionsare themselves equi librium outcomes presumably resulting from the underlying political institutions and political conflict While this last point is important a satisfactory discussion of institutional equilibria necessitates the modeling of political economy interactions and must wait until Part VIII To further illustrate these potential identification problems suppose that climate or geogra phy matter for economic performance In fact a simple scatterplot shows a positive association 124 Chapter 4 Fundamental Determinants of Differences in Economic Performance AGO ARE ARG AUS AUT BEL BFA BGD BGR BHR BHS BOL BRA BWA CAN CHE CHL CHN CIV CMR COG COL CRI CZE DNK DOMDZA ECU EGY ESP ETH FIN FRA GAB GBR GHA GIN GMB GRC GTM GUY HKG HND HTI HUN IDN IND IRL IRN ISL ISR ITA JAM JOR JPN KEN KOR KWT LKA LUX MAR MDG MEX MLI MLT MNG MOZ MWI MYS NER NGA NIC NLD NOR NZL OMN PAK PAN PER PHL POL PRT PRY QAT ROM RUS SAU SDN SEN SGP SLE SLV SUR SWE SYR TGO THA TTO TUN TUR TZA UGA URY USA VEN VNM YEM ZAF ZAR ZMB ZWE 6 8 10 Log GDP per capita 1995 4 6 8 10 Average protection against risk of expropriation 198595 FIGURE 41 Relationship between economic institutions as measured by average expropriation risk between 1985 and 1995 and GDP per capita between latitude the absolute value of distance from the equator and income per capita which is consistent with the views of Montesquieu and other proponents of the geography hypothe sis Interestingly Montesquieu not only claimed that warm climate makes people lazy and thus unproductive but he also asserted that it made them unfit to be governed by democracy Thus according to Montesquieu despotism is the equilibrium political system in warm climates Therefore a potential explanation for the patterns in Figure 41 is that there is an omitted factor geography which explains both economic institutions and economic performance Ignoring this potential third factor would lead to mistaken conclusions Even if Montesquieus claim appears both unrealistic and condescending to our modern sensibilities the general point should be taken seriously the correlations depicted in Figure 41 and for that matter the correlations in Figure 42 do not necessarily reflect causal relationships As noted in the context of the effect of religion or social capital on economic performance these types of scatterplots correlations or their multidimensional version in ordinary least squares regressions cannot establish causality Doubt about the effect of omitted variables will almost always remain even for careful regression analyses How can we overcome the challenge of establishing a causal relationship between eco nomic institutions and economic outcomes The answer to this question is to specify econo metric approaches based on plausible identifying restrictions This can be done by estimating structural econometric models or using more reducedform approaches based on instrumental variable strategies We do not currently know enough about the evolution of economic institu tions and their impact on economic outcomes to be able to specify and estimate fully structural econometric models Thus as a first step we can look at more reducedform evidence that might still be informative about the causal relationship between institutions and economic growth 44 The Effect of Institutions on Economic Growth 125 AGO ARE ARG ARM AUS AUT AZE BDI BEL BEN BFA BGD BGR BHR BHS BLR BLZ BOL BRA BRB BWA CAF CAN CHE CHL CHN CIV CMR COG COL COM CPV CRI CZE DEU DMA DNK DOM DZA ECU EGY ERI ESP EST ETH FIN FJI FRA GAB GBR GEO GHA GIN GMB GRC GRD GTM GUY HKG HND HRV HTI HUN IDN IND IRL IRN ISL ISR ITA JAM JOR JPN KAZ KEN KGZ KNA KOR KWT LAO LCA LKA LSO LTU LUX LVA MAR MDA MDG MEX MKD MLI MLT MNG MOZ MRT MUS MWI MYS NAM NER NGA NIC NLD NOR NPL NZL OMN PAK PAN PER PHL POL PRT PRY QAT ROM RUS RWA SAU SDN SEN SGP SLE SLV SUR SVK SVN SWE SWZ SYR TCD TGO THA TJK TKM TTO TUN TUR TZA UGA UKR URY USA UZB VCT VEN VNM YEM ZAF ZAR ZMB ZWE 6 8 10 Log GDP per capita 1995 00 02 04 06 08 Latitude FIGURE 42 Relationship between latitude distance of capital from the equator and income per capita in 1995 One way of doing so is to learn from history in particular from the natural experiments unusual historical events during which while other fundamental causes of economic growth are held constant institutions change because of potentially exogenous reasons I now discuss lessons from two such natural experiments 441 The Korean Experiment Until the end of World War II Korea was under Japanese occupation Korean independence came shortly after the war The major fear of the United States during this time was the takeover of the entire Korean peninsula either by the Soviet Union or by communist forces under the control of the former guerrilla fighter Kim Il Sung US authorities therefore supported the influential nationalist leader Syngman Rhee who was in favor of separation rather than a united communist Korea Elections in the South were held in May 1948 amid a widespread boycott by Koreans opposed to separation The newly elected representatives proceeded to draft a new constitution and established the Republic of Korea to the south of the 38th parallel The North became the Democratic Peoples Republic of Korea under the control of Kim Il Sung These two independent countries organized themselves in radically different ways and adopted completely different sets of economic and political institutions The North followed the model of Soviet communism and the Chinese Revolution in abolishing private property in land and capital Economic decisions were not mediated by the market but by the communist state The South instead maintained a system of private property and capitalist economic institutions 126 Chapter 4 Fundamental Determinants of Differences in Economic Performance Before these institutional changes North and South Korea shared the same history and cultural roots In fact Korea exhibited an unparalleled degree of ethnic linguistic cultural geographic and economic homogeneity There are few geographic distinctions between the North and South and both share the same disease environment Moreover before the separation the North and the South were at the same level of development If anything there was slightly more industrialization in the North Maddison 2001 estimates that at the time of separation North and South Korea had approximately the same income per capita We can therefore think of the splitting of the Koreas 60 years ago as a natural experiment that can be used to identify the causal influence of institutions on prosperity Korea was split into two with the two halves organized in radically different ways while geography culture and many other potential determinants of economic prosperity were held constant Thus any differences in economic performance can plausibly be attributed to differences in institutions In the 60 years following the split the two Koreas have experienced dramatically diverging paths of economic development By the late 1960s South Korea was transformed into one of the Asian miracle economies experiencing one of the most rapid surges of economic prosperity in history Meanwhile North Korea stagnated By 2000 the level of income per capita in South Korea was 16100 while in North Korea it was only 1000 There is only one plausible explanation for the radically different economic experiences of the two Koreas after 1950 their different institutions led to divergent economic outcomes In this context it is noteworthy that the two Koreas not only shared the same geography but also the same culture so that neither geographic nor cultural differences could have much to do with the divergent paths of the two Koreas Of course one can say that South Korea was lucky while the North was unlucky even though this difference was not due to any kind of multiple equilibria but was a result of the imposition of different institutions Nevertheless the perspective of luck is unlikely to be particularly useful in this context since what is remarkable is the persistence of the dysfunctional North Korean institutions Despite convincing evidence that the North Korean system has been generating poverty and famine the leaders of the Communist Party in North Korea have opted to use all the means available to them to maintain their regime However convincing on its own terms the evidence from this natural experiment is not sufficient for the purposes of establishing the importance of economic institutions as the primary factor shaping crosscountry differences in economic prosperity First this is only one case and in controlled experiments in the natural sciences a relatively large sample is essential Second here we have an example of an extreme case the difference between a marketoriented economy and an extreme communist one Few social scientists today would deny that a lengthy period of totalitarian centrally planned rule has significant economic costs And yet many might argue that differences in economic institutions among capitalist economies or among democracies are not the major factor leading to differences in their economic trajectories To establish the major role of economic institutions in the prosperity and poverty of nations we need to look at a largerscale natural experiment in institutional divergence 442 The Colonial Experiment The Reversal of Fortune The colonization of much of the world by Europeans provides such a largescale natural experiment Beginning in the early fifteenth century and especially after 1492 Europeans conquered many other nations The colonization experience transformed the institutions in many diverse lands conquered or controlled by Europeans Most importantly Europeans imposed different sets of institutions in various parts of their global empire as exemplified most sharply by the contrast of the institutional structure that developed in the northeastern United States based on smallholder private property and democracy versus the institutions in the 44 The Effect of Institutions on Economic Growth 127 AGO ARG AUS BDI BEN BFABGD BHS BLZ BOL BRA BRB BWA CAF CAN CHL CIV CMR COG COL COM CPV CRI DMA DOM DZAECU EGY ERI ETH FJI GAB GHA GIN GMB GRD GTM GUY HKG HND HTI IDN IND JAM KEN KNA LAO LCA LKA LSO MAR MDG MEX MLI MOZ MRT MUS MWI MYS NAM NER NGA NIC NPL NZL PAK PAN PER PHL PRY RWA SDN SEN SGP SLE SLV SUR SWZ TCD TGO TTO TUN TZA UGA URY USA VCT VEN VNM ZAF ZAR ZMB ZWE 6 8 10 Log GDP per capita 1995 0 50 100 Urbanization 1995 percent FIGURE 43 Urbanization and income 1995 Caribbean plantation economies based on repression and slavery As a result while geography was held constant Europeans initiated significant changes in the economic institutions of different societies The impact of European colonialism on economic institutions is perhaps most dramatically conveyed by a single facthistorical evidence shows that there has been a remarkable reversal of fortune in economic prosperity within former European colonies Societies like the Mughals in India and the Aztecs and Incas in the Americas were among the richest civilizations in 1500 yet the nationstates that now exist in their boundaries are among the poorer nations of today In contrast countries occupying the territories of the lessdeveloped civilizations of North America New Zealand and Australia are now much richer than those in the lands of the Mughals Aztecs and Incas The reversal of fortune is not confined to such comparisons To document the reversal more broadly we need a proxy for prosperity 500 years ago Fortunately urbanization rates and population density can serve the role of such proxies Only societies with a certain level of productivity in agriculture and a relatively developed system of transport and commerce can sustain large urban centers and a dense population Figure 43 shows the relationship between income per capita and urbanization fraction of the population living in urban centers with more than 5000 inhabitants in 1995 and demonstrates that even today long after industrialization there is a significant relationship between urbanization and prosperity Naturally high rates of urbanization do not mean that the majority of the population lived in prosperity In fact before the twentieth century urban areas were often centers of poverty and ill health Nevertheless urbanization is a good proxy for average prosperity and closely corresponds to the GDP per capita measures we are using to look at prosperity today Another 128 Chapter 4 Fundamental Determinants of Differences in Economic Performance ARG AUS BGD BLZ BOL BRA CAN CHL COL CRI DOM DZA ECU EGY GTM GUY HKG HND HTI IDN IND JAM LAO LKA MAR MEX MYS NIC NZL PAK PAN PER PHL PRY SGP SLV TUN URY USA VEN VNM 6 8 10 Log GDP per capita 1995 0 5 10 15 20 Urbanization 1500 percent FIGURE 44 Reversal of fortune urbanization in 1500 versus income per capita in 1995 among the former European colonies variable that is useful for measuring preindustrial prosperity is the density of the population which is closely related to urbanization Figures 44 and 45 show the relationship between income per capita today and urbanization rates and log population density in 1500 for the sample of former European colonies I focus on 1500 since it is before European colonization had an effect on any of these societies A strong negative relationship indicating a reversal in the rankings in terms of economic prosperity between 1500 and today is clear in both figures In fact the figures show that in 1500 the temperate areas were generally less prosperous than the tropical ones but this pattern was also reversed by the twentieth century There is something extraordinary and unusual about this reversal A wealth of evidence shows that after the initial spread of agriculture there was remarkable persistence in urbaniza tion and population density for all countries including those that were subsequently colonized by Europeans Extending the data on urbanization to earlier periods shows that both among former European colonies and noncolonies urbanization rates and prosperity persisted for 500 years or longer Though there are prominent examples of the decline and fall of empires such as ancient Egypt Athens Rome Carthage and Venice the overall pattern was one of per sistence Reversal was also not the general pattern in the world after 1500 When we look at Europe as a whole or at the entire world excluding the former European colonies there is no evidence of a similar reversal between 1500 and 1995 There is therefore no reason to think that the pattern in Figures 44 and 45 is some sort of natural reversion to the mean Instead the reversal of fortune among the former European colonies reflects something unusual something related to the intervention that these countries experienced The major intervention of course was related to the change in institutions Not 44 The Effect of Institutions on Economic Growth 129 AGO ARG AUS BDI BEN BFA BGD BHS BLZ BOL BRA BRB BWA CAF CAN CHL CIV CMR COG COL COM CPV CRI DMA DOM DZA ECU EGY ERI ETH GAB GHA GIN GMB GRD GTM GUY HKG HND HTI IDN IND JAM KEN KNA LAO LCA LKA LSO MAR MDG MEX MLI MOZ MRT MWI MYS NAM NER NGA NIC NPL NZL PAK PAN PER PHL PRY RWA SDN SEN SGP SLE SLV SUR SWZ TCD TGO TTO TUN TZA UGA URY USA VCT VEN VNM ZAF ZAR ZMB ZWE 6 8 10 Log GDP per capita 1995 5 0 5 Log population density 1500 FIGURE 45 Reversal of fortune population density in 1500 versus income per capita in 1995 among the former European colonies only did the Europeans impose a different order in almost all countries they conquered there were also tremendous differences among the types of institutions they imposed in the different colonies5 These institutional differences among the former colonies are likely at the root of the reversal in economic fortunes This conclusion is bolstered further when we look at the timing and the nature of the reversal Acemoglu Johnson and Robinson 2002 show that the reversal took place largely in the nineteenth century and appears to be closely connected to industrialization These patterns are clearly inconsistent with the simplest and most common version of the geography hypothesis In 1500 the countries in the tropics were relatively prosperous today it is the reverse Thus it is implausible to base a theory of relative prosperity on the intrinsic poverty of the tropics climate disease environments or other fixed characteristics Nevertheless following Diamond 1997 one could propose what Acemoglu Johnson and Robinson 2002 call a sophisticated geography hypothesis that geography matters but in a timevarying manner For example Europeans created latitudespecific technologies such as heavy metal ploughs that only worked in temperate latitudes and not with tropical soils Thus when Europe conquered most of the world after 1492 they introduced specific technologies that functioned in some places the United States Argentina Australia but not 5 In some instances including those in Central America and India the colonial institutions were built on the precolonial institutions In these cases a major determinant of early institutions was whether Europeans maintained and further developed existing hierarchical institutions such as those in the Aztec Inca or the Mughal empires or whether they introduced or imposed political and economic institutions encouraging broad based participation and investment 130 Chapter 4 Fundamental Determinants of Differences in Economic Performance others Peru Mexico West Africa However the timing of the reversal which was largely in the late eighteenth and nineteenth centuries is inconsistent with the most plausible types of sophisticated geography hypotheses Europeans did bring new technologies but the timing of the reversal implies that the crucial technologies were industrial not agricultural and it is difficult to see why industrial technologies should not function in the tropics and in fact they have functioned quite successfully in tropical Singapore and Hong Kong Similar considerations weigh against the culture hypothesis Although culture changes slowly the colonial experiment was sufficiently radical to have caused major modifications in the cultures of many countries that fell under European rule In addition the destruction of many indigenous populations and immigration from Europe are likely to have created new cultures or at least modified existing ones in major ways Nevertheless the culture hypothesis does not provide a natural explanation for the reversal and has nothing to say about the timing of the reversal Moreover as discussed below econometric models that control for the effect of institutions on income do not show a major effect of religion or culture on prosperity The importance of luck is also limited The different institutions imposed by the Europeans were not random They were instead very much related to the conditions they encountered in the colonies In other words the types of institutions that were imposed and developed in the former colonies were endogenous equilibrium outcomes that we need to study 443 The Reversal and the Institutions Hypothesis Is the reversal of fortune consistent with a dominant role for economic institutions in compar ative development The answer is yes In fact once we recognize the variation in economic institutions created by colonization we see that the reversal of fortune is what the institutions hypothesis predicts The evidence in Acemoglu Johnson and Robinson 2002 shows a close connection be tween initial population density urbanization and the creation of good economic institutions In particular the evidence points out that other things being equal the higher the initial pop ulation density or the greater the initial urbanization the worse were subsequent institutions including both institutions right after independence and also institutions today Figures 46 and 47 illustrate these relationships using the same measure of current economic institutions as in Figure 41 protection against expropriation risk today They document that the rela tively densely settled and highly urbanized colonies ended up with worse institutions while sparsely settled and nonurbanized areas received an influx of European migrants and devel oped institutions protecting the property rights of a broad cross section of society European colonialism therefore led to an institutional reversal in the sense that the previously richer and more densely settled places ended up with worse institutions The institutional reversal does not mean that institutions had been better in the previously more densely settled areas It only implies a tendency for the relatively poorer and less densely settled areas to end up with more growthenhancing institutions than previously rich and more densely settled areas had As discussed in footnote 5 above it is possible that the Europeans did not actively introduce institutions discouraging economic progress in many of these places but inherited them from previous indigenous civilizations The structure of the Mughal Aztec and Inca empires were already very hierarchical with power concentrated in the hands of narrowly based ruling elites These empires were structured to extract resources from the majority of the population for the benefit of a minority Often Europeans simply took over these existing institutions What is important in any case is that in densely settled and relatively developed places it was in the interests of the Europeans to have institutions facilitating the extraction of resources without any respect for the property rights of the majority of the populace In contrast in the sparsely ARG AUS BGD BOL BRA CAN CHL COL CRI DOM DZA ECU EGY GTM GUY HKG HND HTI IDN IND JAM LBY LKA MAR MEX MMR MYS NIC NZL PAK PAN PER PHL PNG PRY SGP SLV TUN URY USA VEN VNM 4 6 8 10 Average protection against risk of expropriation 198595 0 5 10 15 20 Urbanization 1500 percent FIGURE 46 The institutional reversal urbanization in 1500 and economic institutions today among the former European colonies ARG AUS BGD BOL BRA CAN CHL COL CRI DOM DZA ECU EGY GTM GUY HKG HND HTI IDN IND JAM LBY LKA MAR MEX MMR MYS NIC NZL PAK PAN PER PHL PNG PRY SGP SLV TUN URY USA VEN VNM 4 6 8 10 Average protection against risk of expropriation 198595 5 0 5 Log population density 1500 FIGURE 47 The institutional reversal population density in 1500 and economic institutions today among the former European colonies 132 Chapter 4 Fundamental Determinants of Differences in Economic Performance settled areas it was in their interests to develop institutions protecting property rights These incentives led to an institutional reversal The institutional reversal combined with the institutions hypothesis predicts the reversal of fortune relatively rich places ended up with relatively worse economic institutions And if these institutions are important we should see these countries become relatively poor over time Moreover the institutions hypothesis is consistent with the timing of the reversal Recall that the institutions hypothesis links incentives to invest in physical and human capital and in technology to economic institutions and argues that economic prosperity results from these investments Therefore we expect economic institutions to play a more important role in shaping economic outcomes when there are major new investment opportunitiesthus creating a greater need for new entrepreneurs and for the process of creative destruction The opportunity to industrialize was the major investment opportunity of the nineteenth century As documented in Chapter 1 countries that are rich today among both the former European colonies and other countries are those that industrialized successfully during this critical period The timing of the reversal in the late eighteenth and nineteenth centuries is consistent with this perspective The explanation for the reversal that emerges from the discussion so far is one in which the economic institutions in various colonies were shaped by Europeans to serve their own economic interests Moreover because conditions and endowments differed among colonies Europeans created disparate economic institutions which in many cases still persist and continue to shape economic performance Why did Europeans introduce better institutions in previously poor and unsettled areas than in previously rich and densely settled areas Without going into details a number of obvious ideas that have emerged from the research in this area can be mentioned Europeans were more likely to introduce or maintain economic institutions facilitating the extraction of resources in areas where they stood to benefit from this extraction This typically meant areas controlled by a small group of Europeans as well as areas offering resources to be extracted These resources included gold and silver valuable agricultural commodities such as sugar but most importantly what is perhaps the most valuable commodity of all human labor In places with a large indigenous population Europeans could exploit the population in various ways using taxes tributes or employment as forced labor in mines or plantations This type of colonization was incompatible with institutions providing economic or civil rights to the majority of the population Consequently a more developed civilization and a denser population structure made it more profitable for the Europeans to introduce worse economic institutions In contrast in places with little to extract and in sparsely settled places where the Euro peans themselves became the majority of the population it was in their interests to introduce economic institutions protecting their own property rights 444 Settlements Mortality and Development The initial conditions of the colonies emphasized so farindigenous population density and urbanizationare not the only factors that affected the Europeans colonization strategy In addition the disease environments differed markedly among the colonies with obvious conse quences on the attractiveness of European settlement As noted above when Europeans settled they established institutions that they themselves had to live under so whether Europeans could settle had a major effect on the subsequent path of institutional development In other words the disease environment 200 or more years ago especially the prevalence of malaria and yellow fever which crucially affected European mortality likely influenced the paths 44 The Effect of Institutions on Economic Growth 133 of institutional and economic development in the former European colonies If in addition the disease environment of colonial times affects economic outcomes today only through its effect on institutions then this historical disease environment can be used as an exogenous source of variation in current institutions From an econometric point of view this disease environment then corresponds to a valid instrument to estimate the causal effect of economic institutions on prosperity Although mortality rates of potential European settlers could be cor related with indigenous mortality which may affect income today in practice local populations had developed much greater immunity to malaria and yellow fever Acemoglu Johnson and Robinson 2001 present a variety of evidence suggesting that the major effect of European settler mortality is through institutions In particular Acemoglu Johnson and Robinsons argument can be summarized as follows Potential settler mortality Settlements Early institutions Current institutions Current performance That is the European colonization strategy was influenced by the feasibility of settlements Europeans were more likely to develop institutions providing property rights protection and basic political rights to the majority of the population in places where they themselves would settle and become this majority and they were unlikely to settle in lands where they faced very high mortality rates Because the colonial state and institutions persisted to some degree former European colonies that had disease environments more favorable to Europeans are also more likely to have better institutions today Based on this reasoning Acemoglu Johnson and Robinson 2001 use the mortality rates expected by the first European settlers in the colonies as an instrument for current institutions in a sample of former European colonies Their estimates of instrumental variables show a large and robust effect of institutions on economic growth and income per capita Figures 48 and 49 provide an overview of the evidence Figure 48 shows the crosssectional relationship between income per capita and the measure of economic institutions depicted in Figure 41 protection against expropriation risk It shows a strong relationship between the historical mortality risk faced by Europeans and the current extent to which property rights are enforced A bivariate regression yields an R2 of 026 It also shows that there were very large differences in European mortality Countries such as Australia New Zealand and the United States were very healthy and existing evidence suggests that life expectancy in Australia and New Zealand was in fact greater than in Britain In contrast Europeans faced extremely high mortality rates in Africa and parts of Central America and Southeast Asia These differential mortality rates were largely due to tropical diseases such as malaria and yellow fever and at the time it was not understood how these diseases arose or how they could be prevented or cured Figures 48 and 49 already show that if the exclusion restrictionthat the mortality rates of potential European settlers should have no effect on current economic outcomes other than through institutionsis valid then there is a large impact of economic institutions on economic performance This effect is documented in detail in Acemoglu Johnson and Robinson 2001 who present a range of robustness checks confirming this result Their estimates suggest that most of the gap between rich and poor countries today is due to differences in economic institutions For example the evidence suggests that more than 75 of the income gap between relatively rich and relatively poor countries can be explained by differences in their economic institutions as proxied by the security of property rights Equally important the evidence indicates that once the effect of institutions is estimated by this methodology there appears to be no effect of geographic variables latitude whether a country is landlocked and the current disease environment appear to have little effect on current economic outcomes This evidence 134 Chapter 4 Fundamental Determinants of Differences in Economic Performance AGO ARG AUS BFA BGD BHS BOL BRA CAN CHL CIV CMR COG COL CRI DOM DZA ECU EGY ETH GAB GHA GIN GMB GNB GTM GUY HKG HND HTI IDN IND JAM KEN LKA MAR MDG MEX MLI MMR MYS NER NGA NIC NZL PAK PAN PER PNG PRY SDN SEN SGP SLE SLV SUR TGO TTO TUN TZA UGA URY USA VEN VNM ZAF ZAR 4 6 8 10 Average protection against risk of expropriation 198595 2 4 6 8 Log settler mortality FIGURE 48 Relationship between mortality of potential European settlers and current economic institutions again suggests that institutional differences across countries are a major determinant of their economic fortunes while geographic differences are much less important These results also provide an interpretation for why Figure 42 showed a significant correla tion between latitude and income per capita This correlation is accounted for by the association between latitude and the determinants of European colonization strategies Europeans did not have immunity to tropical diseases during the colonial period and thus settler colonies tended other things being equal to be established in temperate latitudes Thus the historical creation of economic institutions was correlated with latitude Without considering the role of eco nomic institutions one would find a spurious relationship between latitude and income per capita However once the influence of economic institutions is controlled for this relationship disappears and there appears to be no causal effect of geography on prosperity today6 445 Culture Colonial Identity and Economic Development One might think that culture played an important role in the colonial experience since Euro peans not only brought new institutions but also their own cultures European culture might have affected the economic development of former European colonies through three different channels First as already mentioned the cultures of former European colonies are likely to have been affected by the identity of the colonizing powers For example the British may have 6 However this conclusion does not imply that geography did not play an important role in the process of economic development before 1500 44 The Effect of Institutions on Economic Growth 135 AGO ARG AUS BDI BEN BFA BGD BHS BLZ BOL BRA BRB CAF CAN CHL CIV CMR COG COL CRI DOM DZA ECU EGY ETH FJI GAB GHA GIN GMB GTM GUY HKG HND HTI IDN IND JAM KEN LAO LKA MAR MDG MEX MLI MRT MUS MYS NER NGA NIC NZL PAK PAN PER PRY RWA SDN SEN SGP SLE SLV SUR TCD TGO TTO TUN TZA UGA URY USA VEN VNM ZAF ZAR 6 8 10 Log GDP per capita 1995 2 4 6 8 Log settler mortality FIGURE 49 Relationship between mortality of potential European settlers and GDP per capita 1995 implanted a superior AngloSaxon culture into colonies such as Australia and the United States relative to the Iberian inheritance in Latin America Second European colonists may have brought a culture work ethic or set of beliefs that were conducive to prosperity in the lands that they conquered Finally Europeans also brought different religions with potentially different implications for prosperity Yet the econometric evidence in Acemoglu Johnson and Robinson 2001 is not consistent with any of these views Similar to the evidence related to geographical variables the econo metric strategy discussed above suggests that once the effect of economic institutions is taken into account the identity of the colonial power the contemporary fraction of Europeans in the population and the proportions of the populations of various religions do not appear to have a direct effect on economic growth and income per capita These econometric results are supported by historical examples Although no Spanish colony has been as successful economically as the United States many former British colonies such as those in Africa India and Bangladesh are poor today It is also clear that the British in no way simply recreated British institutions in their colonies For example by 1619 the North American colony of Virginia had a representative assembly with universal male suffrage something that did not arrive in Britain itself until 1919 Another telling example is that of Providence Island in the Caribbean While the Puritan values are often credited with the arrival of democracy and equality of opportunity in the northeastern United States the Puritan colony in Providence Island quickly became just like any other Caribbean slave colony despite its Puritanical inheritance Similarly even though the seventeenthcentury Dutch had perhaps the best domestic eco nomic institutions in the world their colonies in Southeast Asia ended up with institutions 136 Chapter 4 Fundamental Determinants of Differences in Economic Performance designed for the extraction of resources providing little economic or civil rights to the in digenous population These colonies consequently experienced slow growth relative to other countries Overall the evidence does not appear to be consistent with a major role of geography religion or culture transmitted by the identity of the colonizer or the presence of Europeans Instead differences in economic institutions appear to be the robust causal factor underlying the differences in income per capita across countries Institutions therefore appear to be the most important fundamental cause of income differences and longrun growth 45 What Types of Institutions As already noted the notion of institutions used in this chapter and in much of the literature is rather broad It encompasses different types of social arrangements laws regulations enforcement of property rights and so on One may perhaps rightly complain that we are learning relatively little by emphasizing the importance of such a broad cluster of institutions It is therefore important to try to understand what types of institutions are most important for our purpose Such a study will not only be useful in our empirical analysis of fundamental causes but can provide us with a better sense of what types of models to develop to link fundamental causes to growth mechanics and to ultimate economic outcomes There is relatively little work on unbundling the broad cluster of institutions to understand what specific types of institutions might be important for economic outcomes Much of this type of work remains to be done Here it is useful to briefly mention some recent existing research attempting to distinguish the impact of contracting institutions from the influence of property rights institutions One of the important roles of institutions is to facilitate contracting between lenders and borrowers or between different firms Such contracting is only possible if laws courts and regulations uphold contracts in an appropriate way Let us refer to insti tutional arrangements of this sort that support private contracts as contracting institutions The other cluster of institutions emphasized above relates to those constraining government and elite expropriation Let us refer to these as property rights institution because they po tentially protect the property rights of a broad cross section of society Although in many situations contracting and property rights institutions are intimately linked they are nonethe less conceptually different While contracting institutions regulate horizontal relationships in society between regular citizens property rights institutions are about vertical relationships that is the protection of citizens against the power of elites politicians and privileged groups These two sets of institutions are potentially distinct and can thus have distinct effects Acemoglu and Johnson 2005 investigate the relative roles of these two sets of institutions Their strategy is again to make use of the natural experiments of colonial history What helps this particular unbundling exercise is that in the sample of former European colonies the legal system imposed by colonial powers appears to have a strong effect on contracting institutions but little influence on the available measures of property rights institutions At the same time both mortality rates for potential European settlers and population density in 1500 have a large effect on current property rights institutions and no impact on contracting institutions Using these different sources of variation in the sample of former European colonies it is possible to estimate the separate effects of contracting and property rights institutions The empirical evidence estimating the different sources of variation in colonial history finds that property rights institutions are more important for current economic outcomes than are contracting institutions Countries with greater constraints on politicians and elites and more protection against expropriation by these powerful groups appear to have substantially higher longrun growth rates and higher levels of current income They also have significantly 46 Disease and Development 137 greater investment levels and generate more credit for the private sector In contrast the role of contracting institutions is more limited Once the effects of property rights institutions are controlled for contracting institutions seem to have no impact on income per capita the ratio of investment to GDP and the ratio of private credit to GDP Contracting institutions appear to have some effect on stock market development however These results suggest that contracting institutions affect the form of financial intermediation but have less impact on economic growth and investment It seems that economies can function in the face of weak contracting institutions without disastrous consequences but not in the presence of a significant risk of expropriation from the government or other powerful groups A possible interpretation is that private contracts or other reputationbased mechanisms can at least in part alleviate problems originating from weak contracting institutions For example when it is more difficult for lenders to collect on their loans interest rates increase banks that can monitor effectively play a more important role or reputationbased credit relationships may emerge In contrast property rights institutions relate to the relationship between the state and its citizens When there are no checks on the state politicians and elites private citizens do not have the security of property rights necessary for investment Nevertheless in interpreting the evidence in Acemoglu and Johnson 2005 one should also bear in mind that the sources of variation in income per capita and investment rates identifying the different effects of contracting and property rights institutions relate to the large differences discussed in Chapter 1 It is possible that contracting institutions have modest effects that are hard to detect when looking at countries with 30fold differences in income per capita Therefore this evidence should be interpreted as suggesting that contracting institutions are less important in generating the large differences in economic development compared to property rights institutions not necessarily as suggesting that contracting institutions do not matter for economic outcomes 46 Disease and Development The evidence presented in Section 44 already militates against a major role for geographic factors in economic development One version of the geography hypothesis deserves further analysis however A variety of evidence suggests that unhealthy individuals are less productive and often less successful in acquiring human capital Could the differences in the disease environments across countries have an important effect on economic development Could the burden of disease be a major factor in explaining the very large income differences across countries A recent paper by David Weil 2007 for example argues that the framework used in the previous chapter with physical capital human capital and technology should be augmented by including health capital In other words the aggregate production function may take the form FK H Q A where H denotes efficiency units of labor human capital as conventionally measured while Q is health capital Weil suggests a methodology for measuring the contribution of health capital to productivity from microestimates and argues that differences in health capital emerge as an important factor in accounting for crosscountry differences in income levels The idea that the low productivity of lessdeveloped nations is partly due to the unhealthy state of their workforces has obvious appeal Existing econometric evidence shows that it has some empirical validity as well But does it imply that geographic factors are an important fundamental cause of economic growth Not necessarily As already mentioned the burden of disease is endogenous Todays unhealthy nations are unhealthy precisely because they are poor and are unable to invest in health care clean water and other healthimproving technologies After all much of Europe was unhealthy and suffering from short life expectancy only 200 138 Chapter 4 Fundamental Determinants of Differences in Economic Performance Initially rich countries Initially mediumincome countries Initially poor countries 36 38 4 42 44 Log life expectancy at birth 1920 1940 1960 1980 2000 FIGURE 410 Evolution of life expectancy at birth among initially poor initially middleincome and initially rich countries 19402000 years ago This changed with economic growth In this sense even if health capital is a useful concept and does contribute to accounting for crosscountry income differences it may itself be a proximate cause that is affected by other factors A recent paper by Acemoglu and Johnson 2007 directly investigates the impact of changes in disease burdens on economic development They exploit the large improvements in life expectancy particularly among the relatively poor countries that took place starting in the 1940s These health improvements were the direct consequence of significant international health interventions more effective public health measures and the introduction of new chemicals and drugs More important for the purposes of understanding the effect of disease on economic growth these health improvements were by and large exogenous from the viewpoint of individual nations Moreover their impact on specific countries also varied depending on whether the country in question was affected by the specific diseases for which the cures and drugs had become internationally available The impact of these health improvements was major in fact so significant that it may deserve to be called the international epidemiological transition since it led to an unprecedented improvement in life expectancy in a large number of countries Figure 410 shows this unprecedented convergence in life expectancy by plotting life expectancy in countries that were initially circa 1940 poor middle income and rich It illustrates that while in the 1930s life expectancy was low in many poor and middleincome countries this transition brought their levels of life expectancy close to those prevailing in richer parts of the world As a consequence of these developments health conditions in many parts of the lessdeveloped world today though still in dire need of improvement are significantly better than the corresponding health conditions were in the West at the same stage of development The international epidemiological transition allows a promising empirical strategy to iso late potentially exogenous changes in health conditions The effects of the international epi 46 Disease and Development 139 Initially rich countries Initially mediumincome countries Initially poor countries 6 7 8 9 10 Log GDP per capita 1920 1940 1960 1980 2000 FIGURE 411 Evolution of GDP per capita among initially poor initially middleincome and initially rich countries 19402000 demiological transition on a countrys life expectancy were related to the extent to which its population was initially circa 1940 affected by various specific diseases for example tuber culosis malaria and pneumonia and to the timing of the various health interventions This reasoning suggests that potentially exogenous variation in the health conditions of the country can be measured by calculating a measure of predicted mortality driven by the interaction of baseline crosscountry disease prevalence with global intervention dates for specific diseases Acemoglu and Johnson 2007 show that such measures of predicted mortality have a large and robust effect on changes in life expectancy starting in 1940 but have no effect on changes in life expectancy prior to this date that is before the key interventions This observation suggests that the large increases in life expectancy experienced by many countries after 1940 were in fact related to the global health interventions Not surprisingly Acemoglu and Johnson 2007 find that predicted mortality and the changes in life expectancy that it causes have a fairly large effect on population a 1 increase in life expectancy is related to an approximately 1318 increase in population However somewhat more surprisingly they also find no evidence of a positive effect on GDP per capita Figure 411 provides an aggregated version of this evidence It shows no convergence in income per capita among initially poor initially middleincome and initially rich countries Why did the very significant increases in life expectancy and health not cause improvements in GDP per capita The most natural answer to this question comes from neoclassical growth theory presented in the previous two chapters and in Chapter 8 The firstorder effect of increased life expectancy is to increase population which initially reduces capitallabor and landlabor ratios depressing income per capita This initial decline is later compensated for by higher output as more people enter the labor force However there is no reason to expect a large significant increase in income per capita especially when many of the affected countries 140 Chapter 4 Fundamental Determinants of Differences in Economic Performance are heavily vested in agriculture and experience a decline in landlabor ratios as a result of the rise in population Consequently small beneficial effects of health on productivity may not be sufficient to offset or reverse the negative effects of population pressure on income per capita over periods as long as 50 years or more 47 Political Economy of Institutions First Thoughts The evidence presented in this chapter suggests that institutions are a majorperhaps the most significantfundamental cause of economic growth We must therefore think about why institutions and policies differ across countries to understand why some countries are poor and others are rich I also argue in the Epilogue that understanding institutional changes holds clues about why the process of world economic growth started 200 years or so ago However an explanation of differences in income across countries and over time in terms of institutional differences is also incomplete If as this chapter has documented some in stitutions are conducive to rapid economic growth and others to stagnation why would any society collectively choose institutions that condemn them to stagnation The answer to this question relates to the nature of collective choices in societies Institutions and policies like other collective choices are not taken for the good of the society at large but are a result of political equilibria To understand such political equilibria we need to understand the con flicting interests of different individuals and groups in societies and analyze how they are mediated by different political institutions Thus a proper understanding of how institutions affect economic outcomes and why institutions differ across countries and why they some times change and pave the way for growth miracles requires models of political economy which explicitly study how the conflicting interests of different individuals are aggregated into collective choices Models of political economy also specify why certain individuals and groups may be opposed to economic growth or prefer institutions that eschew growth opportunities The discussion in this chapter therefore justifies the inclusion of a study of political economy as part of any detailed investigation of economic growth Much of the study of economic growth has to be about the structure of models so that we understand the mechanics of economic growth and the proximate causes of income differences But part of this broad study must also confront the fundamental causes of economic growth which relate to policies institutions and other factors that lead to different investment accumulation and innovation decisions 48 Taking Stock This chapter has emphasized the differences between the proximate causes of economic growth related to physical capital accumulation human capital and technology and the fundamental causes which influence the incentives to invest in these factors of production I have argued that many of the questions motivating our study of economic growth must lead us to an investigation of the fundamental causes But an understanding of fundamental causes is most useful when we can link them to the parameters of fully developed models of economic growth to see how they affect the mechanics of growth and what types of predictions they generate The institutions hypothesis which seems to receive support from the evidence presented in this chapter calls for a careful theoretical investigation The institutions view makes sense only when there are groups in society that favor institutions that do not necessarily enhance 49 References and Literature 141 the growth potential of the economy Such groups do so because they do not directly or indirectly benefit from the process of economic growth Thus it is important to develop a good understanding of the distributional implications of economic growth eg how it affects relative prices and relative incomes how it may destroy the rents of incumbents This theoretical understanding of the implications of the growth process then needs to be combined with political economy models of collective decision making to investigate the circumstances under which groups opposed to economic growth can be powerful enough to maintain institutions that are inimical to growth In this chapter my objective has been more limited since many of the more interesting growth models are developed later in the book and I have focused on the broad outlines of a number of alternative fundamental causes of economic growth and on a first look at the long run empirical evidence relevant to these hypotheses I argued that approaches emphasizing institutional differences and differences in policies laws and regulations across societies are most promising for understanding both the current growth experiences of countries and the historical process of economic growth I also emphasized the importance of studying the political economy of institutions as a way of understanding why institutions differ across societies and lead to divergent economic paths 49 References and Literature The early part of this chapter builds on Acemoglu Johnson and Robinson 2005a who discuss the distinction between proximate and fundamental causes and the various different approaches to the fundamental causes of economic growth North and Thomas 1973 appear to be the first to implicitly criticize growth theory for focusing solely on proximate causes and ignoring the fundamental causes of economic growth Diamond 1997 also draws a distinction between proximate and fundamental explanations The model presented in Section 42 draws on Simon 1977 and the more recent work by Michael Kremer 1993 Kremer 1993 argues for the importance of economies of scale and increasing returns to population based on the acceleration in the growth rate of world popu lation Another important argument relating population to technological change is proposed by Ester Boserup 1965 and is based on the idea that increases in population create scarcity inducing societies to increase their productivity Other models that build economies of scale to population and discuss the transition of the world economy from little or no growth to one of rapid economic growth include Galor and Weil 2000 Galor and Moav 2002 and Hansen and Prescott 2002 Some of these papers also try to reconcile the role of population in gener ating technological progress with the later demographic transition Galor 2005 provides an excellent summary of this literature and an extensive discussion McEvedy and Jones 1978 provide a concise history of world population and relatively reliable information going back to 10000 BC Their data indicate that as claimed in the text the total population in Asia has been consistently greater than in Western Europe over this time period The geography hypothesis has many proponents In addition to Montesquieu Niccolo Machiavelli was an early proponent of the importance of climate and geographic character istics Marshall 1890 and Myrdal 1968 are among the economists who have most clearly articulated various versions of the geography hypothesis It has more recently been popularized by Sachs 2001 and Bloom and Sachs 1998 Diamond 1997 offers a more sophisticated version of the geography hypothesis in which the availability of different types of crops and animals as well as the axes of communication within continents influence the timing of set tled agriculture and thus the possibility of developing complex societies Diamonds thesis is 142 Chapter 4 Fundamental Determinants of Differences in Economic Performance therefore based on geographic differences but also relies on institutional factors as intervening variables Scholars emphasizing the importance of various types of institutions in economic devel opment include John Locke Adam Smith John Stuart Mill Arthur Lewis Douglass North and Robert Thomas The recent economics literature includes many models highlighting the importance of property rights for example Skaperdas 1992 Tornell and Velasco 1992 Acemoglu 1995 Grossman and Kim 1995 Hirshleifer 2001 and Dixit 2004 Other models emphasize the importance of policies within a given institutional framework Well known examples of this approach include SaintPaul and Verdier 1993 Alesina and Rodrik 1994 Persson and Tabellini 1994 Krusell and RıosRull 1999 and Bourguignon and Verdier 2000 There is a much smaller literature on endogenous institutions and the effect of these institutions on economic outcomes Surveys of this work can be found in Acemoglu 2007b and Acemoglu and Robinson 2006a The literature on the effect of economic in stitutions on economic growth is summarized and discussed in greater detail in Acemoglu Johnson and Robinson 2005a which also provides an overview of the empirical literature on the topic I return to many of these issues in Part VIII The importance of religion for economic development is most forcefully argued in Max Webers work for example Weber 1930 1958 Many other scholars since then have picked up on this idea and have argued about the importance of religion Prominent examples include the various papers in Harrison and Huntington 2000 and Landes 1998 Landes for example tries to explain the rise of the West based on cultural and religious variables This evidence is criticized in Acemoglu Johnson and Robinson 2005a Barro and McCleary 2003 provide evidence of a positive correlation between the prevalence of religious beliefs and economic growth One has to be careful in interpreting this evidence as showing a causal effect of religion on economic growth since religious beliefs are endogenous both to economic outcomes and to other fundamental causes of income differences The emphasis on the importance of cultural factors or social capital goes back to Banfield 1958 and has recently been popularized by Putnam 1993 The essence of these interpreta tions appears to be related to the role of culture or social capital in ensuring the selection of better equilibria Similar ideas are also advanced in Greif 1994 Many scholars including Veliz 1994 North Summerhill and Weingast 2000 and Wiarda 2001 emphasize the importance of cultural factors in explaining the economic backwardness of Latin American countries Knack and Keefer 1997 and Durlauf and Fafchamps 2005 document positive correlations between measures of social capital and various economic outcomes None of this work establishes a causal effect of social capital because of the potential endogeneity of so cial capital and culture A number of recent papers attempt to overcome these difficulties for example Guiso Sapienza and Zingales 2004 and Tabellini 2007 The discussion of the Puritan colony in the Providence Island is based on Kupperman 1993 The literature on the effect of economic institutions and policies on economic growth is vast Most growth regressions include some controls for institutions or policies and find them to be significant see eg those reported in Barro and SalaiMartin 2004 One of the first papers to examine the crosscountry correlation between property rights measures and economic growth is Knack and Keefer 1995 This literature does not establish a causal effect of institutions on economic performance because of major simultaneity and endogeneity concerns Mauro 1995 and Hall and Jones 1999 present the first instrumentalvariable estimates on the effect of institutions or corruption on longrun economic development The evidence reported here which exploits differences in colonial experience to create an instrumental variables strategy is based on Acemoglu Johnson and Robinson 2001 2002 The urbanization and population density data used here are from Acemoglu Johnson and Robinson 2002 who compiled these data based on work by McEvedy and Jones 1978 410 Exercises 143 Chandler 1987 Bairoch 1988 Bairoch Batou and Chevre 1988 and Eggimann 1999 Further details and econometric results are presented in Acemoglu Johnson and Robinson 2002 The data on mortality rates of potential settlers are from Acemoglu Johnson and Robinson 2001 who compiled the data based on work by Gutierrez 1986 and Curtin 1989 1998 That paper also provides a large number of robustness checks documenting the influence of economic institutions on economic growth and showing that other factors including religion and geography have little effect on longrun economic development once the effect of institutions is controlled for The discussion of the role of leaders on growth draws on Jones and Olken 2005 The details of the Korean experiment and historical references are provided in Acemoglu 2003c and Acemoglu Johnson and Robinson 2005a The discussion of distinguishing the effects of different types of institutions draws on Acemoglu and Johnson 2005 The discussion of the effect of disease on development is based on Weil 2007 and especially on Acemoglu and Johnson 2007 which used the econometric strategy described in the text Figures 410 and 411 are from Acemoglu and Johnson 2007 In these figures initially poor countries are those that are poorer than Spain in 1940 and include Bangladesh Brazil China Ecuador El Salvador Honduras India Indonesia Korea Malaysia Myanmar Nicaragua Pakistan the Philippines Sri Lanka and Thailand Initially rich countries are those that are richer than Argentina in 1940 and include Belgium Netherlands Sweden Denmark Canada Germany Australia New Zealand Switzerland the United Kingdom and the United States Young 2005 investigates the effect of the HIV epidemic in South Africa and reaches a conclusion similar to that reported here though his analysis relies on a calibration of the Solow growth model rather than on econometric estimation 410 Exercises 41 Derive 43 and 44 42 Derive equation 45 Explain how and why the behavior implied for technology by this equation differs from 44 Do you find the assumptions leading to 44 or to 45 more plausible 43 a Show that the models leading to both 44 and 45 imply a constant income per capita over time b Modify 42 to be Lt φYtβ for some β 0 1 Justify this equation and derive the law of motion for technology and income per capita under the two scenarios considered in Section 42 Are the implications of this model more reasonable than those considered in the text PART II TOWARD NEOCLASSICAL GROWTH This part of the book is a preparation for what is to come In some sense Part II can be viewed as the preliminaries for the rest of the book The ultimate purpose is to enrich the basic Solow model by introducing welldefined household preferences and optimization and in the process clarify the relationship between growth theory and general equilibrium theory This will enable us to open the black box of savings and capital accumulation turning these into forward looking investment decisions It will also permit us to make welfare statements about whether the rate of growth of an economy is too slow too fast or just right from a welfaremaximizing Pareto optimality viewpoint The tools introduced in this part of the book are also essential for our study of technology as another forwardlooking investment by firms researchers and individuals However much of this effort will have to wait for Parts III and IV of the book The three chapters in this part present the material necessary to appreciate subsequent chapters in the book The next chapter makes more explicit the relationship between models of economic growth and general equilibrium theory It also highlights some of the assumptions implicit in growth models The two subsequent chapters develop the mathematical tools for dynamic optimization in discrete and continuous time To avoid making these chapters purely about mathematics I use a variety of economic models of some relevance to growth theory as examples and also present some results on equilibrium and optimal growth Nevertheless the material in the next three chapters is significantly more mathematical than anything else in the book except the Appendixes at the end The reader therefore may wish to first consult the Appendixes andor skip some of the proofs presented in the next three chapters in a first reading 5 Foundations of Neoclassical Growth T he Solow growth model is predicated on a constant saving rate It would be more in formative to specify the preference orderings of households individuals as in stan dard general equilibrium theory and derive their decisions from these preferences This specification would enable us both to have a better understanding of the factors that affect savings decisions and also to discuss the optimality of equilibriain other words to pose and answer questions related to whether the competitive equilibria of growth models can be im proved The notion of improvement here is based on the standard concept of Pareto optimality which asks whether some households can be made better off without others being made worse off Naturally we can only talk of households being better off if we have some information about welldefined preference orderings 51 Preliminaries To prepare for this analysis let us consider an economy consisting of a unit measure of infinitelylived households By a unit measure of households I mean an uncountable number of households with total measure normalized to 1 for example the set of households H could be represented by the unit interval 0 1 This abstraction is adopted for simplicity to emphasize that each household is infinitesimal and has no effect on aggregates Nothing in this book hinges on this assumption If the reader instead finds it more convenient to think of the set of households H as a countable set for example H N this can be done without any loss of generality The advantage of having a unit measure of households is that averages and aggregates are the same enabling us to economize on notation It would be even simpler to have H as a finite set of the form 1 2 M While this form would be sufficient in many contexts overlapping generations models in Chapter 9 require the set of households to be infinite Households in this economy may be truly infinitely lived or alternatively they may consist of overlapping generations with full or partial altruism linking generations within the household Throughout I equate households with individuals and thus ignore all possible sources of conflict or different preferences within the household In other words I assume that households have welldefined preference orderings 147 As in basic general equilibrium theory let us suppose that preference orderings can be represented by utility functions In particular suppose that there is a unique consumption good and each household h has an instantaneous utility function given by uhc ht where c ht is the consumption of household h and uh R R is increasing and concave I take the domain of the utility function to be R rather than R so that negative levels of consumption are not allowed Even though some wellknown economic models allow negative consumption this is not easy to interpret in general equilibrium or in growth theory Thus this restriction is sensible in most models The instantaneous utility function captures the utility that an individual or household derives from consumption at time t It is therefore not the same as a utility function specifying a complete preference ordering over all commoditieshere all commodities corresponding to consumption levels at all dates For this reason the instantaneous utility function is sometimes also referred to as the felicity function There are two major assumptions in writing an instantaneous utility function First it supposes that the household does not derive any utility from the consumption of other households so consumption externalities are ruled out Second in writing the instantaneous utility function I have already imposed the condition that overall utility is timeseparable and stationary that is instantaneous utility at time t is independent of the consumption levels at past or future dates and is represented by the same utility function uh at all dates This second feature is important in enabling us to develop tractable models of dynamic optimization Finally let us introduce a third assumption and suppose that households discount the future exponentiallyor proportionally In discrete time and ignoring uncertainty this assumption implies that household preferences or utility starting at time t 0 can be represented as Uhc h1 c h2 c hT T t0β htuhc ht 51 where β h 0 1 is the discount factor of household h and the horizon T could be finite or equal to infinity so that T is allowed Here Uh denotes the utility function of household h defined over the entire stream of consumption levels while uh is still the instantaneous utility function The distinction between these two concepts is important to bear in mind The functional form of the utility function Uh incorporates exponential discounting and time separability It implies that the weight given to tomorrows utility uh is a fraction β h of todays utility and the weight given to the utility the day after tomorrow is a fraction β h2 because they naturally preserve timeconsistent behavior A solution xtTt0 possibly with T to a dynamic optimization problem is timeconsistent if the following is true when xtT is a solution starting at time t t it is a solution to the continuation dynamic optimization problem starting from time t t 0 If a problem is not timeconsistent it is timeinconsistent Timeconsistent problems are much more straightforward to work with and satisfy all the standard axioms of rational decision making Although timeinconsistent preferences may be useful in the modeling of certain behaviors such as problems of addiction or selfcontrol timeconsistent preferences are ideal for the focus in this book since they are tractable relatively flexible and provide a good approximation to reality in the context of aggregative models It is also worth noting that many classes of preferences that do not feature exponential and timeseparable discounting Nevertheless this result is an outcome of strong income effects which can create unintuitive results even in basic consumer theory recall eg Giffen goods Special but approximately realistic preference functions as well as restrictions on the distribution of income across households enable us to rule out arbitrary aggregate excess demand functions To show that the representative household assumption is not as hopeless as Theorem 51 suggests I now present a special but relevant case in which aggregation of individual preferences is possible and enables the modeling of the economy as if the demand side were generated by a representative household To prepare for this theorem consider an economy with a finite number N of commodities and recall that an indirect utility function for household h vhp wh specifies the households ordinal utility as a function of the price vector p p1 pN and the households income wh Naturally any indirect utility function vhp wh has to be homogeneous of degree 0 in p and w a single household A stronger notion the normative representative household would also allow us to use the representative households utility function for welfare comparisons and its introduced later in this section Let us start with the simplest case that leads to the existence of a representative household For concreteness suppose that all households are infinitelylived and identical that is each household has the same discount factor β the same sequence of effective labor endowments eht0 and the same instantaneous utility function ucht where u ℝ ℝ is increasing and concave and cht is the consumption of household h Therefore there really is a representative household in this case Consequently again ignoring uncertainty the demand side of the economy can be represented as the solution to the following maximization problem starting at time t 0 max t0 βt uct where β 0 1 is the common discount factor of all the households and ct is the consumption level of the representative household The economy described so far admits a representative household rather trivially all households are identical In this case the representative households preferences 52 can be used not only for positive analysis eg to determine the level of savings but also for normative analysis such as evaluating the optimality of equilibria The assumption that the economy is inhabited by a set of identical households is not very appealing Instead we would like to know when an economy with heterogeneity can be modeled as if aggregate consumption levels were generated by the optimization decision of a representative household To illustrate the potential difficulties that the as if perspective might encounter let us consider a simple exchange economy with a finite number of commodities and state an important theorem from general equilibrium theory Recall that in an exchange economy the equilibrium can be characterized in terms of excess demand correspondence see Appendix A Let the equilibrium of the economy be represented by the aggregate excess demand function xp when the vector of prices is p The demand side of an economy admits a representative household if xp can be obtained as a solution to the maximization problem of a single household The next theorem shows that this is not possible in general Theorem 51 DebreuMantelSonnenschein Theorem Let ε 0 and N ℕ Consider a set of prices Pe pe ℝN Pipj ε for all i and j and any continuous function x Pe ℝN that satisfies Walrass Law and is homogeneous of degree 0 Then there exists an exchange economy with N commodities and H households where the aggregate excess demand is given by xp over the set Pe Proof See Debreu 1974 or MasColell Whinston and Green 1995 Proposition 17E3 Therefore the fact that excess demands result from aggregating the optimizing behavior of households places few restrictions on the form of these demands In particular recall from basic microeconomics that individual excess demands satisfy the weak axiom of revealed preference and have Slutsky matrices that are symmetric and negative semidefinite These properties do not necessarily hold for the aggregate excess demand function xp Thus without imposing further structure it is impossible to derive xp from the maximization behavior of a single household Theorem 51 therefore raises a severe warning against the use of the representative household assumption In particular let us say that an economy admits a strong representative household if redistribution of income or endowments across households does not affect the demand side The strong representative household applies when preferences take the Gorman form as shown by Theorem 52 Moreover it is straightforward to see that since without the Gorman form the Engel curves of some households have different slopes there exists a specific scheme of income redistribution across households that would affect the aggregate demand for different goods This reasoning establishes the following converse to Theorem 52 Gorman preferences with the same bp for all households are necessary for the economy to admit a strong representative household Notice that instead of the summation Theorem 52 is stated with the integral over the set H to allow for the possibility that the set of households may be a continuum The integral should be thought of as the Lebesgue integral so that when H is a finite or countable set H whdh is indeed equivalent to the summation hH wh Although Theorem 52 is stated for an economy with a finite number of commodities this limitation is only for simplicity and the results in this theorem hold in economies with an infinite number or a continuum of commodities Mostbut importantly not allmacro models assume more than the existence of a representative household First many models implicitly assume the existence of a strong representative household thus abstracting from the distribution of income and wealth among households and its implications for aggregate behavior Second most approaches also impose the existence of a normative representative household not only does there exist a representative household whose maximization problem generates the relevant aggregate demands but also the utility function of this household can be used for welfare analysis subject to the same set of constraints The only difference between the two problems is that in the latter each household has been assigned the same weight Let w R D note that here w is a number whereas w w1 wD is a vector Another important aspect of the standard preferences used in growth theory and macroeconomics concerns the planning horizon of individuals Although some growth models are formulated with finitelylived households see eg Chapter 9 most growth and macro models assume that households have an infinite planning horizon as in 52 or 516 below since probabilities have been substituted in and there is no need to include explicit expectations This argument establishes that fully altruistic behavior within a dynasty socalled dynastic preferences also leads to a situation in which decision makers act as if they have an infinite planning horizon Theorem 54 Representative Firm Theorem Consider a competitive production economy with N N commodities and a countable set F of firms each with a production possibilities set Y R N Let p R N be the price vector in this economy and denote the set of profitmaximizing net supplies of firm f F by Yf p R F so that for any yf Yf p we have p yf p y for all y Yf Theorem 54 implies that when there are no externalities and all factors are priced competitively focusing on the aggregate production possibilities set of the economyor equivalently on the representative firmis without loss of generality naturally assuming that the representative firm acts taking prices as given go from exponential discounting in continuous time to discretetime discounting In particular given a discount rate ρ 0 the discount factor that applies during a time interval of length Δt is βΔt expρΔt where all i N let xh xhijj0 be the consumption bundle of household h and ωh ωh0j0 be its endowment bundle In addition let us assume that feasible xh must belong to some consumption set Xh Rn0 The last requirement implies that the total consumption of each commodity cannot be more than the sum of its total endowment and net production 1 The allocation x y is feasible that is xh Xh for all h H yf Yf for all f F and 2 For every firm f F yf maximizes profits p yf p yf for all yf Yf Theorem 55 First Welfare Theorem I Suppose that x y p is a competitive equilibrium of economy ℰ ℋ ℐ J U ω X Θ with ℋ finite Assume that all households are locally nonsatiated Then x y is Pareto optimal Proof Suppose that x y p is a competitive equilibrium To obtain a contradiction suppose that there exists a feasible ℱ ℱ such that Uhxh Uhxh for all h ℋ and Uhxh Uhxh for all h H where H is a nonempty subset of ℋ Since x y p is a competitive equilibrium it must be the case that for all h ℋ phxhh pxh 517 ph ωh fF θhf yf and for all h H phxh ph ωh fF θhf yf 518 Inequality 518 follows immediately because xh is the utilitymaximizing choice for household h thus if xh is strictly preferred then it cannot be in the budget set Inequality 517 follows with similar reasoning Suppose that it did not hold Then by the hypothesis of local nonsatiation Uh must be strictly increasing in at least one of its arguments let us say the jth component of x Then construct xhe such that xhje xhj and xhke xhk ε for k j For ε sufficiently small xhe is in household hs budget set and yields strictly greater utility than the original consumption bundle xh contradicting the hypothesis that household h is maximizing utility Also note that local nonsatiation implies that Uhxh and thus the righthand sides of 517 and 518 are finite and in particular px ω Now summing 517 over ℋ and 518 over H and combining the two we have p hℋ xh p hℋ ωh fF θhf yf 519 p hℋ ωh fF yf where the second line uses the fact that the sums are finite so that the order of summation can be exchanged and that by the definition of the shares hℋ θhf 1 for all f F Finally since y is profit maximizing at prices p we have p fF yf p fF yf for any yf fF with yf Y for all f F 520 However by feasibility of xh Condition 1 of Definition 51 hℋ xhj hℋ ωh fF yf for all j and therefore by taking the inner products of both sides with p and exploiting 520 and the fact that p 0 we conclude p hℋ xhj p hℋ ωh fF yf which contradicts 519 establishing that any competitive equilibrium allocation x y is Pareto optimal The proof of the First Welfare Theorem is both intuitive and simple The proof is based on two simple ideas First if another allocation Pareto dominates the competitive equilibrium prices then it must be nonaffordable in the competitive equilibrium for at least one household Second profit maximization implies that any competitive equilibrium already maximizes the set of affordable allocations The proof is also simple since it only uses the summation of the values of commodities at a given price vector In particular it makes no convexity assumption However the proof also highlights the importance of the feature that the relevant sums exist and are finite Otherwise the last step would lead to the conclusion that which may or may not be a contradiction The fact that these sums exist in turn follows from two assumptions finiteness of the number of individuals and nonsatiation However as noted runs into problems in the presence of nonconvexities Before stating the theorem recall that the consumption set of each household h ℋ is Xh RN so a typical element of Xh is xh xh1 xh2 xhN where xhj can be interpreted as the finitedimensional vector of consumption of individual h at time t that is xhj xhj1 xhj2 xhjN Similarly a typical element of the production set of firm f F yf is of the form yf yf1 yf2 Let us also define xHT x1H x2H xH 0 0 and yT y1 y2 where the last entries are truncated because there are zero consumption or zero production after some date T It can be verified that limt xHT xH and limt yT y in the product topology see Section A4 in Appendix A Finally in this case xH or y is an Ndimensional vector with a slight abuse of notation I use p xH for an appropriately defined inner product for example p xH j0 pj xHj Theorem 56 First Welfare Theorem II Suppose that x y is a competitive equilibrium of the economy ℰ ℋ ℐ J U ω X Θ with ℋ countably infinite Assume that all households are locally nonsatiated and p ω hℋ j0 pj ωhj Then x y p is Pareto optimal Proof The proof is the same as that of Theorem 55 with a major difference Local nonsatiation does not guarantee that the summations are finite in 519 since the summations are over an infinite number of households In particular 517 and 518 from the proof of Theorem 55 still apply and we have p xh for each h ℋ Moreover by profit maximization p fF yf Now summing 517 over ℋ and 518 over H yields 519 provided that p ω hℋ j0 pj ωhj Then the remaining relations in the proof of Theorem 55 apply and yield a contradiction establishing the desired result wℎ uℎ fS θfhyf Moreover if p wℎ 0 for each h H then economy E has a competitive equilibrium x y p The proof of this theorem involves the application of the Geometric HahnBanach Theorem Theorem A27 It is somewhat long and involved For this reason its proof is provided in the next starred section Here notice that if instead of an infinitedimensional economy we were dealing with an economy with a finite commodity space say with N commodities then parts iiiv of the hypothesis in the theorem would be satisfied automatically by taking T T N In fact this condition is not imposed in the statement of the Second Welfare Theorem in economies with a finite number of commodities Its role in dynamic economies is that changes in allocations that are very far in the future should not have a large effect on utility This condition is naturally satisfied in infinitehorizon economies with discounted utility and separable production structure Intuitively if a sequence xh is strictly preferred to the sequence xh then setting the elements of xh and xh to 0 in the very far and thus heavily discounted future should not change this conclusion since discounting implies that xh could not be strictly preferred to xh because of higher consumption under xh in the arbitrarily far future Similarly if some production vector y is feasible the separable production structure implies that y T which involves zero production after some date T must also be feasible Exercise 513 demonstrates these claims more formally One difficulty in applying this theorem is that Uh may not be defined when the vector xh involves zeros eg when instantaneous utility of consumption is given by log c Exercise 514 shows that the theorem can be generalized to the case in which there exists a sufficiently small positive scalar ε 0 and a sequence εh with each element equal to ε such that h H is restricted to xh ε The conditions for the Second Welfare Theorem are more difficult to satisfy than those for the First Welfare Theorem because of the convexity requirements In many ways it is also the more important of the two theorems While the First Welfare Theorem is celebrated as a formalization of Adam Smiths invisible hand the Second Welfare Theorem establishes the stronger result that any Pareto optimal allocation can be decentralized as a competitive equilibrium An immediate corollary of this property is an existence result since the Pareto optimal allocation can be decentralized as a competitive equilibrium a competitive equilibrium must exist at least for the endowments leading to Pareto optimal allocations The Second Welfare Theorem motivates many macroeconomics to look for the set of Pareto optimal allocations instead of explicitly characterizing competitive equilibria This approach is especially useful in dynamic models in which competitive equilibria can sometimes be quite difficult to characterize or even to specify while the characterization of Pareto optimal allocations is typically more straightforward The real power of the Second Welfare Theorem in dynamic macro models comes when it is combined with a normative representative household Recall that Section 53 shows an equivalence between Pareto optimal allocations and optimal allocations for the representative household In certain models including manybut not allgrowth models studied in this book the combination of a normative representative household and the Second Welfare Theorem enables us to characterize the optimal growth path that maximizes the utility of the representative household and assert that this path corresponds to a competitive equilibrium 176 Chapter 5 Foundations of Neoclassical Growth 510 Taking Stock This chapter introduced the preliminaries necessary for an indepth study of equilibrium and optimal growth theory At some level it can be thought of as an oddsandends chapter introducing the reader to the notions of representative household dynamic optimization welfare theorems and optimal growth However the material here is more than odds and ends since a good understanding of the general equilibrium foundations of economic growth and the welfare theorems is necessary for what is to come in Part III and later The most important messages from this chapter are as follows First the set of models in this book are examples of more general dynamic general equilibrium models It is therefore important to understand which features of the growth models are general in the sense that they do not depend on the specific simplifying assumptions and which results depend on the further simplifying assumptions In this respect the First and the Second Welfare Theorems are essential They show that provided that all product and factor markets are competitive and that there are no externalities in production or consumption and under some relatively mild technical assumptions dynamic competitive equilibria are Pareto optimal and any Pareto optimal allocation can be decentralized as a dynamic competitive equilibrium These results are especially relevant in Part III where the focus is on competitive economies Importantly these results do not directly apply in our analysis of technological change where product markets are monopolistic or in our study of economic development where various market imperfections play an important role Second the most general class of dynamic general equilibrium models are not tractable enough for us to derive sharp results about the process of economic growth For this reason we often adopt a range of simplifying assumptions The most important of these is the repre sentative household assumption which enables us to model the demand side of the economy as if it were generated by the optimizing behavior of a single household We saw how this assumption is generally not satisfied but also how a certain class of preferences the Gorman preferences enable us to model economies as if they admitted a representative household even with arbitrary distributions of wealth and income In addition this chapter introduced the first formulation of the optimal growth problems in discrete and in continuous time These are used as examples in the next two chapters 511 References and Literature This chapter covered a great deal of ground and often many details were omitted for brevity Many readers will be familiar with some of the material in this chapter Deaton and Muellbauer 1980 Hildenbrand and Kirman 1988 and MasColell Whinston and Green 1995 pro vide excellent discussions of the issues related to aggregation and the representative household assumption Some of the original contributions on this topic are contained in Gorman 1953 1959 1976 1980 and Pollak 1971 These and many other relevant results on separability and aggregation appear in the works of W M Terence Gorman Deaton and Muellbauer 1980 provide an excellent discussion of Gormans work and the implications of Gorman preferences Caselli and Ventura 2000 use Gorman preferences in the context of a model of capital accumulation with heterogeneous agents MasColell Whinston and Green also discuss the concepts of positive and normative representative households The concept of nor mative representative household in Theorem 53 is motivated by the use of the representative household assumption in dynamic macroeconomic models which focus on the maximiza tion of the utility of a representative household to characterize all Pareto optimal allocations 511 References and Literature 177 and competitive equilibria This concept is stronger than the one in MasColell Whin ston and Green who define a normative representative household for a given social welfare function The DebreuMantelSonnenschein Theorem Theorem 51 was originally proved by Son nenschein 1972 and then extended by Debreu 1974 and Mantel 1976 Both MasColell Whinston and Green 1995 and Hildenbrand and Kirman 1988 present this theorem and sketch its proof Both Deaton and Muellbauer 1980 and Hildenbrand and Kirman 1988 also show how such aggregation is possible under weaker assumptions on utility functions together with certain restrictions on the distribution of income or endowments Some basic concepts from microeconomic theory were assumed in this chapter and the reader can find a thorough exposition of these in MasColell Whinston and Green 1995 These include Roys Identity used following Theorem 52 and then again in Theorem 53 and Walrass Law the concept of a numeraire and expected utility theory of von Neumann and Morgenstern used throughout the analysis The reader is also referred to Chapter 16 of MasColell Whinston and Green and to Bewley 2007 for clear expositions of the different representation of Pareto optima including the result that every Pareto optimal allocation is a solution to the maximization of the weighted average of utilities of households in the economy The Representative Firm Theorem Theorem 54 presented here is quite straightforward but I am not aware of any discussion of this theorem in the literature or at least in the macroeconomics literature It is important to distinguish the subject matter of this theorem from the Cambridge controversy in early growth theory which revolved around the issue of whether different types of capital goods could be aggregated into a single capital index see eg Wan 1971 The Representative Firm Theorem says nothing about this issue The best reference for the analysis of the existence of competitive equilibria and the welfare theorems with a finite number of households and a finite number of commodities is still Debreus 1959 Theory of Value This short book introduces all mathematical tools necessary for general equilibrium theory and gives a very clean exposition Equally lucid and more modern are the treatments of the same topics in MasColell Whinston and Green 1995 and Bewley 2007 The reader may also wish to consult MasColell Whinston and Green their Chapter 16 for a proof of the Second Welfare Theorem with a finite number of commodities Theorem 57 in this chapter is more general because it covers the case of an infinite number of commodities Both of these books also have an excellent discussion of the necessary restrictions on preferences to allow preferences to be represented by utility functions Mas Colell Whinston and Green their Chapter 19 also gives a very clear discussion of the role of Arrow securities and the relationship between trading at a single point in time and sequential trading The classic reference on Arrow securities is Arrow 1964 Neither Debreu 1959 nor MasColell Whinston and Green 1995 discuss infinite dimensional economies The seminal reference for infinitedimensional welfare theorems is Debreu 1954 Bewley 2007 contains a number of useful results on infinitedimensional economies Stokey Lucas and Prescott 1989 their Chapter 15 present existence and welfare theorems for economies with a finite number of households and countably infinite number of commodities The mathematical prerequisites for their treatment are greater than what has been assumed here but their treatment is both thorough and straightforward once the reader makes the investment in the necessary mathematical techniques The most accessible references for the HahnBanach Theorem which is necessary for a proof of Theorem 57 in infinite dimensional spaces are Luenberger 1969 Kolmogorov and Fomin 1970 and Kreyszig 1978 Luenberger 1969 is also an excellent source for all the mathematical techniques used in Stokey Lucas and Prescott 1989 and also contains much material useful for appreciating continuoustime optimization 512 Exercises 181 b Suppose that the production structure is given by a neoclassical production function where the production vector at time t is only a function of inputs at time t and capital stock chosen at time t 1 that higher capital stock contributes to greater production and there is free disposal Show that the second hypothesis in Theorem 57 which states that for each yf Y f there exists T such that yf T Y f for all T T is satisfied 514 a Show that Theorem 57 does not cover the onegood neoclassical growth model with instan taneous preferences given by uc c1θ 11 θ with θ 1 b For ε 0 construct the sequence ε with each element equal to ε Reformulate and prove a version of Theorem 57 such that that Xh for all h H is restricted to have elements xh ε for ε 0 sufficiently small Hint redefine xhT to have ε rather than 0 after the T th element and reformulate the hypothesis of the theorem accordingly c Show that this modified version of Theorem 57 covers the economy in part a of this exercise 202 Chapter 6 InfiniteHorizon Optimization and Dynamic Programming 661 Basic Equations Consider the functional equation corresponding to Problem 63 V x max yGxUx y βV y for all x X 624 Let us assume throughout that Assumptions 6165 hold Then from Theorem 64 the maximization problem in 624 is strictly concave and from Theorem 66 the maximand is also differentiable Therefore for any interior solution y Int Gx the firstorder conditions are necessary and sufficient for an optimum taking V as given In particular optimal solutions can be characterized by the following convenient Euler equations DyUx y βDV y 0 625 where I use an asterisk to denote optimal values and once again D denotes gradients Recall that in the general case x is a vector not a real number and thus DxU is a vector of partial derivatives I denote the vector of partial derivatives of the value function V evaluated at y by DV y Throughout the rest of the chapter I adopt the convention that DyU or DyUxt xt 1 denotes the gradient vector of U with respect to its last K arguments whereas DxU is the gradient with respect to the first K arguments The set of firstorder conditions in 625 would be sufficient to solve for the optimal policy y if we knew the form of the V function Since this function is determined recursively as part of the optimization problem there is a little more work to do before we obtain the set of equations that can be solved for the optimal policy Fortunately we can use the equivalent of the Envelope Theorem Theorem A31 for dynamic programming and differentiate 624 with respect to the state vector x to obtain DV x DxUx y 626 The reason equation 626 is the equivalent of the Envelope Theorem is that the term DyUx y βDV ydydx ie the effect of a change in y times the induced change in y in response to the change in x is absent from the expression Naturally this is because DyUx y βDV y 0 from 625 Now using the notation y πx to denote the optimal policy function which is singlevalued in view of Assumption 63 and Corollary 61 and the fact that DV y DxV πx ππx we can combine these two equations to write a more conve nient form of the Euler equations expressed simply in terms of the payoff functions DyUx πx βDxUπx ππx 0 627 where DxU represents the gradient vector of U with respect to its first K arguments and DyU represents its gradient with respect to the second set of K arguments Notice that 627 is a functional equation in the unknown function π and characterizes the optimal policy function These equations become even simpler and more transparent in the case where both x and y are real numbers In this case 625 becomes Ux y y βV y 0 628 where V the notes the derivative of the V function with respect to its single argument This equation is intuitive it requires the sum of the marginal gain today from increasing y and the discounted marginal gain from increasing y on the value of all future returns to be equal 66 Applications of Stationary Dynamic Programming 203 to zero For instance as in Example 61 we can think of U as decreasing in y and increasing in x 628 would then require the current cost of increasing y to be compensated by higher values tomorrow In the context of economic growth this condition corresponds to the current cost of reducing consumption to be compensated by higher consumption tomorrow As with 625 the value of higher consumption in 628 is expressed in terms of the derivative of the value function V y which is one of the unknowns Let us now use the onedimensional version of 626 to find an expression for this derivative V x Ux y x 629 Combining 628 and 629 yields the following simple condition Ux πx y β Uπx ππx x 0 where in line with the notation for gradients Ux denotes the derivative of U with respect to its first argument and Uy with respect to the second argument Alternatively explicitly including the time arguments the Euler equation can be written as Uxt xt 1 y β Uxt 1 xt 2 x 0 630 However Euler equation 630 is not sufficient for optimality In addition we need the transver sality condition The transversality condition is essential in infinitedimensional problems because it ensures that there are no beneficial simultaneous changes in an infinite number of choice variables In contrast in finitedimensional problems there is no need for such a con dition since the firstorder conditions are sufficient to rule out possible gains when we change many or all of the control variables at the same time The role that the transversality condition plays in infinitedimensional optimization problems will become more apparent after Theorem 610 is established and after the discussion in Section 662 In the general case the transversality condition takes the form lim t βtDxUxt xt 1 xt 0 631 where denotes the inner product operator In the onedimensional case we have the simpler transversality condition lim t βt Uxt xt 1 x xt 0 632 This condition requires that the product of the marginal return from the state variable x times the value of this state variable does not increase asymptotically at a rate faster than or equal to 1β The next theorem shows that the transversality condition and the Euler equations in 627 are necessary and sufficient to characterize a solution to Problem 62 and therefore to Prob lem 63 Theorem610EulerEquationsandtheTransversalityCondition Let X RK and suppose that Assumptions 6165 hold Then a sequence xt t0 such that xt 1 Int Gxt t 0 1 is optimal for Problem 62 given x0 if and only if it satisfies 627 and 631 218 Chapter 6 InfiniteHorizon Optimization and Dynamic Programming Equation 652 is a remarkable result it shows that the steadystate capitallabor ratio does not depend on household preferences except via the discount factor In particular technology the depreciation rate and the discount factor fully characterize the steadystate capitallabor ratio In addition since f is strictly concave k is uniquely defined Finally since ct c and kt k in the steady state βf k 1 δ 1 and β 1 the transversality condition 651 is automatically satisfied This analysis leads to the following important proposition Proposition 63 In the neoclassical optimal growth model specified in 646 and 647 with Assumptions 1 2 and 3 there exists a unique steadystate capitallabor ratio k given by 652 and starting from any initial k0 0 the economy monotonically converges to this unique steady state that is if k0 k then kt k and if k0 k then kt k Proof Uniqueness and existence were established above To establish monotone conver gence we start with arbitrary initial capital stock k0 and observe that kt 1 skt for all t 0 where s was defined and shown to be nondecreasing in Proposition 61 It must be the case that either k1 sk0 k0 or k1 sk0 k0 Consider the first case Since s is nondecreasing and k2 sk1 we must have k2 k1 By induction kt skt 1 kt 1 skt 2 Moreover by definition kt 0 k Therefore in this case kt t0 is a nondecreasing sequence in a compact set starting with k0 0 Thus it necessarily converges to some limit k 0 which by definition satisfies k sk Since k is the unique steady state corresponding to positive capitallabor ratio this implies that k k and thus kt k Moreover since kt t0 is nondecreasing it must be the case that kt k This argument thus completes the proof for the case k0 k Next consider the case in which k1 sk0 k0 The same argument as above applied in reverse now establishes that kt t0 is a nonincreasing sequence in the compact set 0 k and thus it converges to a unique limit point k In this case there are two candidate values for k k 0 or k k The former is not possible since as Exercise 619 shows Assumption 2 implies that sε ε for ε sufficiently small Thus k k Since kt t0 is nonincreasing in this case we must have k0 k and thus kt t0 k completing the proof Consequently in the optimal growth model there exists a unique steady state and the economy monotonically converges to this unique steady state for example by accumulating more and more capital if it starts with a capitallabor ratio that is too low In addition consumption also monotonically increases or decreases along the path of adjustments to the uniquesteady state as stated in the following proposition Proposition 64 We have that ck defined in Proposition 61 is nondecreasing Moreover if k0 k then the equilibrium consumption sequence satisfies ct c and if k0 k then ct c where c is given by c f k δk Proof See Exercise 617 This discussion illustrates that the optimal growth model is very tractable and shares many features with the Solow growth model for example a unique steady state and global monotonic convergence There is no immediate counterpart of a saving rate since the amount of savings 69 Competitive Equilibrium Growth 219 0 T k k0 FIGURE 61 Turnpike dynamics in a finitehorizon T periods neoclassical growth model starting with initial capitallabor ratio k0 depends on the utility function and changes over time though the discount factor is closely related to the saving rate The convergence behavior of the optimal growth model is both important and remarkable in its simplicity Such convergence results which were first studied in the context of finite horizon economies are sometimes referred to as Turnpike Theorems To understand the meaning of this term suppose that the economy ends at some date T 0 What do optimal growth and capital accumulation look like in this economy The early literature on opti mal growth showed that as T the optimal capitallabor ratio sequence ktT t0 would become arbitrarily close to k as defined by 652 but then in the last few periods it would sharply decline to zero to satisfy the transversality condition recall the discussion of the finitehorizon transversality condition in Section 66 The path of the capitallabor ratio thus resembles a turnpike approaching a highway as shown in Figure 61 see Exercise 618 69 Competitive Equilibrium Growth Our main interest in this book is not optimal growth but equilibrium growth A detailed analysis of competitive equilibrium growth is presented in Chapter 8 For now a brief discussion of how the competitive equilibrium can be obtained from the optimal growth problem is sufficient The Second Welfare Theorem Theorem 57 of the previous chapter implies that the optimal growth path characterized in Section 68 also corresponds to an equilibrium growth path in the sense that it can be decentralized as a competitive equilibrium In fact since we have focused on an economy admitting a representative household the most straightforward competitive allocation would be a symmetric one where all households each with the instantaneous utility 611 Taking Stock 221 Next market clearing immediately implies that 1 rt 1 is given by 653 so the capital labor ratio of the competitive equilibrium is given by βf kt 1 1 δ 1 The steady state then satisfies βf k 1 δ 1 This equation is identical to 652 which characterizes the solution to the optimal growth problem A similar argument establishes that the entire competitive equilibrium path is identical to the optimal growth path Specifically substituting for 1 rt 1 from 653 into 655 yields uct βf kt 1 1 δuct 1 656 which is identical to 650 This condition also implies that given the same initial condition the trajectory of capitallabor ratio in the competitive equilibrium is identical to the behavior of the capitallabor ratio in the optimal growth path see Exercise 621 This behavior is of course exactly what should be expected given the Second and First Welfare Theorems 610 Computation All the results presented here have been about existence of solutions and characterization of the form of the value functions solutions and the properties of the policy functions or optimal plans Dynamic programming techniques are also widely used in explicit numerical compu tations Exercise 63 below provides one useful starting point in this respect In particular the recursive formulation of dynamic programming problems also presents an effective computa tional approach This formulation is particularly useful since as suggested by the discussion in Example 64 only certain special dynamic optimization problems yield closedform solutions Therefore economists like engineers must often use computational tools to obtain qualitative and quantitative insights about solutions to optimization and equilibrium problems The dy namic programming formulation is often the starting point of these computational approaches Space restrictions preclude me from providing a discussion of various computational tools and how dynamic programming methods are used in numerical analysis This omission should not be interpreted as downplaying the importance of computation in the study of economic growth or the usefulness of dynamic programming approaches in computation The reader is encouraged to consult Judd 1998 for an excellent and thorough discussion of computational issues in economics and the role of dynamic programming Ljungqvist and Sargent 2005 also provide a brief introduction to the application of computational methods in macroeconomics 611 Taking Stock This chapter has been concerned with basic dynamic programming techniques for discrete time infinitedimensional problems These techniques are not only essential for the study of economic growth but are also widely used in many diverse areas of macroeconomics and economic dynamics more generally A good understanding of these techniques is essential for an appreciation of the mechanics of economic growth In particular they shed light on how different models of economic growth work how they can be improved and how they can be 222 Chapter 6 InfiniteHorizon Optimization and Dynamic Programming taken to the data For this reason this chapter is part of the main body of the text rather than being relegated to the appendixes at the end This chapter also presented a number of applications of dynamic programming including a preliminary analysis of the onesector optimal growth problem The reader will have already noted the parallels between this model and the basic Solow model discussed in Chapter 2 These parallels are developed further in Chapter 8 I also briefly discussed the decentralization of the optimal growth path and the problem of utility maximization in a dynamic competitive equilibrium It is important to emphasize that the treatment in this chapter has assumed away a number of difficult technical issues First the focus has been on discounted problems which are simpler than undiscounted problems In economics very few situations involve undiscounted objective functions β 1 rather than β 0 1 More important throughout I have assumed that payoffs are bounded and the state vector x belongs to a compact subset of the Euclidean space X These restrictions rule out many interesting problems such as endogenous growth where the state vector grows over time Almost all of the results presented here have equivalents for these cases but these require somewhat more advanced treatments 612 References and Literature At some level the main idea of dynamic programming the Principle of Optimality is straight forward Nevertheless it is also a powerful concept as will be best appreciated once a number of its implications are derived The basic ideas of dynamic programming including the Prin ciple of Optimality were introduced by Richard Bellman in his famous monograph Bellman 1957 Most of the basic results about finite and infinitedimensional dynamic programming problems are contained in this monograph Many of these ideas are also contained in Shap leys 1953 study of stochastic games Shapley analyzed the characterization of equilibrium points of zerosum stochastic games His formulation of these games anticipated what later became known as Markov decision problems which are closely related to dynamic program ming problems Moreover Shapley used ideas similar to the Principle of Optimality and the Contraction Mapping Theorem to show the existence of a unique solution to these dynamic zerosum games A more detailed treatment of Markov decision problems can be found in Put erman 1994 who also discusses the relationship between Shapleys 1953 work the general theory of Markov decision problems and dynamic programming To the best of my knowledge Karlin 1955 was the first to provide a simple formal proof of the Principle of Optimality which is similar to the one presented here Denardo 1967 developed the use of the contraction mappings in the theory of dynamic programming Puter man 1994 contain a more detailed analysis of discounted stochastic dynamic programming problems Blackwell 1965 introduced the Blackwells sufficient conditions for a contraction mapping and applied them in the context of stochastic discounted dynamic programming prob lems The result on the differentiability of the value function was first proved in Benveniste and Scheinkman 1979 The second version of the proof of Theorem 66 follows their approach closely The first version of the proof extends the earlier proof by Mirman and Zilcha 1975 which was only for the neoclassical growth model The most complete treatment of discounted stationary dynamic programming problems is in Stokey Lucas and Prescott 1989 My treatment here is heavily influenced by theirs and borrows much from their insights Relative to their treatment some of the proofs have been simplified and I have limited the analysis to the case with compact sets and bounded payoff functions The reader can find generalizations of Theorems 6166 to certain problems with 71 Variational Arguments 229 Let us first suppose that t1 so that we have a finitehorizon optimization problem Notice that there is also a terminal value constraint xt1 x1 but x1is included as an additional choice variable Thus the terminal value of the state variable x is free In the context of finite horizon economic problems the formulation in which x1is not a choice variable may be simpler see Example 71 but it is more natural to start with the case where the terminal value x1 is free In addition to simplify the exposition throughout I assume that f and g are continuously differentiable functions of x y and t and I simply state this as f and g are continuously differentiable The challenge in characterizing the optimal solution to this problem lies in two features 1 We are choosing a function y 0 t1 Y rather than a vector or a finitedimensional object 2 The constraint takes the form of a differential equation rather than a set of inequalities or equalities These features make it difficult for us to know what type of optimal policy to look for For example y may be a highly discontinuous function It may also hit the boundary of the feasible setthus corresponding to a corner solution Fortunately in most economic problems there is enough structure to make solutions continuous functions Moreover in most macroeconomic and growth applications the Inada conditions eg Assumption 2 from Chapter 2 ensure that solutions to the relevant dynamic optimization problems lie in the interior of the feasible set These features considerably simplify the characterization of the solution In fact when y is a continuous function of time and lies in the interior of the feasible set it can be characterized by using variational arguments similar to those developed by Euler Lagrange and others in the context of the theory of calculus of variations Since these tools are not only simpler but also more intuitive than the optimal control approach I start with these variational arguments The variational approach simplifies the above maximization problem by first assuming that there exists a continuous solution function ˆy that lies everywhere in the interior of the set Y with corresponding state variable ˆx everywhere in the interior of X It then characterizes the properties of this solution see Exercise 733 More formally let us assume that ˆxt ˆyt is an admissible pair such that ˆy is continu ous on 0 t1 ˆxt Int X and ˆyt Int Y for all t or more simply ˆxt ˆyt Int X Y and that we have Wˆxt ˆyt Wxt yt for any other admissible pair xt yt The important and stringent assumption here is that ˆxt ˆyt is a solution that never hits the boundary and does not involve discontinuities Even though this feature is true of optimal controls in most economic applications in purely mathematical terms it is a strong assumption Recall for example that the previous chapter did not make such an assumption and instead started with a result on the existence of solutions and then proceeded to characterizing the properties of this solution eg continuity and differentiability of the value function However the problem of continuoustime optimization is sufficiently difficult that proving existence of solutions is not a trivial matter I return to this issue below but for now I follow the standard practice and assume that an interior continuous solution ˆxt ˆyt Int X Y exists Note 3 In addition the calculus of variations approach compares this candidate solution to other continuous paths In the optimal control approach used in Theorem 79 below the candidate path is compared to any other admissible path in the sense of footnote 2 234 Chapter 7 An Introduction to the Theory of Optimal Control Now using this condition and differentiating 714 yields a differential equation in consump tion This differential equation derived in the next chapter in a somewhat more general context is the key consumption Euler equation in continuous time Leaving the derivation of this equa tion to the next chapter we can simply integrate 715 to obtain λt λ0 exprt Combining this equation with the firstorder condition for consumption yields a straight forward expression for the optimal consumption level at time t ˆct u1rλ0 expρ rt where u1 is the inverse function of the marginal utility u This inverse exists and is strictly decreasing in view of the fact that u is strictly concave This equation therefore implies that when ρ r so that the discount factor and the rate of return on assets are equal the individual will have a constant consumption profile When ρ r the argument of u1 is increasing over time so consumption must be declining Thus when the individual discounts the future more heavily than the rate of return she wishes to have a frontloaded consumption profile In contrast when ρ r the opposite reasoning applies and she chooses a backloaded consumption profile These are naturally identical to the conclusions reached in the discrete time intertemporal consumer optimization problem in Example 65 in particular 640 The only variable left to determine to completely characterize the consumption profile is the initial value of the costate variable and thus the initial value of consumption This comes from the observation that the individual will run down all her assets by the end of her planning horizon that is a1 0 Using the consumption rule we have at rat w u1rλ0 expρ rt The initial value of the costate variable λ0 then has to be chosen such that a1 0 You are asked to complete the details of this step in Exercise 76 Example 71 applied the results of Theorem 72 It may at first appear that Theorem 71 is more convenient to use than Theorem 72 since it enables us to directly formulate the problem as one of dynamic optimization rather than first having to guess the terminal value of the state variable a1 as we did in Example 71 However as the continuation of the previous example illustrates this is not necessarily the case Example 71 continued Let us try to apply Theorem 71 to the economic environment in Example 71 The firstorder necessary conditions still give λt λ0 exprt However since λ1 0 this equation holds only if λt 0 for all t 0 1 But the necessary conditions still imply the Euler equation expρtuˆct λt which cannot be satisfied since u 0 Thus when the terminal value of the assets a1 is a choice variable there exists no solution at least no solution with an interior continuous control How is this possible The answer is that Theorem 71 cannot be applied to this problem because there is an additional constraint that at 0 We would need to consider a version of Theorem 71 with inequality constraints The necessary conditions with inequality constraints are somewhat 72 The Maximum Principle A First Look 235 more difficult to work with Using a little bit of economic reasoning to observe that the terminal value of the assets must be equal to zero and then applying Theorem 72 simplifies the analysis considerably This discussion highlights that it may also be useful to have a version of Theorem 72 in which the terminal condition is specified as an inequalityas xt1 x1 rather than as xt1 x1 This alternative is presented next Theorem 73 Necessary Conditions III Consider the problem of maximizing 72 subject to 73 and to xt yt X Y for all t x0 x0 and xt1 x1 with f and g continuously differentiable Suppose that this problem has an interior continuous solution ˆxt ˆyt Int X Y Then there exists a continuously differentiable costate function λ defined over t 0 t1 such that 73 711 and 712 hold and moreover λt1xt1 x1 0 Proof See Exercise 79 72 The Maximum Principle A First Look 721 The Hamiltonian and the Maximum Principle By analogy with the Lagrangian a more economical way of expressing Theorem 72 is to construct the Hamiltonian Ht xt yt λt f t xt yt λtgt xt yt 716 I often write Ht x y λ for the Hamiltonian to simplify notation5 Since f and g are continuously differentiable so is H Denote the partial derivatives of the Hamiltonian with respect to xt yt and λt by Hx Hy and Hλ respectively Theorem 72 then immediately leads to the following result Theorem 74 Simplified Maximum Principle Consider the problem of maximizing 72 subject to 73 and 74 with f and g continuously differentiable Suppose that this problem has an interior continuous solution ˆxt ˆyt Int X Y Then there exists a con tinuously differentiable function λt such that the optimal control ˆyt and the corresponding path of the state variable ˆxt satisfy the following necessary conditions Hyt ˆxt ˆyt λt 0 for all t 0 t1 717 λt Hxt ˆxt ˆyt λt for all t 0 t1 718 and xt Hλt ˆxt ˆyt λt for all t 0 t1 719 5 More generally the Hamiltonian should be written as Ht x y λ λ0f t xt yt λtgt xt yt for some λ0 0 In some pathological cases λ0 may be equal to 0 However in all economic applications this will not be the case and we will have λ0 0 When λ0 0 it can be normalized to 1 without loss of any generality Thus the definition of the Hamiltonian in 716 is appropriate for economic applications 236 Chapter 7 An Introduction to the Theory of Optimal Control with x0 x0 and λt1 0 where the Hamiltonian Ht x y λ is defined in 716 Moreover the Hamiltonian Ht x y λ also satisfies the Maximum Principle that Ht ˆxt ˆyt λt Ht ˆxt y λt for all y Y for all t 0 t1 For notational simplicity in 719 I wrote xt instead of ˆxt d ˆxtdt The latter notation is rather cumbersome and I refrain from using it as long as the context makes it clear that xt stands for this expression6 Theorem 74 is a simplified version of the celebrated Maximum Principle of Pontryagin A more general version of the Maximum Principle is given below For now a couple of features are worth noting 1 As in the usual constrained maximization problems a solution is characterized jointly with a set of multipliers here the costate variable λt and the optimal path of the control and state variables ˆyt and ˆxt 2 Again as with the Lagrange multipliers in the usual constrained maximization problems the costate variable λt is informative about the value of relaxing the constraint at time t In particular λt is the value of an infinitesimal increase in xt at time t see Section 734 3 With this interpretation it makes sense that λt1 0 is part of the necessary conditions After the planning horizon there is no value to having more or less x This is therefore the finitehorizon equivalent of the transversality condition in the previous chapter As emphasized above Theorem 74 gives necessary conditions for an interior continuous solution However we do not know whether such a solution exists Moreover these necessary conditions may characterize a stationary point rather than a maximum or simply a local rather than a global maximum Therefore a sufficiency result is even more important in this context than in finitedimensional optimization problems Sufficiency is again guaranteed by imposing concavity The following theorem first proved by Mangasarian shows that concavity of the Hamiltonian ensures that conditions 717719 are not only necessary but also sufficient for a maximum Theorem 75 Mangasarians Sufficiency Conditions Consider the problem of max imizing 72 subject to 73 and 74 with f and g continuously differentiable Define Ht x y λ as in 716 and suppose that an interior continuous pair ˆxt ˆyt Int X Y exists and satisfies 717719 Suppose also that X Y is a convex set and given the result ing costate variable λt Ht x y λ is jointly concave in x y X Y for all t 0 t1 Then the pair ˆxt ˆyt achieves the global maximum of 72 Moreover if Ht x y λ is strictly concave in x y for all t 0 t1 then the pair ˆxt ˆyt is the unique solution to 72 6 Conditions 718 and 719 also clarify why H is referred to as a Hamiltonian Given vectors x and z a Hamiltonian dynamical system is a dynamical system set of differential equations with a representation of the form x DzHx z and z DxHx z for some function H The Hamiltonian function H then plays the role of potential energy and is constant along the solution trajectories of this dynamical system see eg Perko 2001 If H in Theorem 74 were independent of time it would indeed be such a function and the resulting dynamical system would be a Hamiltonian system This is generally not the case when there is dependence on t that is discounting 240 Chapter 7 An Introduction to the Theory of Optimal Control Note also that various conditions in this theorem or equivalently in the onedimensional Theorem 74 can be relaxed further For example the requirement that ˆxt ˆyt Int X Y is not necessary and when either the state or the control variables take boundary values there may be jumps in the control variables and the Hamiltonian may not be differentiable everywhere see below These possibilities can be incorporated by allowing ˆyt to be only piecewise continuous Since in most economic applications both state and control variables are interior and the corresponding Hamiltonian is differentiable everywhere the form of Theorem 77 stated here is sufficient for most problems of interest The sufficiency conditions provided above also have straightforward generalizations which are presented next Theorem78SufficiencyConditionsForMultivariateProblems Consider the prob lem of maximizing 725 subject to 726 and 727 with f and G continuously differentiable Define Ht x y λ as in 728 and suppose that an interior continuous pair ˆxt ˆyt Int X Y satisfies 729731 If X is convex and Mt x λ maxytY Ht x y λ is concave in x X for all t 0 t1 then the pair ˆxt ˆyt achieves the global maximum of 725 Moreover if Mt x λ is strictly concave in x then the pair ˆxt ˆyt is the unique solution to 725 Proof See Exercise 712 723 Limitations The limitations of the results presented so far are obvious First the existence of a continuous and interior solution to the optimal control problem has been assumed Second and equally important the analysis has focused on the finitehorizon case whereas the study of growth models requires us to solve infinitehorizon problems To deal with both of these issues we need to look at the more modern theory of optimal control This is done in the next section 73 InfiniteHorizon Optimal Control The results presented so far are most useful in developing an intuition for how dynamic optimization in continuous time works While some problems in economics require finite horizon optimal control most economic problemsincluding almost all growth modelsare more naturally formulated as infinitehorizon problems This is obvious in the context of economic growth but is also the case in repeated games political economy or industrial organization where even though individuals may have finite expected lives the end date of the game or of their lives may be uncertain For this reason the canonical model of optimization in economic problems is the infinitehorizon one In this section I provide necessary and sufficient conditions for optimality in infinitehorizon optimal control problems Since these are the results that are most often used in economic applications I simplify the exposition and state these results for the case in which both the state and the control variables are one dimensional The more general multivariate case is discussed in Section 76 when I return to the issue of existence of solutions and to the properties of the value functions 731 The Basic Problem Necessary and Sufficient Conditions Let us focus on infinitehorizon control with a single control and a single state variable For reasons that will be explained below it is useful to generalize the terminal value constraint 73 InfiniteHorizon Optimal Control 243 Two features in this version of the Principle of Optimality are noteworthy First in contrast to the similar equation 63 in the previous chapter it may appear that there is no discounting in 738 This is not the case since the discounting is embedded in the instantaneous payoff function f and is thus implicit in V t1 ˆxt1 Second Lemma 71 may appear to contradict the discussion of time consistency in Chapter 5 since this lemma is stated without additional as sumptions that ensure time consistency The important point here is that in the timeconsistency discussion the decision maker considered updating his or her continuation plan from t1onward at date t1 In contrast Lemma 71 considers the optimality of the plan after t1 at time t0 The issue of time consistency that is whether the individual would like to change his or her plan at date t1 is discussed further in Exercise 722 I next state the main result on necessary conditions for infinitehorizon optimal control problems In this theorem I also slightly relax the assumption that the optimal control ˆyt is continuous Theorem 79 InfiniteHorizon Maximum Principle Suppose that the problem of max imizing 732 subject to 733 and 734 with f and g continuously differentiable has a piecewise continuous interior solution ˆxt ˆyt Int X Y Let Ht x y λ be as de fined in 716 Then given ˆxt ˆyt the Hamiltonian Ht x y λ satisfies the Maximum Principle Ht ˆxt ˆyt λt Ht ˆxt yt λt for all yt Y and for all t R Moreover for all t R for which ˆyt is continuous the following necessary conditions are satisfied Hyt ˆxt ˆyt λt 0 739 λt Hxt ˆxt ˆyt λt 740 and xt Hλt ˆxt ˆyt λt with x0 x0 and lim t btxt x1 741 The proof of this theorem is relatively long and is provided later in this section For now notice that when a solution of the specified form exists it satisfies the Maximum Principle Thus in some ways Theorem 79 can be viewed as stronger than the theorems presented in the previous chapter especially since it does not impose compactnesstype conditions Nevertheless this theorem only applies when the maximization problem has a piecewise continuous solution ˆyt In addition Theorem 79 states that if the optimal control ˆyt is a continuous function of time conditions 739741 are satisfied everywhere Since ˆyt is a piecewise continuous function the optimal control may include discontinuities but these will be relatively rarein particular it will be continuous most of the time The added generality of allowing discontinuities is somewhat superfluous in most economic applications because economic problems often have enough structure to ensure that ˆyt is indeed a continuous function of t Consequently in most economic problems and in all models studied in this book it is sufficient to focus on the necessary conditions 739741 The necessary conditions in Theorem 79 can also be expressed in the form of the socalled HamiltonJacobiBellman HJB equation which is analogous to the dynamic programming formulation in the previous chapter 74 More on Transversality Conditions 251 and lim t kt 0 where c kα δk and k αδ11α In other words c is the maximum level of consumption that can be achieved in the steady state of this model and k is the corresponding steadystate level of capital This way of writing the objective function makes sure that the integral converges and takes a finite value since for any ε 0 ct cannot exceed c ε forever The Hamiltonian is straightforward to construct it does not explicitly depend on time and takes the form Hk c λ log ct log c λtktα ct δkt and implies the following necessary conditions Hck c λ 1 ct λt 0 and Hkk c λ λtαktα1 δ λt It can be verified that any optimal path must feature ct c as t This condition however implies that lim t λt 1 c 0 and lim t kt k Now recall from Theorem 73 that the finitehorizon transversality condition in this case would have been λt1kt1 0 whereas here limt λtkt kc 0 Therefore the equivalent of the finitehorizon transversality condition does not hold It can be verified however that along the optimal path the following condition holds instead lim t Hkt ct λt 0 The next theorem shows that this equation is indeed one version of the transversality condition for infinitehorizon optimization problems Theorem 712 Transversality Condition for InfiniteHorizon Problems Suppose that the problem of maximizing 732 subject to 733 and 734 with f and g continuously differentiable has a piecewise continuous optimal control ˆyt Int Yt with a correspond ing path of state variable ˆxt Int Xt Let V t xt be the value function defined in 735 Suppose that V t ˆxt is differentiable in x and t for t sufficiently large and that limt V t ˆxtt 0 Let Ht x y λ be given by 716 Then the pair ˆxt ˆyt satisfies the necessary conditions 739741 and the transversality condition lim t Ht ˆxt ˆyt λt 0 756 76 Existence of Solutions Concavity and Differentiability 259 control problem is strictly concave ˆyt must be continuous This result is stated and proved in the next corollary Corollary 71 Suppose that the hypotheses in Theorem 714 are satisfied Mt x μ is strictly concave in x for all t and Y is compact Then ˆyt is a continuous function of t on R Proof Given the strict concavity of Mt x μ Theorem 714 established the uniqueness of ˆxt ˆyt Now take some ˆt R and any sequence tn in R converging to ˆt Since Y is compact the corresponding sequence ˆytn converges to some ˆy Theorem A7 We have that ˆxt and μt given by the differential equations in 766 and 767 are continuous and thus ˆxtn and μtn converge to ˆxˆt and μˆt Moreover by the Maximum Principle ˆHtn ˆxtn ˆytn μtn ˆHtn ˆxtn y μtn for all y Y Using the fact that ˆH is also continuous and taking limits we obtain ˆHˆt ˆxˆt ˆy μˆt ˆHˆt ˆxˆt y μˆt for all y Y Since ˆxt ˆyt is unique this implies that ˆyˆt ˆy and therefore ˆyt is continuous at ˆt Since this is true for any ˆt R ˆyt is continuous on R Although Corollary 71 is useful it should be noted that it does not provide primitive conditions for the existence of a continuous optimal control It is stated and proved under the hypothesis that there exists a pair ˆxt ˆyt satisfying 765768 Conditions on objective and constraint functions that guarantee the existence of a solution are presented in the next section 76 Existence of Solutions Concavity and Differentiability The theorems presented so far characterize the properties of a solution to a continuoustime maximization problem The natural question of when a solution exists has not been posed or answered This omission might appear curious since in both finitedimensional and discrete time infinitehorizon optimization problems studied in the previous chapter the analysis starts with existence theorems There is a good reason for this omission however In continuous time optimization problems establishing the existence of solutions is considerably more difficult than the characterization of solutions I now present the general theorem on existence of solutions to continuoustime optimization problems and two additional results providing conditions under which the value function V t x defined in 735 and Lemma 71 is concave and differentiable The reader may have already wondered how valid the approach of using the necessary conditions provided so far which did not verify the existence of a solution would be in practice This concern is important and ordinarily such an approach would open the door for potential mistakes One line of defense however is provided by the sufficiency theorems for example Theorems 711 or 714 for infinitehorizon problems If given a continuoustime optimization problem we find an admissible pair ˆxt ˆyt that satisfies the necessary conditions eg those in Theorem 79 and we can then verify that the optimization problem satisfies the conditions in either of Theorems 711 or 714 then we must have characterized a solution and can dispense with an existence theorem Therefore the sufficiency results contained in these theorems enable us to bypass the step of checking for the existence of a solution or it amounts to proof by construction Nevertheless this approach is only valid when the problem possesses sufficient concavity to satisfy the conditions of Theorems 711 or 714 For 78 The qTheory of Investment and SaddlePath Stability 271 Several interesting economic features emerge from this equation First as φI tends to zero it can be verified that It diverges meaning that investment jumps to a particular value In other words it can be shown that this value is such that the capital stock immediately reaches its steadystate value see Exercise 728 This result is intuitive As φI tends to zero φI becomes linear In this case adjustment costs simply increase the cost of investment linearly and do not create any need for smoothing In contrast when φIt 0 there is a motive for smoothing It takes a finite value and investment adjusts slowly Therefore as claimed above adjustment costs lead to a smoother path of investment The behavior of investment and capital stock can now be analyzed using the differential equations 784 and 788 First it can be verified easily that there exists a unique steady state solution with K 0 This solution involves a level of capital stock K for the firm and investment just sufficient to replenish the depreciated capital I δK This steadystate level of capital satisfies the firstorder condition corresponding to the righthand side of 788 being equal to zero f K r δ1 φδK This firstorder condition differs from the standard modified golden rule condition which requires the marginal product of capital to be equal to the interest rate plus the depreciation rate because an additional cost of having a higher capital stock is that there must be more investment to replenish depreciated capital This is captured by the term φδK Since φ is strictly convex and f is strictly concave and satisfies the Inada conditions from Assumption 2 there exists a unique value of K that satisfies this condition The analysis of dynamics in this case requires somewhat different ideas than those used in the basic Solow growth model compare Theorems 24 and 25 In particular instead of global stability in the KI space the correct concept is one of saddlepath stability The reason for this is that instead of an initial value constraint I0 is pinned down by a boundary condition at infinity that is to satisfy the transversality condition lim texprtqtKt 0 Thus in the context of the current theory with one state and one control variable we should have a onedimensional manifold a curve along which capitalinvestment pairs tend toward the steady state This manifold is also referred to as the stable armThe initial value of investment I0 will then be determined so that the economy starts along this curve In fact if any capital investment pair rather than only pairs along this curve were to lead to the steady state we would not know how to determine I0 in other words there would be an indeterminacy of equilibria Mathematically rather than requiring all eigenvalues of the linearized system to be negative what we require now is saddlepath stability which involves the number of strictly negative eigenvalues to be the same as the number of state variables This notion of saddlepath stability is central in most growth models Let us now make these ideas more precise by considering the following generalizations of Theorems 24 and 25 see Appendix B Theorem 718 SaddlePath Stability in Linear Systems Consider the following lin ear differential equation system xt Axtb 789 with initial value x0 where xt Rn for all t and A is an n n matrix Let x be the steady state of the system given by Ax b 0 Suppose that m n of the eigenvalues of A have 272 Chapter 7 An Introduction to the Theory of Optimal Control negative real parts Then there exists an mdimensional subspace M of Rn such that starting from any x0 M the differential equation 789 has a unique solution with xt x Theorem 719 SaddlePath Stability in Nonlinear Systems Consider the following nonlinear autonomous differential equation xt Gxt 790 where G Rn Rn and suppose that G is continuously differentiable with initial value x0 Let x be a steady state of this system given by Gx 0 Define A DGx where DGx is the Jacobian of G at x Suppose that m n of the eigenvalues of A have strictly negative real parts and the rest have strictly positive real parts Then there exists an open neighborhood of x Bx Rn and an mdimensional manifold M Bx such that starting from any x0 M the differential equation 790 has a unique solution with xt x Put differently these two theorems state that when only a subset of the eigenvalues have negative real parts a lowerdimensional subset of the original space leads to stable solutions Fortunately in this context this is exactly what we require since I0 should adjust to place us on exactly such a lowerdimensional subspace manifold of the original space Armed with these theorems we can now investigate the transitional dynamics in the qtheory of investment To see that the equilibrium tends to this steadystate level of capital stock let us plot 784 and 788 in the KI space Figure 71 The curve corresponding to K 0 784 is upward sloping since a greater level of capital stock requires more investment to replenish the depreciated capital Above this curve there is more investment than necessary for replenishment so that K 0 Below this curve K 0 On the other hand the curve corresponding to I 0 788 can be nonmonotonic Nevertheless it is straightforward to verify that in the neighborhood of the steady state it is downward sloping see Exercise 728 To the right of this curve f K is lower and thus I 0 To its left I 0 The resulting phase diagram and the onedimensional stable curve manifold often referred to as the stable arm are shown in Figure 71 Next we see that starting with an arbitrary level of capital stock K0 0 the unique solution involves an initial level of investment I0 0 followed by convergence to the steady state investment level of δK along the stable arm In particular it can easily be shown that when K0 K I0 I and it monotonically decreases toward I see Exercise 728 This conlcusion is intuitive Adjustment costs discourage large values of investment thus the firm cannot adjust its capital stock to its steadystate level immediately However because of diminishing returns the benefit of increasing the capital stock is greater when the level of capital stock is low Therefore initially the firm is willing to incur greater adjustment costs to increase its capital stock and I0 is high As capital accumulates and Kt K0 the benefit of boosting the capital stock declines and the firm also reduces investment toward the steadystate investment level There are two ways of seeing why the solution corresponding to the stable arm in Figure 71the one starting with K0 I0 and converging to K I is the unique solution The first way which is more rigorous and straightforward is to use Theorem 714 As noted above the conditions of this theorem hold in this problem Thus we know that a path of capital and investment that satisfies the necessary conditions ie a path starting with K0 I0 and converging to K I is the unique optimal path By implication other paths for example those that start in I 0 or I 0 in Figure 71 cannot be optimal 274 Chapter 7 An Introduction to the Theory of Optimal Control for an interior continuous solution However when I0 0 we are no longer in the interior of the feasibility set for the control variable here R Despite this potential problem this argument is often used in many different contexts including in the analysis of the neoclassical growth model Nevertheless the same result can be established more rigorously and the conclusion from this argument is valid see Exercise 729 in this chapter and Exercise in 814 in the next chapter for the neoclassical growth model Let us next turn to the qtheory aspects James Tobin argued that the value of an extra unit of capital to the firm divided by its replacement cost is a measure of the value of investment to the firm In particular when this ratio is high the firm would like to invest more In steady state the firm settles where this ratio is 1 or close to 1 In our formulation the costate variable qt measures Tobins q To see this let us denote the current maximized value of the firm when it starts with a capital stock of Kt by V Kt The same arguments as above imply that V Kt qt 791 so that qt measures exactly by how much a 1dollar increase in capital raises the value of the firm In steady state qt 0 so that q f Kr δ which is approximately equal to 1 when φδK is small Nevertheless away from the steady state qt can differ significantly from this amount When it is greater this signals that there is need for further investments Therefore in this model Tobins q or alternatively the costate variable qt signals when investment demand is high The qtheory of investment is one of the workhorse models of macroeconomics and finance since proxies for Tobins q can be constructed using stock market prices and book values of firms When stock market prices are greater than book values this corresponds to periods in which the firm in question has a high Tobins qmeaning that the value of installed capital is greater than its replacement cost which appears on the books Nevertheless whether this approach is satisfactory in practice is intensely debated in part because Tobins q does not contain all the relevant information when there are irreversibilities or fixed costs of investment and also perhaps more importantly what is relevant in theory and in practice is the marginal q which corresponds to the marginal increase in value as suggested by 791 However in the data most measures compute average q The discrepancy between these two concepts can be large 79 Taking Stock This chapter has reviewed the basic tools of dynamic optimization in continuous time By its nature the chapter has been technical The material covered here may have been less familiar to many readers than the discretetime optimization methods presented in Chapter 6 Part of the difficulty arises from the fact that optimization in continuous time is with respect to functions even when the horizon is finite rather than with respect to vectors or infinite sequences as in the discretetime case This introduces a range of complications and some technical difficulties which are not of great interest in the context of economic applications As a result this chapter has provided an overview of the main results with an emphasis on those that are most useful in economic applications together with some of the proofs These proofs are included to provide readers with a sense of where the results come from and to develop a better intuition for the results 710 References and Literature 275 It is useful to recap the main approach developed in this chapter Most of the problems in economic growth and macroeconomics require the use of discounted infinitehorizon optimal control methods Theorem 713 provides necessary conditions for an interior continuous solution to such problems Theorem 714 provides sufficient conditions related to the concavity of the maximized Hamiltonian for such a solution to be a global or unique global maximum these conditions require the existence of a candidate solution since they use information on the costate variable of this solution More importantly the conditions in Theorem 714 are more straightforward to verify than those in Theorem 713 in particular than Assumption 71 Therefore the following strategy is used in the rest of this book 1 Start with the necessary conditions in Theorem 713 to construct a candidate solution which can be done even when Assumption 71 is not satisfied 2 Once a candidate path has been located verify that the concavity conditions in Theorem 714 are satisfied If they are then we have located a path that is optimal If in addition the maximized Hamiltonian is strictly concave then this solution is unique It is also worth noting that while the basic ideas of optimal control may be a little less familiar than those of discretetime dynamic programming these methods are used in much of growth theory and in other areas of macroeconomics Moreover while some problems naturally lend themselves to analysis in discrete time other problems become easier in continuous time Some argue that this is indeed the case for growth theory Regardless of whether one agrees with this assessment it is important to have a good command of both discretetime and continuoustime models in macroeconomics since it should be the context and economic questions that dictate which type of model is used not the force of habit This consideration motivated my choice of giving roughly equal weight to the two sets of techniques There is another reason for studying optimal control The most powerful theorem in optimal control Pontryagins Maximum Principle is as much an economic result as a mathematical one As discussed in this chapter the Maximum Principle has a very natural interpretation both in terms of maximizing flow returns plus the value of the stock and in terms of an asset value equation for the value of the maximization problem These economic intuitions are not only useful in illustrating the essence of this mathematical technique but they also provide a useful perspective on a large set of questions that involve the use of dynamic optimization techniques in macroeconomics labor economics finance and other fields This chapter also concludes our exposition of the foundations of growth theory which comprised general equilibrium foundations of aggregative models as well as an introduction to the mathematical tools necessary for dynamic economic analysis I next turn to economically more substantive issues 710 References and Literature The main material covered in this chapter is the topic of many excellent applied mathematics textbooks The purpose here has been to provide a review of the results that are most relevant for economists together with simplified versions of the most important proofs The first part of the chapter is closer to the theory of the calculus of variations because it makes use of variational arguments combined with continuity properties Nevertheless most economists do not need to study the calculus of variations in detail both because it has been superseded by optimal control theory and because most of the natural applications of the calculus of variations are in physics and the other natural sciences The interested reader can look at Gelfand and 276 Chapter 7 An Introduction to the Theory of Optimal Control Fomin 2000 Chiang 1992 provides a readable and simple introduction to the calculus of variations with economic examples The theory of optimal control was originally developed by Pontryagin et al 1962 For this reason the main necessary condition is also referred to as the Pontryagins Maximum Principle The type of problem considered here and in economics more generally is referred to as the Lagrange problem in optimal control theory The Maximum Principle is generally stated either for the somewhat simpler socalled Meyer problem or the more general Bolza problem though all of these problems are essentially equivalent and when the problem is formulated in vector form one can easily go back and forth between these different problems by simple transformations A more modern approach which underlies the necessary conditions used in infinitehorizon problems is developed in Rockefeller 1971 There are several books with varying levels of difficulty dealing with optimal control Many of these books are not easy to read but are also not entirely rigorous in their proofs An excellent source that provides an advanced and complete treatment is Fleming and Rishel 1975 The first part of this book provides a complete but rather different proof of Pontryagins Maximum Principle and various applications This book also provides a number of theorems on existence and continuity of optimal controls though for more specialized problems than those covered in Theorem 715 or necessary for economic applications The proof of existence of solutions in Section 76 combines certain ideas from Baums 1976 proof which in turn extends Cesaris 1966 classic proof of existence of solutions to infinitehorizon problems with part of the proof in Fleming and Rishel 1975 Chapter 3 In particular the last part of Theorem 715 which established the measurability of control ˆyt can be shown in greater detail using a similar line of argument to that in Fleming and Rishel which involves the use of Lusins Theorem In the economics literature existence theorems are provided in Magill 1981 and Romer 1986b but under somewhat more restrictive conditions and using a different method of proof A deeper understanding of sufficient conditions for existence of solutions and the structure of necessary conditions can be gained from the excellent but abstract and difficult book by Luenberger 1969 The results in this book are general enough to cover both discretetime and continuoustime dynamic optimization Luenberger also gives a very good sense of why maximization in function spaces is different from finitedimensional maximization and when such infinitedimensional maximization problems may fail to have solutions The main theorems in the infinitehorizon case Theorems 79 713 711 and 714 have been presented with the terminal value constraint limt btxt x1 This is important since the constraint on household assets in the competitive equilibrium of the neoclassical growth model the noPonzi condition takes this form The standard results with terminal value constraints of the form limt xt x1 cannot be applied directly Many authors seem to use the following reasoning ignore the terminal value constraint apply the Maximum Principle and then if necessary use the terminal value constraint at the end While this procedure typically gives the right answer it is not mathematically correct The Maximum Principle cannot be applied in economic problems without a terminal value constraint since in that case a solution typically fails to exist see eg Exercise 82 in the next chapter Therefore the application of the Maximum Principle to these problems is vacuous A slight strengthening of the terminal value constraints of the Maximum Principle circumvents this problem Note also that in contrast to the standard practice in economic growth and macroeconomics the emphasis here has been mostly on the sufficiency results for concave problems in particular on Theorem 714 This approach has been taken because the standard form of the Maximum Principle only gives necessary conditions for interior continuous solutions But it is not easy to verify that such a solution exists Since most problems in economics are concave Theorem 714 710 References and Literature 277 or other sufficiency results eg Theorems 75 76 78 or 711 are easy to apply and enable us to verify that a candidate solution that satisfies the Maximum Principle is indeed a solution and achieves the global optimum It should also be noted that all of the sufficiency results here have been stated and proved assuming that the control function yt or yt is continuous The logic of the proof is very similar when these functions are piecewise continuous and a formal proof along these lines is provided in Seierstad and Sydsaeter 1977 Books that develop the theory of optimal control with economic applications may be more accessible for economists The best reference is Seierstad and Sydsaeter 1987 This book is not as rigorous as Fleming and Rishel 1975 and in fact does not contain detailed proofs Nevertheless it does provide a number of useful results and is more interesting to read for economists It also shows how the results can be applied to economic problems Other references in economics are Kamien and Schwartz 1981 and Leonard and Van Long 1992 Another classic book is Arrow and Kurz 1970 which covers the same material and also presents rich economic insights on growth theory and related problems This book also states Arrows Sufficiency Theorem which first appeared in Arrow 1968 This theorem strengthens Mangasarians Sufficiency Theorem stated in Theorem 75 which appears in Mangasarian 1966 Two recent books on applications of optimal control in economics Weitzman 2003 and Caputo 2005 are somewhat more accessible My treatment of the sufficiency results here is similar to Caputo 2005 Weitzman 2003 provides a lively discussion of the applications of the Maximum Principle especially in the context of environmental economics and the depletion of natural resources There is some confusion in the literature over the role of the transversality condition The example provided in Section 74 shows that the stronger transversality condition which is very useful in many applications does not always hold Halkin 1974 was the first to provide an example to show this failure The weaker form of the transversality condition 756 was derived in Michel 1982 His results are similar to those of Theorem 712 though Theorem 712 is stated under stronger assumptions Michel instead considers stationary problems assumes that the payoff function is nonnegative and imposes an additional technical assumption that is not easy to verify Michel 1982 also provides another set of sufficient conditions for the stronger transversality condition 769 More general weaker transversality conditions appropriate for economic models are presented in Benveniste and Scheinkman 1982 and Araujo and Scheinkman 1983 Theorem 714 is stated under somewhat different and easy to check assumptions The original economic interpretation of the Maximum Principle appeared in Dorfman 1969 The interpretation here builds on the discussion by Dorfman but also extends it based on the noarbitrage interpretation of asset values in the HJB equation This interpretation of HJB is well known in many areas of macroeconomics and labor economicsis Weitzman 2003 also provides an economic interpretation for the Maximum Principle related to the HJB equation The classic reference for exploitation of a nonrenewable resource is Hotelling 1931 Weitzman 2003 provides a detailed treatment and a very insightful discussion Dasgupta and Heal 1979 and Conrad 1999 are also useful references for applications of similar ideas to sustainability and environmental economics Classic references on investment with costs of adjustment and the qtheory of investment include Tobin 1969 and Hayashi 1982 Detailed treatments of the qtheory of investment can be found in any graduatelevel macroeconomics textbook for example Blanchard and Fischer 1989 or Romer 2006 as well as in Dixit and Pindycks 1994 book on investment under uncertainty and Caballeros 1999 survey Caballero 1999 also includes a critique of the qtheory PART III NEOCLASSICAL GROWTH T his part of the book covers the basic workhorse models of economic growth I start with the infinitehorizon neoclassical growth model which has already been discussed in the previous three chapters A closely related model is the baseline overlapping generations model of Samuelson and Diamond which is the topic of Chapter 9 Despite the similarities between the two models they have quite different normative and positive implications and each model may be appropriate for different sets of issues It is therefore important to discuss both in detail This part of the book also presents an introduction to models that endogenize human capital investments Human capital plays an increasingly important role in the analysis of economic growth and in macroeconomics These models allow us to study the interactions between human capital and growth and to link macroeconomic approaches to growth to microdata on schooling and returns to education Finally Chapter 11 introduces the simplest models of sustained economic growth These are contained in this part of the book rather than the next because they are models of sustained growth without technological change Despite their simplicity these models lead to a number of important economic insights and provide a good introduction to the issues discussed in the Part IV of the book 8 The Neoclassical Growth Model W e are now ready to start our analysis of the standard neoclassical growth model also known as the Ramsey or CassKoopmans model This model differs from the Solow model in only one crucial respect it explicitly models the consumer side and endogenizes savings In other words it introduces household optimization Beyond its use as a basic growth model this model has become a workhorse for many areas of macroeconomics including the analysis of fiscal policy taxation business cycles and even monetary policy Since both the basic equilibrium and optimal growth models in discrete time have already been presented as applications of dynamic programming in Chapter 6 much of this chapter focuses on the continuoustime neoclassical growth model Section 86 provides the charac terization of the competitive equilibrium in discrete time 81 Preferences Technology and Demographics 811 Basic Environment Consider an infinitehorizon economy in continuous time and suppose that the economy admits a normative representative household as defined in Theorem 53 with instantaneous utility function uct 81 The following standard assumption on this utility function is maintained throughout the book unless stated otherwise Assumption 3 Neoclassical Preferences The instantaneous utility function uc is defined on R or R0 It is strictly increasing concave and twice differentiable with derivatives uc 0 and uc 0 for all c in the interior of its domain More explicitly the reader may wish to suppose that the economy consists of a set of identical households with measure normalized to 1 Each household has an instantaneous 287 81 Preferences Technology and Demographics 289 where as before kt Kt Lt 84 Competitive factor markets then imply that the rental rate of capital and the wage rate at time t are given by respectively Rt FKKt Lt f kt 85 and wt FLKt Lt f kt ktf kt 86 The demand side is somewhat more complicated since each household solves a continuous time optimization problem in deciding how to use their assets and allocate consumption over time To prepare for this let us denote the asset holdings of the representative household at time t by At Then the law of motion for the total assets of the household is At rtAt wtLt ctLt 87 where ct is consumption per capita of the household rt is the riskfree market rate of return on assets and wtLt is the flow of labor income earnings of the household Defining per capita assets as at At Lt dividing 87 by Lt substituting for the definition of at and using the fact that Lt grows at the rate n see 82 the law of motion of per capita assets is obtained as at rt nat wt ct 88 In practice household assets can consist of claims to capital stock Kt which the households rent to firms and government bonds Bt In models with uncertainty households would have a portfolio choice between the capital stock of the corporate sector and riskless bonds typically assumed to be supplied by the government Bonds play an important role in models with incomplete markets allowing households to smooth idiosyncratic shocks Since these bonds are in zero net supply in the aggregate Bt 0 and thus market clearing implies that assets per capita must be equal to the capital stock per capita That is at kt 89 Because there is no uncertainty here I ignore government bonds until Chapter 171 Since household assets are the same as the capital stock and capital depreciates at the rate δ the market rate of return on assets is rt Rt δ 810 1 In particular if bonds were present by a noarbitrage argument they would have exactly the same rate of return as capital and thus would be redundant 82 Characterization of Equilibrium 293 82 Characterization of Equilibrium 821 Definition of Equilibrium Let us now define an equilibrium in this dynamic economy I provide two definitions each emphasizing different aspects of the nature of the equilibrium In what follows I typically make use of the second definition though the first one is particularly useful in clarifying what a competitive equilibrium corresponds to conceptually As background for the first definition recall that we have described the environment in terms of demographics preferences and technology Given this description we can ask the question of how resources should be allocated in this environment One way of doing this is by vesting all power to allocate resources in a single body for example a social planner or in less fortunate situations a dictator The optimal growth problem already introduced in the previous two chapters and discussed further in Section 83 focuses on the allocation of resources by a social planner wishing to maximize the utility of the representative household The competitive equilibrium instead imposes a different set of institutionscompetitive markets for factors and goods and private ownership of capital and labor It then allows households to make their own choices given market prices The first definition states this explicitly Definition 81 A competitive equilibrium of the neoclassical growth model consists of paths of consumption capital stock wage rates and rental rates of capital Ct Kt wt Rt t0 such that the representative household maximizes its utility given initial asset holdings capital stock K0 0 and taking the time path of prices wt Rt t0 as given firms maximize profits taking the time path of factor prices wt Rt t0 as given and factor prices wt Rt t0 are such that all markets clear This definition states that households and firms act in a pricetaking manner and that competitive markets clear While Definition 81 emphasizes the important conceptual aspects of a competitive equilibrium it is often more mathematically convenient to define an equilibrium by incorporating some of the equilibrium relationships This is done in the next definition which imposes the equations that the factor prices wt Rt t0 must satisfy In addition this definition expresses the key objects in terms of per capita terms which also facilitates further characterization Definition 82 A competitive equilibrium of the neoclassical growth model consists of paths of per capita consumption capitallabor ratio wage rates and rental rates of capital ct kt wt Rt t0 such that factor prices wt Rt t0 are given by 85 and 86 and the representative household maximizes 83 subject to 88 and 814 given initial per capita asset holdings capitallabor ratio k0 0 and factor prices wt Rt t0 with the rate of return on assets rt given by 810 Since this definition of equilibrium already incorporates some of the equilibrium behavior one might have a preference for Definition 81 on theoretical grounds Nevertheless definitions of equilibria similar to Definition 82 are often more convenient to work with and are more widely used because they explicitly state the equations corresponding to the equilibrium and thus facilitate the characterization of allocations that solve the specified maximization problem subject to the relevant constraints In the remainder of the book I follow the standard practice of using definitions of equilibria similar to Definition 82 though the reader should bear in mind that it is derived from the more primitive Definition 81 by incorporating some of the equilibrium conditions 300 Chapter 8 The Neoclassical Growth Model Proposition 81 In the neoclassical growth model described in Section 81 with Assump tions 1 2 3 and 4 the equilibrium is Pareto optimal and coincides with the optimal growth path maximizing the utility of the representative household 84 SteadyState Equilibrium As in Chapter 2 a steadystate equilibrium is defined as an equilibrium path in which the capitallabor ratio consumption and output are constant The steadystate equilibrium and also by the equivalence between the two problems the stationary solution to the optimal growth problem is straightforward to characterize Steady state requires that consumption per capita is constant thus ct 0 From 828 this expression implies that regardless of the exact utility function as long as f k 0 we must have a capitallabor ratio k that satisfies f k ρ δ 835 which is the equivalent of the steadystate relationship in the discretetime optimal growth model5 Equation 835 pins down the steadystate capitallabor ratio as a function only of the production function the discount rate and the depreciation rate The steadystate condition 835 corresponds to the modified golden rule rather than to the golden rule in the Solow model see Exercise 812 The modified golden rule involves a level of the capital stock that does not maximize steadystate consumption because earlier consumption is preferred to later consumption This preference is because of discounting which means that the objective is not to maximize steadystate consumption but instead involves giving a higher weight to earlier consumption Note also at this point that Assumption 4 ρ n and 835 together imply that the steady state interest rate is r f k δ n 836 and thus satisfies the natural requirement that r n Since in steady state the wage rate is w f k kf k it can also be verified that households have finite wealth at all points in time Given k the steadystate consumption level is also straightforward to determine as c f k n δk 837 which is similar to the consumption level in the basic Solow model Moreover given Assump tion 4 a steady state where the capitallabor ratio and thus output are constant necessarily satisfies the transversality condition This analysis therefore establishes the following result Proposition 82 In the neoclassical growth model described in Section 81 with Assump tions 1 2 3 and 4 the steadystate equilibrium capitallabor ratio k is uniquely determined 5 In addition there again exists another economically uninteresting steady state at k 0 As in Chapter 2 I ignore this steady state throughout Moreover as in Chapter 2 starting with any k0 0 the economy will always tend to the steadystate capitallabor ratio k given by 835 84 SteadyState Equilibrium 301 by 835 and is independent of the instantaneous utility function The steadystate consump tion per capita c is given by 837 As with the basic Solow growth model there are also several straightforward comparative static results that show how the steadystate values of capitallabor ratio and consumption per capita change with the underlying parameters For this reason let us again parameterize the production function as f k A f k where A 0 so that A is again a shift parameter with greater values corresponding to greater productivity of factors Since f k satisfies the regularity conditions imposed above so does f k Proposition 83 Consider the neoclassical growth model described in Section 81 with Assumptions 1 2 3 and 4 and suppose that f k A f k Denote the steadystate level of the capitallabor ratio by k A ρ n δ and the steadystate level of consumption per capita by c A ρ n δ when the underlying parameters are A ρ n and δ Then kA ρ n δ A 0 kA ρ n δ ρ 0 kA ρ n δ n 0 and kA ρ n δ δ 0 cA ρ n δ A 0 cA ρ n δ ρ 0 cA ρ n δ n 0 and cA ρ n δ δ 0 Proof See Exercise 817 The new results here relative to the basic Solow model concern the comparative statics with respect the discount rate ρ In particular instead of the saving rate it is now the discount rate that affects the rate of capital accumulation There is a close link between the discount rate in the neoclassical growth model and the saving rate in the Solow model Loosely speaking a lower discount rate implies greater patience and thus greater savings In the model without technological progress the steadystate saving rate can be computed as s n δk f k 838 where k is the steadystate capitallabor ratio given in 835 Exercise 819 investigates the relationship between the discount rate the saving rate and the steadystate per capita consumption level A further interesting result is that the rate of population growth has no impact on the steadystate capitallabor ratio which contrasts with the basic Solow model Exercise 816 shows that this result depends on the way in which intertemporal discounting takes place Another important result which is more general is that k and thus c do not depend on the instantaneous utility function u The form of the utility function only affects the transitional dynamics but has no impact on steady states This is because the steady state is determined by the modified golden rule This result is not true in the presence of technological change and sustained growth 312 Chapter 8 The Neoclassical Growth Model It is also useful to briefly look at an example with CobbDouglas technology Example 82 Consider the model with CRRA utility and laboraugmenting technological progress at the rate g Dropping time dependence to simplify notation the production function is F K AL Kα AL1α and thus f k kα so that r αkα1 δ The Euler equation written in terms of normalized consumption then becomes d cdt c 1 θ αkα1 δ ρ θg and the accumulation equation can be written as k k kα1 δ g n c k Now define z ck and x kα1 which implies that xx α 1 kk Therefore we have x x 1 αx δ g n z 855 and also z z d cdt c k k which implies that z z 1 θ αx δ ρ θg x δ g n z 1 θ α θx 1 θδ θn ρ θ z 856 The two differential equations 855 and 856 together with the initial condition x0 and the transversality condition completely determine the dynamics of the system In Exercise 824 you are asked to complete this example for the special case in which θ 1 log preferences 88 The Role of Policy In the model developed in Section 87 the rate of growth of per capita consumption and output per worker per capita are determined exogenously by the growth rate of laboraugmenting technological progress The level of income on the other hand depends on the intertemporal elasticity of substitution 1θ the discount rate ρ the depreciation rate δ the population growth rate n and naturally the form of the production function f Returning to the proximate causes of crosscountry differences in income per capita and growth this model gives us a way of understanding those differences in terms of preference and technology parameters As discussed in Chapter 4 we also wish to link the proximate causes of economic growth to potential fundamental causes The intertemporal elasticity of 812 Taking Stock 317 capita across countries are unlikely to be accounted for solely by differences in capital per worker To explain such large differences in income per capita across countries we need sizable differences in the efficiency with which these factors are being used in different countries Such efficiency differences are not present in this model Therefore the simplest neoclassical model does not generate sufficient differences in capitallabor ratios to explain crosscountry income differences Nevertheless many economists have tried and still try to use versions of the neoclassical model to go further The motivation is simple If instead of using α 13 we take α 23 the ratio of incomes in the two countries would be Yτ Yτ 82 64 Thus if the responsiveness of capital or other factors to policy distortions were higher than that implied by the neoclassical growth model with α 13 eg corresponding to the case where α 23 then the predicted differences across countries can be made much larger How could we have a model in which α 23 Such a model must have additional accumulated factors while still keeping the share of capital income in national product at roughly 13 One possibility might be to include human capital see Chapter 10 However the discussion in Chapter 3 showed that human capital differences appear to be insufficient to explain a large portion of the income per capita differences across countries Another possibility is to introduce other types of capital or perhaps technology that responds to distortions in the same way as capital While these are all logically possible a systematic analysis of these issues requires models of endogenous technology which is our focus in the next part of the book 811 Extensions There are many empirically and theoretically relevant extensions of the neoclassical growth model I do not present them here for the sake of brevity The most important ones are presented as exercises instead In particular Exercise 833 endogenizes the labor supply decisions of households by introducing leisure in the utility function The model presented in this exercise is particularly important since it corresponds to the version of the neoclassical growth model most often employed in shortrun and mediumrun macroeconomic analyses This exercise also shows that further restrictions on the form of the utility function need to be imposed to preserve balanced growth in this case Exercise 834 further studies models that incorporate government expenditures and taxation Exercise 836 looks at the behavior of the basic neoclassical growth model with a free capital account representing borrowing and lending opportunities for the economy at some exogenously given international interest rate r Exercise 837 combines the costs of adjustments in investment as in the qtheory with the basic neoclassical model Finally Exercise 838 looks at a version of the neoclassical model with multiple sectors 812 Taking Stock This chapter presented arguably the most important model in macroeconomics the onesector neoclassical growth model Recall that our study of the basic models of economic growth started in Chapter 2 with the Solow growth model We saw that while this model gives a number of important insights it treats much of the mechanics of economic growth as a black box Growth can only be generated by technological progress unless we are in the special 318 Chapter 8 The Neoclassical Growth Model AK model without diminishing returns to capital but technological progress is outside the model The next important element in determining crosscountry differences in income is the saving rate but in the Solow growth model the saving rate was also taken as exogenous The major contribution of the current chapter has been to open the black box of savings and capital accumulation by specifying the preferences of households Consequently we can link saving rates to preferences technology and prices in the economy Moreover as Exercise 839 shows the implications of policy on equilibrium quantities are different in the neoclassical model than in the Solow growth model with exogenously specified saving rates Another major advantage of the neoclassical growth model is that because preferences are explicitly specified equilibrium and optimal growth can be compared Perhaps the most important contribution of this model is that it paves the way for further analysis of capital accumulation human capital investments and endogenous technological progress which is our topic in the next few chapters starting with the analysis of human capital in Chapter 10 Therefore this chapter is the first and perhaps conceptually the most important step toward a systematic study of economic growth It provides us with the mathematical and conceptual tools necessary for modeling capital accumulation human capital accumulation and technological change endogenously Did our study of the neoclassical growth model generate new insights about the sources of crosscountry income differences and economic growth relative to the Solow growth model The answer here is largely no While the current model is an important milestone in the study of the mechanics of economic growth as with the Solow growth model the focus is on the proximate causes of these differenceswe are still looking at differences in saving rates investments in human capital and technology perhaps as determined by preferences and other dimensions of technology eg the rate of laboraugmenting technological change It is therefore important to bear in mind that this model by itself does not enable us to answer questions about the fundamental causes of economic growth What it does however is to clarify the nature of the economic decisions so that we are in a better position to ask such questions 813 References and Literature The neoclassical growth model goes back to Frank Ramseys 1928 classic article and for that reason is often referred to as the Ramsey model Ramseys model was very similar to the standard neoclassical growth model except that it did not feature discounting Another early optimal growth model was presented by John von Neumann 1945 focusing on the limiting behavior of the dynamics in a linear model The current version of the neoclassical growth model is most closely related to the analysis of optimal growth by David Cass 1965 and Tjalling Koopmans 1965 An excellent discussion of optimal growth is provided in Arrow and Kurzs 1970 volume All growth and macroeconomic textbooks cover the neoclassical growth model Ljungqvist and Sargent 2005 Chapter 14 provides an introductory treatment of the neoclassical growth model in discrete time Barro and SalaiMartin 2004 Chapter 2 provides a detailed treatment focusing on continuoustime models Blanchard and Fischer 1989 Chapter 2 and Romer 2006 Chapter 2 also present the continuoustime version of the neoclassical growth model These books use the necessary conditions implied by the Maximum Principle including the strong version of the transversality condition and characterize utilitymaximizing consumption behavior The typical approach is to first ignore the noPonzi condition and then rule out paths that violate this condition As also pointed out in the previous chapter more care is 9 Growth with Overlapping Generations A key feature of the neoclassical growth model of the previous chapter is that it admits a normative representative household This model provides us with a tractable framework for the analysis of capital accumulation Moreover it enables us to appeal to the First and Second Welfare Theorems to establish the equivalence between equilibrium and optimum growth problems In many situations however the assumption of a representative household is not appropriate One important set of circumstances that may require departure from this assumption is in the analysis of an economy in which new households arrive or are born over time The arrival of new households in the economy is not only a realistic feature but it also introduces a range of new economic interactions In particular decisions made by older generations will affect the prices faced by younger generations These economic interactions have no counterpart in the neoclassical growth model They are most succinctly captured in the overlapping generations OLG models introduced and studied by Paul Samuelson and later by Peter Diamond These models are useful for a number of reasons First they capture the potential interac tion of different generations of individuals in the marketplace Second they provide a tractable alternative to the infinitehorizon representative agent models Third some of their key impli cations are different from those of the neoclassical growth model Fourth the dynamics of capital accumulation and consumption in some special cases of these models are quite similar to the basic Solow model rather than to the neoclassical model Finally they generate new insights about the role of national debt and social security in the economy I start with an illustration of why the First Welfare Theorem cannot be applied in OLG models I then discuss the baseline OLG model and a number of applications of this framework Finally I present the OLG model in continuous time The latter model originally developed by Menahem Yaari and Olivier Blanchard and also referred to as the perpetual youth model is a tractable alternative to the basic OLG model This model is also used in the context of human capital investments in the next chapter 327 328 Chapter 9 Growth with Overlapping Generations 91 Problems of Infinity This section illustrates why the First Welfare Theorem does not apply to OLG models using an abstract general equilibrium economy introduced by Karl Shell This model is interesting in part because it is closely related to the baseline OLG model of Samuelson and Diamond which is presented in the next section Consider the following static economy with a countably infinite number of households each denoted by i N and a countably infinite number of commodities denoted by j N Assume that all households behave competitively alternatively we can assume that there are M households of each type where M is a large number Household i has preferences given by ui ci i ci i1 where ci j 0 denotes the consumption of the jth type of commodity by household i These preferences imply that household i enjoys the consumption of the commodity with the same index as its own and the next indexed commodity eg the household indexed by 3 only derives utility from the consumption of goods indexed by 3 and 4 The endowment vector ω of the economy is as follows each household has one unit endowment of the commodity with the same index as its own Let us choose the price of the first commodity as the numeraire so that p0 1 A competitive equilibrium in this economy is defined in the usual manner eg Definition 51 in Chapter 5 The following proposition characterizes a competitive equilibrium see Exercise 91 for the uniqueness of equilibrium Proposition 91 In the abovedescribed economy the price vector p such that pj 1 for all j N is a competitive equilibrium price vector and induces an equilibrium with no trade denoted by x Proof At p each household has income equal to 1 Therefore the budget constraint of household i can be written as ci i ci i1 1 Thus consuming its own endowment is optimal for each household establishing that the price vector p and no trade x constitute a competitive equilibrium However the competitive equilibrium in Proposition 91 is not Pareto optimal To see this consider the following alternative allocation xi for i N In this allocation each household i i consumes one unit of good j i Household i consumes one unit of good j i and one unit of good j i 1 Finally each household i i consumes one unit of good i 1 In other words household i consumes its own endowment and that of household i 1 while all other households indexed i i consume the endowment of the neighboring household i 1 while the consumption bundles of all households i i are the same as in x In this allocation all households with i i are as well off as in the competitive equilibrium p x and household i is strictly better off This argument establishes the following proposition Proposition 92 In the abovedescribed economy the competitive equilibrium at p x is not Pareto optimal In fact it is also straightforward to construct alternative allocations that make more than a single agent strictly better off relative to the equilibrium allocation x see Exercise 91 So why does the First Welfare Theorem not apply in this economy Recall that the first version of this theorem Theorem 55 is for an economy with a finite number of households whereas 330 Chapter 9 Growth with Overlapping Generations general separable utility function for individuals born at date t of the form Utc1t c2t 1 uc1t βuc2t 1 91 where u R R satisfies the conditions in Assumption 3 from Chapter 8 c1t denotes the consumption of an individual born at time t when young at date t and c2t 1 is this individuals consumption when old at date t 1 Also β 0 1 is the discount factor There is no need to distinguish among different individuals of the same generation and I do not do so to economize on notation Factor markets are competitive Individuals can only work in the first period of their lives and they supply one unit of labor inelastically earning the equilibrium wage rate wt Suppose also that there is exponential population growth and in particular the size of generation t born at time t is Lt 1 ntL 0 92 The production side of the economy is the same as before characterized by a set of competitive firms and it is represented by a standard constant returns to scale aggregate production function satisfying Assumptions 1 and 2 from Chapter 2 Yt F Kt Lt which uses the fact that employment at time t is equal to the size of the group at this date Lt To simplify the analysis let us assume that δ 1 so that capital fully depreciates after use see Exercise 94 Thus again defining k KL the gross rate of return to saving which equals the rental rate of capital is given by 1 rt Rt f kt 93 where f k F k 1 is the standard per capita production function As usual the wage rate is wt f kt ktf kt 94 922 Consumption Decisions Let us start with the individual consumption decisions Savings by an individual of generation t st are determined as a solution to the following maximization problem max c1tc2t1st uc1t βuc2t 1 subject to c1t st wt and c2t 1 Rt 1st where I am using the convention that young individuals rent their savings as capital to final good producers at the end of time t and receive the return at time t 1 after production is carried 92 The Baseline Overlapping Generations Model 331 out1 The gross rate of return they receive on their savings is Rt 1 1 rt 1 The second constraint incorporates the notion that individuals only spend money on their own end oflife consumption since there is no altruism or bequest motive There is no need to introduce the additional constraint that st 0 since negative savings would violate the secondperiod budget constraint given that c2t 1 0 Since the utility function u is strictly increasing Assumption 3 both constraints hold as equalities Therefore the firstorder condition for a maximum can be written in the familiar form of the consumption Euler equation recall Chapter 6 eg 645 uc1t βRt 1uc2t 1 95 Moreover since the problem of each individual is strictly concave this Euler equation is sufficient to characterize an optimal consumption path given market prices Combining this equation with the budget constraint we obtain the following implicit function that determines savings per person as st s wt Rt 1 96 where s R2 R is strictly increasing in its first argument and may be increasing or decreasing in its second argument see Exercise 95 Total savings in the economy is equal to St stLt where Lt denotes the size of generation t who are saving for time t 1 Since capital depreciates fully after use and all new savings are invested in the only productive asset of the economycapitalthe law of motion of the capital stock is given by Kt 1 Ltswt Rt 1 97 923 Equilibrium A competitive equilibrium in the OLG economy can be defined as follows Definition 91 A competitive equilibrium can be represented by sequences of aggregate capital stocks household consumption and factor prices Kt c1t c2t Rt wt t0 such that the factor price sequence Rt wt t0 is given by 93 and 94 individual consumption decisions c1t c2t t0 are given by 95 and 96 and the aggregate capital stock Kt t0 evolves according to 97 A steadystate equilibrium is defined in the usual fashion as an equilibrium in which the capitallabor ratio k KL is constant 1 Here we could have used a number of different conventions all with identical results For example it could be assumed that the young keep their savings from time t until the beginning of time t 1 and at that point they rent this as capital to final good producers Or alternatively one could introduce another set of competitive firms transforming savings in terms of date t commodities to date t 1 commodities In this case the young would use these firms to transfer resources from t to t 1 The convention used in the text is the simplest 340 Chapter 9 Growth with Overlapping Generations These funds are invested in the only productive asset of the economy the capital stock and the workers receive the returns given by Rt 1dt when they are old Thus the individual maximization problem under a fully funded social security system becomes max c1tc2t1st uc1t βuc2t 1 subject to c1t st dt wt and c2t 1 Rt 1 st dt for a given choice of dt by the government Notice that now the total amount invested in capital accumulation is st dt 1 nkt 1 It is also no longer the case that individuals always choose st 0 since they have the income from social security Therefore this economy can be analyzed under two alternative assumptions with the constraint that st 0 and without it It is clear that as long as st is free the competitive equilibrium applies regardless of the sequence of feasible social security payments dt t0 When st 0 is imposed as a constraint then the competitive equilibrium applies if given the sequence dt t0 the privately optimal saving sequence st t0 is such that st 0 for all t Proposition 97 Consider a fully funded social security system in the abovedescribed environment whereby the government collects dt from young individuals at date t 1 Suppose that st 0 for all t If given the feasible sequence dt t0 of social security payments the utilitymaximizing sequence of savings st t0 is such that st 0 for all t then the set of competitive equilibria without social security is the same as the set of competitive equilibria with social security 2 Without the constraint st 0 given any feasible sequence dt t0 of social security payments the set of competitive equilibria with social security is identical to the set of competitive equilibria without social security Proof See Exercise 913 This result is intuitive the amount dt taken out by the government is fully offset by a decrease in st as long as individuals are accumulating enough savings or always when there are no constraints to force positive savings privately Exercise 914 shows that even when there is the restriction that st 0 a fully funded social security program cannot lead to a Pareto improvement 952 Unfunded Social Security The situation is different with unfunded social security Now the government collects dt from the young at time t and distributes it to the current old with per capita transfer bt 1 n dt which takes into account that there are more young than old because of population growth Therefore the individual maximization problem becomes max c1tc2t1st uc1t βuc2t 1 95 Role of Social Security in Capital Accumulation 341 subject to c1t st dt wt and c2t 1 Rt 1st 1 n dt 1 for a given feasible sequence of social security payment levels dt t0 In this environment the rate of return on social security payments is n rather than rt 1 Rt 1 1 because unfunded social security is a pure transfer system Only strather than st dt as in the funded schemegoes into capital accumulation This observation is the basis of the claim that unfunded social security systems discourage aggregate savings Consequently unfunded social security reduces capital accumulation Discouraging capital accumulation can have negative consequences for growth and welfare In fact the empirical evidence in Chapters 14 suggest that there are many societies in which the level of capital accumulation is suboptimally low In contrast in the present model reducing aggregate savings and capital accumulation may be lead to a Pareto improvement when the economy exhibits dynamic inefficiency and overaccumaltion More specifically suppose that individuals of generation t can choose how much to con tribute to unfunded social security ie dt is a choice variable Whatever they contribute is given to the current old generation as consumption and they receive 1 n dollars for every dollar invested when they become old themselves In this case there would be no investment in physical capital until rt 1 n Thus the unfunded social security system would increase the interest rate enough so that the economy is no longer in the dynamic inefficiency region This analysis establishes the following proposition Proposition 98 Consider the abovedescribed OLG economy and suppose that the de centralized competitive equilibrium is dynamically inefficient Then there exists a feasible sequence of unfunded social security payments dt t0 that leads to a competitive equilib rium starting from any date t that Pareto dominates the competitive equilibrium without social security Proof See Exercise 916 Unfunded social security reduces overaccumulation and improves the allocation of re sources The similarity between the way in which unfunded social security achieves a Pareto improvement in the OLG model and the way in which the Pareto optimal allocation was decen tralized in the example economy of Section 91 is evident In essence unfunded social security transfers resources from future generations to the initial old generation and when designed appropriately it can do so without hurting future generations Once again this result depends on dynamic inefficiency when there is no dynamic inefficiency any transfer of resources and any unfunded social security program makes some future generation worse off You are asked to prove this result in Exercise 917 Another interesting aspect of unfunded social security is also worth noting With this type of social security system the government is essentially running a Ponzi game or pyramid scheme Each generation sacrifices an amount d when young and receives 1 n d from the current young when they are old This pattern is typical of a pyramid scheme In the previous chapter such schemes were ruled out so why are they possible and in fact desirable here The answer is related to the fact that in the neoclassical growth model there exists a representative household whose utility maximization decision ensures that the economy is never in the dynamic inefficiency region In particular the transversality conditionor equivalently the finiteness of the utilityof the representative household rules out equilibria 342 Chapter 9 Growth with Overlapping Generations where r n recall 836 This is no longer the case in the OLG economy and unfunded social security is one way of running a Pareto improving pyramid scheme in an economy with dynamic inefficiency Interestingly it is not the only such scheme possible When r n the equilibrium allows for a range of bubbles that can play the same role as unfunded social security We say that there is a bubble when an asset trades at a value greater than its intrinsic value A bubble on any asset could play the same role as unfunded social security because it can create a way of transferring resources across dates The maximum rate of return on any bubble is n which is also the maximum rate of return on unfunded social security When there is dynamic inefficiency and r n a bubble provides a better way of transferring resources across dates than investing in physical capital A simple example of a bubble that can play this role is fiat money which has no intrinsic value But all agents might expect fiat money to appreciate over time so that the purchasing power of fiat money increases by some factor 1 n at each period In this case giving a limited amount of fiat money to some generation would also play the same role as unfunded social security Equivalently however the same role can be played by other assets generating similar bubbles Finally it is interesting to note that if the OLG economy has a family structure so that future generations are linked to previous generations as members of a particular family or dynasty withinfamily transfers eg supported by social norms or repeated game punishment strategies see Appendix C could play the same role In this case we would see withinfamily transfers which could improve the allocation of resources and these transfers could be supported even though there is no altruism across family members 96 Overlapping Generations with Impure Altruism Section 53 in Chapter 5 demonstrated that altruism within families eg of parents toward their offspring can lead to a structure of preferences identical to those of the representative household in the neoclassical growth model In contrast this chapter has so far ignored altruistic preferences to emphasize the effect of finite lives and the economic implications of the arrival of new agents in the economy As briefly noted in Section 53 the exact form of altruism within a family determines whether the representative household assumption is a good approximation to the preference side of the economy In particular a potentially empirically relevant form of altruism is one in which parents care about certain dimensions of the consumption vector of their offspring instead of their total utility This type of preference is often referred to as impure altruism to distinguish it from the pure altruism discussed in Section 53 One particular type of impure altruism commonly referred to as warm glow preferences plays an important role in many growth models because of its tractability Warm glow preferences assume that parents derive utility from the warm glow of their bequest rather than from their offsprings utility or consumption These preferences constitute another convenient alternative to the neoclassical growth and the baseline OLG models This alternative has some clear parallels to the canonical OLG model of the last section since it also leads to equilibrium dynamics similar to those of the Solow growth model Given the importance of this class of preferences in many applied growth models it is useful to review them briefly These preferences are also used in the next chapter and in Chapter 21 Suppose that the production side of the economy is given by the standard neoclassical production function satisfying Assumptions 1 and 2 from Chapter 2 Let us write this in per capita form as f k The economy is populated by a continuum of individuals with measure normalized to 1 Each individual lives for two periods childhood and adulthood In the second period of his life each individual begets an offspring works and then his life comes to an end For simplicity let 346 Chapter 9 Growth with Overlapping Generations Individual is flow budget constraint can be written as ait 1 1 rtait cit wt zit 932 which is similar to the standard flow budget constraint for example 654 in Chapter 6 Recall that the gross rate of return on savings is 1 rt The only difference from the standard budget constraint is the additional term zit which reflects transfers to the individual The reason these transfers are introduced is as follows since individuals face an uncertain time of death there may be accidental bequests In particular individuals typically come to the end of their lives while their asset positions are positive When this happens one possibility is that the accidental bequests might be collected by the government and redistributed equally across all households in the economy In this case zit would represent these receipts for individual i However this modeling assumption would require that we impose a constraint of the form ait 0 to prevent individuals from accumulating debts by the time their lives come to an end An alternative which avoids this additional constraint and makes the model more tractable has been proposed and studied by Menahem Yaari and Olivier Blanchard This alternative involves introducing life insurance or annuity markets where competitive life insurance firms make payments to individuals as a function of their asset levels in return for receiving their positive assets when they die The term zt captures these annuity payments In particular imagine the following type of life insurance contract a company would make a payment equal to z at to an individual as a function of her asset holdings during every period in which she is alive3 When the individual dies all her assets go to the insurance company The fact that the payment level z at depends only on the asset holdings of the individual and not on her age is a consequence of the perpetual youth assumptionthe conditional expectation of further life is independent of when the individual was born in fact it is independent of everything else in the model The profits of a particular insurance company contracting with an individual with asset holding equal to at at time t are π a t 1 ν z a νa With free entry insurance companies should make zero expected profits in terms of net present discounted value which requires that π at t 0 for all t and a thus we have z at ν 1 ν at 933 The other important element of the model is the evolution of demographics Since each agent faces a probability of death equal to ν at every date there is a natural force toward decreasing population We assume however that there are also new agents who are born at every date In contrast to the basic neoclassical growth model suppose that these new agents are not born into a dynasty instead they become separate households themselves We assume that when the population at time t is Lt there are nLt new households born Consequently the evolution of the total population is given by Lt 1 1 n νLt 934 3 The reader might note that this contract is the opposite of the most common type of life insurance contract where individuals make payments for their families to receive payments after their deaths These types of insurance contracts are not useful in the current model since individuals do not have offspring or are not altruistic toward them 354 Chapter 9 Growth with Overlapping Generations and feature overaccumulationa steadystate capitallabor ratio greater than the golden rule capitallabor ratio We have also seen how an unfunded social security system can reduce aggregate savings and thus ameliorate the overaccumulation problem The important role that unfunded social security or national debt plays in the OLG model has made this model a workhorse for analysis of transfer programs and fiscal policies Our analysis of perpetual youth models especially Yaari and Blanchards continuoustime perpetual youth model further clarified the roles of the path of labor income finite horizons and arrival of new individuals in generating overaccumulation In particular this model shows that the declining path of labor income is important for the overaccumulation result the SamuelsonDiamond twoperiod model is an extreme case since there is no labor income in the second period of the life of the individual But perhaps the more important insight generated by these models is that what matters is not the finite horizons per se but the arrival of new individuals While overaccumulation and dynamic inefficiency have dominated much of the discussion of OLG models in the literature one should not overemphasize the importance of dynamic inefficiency As discussed in Chapter 1 the major question of economic growth is why so many countries have so little capital for their workers and why the process of economic growth and capital accumulation started only in the past 200 years It is highly doubtful that overaccumulation is a major problem for most countries in the world The models presented in this chapter are useful for another reason however They sig nificantly enrich our arsenal in the study of the mechanics of economic growth and capital accumulation All three of the models presented in this chapter the baseline OLG model the OLG model with impure altruism and the perpetual youth model are tractable and useful vehi cles for the study of economic growth in a variety of circumstances For example the first two lead to equilibrium dynamics similar to those of the baseline Solow growth model but without explicitly imposing an exogenously constant saving rate The latter model on the other hand allows an analysis of equilibrium dynamics similar to that of the basic neoclassical growth model but also incorporates finite lives and overlapping generations which are essential in many problems for example in the context of human capital investments studied in the next chapter In summary this chapter has provided us with new modeling tools and different perspectives for the analyses of capital accumulation aggregate saving and economic growth Although these perspectives do not directly offer fresh answers to the questions of why countries grow and why some countries are much poorer than others they will be useful in developing such answers in subsequent chapters 910 References and Literature The baseline OLG model with twoperiod lived agents is due to Samuelson 1958 and Diamond 1965 A related model appears in French in the work of Maurice Allais Blanchard and Fischer 1989 Chapter 3 provide an excellent textbook treatment of the baseline OLG model Some textbooks use this setup as the main workhorse macroeconomic model for example McCandless and Wallace 1991 Azariadis 1993 and De La Croix and Michel 2002 See Galor and Ryder 1989 on the multiplicity of steadystate equilibria in the OLG model and Galor 1996 for a discussion of the similarities between the Solow growth model and the OLG model recall also Exercise 213 in Chapter 2 The economy studied in Section 91 is due to Shell 1971 The source of inefficiency in the OLG model is much discussed in the literature Shells 1971 example economy in Section 91 911 Exercises 355 provides the clearest intuitive explanation for why the First Welfare Theorem does not apply A lucid discussion is contained in Bewley 2007 The issues of dynamic inefficiency in OLG models are discussed in Samuelson 1958 and Diamond 1965 A more complete treatment without restricting attention to steady states is provided in Cass 1972 in the text I simplified the discussion of dynamic inefficiency by focusing on steady states The role of unfunded social security when there is dynamic inefficiency is discussed in Samuelson 1975 while the role of national debt in the same context is studied in Diamond 1965 Samuelson 1958 also notes how fiat money can play this role and this point is further developed in Wallace 1980 and Weil 1987 See Blanchard 1979 Tirole 1985 and Gilles and LeRoy 1992 for some of the early important work on bubbles in OLG models Tirole 1982 on the importance of infinite horizon for the possibility of bubbles and Ventura 2002 for the relationship between asset bubbles and capital flows The model of OLG with impure altruism is due to Andreoni 1989 This model has been used extensively in the economic growth and economic development literatures especially for the analysis of equilibrium dynamics in the presence of imperfect capital markets Wellknown examples include the models by Banerjee and Newman 1991 1993 Galor and Zeira 1993 Aghion and Bolton 1997 and Piketty 1997 which are studied in Chapter 21 I am not aware of an analysis of the dynamics of wealth inequality with perfect markets in this economy along the lines of the model presented in Section 96 even though the analysis is quite straightforward A similar analysis of wealth inequality dynamics is included in Stiglitzs 1969 model but he assumes that each household can only use its savings in its own diminishing return technology thus creating a strong force toward convergence of incomes The continuoustime perpetual youth model is due to Yaari 1965 and Blanchard 1985 The discretetime version of this model was presented to facilitate the transition to the continuoustime version My treatment of the continuoustime version closely followed Blan chard 1985 The importance of the path of labor income is emphasized and analyzed in Blanchard 1985 The importance of new arrivals in the market is emphasized and explained in Weil 1989 Models with OLG and finite lives are used extensively in the analysis of Ricardian Equivalence introduced in Exercise 835 Blanchard 1985 includes extensive discussions of this issue 911 Exercises 91 a Prove that the allocation characterized in Proposition 91 is the unique competitive equilib rium allocation b Show that in addition to the allocations xi discussed in Proposition 93 it is possible to construct an allocation xi1i2 for i1 i2 N that makes all individuals with index i i1 i2 strictly better off and all other individuals as well off as in allocation x 92 Show that the allocation xi in Proposition 93 can also be decentralized as a competitive equi librium with the price vector p such that pj 1 for all j i and pj ρji1 for j i with ρ 0 1 93 Consider the following variant of the economy in Section 91 The utility of the individual indexed i j is ucj βucj 1 where β 0 1 and each individual has one unit of the good with the same index as his own a Define a competitive equilibrium for this economy 10 Human Capital and Economic Growth T his chapter investigates the role of human capital in economic growth and in cross country income differences Our main purpose is to understand which factors affect human capital investments and how these influence the process of economic growth and economic development Human capital refers to all the attributes of workers that potentially increase their productivity in all or some productive tasks The term was coined because many of these attributes are accumulated by workers through investments Human capital theory developed primarily by Becker 1965 and Mincer 1974 is about the role of human capital in the production process and about the incentives to invest in skills including prelabor market investments in the form of schooling and onthejob investments in the form of training It would not be an exaggeration to say that this theory is the basis of much of labor economics and plays an equally important role in macroeconomics The literature on education and other types of human capital investments is vast so only parts of this literature that are relevant to the main focus of this book are covered here Other important connections between human capital and economic growth especially those related to its effect on technological progress and its role in economic takeoff are discussed later in the book 101 A Simple Separation Theorem Let us start with the partial equilibrium schooling decisions and establish a simple result sometimes referred to as a separation theorem for human capital investments I set up the basic model in continuous time for simplicity Consider the schooling decision of a single individual facing exogenously given prices for human capital Throughout I assume that there are perfect capital markets The separation theorem shows that with perfect capital markets schooling decisions maximize the net present discounted value of the individual and can thus be separated from consumption decisions I return to human capital investments with imperfect capital markets in Chapter 21 In particular consider an individual with an instantaneous utility function uc that satisfies Assumption 3 from Chapter 8 Suppose that the individual has a planning horizon of T where T is allowed discounts the future at the rate ρ 0 and faces a constant flow rate of death equal to 359 364 Chapter 10 Human Capital and Economic Growth Let us next set up the currentvalue Hamiltonian which in this case takes the form Hh s μ 1 stht μtφstht δhht where H is used to denote the Hamiltonian to avoid confusion with human capital The necessary and sufficient conditions for this problem are see Exercise 105 Hsh s μ ht μthtφstht 0 Hhh s μ 1 st μtstφstht δh r νμt μt and lim t expr νtμtht 0 1014 To solve for the optimal path of human capital investments let us adopt the following transformation of variables xt stht Instead of st or μt and ht we can look at the dynamics of the optimal path in xt and ht The first condition in 1014 implies that 1 μtφxt 1015 while the second necessary condition can be expressed as μt μt r ν δh stφxt 1 st μt Substituting for μt from 1015 and simplifying yields μt μt r ν δh φxt 1016 The steadystate stationary solution of this optimal control problem involves μt 0 and ht 0 and thus implies that x φ1r ν δh 1017 where φ1 is the inverse function of φ which exists and is strictly decreasing since φ is strictly concave Equation 1017 shows that x sh will be higher when the interest rate is low when the life expectancy of the individual is high and when the rate of depreciation of human capital is low To determine s and h separately we set ht 0 in the human capital accumulation equation 1013 which gives h φx δh φφ1r ν δh δh 1018 366 Chapter 10 Human Capital and Economic Growth 0 t h h0 ht FIGURE 102 Time path of human capital investments in the simplified BenPorath model h0 0 h s0 jumps to the level necessary to ensure s0h0 x From then on ht increases and st decreases so as to keep stht x Therefore the pattern of human capital investments implied by the BenPorath model is one of high investment at the beginning of an individuals life followed by lower investments later on In our simplified version of the BenPorath model this process takes place smoothly In the original BenPorath model which involves the use of other inputs in the production of human capital and finite horizons the constraint st 1typically binds early in the life of the individual and the interval during which st 1 can be interpreted as fulltime schooling After fulltime schooling the individual starts working st 1 But even on the job the individual continues to accumulate human capital st 0 which can be interpreted as spending time in training programs or allocating some time on the job to learning rather than to production Moreover because the horizon is finite if the Inada conditions were relaxed the individual could prefer to stop investing in human capital at some point As a result the time path of human capital generated by the standard BenPorath model may be humpshaped with a possibly declining portion at the end see Exercise 107 Instead the path of human capital and the earning potential of the individual in the current model is always increasing as shown in Figure 102 The importance of the BenPorath model is twofold First it emphasizes that schooling is not the only way in which individuals can invest in human capital and there is a continuity between schooling and other investments in human capital Second it suggests that in societies where schooling investments are high we may also expect higher levels of onthejob investments in human capital Thus there may be systematic mismeasurement of the amount or quality of human capital across societies 368 Chapter 10 Human Capital and Economic Growth arguments Let us denote its derivatives by fk fh fkh and so on Throughout I assume that physical and human capital are complementary that is fkhk h 0 for all k h 0 Physical and human capital per capita evolve according to the following two differential equations kt ikt δkkt 1021 and ht iht δhht 1022 where ikt and iht are the investment levels in physical and human capital respectively while δk and δh are the depreciation rates of these two capital stocks The resource constraint for the economy expressed in per capita terms is ct ikt iht f kt ht for all t 1023 Since the environment described here is similar to the standard neoclassical growth model equilibrium and optimal growth coincide For this reason I focus on the optimal growth problem the competitive equilibrium is discussed in Exercise 1012 The optimal growth prob lem involves the maximization of 1020 subject to 1021 1022 and 1023 The solution to this maximization problem can again be characterized by setting up the currentvalue Hamiltonian and using Theorems 713 and 714 To simplify the analysis observe first that since uc is strictly increasing 1023 will always hold as equality Then substitute for ct using this constraint and write the currentvalue Hamiltonian as Hkt ht ikt iht μkt μht uf kt ht iht ikt 1024 μhtiht δhht μktikt δkkt where now there are two control variables ikt and iht and two state variables kt and ht as well as two costate variables μkt and μht corresponding to the two constraints 1021 and 1022 The candidate solution from Theorem 713 satisfies Hikkt ht ikt iht μkt μht uct μkt 0 Hihkt ht ikt iht μkt μht uct μht 0 Hkkt ht ikt iht μkt μht fkkt htuct μktδk ρμkt μkt Hhkt ht ikt iht μkt μht fhkt htuct μhtδh ρμht μht lim t expρtμktkt 0 lim t expρtμhtht 0 The last two are the two transversality conditions since there are two state variables and two costate variables It can next be verified that Hkt ht ikt iht μkt μht is concave given the costate variables μkt and μht so that Theorem 714 can be applied to conclude that these conditions indeed generate an optimal path see Exercise 108 374 Chapter 10 Human Capital and Economic Growth Using 1036 this expression becomes κt 1γ 1af κt 1 κt 1f κt 1 1 ηf κtγ 1af κt κtf κt 1037 A steady state as usual involves a constant effective capitallabor ratio κt κ for all t Substituting this into 1037 yields κ 1 ηf κ 1038 which defines the unique positive steadystate effective capitallabor ratio κ since f is strictly concave Proposition 102 In the OLG economy with physical and human capital described above there exists a unique steady state with positive activity and the effective capitallabor ratio κ is given by 1038 This steadystate equilibrium is also typically stable but some additional conditions need to be imposed on f and γ to ensure stability see Exercise 1018 An interesting implication of this equilibrium is that the capitalskill kh complementarity in the production function F implies that a certain target level of physical to human capital ratio κ has to be reached in equilibrium In other words physical capital will not be too abundant relative to human capital and neither will human capital be excessive relative to physical capital Consequently this model also limits equilibrium imbalances between physical and human capital A possible and arguably attractive way of introducing such imbalances is to depart from perfectly competitive labor markets This is also useful for illustrating how the role of human capital can be quite different in models with imperfect labor markets 106 Physical and Human Capital with Imperfect Labor Markets In this section I analyze the implications of labor market frictions that lead to factor prices that differ from the ones used so far in particular prices that deviate from the competitive pricing formula 1032 The literature on labor market imperfections is vast and my purpose here is not to provide an overview For this reason I adopt the simplest representation In particular imagine that the economy is identical to that described in the previous section except that there is a continuum of firms with measure normalized 1 as well as a continuum of individuals also with measure 1 at any point in time and each firm can only hire one worker Let us first suppress time dependence to simplify notation Then the production function of each firm can be written as yj Fkj hi where yj refers to the output of firm j kj is its capital stock equivalently capital per worker since the firm is hiring only one worker and hi is the human capital of worker i employed by the firm This production function again satisfies Assumptions 1 and 2 The main departure from the models analyzed so far is in the structure for the labor market which is summarized next 1 Firms choose their physical capital level irreversibly incurring the cost Rkj where R is the market rate of return on capital and simultaneously workers choose their human capital level irreversibly 107 Human Capital Externalities 379 Intuitively each firm expects the average worker that it will be matched with to have higher human capital and since physical and human capital are complements it is more profitable for each firm to increase its physical capital investment Greater investments by firms in turn raise Fˆk h for each h in particular for ˆh2ˆk Since the earnings of type 2 workers is equal to λFˆk ˆh2ˆk their earnings also increase as a result of the response of firms to the change in the composition of the workforce These interactions correspond to human capital externalities because greater human capital investments by one group of workers increase the earnings of the remaining workers In fact human capital externalities in this economy are even stronger because the increase in ˆk also raises Fˆk ˆh2ˆkh and thus encourages further investments by type 2 workers This discussion is summarized in the following result Proposition 106 The positive activity equilibrium described in Proposition 103 exhibits human capital externalities in the sense that an increase in the human capital investments of a group of workers raises the earnings of the remaining workers 107 Human Capital Externalities The previous section illustrated how a natural form of human capital externalities can emerge in the presence of capitalskill complementarities combined with labor market imperfections This channel is not the only one through which human capital externalities may arise Many economists believe that the human capital stock of the workforce creates a direct nonpecu niary technological spillover on the productivity of each worker In The Economy of Cities Jane Jacobs 1970 for example argues for the importance of human capital externalities and suggests that the concentration of economic activity in cities is partly a result of these exter nalities and also acts as an engine of economic growth because it facilitates the exchange of ideas among workers and entrepreneurs In the growth literature a number of wellknown pa pers including those by Robert Lucas 1988 and Azariadis and Drazen 1990 suggest that such technological externalities are important and play a major role in the process of economic growth Human capital externalities are interesting in their own right For example when such external effects are present the competitive price system is likely to be inefficient Human cap ital externalities are also important for our understanding of the sources of income differences across countries The discussion of the contribution of physical and human capital to cross country income differences in Chapter 3 showed that differences in human capital are unlikely to account for a large fraction of crosscountry income differences unless external effects are important At this point it is therefore useful to briefly review the empirical evidence on the extent of human capital externalities Early work in the areain particular the paper by James Rauch 1993tried to measure the extent of human capital externalities by estimating quasi Mincerian wage regressions with the major difference that average human capital of workers in the local labor market is also included on the righthand side More specifically Rauch estimated models of the following form log Wjm XT jmβ γpSjm γeSm where Xjm is a vector of controls Sjm is the years of schooling of individual j working in labor market m and Sm is the average years of schooling of workers in labor market m Without this last term this equation would be similar to the standard Mincerian wage regressions discussed in Section 102 and we would expect an estimate of the private return 380 Chapter 10 Human Capital and Economic Growth to schooling γp between 6 and 10 When the average years of schooling Sm is also included in the regression its coefficient γe measures the external return to schooling in the same units For example if γe is estimated to be of the same magnitude as γp we would conclude that external returns to schooling are as important as private returns which would correspond to very large externalities Rauch estimated significant external returns with the magnitude of the external returns often exceeding that of the private returns External returns of this magnitude would imply that human capital differences could play a much more important role as a proximate source of crosscountry differences in income per capita than implied by the computations in Chapter 3 However Rauchs regressions exploited differences in average schooling levels across cities which could reflect many other factors that also directly affect wages For example wages are much higher in New York City than Ames Iowa but this difference is not only the result of the higher average education of New Yorkers A more convincing estimate of external returns necessitates a source of exogenous variation in average schooling Acemoglu and Angrist 2000 exploited differences in average schooling levels across states and cohorts resulting from changes in compulsory schooling and child labor laws These laws appear to have had a large effect on schooling especially at the high school margin Exploiting changes in average schooling in state labor markets driven by these law changes Acemoglu and Angrist estimate external returns to schooling that are typically about 12 and are statistically insignificant compared to private returns of about 10 These results suggest that there are relatively small human capital externalities in local labor markets This result is confirmed by Duflo 2004 using Indonesian data and by Ciccone and Peri 2006 on US data2 Overall the evidence appears to suggest that local human capital externalities are not very large and calibration exercises such as those in Chapter 3 that ignore these externalities are unlikely to lead to significant downward bias in the contribution of human capital to crosscountry income differences The qualification local in the above discussion has to be emphasized These estimates focus on local externalities originally emphasized by Jacobs Nevertheless if a few very talented scientists and engineers or other very skilled workers generate ideas that are then used in other parts of the country or even in the world economy there may exist significant global human capital externalities Such global external effects would not be captured by the currently available empirical strategies Whether such global human capital externalities are important is an interesting area for future research 108 The NelsonPhelps Model of Human Capital The discussion in this chapter so far has focused on the productivityenhancing role of human capital emphasized by Becker and Mincers seminal analyses This is arguably the most important role of human capital An alternative perspective on human capital is provided by Richard Nelson and Edmund Phelps in their short and influential paper Nelson and Phelps 1966 and also by Ted Schultz 1964 1975 According to this perspective the major role of human capital is not to increase productivity in existing tasks but to enable workers to cope with change disruptions and especially new technologies The NelsonPhelps view of human 2 Moretti 2004 also estimates human capital externalities and he finds larger effects This may be because he focuses on college graduation but it also partly reflects the fact that the source of variation that he exploits changes in age composition and the presence of landgrant colleges may have other effects on average earnings in an area 108 The NelsonPhelps Model of Human Capital 381 capital has played an important role in a variety of different literatures and features in a number of growth models Here I provide a simple presentation of the main ideas along the lines of their original model and a discussion of how this new dimension of human capital may enrich our view of its role in economic growth and development This model also acts as a stepping stone toward our study of technology adoption in Part VI Consider the following continuoustime model to illustrate the basic ideas Suppose that output in the economy is given by Yt AtL 1044 where L is the constant labor force supplying its labor inelastically and At is the technology level of the economy There is no capital and thus no capital accumulation decision and also no labor supply margin The only variable that changes over time is technology At Suppose that the world technological frontier is given by AFt This frontier might cor respond to technology in some other country or perhaps to the technological knowhow of scientists that has not yet been applied to production processes Suppose that AFt evolves exogenously according to the differential equation AFt AFt gF with initial condition AF0 0 Let the human capital of the workforce be denoted by h Notice that this human capital does not feature in the production function 1044 This case is an extreme one in which human capital does not play any productivityenhancing role Instead the only role of human capital in the current model is to facilitate the implementation and use of frontier technology in the production process In particular the evolution of the technology level of the country in question At is governed by the differential equation At gAt φhAFt with initial condition A0 0 AF0 The parameter g is strictly less than gF and measures the growth rate of technology At resulting from learningbydoing or other sources of productivity growth But first term is only one source of improvement in technology The other one comes from the second term and can be interpreted as improvements in technology because of implementation and adoption of frontier technologies The extent of the second source of improvement is determined by the average human capital of the workforce h The second source captures the abovementioned role of human capital in the context of adoption and adaptation of technology In particular suppose that φ is nondecreasing and satisfies φ0 0 and φh gF g 0 for all h h where h 0 This specification implies that the human capital of the workforce regulates the ability of the economy to cope with new developments embedded in the frontier technologies if the workforce has no human capital there is no adoption or implementation of frontier technologies and At grows at the rate g If in contrast h h there is rapid adoption of frontier technologies Since AFt expgFtAF0 the differential equation for At can be written as At gAt φhAF0 expgFt 109 Taking Stock 383 and human capital studied in Section 104 models and quantifies this effect It also provides a tractable framework in which physical and human capital investments can be studied Nev ertheless any effect of human capital differences resulting from differences in distortions or policies across countries should have shown up in the measurements in Chapter 3 The find ings there suggest that human capital differences though important can only explain a small fraction of crosscountry income differences unless there is a significant mismeasurement of the impact of human capital on productivity The second important issue connected to the role of human capital relates to the mea surement of the contribution of education and skills to productivity A possible source of mismeasurement of these effects is the presence of human capital externalities There are many compelling reasons why significant pecuniary or technological human capital externalities may exist Section 106 illustrated how capitalskill complementarities in imperfect labor markets can lead to pecuniary externalities Nevertheless existing evidence suggests that the extent of human capital externalities is rather limitedwith the important caveat that there might be global externalities that remain unmeasured Specific channels through which global external ities may arise are RD and technological progress which are the topics of the next part of the book An alternative source of mismeasurement of the contribution of human capital is differences in human capital quality There are significant differences in school and teacher quality even within a narrow geographical area so we may expect much larger differences across countries In addition most available empirical approaches measure human capital dif ferences across countries by using differences in formal schooling But the BenPorath model analyzed in Section 103 suggests that human capital continues to be accumulated even af ter individuals complete their formal schooling When human capital is highly rewarded we expect both higher levels of formal schooling and greater levels of onthejob investment Con sequently the BenPorath model suggests that there might be higher quality of human capital or greater amounts of unmeasured human capital in economies where the level of formal schooling is high If this is the case the empirical measurements reported in Chapter 3 may understate the contribution of human capital to productivity The exploration of this issue is an interesting area for future research The third set of novel issues raised by the modeling of human capital is the possibility of an imbalance between physical and human capital Empirical evidence suggests that physical and human capital are complementary Thus productivity will be high when the correct bal ance is achieved between physical and human capital Could equilibrium incentives lead to an imbalance whereby too much or too little physical capital is accumulated relative to human capital We saw that such imbalances are unlikely or rather short lived in models with com petitive labor markets However the analysis in Section 106 shows that they become a distinct possibility when factor prices do not necessarily reflect marginal products as in labor markets with frictions The presence of such imbalances might increase the impact of human capital on aggregate productivity The final issue relates to the role of human capital in technological change and technology adoption Section 108 presented the NelsonPhelps view of human capital which emphasizes the role of skills in facilitating the adoption and implementation of new technologies While this perspective is likely to be important in a range of situations it seems that in the absence of significant external effects this particular role of human capital should also not lead to a major mismeasurement of the contribution of human capital to aggregate productivity especially in the types of exercises reported in Chapter 3 This chapter contributes to our quest toward understanding the sources of economic growth and crosscountry income differences and offers a useful framework for understanding both physical and human capital accumulation decisions Our next task is to develop models for the other major proximate source of economic growth and income differences technology 384 Chapter 10 Human Capital and Economic Growth 1010 References and Literature The concept of human capital is due to Gary Becker 1965 Ted Schultz 1965 and Jacob Mincer 1974 The standard models of human capital used extensively in labor and other areas of economics have been developed by Becker 1965 Yoram BenPorath 1967 and Mincer 1974 These models have been the basis of the first three sections of this chapter Recently there has been a renewed interest in the BenPorath model among macroeconomists Recent contributions include Heckman Lochner and Taber 1998 Guvenen and Kuruscu 2006 and Manuelli and Seshadri 2006 These models make parametric assumptions CobbDouglas functional forms and try to gauge the quantitative implications of the BenPorath model for crosscountry income differences and for the evolution of wage inequality Caselli 2005 on the other hand argues that quality differences are unlikely to increase the contribution of human capital to aggregate productivity There is a large literature on returns to schooling As noted in the text and in Chapter 3 this literature typically finds that one more year of schooling increases earnings by about 610 see eg the survey in Card 1999 There is also a large literature on capitalskill complementarity The idea was first put forward and empirically supported in Griliches 1969 Katz and Autor 2000 summarize more recent evidence on capitalskill complementarities Technological human capital externalities are emphasized in Jacobs 1970 Lucas 1988 and Azariadis and Drazen 1990 while pecuniary human capital externalities were first dis cussed by Marshall 1890 who argued that increasing the geographic concentration of spe cialized inputs increases productivity since the matching between factor inputs and industries is improved Models of pecuniary human capital externalities are constructed in Acemoglu 1996 1997a The model with capitalskill complementarity and labor market imperfections is based on Acemoglu 1996 In that paper I provided a more detailed and microfounded model leading to similar results to those presented in Section 106 and derived the results on pecuniary externalities and human capital externalities discussed here The empirical literature on human capital externalities includes Rauch 1993 Acemoglu and Angrist 2000 Duflo 2004 Moretti 2004 and Ciccone and Peri 2006 The role of human capital in adapting to change and implementing new technologies was first suggested by Schultz 1975 in the context of agricultural technologies he emphasized the role of ability rather than human capital and stressed the importance of disequilibrium situations Nelson and Phelps 1966 formulated the same ideas and presented a simple model similar to that presented in Section 108 Foster and Rosenzweig 1995 provide evidence consistent with this role of human capital Benhabib and Spiegel 1994 and Aghion and Howitt 1998 also include extensive discussions of the NelsonPhelps view of human capital Recent macroeconomic models that feature this role of human capital include Galor and Tsiddon 1997 Greenwood and Yorukoglu 1997 Caselli 1999 Galor and Moav 2000 and Aghion Howitt and Violante 2004 1011 Exercises 101 Formulate state and prove the Separation Theorem Theorem 101 in an economy in discrete time 102 a Consider the environment discussed in Section 101 Write the flow budget constraint of the individual as at rat ct Wt 1011 Exercises 385 and suppose that there are credit market imperfections so that at 0 Construct an example in which Theorem 101 does not apply Can you generalize this example to the case in which the individual can save at the rate r but can only borrow at the rate r r b Now modify the environment in part a so that the instantaneous utility function of the individual is uct 1 lt where lt denotes total hours of work and labor supply at the market is equal to lt st so that the individual has a nontrivial leisure choice Construct an example in which Theorem 101 does not apply 103 Derive 109 from 108 104 Consider the model presented in Section 102 and suppose that the effective discount rate r varies across individuals eg because of credit market imperfections Show that individuals facing a higher r would choose lower levels of schooling What happens if you estimate the wage regression similar to 1012 in a world in which the source of disparity in schooling is differences in discount rates across individuals 105 Verify that Theorems 713 and 714 from Chapter 7 can be applied to the BenPorath and lead to 1014 as necessary and sufficient conditions for an optimal path of human capital investments Hint use a similar argument to that in Section 77 in Chapter 7 106 Consider the following variant of the BenPorath model in which the human capital accumulation equation is given by ht stφht δhht where φ is strictly increasing differentiable and strictly concave with st 0 1 Assume that individuals are potentially infinitely lived and face a Poisson death rate of ν 0 Show that the optimal path of human capital investments involves st 1 for some interval 0 T and then st s for t T 107 Modify the BenPorath model studied in Section 103 as follows Assume that the horizon is finite and suppose that φ0 Also suppose that φh0 δh1 expδhT where recall that δh is the rate of depreciation of human capital a Provide the necessary conditions for an interior solution Highlight how these necessary conditions should be modified to allow for corner solutions where st might take the value of 0 or 1 b Show that under these conditions the optimal path of human capital accumulation involves an interval 0 t of fulltime schooling with st 1for all t 0 t where t 0 followed by another interval of onthejob investment st 0 1 and finally an interval of no human capital investment that is st 0 for all t t T where t T Hint Suppose that the first part of the claim is not true and show that in this case the necessary conditions must hold as equality Combining the two necessary conditions derive a firstorder linear nonautonomous differential equation for the costate variable λt and solve this differential equation with the boundary condition λT 0 Then show that given the implied value for λ0 and the inequality above the necessary conditions at t 0 cannot be satisfied Next use the assumption that φ together with the fact that the costate variable λt is continuous and must satisfy λT 0 to prove that st must be equal to zero for some interval T ξ T Finally using these intermediate steps conclude that st must take intermediate values before this final interval 386 Chapter 10 Human Capital and Economic Growth c How do the earnings of the individual evolve over the life cycle d How would you test the implications of this model 108 Prove that the currentvalue Hamiltonian in 1024 is jointly concave in kt ht ikt iht 109 Prove that 1025 implies the existence of a relationship between physical and human capital of the form h ξk where ξ is uniquely defined strictly increasing and differentiable 1010 a Prove Proposition 101 b Show that the differential equation for consumption growth alternatively could have been written as ct ct 1 εuctfhkt ξkt δh ρ 1011 Derive 1026 1012 Consider the neoclassical growth model with physical and human capital discussed in Section 104 a Specify the consumer maximization problem in this economy b Define a competitive equilibrium specifying firm optimization and market clearing condi tions c Characterize the competitive equilibrium and show that it coincides with the solution to the optimal growth problem 1013 Introduce laboraugmenting technological progress at the rate g into the neoclassical growth model with physical and human capital discussed in Section 104 a Define a competitive equilibrium b Determine transformed variables that remain constant in a steadystate allocation c Characterize the steadystate equilibrium and the transitional dynamics d Why does faster technological progress lead to more rapid accumulation of human capital 1014 Characterize the optimal growth path of the economy in Section 104 subject to the additional constraints that ikt 0 and iht 0 1015 Prove that as long as Yt FKt Ht satisfies Assumptions 1 and 2 see Chapter 2 the inequality in 1030 holds 1016 Show that the equilibrium dynamics in Section 105 remain unchanged if δ 1 1017 Derive 1033 and 1034 1018 Provide conditions on f and γ such that the unique steadystate equilibrium in the model of Section 105 is locally stable 1019 Analyze the economy in Section 106 under the closedeconomy assumption Show that an increase in a1 for group 1 workers now creates a dynamic externality in the sense that current output increases which leads to greater physical and human capital investments in the next period 1020 Prove Proposition 105 11 FirstGeneration Models of Endogenous Growth T he models presented so far focus on physical and human capital accumulation and generate growth because of exogenous technological progress While such models are useful in thinking about sources of income differences among countries that have free access to the same set of technologies they do not generate sustained longrun growth of the country or of the world economy and have relatively little to say about sources of technology differences A systematic analysis of both crosscountry income differences and the process of world economic growth requires models in which technology choices and technological progress are endogenized This topic is discussed in Part IV While models in which technology evolves as a result of firms and workers decisions are most attractive in this regard sustained economic growth is possible in the neoclassical model as well I end this part of the book by investigating sustained endogenous economic growth in neoclassical or quasineoclassical models We have already encountered the AK model in Chapter 2 This model relaxed one of the key assumptions on the aggregate production function of the economy Assumptions 1 and 2 from Chapter 2 and prevented diminishing returns to capital Consequently continuous capital accumulation could act as the engine of sustained economic growth This chapter starts with a neoclassical version of the AK model which not only shows the possibility of endogenous growth in the neoclassical growth model but also provides us with a tractable model that has applications in diverse areas This model is not without shortcomings however The most important one is that capital is the only or essentially the only factor of production and asymptotically the share of national income accruing to capital tends to 1 I then present two different twosector endogenous growth models which behave very similarly to the baseline AK model but avoid this counterfactual prediction The first of these models incorporates physical and human capital accumulation and is thus a close cousin of the neoclassical growth model with physical and human capital studied in Section 104 The second which builds on the work by Rebelo 1991 is a substantially richer model and is also interesting since it allows investment and consumption goods sectors to have different capital intensities I conclude this chapter with a presentation of Paul Romers 1986a article that started the endogenous growth literature and rejuvenated the interest in economic growth among economists While Romers objective was to model technological change he achieved this by introducing technological spilloverssimilar to those we encountered in Chapter 10 387 111 The AK Model Revisited 391 Since A δθ 1θ1 ρθ1 n 0 the second term in this expression converges to zero as t But the first term is a constant Thus the transversality condition can only be satisfied if κ 0 Therefore 1114 implies that kt A δθ 1θ1 ρθ1 n1c0 expθ1A δ ρt 1115 k0 expθ1A δ ρt where the second line immediately follows from the fact that capital is equal to k0 at t 0 Therefore capital and output grow at the same rate as consumption Equation 1115 pins down the initial level of consumption per capita as c0 A δθ 1θ1 ρθ1 nk0 1116 Note that in this simple AK model growth is not only sustained but is also endogenous in the sense of being affected by underlying parameters For example consider an increase in the discount rate ρ Recall that in the Ramsey model such a change only influenced the level of income per capitait could have no effect on the growth rate which was determined by the exogenous laboraugmenting rate of technological progress Here it is straightforward to verify that an increase in ρ reduces the growth rate households become less patient and the rate of capital accumulation declines Since capital accumulation is the engine of growth the equilibrium rate of growth will decline Similarly changes in A and θ affect the levels and growth rates of consumption capital and output Finally let us calculate the equilibrium saving rate It is defined as total investment which is equal to increase in capital plus replacement investment divided by output Thus we have s Kt δKt Yt ktkt n δ A A ρ θn θ 1δ θA 1117 where the last equality exploits the fact that ktkt A δ ρθ This equation implies that the saving rate which was taken as constant and exogenous in the basic Solow model is again constant over time but now depends on preferences and technology Proposition 111 Consider the AK economy with a representative household with prefer ences given by 111 and the production technology given by 116 Suppose that condition 1112 holds Then there exists a unique equilibrium path in which consumption capital and output per capita all grow at the same rate g A δ ρθ 0 starting from any initial positive capital stock per capita k0 0 and the saving rate is given by 1117 One important implication of the AK model is that since all markets are competitive there is a representative household and there are no externalities the competitive equilibrium will be Pareto optimal This can be proved either using the First Welfare Theorem Theorem 56 or by directly constructing the optimal growth solution 392 Chapter 11 FirstGeneration Models of Endogenous Growth Proposition 112 Consider the AK economy with a representative household with prefer ences given by 111 and the production technology given by 116 Suppose that condition 1112 holds Then the unique competitive equilibrium is Pareto optimal Proof See Exercise 112 1114 The Role of Policy It is straightforward to incorporate policy differences into this framework and investigate their implications for the equilibrium growth rate Suppose that there is a tax rate equal to τ on capital income as in Chapter 8 The budget constraint of the representative household then becomes at 1 τrt nat wt ct 1118 Repeating the analysis above immediately implies that this tax adversely affects the growth rate of the economy which becomes see Exercise 115 g 1 τA δ ρ θ 1119 Moreover it can be calculated that the saving rate is s 1 τA ρ θn 1 τ θδ θA 1120 which is a decreasing function of τ provided that A δ 0 Therefore in this model the equilibrium saving rate responds endogenously to policy In addition since the saving rate is constant differences in policies lead to permanent differences in the rate of capital accumu lation This observation has an important implication While in the baseline neoclassical growth model even reasonably large differences in distortions eg eightfold differences in τ could only have limited effects on differences in income per capita here even small differences in τ can have very large effects In particular consider two economies with the same technology and preferences but with different constant tax rates on capital income τ and τ τ Then for any τ τ lim t Yτ t Yτ t 0 where Yτ t denotes aggregate output in the economy with tax τ at time t Therefore even small policy differences can have very large effects in the long run So why does the literature focus on the inability of the standard neoclassical growth model to generate large differences rather than the possibility that the AK model can generate arbitrarily large differences The reason is twofold first as noted above the AK modelwith no diminishing returns and the share of capital in national income asymptoting to 1is not viewed as a good approximation to reality Second and related to the discussion in Chapter 1 most economists believe that the relative stability of the world income distribution in the postwar era makes it more attractive to focus on models in which there is a stationary world income distribution rather than on models in which small policy differences can lead to permanent growth differences Whether this last belief is justified is in part an empirical question 394 Chapter 11 FirstGeneration Models of Endogenous Growth Once again using Theorem 713 we can generate the following candidate solution to this maximization problem see Exercise 118 μat μht μt for all t wt δh rt for all t ct ct 1 θ rt ρ for all t 1125 Intuitively there are no constraints on human and physical capital investments thus the shadow values of these two different types of investments have to be equal at all points in time as stated in the first condition in 1125 This in turn yields the second condition in 1125 equating the rates of return on human and physical capital The third condition is the standard Euler equation It can be verified that the currentvalue Hamiltonian is concave and satisfies the sufficiency conditions in Theorem 714 Therefore a solution to the conditions in 1125 necessarily solves the representative households maximization problem Moreover with the same argument as in Exercise 811 this solution is unique Combining 1125 with 1124 yields f kt δk f kt ktf kt δh for all t Since the lefthand side is decreasing in kt while the righthand side is increasing the effective capitallabor ratio must satisfy kt k for all t Proposition 113 Consider the AK economy with physical and human capital with pref erences given by 111 and the production technology given by 1121 Let k be given by f k δk f k kf k δh 1126 Suppose that f k ρ δk 1 θf k δ δk Then in this economy there exists a unique equilibrium path in which consumption human capital physical capital and output all grow at the same rate g f k δk ρθ 0 starting from any initial conditions where k is given by 1126 The share of capital in national income is constant and less than 1 at all times Proof See Exercise 119 The advantage of the economy studied here compared to the baseline AK model is that it generates a stable factor distribution of income with a significant fraction of national income accruing to labor as rewards to human capital Consequently the current model cannot be criticized on the basis of generating counterfactual results on the capital share of GDP A similar analysis to that in the previous section also shows that the current model generates longrun differences in growth rates from small policy differences Therefore it can account for arbitrarily large differences in income per capita across countries Nevertheless it does so partly by generating large human capital differences across countries As such the empirical mechanism through which these large crosscountry income differences are generated may again not fit with the empirical patterns discussed in Chapter 3 Moreover given substantial differences in policies across economies in the postwar period like the baseline AK economy the current model suggests significant changes in the world income distribution whereas the evidence in Chapter 1 points to a relatively stable postwar world income distribution 113 The TwoSector AK Model 395 113 The TwoSector AK Model The models studied in the previous two sections are attractive in many respects they generate sustained growth and the equilibrium growth rate responds to policy underlying preferences and technology Moreover these are very close cousins of the neoclassical model In fact as argued there the endogenous growth equilibrium is Pareto optimal One unattractive feature of the baseline AK model of Section 111 is that all national income accrues to capital Essentially it is a onesector model with only capital as the factor of production This limitation makes it difficult to apply this model to realworld situations The model in the previous section avoids this problem but at some level it does so by creating another factor of production that accumulates linearly so that the equilibrium structure is again equivalent to the onesector AK economy Therefore in some deep sense the economies of both sections are onesector models Another important shortcoming in addition to this onesector property is that these models do not delineate the key feature driving sustained growth What is important for sustained growth is not that the production technology is AK but instead the related feature that the accumulation technology is linear In this section I discuss a richer two sector model of neoclassical endogenous growth based on Rebelo 1991 This model generates constant factor shares in national income without introducing human capital accumulation It also illustrates the role of differences in the capital intensity of the production functions of consumption and investment goods The preference and demographics are the same as in Section 111 in particular 111 115 apply as before but with a slightly different interpretation for the interest rate in 114 as discussed below Moreover to simplify the analysis suppose that there is no population growth that is n 0 and that the total amount of labor in the economy L is supplied inelastically The main difference is in the production technology Rather than a single good used for consumption and investment let us now envisage an economy with two sectors The first sector produces consumption goods with the following technology Ct BKCtαLCt1α 1127 where the subscript C denotes that these are capital and labor used in the consumption sector which has a CobbDouglas technology In fact the CobbDouglas assumption here is quite important in ensuring that the share of capital in national income is constant see Exercise 1112 The capital accumulation equation is given by Kt It δKt where It denotes investment Investment goods are produced in the second sector which has a different technology from 1127 It AKIt 1128 The distinctive feature of the technology for the investment goods sector 1128 is that it is linear in the capital stock and does not feature labor This assumption is an extreme version of one often made in twosector models that the investment good sector is more capital intensive than the consumption good sector In the data there seems to be some support for this assumption though the capital intensities of many sectors have been changing over time as the nature of consumption and investment goods has changed 398 Chapter 11 FirstGeneration Models of Endogenous Growth Moreover with the same arguments as in Section 112 it can be shown that there are no transitional dynamics in this economy This analysis establishes the following proposition Proposition 114 In the abovedescribed twosector neoclassical economy there exists a unique equilibrium where for any K0 0 consumption and labor income grow at the constant rate given by 1134 while the capital stock grows at the constant rate given by 1133 Policy analysis in this model is similar to that in the basic AK model taxes on investment income or other policies that discourage investment will depress growth One important implication of this model that differs from the neoclassical growth model is that there is continuous capital deepeningCapital grows at a faster rate than consumption and output Whether this feature is realistic is debatable The Kaldor facts discussed in Chapter 2 include constant capitaloutput ratio as one of the requirements of balanced growth The balanced growth here does not have this feature For much of the twentieth century the capitaloutput ratio appears to have been constant but it has been increasing steadily over the past 30 years Part of the reason is relative price adjustments New capital goods are of higher quality which needs to be incorporated in calculating the capitaloutput ratio These calculations have only been performed in the recent past which may explain why capital output ratio has been constant in the earlier part of the twentieth century but not recently Thus it is not clear whether a constant or an increasing capitaloutput ratio is a better approximation to reality 114 Growth with Externalities The model that started much of endogenous growth theory and revived economists interest in economic growth was presented in Paul Romers 1986a paper Romers objective was to model the process of knowledge accumulation He realized that this would be difficult in the context of a competitive economy His initial solution later updated and improved in his and others work during the 1990s was to consider knowledge accumulation to be a byproduct of capital accumulation In other words Romer introduced technological spillovers similar to the human capital externalities discussed in Chapter 10 While arguably crude this approach captures an important dimension of knowledge namely that knowledge is a largely nonrival goodonce a particular technology has been discovered many firms can make use of this technology without preventing others from using the same knowledge Nonrivalry does not imply knowledge is also nonexcludable which would make it a pure public good A firm that discovers a new technology may use patents or trade secrecy to prevent others from using it for example to gain a competitive advantage These issues are discussed in Part IV of the book For now it suffices to note that some of the important characteristics of knowledge and its role in the production process can be captured in a reducedform way by introducing technological spillovers I next discuss a version of the model in Romers 1986a paper that introduces such technological spillovers as the engine of economic growth While the type of technological spillovers used in this model are unlikely to be the engine of sustained growth in practice the model is a good starting point for our analysis of endogenous technological progress since its similarity to the baseline AK economy makes it a tractable model of knowledge accumulation 400 Chapter 11 FirstGeneration Models of Endogenous Growth discussed in detail in Part IV this property is a very common feature of models of endogenous growth This feature also highlights that in this class of models we can no longer appeal to the Representative Firm Theorem Theorem 54 Thus I specified the production function and equilibrium behavior of each firm in the economy More generally Theorem 54 applies when there are no externalities and all firms are pricetaking whereas almost all models of endogenous technologystarting with the Romer model in this sectioninvolve either technological externalities or monopolistic competition Substituting for 1136 into 1135 and using the fact that all firms are functioning at the same capitaleffective labor ratio and that F is homogeneous of degree 1 the production function of each firm can be written as Yt FKt BKtL Since the measure of firms is equal to 1 this equation also gives aggregate output Using the fact that F is homogeneous of degree 1 we can write Yt Kt F1 BL f L Output per capita is therefore yt Yt L Yt Kt Kt L kt f L where again kt KtL is the capitallabor ratio in the economy Marginal products and factor prices can then be expressed in terms of the normalized production function now f L wt Kt f L 1137 and the rental rate of capital is constant at Rt R f L L f L 1138 1142 Equilibrium A competitive equilibrium is defined similarly to that in the neoclassical growth model as a path of consumption and capital stock for the economy Ct Kt t0 that maximizes the utility of the representative household and wage and rental rates wt Rt t0 that clear markets The important feature is that because the knowledge spillovers in 1136 are external to each firm equilibrium factor prices are given by 1137 and 1138that is they do not price the role of the capital stock in increasing future productivity Since the market rate of return is rt Rt δ it is also constant The usual consumption Euler equation 114 then implies that consumption must grow at the constant rate given by 114 Growth with Externalities 401 g C 1 θ f L L f L δ ρ 1139 It is also clear that capital grows at the same rate as consumption so the rate of capital output and consumption growth are all given by 1139 see Exercise 1115 Let us assume that f L L f L δ ρ 0 1140 so that there is positive growth but the growth is not fast enough to violate the transversality condition finiteness of utility 1 θ f L L f L δ ρ 1141 Proposition 115 Consider the Romer model with physical capital externalities Suppose that conditions 1140 and 1141 are satisfied Then there exists a unique equilibrium path where starting with any level of capital stock K0 0 capital output and consumption grow at the constant rate 1139 Proof Much of this proposition is proved in the preceding discussion You are asked to verify the transversality conditions and show that there are no transitional dynamics in Exercise 1116 This model therefore provides us with the first example of endogenous technological change The technology of the economy At as given in 1136 evolves endogenously as a result of the investment decisions of firms Consequently the growth rate of the economy is endogenous even though none of the firms purposefully invest in research or acquiring new technologies Population must be constant in this model because of a scale effect Since f L L f L is always increasing in L by Assumption 1 a higher population labor force L leads to a higher growth rate The scale effect refers to this relationship between population and the equilibrium rate of economic growth If population were growing then the economy would not admit a steady state BGP and the growth rate of the economy would increase over time with output reaching infinity in finite time thus violating the finiteness of household utility and the transversality condition The implications of positive population growth are discussed further in Exercise 1118 Scale effects and how they can be removed are discussed in detail in Chapter 13 1143 Pareto Optimal Allocations Given the presence of externalities it is not surprising that the decentralized equilibrium characterized in Proposition 115 is not Pareto optimal To characterize the allocation that maximizes the utility of the representative household let us again set up the currentvalue Hamiltonian and look for a candidate path that satisfies the conditions in Theorem 713 see Exercise 1117 The per capita accumulation equation for this economy can be written as kt f Lkt ct δkt The currentvalue Hamiltonian is ˆHk c μ ct1θ 1 1 θ μt f Lkt ct δkt 115 Taking Stock 403 connected to technological progress Except for the Romer model of Section 114 the models studied in this chapter do not feature technological progress This omission does not imply that they are necessarily inconsistent with the data As already noted in Chapter 3 there is a lively debate about whether the observed total factor productivity growth is partly a result of the mismeasurement of inputs If so it could be that much of what we measure as technological progress is in fact capital deepening which is the essence of economic growth in the AK model and its variants Consequently the debate about the measurement of total factor productivity has important implications for what types of models we should use for thinking about world economic growth and crosscountry income differences In the final analysis however it seems unlikely that some form of technological progress has not played an important role in the process of economic growth over the past 200 years The discussion in this chapter has also revealed another important tension Chapters 3 and 8 demonstrated that the neoclassical growth model or the simpler Solow growth model has difficulty in generating the very large income differences across countries that we observe in the data Even if we choose quite large differences in crosscountry distortions eg eightfold differences in effective tax rates the implied steadystate differences in income per capita are relatively modest As noted before this observation has generated a large literature that seeks reasonable extensions of the neoclassical growth model in order to derive more elastic responses to policy distortions and so provide a better mapping of these models to differences across countries The models presented in this chapter like those that we will encounter in the next part of the book suffer from the opposite problem They imply that even small differences in policies technological opportunities or other characteristics of societies lead to permanent differences in longrun growth rates Consequently these models can explain very large differences in living standards from small policy institutional or technological differences But this ability is both a blessing and a curse The byproduct of generating large crosscountry differences from small policy or technological differences is that these models also predict an everexpanding world income distributioncountries with different characteristics should grow at permanently different rates The relative stability of the world income distribution in the postwar era pointed out in Chapter 1 is then a challenge to the baseline endogenous growth models Although one can debate whether endogenous growth models with each country growing at a potentially different longrun rate are a better approximation to postwar data than models in which there is a stable world income distribution at some level this debate is not particularly interesting First there is more to understanding the nature of the growth process and the role of technological progress than simply looking at the postwar data As illustrated in Chapter 1 the era of divergence is not the past 60 years but the nineteenth century Therefore we should not just focus on postwar data but also confront our growth models with historical data These data are both richer and more informative about the era when the divergence across countries began Second as discussed in Chapters 18 and 19 most economies do not generate their own technology by RD but largely import or adopt these technologies from more advanced nations or from the world technology frontier They also engage in substantial trade with other coun tries Once technological financial and trade interdependences across countries are modeled the sharp distinction between models of exogenous and endogenous growth disappears This point again reiterates the potential pitfalls in modeling each country as an island especially when we wish to map these models to data Having noted the importance of understanding inter dependences across nations in Part IV I follow the established literature and develop the models of endogenous technological progress without international interdependences only returning to these themes in Chapters 18 and 19 PART IV ENDOGENOUS TECHNOLOGICAL CHANGE T his part of the book focuses on models of endogenous technological change Chapter 12 discusses various approaches to technological change and provides a brief overview of some workhorse models from the literature on industrial organization Chapters 13 and 14 present the baseline endogenous technological progress models developed by Romer Grossman and Helpman and Aghion and Howitt Chapter 15 considers a richer class of models in which the direction of technological changefor example which factors technological change will augment or complementis also endogenous The models presented in this part of the book are useful for two related purposes First by making technological progress respond to incentives market structure and policies they allow us to develop a more satisfactory framework for the study of crosscountry and overtime differences in economic performance Second they provide a tractable approach to modeling sustained growth in which technological progress acts as the engine of longrun growth 12 Modeling Technological Change W e have so far investigated models of economic growth of the exogenous and endoge nous varieties But economic growth has not resulted from technological change It has been exogenous sustained by linear capital accumulation or taken place as a byproduct of knowledge spillovers Since our purpose is to understand the process of eco nomic growth models in which growth results from technological progress and technological change itselfas a consequence of purposeful investments by firms and individualsare much more attractive These models not only endogenize technological progress but they also relate the process of technological change to market structure and to policies concerning antitrust competition and intellectual property rights They also enable us to discuss issues of directed technological change In this chapter I begin with a brief discussion of different conceptions of technological change and provide some foundations for the models that come later 121 Different Conceptions of Technology 1211 Types of Technological Change The literature on technological change often distinguishes among different types of innova tions A first common distinction is between process and product innovations The latter refers to the introduction of a new product eg the introduction of the first DVD player The former is concerned with innovations that reduce the costs of production of existing products eg the introduction of new machines to produce existing goods Models of process and product in novations are often mathematically similar Nevertheless the distinction between the two types of innovations is still useful in mapping these theories to data Process innovations that introduce higher quality versions of existing products or generate a lower cost technology to produce an existing product might be more important in practice than innovations reducing costs in production processes The introduction of a better DVD player and the innovation to manufacture an existing DVD player at a lower cost would be 411 412 Chapter 12 Modeling Technological Change examples of such process innovations These innovations typically lead to the replacement of older vintages of the same good or machine and to potential competition between existing producers and the innovator In this context one might additionally wish to distinguish between the introduction of a higher quality DVD player and the production of a cheaper DVD player because heterogeneous consumers may have differential willingness to pay for quality than for quantity Issues of differential willingness to pay for quality are important in the theory of industrial organization and for constructing accurate qualityadjusted price indices However most growth models represent the consumer side by a representative household and implicitly assume perfect substitution between quality and quantity These features create a close connection between process innovations that increase the quality of existing products and those that reduce the costs of production The following example illustrates why in the context of typical growth models quality improvements and cost reductions are essentially equivalent Example121 Consider an economy admitting a representative household with preferences Uqcq y q where y stands for a generic good perhaps representing all other goods and c is a particular consumption good available in different qualities Here cq denotes the amount consumed of the vintage of quality q The utility function is also conditioned on q This specification with q multiplying cq implies that quality and quantity are perfect substitutes so that higher quality products increase the effective units of consumption This assumption is typical in growth models though it is clearly restrictive the consumption use of five 1GHz computers would not give the same services as the use of a single 5GHz computer Let the budget constraint of the representative household be pqcq y m where pq is the price of the good of quality vintage q the price of the generic good is normalized to 1 and m denotes the resources available to the consumer The problem of the household can then be equivalently written as max xqy Uxq y q subject to pq q xq y m where xq qcq corresponds to the effective units of consumption of good c It is straight forward to see from this problem formulation that proportional increases in quality q and declines in the price pq have the same effects on the effective units of consumption and on welfare This observation justifies the claim above that in many models process innovations reducing costs of production and quality improvements have identical effects Another important distinction in the technological change literature is between macro and micro innovations see Mokyr 1990 The first refers to radical innovations including the introduction of generalpurpose technologies such as electricity or the computer which potentially change the organization of production in many different product lines In contrast micro innovations refer to the more common innovations that introduce newer models of existing products improve the quality of a certain product line or simply reduce costs Most of the innovations modeled below can be viewed as micro innovations though most endogenous technology models do not make an explicit distinction between micro and macro innovations Empirically it appears that micro innovations are responsible for most productivity growth 121 Different Conceptions of Technology 413 though they often build upon some macro or generalpurpose innovation such as the invention of electricity or the microchip see the evidence and discussion in Abernathy 1978 and Freeman 1982 1212 A Production Function for Technology A potentially confusing issue in the study of technological progress is how to conceptualize the menu of technologies available to firms or individuals Since our purpose is to develop models of endogenous technology firms andor individuals must have a choice among different types of technologies and greater effort research spending and investment should lead to the invention of better technologies These requirements imply that there must exist a meta production function a production function over production functions that determines how new technologies are generated as a function of inputs In what follows I refer to this meta production function as the innovation possibilities frontier or as the RD production function While a meta production function may appear natural to some there are various economists and social scientists who do not find this approach compelling Their argument against the production function approach to technology is that by its nature innovation includes the discovery of the unknown how could we put the unknown in the context of a production function where inputs go in and outputs come out in a deterministic fashion Although this question has some descriptive merit in the sense that describing the discovery of new technologies with a production function obscures some important details of the inno vation process the concern is largely irrelevant There is no reason to assume that the meta production function for technology is deterministic Both the success of a research project and the quality of the research output conditional on success can be uncertain corresponding to a meta production function with stochastic output Therefore the production function approach to technology is not particularly restrictive as long as uncertain outcomes are allowed and we are willing to assume that individuals can make calculations about the effect of their actions on the probability of success and quality of the research project Naturally some observers may argue that such calculations are not possible But without such calculations we would have little hope of modeling the process of technological change or technology adoption Since our objective is to model purposeful innovations assuming that individuals and firms can make such calculations is natural and this assumption is equivalent to assuming the existence of a meta production function for technologies 1213 Nonrivalry of Ideas Another important aspect of technology is emphasized in Paul Romers work As already dis cussed in the previous chapter Romers 1986a first model of endogenous growth introduced increasing returns to scale to physical capital accumulation The justification for this assump tion was that the accumulation of knowledge could be considered a byproduct of the economic activities of firms Later work by Romer which will be studied in the next chapter took a very different approach to modeling the process of economic growth but the same key idea is present in both his early and later work the nonrivalry of ideas matters By nonrivalry Romer means that the use of an idea by one producer to increase efficiency does not preclude its use by others While the same unit of labor or capital cannot be used by multiple producers the same idea can be used by many potentially increasing everybodys productivity Let us consider a production function of the form FK L A with A denoting technology Romer argues that an important part of this technology is the ideas or blueprints concerning how to produce new goods how to increase quality or how to reduce costs 414 Chapter 12 Modeling Technological Change Economists are generally comfortable assuming that the production function FK L A exhibits constant returns to scale in capital and labor K and L and I adopted this assumption throughout the first three parts of the book For example replication arguments can be used to justify this type of constant returns to scale unless land is an important factor of production when capital and labor double the society can always open a replica of the same production facility and in the absence of externalities this new facility will at least double output Romer argues that endogenizing A naturally leads to increasing returns to scale to all three inputs K L and A To understand why nonrivalry is important here imagine that A is like any other input Then the replication argument would require the new production facility to replicate A as well and thus we should expect constant returns to scale when we vary all three inputs K L and A Instead when ideas are nonrival the new production facility does not need to recreate or replicate A because it is already available for all firms to use Then FK L A will exhibit constant returns in K and L and increasing returns to scale in K L and A Therefore the nonrivalry of ideas and increasing returns are intimately linked This has motivated Romer and others to develop endogenous growth models with various conceptions of technology during the 1980s and 1990s But the nonrivalry of ideas and the resulting increasing returns to scale have been a central element in most of these models Another important implication of the nonrivalry of ideas is the market size effect If once discovered an idea can be used as many times as one wishes then the size of its potential market will be a crucial determinant of whether it is profitable to implement it and whether to research it in the first place This idea is well captured by a famous quote from Matthew Boulton James Watts business partner who wrote to Watt It is not worth my while to manufacture your engine for three countries only but I find it very well worth my while to make it for all the world quoted in Scherer 1984 p 13 To see why nonrivalry is related to the market size effect imagine another standard rival input that is also essential for production A greater market size does not necessarily induce firms to use this alternate input more intensively since a greater market size and thus greater sales means that more of this input has to be used It is the fact that nonrival ideas can be embedded in as many units as desired without incurring further costs that makes the market size effect particularly important In the next section I discuss some empirical evidence on the importance of the market size effect Nevertheless the nonrivalry of ideas does not make ideas or innovations pure public goods Recall that pure public goods are both nonrival and nonexcludable While some discoveries may be by their nature nonexcludable eg the discovery that providing excessively high powered incentives to CEOs in the form of stock options leads to counterproductive incentives and cheating most discoveries can be made partly excludable by patenting An important aspect of the process of technological change is the protection of intellectual discoveries from rivals For this reason intellectual property rights protection and patent policy often play an important role in models of technological progress 122 Science and Profits Another major question for the economic analysis of technological change is whether innova tion is mainly determined by scientific constraints and stimulated by scientific breakthroughs in particular fields or whether it is at least in part driven by profit motives Historians and economists typically give different answers to this question Many historical accounts of technological change come down on the side of the sciencedriven view emphasizing the autonomous progress of science and how important breakthroughsperhaps macro innova 122 Science and Profits 415 tions as discussed abovehave taken place as scientists build on one anothers work with little emphasis on profit opportunities For example in his History of Modern Computing Ceruzzi emphasizes the importance of a number of notable scientific discoveries and the role played by certain talented individuals rather than profit motives and the potential market for computers He points out for example how important developments took place despite the belief of many important figures in the development of the computer such as Howard Aiken that there would not be a demand for more than a handful of personal computers in the United States Ceruzzi 2003 p 13 Many economic historians eg Rosenberg 1976 similarly argue that a key de terminant of innovation in a particular field is the largely exogenous growth of scientific and engineering knowledge in that field In contrast most economists believe that profit opportunities play a much more important role and that the demand for innovation is the key to understanding the process of technological change John Stuart Mill provides an early and clear statement of this view in his Principles of Political Economy when he writes The labor of Watt in contriving the steamengine was as essential a part of production as that of the mechanics who build or the engineers who work the instrument and was undergone no less than theirs in the prospect of a renumeration from the producers quoted in Schmookler 1966 p 210 In fact profits were very much in the minds of James Watt and his business partner Matthew Boulton as the previous quote illustrates James Watt also praised the patent system for the same reasons arguing that an engineers life without patent was not worthwhile quoted in Mokyr 1990 p 248 The view that profit opportunities are the primary determinant of innovation and invention is articulated by Griliches and Schmookler 1963 and then most forcefully and eloquently by Schmooklers seminal study Invention and Economic Growth Schmookler 1966 p 206 writes that invention is largely an economic activity which like other economic activities is pursued for gain Schmookler concludes from his analysis of innovations in petroleum refining papermaking railroad construction and farming that there is no evidence that past breakthroughs have been the major factor in new innovations He Schmookler 1966 p 199 argues Instead in hundreds of cases the stimulus was the recognition of a costly problem to be solved or a potentially profitable opportunity to be seized If potential profits are a main driver of technological change then the market size that will be commanded by new technologies or products will be a key determinant of innovations A greater market size increases profits and makes innovation and invention more desirable To emphasize this point Schmookler called two of his chapters The amount of invention is governed by the extent of the market Schmooklers argument is most clearly illustrated by the example of the horseshoe He documented that there was a very high rate of innovation throughout the late nineteenth and early twentieth centuries in the ancient technology of horseshoe making and no tendency for inventors to run out of additional improvements On the contrary inventions and patents increased because demand for horseshoes was high Innovations came to an end only when the steam traction engine and later internal combustion engine began to displace the horse Schmookler 1966 p 93 The classic study by Griliches 1957 on the spread of hybrid seed corn in US agriculture also provides support for the view that technological change and technology adoption are closely linked to profitability and market size A variety of more recent papers also reach similar conclusions Newell Jaffee and Stavins 1999 show that between 1960 and 1980 the typical air conditioner sold at Sears became significantly cheaper but not much more energyefficient On the other hand between 1980 416 Chapter 12 Modeling Technological Change and 1990 there was little change in costs but air conditioners became much more energy efficient which they argue was a response to higher energy prices This example provides a clear case of the pace and type of innovation responding to profit incentives In a related study Popp 2002 documents evidence consistent with this pattern and finds a strong positive correlation between patents for energysaving technologies and energy prices Evidence from the pharmaceutical industry also illustrates the importance of profit incen tives and especially of the market size on the rate of innovation Finkelstein 2004 exploits three different policy changes affecting the profitability of developing new vaccines against six infectious diseases She finds that increases in vaccine profitability resulting from these policy changes are associated with a significant increase in the number of clinical trials to develop new vaccines against the relevant diseases Acemoglu and Linn 2004 look at demographi cally driven exogenous changes in the market size for drugs and find a significant response in the rate of innovation to these changes in market sizes Overall existing evidence suggests that market size is a major determinant of innovation incentives and the amount and type of technological change This evidence motivates the types of models presented below in which technological change is an economic activity and responds to profit incentives 123 The Value of Innovation in Partial Equilibrium Let us now turn to the analysis of the value of innovation and RD to a firm The equilibrium value of innovation and the difference between this private value and the social value defined as the value to a social planner internalizing externalities plays a central role in our analysis As emphasized at the beginning of the book economic growth is a process we can only understand in the context of dynamic general equilibrium analysis Nevertheless it is useful to start our investigation of the value of innovation in partial equilibrium where much of the industrial organization literature starts Throughout this section I consider a single industry Firms in this industry have access to an existing technology to produce one unit of the product at the marginal cost ψ 0 in terms of some numeraire The demand side of the industry is modeled with a demand curve Q Dp where p is the price of the product and Q is the demand at this price Throughout I assume that Dp is strictly decreasing differentiable and satisfies the following conditions Dψ 0 and εDp pDp Dp 1 The first condition ensures that there is positive demand when price is equal to marginal cost and the second ensures that the elasticity of demand εDp is greater than 1 so that there always exists a welldefined profitmaximizing monopoly price Moreover this elasticity is less than infinity so that the monopoly price is above marginal cost In this chapter as in the rest of the book when there are economies with monopolistic or oligopolistic competition equilibrium refers to Nash Equilibrium or Subgame Perfect Nash Equilibrium when the game in question is dynamic A brief review of these concepts is contained in Appendix C 418 Chapter 12 Modeling Technological Change For example the cost of innovation μ could be arbitrarily small but still positive and the productivity gain from innovation λ could be arbitrarily large 1232 Some Caveats The above example illustrates the problem of innovation under pure competition The main problem is the inability of the innovator to exclude others from using this innovation One way of ensuring such excludability is via the protection of intellectual property rights or a patent system which will create ex post monopoly power for the innovator This type of intellectual property right protection is present in most countries and plays an important role in many of the models we study below Before embarking on an analysis of the implications of ex post monopoly power of innova tors some caveats are worth noting First even without patents trade secrecy may be sufficient to provide some incentives for innovation Second firms may engage in innovations that are only appropriate for their own firm making their innovations de facto excludable For exam ple imagine that at the same cost the firm can develop a new technology that reduces the marginal cost of production by only λ λ But this technology is specific to the needs and competencies of the current firm and cannot be used by any other or alternatively λλ is the proportional cost of making the innovation excludable The adoption of this technology may be profitable for the firm since the specificity of the innovation acts exactly like patent protec tion see Exercise 125 Therefore some types of innovations in particular those protected by trade secrecy can be undertaken under pure competition Finally a number of authors have recently argued that innovations in competitive markets are possible One strand of the literature shows that competitive growth may originate because firms are able to replicate new technologies eg copy software or compact discs and sell them to competitors during a certain interval of time before being imitated by others see eg Boldrin and Levine 2003 Another strand incorporates diminishing returns at the firm level which creates profits and potential innovation incentives even for pricetaking firms see eg Hellwig and Irmen 2001 This recent work on competitive growth constitutes a promising direction for future research though existing models generate innovations and sustained growth in competitive equilibria only under somewhat special assumptions 1233 Innovation and Ex Post Monopoly Let us now return to the simple environment introduced above and suppose that if firm 1 undertakes a successful innovation it can obtain a fully enforced patent Firm 1 then has a better technology than the rest of the firms and possesses ex post monopoly power This monopoly power enables the firm to earn profits from the innovation potentially encouraging its research activity in the first place This is the basis of the claim by Schumpeter Arrow Romer and others that there is an intimate link between ex post monopoly power and innovation Let us now analyze this situation in a little more detail It is useful to separate two cases 1 Drastic innovation a drastic innovation corresponds to a sufficiently high value of λ such that firm 1 becomes an effective monopolist after the innovation To determine which values of λ lead to a situation of this sort let us first suppose that firm 1 does indeed act like a monopolist Then chooses its price to maximize πI 1 Dpp λ1ψ μ 123 The Value of Innovation in Partial Equilibrium 419 Clearly this maximization gives the following standard monopoly pricing formula see Exercise 121 pM λ1ψ 1 εDpM1 122 We say that the innovation is drastic if pM ψ It is clear that this is the case when λ λ 1 1 εDpM1 When the innovation is drastic firm 1 can set its unconstrained monopoly price pM and capture the entire market 2 Limit pricing when the innovation is not drastic so that pM ψ or alternatively when λ λ the unique equilibrium involves limit pricing where firm 1 sets the price p1 ψ so as to make sure that it still captures the entire market if in this case it were to set p1 pM other firms can profitably undercut firm 1 This type of limit pricing arises in many situations In this case limit pricing results from process innovations by some firms that now have access to a better technology than their rivals Alternatively it can also arise when a fringe of potential entrants can imitate the technology of a firm either at some cost or with lower efficiency and the firm may be forced to set a limit price to prevent the fringe from stealing its customers Proposition 121 Consider the abovedescribed industry Suppose that firm 1 undertakes an innovation reducing the marginal cost of production from ψ to λ1ψ If pM ψ or if λ λ then it sets the unconstrained monopoly price p1 pM and makes profits ˆπI 1 DpMpM λ1ψ μ 123 If pM ψ if λ λ then firm 1 sets the limit price p1 ψ and makes profits πI 1 Dψλ1λ 1ψ μ ˆπI 1 124 Proof The proof of this proposition involves solving for the equilibrium of an asymmetric cost Bertrand competition game While this is standard it is useful to repeat the argument to emphasize that as claimed before the proposition all demand must be met by the lowcost firm Exercise 122 asks you to work through the steps of the proof The fact that ˆπI 1 πI 1 is intuitive since the former refers to the case where λ is greater than λ whereas in the latter firm 1 has a sufficiently low λ that it is forced to charge a price lower than the profitmaximizing monopoly price Note further that ˆπI 1 and πI 1 also correspond to the value of innovation to firm 1 since without innovation it would make zero profits Both of these expressions can be strictly positive so that with ex post monopoly innovation is potentially profitable This situation corresponds to one in which we start with pure competition but one of the firms undertakes an innovation to escape competition and gains ex post monopoly power The fact that the ex post monopoly power is important for innovation incentives is consistent with Schumpeters emphasis on the role of monopoly in generating innovations Let us next contrast the value of innovation for firm 1 in these two regimes to the social value of innovation which is still given by 121 Moreover we can also compare social values in the 124 The DixitStiglitz Model and Aggregate Demand Externalities 427 the right objective function for firms because an allocation in which firms do not maximize profits and instead act in the way that a social planner would like them to act cannot be an equilibrium To see this note that the representative household itself takes prices as given for example it represents a large number of identical pricetaking households If some firms did not maximize profits then the households would refuse to hold the stocks of these firms in their portfolios and there would be entry by other profitmaximizing firms instead Thus as long as the representative household or the set of households on the consumer side act as price takers as has been assumed to be the case throughout profit maximization is the only consistent strategy for the monopolistically competitive firms The only caveat to this arises from a different type of deviation on the production side In particular a single firm may buy all monopolistically competitive firms and act as the single producer in the economy This firm might then ensure an allocation that makes the representative household better off relative to the equilibrium allocation considered here while also increasing its profits Nevertheless I ignore this type of deviation for two reasons First as usual we are taking the market structure as given and the market structure here is monopolistic competition not pure monopoly by a single firm A single firm owning all production units would correspond to an entirely different market structure with much less realism and relevance to the issues studied here Second different firms typically specialize in different sectors of the economy and it is generally impossible for a single monopolist to operate all of the economic activities at once Finally in a related model Acemoglu and Zilibotti 1997 show that a single firm owning all production units cannot be an equilibrium in the presence of free entry This issue is discussed further at the end of Chapter 17 Given these considerations throughout the book I assume that firms are profit maximizing 1244 Limit Prices in the DixitStiglitz Model We have already encountered how limit prices can arise in Section 123 when process inno vations are nondrastic relative to the existing technology Another reason limit prices can arise is because of the presence of a competitive fringe of firms that can imitate the technology of monopolists Such a competitive fringe is straightforward to incorporate into the DixitStiglitz model and will be useful in later chapters as a way of parameterizing competitive pressures Let us assume that there is a large number of fringe firms that can imitate the technology of the incumbent monopolists Suppose that this imitation is equivalent to the production of a similar good and is not precluded by patents It may be reasonable to assume that the imitating firms will be less efficient than those who have invented the variety in question and produced it for a while A simple way of capturing this would be to assume that while the monopolist creates a new variety by paying the fixed cost μ and then having access to a technology with the marginal cost of production of ψ the fringe firms do not pay any fixed costs but can only produce with a marginal cost of γ ψ where γ 1 Similar to the analysis in the Section 123 if γ εε 1 then the fringe firms are sufficiently unproductive that they cannot profitably produce even when the monopolists charge the unconstrained monopoly price given in 1215 Instead when γ εε 1 the monopolists are forced to charge a limit price The same arguments as in the Section 123 establish that this limit price must take the form p γ ψ ε ε 1ψ 126 Taking Stock 429 household would be facing consumption risk if it invested in this project In particular the maximization problem that determines how much it should invest is a solution to the following expected utility maximization max x0y1 puy x puy Rx The firstorder condition of this problem implies that the optimal amount of investment in the risky research activity is given by uy x uy Rx pR 1 p The assumption limc0 uc implies that x y thus less than the full endowment of the individual will be invested in the research activity even though this project has positive expected returns Intuitively the household requires a risk premium to bear the consumption risk associated with the risky investment Next imagine a situation in which many different firms can invest in similar risky research ventures Suppose that the success or failure of each project is independent of the others Imagine that the household invests an amount xN in each of N projects The Strong Law of Large Numbers implies that as N a fraction p of these projects will be successful and the remaining fraction 1 p will be unsuccessful Therefore the household receives almost surely a utility of uy p1 R 1x Since 1 R 1p this is strictly increasing in x and implies that the household prefers to invest all of its endowment in the risky projects that is x y Therefore the ability to hold a balanced portfolio of projects with independently distributed returns allows the household to diversify the risks and act in a riskneutral manner A similar logic applies in many of the models presented in the remainder of the book even though individual firms have stochastic returns the representative household holds a balanced portfolio of all the firms in the economy and diversifies idiosyncratic risks This observation also implies that the objective of each firm is to maximize expected profits without a risk premium 126 Taking Stock This chapter has reviewed several conceptual and modeling issues related to the economics of RD I discussed why ex post monopoly power is important in creating incentives for inno vation how innovation incentives differ between competitive firms and monopolies and how these compare to the social value of innovation In this context I emphasized the importance of the appropriability effect which implies that the private value of innovation often falls short of its social value because even with ex post monopoly power an innovating firm is not able to appropriate the entire consumer surplus created by a better product or a cheaper process I also discussed Arrows replacement effect which implies that incumbent monopolists typi cally have weaker incentives for innovation than do entrants Despite the appropriability effect the amount of innovation in equilibrium can be excessive because of another countervailing force the business stealing effect which encourages firms to undertake innovations to become the new monopolist and take over steal the monopoly rents Therefore whether there is too 430 Chapter 12 Modeling Technological Change little or too much innovation in equilibrium depends on the market structure and the parameters of the model This chapter has also introduced the DixitStiglitz model which plays an important role in the analysis of the next few chapters This model offers a simple formalization of Chamberlins approach to monopolistic competition in which each firm has some monopoly power but free entry ensures that all firms or the marginal entrants make zero profits The DixitStiglitz model is particularly tractable because the markup charged by monopolists is independent of the number of competing firms This property makes it an ideal model to study endogenous growth because it enables innovation to remain profitable even when the number of products or the number of machines increases continuously 127 References and Literature The literature on RD in industrial organization is vast My purpose in this chapter has not been to review this literature but to highlight the salient features that are used in the remainder of the book The reader who is interested in this area can start with Tirole 1988 Chapter 10 which contains an excellent discussion of the contrast between the private and the social values of innovation A more uptodate reference that surveys the recent developments in the economics of innovation is Scotchmer 2005 The classic reference on the private and social values of innovation is Arrow 1962 Schumpeter 1934 was the first to emphasize the role of monopoly in RD and innovation The importance of monopoly power for innovation and the implications of the nonrival nature of ideas are discussed in Romer 1990 1993 Most of the industrial organization literature also emphasizes the importance of ex post monopoly power and patent systems in providing incentives for innovation See for example Scotchmer 2005 This perspective has recently been criticized by Boldrin and Levine 2003 The idea of creative destruction was also originally developed by Schumpeter 1942 Models of creative destruction in the industrial organization literature include Reinganum 1981 1985 Similar models in the growth literature are developed in Aghion and Howitt 1992 1998 Chamberlin 1933 is the classic reference on monopolistic competition The socalled DixitStiglitz model is developed in Dixit and Stiglitz 1977 and in Spence 1976 This model was first used for an analysis of RD in Dasgupta and Stiglitz 1980 An excellent exposition of the DixitStiglitz model is provided in Matsuyama 1995 Tirole 1988 also discusses the DixitStiglitz model as well as other models of product innovation including Salops 1979 model which is presented in Exercise 1214 A stimulating general discussion of issues of innovation and the importance of market size and profit incentives is provided in Schmookler 1966 Recent evidence on the effect of market size and profit incentives on innovation is discussed in Newell Jaffee and Stavins 1999 Popp 2002 Finkelstein 2004 and Acemoglu and Linn 2004 Mokyr 1990 contains an excellent history of innovation Freeman 1982 also provides a survey of the qualitative literature on innovation and discusses the different types of innovation The rest of this part of the book like this chapter focuses on monopolistic environments in which the appropriate equilibrium concept is not the competitive equilibrium but one that incorporates gametheoretic interactions Since all games in this book have complete information the appropriate notion of equilibrium is the standard Nash Equilibrium concept or when the game is multistage or dynamic it is the Subgame Perfect Equilibrium or the Markov Perfect Equilibrium In these situations equilibrium always refers to Nash Subgame Perfect or Markov Perfect Equilibrium The treatment here presumes that the reader is familiar with 436 Chapter 13 Expanding Variety Models denotes the profits of the monopolist producing machine ν at time t xν t and pxν t are the profitmaximizing choices for the monopolist and rt is the market interest rate at time t1 Alternatively assuming that the value function is differentiable in time this equation could be written in the form of a HJB equation as in Theorem 710 in Chapter 7 rtV ν t V ν t πν t 138 Exercise 131 provides a different derivation of this equation than in Theorem 710 1312 Characterization of Equilibrium An allocation in this economy is defined by the following objects time paths of consumption levels aggregate spending on machines and aggregate RD expenditure Ct Xt Zt t0 time paths of available machine varieties Nt t0 time paths of prices and quantities of each machine pxν t xν t ν0Ntt0 and time paths of inter est rates and wage rates rt wt t0 An equilibrium is an allocation in which all monopolists research firms choose pxν t xν t ν0Ntt0 to maximize the discounted value of profits the evolution of Nt t0 is determined by free entry the evolution of rt wt t0 is consistent with market clearing and the evolution of Ct Xt Zt t0 is consistent with household maximiza tion Note that this equilibrium is not competitive since machine producers have market power Let us start with the firm side Since 136 defines isoelastic demands the solution to the maximization problem of any monopolist ν 0 Ntinvolves setting the same price in every period see Exercise 132 pxν t ψ 1 β for all ν and t All monopolists thus charge a constant rental rate equal to a markup over their marginal cost of production ψ Let us normalize the marginal cost of machine production to ψ 1 β so that pxν t px 1 for all ν and t 139 Profitmaximization also implies that each monopolist rents out the same quantity of machines in every period equal to xν t L for all ν and t 1310 This gives monopoly profits as πν t βL for all ν and t 1311 Equation 1311 implies that each monopolist sells the same amount of machines charges the same price and makes the same amount of profits at all points in time 1 As usual the interest rate rt is determined from the prices of zero net supply Arrow securities that households can trade to transfer consumption across dates The aggregate economy can only transfer resources across dates by changing the stock of machine varieties Nt 438 Chapter 13 Expanding Variety Models Ct Xt Zt Nt t0 such that 133 137 1314 1315 1316 and 1317 are satisfied time paths of prices and quantities of each machine px ν t xν t νNtt0 that satisfy 139 and 1310 and time paths of interest rate and wages rt wt t0 such that 1313 and 1316 hold A balanced growth path BGP is an equilibrium path where consumption Ct and output Yt grow at a constant rate Equation 1312 then implies that Nt must also grow at a constant rate in a BGP A BGP can alternatively be referred to as a steady state since it is a steady state in transformed variables even though the original variables grow at a constant rate 1313 Balanced Growth Path A BGP requires that consumption grows at a constant rate say g C This is possible from 1316 only if the interest rate is constant Let us therefore look for an equilibrium allocation in which rt r for all t where the asterisk refers to BGP values Since profits at each date are given by 1311 and the interest rate is constant 138 implies that V t 0 Substituting this in either 137 or in 138 we obtain V βL r 1318 This equation is intuitive a monopolist makes a flow profit of βL and along the BGP this profit is discounted at the constant interest rate r Let us next suppose that the freeentry condition 1314 holds as an equality in which case we also have ηβL r 1 This equation pins down the BGP interest rate r as r ηβL The consumption Euler equation 1316 then implies that the rate of growth of consumption in BGP must be given by g C Ct Ct 1 θ r ρ 1319 Moreover it can be verified that the currentvalue Hamiltonian for the representative house holds maximization problem is concave Thus this condition together with the transversality condition is sufficient to characterize the unique optimal consumption plan of the representa tive household recall Theorem 714 in Chapter 7 and Exercise 811 in Chapter 8 In a BGP consumption cannot grow at a different rate than total output see Exercise 136 thus the growth rate of output in the economy must be g g C 131 The LabEquipment Model of Growth with Input Varieties 439 Given the BGP interest rate the longrun growth rate of the economy is then obtained as g 1 θ ηβL ρ 1320 Let us next assume that ηβL ρ and 1 θηβL ρ 1321 The first inequality ensures that g 0 while the second one ensures that the representative households utility is finite and the transversality condition is satisfied Proposition 131 Suppose that condition 1321 holds Then in the abovedescribed lab equipment expanding input variety model there exists a unique BGP in which technology output and consumption all grow at the same rate g given by 1320 Proof The preceding discussion establishes all the claims in the proposition except that the transversality condition holds Exercise 134 verifies this and also shows that the resource constraint 133 is satisfied with positive consumption at all points An important feature of this class of endogenous technological progress models is the presence of the scale effect which was already discussed in Section 114 in the context of Romers 1986a model the larger is L the greater is the growth rate The scale effect comes from a very strong form of the market size effect discussed in the previous chapter The increasing returns to scale nature of the technology eg as highlighted in 1312 is responsible for this strong form of the market size effect and thus for the scale effect We will see in Section 155 that it is possible to have variants of the current model that feature the market size effect but not the scale effect 1314 Transitional Dynamics It is straightforward to see that there are no transitional dynamics in this model To derive this result let us go back to the value function for each monopolist in 138 Substituting for profits from 1311 noting that V ν t is independent of ν and denoting it by V t this equation implies that rtV t V t βL The key observation is that positive growth of output at any point implies that ηV t 1for all t In other words if ηV t 1for t t ε t ε for some t and ε 0 then ηV t 1for all t Moreover given 1321 zero growth at all points is not possible and thus we must have ηV t 1at least for some interval of time see Exercise 135 Then differentiating ηV t 1 with respect to time yields V t 0 for all t which is only consistent with rt r for all t thus yielding rt ηβL for all t Proposition132 Suppose that condition 1321 holds and that the initial technology stock is N0 0 Then there exists a unique equilibrium path In this equilibrium technology output and consumption always grow at the rate g as in 1320 Proof See Exercise 135 442 Chapter 13 Expanding Variety Models Proposition 133 In the abovedescribed expanding input variety model the decentralized equilibrium is always Pareto suboptimal Moreover provided that 1 θη1 β1ββL ρ starting with any N0 0 the Pareto optimal allocation involves a constant growth rate of gS 1 θ η1 β1ββL ρ which is strictly greater than the equilibrium growth rate g given in 1320 Proof See Exercises 139 and 1310 Intuitively the Pareto optimal growth rate is greater than the equilibrium growth rate because of the greater social value of innovations This greater social value stems from the fact that the Pareto optimal allocation involves no markups and thus uses the available set of machines more intensively So the source of inefficiency in equilibrium is related to the pecuniary externality resulting from monopoly markups which affect the set of traded commodities and thus the rate of growth of machines and technology Other models of endogenous technological progress incorporate technological spillovers and thus generate inefficiencies both because of the pecuniary externality isolated here and because of the technological spillovers 1316 Policy in Models of Endogenous Technological Progress The divergence between the decentralized equilibrium and the Pareto optimal allocation intro duces the possibility of Pareto improving policy interventions There are two natural alterna tives to consider 1 Subsidies to research by subsidizing research the government can increase the growth rate of the economy and this can be turned into a Pareto improvement if taxation is not distortionary and in the presence of heterogeneity if there is appropriate redistribution of resources so that all parties benefit 2 Subsidies to machines andor inputs inefficiencies also arise because the decentralized economy is not using as many units of the machines andor inputs because of the monopoly markup so subsidies to the use of machines by the final good sector would also increase the growth rate Moreover as in the firstgeneration endogenous growth models a variety of different policy interventions including taxes on investment income and subsidies of various forms have growth effects rather than merely level effects see eg Exercise 1312 Naturally once we start thinking of policy as potentially closing the gap between the decentralized equilibrium and the Pareto optimal allocation we also have to consider the objectives of policy makers and this brings us to issues of political economy which are the subject matter of Part VIII For that reason I do not present a detailed discussion of optimal policy leaving some discussion to Exercises 13111313 Nevertheless it is useful to briefly discuss the role of competition policy in models of endogenous technological progress Recall that the profitmaximizing price that the monopolist charges for machines is px ψ1 β Imagine instead that a fringe of competitive firms can copy the innova tion of any monopolist but they are not able to produce at the same level of costs because the inventor has more knowhow In particular as in the previous chapter suppose that in stead of a marginal cost ψ the fringe companies have marginal cost of γ ψ with γ 1 If γ 11 β then this fringe is not a threat to the monopolist since the monopolist could set its ideal profitmaximizing markup and the fringe would not be able to enter without making losses However if γ 11 β then the fringe would prevent the monopolist from setting 131 The LabEquipment Model of Growth with Input Varieties 443 its ideal monopoly price In particular in this case the monopolist would be forced to set a limit price of px γ ψ 1324 which has an explanation identical to the limit price in the previous chapter3 When the monopolist charges this limit price its profits per unit would be γ 1ψ γ 11 β which is less than monopoly profits per unit in the absence of a competitive fringe What does this imply for economic growth It is straightforward to see that in this case the economy would grow at a slower rate For example in the baseline model with the labequipment technology this growth rate would be see Exercise 1314 ˆg 1 θ ηγ 1βγ 11 β1ββL ρ which is less than g given in 1320 Therefore in this model greater competition which reduces markups and thus static distortions also reduces longrun growth This result might at first appear to be counterintuitive since the monopoly markup may be thought to be the key source of inefficiency and greater competition lower γ reduces this markup Nevertheless as mentioned above inefficiency results both because of monopoly markups and because the set of available machines may not be chosen appropriately As γ declines monopoly markups decline but the problem of underprovision of machines becomes more severe This is because when monopoly profits are reduced incentives for research are also reduced Since γ can also be interpreted as a parameter of antitrust competition policy this result implies that in the baseline endogenous technological change models stricter antitrust policy reduces economic growth Welfare is not the same as growth and some degree of competition to reduce prices below the unconstrained monopolistic level might be useful for welfare depending on the discount rate of the representative household Essentially with a lower markup households will enjoy a higher level of consumption for a given level of N but they will also suffer slower consumption growth slower growth of N The tradeoff between these two opposing effects depends on the discount rate of the representative household see Exercise 1314 Similar results apply when we consider patent policy In practice patents are for limited durations In the baseline model we assumed that patents are perpetual once a firm invents a new good it has a fully enforced patent forever If patents are strictly enforced then this might prevent the competitive fringe from competing restoring the growth rate of the economy to 1320 And even in the absence of a competitive fringe we can imagine that once the patent runs out the firm ceases to make profits on its innovation In this case it can easily be shown that growth is maximized by having patents last as long as possible but there is again a welfare tradeoff The results of this baseline endogenous technology model on the effects of competition and patent duration on growth are extreme partly because this model does not incorporate 3 More specifically if the price of the monopolist were higher than γ ψ the fringe could undercut the price of the monopolist take over the market and make positive profits If it were below this amount then the monopolist could increase its price toward the unconstrained monopoly price and make more profits Thus there is a unique equilibrium price given by 1324 444 Chapter 13 Expanding Variety Models rich competitive interactions among firms The quality competition Schumpeterian models introduced in the next chapter allow a more nuanced analysis of the effects of competition and patents on innovation and economic growth 132 Growth with Knowledge Spillovers In the previous section growth resulted from the use of final output for RD At some level this is similar to that of Rebelos 1991 model of sustained growth discussed in Chapter 11 since the innovation possibilities frontier RD technology is linear in accumulable factors An alternative is to have scarce factors used in RD In other words instead of the lab equipment specification researchers and scientists are now the key creators of RD The lab equipment model generated sustained economic growth by investing more and more resources in the RD sector This is impossible with scarce factors since by definition a sustained increase in the use of these factors in the RD sector is not possible Consequently with this alternative specification there cannot be endogenous growth unless there are knowledge spillovers from past RD making the scarce factors used in RD increasingly productive over time In other words we now need current researchers to stand on the shoulder of past giants The original formulation of the endogenous technological change model by Romer 1990 relied on this type of knowledge spillovers While such knowledge spillovers might be important in practice the labequipment model studied in the previous section was a better starting point because it clearly delineated the role of technology accumulation and showed that growth need not be generated by technological externalities or spillovers Since knowledge spillovers play an important role in many models of economic growth it is useful to see how the baseline model of endogenous technological progress works in the presence of such spillovers I now present the simplest version of the endogenous technological change model with knowledge spillovers The environment is identical to that of the previous section with the exception of the innovation possibilities frontier which now takes the form Nt ηNtLRt 1325 where LRt is labor allocated to RD at time t The term Nt on the righthand side captures spillovers from the stock of existing ideas The greater is Nt the more productive is an RD worker Notice that 1325 requires that these spillovers are proportional or linear This linearity is the source of endogenous growth in the current model In 1325 LRt is research employment which comes out of the regular labor force An alternative which was originally used by Romer 1990 would be to suppose that only skilled workers or scientists can work in the knowledgeproduction RD sector Here I use the assumption that a homogeneous workforce is employed both in the RD sector and in the final good sector Competition between the production and the RD sectors for workers then ensures that the cost of workers to the research sector is given by the wage rate in the final good sector The only other change is that now the total labor input employed in the final good sector represented by the production function 132 is LEt rather than L since some of the workers are employed in the RD sector Labor market clearing requires that LRt LEt L Aggregate output is then given by Yt 1 1 β NtLEt 1326 132 Growth with Knowledge Spillovers 445 and profits of monopolists from selling their machines are πt βLEt 1327 The net present discounted value of a monopolist for a blueprint ν is still V ν t as in 137 or 138 with the flow profits given by 1327 However the freeentry condition is no longer the same as that which followed from 134 Instead 1325 implies the following freeentry condition when there is positive research ηNtV ν t wt 1328 The lefthand side of 1328 is the return from hiring one more worker for RD The term Nt is on the lefthand side because higher Nt translates into higher productivity of RD workers The righthand side is the flow cost of hiring one more worker for RD wt The equilibrium wage rate must be the same as in the labequipment model of the previous section in particular wt βNt1 β as in 1313 Moreover balanced growth again requires that the interest rate be constant at some level r Using these observations together with the freeentry condition 1328 BGP requires that ηNtβLEt r β 1 β Nt 1329 Hence the BGP equilibrium interest rate must be r 1 βηL E where L E is the number of workers employed in production in BGP given by L E L L R The fact that the number of workers in production must be constant in BGP follows from 1329 Now using the Euler equation of the representative household 1316 we obtain Ct Ct 1 θ 1 βηL E ρ g for all t 1330 To complete the characterization of the BGP equilibrium the BGP level of employment L E needs to be determined The innovation possibilities frontier 1325 implies NtNt ηL R ηL L E Moreover by definition the BGP growth rate of consumption must be equal to the rate of technological progress thus g NtNt This implies that the BGP level of employment is uniquely pinned down as L E θηL ρ 1 βη θη 1331 The rest of the analysis is unchanged It can also be verified that there are again no transitional dynamics in the decentralized equilibrium see Exercise 1317 Proposition134 Consider the abovedescribed expanding input variety model with knowl edge spillovers and suppose that 1 θ1 βηL E ρ 1 βηL E 1332 where L E is the number of workers employed in production in BGP given by 1331 Then starting from any initial level of technology stock N0 0 there exists a unique equilibrium 133 Growth without Scale Effects 447 Second in contrast to the knowledge spillover model studied in the previous section the RD sector only admits limited knowledge spillovers and 1325 is replaced by Nt ηNtφLRt 1334 where φ 1 and LRt is labor allocated to RD activities at time t Labor market clearing requires that LEt LRt Lt 1335 where LEt is the level of employment in the final good sector and Lt is population at time t The key assumption for the model is that φ 1 The case where φ 1 is the one analyzed in the previous section and as mentioned above with population growth it would lead to an exploding path and to infinite utility for the representative household Aggregate output and profits are given by 1326 and 1327 as in the previous section An equilibrium is also defined similarly Let us focus on the BGP where a constant fraction of workers are allocated to RD and the interest rate and the growth rate are constant Suppose that this BGP involves positive growth so that the freeentry condition holds as equality Then provided that r n the BGP freeentry condition can be written as see Exercise 1318 ηNtφ βLEt r n wt 1336 As before the equilibrium wage is determined by the production side and is given by 1313 Combining 1313 with 1336 gives the following freeentry condition ηNtφ11 βLEt r n 1 Now differentiating this condition with respect to time we have φ 1 Nt Nt LEt LEt 0 Since in BGP the fraction of workers allocated to research is constant LEtLEt n Thus the BGP growth rate of technology is given by g N Nt Nt n 1 φ 1337 Using 1312 1337 implies that total output grows at the rate g N n But now there is population growth so consumption per capita grows only at the rate g C g N n 1 φ 1338 The consumption Euler equation then determines the BGP interest rate as r θg N ρ θn 1 φ ρ Proposition 135 Suppose that ρ 1 φ θn1 φ Then in the expanding input variety model with limited knowledge spillovers there exists a unique BGP in which technology 450 Chapter 13 Expanding Variety Models where wtcν t is the total expenditure of the firm to produce a total quantity of cν t given the production function 1341 and the wage rate at time t wt while pcν tcν t is its revenue consistent with the demand function 1344 The maximization of the net present discounted value again requires profit maximization at every instant Since each monopolist faces the isoelastic demand curve given in 1344 the profitmaximizing monopoly price is pcν t ε ε 1wt for all ν and t All firms thus charge the same price produce the same amount and employ the same amount of labor At time t there are Nt products so the labor market clearing condition 1343 implies that cν t lν t LEt Nt for all ν and t 1347 where LEt L LRt The instantaneous profits of each monopolist are πν t pcν tcν t wtcν t 1 ε 1 LEt Nt wt for all ν and t 1348 Since prices sales and profits are equal for all monopolists we can simplify notation by letting V t V ν t for all ν and t In addition since cν t ct for all ν we have Ct Nt ε ε1ct LEtNt 1 ε1 1349 where the second equality uses 1347 Labor demand comes from the research sector as well as from the final good producers Labor demand from research can again be determined using the freeentry condition Assuming that there is positive research so that the freeentry condition holds as equality this demand takes the form ηNtV t wt 1350 Combining 1350 with 1348 yields πt 1 ε 1LEtηV t 1351 where πt denotes the profits of all monopolists at time t In BGP where the fraction of the workforce working in research is constant 1351 implies that profits and the net present discounted value of monopolists must grow at the same rate Let us denote the BGP growth rate of the number of products Nt by gN the growth rate of profit and values by gV and the growth rate of wages by gw Moreover given the choice of numeraire the consumption growth rate in this economy g must equal the wage growth rate gw The freeentry condition 1350 then implies that g gN gV Given these constant growth rates and the constant BGP interest rate r 1348 implies that in BGP V t πt r g gN 1352 134 Growth with Expanding Product Varieties 451 Intuitively at time t profits are equal to πt Subsequently because of product expansion the number of employees per product decreases at the rate gN reducing profits and wages increase at the rate g increasing profits Taking into account discounting at the rate r yields 1352 Now combining 1351 with 1352 gives r η ε 1L L R g gN with L R denoting the BGP level of employment in the research sector Combined with the RD sector production function 1342 this equation gives the growth rate of products as Nt Nt gN ηL R Then from 1346 the BGP growth rate is g r ρ Combining this expression with the previous two equations we obtain the BGP level of research employment as L R ηL ε 1ρ ηε 1353 Let us assume that L R 0 ie ηL ε 1ρ so that there is positive growth otherwise the freeentry condition would hold as inequality and there would be zero growth Moreover from 1349 g gNε 1 and therefore we have g ηL ε 1ρ εε 1 1354 Finally since r g because of logarithmic preferences household utility is always finite and the relevant transversality condition is satisfied Proposition 136 Suppose that ηL ε 1ρ Then there exists a unique BGP in which aggregate consumption expenditure Ct and the wage rate wt grow at the rate g given by 1354 In this equilibrium there is growth of real income even though the production function of each good remains unchanged This is because while there is no process innovation reducing costs or improving quality the number of products available to households expands because of product innovations Since the utility function of the representative household 1339 exhibits love for variety the expanding variety of products increases utility What happens to income depends on the choice of numeraire The natural numeraire is the one setting the ideal price index 1345 equal to 1 which amounts to measuring incomes in similar units at different dates With this choice of numeraire real incomes grow at the same rate as Ct ie at the rate g Exercise 1324 further highlights the similarity between the expanding product and machine variety models Exercise 1327 shows that as in other expanding variety models there are no transitional dynamics in the current model and the equilibrium is again Pareto suboptimal Finally it can be verified that there is again a scale effect here This discussion then reveals that whether one wishes to use the expanding input variety or the expanding product 452 Chapter 13 Expanding Variety Models model is mostly a matter of taste and perhaps one of context Both models lead to similar structures of equilibria similar effects of policy on equilibrium growth and similar welfare properties 135 Taking Stock In this chapter we had our first look at models of endogenous technological progress The distinguishing feature of these models is the fact that profit incentives shape RD spending and investments which in turn determines the rate at which the technology of the economy evolves over time At some level there are many parallels between the models studied here and the Romer 1986a model of growth with externalities discussed in Section 114 both have a mathematical structure similar to the neoclassical AK models constant longrun growth rate no transitional dynamics both generate endogenous growth as a function of preferences and policies and in both technological or pecuniary externalities make the equilibrium growth rate less than the Pareto optimal growth rate Nevertheless the fundamental difference between the Romer 1986a model and the endogenous technological change model should not be understated While one may interpret the Romer 1986a model as involving knowledge accumulation this accumulation is not the outcome of purposeful economic activityit is a byproduct of other decisions in this particular instance those involving individual physical capital accumulation While such a model may endogenize technology it does so without explicitly specifying the costs and benefits of investing in new technologies Since as discussed in Chapter 3 technology differences across countries are likely to be important in accounting for their income differences understanding the sources of technology differences is an essential part of our effort to understand the mechanics of economic growth In this respect the models presented in this chapter constitute a major improvement over those in Chapter 11 The models studied in this chapter like those of the previous chapter emphasize the importance of profits in shaping technology choices We have also seen the role of monopoly power and patent length on the equilibrium growth rate In addition the same factors that influence the equilibrium growth rate in the neoclassical AK model also affect equilibrium economic growth These include the discount rate ρ as well as taxes on capital income or corporate profits Nevertheless the effect of the market structure on equilibrium growth and innovation rates is somewhat limited in the current models because the DixitStiglitz structure and expanding product or input varieties limit the extent to which firms can compete with one another The models of quality competition in the next chapter feature a richer interaction between market structure and equilibrium growth An important shortcoming of the models in this chapter and those in the next two chapters should also be noted at this point In these models the technology stock of a society is determined only by its own RD Thus technological differences result simply from RD differences In our current world of relatively free knowledge flows many countries not only generate technological knowhow from their own RD but also benefit from the advances in the world technology frontier Consequently in practice technology adoption decisions and the patterns of technology diffusion may be equally important as or more important than RD toward the invention of new technologies see Chapter 18 Therefore the major contribution of the approaches studied in this chapter to our understanding is not necessarily in pinpointing the exact source of technology differences across countries but in their emphasis on the endogenous nature of technology and in the perspective they provide for modeling technological investments In addition even if technology adoption and imitation may be 137 Exercises 455 139 Consider the expanding input variety model of Section 131 Show that it is possible for the equilibrium allocation to satisfy the transversality condition while the optimal growth path may violate it Interpret this result Does it imply that the solution to the optimal growth problem is less compelling Show that when the condition in Proposition 133 is satisfied the optimal growth path satisfies the transversality condition and yields a finite level of utility for the representative household 1310 Complete the proof of Proposition 133 in particular showing that the Pareto optimal allocation always involves a constant growth rate and no transitional dynamics 1311 Consider the expanding input variety model of Section 131 a Suppose that a benevolent government has access only to research subsidies which can be financed by lumpsum taxes Can these subsidies be chosen to ensure that the equilibrium growth rate is the same as the Pareto optimal growth rate Can they be used to replicate the Pareto optimal equilibrium path Would it be desirable for the government to use subsidies to achieve the Pareto optimal growth rate from the viewpoint of maximizing social welfare at time t 0 b Suppose that the government now has access only to subsidies to machines which can again be financed by lumpsum taxes Can these be chosen to induce the Pareto optimal growth rate Can they be used to replicate the Pareto optimal equilibrium path c Will the combination of subsidies to machines and subsidies to research be better than either of these two policies by themselves 1312 Consider the expanding input variety model of Section 131 and assume that corporate profits are taxed at the rate τ a Characterize the equilibrium allocation b Consider two economies with identical technologies and identical initial conditions but with different corporate tax rates τ and τ Determine the relative income of these two economies as a function of time 1313 Consider the expanding input variety model of Section 131 with one difference A firm that invents a new machine receives a patent which expires at the Poisson rate ι Once the patent expires that machine is produced competitively and is supplied to final good producers at marginal cost a Characterize the BGP equilibrium in this case and show how the growth rate depends on ι Hint notice that there will be two different machine varieties supplied at different prices b Characterize the transitional dynamics Hint show that the growth rate of consumption is constant but output growth is not c What is the value of ι that maximizes the equilibrium rate of economic growth d Show that a policy of ι 0 does not necessarily maximize social welfare at time t 0 1314 Consider the formulation of competition policy in Section 1316 a Characterize the equilibrium fully b Write down the welfare of the representative household at time t 0 in this equilibrium c Maximize the welfare function derived in part b by choosing a value of γ d Why is the optimal value of γ not equal to some γ 11 β Provide an interpretation in terms of the tradeoff between level and growth effects e What is the relationship between the optimal value of γ and ρ 14 Models of Schumpeterian Growth T he previous chapter presented the basic endogenous technological change models based on expanding input machine or product varieties The advantage of these models is their tractability While the expansion of the variety of machines used in production captures certain aspects of the economics of innovation most process innovations in practice either increase the quality of an existing product or reduce the costs of production Therefore typical process innovations have a number of distinct features compared to the horizontal innovations of the previous chapter For example in the expanding machine variety model a newly invented computer is used alongside all previous vintages of computers in reality a newly invented computer often replaces existing vintages Thus models of expanding machine variety may not provide a good description of innovation dynamics in practice because they do not capture the competitive aspect of innovations These competitive aspects bring us to the realm of Schumpeterian creative destruction in which economic growth is driven at least in part by new firms replacing incumbents and new machines and products replacing old ones For this reason the models discussed in this chapter are referred to as Schumpeterian growth models My purpose in this chapter is to develop tractable models of Schumpeterian growth Schumpeterian growth raises a number of novel and important issues First in contrast to the models of expanding varieties there may be direct price competition among various producers with different vintages of quality or different costs of producing the same product This competition affects both the description of the growth process and several of its central implications For example market structure and antitrust policy can play potentially richer roles in models exhibiting this type of price competition Second competition between incumbents and entrants brings the replacement and business stealing effects discussed in Chapter 12 to the forefront and raises the possibility of excessive innovation This description suggests that a number of new and richer issues arise in the context of Schumpeterian growth models One may then expect models of Schumpeterian models to be significantly more complicated than expanding varieties models This is not necessarily the case however This chapter presents the basic models of competitive innovations first proposed by Aghion and Howitt 1992 and then further developed by Grossman and Helpman 458 141 A Baseline Model of Schumpeterian Growth 465 This analysis establishes the following proposition Proposition 141 Consider the model of Schumpeterian growth described above Sup pose that ληβL ρ 1 θλ 1ηβL Then there exists a unique BGP in which average quality of machines output and consumption grow at rate g given by 1423 The rate of innovation is gλ 1 Proof Most of the proof is given in the preceding analysis Exercise 144 asks you to check that the BGP equilibrium is unique and satisfies the transversality condition The above analysis illustrates that the mathematical structure of the model is similar to those analyzed in the previous chapter Nevertheless creative destructionthe process of incumbent monopolists being replaced by new entrantsis an important new element and provides a different interpretation of the growth process I return to some of the applications of creative destruction below Before doing this let us briefly look at transitional dynamics in this economy Similar arguments to those used in the previous chapter establish the following result Proposition142 In the model of Schumpeterian growth described above starting with any average quality of machines Q0 0 there are no transitional dynamics and the equilibrium path always involves constant growth at the rate g given by 1423 Proof See Exercise 145 As noted above only the average quality of machines Qt matters for the allocation of resources Moreover the incentives to undertake research are identical for two machine varieties ν and ν with different quality levels qν t and qν t Thus there are no differential incentives for RD in more and less advanced machines Both features are a result of the functional form in 143 Exercise 1414 investigates the conditions under which these results may not apply Nevertheless the specification chosen in this section is appealing since it implies that research is directed toward a broad range of machines rather than a specific subset of machines 1414 Pareto Optimality This equilibrium like that of the endogenous technology model with expanding varieties is Pareto suboptimal The first reason for this is the appropriability effect which results because monopolists are not able to capture the entire social gain created by an innovation However Schumpeterian growth also introduces the business stealing effect discussed in Chapter 12 Consequently the equilibrium rate of innovation and growth can now be too high or too low relative to the social optimum Let us proceed as in the previous chapter first deriving the quantities of machines that are used in the final good sector in the optimal growth allocation for given Qt In this allocation there are no markups and thus xSν t q ψ1βL 1 β1βL 1424 Substituting 1424 into 143 we obtain Y St 1 β1βQStL 141 A Baseline Model of Schumpeterian Growth 467 to the expanding varieties models the Pareto optimal growth rate is not always greater than the equilibrium growth rate This can be seen by comparing gS to g in 1423 In particular when λ is very large gS g and there is insufficient growth in the decentralized equilibrium For example as λ we have gSg 1 β1β 1 In contrast to obtain an example in which there is excessive growth in equilibrium suppose that θ 1 β 09 λ 13 η 1 L 1 and ρ 038 In this case it can be verified that gS 0 while g 018 gS3 This example illustrates the counteracting influences of the appropriability and business stealing effects discussed above The following proposition summarizes this result Proposition143 In the model of Schumpeterian growth described above the decentralized equilibrium is Pareto suboptimal The equilibrium may have a higher or lower rate of growth than the Pareto optimal allocation It is also straightforward to verify that as in the models of the previous chapter there is a scale effect and thus population growth would lead to an exploding growth path Exercise 1411 asks you to construct a Schumpeterian growth model without scale effects 1415 Policy in Models of Schumpeterian Growth As in the previous chapter antitrust policy patent policy and taxation affect equilibrium growth For example two economies that tax corporate incomes at different rates grow at different rates Nevertheless the current model may be more appropriate for conducting policy analysis than the expanding varieties models In these previous models there was no reason for any agent in the economy to support distortionary taxes4 In contrast the fact that growth here takes place through creative destruction implies that there is an inherent conflict of interest and certain types of distortionary policies may have a natural constituency To illustrate this point which is to be discussed in greater detail in Part VIII of the book suppose that there is a tax τ imposed on RD spending This has no effect on the profits of existing monopolists and only influences their net present discounted value via replacement Since taxes on RD discourage RD there will be replacement at a slower rate that is BGP RD effort z falls A slower replacement rate directly increases the steadystate value of all monopolists given by 1418 In particular the value of a monopolist with a machine of quality q is V q βqL rτ zτ where the equilibrium interest rate and the replacement rate have been written as functions of τ With the tax rate on RD the freeentry condition 1414 becomes V q 1 τ λη q This equation shows that V q is increasing in the tax rate on RD τ Combining the previous two equations it can be seen that in response to a positive rate of taxation rτ zτ must adjust downward so that the value of current monopolists increases consistent with 3 Notice that the combination of β 09 and λ 13 is consistent with 145 which was used in deriving the equilibrium growth rate g 4 Naturally one can enrich these models so that tax revenues are distributed unequally across agents for example with taxes on capital distributed to workers In this case even in the basic neoclassical growth model some groups could prefer distortionary taxes Such models is discussed in Part VIII of the book 468 Chapter 14 Models of Schumpeterian Growth the previous equation Intuitively when the costs of RD are raised because of tax policy the value of a successful innovation V q must increase to satisfy the freeentry condition This can only happen through a decline in the effective discount rate rτ zτ A lower effective discount rate in turn is achieved by a decline in the equilibrium growth rate of the economy which now takes the form gτ 1 τ1ληβL ρ θ λ 11 This growth rate is strictly decreasing in τ But as the previous expression shows incumbent monopolists benefit from an increase in τ and would be in favor of such a growthretarding policy Therefore an important advantage of models of Schumpeterian growth is that they provide us with clues about why some societies may adopt policies that reduce the equilibrium growth rate Since taxing RD by new entrants benefits incumbent monopolists when incumbents are politically powerful such distortionary taxes can emerge in the political economy equilibrium even though they are not in the interest of the society at large 142 A OneSector Schumpeterian Growth Model The model of Schumpeterian growth presented in the previous section was designed to maxi mize the parallels between this class of models and those based on expanding varieties I now discuss a model more closely related to the original Aghion and Howitt 1992 paper There are two major differences from the previous section First there is only one sector experiencing quality improvements rather than a continuum of machine varieties Second the innovation possibilities frontier uses a scarce factor labor as in the model of knowledge spillovers in Section 132 of the previous chapter Since there are many parallels between this model and the one in the previous section I only provide a brief exposition 1421 The Basic AghionHowitt Model The household side is the same as before with the only difference being that we now assume households are risk neutral so that the interest rate is determined as r ρ at all points in time Population is again equal to L and labor is supplied inelastically The aggregate production function of the unique final good is Yt 1 1 β xt q1βqtLEtβ 1428 where qt is the quality of the unique machine used in production and is written in the labor augmenting form for simplicity xt q is the quantity of this machine used at time t and LEt denotes the amount of labor used in production at time t which is less than L since LRt workers are employed in the RD sector Market clearing requires that LEt LRt L Once invented a machine of quality qt can be produced at the constant marginal cost ψ in terms of the final good Let us again normalize ψ 1 β The innovation possibilities frontier now involves labor being used for RD In particular each worker employed in the RD sector 143 Innovation by Incumbents and Entrants 473 is again quality improvements but these are driven by two types of innovations 1 innovation by incumbents and 2 creative destruction by entrants Let qν t be the quality of machine line ν at time t In particular the quality ladder for each machine variety again takes the form qν t λnνtqν s for all ν and t where λ 1 and nν t now denotes the number of incremental innovations on this machine line between times s t and t where time s is the date at which this particular type of technology was first invented and qν s refers to its quality at that point The incumbent has a fully enforced patent on the machines that it has developed though this patent does not prevent entrants leapfrogging the incumbents machine quality I assume that at time t 0 each machine line starts with some quality qν 0 0 and is owned by an incumbent Incremental innovations can only be performed by the incumbent producer So we can think of those as tinkering innovations that improve the quality of the machine More specifically if the current incumbent spends an amount zν tqν t of the final good for incremental innovation on a machine of quality qν t it has a flow rate of innovation equal to φzν t for φ 0 The resulting new machine is of quality λqν t Alternatively a new firm entrant can undertake RD to innovate over the existing ma chines in machine line ν at time t If the current quality of machine is qν t then by spending one unit of the final good this new firm innovates at the flow rate ηˆzν tqν t where η is a strictly decreasing continuous and differentiable function and ˆzν t is RD ex penditure by new entrants for machine line ν at time t Incumbents also have access to the same technology for radical innovation as the entrants However Arrows replacement effect implies that incumbents would never use this technology since entrants make zero profits from this technology the profits of incumbents would be negative see Exercise 1424 Incumbents still find it profitable to use the technology for incremental innovations which is not available to entrants The presence of the strictly decreasing function η which was also used in Section 1422 captures the fact that when many firms are undertaking RD to replace the same machine line they are likely to try similar ideas thus there will be some amount of external diminishing returns new entrants are fishing out of the same pond Since each entrant attempting RD on this line is potentially small they all take ηˆzν t as given Throughout I assume that zηz is strictly increasing in z so that greater aggregate RD toward a particular machine line increases the overall probability of discovering a superior machine I also suppose that ηz satisfies the following Inadatype assumptions lim z ηz 0 and lim z0 ηz 1434 An innovation by an entrant leads to a new machine of quality κqν t where κ λ Therefore innovations by entrants are more radical than those of incumbents Existing empirical evidence from studies of innovation support the notion that innovations by new entrants are more significant or radical than those of incumbents7 Whether the entrant was a previous incumbent on this specific machine line or whether it is currently an incumbent in some other machine line does not matter for its technology of innovation 7 Nevertheless it may take a while for the successful entrants to realize the full productivity gains from these innovations and the treatment here abstracts from this aspect 143 Innovation by Incumbents and Entrants 475 Moreover since the labor market is competitive the wage rate at any point in time is given by 1412 as before Let us next determine RD effort levels by incumbents and entrants To do this let us write the net present value of a monopolist with the highest quality of machine q at time t in machine line ν This value satisfies the standard HJB equation rtV ν t q V ν t q max zνtqoπν t q zν t qqν t 1440 φzν t qV ν t λq V ν t q ˆzν t qηˆzν t qV ν t q where ˆzν t qηˆzν t q is the rate at which radical innovations by entrants occur in sector ν at time t and φzν q q is the rate at which the incumbent improves its technology The first term in 1440 πν t q is the flow of profits given by 148 while the second term is the expenditure of the incumbent for improving the quality of its product The second line includes changes in the value of the incumbent due to innovation either by itself at the rate φzν t q the quality of its product increases from q to λq or by an entrant at the rate ˆzν t qηˆzν t q the incumbent is replaced and receives zero value from then on8 The value function is written with a maximum on the righthand side since zν t q is a choice variable for the incumbent Free entry by entrants implies a freeentry condition similar to 1414 in Section 141 ηˆzν t qV ν t κq qν t ˆzν t q 0 and ηˆzν t qV ν t κq qν t if ˆzν t q 0 1441 which takes into account that by spending an amount qν t the entrant generates a flow rate of innovation equal to ηˆz leading a product of quality κq thus earning the value ηˆzν t qV ν t κq In addition the incumbents choice of RD effort implies a similar complementary slack ness condition φV ν t λq V ν t q qν t zν t q 0 and φV ν t λq V ν t q qν t if zν t q 0 1442 Finally household maximization implies the familiar Euler equation and the transversality condition given by 1415 and 1416 as before As usual in the BGP we have rt r from 1415 Moreover zν t q zq and ˆzν t q ˆzq These together imply that in the BGP V ν t q 0 and V ν t q V q Furthermore since profits and costs are both proportional to quality q we can also see that ˆzq ˆz and V q vq Exercise 1423 in fact shows that ˆzν t q ˆzt and V ν t q vtq in any equilibrium even outside the BGP These results enable a straight forward characterization of the BGP and the dynamic equilibrium9 Let us first look for an interior BGP equilibrium This solution implies that incumbents undertake research and thus φV ν t λq V ν t q qν t 1443 8 The fact that the incumbent receives a zero value from then on follows from the assumption that a previous incumbent has no advantage relative to other entrants in competing for another round of innovations 9 While ˆzq ˆz for all q it is not necessarily true that zq z for all q In fact as we will see the equilibrium only pins down the average RD intensity of incumbents 488 Chapter 14 Models of Schumpeterian Growth undertake less RD Intuitively the benefits of further RD investments are decreasing in the technology gap since greater values of the technology gap translate into smaller increases markups and profits recall 1469 The fact that leaders who are sufficiently ahead of their competitors undertake little RD is the main reason composition effects play an important role in this model For example all else equal closing the technology gaps between leaders and followers increases RD spending and equilibrium growth Proposition 149 In any steadystate equilibrium z n1 z n for all n 1 and moreover z n1 z n if z n 0 Furthermore z 0 z 1 and z 0 z 1 Proof From 1481 δn1 vn1 vn vn vn1 δn 1489 is sufficient to establish that z n1 z n Let us write ρvn max zn 1 λn ωGzn znvn1 vn z 1 κv0 1490 where ρ ρ z 1 κ Since z n1 z n and z n1 are maximizers of the value functions vn1 vn and vn1 respectively 1490 implies ρvn1 1 λn1 ωGz n1 z n1vn2 vn1 z 1 κv0 ρvn 1 λn ωGz n1 z n1vn1 vn z 1 κv0 ρvn 1 λn ωGz n1 z n1vn1 vn z 1 κv0 ρvn1 1 λn1 ωGz n1 z n1vn vn1 z 1 κv0 1491 Now taking differences with ρvn and using the definition of δn yields ρδn1 λn1 λ1 z n1δn2 δn1 ρδn λn11 λ1 z n1δn1 δn Therefore ρ z n1δn1 δn kn z n1δn2 δn1 where kn λ 12λn1 0 Now to obtain a contradiction suppose that δn1 δn 0 From the previous equation this implies that δn2 δn1 0 since kn is strictly positive Repeating this argument successively we have that if δn1 δn 0 then δn1 δn 0 for all n n However we know from Proposition 148 that vn n0 is strictly increasing and converges to a constant v Thus δn1 δn 0 for sufficiently large n with δn 0 which contradicts the hypothesis that δn1 δn 0 for all n n 0 and establishes that z n1 z n To see that the inequality is strict when z n 0 it suffices to note that 1489 has already been established that is δn1 δn 0 Thus if 1481 has a positive solution then we necessarily have z n1 z Proof of zn 0 z 1 1479 can be written as ρv0 ωGz 0 z 0v1 v1 2v0 1492 145 Taking Stock 489 We have v0 0 from Proposition 148 Suppose v0 0 Then 1492 implies z 0 0 and v1 v1 2v0 0 or 1493 v1 v0 v0 v1 This inequality combined with 1483 and 1482 yields z 0 z 1 Suppose next that v0 0 The Inequality 1493 now holds as a weak inequality and implies that z 0 z 1 Moreover since G is strictly convex and z 0 is given by 1483 1492 implies z 0 0 and thus z 1 0 Proof of z 0 z 1 See Exercise 1432 This proposition therefore shows that the greatest amount of RD is undertaken in neck andneck industries This explains why composition effects can increase aggregate innovation Exercise 1433 shows how a relaxation of IPR protection can increase growth So far I have not provided a closedform solution for the growth rate in this economy It turns out that this is generally not possible because of the endogenous market structure in these types of models Nevertheless it can be proved that a steadystate equilibrium exists in this economy though the proof is somewhat more involved and does not generate additional insights for our purposes see Acemoglu and Akcigit 2006 An important feature is that equilibrium markups are endogenous and evolve over time as a function of competition between the firms producing in the same product line More importantly Proposition 149 implies that when a particular firm is sufficiently ahead of its rival it undertakes less RD Therefore this model in contrast to the baseline Schumpeterian model and also contrary to all expanding varieties models implies that greater competition eg that generated by closing the gap between the followers and leaders may increase growth because it induces the leaders to undertake more RD to escape the competition from the followers Similarly the model can be extended to incorporate different market structures and entry barriers and in this case the effect of competition on growth can be positive or negative 145 Taking Stock This chapter presented the basic Schumpeterian model of economic growth Schumpeterian growth incorporates the process of creative destruction where new products or machines replace older models and new firms replace incumbent producers The baseline model features process innovations leading to quality improvements The description of economic growth that emerges from this model is in many ways more realistic than the expanding variety models In particular technological progress does not always correspond to new products or machines complementing existing ones but instead involves the creation of higher quality producers replacing incumbents Arrows replacement effect discussed in Chapter 12 implies that there is a strong incentive for new entrants to undertake research because the new higherquality products will replace the products of incumbents leading to Schumpeterian creative destruction as the engine of economic growth Even though the description of economic growth in this model is richer the mathematical structure turns out to be quite similar to the models with expanding varieties In reduced form the model again resembles an AK economy An important difference is that now the growth rate of the economy through the rate of replacement of old products affects the value of innovation A major insight of Schumpeterian models is that growth comes with potential conflicts of interest The process of creative destruction destroys the monopoly rents of previous incum bents This raises the possibility that distortionary policies may arise endogenously as a way of protecting the rents of politically powerful incumbents Models of creative destruction thus 490 Chapter 14 Models of Schumpeterian Growth naturally raise the political economy issues that are central for understanding the fundamental causes of economic growth and provide us with insights about both the endogenous nature of technology and the potential resistance to technological change Schumpeterian models also enable us to make greater contact with the industrial organi zation of innovation The process of creative destruction implies that market structures may be evolving endogenously over time Nevertheless the baseline Schumpeterian models have a number of shortcomings and addressing these is an interesting and important area for future research An important discrepancy between the baseline models and the data is that while the models predict all productivity growth should come from creative destruction and entry in the data much of it comes from the incumbent firms and plants Section 143 provided a first look at how the baseline models can be extended to account for these patterns and to provide a richer framework for the analysis of the industrial organization of innovation A second important shortcoming of the baseline models is that they predict that markups are constant and there is always a single firm supplying the entire market These implications can also be relaxed by considering a richer framework for example by allowing cumulative or stepbystep innova tion and competition between multiple firms that engage in innovation Section 144 showed how the baseline model can be extended in this direction Perhaps more interestingly in mod els that incorporate different aspects of the industrial organization of innovation the effects of competition and patent protection on economic growth are potentially richer This observation suggests that extensions of Schumpeterian models might provide a useful framework for the analysis of a range of policies including antitrust licensing and IPR policies on growth 146 References and Literature The baseline model of Schumpeterian growth presented in Section 141 is based on the work by Aghion and Howitt 1992 Similar models have also been developed by Segerstrom Anant and Dinopoulos 1990 and Grossman and Helpman 1991ab Aghion and Howitt 1998 provide an excellent survey of many Schumpeterian models of economic growth and numerous extensions The specific modeling assumptions made in the presentation here draw on Acemoglu 1998 The original Aghion and Howitt 1992 approach is very similar to that used in Section 142 Aghion and Howitt 1992 also discuss uneven growth and potential growth cycles which were presented in Section 142 Uneven growth and cycles are also possible in other models of endogenous technology as shown by Matsuyama 1999 I only discussed the possibility of such cycles in the context of Schumpeterian growth since the forces leading to such cycles are more pronounced in these models The effect of creative destruction on unemployment was first studied in Aghion and Howitt 1994 The implications of creative destruction for firmspecific investments are discussed in Francois and Roberts 2003 and in Martimort and Verdier 2004 The model in Section 143 draws on Acemoglu 2008b and is a first attempt to introducing productivity growth driven both by incumbents and by entrants Klette and Kortum 2004 construct a related model of firm and aggregate innovation dynamics based on expanding product varieties Klepper 1996 documents various facts about the firm size entry and exit and innovation Stepbystep or cumulative innovations have been analyzed in Aghion Harris and Vickers 1999 and Aghion et al 2001 The model presented here is a simplified version of Acemoglu and Akcigit 2006 The notion of MPE used in Section 144 is further discussed in Appendix C and in Fudenberg and Tirole 1994 494 Chapter 14 Models of Schumpeterian Growth 1421 Consider the model discussed in Section 1422 a Choose a functional form for η such that equations 1432 have solutions L1 R and L2 R L1 R Explain why when such solutions exist there is an equilibrium with twoperiod endogenous cycles b Show that even when such solutions exist there also exists a steadystate equilibrium with constant research c Show that when such solutions do not exist there exists an equilibrium that exhibits oscilla tory transitional dynamics converging to the steady state in part b 1422 Show that the qualitative results in Section 1422 generalize when there is a single firm under taking research thus internalizing the effect of LR on ηLR 1423 This exercise sketches the proof of Proposition 145 a Note that in an interior BGP where φV ν t λq V ν t q q V must be linear in q and thus V ν t q vq as used in the text Given this observation show that ˆz is uniquely determined by 1445 and is strictly positive and 1448 gives the unique BGP growth rate which is also strictly positive Next use 1448 and 1452 to show that z is also strictly positive Finally show that the transversality condition 1416 is satisfied when 1453 holds b Now show that the interior BGP from part a also gives the unique dynamic equilibrium path First show that when 1443 holds V ν t q is everywhere linear in q and thus can be written as V ν t q vtq for some function vt Therefore from 1443 φλ 1vt 1 for all t Differentiating this equation with respect to time establish that 1441 must hold as equality so that ηˆzν t κ1qvt 1 for all t From this conclude that rtv βL ˆzηˆzv for all t and thus all variables must immediately take their BGP values rt r and ˆzt ˆz for all t Second sketch the argument for the case in which 1443 does not hold for some ν N 0 1 q and t Hint use 1440 to derive a differential equation for ˆzν t q and show that the unique steady state of this differential equation is the BGP allocation above and this steady state is unstable 1424 Suppose that in the model of Section 143 incumbents also have access to the radical innovation technology used by entrants Show that there cannot exist an equilibrium where incumbents undertake positive RD with this technology 1425 Set up the social planners problem of maximizing the utility of the representative household in Section 143 a Show that this maximization problem corresponds to a concave currentvalue Hamiltonian and derive the unique solution to this problem Show that this solution involves the consump tion of the representative household growing at a constant rate at all points b Show that the social planner may choose higher growth because she avoids the monopoly markup over machines Alternatively she may choose lower entry because of the negative externality in the research process Give numerical examples in which the growth rate in the Pareto optimal allocation is greater than or less than the decentralized growth rate 1426 Consider the model of Section 143 and suppose that the RD technology of the incumbents for innovation is such that if an incumbent with a machine of quality q spends an amount zq for incremental innovations then the flow rate of innovation is φz and this innovation again increases the quality of the incumbents machine to λq Assume that φz is strictly increasing strictly concave differentiable and satisfies limz0 φz and limz φz 0 a Focus on steadystate BGP equilibria and conjecture that V q vq Using this conjecture show that incumbents choose RD intensity z such that λ 1v φz Combining this equation with the freeentry condition for entrants and the equation for growth rate given by 1452 show that there exists a unique BGP equilibrium under the conjecture that V q is linear 496 Chapter 14 Models of Schumpeterian Growth determined as a draw from the uniform distribution 0 c1 and the marginal cost of the second duopolist is determined as an independent draw from 0 c2 Both cost realizations are observed and then prices are set Demand is given by Q A P with A 2 maxc1 c2 a Characterize the equilibrium pricing strategies and calculate expected ex ante profits of the two duopolists b Now imagine that both duopolists start with a cost distribution 0 cand can undertake RD at cost μ If they do with probability η their cost distribution shifts to 0 c α where α c Find the conditions under which one of the duopolists invests in RD and the conditions under which both do c What happens when c declines Interpreting the decline in c as increased competition discuss the effect of increased competition on innovation incentives Why is the answer different from that implied by the baseline endogenous technological change models of expanding varieties or Schumpeterian growth 15 Directed Technological Change T he previous two chapters introduced the basic models of endogenous technological change These models provide us with a tractable framework for the analysis of aggregate technological change but focus on a single type of technology Even when there are multiple types of machines they all play the same role in increasing aggregate productivity There are two important respects in which these models are incomplete First technological change in practice is often not neutral it benefits some factors of production and some agents in the economy more than others Only in special cases such as in economies with Cobb Douglas aggregate production functions can these types of biases be ignored The study of why technological change is sometimes biased toward certain factors or sectors is both important for understanding the nature of endogenous technology and also for clarifying the distributional effects of technological change which determine which groups embrace new technologies and which oppose them Second limiting the analysis to only one type of technological change potentially obscures the different competing effects that determine the nature of technological change The purpose of this chapter is to extend the models of the last two chapters to consider directed technological change which endogenizes the direction and bias of new technologies that are developed and adopted Models of directed technological change not only generate new insights about the nature of endogenous technological progress but also enable us to ask and answer new questions about recent and historical technological developments I start with a brief discussion of a range of economic problems in which considering the endogenous bias of technology is important and also present some of the general economic insights that are important in models of directed technological change The main results are presented in Section 153 The rest of the chapter generalizes these results and presents a few of their applications Section 156 uses these models to return to the question raised in Chapter 2 concerning why technological change might take a purely laboraugmenting Harrodneutral form Section 158 presents an alternative approach to this question suggested by Jones 2005 497 498 Chapter 15 Directed Technological Change 151 Importance of Biased Technological Change To see the potential importance of the biased technological change let us first review a number of examples 1 Perhaps the most important example of biased technological change is the socalled skillbiased technological changewhich plays an important role in the analysis of recent changes in the wage structure Figure 151 plots a measure of the relative supply of skills defined as the number of college equivalent workers divided by noncollege equivalents and a measure of the return to skills the college premium It shows that over the past 60 years the US relative supply of skills has increased rapidly but there has been no tendency for the returns to college to fallon the contrary there has been an increase in the college premium over this time period The standard explanation for this pattern is that new technologies over the postwar period have been skill biased In fact at some level this statement is tautological if skilled and unskilled workers are imperfect substitutes an increase in the relative supply of skills without some countervailing skillbiased changes in demand will necessarily reduce the skill premium The figure also shows that beginning in the late 1960s the relative supply of skills increased more rapidly than before compare the slope of the relative supply curve before and after 1969 Starting in the late 1970s the skill premium also increases more rapidly than before The standard explanation for this increase is an acceleration in the skill bias of technological change that happens to be coincidental with or following shortly after the significant changes in the relative supply of skills An obvious question concerns why technological changes have been skillbiased over the past 60 or even 100 years Relatedly why does it appear that skillbiased technological change accelerated starting in the 1970s precisely when the supply Wage premium Relative skill supply 00 02 04 06 08 Relative supply of college skills 03 04 05 06 College wage premium 1939 1949 1959 1969 1979 1989 1999 FIGURE 151 Relative supply of college graduates and the college premium in the US labor market 151 Importance of Biased Technological Change 499 of skills increased rapidly While some economists are happy to treat the bias of technological change as exogenous this is not entirely satisfactory We have seen that understanding the endogenous nature of technology is important for our study of crosscountry income differences and the process of modern economic growth It is unlikely that while the amount of aggregate technological change is endogenous the bias of technological change is entirely exogenous It is therefore important to study the determinants of endogenous bias of technological change and ask why technological change has become more skillbiased in recent decades 2 This conclusion is strengthened when we look at the historical process of technological change In contrast to the developments during recent decades technological changes during the eighteenth and nineteenth centuries appear to have been unskillbiased The artisan shop was replaced by the factory and later by interchangeable parts and the assembly line Products previously manufactured by skilled artisans started to be produced in factories by workers with relatively few skills and many previously complex tasks were simplified reducing the demand for skilled workers Mokyr 1990 p 137 summarizes this process as follows First in firearms then in clocks pumps locks mechanical reapers typewriters sewing machines and eventually in engines and bicycles interchangeable parts technology proved superior and replaced the skilled artisans working with chisel and file Even though the types of skills valued in the labor market during the nineteenth century were different from those supplied by college graduates in todays labor mar kets the juxtaposition of technological change biased toward college graduates in the recent past and biased against the most skilled workers of the time in the nineteenth century is both puzzling and intriguing It raises the question why was technological change which has been generally skillbiased over the twentieth century biased toward unskilled workers in the nineteenth century 3 As another example consider the potential effect of labor market conditions on tech nological change Beginning in the late 1960s and the early 1970s both unemployment and the share of labor in national income increased rapidly in a number of continental European countries During the 1980s unemployment continued to increase but the labor share started a steep decline and in many countries it fell below its initial level Blanchard 1997 interprets the first phase as the response of these economies to a wage push by workers and the second phase as a possible consequence of capitalbiased tech nological changes Is there a connection between capitalbiased technological changes in European economies and the wage push preceding it 4 The other obvious example of the importance of directed technological change is the common restriction to Harrodneutral purely laboraugmenting technological progress in growth models Recall from Chapters 2 and 8 that if technological change is not laboraugmenting equilibrium growth will not be balanced But a range of evi dence suggests that modern economic growth has been relatively balanced Is there any reason to expect technological change to be endogenously laboraugmenting This chapter shows that a framework of directed technological change can provide potential answers to these questions The main insight is to think of profit incentives as affecting not only the amount but also the direction of technological change Before presenting detailed models let us review the basic arguments which are quite intuitive 500 Chapter 15 Directed Technological Change Imagine an economy with two different factors of production say L and H correspond ing eg to unskilled and skilled workers and two different types of technologies that can complement augment either one or the other factor We would expect that when the prof itability of Haugmenting technologies is greater than the Laugmenting technologies more of the former type will be developed by profitmaximizing research firms What determines the relative profitability of developing different technologies The answer to this question sum marizes most of the economics in the models of directed technological change Two potentially counteracting effects shape the relative profitabilities of different types of technologies 1 The price effect there are stronger incentives to develop technologies when the goods produced by these technologies command higher prices 2 The market size effect it is more profitable to develop technologies that have a larger market eg for the reasons discussed in Chapter 12 An important result of the analysis in this chapter is that this market size effect is powerful enough to outweigh the price effect In fact under fairly general conditions the following two results hold Weak Equilibrium Relative Bias An increase in the relative supply of a factor always induces technological change that is biased in favor of this factor Strong Equilibrium Relative Bias If the elasticity of substitution between factors is sufficiently large an increase in the relative supply of a factor induces sufficiently strong technological change bias toward this factor that the endogenoustechnology relative demand curve becomes upward sloping To explain these concepts in a little more detail suppose that the inverse relative demand curve takes the form wHwL DHL A where wHwL is the relative price of the H factor relative to the L factor HL is the relative supply of the H factor and A R is a technology term for now taken to be onedimensional for simplicity Technology A is H biased if D is increasing in A so that a higher A increases the relative demand for the H factor Standard microeconomic theory implies that D is always decreasing in HL Equilibrium bias concerns the behavior of A as HL changes so let us write this as AHL As a normalization suppose that A is Hbiased so that DHL A is increasing in A Weak equilibrium bias then corresponds to AHL being increasing nondecreasing in HL Strong equilibrium bias on the other hand implies that AHL is sufficiently responsive to an increase in HL that the total effect of the change in relative supply HL is to increase wHwL In other words let the endogenoustechnology relative demand curve be wHwL DHL AHL DHL Then strong equilibrium bias corresponds to this endogenoustechnology relative demand curve D being increasing At first both the weak and the strong equilibrium bias results appear surprising However they become quite intuitive once the logic of directed technological change is understood Moreover they have a range of important implications In particular Section 1533 shows how the weak and the strong relative bias results provide us with potential answers to the questions posed at the beginning of this section 152 Basics and Definitions Before studying directed technological change it is useful to clarify the difference between factoraugmenting and factorbiased technological changes which are sometimes confused in the literature Suppose that the production side of the economy can be represented by an aggregate production function 153 Baseline Model of Directed Technological Change 511 0 Relative supply of skills Skill premium ET2 0 ET1 Endogenous technology demand ET1 Constant technology demand CT Endogenous technology demand ET2 CT FIGURE 153 The relationship between the relative supply of skills and the skill premium in the model of directed technical change There is an obvious analogy between this result and Samuelsons LeChatelier Principle which states that longrun demand curves which apply when all factors can adjust must be more elastic than the shortrun demand curves which hold some factors constant We can think of the endogenoustechnology demand curve as adjusting the factors of production corresponding to technology However the analogy is imperfect because the effects here are caused by general equilibrium changes while the LeChatelier Principle focuses on partial equilibrium effects In fact in basic producer theory with or without the LeChatelier effects all demand curves must be downward sloping whereas here ET2 which applies when the conditions of Proposition 154 hold is upward sloping higher levels of relative supply of skills correspond to higher skill premiums ωET 2 is greater than ω0 in Figure 153 A complementary intuition for this result can be obtained by going back to the importance of the nonrivalry of ideas discussed in Chapter 12 Here as in the basic endogenous technology models of the last two chapters the nonrivalry of ideas leads to an aggregate production function that exhibits increasing returns to scale in all factors including technologies It is this increasing returns to scale that leads to potentially upwardsloping relative demand curves Put differently the market size effect which results from the nonrivalry of ideas and is at the root of aggregate increasing returns can create sufficiently strong induced technological change to increase the relative marginal product and the relative price of the factor that has become more abundant 1533 Implications One of the most interesting applications of Propositions 153 and 154 is to changes in the wage structure and the skill premium For this application suppose that H stands for college educated workers In the US labor market the skill premium has shown no tendency to decline 512 Chapter 15 Directed Technological Change despite a large increase in the supply of collegeeducated workers On the contrary following a brief period of decline during the 1970s in the face of the large increase in the supply of college educated workers the skill college premium has increased sharply throughout the 1980s and 1990s to reach a level not experienced in the postwar era Figure 151 above showed these general patterns The most popular explanation for these patterns is skillbiased technological change For example computers or new information technologies IT are argued to favor skilled workers relative to unskilled workers But why should the economy adopt and develop more skillbiased technologies throughout the past 30 years or more generally throughout the entire twentieth century This question becomes more relevant once we remember that during the nineteenth century many of the technologies that were fueling economic growth such as the factory system and the spinning and weaving innovations were unskillbiased rather than skillbiased Thus in summary the following stylized facts are relevant 1 secular skillbiased technological change increasing the demand for skills throughout the twentieth century 2 possible acceleration in skillbiased technological change over the past 25 years and 3 a range of important technologies biased against skilled workers during the nineteenth century Propositions 153 and 154 provide us with a framework for thinking about these issues 1 According to Propositions 153 and 154 the increase in the number of skilled workers that has taken place throughout the twentieth century should cause steady skillbiased technological change Therefore models of directed technological change offer a natural explanation for the secular skillbiased technological developments of the past century 2 The more rapid increase in the number of skilled workers over the past 25 years shown in Figure 151 should also induce an acceleration in skillbiased technological change If σ 2 and Proposition 154 applies then this acceleration can also lead to a rapid increase in the skill premium How this class of models might account for the dynamics of factor prices in the face of endogenously changing technologies is discussed later in this section 3 Can the framework also explain the prevalence of skillreplacinglaborbiased techno logical change in the late eighteenth and nineteenth centuries While we know less about both changes in relative supplies and technological developments during these historical periods available evidence suggests that there were large increases in the number of unskilled workers available to be employed in the factories Bairoch 1988 p 245 for example describes this rapid expansion of unskilled labor in the cities as follows between 1740 and 1840 the population of England went up from 6 mil lion to 157 million while the agricultural labor force represented 6070 of the total work force in 1740 by 1840 it represented only 22 Habakkuks wellknown account of nineteenthcentury technological development 1962 pp 136137 also em phasizes the increase in the supply of unskilled labor in English cities and attributes it to a variety of factors First technical changes in agriculture increased the supply of labor available to industry p 137 Second population was increasing very rapidly p 136 Third labor reserves of rural industry came to the cities Fourth there was a large influx of labor from Ireland p 137 In addition to accounting for the recent skillbiased technological developments and for the historical technologies that appear to have been biased toward unskilled workers this framework also gives a potential interpretation for the dynamics of the college premium during 153 Baseline Model of Directed Technological Change 513 0 Skill premium Longrun relative demand for skills Longrun premium Shortrun response Initial premium Exogenous shift in relative supply FIGURE 154 Dynamics of the skill premium in response to an exogenous increase in the relative supply of skills with an upwardsloping endogenoustechnology relative demand curve the 1970s and 1980s It is reasonable to presume that NHNL changes slowly as a result of the gradual buildup and development of new technologies as the analysis of transitional dynamics in Proposition 152 shows In this case a rapid increase in the supply of skills first reduces the skill premium as the economy moves along a constant technology constant NHNL curve as shown in Figure 154 After a while technology starts adjusting and the economy moves back to the upwardsloping relative demand curve with a relatively sharp increase in the college premium This approach can therefore explain both the decline in the college premium during the 1970s and its subsequent large surge and relates both of these phenomena to the large increase in the supply of skilled workers If on the other hand σ 2 then the longrun relative demand curve is downward sloping though again it is shallower than the shortrun relative demand curve Following the increase in the relative supply of skills there is again an initial decline in the college premium and as technology starts adjusting the skill premium increases But it ends up below its initial level Figure 155 Consequently a model of directed technological change can shed light both on the secular skill bias of technology and on the relatively shortrun changes in technologyinduced factor prices Before discussing other implications of these results a couple of further issues are worth noting First Proposition 154 shows that upwardsloping relative demand curves arise only when σ 2 In the context of substitution between skilled and unskilled workers an elasticity of substitution much higher than 2 is unlikely Most estimates put the elasticity of substitution between 14 and 2 Section 154 shows that whether or not σ 2 is not critical for this result what is necessary for upwardsloping relative demand curves is that σ should be greater than a certain threshold see in particular Proposition 158 Second we would like to understand the relationship between the market size and the scale effects in particular whether the results on 514 Chapter 15 Directed Technological Change 0 Skill premium Longrun relative demand for skills Longrun premium Shortrun response Initial premium Exogenous shift in relative supply FIGURE 155 Dynamics of the skill premium in response to an increase in the relative supply of skills with a downwardsloping endogenoustechnology relative demand curve induced technological change are an artifact of the scale effect which many economists do not view as an attractive feature of endogenous technological change models Section 155 shows that this is not the case and exactly the same results apply when scale effects are removed Third we would like to apply these ideas to investigate whether there are reasons for technological change to be endogenously laboraugmenting in the neoclassical growth model This issue is investigated in Section 156 Finally it is also useful to contrast the equilibrium allocation to the Pareto optimal allocation which is done in Exercise 156 This exercise shows that the qualitative results here including the weak and the strong equilibrium bias results also hold in the Pareto optimal allocation 154 Directed Technological Change with Knowledge Spillovers I now consider the directed technological change model of the previous section with knowledge spillovers This exercise has three purposes First it shows how the main results on the direction of technological change can be generalized to a model using the other common specification of the innovation possibilities frontier Second this analysis shows that the strong bias result in Proposition 154 can hold under somewhat weaker conditions Third this formulation is essential for the study of laboraugmenting technological change in Section 156 The labequipment specification of the innovation possibilities frontier is special in one respect it does not feature state dependence State dependence refers to the phenomenon in which the path of past innovations affects the relative costs of different types of innovations The labequipment specification implied that RD spending always leads to the same increase in the number of Laugmenting and Haugmenting machines I now introduce a specification 154 Directed Technological Change with Knowledge Spillovers 515 with knowledge spillovers which allows for state dependence Recall that as discussed in Section 132 in Chapter 13 when there are scarce factors used for RD then growth cannot be sustained by continuously increasing the amount of these factors allocated to RD Therefore to achieve sustained growth these factors need to become more and more productive over time because of spillovers from past research Here for simplicity let us assume that RD is carried out by scientists and that there is a constant supply of scientists equal to S Exercise 1517 shows that the results are similar when workers can be employed in the RD sector With only one sector the analysis in Section 132 indicates that sustained endogenous growth requires NN to be proportional to S With two sectors there is instead a variety of specifications with different degrees of state dependence because productivity in each sector can depend on the state of knowledge in both sectors A flexible formulation is the following NLt ηLNLt1δ2NHt1δ2SLt and NHt ηHNLt1δ2NHt1δ2SHt 1531 where δ 1 and SLt is the number of scientists working to produce Laugmenting machines while SHt denotes the number of scientists working on Haugmenting machines Clearly market clearing for scientists requires that SLt SHt S 1532 In this specification δ measures the degree of state dependence when δ 0 there is no state dependence NHSH NLSL ηHηL regardless of the levels of NL and NH because both NL and NH create spillovers for current research in both sectors In this case the results are identical to those in the previous section In contrast when δ 1 there is an extreme amount of state dependence In this case NHSH NLSL ηHNHηLNL so an increase in the stock of Laugmenting machines today makes future laborcomplementary innovations cheaper but has no effect on the cost of Haugmenting innovations This discussion clarifies the role of the parameter δ and the meaning of state dependence In some sense state dependence adds another layer of increasing returns this time not for the entire economy but for specific technology lines In particular a significant amount of state dependence implies that when NH is high relative to NL it becomes more profitable to undertake more NHtype innovations With this formulation of the innovation possibilities frontier the freeentry conditions become see Exercise 157 ηLNLt1δ2NHt1δ2VLt wSt ηLNLt1δ2NHt1δ2VLt wSt if SLt 0 1533 and ηHNLt1δ2NHt1δ2VHt wSt ηHNLt1δ2NHt1δ2VHt wSt if SHt 0 1534 where wSt denotes the wage of a scientist at time t When both of these freeentry conditions hold BGP technology market clearing implies ηLNLtδπL ηHNHtδπH 1535 where δ captures the importance of state dependence in the technology market clearing con dition and profits are not conditioned on time since they refer to the BGP values which are 154 Directed Technological Change with Knowledge Spillovers 517 In contrast to the model with the labequipment technology transitional dynamics do not always take the economy to the BGP equilibrium however This is because of the additional increasing returns to scale mentioned above With a high degree of state dependence when NH0 is very high relative to NL0 it may no longer be profitable for firms to undertake further RD directed at laboraugmenting Laugmenting technologies Whether this is so depends on a comparison of the degree of state dependence δ and the elasticity of substitution σ The elasticity of substitution matters because it regulates how prices change as a function of the composition of technology and thus determines the strength of the price effect on the direction of technological change The next proposition analyzes the transitional dynamics in this case Proposition 156 Consider the directed technological change model with knowledge spillovers and state dependence in the innovation possibilities frontier Suppose that σ 1δ Then starting with any NH0 0 and NL0 0 there exists a unique equilibrium path If NH0NL0 NHNL as given by 1536 then ZHt 0 and ZLt 0 until NHtNLt NHNL If NH0NL0 NHNL then ZHt 0 and ZLt 0 until NHtNLt NHNL If σ 1δ then starting with NH0NL0 NHNL the economy tends to NHtNLt as t and starting with NH0NL0 NHNL it tends to NHtNLt 0 as t Proof See Exercise 1511 Of greater interest for the focus here are the results on the direction of technological change The first result on weak equilibrium bias immediately generalizes from the previous section Proposition 157 Consider the directed technological change model with knowledge spillovers and state dependence in the innovation possibilities frontier Then there is always weak equilibrium relative bias in the sense that an increase in HL always induces relatively Hbiased technological change Proof See Exercise 1512 While the results regarding weak bias have not changed inspection of 1537 shows that it is now easier to obtain strong equilibrium relative bias Proposition 158 Consider the directed technological change model with knowledge spillovers and state dependence in the innovation possibilities frontier Then if σ 2 δ there is strong equilibrium relative bias in the sense that an increase in HL raises the relative marginal product and the relative wage of the H factor compared to the L factor Intuitively the additional increasing returns to scale coming from state dependence makes strong bias easier to obtain because the induced technology effect is stronger When a particular factor say H becomes more abundant this encourages an increase in NH relative to NL in the case where σ 1 State dependence makes further increases in NH more profitable culminating in a larger effect on NHNL Since with σ 1 greater values of NHNL increase the relative price of factor H compared to L this tends to make the strong bias result more likely Returning to the discussion of the implications of the strong bias results for the behavior of the skill premium in the US labor market Proposition 158 implies that values of the elasticity of substitution between skilled and unskilled labor significantly less than 2 may be sufficient to generate strong equilibrium bias How much lower than 2 the elasticity of substitution can be 156 Endogenous LaborAugmenting Technological Change 519 Proposition 159 Consider the directed technological change model with no scale effects described above Then there is always weak equilibrium relative bias meaning that an increase in HL always induces relatively Hbiased technological change Moreover if σ 2 λ then there is strong equilibrium relative bias in the sense that an increase in HL raises the relative marginal product and the relative wage of the H factor compared to the L factor 156 Endogenous LaborAugmenting Technological Change One of the advantages of the models of directed technological change is that they allow us to investigate why technological change might be purely laboraugmenting as required for balanced growth recall Theorem 26 in Chapter 2 This section shows that models of directed technological change create a natural reason for technology to be more laboraugmenting than capitalaugmenting However under most circumstances the resulting equilibrium is not purely laboraugmenting and as a result a BGP fails to exist Nevertheless in one important special case the model delivers longrun purely laboraugmenting technological changes exactly as in the neoclassical growth model thus providing a rationale for one of the strong assumptions of the standard growth models In thinking about laboraugmenting technological change it is useful to consider a two factor model with H corresponding to capital that is Ht Kt in the aggregate production function 153 Correspondingly let us use NL and NK to denote the varieties of machines in the two sectors Let us also simplify the discussion by assuming that there is no depreciation of capital so that the price of capital Kt is equal to the interest rate rt Let us first note that in the context of capitallabor substitution the empirical evidence suggests that an elasticity of substitution of σ 1 is much more plausible whereas in the case of substitution between skilled and unskilled labor the evidence suggests that σ 1 An elasticity less than 1 is not only consistent with the available empirical evidence but it is also economically plausible For example with the CES production function an elasticity of substitution between capital and labor greater than 1 would imply that production is possible without labor or without capital which appears counterintuitive Now recall that when σ 1 factoraugmenting and factorbiased technologies are reversed Therefore laboraugmenting technological change corresponds to capitalbiased technological change Then the question becomes Under what circumstances would the economy generate relatively capitalbiased technological change And when will the equilibrium technology be sufficiently capital biased that it corresponds to Harrodneutral technological change What distinguishes capital from labor is the fact that it accumulates In other words most growth models feature some type of capital deepening with KtL increasing as the economy grows Then in contrast to the analysis so far which looked at the effect of onetime changes in relative supplies the focus must now be on the implications of continuous changes in the relative supply of capital on technological change In light of this observation the answer to the first question above is straightforward capital deepening combined with Proposition 153 implies that technological change should be more labor than capitalaugmenting The next proposition summarizes the main idea of the previous paragraph For simplic ity this proposition treats the increase in KtL as a sequence of onetime increases full equilibrium dynamics are investigated in the next two propositions 522 Chapter 15 Directed Technological Change elasticity of substitution between capital and labor that is less than 1 induces the economy to strive toward a balanced allocation of effective capital and labor units where effective here refers to capital and labor units augmented with their complementary technologies Since capital accumulates at a constant rate a balanced allocation implies that the productivity of labor should increase faster in particular the economy should converge to an equilibrium path with purely laboraugmenting technological progress 157 Generalizations and Other Applications The results presented so far rely on a range of specific assumptions that are inherent in en dogenous technological change models eg DixitStiglitz preferences and linear structure to ensure sustained growth One may naturally wonder whether the results on weak and strong equilibrium biases generalize to situations in which these assumptions are relaxed The answer is broadly yes In Acemoglu 2007a I show that as long as only factoraugmenting techno logical changes are possible the main results presented here also apply in an environment in which production and cost functions take more general forms In particular in this general en vironment there is always weak relative equilibrium bias in response to increases in relative supplies and there will be strong equilibrium bias when the elasticity of substitution is suffi ciently high However once we allow for a richer menu of technological changes these results do not necessarily hold Nevertheless the essence of the results is much more general In Ace moglu 2007a I define the complementary notions of weak and strong absolute equilibrium biases which refer to whether the equilibrium price of a factor changes as the supply of that factor changes rather than the price of a factor relative to the price of another factor which is what I have focused on in this chapter Under very weak regularity assumptions there is always weak absolute equilibrium bias in the sense that an increase in the supply of a factor always induces technological change biased in favor of that factor Moreover even though standard producer theory implies that an increase in the supply of a factor should reduce its price under plausible assumptions the induced technology effect can be strong enough that the price of the factor that has become more abundant can increase In this case there is strong absolute equilibrium bias and the general equilibrium demand curves for factors are upward sloping Since these results require additional notation and somewhat different mathematical arguments I do not present them here It is also useful to briefly discuss a number of other important applications of the models of directed technological change To save space these are left as exercises In particular Exercise 1519 shows how this model can be used to shed light on the famous Habakkuk hypothesis in economic history which relates the rapid technological progress in nineteenth century United States to relative labor scarcity Despite the importance of this hypothesis in economic history there have been no compelling models of this process This exercise shows why neoclassical models may have difficulties in explaining these patterns and how a model of directed technological change can account for this phenomenon as long as the elasticity of substitution between capital and labor is less than 1 Exercise 1520 shows the effects of international trade on the direction of technological change It highlights that international trade often affects the direction in which new technolo gies are developed and this often works through the price effect emphasized above Exercise 1526 returns to the discussion of the technological change and unemployment experiences of continental European countries discussed above It shows how a wage push shock can first increase equilibrium unemployment and then induce endogenous capital biased technological change which reduces the demand for employment further increasing unemployment 158 An Alternative Approach to LaborAugmenting Technological Change 523 Finally Exercise 1527 shows how the relative supply of factors can be endogenized and studies the twoway causality between relative supplies and relative technology 158 An Alternative Approach to LaborAugmenting Technological Change The models presented so far in this chapter are all based on the basic directed technological change framework developed in Acemoglu 1998 2002a Section 156 showed how this approach can be used to provide conditions under which technological change is endogenously laboraugmenting An alternative approach to this problem is suggested in a recent paper by Jones 2005 I now briefly discuss this alternative approach The models developed so far treat the different types of technologies eg NL and NH in the previous sections as state variables Thus shortrun production functions correspond to the production possibilities sets for given state variables while longrun production functions apply when technology state variables also adjust Jones proposes a different approach building on a classic paper by Houthakker 1955 Houthakker suggested that the aggregate production function should be derived as the upper envelope of different ideas or activities Each technique or activity corresponds to a particular way of combining capital and labor thus to a Leontief production function of these two factors of production However when a producer has access to multiple ways of combining capital and labor the resulting envelope is different than Leontief In a remarkable result Houthakker showed that if the distribution of techniques is given by the Pareto distribution defined formally below this upper envelope of a large number of activities corresponds to a CobbDouglas production function Houthakker thus suggested a justification for CobbDouglas production functions based on activity analysis Jones builds on and extends these insights He argues that the longrun production function should be viewed as the upper envelope of a large number of ideas generated over time At a given point in time the set of ideas that the society has access to is fixed and these ideas determine the shortrun production function of the economy In the long run however the society generates more ideas either exogenously or via RD and the longrun production function is obtained as the upper envelope of this expanding set of ideas Using a combination of Pareto distribution and Leontief production possibilities for a given idea Jones shows that there is a major difference between shortrun and longrun production functions In particular as in Houthakkers analysis the longrun production function takes a CobbDouglas form and implies a constant share of capital in national income However this is not necessarily the case for shortrun production functions Then with an argument similar to that in Section 156 the economy adjusts from the shortrun to longrun production functions by undergoing a form of laboraugmenting technological change I now provide a brief sketch of Joness model focusing on the main economic insights As pointed out above the key building block of Joness model are ideas An idea is a technique for combining capital and labor to produce output At any given point in time the economy has access to a set of ideas Let us denote the set of possible ideas by I and the set of ideas available at time t by It I Each idea i I is represented by a vector ai bi The essence of the model is to construct the production possibilities set of the economy from the set of available ideas To do this we first need to specify how a given idea is used for production Let us suppose that there is a single final good Y that can be produced using any idea i I with a Leontief production function given by Yt minbiKt aiLt 1549 526 Chapter 15 Directed Technological Change so that the longrun distribution of output appropriately normalized converges asymptotically to a Frechet distribution Then as Nt becomes large ie as t and more ideas are discovered the longrun global production function behaves approximately as Yt Nt εtγ NtKtβLtα 1 αβ 1556 where εt is a random variable drawn from a Frechet distribution The intuition for this result is similar to Houthakkers result that aggregation over different units producing with techniques drawn independently from a Pareto distribution leads to a CobbDouglas production func tion The implications are different however In particular since the longrun production function behaves approximately as CobbDouglas it implies that factor shares must be constant in the long run However the shortrun production function for a finite number of ideas is not CobbDouglas Therefore as Nt increases the production function evolves endogenously toward the CobbDouglas limit with constant factor shares and as in the analysis in Section 156 this means that technological change must ultimately become purely laboraugmenting Therefore Joness model shows that insights related to Houthakkers derivation of a static production function also imply that the shortrun production function evolves endogenously on average with laboraugmenting technological change dominating the limiting behavior and making sure that the economy in the long run acts as if it has a CobbDouglas production function Although this idea is interesting and the Pareto distribution appears in many important con texts and has various desirable properties it is not clear whether it provides a compelling reason for technological change to be laboraugmenting in the long run Laboraugmenting technological change should be an equilibrium outcome resulting from the research and inno vation incentives of firms and workers The directed technological change models emphasized how these incentives play out under various equilibrium scenarios In the current model the CobbDouglas production function arises purely from aggregation There are no equilibrium interactions price or market size effects Related to this the unit of analysis is unclear The same argument can be applied to a single firm to an industry or to a region Thus if we are happy with this argument for the economy as a whole we may also wish to apply it to firms industries and regions concluding that the longrun production function of every unit of production or every firm industry and region should be CobbDouglas However existing evidence indicates that there are considerable differences in the production functions across industries and they cannot be well approximated by CobbDouglas production functions see the overview of the evidence on industry and aggregate production functions in Acemoglu 2003a This suggests that it would be interesting to combine the aggregation of different ac tivities or ideas as in Houthakkers and Joness papers with equilibrium interactions which might delineate at what level the aggregation should take place and why it may apply to some economies but not necessarily to single firms or industries 159 Taking Stock This chapter introduced the basic models of directed technological change These approaches differ from the endogenous technological change models of the previous two chapters because they not only determine the evolution of aggregate technology but also the direction and bias of technological change Models of directed technological change enable us to investigate a range of new questions These include the sources of skillbiased technological change over the past 100 years the causes of acceleration in skillbiased technological change during more recent decades the causes of unskilledbiased technological developments during the nineteenth 1510 References and Literature 527 century the impact of international trade on the direction of technological change and the relationship between labor market institutions and the types of technologies that are developed and adopted Last but not least they also enable an investigation of why technological change in neoclassicaltype models may be largely laboraugmenting A relatively simple class of directed technological change models can shed light on all these questions These models are quite tractable and allow closedform solutions for equilibrium relative technologies and longrun growth rates Their implications for the empirical questions mentioned above stem from two important and perhaps at first surprising results which we can refer to as weak equilibrium bias and strong equilibrium bias results The first states that under fairly weak assumptions an increase in the relative supply of a factor always induces endogenous changes in technology that are relatively biased toward that factor Consequently any increase in the ratio of skilled to unskilled workers or in the capitallabor ratio has major implications for the relative productivities of these factors The more surprising result is that for the strong equilibrium bias which states that contrary to basic producer theory relative demand curves can slope up In particular if the elasticity of substitution between factors is sufficiently high a greater relative supply of a factor causes sufficiently strong induced techno logical change to make the resulting relative price of the more abundant factor increase In other words the longrun endogenoustechnology relative demand curve becomes upward slop ing The possibility that relative demand curves may be upward sloping not only has a range of important empirical implications but also illustrates the strength of induced technology effects This chapter also presented a number of applications of these ideas to several empirically important areas Models of directed technological change are in their infancy and there are many theoretical dimensions in which further developments are possible Perhaps more importantly there are also numerous applications of these ideas Finally the models in this chapter have further emphasized that technology should not be thought of as a black box instead it should be modeled as the outcome of decisions by firms individuals and other agents in the economy This implies that profit incentives play a major role both in the aggregate rate of technological progress and in the biases of the technologies that are being developed and adopted 1510 References and Literature Models of directed technological change were developed in Acemoglu 1998 2002a 2003ab 2007a Kiley 1999 and Acemoglu and Zilibotti 2001 These papers use the term di rected technical change but here I used the related term directed technological change to emphasize continuity with the models of endogenous technological change studied in the pre vious chapters The framework presented here builds on Acemoglu 2002a A more general framework without functional form restrictions is presented in Acemoglu 2007a Other papers modeling the direction of technological change include Xu 2001 Gancia 2003 Thoenig and Verdier 2003 Ragot 2003 Duranton 2004 Benabou 2005 Caselli and Coleman 2005 and Jones 2005 Models of directed technological change are closely related to the earlier literature on induced innovation The induced innovation literature was started indirectly by Hicks who in The Theory of Wages 1932 pp 12425 argued A change in the relative prices of the factors of production is itself a spur to invention and to invention of a particular kinddirected to economizing the use of a factor which has become relatively expensive An important paper by Kennedy 1964 introduced the concept of innovation possibilities frontier and argued that it is the form of this frontierrather than the shape of a given neoclassical production functionthat determines the factor distribution of income Kennedy furthermore argued 528 Chapter 15 Directed Technological Change that induced innovations would push the economy to an equilibrium with a constant relative factor share see also Drandakis and Phelps 1965 and Samuelson 1965 Around the same time Habakkuk 1962 published his important treatise American and British Technology in the Nineteenth Century The Search for LaborSaving Inventions where he argued that labor scarcity and the search for laborsaving inventions were central determinants of technological progress The flavor of Habakkuks argument was one of induced innovations labor scarcity increased wages which in turn encouraged laborsaving technological change Nevertheless neither Habakkuk nor the induced innovation literature provided microfounded approaches to technological change or technology adoption For example in Kennedys specification the production function at the firm level exhibited increasing returns to scale because in addition to factor quantities firms could choose technology quantities but this increasing returns to scale was not taken into account in the analysis Similar problems are present in other earlier works as well It was also not clear who undertook the RD activities and how they were financed and priced These shortcomings reduced the interest in this literature The analysis in Acemoglu 1998 and the subsequent work in this area instead starts from the explicit microfoundations of the endogenous technological change models discussed in the previous two chapters The presence of monopolistic competition avoids the problems that the induced innovations literature had with increasing returns to scale Acemoglu 2002a b shows that the specific way in which endogenous technological change is modeled does not affect the major results on the direction of technological change This is also illustrated in Exercises 1518 and 1528 In addition even though the focus here has been on technological progress in Acemoglu 2007a I show that all results generalize to models of technology adoption There I also introduce the alternative concepts of weak absolute bias and strong absolute bias which focus on the marginal product of a factor rather than on the relative marginal product and I prove that there are considerably more general theorems on weak and strong absolute biases In the text here I refer to weak relative bias and strong relative bias to distinguish the results here from the absolute bias results The results in Acemoglu 2007a also show that the CES aggregator used here is unnecessary for the results Nevertheless I have kept the CES structure to simplify the exposition Changes in US wage inequality over the past 60 years are surveyed in Autor Katz and Krueger 1998 Katz and Autor 2000 and Acemoglu 2002b The last paper also discusses how models on directed technological change can provide a good explanation for changes in wage inequality over the past 100 years and also for changes in the direction of technological change in the US and UK economies over the past 200 years There are many studies estimating the elasticity of substitution between skilled and unskilled workers The estimates are typically between 14 and 2 See for example Katz and Murphy 1992 Angrist 1995 Krusell et al 1999 A number of these estimates are summarized and discussed in Hammermesh 1993 and Acemoglu 2002b Evidence that nineteenthcentury technologies were generally labor complementary un skilled biased is provided in James and Skinner 1985 and Mokyr 1990 while Goldin and Katz 1998 argue the same for a range of important early twentiethcentury technologies Blanchard 1997 discusses the persistence of European unemployment and argues that the phase during the 1990s can only be understood by changes in technology reducing demand for highcost labor This idea is the basis of Exercise 1526 Caballero and Hammour 1999 provide an alternative and complementary explanation to that suggested here Acemoglu 2003b suggested that increased international trade can cause endogenous skill biased technological change Exercise 1520 is based on this idea Variants of this story have been developed by Xu 2001 Gancia 2003 and Thoenig and Verdier 2003 The model of longrun purely laboraugmenting technological change presented in Section 156 was first proposed in Acemoglu 2003a and the model presented here is a simplified 1511 Exercises 533 πht 1 ε 1 vtH nt mt and πlt 1 ε 1 wtL mt Interpret these equations Why is the condition that vt is sufficiently larger than wt necessary b Assume moreover that a firm that undertakes RD to replace the skillintensive good has an equal probability of replacing any of the existing n m skillintensive goods Define a BGP as an allocation where n and m grow at the same rate g Show that this condition implies that output and wages of skilled and unskilled workers must grow at the rate gε 1 Hint use the equation for the numeraire setting the price of the final good equal to 1 at each date c Show that in BGP the wages of scientists also grow at the same rate as the wages of skilled and unskilled workers d Show that the BGP must satisfy the following condition bnμ12δ vH r 2 εg1 ε μg1 μ bm wL r 2 εg1 ε where μ mn Hint note that a monopolist producing a laborintensive good will never be replaced but a monopolist producing a skillintensive good faces a constant flow rate of being replaced also use the fact that mn m gμ1 μ e Using demands over varieties ie yν tyν t pν tpν tε characterize the BGP path level of μ What is the effect of an increase in HL on μ Interpret f Why was it necessary to impose δ 1in the innovation possibilities frontier Briefly discuss how the analysis would change if δ 1 1529 Consider the model presented in Section 158 a Show that if capital and labor are allocated in competitive markets in general more than one technique is used in equilibrium Hint construct an example in which there are three ideas i 1 2 and 3 such that when only one can be used it is i 1 but output can be increased by allocating some of labor and capital to ideas 2 and 3 b Show that in this case the aggregation result used in Section 158 does not apply 1530 Suppose that y has a Pareto distribution given by Gy 1 Byα Determine the variance of y and show that it may be infinite 1531 Suppose that y has a Pareto distribution given by Gy 1 Byα with α 1 Show that Ey y y α α 1y What happens if α 1 PART V STOCHASTIC GROWTH T his part of the book focuses on stochastic growth models and provides a brief intro duction to basic tools of stochastic dynamic optimization Stochastic growth models are useful for two related reasons First a range of interesting growth problems involve either aggregate uncertainty or nontrivial individuallevel uncertainty interacting with investment de cisions and the growth process Some of these models are discussed in Chapter 17 Second the stochastic neoclassical growth model has a wide range of applications in macroeconomics and in other areas of dynamic economic analysis Various aspects of the stochastic neoclassical growth model are discussed in the next two chapters The study of stochastic models requires us to extend the dynamic optimization tools of Chapters 6 and 7 to an environment in which either returns or constraints are uncertain governed by probability distributions1 Unfortu nately dynamic optimization under uncertainty is considerably harder than the nonstochastic optimization The generalization of continuoustime methods to stochastic optimization re quires fairly advanced tools from measure theory and stochastic differential equations While continuoustime stochastic optimization methods are very powerful they are not used widely in macroeconomics and economic growth and here I focus on discretetime stochastic mod els Thus the next chapter includes the most straightforward generalization of the discretetime dynamic programming techniques presented in Chapter 6 to stochastic environments A fully rigorous development of stochastic dynamic programming also requires further mathemati cal investment than is typically necessary in most courses on macroeconomics and economic growth To avoid a heavy dose of new mathematical tools in particular a lengthy detour into measure theory at this stage of the book the next chapter develops the basics of stochastic dynamic programming without measure theory 1 Throughout I do not draw a distinction between risk and uncertainty Some economists follow Frank Knight and identify risk with situations in which there is a known probability distribution of events and uncertainty with situations in which such a probability distribution cannot be specified While Knightian uncertainty may be important in a range of situations given the set of models being studied here there is little cost of following the standard practice of using the word uncertainty interchangeably with risk 16 Stochastic Dynamic Programming T his chapter provides an introduction to basic stochastic dynamic programming To avoid the use of measure theory in the main body of the text I first focus on economies in which stochastic variables take finitely many values This restriction enables us to use Markov chains instead of general Markov processes to represent uncertainty Since many commonly used stochastic processes such as those based on normal or uniform distributions fall outside this class I then indicate how the results can be generalized to situations in which stochastic variables can be represented by continuousor a mixture of continuous and discreterandom variables Throughout my purpose is to provide a basic understanding of the tools of stochastic dynamic programming and how they can be used in dynamic macroeconomic models For this reason I make a number of judicious choices rather than attempting to provide the most general results Throughout I focus on stationary problems that is the equivalents of Problems 62 and 63 in Chapter 6 Analogues of Theorems 611 and 612 which applied to nonstationary optimization problems under certainity can be proved using the same arguments in the stochastic case and I omit these results to save space 161 Dynamic Programming with Expectations I use a notation similar to that in Chapter 6 Let us first introduce the stochastic random variable zt Z z1 zN with z1 z2 zN Note that the set Z is finite and thus compact which simplifies the analysis considerably Let the instantaneous payoff at time t be Uxt xt 1 zt 161 where xt X RK for some K 1 and U X X Z R Equation 161 extends the payoff function in Chapter 6 which took the form Uxt xt 1 by making payoffs directly a function of the stochastic variable zt As usual returns are discounted by some discount factor β 0 1 xt again denotes the state variables state vector and xt 1 the control variables control vector at time t The initial values of the state vector x0 and of stochastic variable z0 are taken as given 537 161 Dynamic Programming with Expectations 539 As usual ct denotes per capita consumption at time t and u is the instantaneous utility function The maximand in this problem differs from those studied so far only because of the presence of the expectations operator E0 which stands for expectations conditional on information available at the beginning of time t 0 Expectations are necessary here because the future values of consumption per capita are stochastic as they depend on the realizations of future z values In particular suppose that the production function per capita takes the form yt f kt zt where kt again denotes the capitallabor ratio and zt Z z1 zN represents a stochastic variable that affects how much output is produced with a given amount of inputs The most natural interpretation of zt in this context is as a stochastic TFP term The resource constraint written as an equality takes the form kt 1 f kt zt 1 δkt ct 162 and kt 0 for all t with k0 0 given Again δ represents the depreciation rate This formulation implies that at the time consumption ct is chosen the random variable zt has been realized Thus ct depends on the realization of zt and in fact on the entire history of zt In particular let us define zt z1 zt as the history of zt up to date t As a convention this history does not include z0 which is taken as given and this ensures that zt indeed has t elements In particular let Zt Z Z the ttimes product so that zt Zt For given k0 the level of consumption at time t can be most generally written as ct czt which simply states that consumption at time t is a function of the entire sequence of random variables observed up to that point Clearly consumption at time t cannot depend on future realizations of the random variablethose values have not yet been realized A function of the form ct czt is thus natural Nevertheless not all functions czt could be admissible as feasible plans because they may violate the resource constraints I return shortly to additional restrictions to ensure feasibility There is also no point in making consumption a function of the history of capital stocks at this stage since those are endogenously determined by the choice of past consumption levels and by the realization of past stochastic variables When we turn to the recursive formulation of this problem we will write consumption as a function of the current capital stock and the current value of the stochastic variable In terms of 161 here xt kt so that xt 1 kt 1 f kt zt 1 δkt czt kzt where the second line simply uses the resource constraint with equality and the third line defines the function kzt With this notation feasibility is easier to express since kt 1 kzt 167 References and Literature 561 trial organization once augmented by the possibility that firms are uncertain about future demand andor productivity Exercise 1615 considers this case 3 Optimal Stopping Problems the search model discussed in Section 1652 is an example of an optimal stopping problem More general optimal stopping problems can also be set up and analyzed as stochastic dynamic programming problems Exercise 1616 considers an example of such an optimal stopping problem 166 Taking Stock The material in this chapter is technical in nature and is more useful for its applications than for its own sake It has widespread applications in macroeconomics and economic growth The stochastic neoclassical growth model presented in the next chapter utilizes the methods developed here In addition to presenting the basic tools of stochastic dynamic programming this chapter has presented two important economic models The first the stochastic permanent income hypothesis model is one of the most famous macroeconomic models and has led both to a large theoretical and empirical literature The early empirical literature focused on excess sensitivity tests as discussed in Section 1651 using aggregate data The more recent literature focuses on micro and panel data to derive sharper results about the behavior of individual consumption The other substantial model introduced in this chapter is the searchforideas model in Section 1652 which is adapted from McCalls 1970 labor market search model McCalls model is the basis of much of the modern equilibrium theory of unemployment While the model here has been cast in terms of searching for ideas the reader can easily adapt it to unemployment and use it as an introduction to equilibrium unemployment theory see Exercise 1613 In addition some of the other applications mentioned above and treated in the exercises including the asset pricing model based on Lucas 1978 and the model of investment under uncertainty are widely used in other areas of macroeconomics 167 References and Literature Most of the references from Chapter 6 are relevant to stochastic dynamic programming as well The reader may want to look at Howard 1960 Blackwell 1965 and Puterman 1994 for advanced treatments The most complete treatment of discounted stochastic dynamic pro gramming problems with economic applications is in Stokey Lucas and Prescott 1989 This chapter covers the same material as Stokey Lucas and Prescott though at a slightly less tech nical level In particular I presented all the major results of stochastic dynamic programming without introducing measure theory A thorough study of stochastic dynamic programming re quires a nontrivial investment in these methods The reader should consult Stokey Lucas and Prescott 1989 Chapters 813 who present a more measuretheoretic approach and develop the necessary material on Markov processes The reader may also wish to consult Rudin 1976 or the very lively and readable treatment in Williams 1991 for some of the basic definitions and results in measure theory used in the discussion of Markov processes These references also provide a formal definition of the Lebesgue integral which I used informally a number of times throughout the text A slightly more advanced but excellent treatment of measure theory is contained in Royden 1994 Williams 1991 also contains an excellent introductory treatment of martingales which were mentioned in Section 165 17 Stochastic Growth Models I n this chapter I present four models of stochastic growth emphasizing different aspects of the interaction between growth and uncertainty The first is the baseline neoclassical growth model with complete markets augmented with stochastic productivity shocks first studied by Brock and Mirman 1972 This model is not only an important generalization of the baseline neoclassical growth of Chapter 8 but also provides the starting point of the influential Real Business Cycle models which are used extensively for the study of a range of short and mediumrun macroeconomic questions I present this model and some of its implications in the next three sections The baseline neoclassical growth model incorporates complete markets in the sense that households and firms can trade using any ArrowDebreu commodity In the presence of uncertainty this implies that a full set of contingent claims is traded competitively For example a household can buy an asset that pays one unit of the final good after a prespecified history The presence of complete marketsor the full set of contingent claimsimplies that households can fully insure themselves against idiosyncratic risks The source of interesting uncertainty in these models is aggregate shocks For this reason the standard neoclassical growth model under uncertainty does not even introduce idiosyncratic shocks had they been present they would have been diversified away This discussion shows the importance of contingent claims in the basic neoclassical model under uncertainty Moreover trading in contingent claims is not only sufficient but it is essentially also necessary for the representative household assumption to hold in environments with uncertainty This result is illustrated in Section 174 which considers a model in which households cannot use contingent claims and can only trade in riskless bonds This model which builds on Bewleys seminal work in the 1970s and the 1980s explicitly prevents risk sharing across households and thus features incomplete marketsin particular one of the most relevant types of market incompleteness for macroeconomic questions preventing the sharing or diversification of idiosyncratic risk Households face a stochastic stream of labor income and can only achieve consumption smoothing by selfinsurance that is by borrowing and lending at a market interest rate Like the OLG model of Chapter 9 the Bewley model does not admit a representative household The Bewley model is not only important in illustrating the role of contingent claims in models under uncertainty but also because it is a tractable model for the study of a range of macroeconomic questions related to risk income fluctuations and policy Consequently over the past decade or so it has become another workhorse model for macroeconomic analysis The last two sections Sections 175 and 176 turn to stochastic OLG models The first presents a simple extension of the canonical OLG model that includes stochastic elements 566 171 The BrockMirman Model 567 Section 176 shows how stochastic growth models can be useful in understanding the process of takeoff from low growth to sustained growth which was discussed in Chapter 1 A notable feature of the longrun experience of many societies is that the early stages of economic development were characterized by little growth in income per capita and by frequent economic crises The process of takeoff not only led to faster growth but also to a more steady less variable growth process An investigation of these issues requires a model of stochastic growth Section 176 presents a model that provides a unified framework for the analysis of the variability of economic performance and takeoff The key feature is the tradeoff between investment in risky activities and safer activities with lower returns At the early stages of development societies do not have enough resources to invest in sufficiently many activities to achieve diversification and are thus forced to bear considerable risk As a way of reducing this risk they also invest in lowreturn safe activities such as a storage or safe technology and low yield agricultural products The result is an equilibrium process that features a lengthy period of slow or no growth associated with high levels of variability in economic performance An economy can escape this stage of development and take off into sustained growth only when its risky investments are successful for a number of consecutive periods When this happens the economy achieves better diversification and also better risk management through more developed financial markets Better diversification reduces risk and also enables the economy to channel its investments in higher return activities increasing its productivity and growth rate Thus this simple model of stochastic growth presents a stylistic account of the process of takeoff from low and variable growth to sustained and steady growth The model I use to illustrate these ideas features both a simple form of stochastic growth and endogenously incomplete markets I therefore use this model to show how some simple ideas from Markov processes can be used to characterize the stochastic equilibrium path of a dynamic economy and to highlight potential inefficiencies that can arise in models with endogenous incomplete markets Finally this model gives us a first glimpse of the relationship between financial development and economic growth a topic that is discussed more extensively in Chapter 21 171 The BrockMirman Model The first systematic analysis of economic growth with stochastic shocks was undertaken by Brock and Mirman in their 1972 paper Brock and Mirman focused on the optimal growth problem and solved for the social planners maximization problem in a dynamic neoclassical environment with uncertainty Since with competitive and complete markets the First and Second Welfare Theorems still hold the equilibrium growth path is identical to the optimal growth path Nevertheless the analysis of equilibrium growth is more involved and also introduces a number of new concepts I start with the BrockMirman approach and then discuss competitive equilibrium growth under uncertainty in the next section The economy is similar to the baseline neoclassical growth model studied in Chapters 6 and 8 It is in discrete time and the aggregate production function is now given by Yt FKt Lt zt 171 where zt denotes a stochastic aggregate productivity term affecting how productive a given combination of capital and labor is in producing the unique final good of the economy Let us suppose that zt follows a Markov chain with values in the set Z z1 zN Many applications of the neoclassical growth model under uncertainty also assume that the stochastic shock is a laboraugmenting productivity term so that the aggregate production function takes the form Yt FKt ztLt though for the analysis here we do not need to impose 174 Growth with Incomplete Markets The Bewley Model 585 Recall from Chapters 6 and 8 that the neoclassical growth model with complete markets and no uncertainty implies that there exists a unique steady state in which βR 1 that is f k β1 1 δ 1732 where k refers to the capitallabor ratio of the neoclassical growth model under certainty In the Bewley economy 1732 is no longer true Proposition 176 In any stationary equilibrium of the Bewley economy the stationary equilibrium capitallabor ratio k is such that f k β1 1 δ 1733 and k k 1734 where k is the capitallabor ratio of the neoclassical growth model under certainty Proof Suppose f k β1 1 δ Then the result in Exercise 1611 from the previous chapter implies that each households expected consumption is strictly increasing Thus av erage consumption in the population which is deterministic is strictly increasing and would tend to infinity This is not possible in view of Assumption 2 which implies that aggregate resources must always be finite This argument establishes 1733 Given this result 1734 immediately follows from 1732 and from the strict concavity of f Assumption 1 Intuitively the interest rate in the incomplete markets economy is depressed relative to the neoclassical growth model with certainty because each household has an additional self insurance or precautionary incentive to save These additional savings increase the capital labor ratio and reduce the equilibrium interest rate Interestingly therefore the Bewley econ omy like the OLG model of Chapter 9 leads to a higher capital intensity of production than the standard neoclassical growth model Observe that in both cases the lack of a representative household plays an important role in this result While the Bewley model is an important workhorse for macroeconomic analysis two of its features may be viewed as potential shortcomings First as already remarked in the context of the OLG model the source of inefficiency coming from overaccumulation of capital is unlikely to be important for explaining income per capita differences across countries Thus the Bewley model is not interesting because of the greater capitallabor ratio that it generates Instead it is important as an illustration of how an economy might exhibit a stationary equilibrium in which aggregates are constant while households have uncertain and fluctuating consumption and income profiles It also emphasizes the role of idiosyncratic risks in the context of the neoclassical growth model Issues of individual risk bearing are important in the context of economic development as shown in Section 176 below and also in Chapter 21 Second the incomplete markets assumption in this model may be extreme In practice when their incomes are low households may receive transfers either because they have entered into some form of private insurance or because of governmentprovided social insurance Instead the current model exogenously assumes that there are no insurance possibilities Models in which the lack of insurance opportunities are derived from microfoundations eg from moral hazard or adverse selection or models in which the set of active markets is determined endogenously would be more satisfactory While models of limited insurance due to moral hazard or adverse selection are beyond the scope of this book I present an economic growth model with endogenously incomplete markets in Section 176 588 Chapter 17 Stochastic Growth Models Another noteworthy feature of this model is that together with the stochastic Solow model discussed in Exercise 173 and the specific form of the neoclassical growth model in Example 171 it provides a much simpler model of stochastic growth than the neoclassical growth model under uncertainty In the OLG model with log preferences and the Solow model this is because saving decisions are myopic and remain unaffected by the distribution of stochastic shocks or even their realizations Thus for a range of macroeconomic questions these more myopic models or the simple neoclassical model of Example 171 might provide tractable alternatives to the full neoclassical growth model under uncertainty 176 Risk Diversification and Growth In this section I present a stochastic model of longrun growth based on Acemoglu and Zilibotti 1997 This model is useful for two distinct purposes First because it is simpler than the baseline neoclassical growth model under uncertainty it enables a complete characterization of the stochastic dynamics of growth and shows how simple ideas from the theory of Markov processes can be used in the context of the study of economic growth Second and more important this model introduces a number of issues in the theory of longrun growth In particular I have so far focused on models with balanced growth and relatively wellbehaved transitional dynamics The experience of economic growth over the past few thousand years has been much less orderly than implied by these models however Until about 200 years ago growth in income per capita was relatively rare Sustained growth in income per capita is a relatively recent phenomenon Before this takeoff into sustained growth societies experienced periods of growth followed by large slumps and crises Acemoglu and Zilibotti 1997 Imbs and Wacziarg 2003 and Koren and Tenreyro 2007 document that even today richer countries have much more stable growth performances than less developed economies which suffer from higher variability in their growth rates In many ways this pattern of relatively risky growth and low productivity followed by a process of capitaldeepening financial development and better risk management is a major characteristic of the history of economic growth The famous economic historian Fernand Braudel 1973 p xi describes the start of economic growth in Western Europe as follows The advance occurred very slowly over a long period and was broken by sharp recessions The right road was reached and thereafter never abandoned only during the eighteenth century and then only by a few privileged countries Thus before 1750 or even 1800 the march of progress could still be affected by unexpected events even disasters In the model I present here these patterns arise endogenously because the extent to which the economy can diversify risks by investing in imperfectly correlated activities is limited by the amount of capital it possesses As the amount of capital increases the economy achieves better diversification and faces fewer risks The resulting equilibrium process thus generates greater variability and risk at the early stages of development and these risks are significantly reduced after the economy manages to take off into sustained growth Moreover the desire of households to avoid risk makes them invest in lower return less risky activities during the early stages of development thus the growth rate of the economy is endogenously limited during this pretakeoff stage In addition in this model economic development goes handin hand with financial development as greater availability of capital enables better risk sharing through asset markets Finally because the model is one of endogenously incomplete markets it also enables us to show that pricetaking behavior by itself is not sufficient to guarantee 590 Chapter 17 Stochastic Growth Models 0 nt 1 j n D st Mj Int Int FIGURE 172 Minimum size requirements Mj of different sectors and demand for assets I n minimum size requirement increases linearly Figure 172 shows the minimum size require ments thick line This figure is used to illustrate the determination of the set of open sectors once the equilibrium investments are specified It is worth noting that there are three important features introduced so far 1 Risky investments have a higher expected return than the safe investment which is captured by the assumption that Q q 2 The output of the risky investments of the intermediate sectors are imperfectly corre lated so that there is safety in numbers 3 The mathematical formulation here implies a simple relationship between investments and returns As already hinted above if a household holds a portfolio consisting of an equiproportional investment I in all sectors j J 0 1 and the Lebesgue measure of the set J is p then the portfolio pays the return QI with probability p and it pays nothing with probability 1 p The first two features imply that if the aggregate production set of this economy had been convex for example because D 0 all households would have invested an equal amount in all intermediate sectors and diversified all risks without sacrificing any of the high returns However in the presence of nonconvexities as captured by the minimum size requirements there is a tradeoff between insurance and high productivity Let us next turn to the preferences of households Recall that each generation has size normalized to 1 Consider a household from a generation born at time t The preferences of this household are given by 176 Risk Diversification and Growth 601 0 j 1 j n D MnsKt nsKt Mj Mj FIGURE 175 The Pareto optimal portfolio allocation This proposition implies that when the economy has not achieved full diversification the social planner will open more sectors than the decentralized equilibrium She will finance these additional sectors by deviating from the balanced portfolio which is always a feature of the equilibrium allocation In other words she will invest less in the sectors without the minimum size requirement The Pareto optimal allocation of funds is shown in Figure 175 The deviation from the balanced portfolio implies that the social planner is implicitly cross subsidizing the sectors with high minimum size requirements at the expense of sectors with low or no minimum size requirements This is because starting with a balanced portfolio opening a few more sectors always benefits all households who will be able to achieve better risk diversification The only way the social planner can achieve this is by implicitly taxing sectors that have low or no minimum size requirements so that they have lower investments and subsidizing the marginal sectors with high minimum size requirements Why does the decentralized equilibrium not achieve the same allocation There are two complementary ways of providing the intuition for this The first is that a marginal dollar of investment by a household in a sector with a high minimum size requirement creates a pecuniary externality because this investment makes it possible for the sector to be active and to provide better risk diversification possibilities to all other households However each household taking equilibrium prices as given ignores this pecuniary externality and tends to underinvest in marginal sectors with high minimum size requirements Thus the source of inefficiency is that each household ignores its impact on others diversification opportunities The second intuition for this result is related Because households take the set of prices as given and in equilibrium Pj t 1 for all open sectors they will always hold a balanced portfolio However the Pareto optimal allocation involves crosssubsidization across sectors in a nonbalanced portfolio Market prices do not induce the households to hold the right portfolio At this point the reader may wonder why the First Welfare Theorem does not apply in this environment especially since all households are price takers This because the equilibrium 602 Chapter 17 Stochastic Growth Models here does not correspond to an ArrowDebreu equilibrium In particular this is an equilibrium for an economy with endogenously incomplete markets where the set of open markets is deter mined by a zero profit freeentry condition All commodities that are traded in equilibrium are priced competitively but there is no competitive pricing for commodities that are not traded Instead in an ArrowDebreu equilibrium all commodities even those that are not traded in equilibrium are priced and in fact a potential commodity would not be traded in equilibrium only if its price were equal to zero and at zero prices there were excess supply In this sense the equilibrium characterized here is not an ArrowDebreu equilibrium In fact it can be ver ified that such an ArrowDebreu equilibrium does not exist in this economy because of the nonconvexity of the production possibilities set Instead the equilibrium concept used here is a more natural competitive equilibrium notion it requires that all commodities that are traded in equilibrium are priced competitively and then determines the set of traded commodities by a freeentry condition Some additional discussion of this equilibrium concept is provided in the References and Literature section below 1765 Inefficiency with Alternative Market Structures Would the market failure in portfolio choices be overcome if some financial institution could coordinate households investment decisions Imagine that rather than all households acting in isolation and ignoring their impact on one anothers decisions funds are intermediated through a financial coalitionintermediary This intermediary can collect all the savings and offer to each saver a complex security as different from an Arrow security that pays QI Sj t qXSt in each state j where I Sj t and XSt are as in the optimal portfolio Holding this security would make each household better off compared to the equilibrium Although from this discussion it may appear that the inefficiency identified here may not be robust to the formation of more complex financial institutions this is not the case The remarkable result is that unless some rather strong assumptions are made about the set of contracts that a financial intermediary can offer equilibrium allocations resulting from competition among intermediaries are identical to the equilibrium allocation in Proposition 177 A full analysis of this issue is beyond our current scope but a brief discussion gives the flavor Let us model more complex financial intermediaries as intermediarycoalitions that is as sets of households who join their savings together and invest in a particular portfolio of intermediate sectors Such coalitions may be organized by a specific household and if it is profitable for other households to join the coalition the organizer of the coalition can charge a premium or a joining fee thus making profits Let us assume that there is free entry into financial intermediation or coalitionbuilding so that any household can attempt to exploit profit opportunities if there are any Let us also impose some structure on how the timing of financial intermediation works and also how households can participate in different coalitions Let us adopt the following assumptions 1 Coalitions maximize a weighted utility of their members at all points in time In particular a coalition cannot commit to a path of action that will be against the interests of its members in the continuation game 2 Coalitions cannot exclude other households from investing in a particular project The following result is established in Acemoglu and Zilibotti 1997 Proposition 1712 The set of equilibria of the financial intermediation game described above is always nonempty and all equilibria have the same structure as those characterized in Section 1762 and Proposition 177 177 Taking Stock 603 I do not provide a proof of this proposition since a formal statement and the proof require additional notation But the intuition is straightforward as shown in Proposition 1711 the Pareto optimal allocation involves a nonbalanced portfolio and crosssubsidization across different sectors Thus the shadow price of investing in some sectors should be higher than in others even though the cost of investing in each sector is equal to 1 in terms of date t final goods These differences in shadow prices then support a nonbalanced portfolio Recall also that it is the sectors with no or low minimum size requirements that are being implicitly taxed in this allocation This kind of crosssubsidization is difficult to sustain because each household can deviate by slightly reducing its investments in coalitionsintermediaries that engage in crosssubsidization and undertake investments on the side to move its portfolio toward a balanced one by investing in sectors with no or low minimum size requirements At the end only allocations without crosssubsidization that is those as in Proposition 177 can survive as equilibria The most important implication of this result is that even with unrestricted financial in termediaries or coalitions the inefficiency resulting from endogenously incomplete markets cannot be prevented The key economic force is that each household creates a positive pecu niary externality by holding a nonbalanced portfolio but in a decentralized equilibrium each household wishes to and can easily move toward a balanced portfolio undermining efforts to sustain the efficient allocation 177 Taking Stock This chapter presented a number of different models of stochastic growth My selection of topics was geared toward achieving two objectives First I introduced a number of workhorse models of macroeconomics such as the neoclassical growth model under uncertainty and the basic Bewley model These models are not only useful for the analysis of economic growth but also have a wide range of applications in the macroeconomics literature Second the model in Section 176 demonstrated how stochastic models can significantly enrich the analysis of economic growth and economic development In particular this model showed how a simple extension of our standard models can generate an equilibrium path in which economies spend a long time with low productivity and suffer frequent crises They take off into sustained and steady growth once they receive a sequence of favorable realizations The takeoff process not only reduces volatility and increases growth but is also associated with better management of risk and greater financial development Though stylistic this model provides a good approximation to the economic development process that much of Western Europe underwent over the past 700 years or so It also emphasizes the possibility that luck may have played an important role in the timing of takeoff and perhaps even in determining which countries were early industrializers Therefore this model provides an attractive formalization of the luck hypothesis discussed in Chapter 4 Nevertheless underlying the equilibrium in this model is a set of market institutions that enable trade and investment in competitive markets Thus my interpretation is that the current model shows how random elements and luck can matter for the timing of takeoff among countries that satisfy the major prerequisites for modern growth This could account for some of the currentday crosscountry income differences and may also provide important insights about the beginning of the process of sustained growth However institutional factorswhich determine whether those prerequisites are satisfied are more important for understanding why some parts of the world did not take off during the nineteenth century and have not yet embarked on a path of sustained and steady growth These are topics that are discussed in the rest of the book 604 Chapter 17 Stochastic Growth Models Section 176 also introduced a number of important ideas related to incomplete markets The Bewley model presented in Section 174 is a prototypical incomplete markets model and as with most incomplete markets models in the literature it takes the set of markets that are open as given In contrast the model in Section 176 incorporates endogenously incomplete markets The fact that the set of open markets the set of traded commodities is determined in equilibrium with a freeentry condition can lead to a novel Pareto inefficiency due to pecuniary externalities even though all households take prices as given Although this type of Pareto inefficiency is different from those highlighted so far there are some important parallels between the phenomena of an insufficient number of markets being open in this model and too few machine varieties being introduced in the baseline endogenous technological change model of Chapter 13 178 References and Literature The neoclassical growth model under uncertainty presented in Section 171 was first analyzed by Brock and Mirman 1972 Because the optimal growth problem is considerably easier than the study of equilibrium growth under uncertainty most analyses in the literature focus on the optimal growth problem and then appeal to the Second Welfare Theorem Stokey Lucas and Prescott 1989 provide an example of this approach An analysis of the full stochastic dynam ics of this model requires a more detailed discussion of the general theory of Markov processes Space restrictions preclude me from presenting these tools The necessary material can be found in Stokey Lucas and Prescott 1989 Chapters 8 11 12 and 13 or the reader can look at Futia 1982 for a more compact treatment More advanced and complete treatments are presented in Gikhman and Skorohod 1974 or Ethier and Kurtz 1986 The tools in Stokey Lucas and Prescott 1989 are sufficient to prove that the optimal path of capitallabor ratio in the neo classical growth model under uncertainty converges to a unique invariant distribution and they can also be used to prove the existence of a stationary equilibrium in the Bewley economy The first systematic analysis of competitive equilibrium under uncertainty is provided in Lucas and Prescott 1971 Ljungqvist and Sargent 2005 Chapter 12 provides an excellent textbook treatment The material in Section 172 is similar to Ljungqvist and Sargents treat ment but is somewhat more detailed The RBC literature is enormous and Section 173 only scratches the surface The clas sic papers in this literature are Kydland and Prescott 1982 and Long and Plosser 1983 Ljungqvist and Sargent 2005 again provides a good introduction The collection of papers in Cooley 1995 is an excellent starting point and provides a range of tools for theoretical and quantitative analysis Blanchard and Fischer 1989 summarizes various critiques of the RBC approach The interested reader is also referred to the exchange between Edward Prescott and Lawrence Summers Prescott 1986 Summers 1986 and to the review of the more recent literature in King and Rebelo 1999 Section 174 presents the incomplete markets model first introduced by Truman Bewley 1977 1980 This model has become one of the workhorse models of macroeconomics and has been used for analysis of business cycle dynamics income distribution optimal fiscal policy monetary policy and asset pricing A more modern treatment is provided in Aiyagari 1994 though the published version of the paper does not contain the proofs of the main results The reader is referred to Bewley 1977 1980 and to the working paper version of Aiyagaris paper Aiyagari 1993 for more details on some of the propositions stated in Section 174 as well as a proof of existence of a stationary equilibrium Krusell and Smith 1998 2005 among others have used this model for business cycle analysis and have also provided new quantitative tools for the study of incomplete market economies 179 Exercises 605 Section 176 builds on Acemoglu and Zilibotti 1997 and more details on some of the re sults stated in this section are provided there Evidence on the relationship between economic development and volatility is provided in Acemoglu and Zilibotti 1997 Imbs and Wacziarg 2003 and Koren and Tenreyro 2007 Ramey and Ramey 1995 also provide related ev idence The concept of decentralized equilibrium used in this model is not ArrowDebreu Instead it imposes pricetaking behavior in all open markets and determines the set of open markets via a freeentry condition This equilibrium concept is natural and is used in various different contexts in general equilibrium theory see for example Hart 1979 Makowski 1980 and Allen and Gale 1991 179 Exercises 171 Proposition 172 shows that kt 1 is increasing in kt and zt Provide sufficient conditions such that ct is also increasing in these variables 172 Consider the neoclassical growth model under uncertainty analyzed in Section 171 and assume that zt is realized after ct and kt 1 are chosen a Show that if zt is distributed independently across periods the choice of capital stock and consumption in this economy is identical to that in a neoclassical growth model under certainty with a modified production function Explain the intuition for this result b Now suppose that zt is not distributed independently across periods Establish the equiva lent of Proposition 171 How does the behavior in this economy differ from the neoclassical growth model under uncertainty in Section 171 173 Consider the same production structure as in Sections 171 and 172 but assume that regardless of the level of the capital stock and the realization of the stochastic variable each household saves a constant fraction s of its income Characterize the stochastic law of motion of this economy How does equilibrium behavior in this economy differ from that in the canonical neoclassical growth model under uncertainty 174 Consider the neoclassical growth model under uncertainty in Section 171 a Provide conditions under which πk z is strictly increasing in both of its arguments b Show that when the conditions in part a hold the capitallabor ratio can never converge to a constant unless z has a degenerate distribution always taking the same value 175 Consider Example 171 a Prove that 1710 cannot be satisfied for any B0 0 b Conjecture that the value function for this example takes the form V k z B2 B3 log k B4 log z Verify this guess and compute the parameters B2 B3 and B3 176 Show that the policy function in Example 171 πk z βαzkα applies when z follows a general Markov process rather than a Markov chain Hint instead of the summation replace the expectations sign with an appropriately defined integral and cancel terms under the integral sign 177 a Consider the economy analyzed in Example 171 with 0 z1 zN Characterize the limiting invariant distribution of the capitallabor ratio and show that the stochastic corre spondence of the capital stock can be represented by Figure 171 in Section 175 Use this figure to show that the capitallabor ratio k always grows when it is sufficiently small and always declines when it is large b Consider the special case where z takes two values zh and zl with each value persisting with probability q 12 and switching to the other value with probability 1 q Show that as q 1 the behavior of the capitallabor ratio converges to its equilibrium in the neoclassical growth model under certainty 606 Chapter 17 Stochastic Growth Models 178 Consider the economy studied in Example 171 but suppose that δ 1 Show that in this case there does not exist a closedform expression for the policy function πk z 179 Write the maximization problem of the social planner explicitly as a sequence problem with output capital and labor following different histories interpreted as different ArrowDebreu commodities Using this formulation carefully show that all of the conditions of Theorem 57 are satisfied so that the optimal growth path can be decentralized as a competitive equilibrium 1710 Consider an extended version of the neoclassical growth model under uncertainty such that the instantaneous utility function of the representative household is uc b where b is a random variable following a Markov chain a Set up and analyze the optimal growth problem in this economy Show that the optimal consumption sequence satisfies a modified stochastic Euler equation b Prove that Theorem 57 can be applied to this economy and the optimum growth path can be decentralized as a competitive growth path 1711 Explain why in Section 1651 in the previous chapter the Lagrange multiplier λyt was con ditioned on the entire history of labor income realizations while in the formulation of the com petitive equilibrium with a full set of ArrowDebreu commodities contingent claims in Section 172 there is a single multiplier λ associated with the lifetime budget constraint 1712 Consider the model of competitive equilibrium in Section 172 Repeat the analysis of the competitive equilibrium of the neoclassical growth model under uncertainty by assuming that instead of a price for buying and selling capital goods in each state R0zt there is a market for renting capital goods Let the rental price of capital goods in terms of date 0 final good be R0zt when the sequence of stochastic variables is zt Characterize the competitive equilibrium and show that it is equivalent to that obtained in Section 172 Explain why the two formulations give identical results 1713 Prove Proposition 173 Hint use Theorem 168 together with 176 and 1722 and then show that the lifetime budget constraint 1711 implies 177 1714 Characterize the competitive equilibrium path of the neoclassical growth model under uncertainty analyzed in Section 172 with sequential trading using the sequence rather than the recursive formulation of the households maximization problem 1715 Show that Theorems 161167 can be applied to V a z defined in 1724 and establish that V a z is continuous strictly increasing in both of its arguments concave and differentiable in a 1716 Derive 1727 1717 Prove Proposition 174 1718 Consider the RBC model presented in Section 173 and suppose that the production function takes the form FK zAL with both z and A corresponding to laboraugmenting technological productivity terms Suppose that z follows a Markov chain and At 1 1 gAt is an exogenous and deterministic productivity growth process Set up the social planners problem in this case What restrictions do we need to impose on uC L to ensure that the optimal growth path corresponds to a BGP where labor supply does not with probability 1 go to zero or infinity 1719 In Example 172 suppose that the utility function of the representative household is uC L log C hL where h is a continuous decreasing and concave function Show that the equilibrium level of labor supply is constant and independent of the level of capital stock and the realization of the productivity shock 1720 Explain why in the Bewley model of Section 174 the budget constraint of the household must hold along all sample paths Compare the resulting constraint 1730 to 1711 in Section 172 1721 Prove Proposition 175 PART VI TECHNOLOGY DIFFUSION TRADE AND INTERDEPENDENCES O ne of the most important shortcomings of the models presented so far is that each country is treated as an isolated island that does not interact with the rest of the world This is problematic for at least two reasons The first is related to the technological interdependences across countries and the second to international trade in commodities and in assets In this part of the book I investigate the implications of technological and trade interdependences on the process of economic growth The models presented so far treat technology either as exogenous or as endogenously generated within the boundaries of the economy in question We have already seen how allowing for endogeneity of technology provides new and important insights about the process of growth But should we think of the potential technology differences between Portugal and Nigeria as resulting from lower RD in Nigeria The answer to this question is most probably no Nigeria like most lessdeveloped or developing countries imports many of its technologies from the rest of the world The same is the case for Portugal despite its substantially more developed economy This observation suggests that a framework in which frontier technologies in the world are produced in the United States or other advanced economies and then copied or adopted by other follower countries provides a better approximation to reality Therefore to understand technology differences between advanced and developing economies we should focus not only or not even primarily on differential rates of endogenous technology generation in these economies but also on their decisions concerning technology adoption and efficient technology use While the exogenous growth models of Chapters 2 and 8 have this feature they too have important shortcomings First technology is entirely exogenous so interesting economic de cisions only concern investment in physical capital There is a conceptually and empirically compelling sense in which technology is different from physical capital and also from human capital so we would like to understand sources of differences in technology arising endoge nously across countries Thus the recognition that technology adoption from the world frontier matters is not the same as accepting that the Solow or the neoclassical growth models are the 610 Part VI Technology Diffusion Trade and Interdependences best vehicles for studying crosscountry income differences Second while the emphasis on technology adoption makes the process of growth resemble the exogenous growth models of Chapters 2 and 8 technological advances at the world level are unlikely to be manna from heaven Instead economic growth at the world level results either from the interaction of the adoption and RD decisions of all countries or perhaps from the innovations by frontier econ omies Thus models in which the growth rate at the world level is endogenous and interacts and coexists with technology adoption may provide a better approximation to reality and a better framework for the analysis of the mechanics of economic growth We will also see that international trade may play the same role of linking growth across countries while allowing for endogenous world growth In Chapter 18 I start with models of technology adoption and investigate the factors affecting the speed and nature of technology adoption In addition to factors slowing down technology diffusion and the importance of barriers against new technologies I discuss the role of whether technologies from the world frontier are appropriate for the needs of less developed countries Recall also that technology differences not only reflect differences in techniques used in production but also differences in the organization of production affecting the efficiency with which existing factors of production are utilized A satisfactory theory of technology differences among countries must therefore pay attention to barriers to technology adoption and to potential inefficiencies in the organization of production leading to apparent technology differences across countries Chapter 18 also provides a simple model of inefficient technology adoption resulting from contracting problems among firms The second major element missing from our analysis so far international trade and interna tional capital flows is discussed in Chapter 19 International trade in commodities and assets links the economic fortunes of the countries in the world as well For example economies with low capitallabor ratios may be able to borrow internationally which would change equilib rium dynamics Similarly and perhaps more importantly less productive countries that export certain goods to the world economy will be linked with other economies because of changes in relative pricesbecause of changes in their terms of trade This type of termsoftrade ef fects may also work toward creating a framework in which while the world economy grows endogenously the growth rates of each country is linked to those of others through trading re lationships Finally I emphasize the connections between international trade and technology adoption in particular emphasizing how trade and the international product cycle facilitate technology diffusion Throughout the rest of the book including this part my treatment will be less comprehensive than in the previous chapters In particular to economize on space I will be more selective in the range of models covered focusing on the models that I believe provide the main insights in an economical fashion I leave many alternative models and approaches to the discussion of the literature at the end or to exercises In addition I make somewhat greater use of simplifying assumptions and leave to exercises the proofs of results that are similar to those provided so far and the relaxation of some of the simplifying assumptions 18 Diffusion of Technology I n many ways the problem of innovation ought to be harder to model than the problem of technology adoption Nevertheless the literature on economic growth and development has made more progress on models of innovation such as those we discussed in Chapters 13 15 than on models of technology diffusion This is in part because the process of technology adoption involves many challenging features First even within a single country we observe considerable differences in the technologies used by different firms in the same narrowly defined industry Second and relatedly it is difficult to explain how in the globalized world in which we live some countries fail to import and use technologies that would significantly increase their productivity In this chapter I begin the study of these questions Since potential barriers to technology adoption are intimately linked to the analysis of the political economy of growth I return to some of these themes in Part VIII of the book For now the emphasis is on how technological interdependences change the mechanics of economic growth and can thus enrich our understanding of the potential sources of crosscountry income differences and economic growth over time I first provide a brief overview of some of the empirical patterns pertaining to technology adoption and diffusion within countries and industries and how this appears to be important for withinindustry productivity differences I then turn to a benchmark model of world equilibrium with technology diffusion which provides a reducedform model for analyzing the slow diffusion of technological knowhow across countries I then enrich this model by incorporating investments in RD and technology adoption Next I discuss issues of appropriate technology and finally I turn to the impact of contractual imperfections on technology adoption decisions Throughout this chapter the only interaction among countries is through technological exchange and there is no international trade in goods or assets 181 Productivity Differences and Technology Let us first start with a brief overview of productivity and technology differences within countries This overview will help us place the crosscountry differences in productivity and technology into perspective The most important lesson from the withincountry studies is that productivity and technology differences are ubiquitous even across firms within narrow sectors in the same country 611 612 Chapter 18 Diffusion of Technology 1811 Productivity and Technology Differences within Narrow Sectors A large literature uses longitudinal microdata often for the manufacturing sector to study labor and TFP differences across plants within narrow sectors eg three or fourdigit man ufacturing sectors For our focus the most important pattern that emerges from these studies is that even within a narrow sector of the US economy there are significant differences in productivity across plants with an approximately two or threefold difference between the top and the bottom of the distribution see eg the survey in Bartelsman and Doms 2000 for a summary of various studies and estimates In addition these productivity differences appear to be highly persistent eg Baily Hulten and Campbell 1992 There is little consensus on the causes of these differences Many studies find a correlation between plant productivity and plant or firm size various measures of technology in particular IT technology capital intensity the skill level of the workforce and management practices eg Davis and Haltiwanger 1991 Doms Dunne and Troske 1997 Black and Lynch 2005 Nevertheless since all of these features are choice variables for firms these correlations cannot be taken to be causal Thus to a large extent the determinants of productivity differences across plants are still unknown In this light it should not appear as a surprise that there is no consensus on the determinants of crosscountry differences in productivity Nevertheless the existing evidence suggests that technology differences are an important factor at least as a proximate cause for productivity differences For example Doms Dunne and Troske 1997 and Haltiwanger Lane and Spletzer 1999 document significant technol ogy differences across plants within narrow sectors Interestingly as emphasized by Doms Dunne and Troske 1997 and Caselli and Coleman 2001a a key determinant of technology adoption decisions seems to be the skill level of the workforce of the plant often proxied by the share of nonproduction workers though adoption of new technology does not typically lead to a significant change in the skill level of the employees of the plant These results sug gest that consistent with some of the models discussed in Chapters 10 and 15 differences in the availability of skills and skilled workers might be an important determinant of technology adoption and development The distribution of productivity across firms appears to be related to the entry of new and moreproductive plants and the exit of lessproductive plants For example consistent with the basic Schumpeterian models of economic growth discussed in Chapter 14 Bartelsman and Doms 2000 and Foster Haltiwanger and Krizan 2000 document that entry of new plants makes an important contribution to industry productivity growth Nevertheless entry and exit appear to account for only about 25 of average TFP growth with the remaining productivity improvements accounted for by continuing plants This suggests that models in which firms continually invest in technology and productivity as in the models in Sections 143 and 144 in Chapter 14 may be important for understanding the productivity differences across firms and plants and also for the study of crosscountry productivity differences 1812 Diffusion of New Technologies A key implication of the sectoral studies is that despite our presumption that technology and knowhow are freely available and can be adopted easily there are considerable technology and productivity differences among firms operating under similar circumstances In addition new and more productive technologies once they arrive on the scene diffuse and are gradually adopted by more firms and plants The literature on technology diffusion studies this process of adoption of new technologies As one might expect there are parallels between the issue of technology diffusion across countries and slow technology diffusion across firms Let us then briefly overview the main findings of the technology diffusion literature 182 A Benchmark Model of Technology Diffusion 613 The classic paper in this area is Grilichess 1957 study of the adoption of hybrid corn in the United States Griliches showed that the more productive hybrid corn diffused only slowly in US agriculture and that this diffusion was affected by the local economic conditions of different areas Consistent with the theoretical models presented so far the likelihood of adoption appears to be related to the productivity contribution of the hybrid corn in a particular area the market size and the skill level of the workforce in the area The importance of these factors has been found in other studies as well Another important result of Grilichess study was to uncover the famous Sshape of diffusion whereby a particular technology first spreads slowly and then once it reaches a critical level of adoption it starts spreading much more rapidly Finally once a large fraction of the target population adopts the technology the rate of adoption again declines The overall pattern thus approximates an S curve or a logistic function The important lesson for our focus here is that productivity and technology differences are not only present across countries but also within countries Moreover even within coun tries better technologies are not immediately adopted by all firms Nevertheless the causes of withincountry and crosscountry productivity and technology differences might be different and despite the presence of withincountry differences the significant crosscountry differ ences are a major puzzle For example withincountry productivity differences might be due to differences in managerial entrepreneurial ability or related to the success of the match between the manager and the technology or the product These types of explanations are unlikely to account for why almost all firms in many lessdeveloped countries are much less productive than the typical firms in the United States and other advanced economies or why the distribution of firmlevel productivity is very different across countries Motivated by the evi dence briefly surveyed here I discuss both models in which technology diffuses slowly across countries and models in which productivity differences may remain even when instantaneous technology diffusion and adoption are possible 182 A Benchmark Model of Technology Diffusion 1821 A Model of Exogenous Growth In the spirit of providing the main insights with the simplest possible models let us return to the Solow growth model of Chapter 2 Suppose that the world economy consists of J countries indexed by j 1 J each with access to an aggregate production function for producing a unique final good Yjt FKjt AjtLjt where Yjt is the output of this unique final good in country j at time t and Kjt and Ljt are the capital stock and labor supply respectively Finally Ajt is the technology of this economy which is both countryspecific and timevarying In line with the result in Theorem 26 in Chapter 2 technological change has already been assumed to be purely laboraugmenting Harrodneutral in form In addition F satisfies the standard neoclassical assumptions that is Assumptions 1 and 2 from Chapter 2 Throughout this chapter and the next whenever the world economy consists of J countries I assume that J is large enough so that each country is small relative to the rest of the world and thus it ignores its effect on world aggregates1 1 We can think of J as a large finite number or consider the limit where J Alternatively we could have assumed that there is a continuum rather than a countable number of countries None of the results in this and the next chapter depend on whether the number of countries is a continuum or finite Throughout I work with a finite number of countries to simplify the exposition 616 Chapter 18 Diffusion of Technology Once we solve for the law of motion of ajt this is simply a function of time making 181 a simple nonautonomous differential equation Let us start the analysis with the steadystate world equilibrium A world equilibrium is defined as an allocation kjt ajtt0J j1 such that 181 and 184 are satisfied for each j 1 J and for all t starting with the initial conditions kj0 aj0J j1 A steady state world equilibrium is then defined as a steady state of this equilibrium path that is an equilibrium with kjt ajt 0 for each j 1 J The steadystate equilibria studied in this chapter exhibit constant growth so I could have alternatively referred to them as balanced growth path equilibria Throughout I use the term steadystate equilibrium for consistency2 Proposition 181 In the abovedescribed model there exists a unique steadystate world equilibrium in which income per capita in all countries grows at the same rate g 0 Moreover for each j 1 J we have a j σj σj g λj 185 and k j is uniquely determined by sj f k j k j nj g δ The steadystate world equilibrium k j a jJ j1 is globally stable in the sense that starting with any strictly positive initial values kj0 aj0J j1 the equilibrium path kjt ajtJ j1 converges to k j a jJ j1 Proof First solve 181 and 184 for each j 1 J imposing the steadystate con dition that kjt ajt 0 This yields a unique solution establishing the uniqueness of the steadystate equilibrium Then standard arguments show that the steady state a j of the differ ential equation for ajt is globally stable Using this result the global stability of the steady state of the differential equation for kjt follows straightforwardly Exercise 184 asks you to complete the details of this proof Several features of this world equilibrium are noteworthy First there is a unique steady state world equilibrium and it is globally stable This enables us to perform simple comparative static and comparative dynamic exercises see Exercise 185 Second and most importantly despite differences in saving rates and technology absorption rates across countries income per capita in all economies grows at the same rate which is equal to the growth rate of the world technology frontier g Equation 183 clarifies the reason for this the rate of technology diffusion absorption is higher when the gap between the world technology frontier and the technology level of a particular country is greater Thus there is a force pulling backward economies toward the technology frontier and in steady state this force is powerful enough to ensure that all countries grow at the same rate Does this imply that all countries will converge to the same level of income per capita The answer is clearly no Differences in saving rates and absorption rates translate into level differences instead of growth rate differences across countries For example a society with a low level of σj initially grows less than others until it is sufficiently behind the world technology frontier At this point it will also grow at the world rate g This discussion 2 In the remainder I sometimes write kjt ajtt0 instead of kjt ajt t0 to simplify the notation 618 Chapter 18 Diffusion of Technology exists a unique steadystate world equilibrium where for each j a j is given by 185 and k j is uniquely determined by f k j ρ δ θg and consumption per capita in each country grows at the rate g 0 Moreover the steadystate world equilibrium is globally saddlepath stable starting with any strictly positive initial values kj0 aj0J j1 the equilibrium path kjt ajt cjtJ j1 converges to k j a j c jJ j1 where c j is the steadystate consumption to effective labor ratio in economy j Proof We can first show that a j can be determined from the differential equation 184 without reference to any other variables and satisfies 185 The consumption Euler equations and the dynamics of capital accumulation are the same as in the baseline neoclassical growth model taking into account that in steady state gjt g To complete the proof of the propo sition we need to show the stability of a j and then taking into account the behavior of gjt we must establish the saddlepath stability of k j using the same type of analysis as in Chap ter 8which is slightly more complicated here because the differential equation for capital accumulation is not autonomous You are asked to complete these details in Exercise 188 This proposition shows that all qualitative results of the benchmark model of technology diffusion apply regardless of whether we assume constant saving rates or dynamic household maximization as long as we ensure that the growth rate is not so high as to lead to infinite utility and violate the transversality condition Naturally an equilibrium now corresponds not only to paths of kjt ajt but also includes the time path of consumption to effective labor cjt Consequently the appropriate notion of stability is saddlepath stability which the equilibrium in Proposition 183 satisfies 1823 The Role of Human Capital in Technology Diffusion The model presented above is in part inspired by the classic paper by Richard Nelson and Edmund Phelps 1966 which was already discussed in Chapter 10 Recall that the Becker Mincer view emphasizes how human capital increases the productivity of the labor hours supplied by an individual While this approach allows the effect of human capital to be different in different tasks in most applications it is presumed that greater human capital translates into higher productivity in all or most tasks with the set of productive tasks typically taken as given In contrast Nelson and Phelps and Ted Schultz emphasize the role of human capital in facilitating the adoption of new technologies and adaptation to changing environments In terms of the model described above the simplest way of capturing this argument is to posit that the parameter σj is a function of the human capital of the workforce The greater is the human capital of the workforce the higher is the absorption capacity of the economy If so high human capital societies will be richer because as shown in Proposition 182 economies with higher σj have higher steadystate levels of income While this modification leaves the mathematical exposition of the model unchanged the implications for how we view the growth experiences of societies with different levels of human capital are potentially quite distinct from the BeckerMincer approach or at the very least from the simplest version of the BeckerMincer approach The latter approach suggests that we can approximate the role of human capital in economic development by carefully accounting for its role in the aggregate production function This in turn can be done by estimating individual returns to schooling and returns to other dimensions of human capital in the labor market The 624 Chapter 18 Diffusion of Technology the next I focus on how technology differences and income gaps can remain substantial even with free flow of ideas A first possibility is that productivity differences may remain even if all differences in tech niques disappear because production is organized differently and the extent of inefficiencies in production may vary across countries The next section discusses this possibility Another important idea is that technologies of the world technology frontier may be inappropriate to the needs of specific countries so that importing the most advanced frontier technologies may not guarantee the same level of productivity for all countries At some level this idea is both simple and attractive Technologies and skills consist of bundles of complementary attributes and these bundles vary across countries so that there is no guarantee that a new technology that works well given the skills and competences in the United States or Switzerland will also do so in Nigeria or Turkey Nevertheless without specifying these attributes that make some technologies work well in certain nations and not in others this story has little explanatory power In this section I present three versions of this story that may have some theoretical and empirical appeal First I discuss how differences in exogenous eg geographic conditions may make the same set of technologies differentially productive in different areas Second I show how differences in capital intensity across countries may change the appropriateness of different types of technologies Finally most of this section is devoted to the implications of differences in skill supplies across countries for the appropriateness of frontier technologies to developing economies In this context I show how the degree of appropriateness of technolo gies may arise endogenously in the world equilibrium and also present a model of economic growth in which labor has to be allocated across different sectors which is of independent interest 1841 Inappropriate Technologies The idea of inappropriate technologies can be best illustrated by an example on health in novations Suppose that productivity in country j at time t Ajt is a function of whether there are effective cures against certain diseases affecting their populations Suppose that there are two different diseases heart attack and malaria Countries j 1 J are affected by malaria and not by heart attacks while j J 1 J are affected by heart attacks and not by malaria If the disease affecting country j has no cure then productivity in that country is given by Ajt A while when a cure against this disease is introduced then Ajt A Now imagine that a new cure against heart attacks is discovered and becomes freely available to all countries Consequently the productivity in countries j J 1 J increases from A to A but productivity in countries j 1 J remains at A This simple example illustrates how technologies of the world frontier may be inappropriate to the needs of some countries in this case the J countries affected by malaria In fact in this extreme case a technological advance that is freely available to all countries in the world increases productivity in a subset of the countries and creates crosscountry income differences Is there any reason to expect that issues of this sort might be important The answer is both yes and no There are natural reasons to expect that new technologies should be optimized for the conditions and the needs of OECD countries because these countries are both the largest market for new technologies and the producers of much of new world knowhow see Section 1843 below Nevertheless other than the issue of disease prevention there are not many obvious fixed country characteristics that will create this type of inappropriateness Instead the issue of appropriate technology is much more likely to be important in the context of whether new technologies increasing productivity via process and product innovations function 634 Chapter 18 Diffusion of Technology When all investment levels are identical and equal to x output is q Nκ1x Since a total of NX Nx inputs are used in the production process a natural measure of productivity is output divided by total input use P Nκ In the case of complete contracts this productivity level is P Nκ which is increasing in the level of technology The next proposition summarizes this analysis Proposition 189 Consider the abovedescribed model take A as given and suppose that there are complete contracts Then there exists a unique SPE with technology and investment levels N 0 and x 0 given by 1836 and 1837 respectively Furthermore in this SPE N A 0 x A 0 and N α x α 0 Proof See Exercise 1827 In the case of complete contracts the size of the market which corresponds to A and from the viewpoint of the individual firm is exogenous has a positive effect on investments by suppliers of inputs and productivity because a greater market size makes both suppliers and the producers investments more productive The other noteworthy implication of this proposition is that under complete contracts the level of technology and thus productivity do not depend on the elasticity of substitution between inputs 11 α 1853 Equilibrium under Incomplete Contracts Let us next consider the same environment under incomplete contracts We model the im perfection of the contracting institutions by assuming that there exists μ 0 1 such that for every input j investments in activities 0 i μ are observable and verifiable and there fore contractible while investments in activities μ i 1 are not contractible Consequently a contract stipulates investment levels xi j for the μ contractible activities but does not specify the investment levels in the remaining 1 μ noncontractible activities Instead suppliers choose their investments in noncontractible activities in anticipation of the ex post distribution of revenue and they may decide to withhold their services in these activities from the firm Economies with weak contracting institutions have a low μ and thus feature only a small set of tasks that are contractible whereas more developed contracting institutions correspond to high levels of μ The ex post distribution of revenues in activities that are not ex ante contractible is de termined by multilateral bargaining between the firm and its suppliers The exact bargaining protocol determines investment incentives of suppliers and the profitability of investment for the firm First consider the timing of events The firm adopts a technology N and offers a contract xci jμ i0 τj for every input j 0 N where xci j is an investment level in a contractible activity and τj is an upfront payment to supplier j The payment τj can be positive or negative Potential suppliers decide whether to apply for the contracts Then the firm chooses N suppliers one for each input j All suppliers j 0 N simultaneously choose investment levels xi j for all i 0 1 In the contractible activities i 0 μ the suppliers invest xi j xci j The suppliers and the firm bargain over the division of revenue and at this stage suppliers can withhold their services in noncontractible activities 642 Chapter 18 Diffusion of Technology 186 Taking Stock This chapter presented models of technology differences across societies While the baseline endogenous growth models such as those studied in Part IV are useful in understanding the incentives of research firms to create new technologies and can generate different rates of tech nological change across different economies two factors suggest that a different perspective is necessary for understanding technology differences across nations First technology and pro ductivity differences do not only exist across nations but are ubiquitous within countries Even within narrowly defined sectors there are substantial productivity differences across firms and only a small portion of these differences can be attributed to differences in capital intensity of production This withincountry pattern suggests that technology adoption and use decisions of firms are complex and new technologies only diffuse slowly across firms This pattern gives us some clues about potential sources of productivity and technology differences across nations and suggests that a slow process of technology diffusion across countries may not be unreason able Second while the United States Germany or Japan can be thought of as creating their own technologies via the process of RD most countries in the world are technology importers rather than technology leaders This is not to deny that some firms in these societies do engage in RD nor to imply that a number of important technologies most notably those related to the Green Revolution have been invented in developing countries These exceptions notwith standing adoption of existing frontier technologies appears more important for most firms in developing countries than the creation of entirely new technologies This perspective also suggests that a detailed analysis of technology diffusion and technology adoption decisions is necessary for obtaining a good understanding of productivity and technology differences across countries Several important lessons have emerged from our study in this chapter First we can make considerable progress in understanding technology and productivity differences across nations by positing a slow process of technology transfer across countries Such an approach enables us to have a tractable model of technology differences across countries An important element of most models of technology diffusion is a builtin advantage for countries or firms that are relatively behind since there is a larger gap for them to close it is relatively easier for them to close it This catchup advantage for backward economies ensures that models of slow technology diffusion lead to differences in income levels not necessarily in growth rates In other words the canonical model of technology diffusion implies that countries that create barriers against technology diffusion or those that are slow in adopting new technologies for other reasons will be poor but they eventually converge to the growth rate of the frontier economies Thus a study of technology diffusion enables us to develop a model of world income distribution in which the position of each country in the world income distribution is determined by their ability to absorb new technologies from the world frontier This theoretical machinery is also useful for developing a framework in which while each country may act as a neoclassical exogenous growth economy importing its technology from the world frontier the entire world behaves as an endogenous growth economy with its growth rate determined by the investment in RD decisions of all firms in the world This class of models becomes particularly useful when we wish to think of the joint process of world growth and world income distribution across countries Such models also emphasize that much is lost in terms of insights when we focus on the baseline neoclassical growth model in which each country is treated as an isolated island that does not interact with others in the world Technological interdependences across countries implies that we should often consider the world equilibrium not simply the equilibrium of each country on its own While slow diffusion of existing technologies across countries is reasonable in the global ized world we live in today it is becoming increasingly easier for firms to adopt technologies 187 References and Literature 643 that have already been tried and implemented in other parts of the world Once we allow a relatively rapid diffusion of technologies does there remain any reason for technology or pro ductivity differences across countries beyond differences in physical and human capital The second part of the chapter has argued that the answer to this question is also yes and is related to the appropriateness of technologies and to differences in contracting institutions that affect technology adoption and productivity Issues of appropriateness imply that a given technology does not have the same impact on the productivity in all economies because it may be a better match to the conditions or to the factor proportions of some countries than of others A particularly important channel of inappropriateness is the potential mismatch between technologies developed at the world frontier and the skills of the adopting countrys workforce A technologyskill mismatch can lead to large endogenous productivity differences If the types of technologies developed at the world frontier were random the possibility of the technologyskill mismatch creating a significant gap between rich and poor nations would be a mere possibility no more However there are reasons to suspect that technologyskill mismatch may be more important because of the organization of the world technology market Two features are important here First the majority of frontier technologies are developed in a few rich countries Second the lack of effective IPR enforcement implies that technology firms in advanced countries target the needs of their own markets This creates a powerful force for new technologies that are appropriate to designed for the needs of the advanced nations and thus are typically inappropriate to the factor proportions of developing nations In particular new technologies will often be too skill biased to be effectively used in developing countries This source of inappropriateness of technologies can create a large endogenous technology and income gap across nations Finally this chapter has also emphasized that productivity differences do not simply stem from differences in the use of different techniques of production but also from differences in production organization around the world A key reason for such differences is the institutions and policies in place in different parts of the world The last part of the chapter showed how contracting institutions affecting what types of contracts firms can write with their suppliers can have an important effect on their technology adoption decisions and thus on crosscountry differences in productivity Contracting institutions are only one of many potential organiza tional differences across countries that might impact equilibrium productivity Other sources of differences in the organization of production and technology are discussed in Chapter 21 187 References and Literature The large literature documenting productivity and technology differences across firms and the patterns of technology diffusion was discussed in Section 181 and the relevant references can be found there The simple model of technology diffusion presented in Section 182 is inspired by Gerschenkrons 1962 essay and by Nelson and Phelpss 1966 seminal paper and by Schultz 1975 The NelsonPhelps approach which was discussed in greater detail in Chapter 10 has been important in a number of recent papers Benhabib and Spiegel 1994 reinterpret and modify Barrostyle growth regressions in light of NelsonPhelpss view of human capital Aghion and Howitt 1998 also provide a similar reinterpretation of growth regressions Greenwood and Yorukoglu 1997 Caselli 1999 Galor and Moav 2000 and Aghion Howitt and Violante 2004 provide models inspired by the NelsonPhelpsSchultz view of human capital and applied to understanding the recent increase in the returns to skills in the United States and other OECD economies The model in Section 183 is inspired by Howitt 2000 but is different in a number of important respects Howitt constructs a model of Schumpeterian growth rather than the baseline 19 Trade and Growth T he previous chapter discussed how technological linkages across countries and technol ogy adoption decisions lead to a pattern of interdependent growth across countries This chapter studies world equilibria with international trade in financial assets or commodi ties I start with growth in economies that can borrow and lend internationally and discuss how this affects crosscountry income differences and growth dynamics I then turn to the growth implications of international trade in commodities Our first task is to construct models of world equilibria that feature both international trade in commodities or intermediate goods and economic growth The exact interactions between trade and growth depend on the nature of trade that countries engage in I try to provide an overview of these different interactions I start with a model in which trade is of the HeckscherOhlin type that is it originates only because of differences in factor abundance across countries and growth is driven by capital accumulation I then turn to a model of Ricardian type where trade is driven by technological comparative advantage A key difference between these two approaches concerns whether the prices of the goods that a country supplies to the world are affected by its own production and accumulation decisions These models shed new light on the patterns of interdependences across countries for example showing that growth in one country cannot be analyzed in isolation from the growth experiences of other nations in the world Our second task is to turn to a central question of the literature on trade and growth whether international trade encourages economic growth The answer to this question also depends on exactly how trade is modeled as well as on what the source of economic growth is in particular learningbydoing versus innovation Throughout the emphasis is on the importance of considering the world equilibrium rather than that of a closed economy in isolation 191 Growth and Financial Capital Flows In a globalized economy if the rates of return to capital differ across countries we would expect capital to flow toward areas where its rate of return is higher This simple observation has a number of important implications for growth theory First it implies a different pattern of economic growth in a financially integrated world Our first task in this section is to illustrate the implications of international capital flows for economic growth and show how they significantly change transitional dynamics in the basic neoclassical growth model Our second 648 652 Chapter 19 Trade and Growth The next proposition focuses on the steadystate world equilibrium Proposition 192 Suppose that Assumption 4 in Chapter 2 is satisfied Then in the world economy with free flows of capital there exists a unique steadystate world equilibrium in which output capital and consumption per capita in all countries grow at the rate g and effective capitallabor ratios are given by k j k f 1ρ δ θg for all j 1 J Moreover in the steadystate equilibrium lim t ajt 0 for all j 1 J Proof See Exercise 191 This result is intuitive with free capital flows the world economy is integrated This integrated world economy has a unique steadystate equilibrium similar to that in the standard neoclassical growth model The steadystate equilibrium not only determines the effective capitallabor ratio and its growth rate but also the distribution of the available capital across different countries in the world economy Even though this proposition is intuitive its proof requires some care to ensure that no country runs a Ponzi scheme the absence of a Ponzi scheme requires that the change in normalized asset position of each country and each household within each country ajt for each j asymptote to zero This last feature is no longer the case when the model is extended so that countries differ according to their discount rates see Exercise 192 Let us next consider the transitional dynamics of the world economy The analysis of transitional dynamics is simplified by the fact that the world behaves as an integrated economy rather than an independent collection of economies see Exercise 192 Proposition 193 In the world equilibrium of the economy with free flows of capital there exists a unique equilibrium path kjt cjt ajtt0J j1 that converges to the steadystate world equilibrium Along this equilibrium path kjtkjt 1 and cjtcjt constant for any two countries j and j Proof See Exercise 193 Intuitively the integrated world economy acts as if it has a single neoclassical aggregate production function thus the characterization of the dynamic equilibrium path and of transi tional dynamics from Chapter 8 applies In addition Proposition 191 implies that kjtkjt is constant and the consumption Euler equations imply that cjtcjt must also be con stant Therefore both production and consumption in each economy grow in tandem Notice however that Proposition 193 does not state that cjt cjt even though kjt kjt This is because while GDP and production per capita across countries are equalized Gross National Product GNP need not be equalized because different countries could have unequal asset positions This point is further emphasized in Exercise 192 The following is an important corollary to Proposition 193 Corollary 191 Consider the world economy with free flows of capital Suppose that at time t a fraction λ of the capital stock of country j is destroyed Then capital flows immediately to this country ajt to ensure that kjtkjt 1 for all t t and for all j j Proof This corollary is a direct implication of Propositions 191 and 193 The latter implies that there exists a unique globally stable equilibrium while the former implies that for all t kjtkjt 1 This is only possible if there is an immediate inflow of capital into country j 192 Why Does Capital Not Flow from Rich to Poor Countries 653 This result implies that in the world economy with free flows of capital there are only transitional dynamics for the aggregate world economy but no transitional dynamics separately for each country in particular kjtkjt 1 for all t and any j and j This is intuitive since international capital flows ensure that each country has the same effective capitallabor ratio thus dynamics resulting from slow capital accumulation are removed The corollary therefore implies that any theory emphasizing the role of transitional dynamics in explaining the evolution of crosscountry income differences must implicitly limit the extent or the speed of international capital flows The evidence on this point is mixed While the amount of gross capital flow in the world economy is large the FeldsteinHorioka puzzle which is discussed below highlights that countries that save more also tend to invest more One reason for this might be the potential risk of sovereign default by countries that borrow significant amounts from the world financial markets Exercise 194 investigates this issue Although the implications of this corollary for crosscountry patterns of divergence can be debated its implications for crossregional convergence are clear crossregional patterns of convergence cannot be related to slow capital accumulation as in the baseline neoclassical growth model see Exercise 195 192 Why Does Capital Not Flow from Rich to Poor Countries The model studied in the previous section provides us with a framework to answer the question posed above and in the title of this section In the basic Solow and neoclassical growth models a key source of crosscountry income differences is capitallabor ratios For example if we consider a world economy in which all countries have access to the same technology and there are no human capital differences the only reason one country would be richer than another is differences in capitallabor ratios But if two countries with the same production possibilities set differ in terms of their capitallabor ratios then the rate of return to capital will be lower in the richer economy and there will be incentives for capital to flow from rich to poor countries I now discuss the reasons that capital may not flow from societies with higher capitallabor ratios to those with greater capital scarcity 1921 Capital Flows under Perfect International Capital Markets One potential answer to the question posed above is provided by the analysis in the previous section With perfect international capital markets capital flows equalize effective capitallabor ratios But this does not imply equalization of capitallabor ratios This result which follows directly from the analysis in the previous section is stated in the next proposition Note that this result does not give a complete answer to our question since it takes productivity differences across countries as given Nevertheless it explains how given these productivity differences there is no compelling reason to expect capital to flow from rich to poor countries Proposition194 Consider a world economy with identical neoclassical preferences across countries and free flows of capital Suppose that countries differ according to their productivi ties the Ajs Then there exists a unique steadystate equilibrium in which capitallabor ratios differ across countries in particular effective capitallabor ratios the kjs are equalized Proof See Exercise 197 There is thus no reason to expect capital flows when countries differ according to their productivities The more productive countries should have higher capitallabor ratios To the 654 Chapter 19 Trade and Growth extent that two countries j and j have different levels of productivity Ajt and Ajt Ajt their capitallabor ratios should not be equalized instead country j should have a higher capitallabor ratio than that of j Consequently capital need not flow from rich to poor countries This explanation is similar to that suggested in Lucas 1990 except that Lucas also linked differences in Ajs to differences in human capital and in particular to human capital externalities Instead Proposition 194 emphasizes that any source of differences in Ajs generates this pattern The reader would be right to object at this point that this is only a proximate answer to the question since it provides no reason for why productivity differs across countries This objection is largely correct Nevertheless Proposition 194 is still useful since it suggests a range of explanations for the lack of capital flows from rich to poor countries that do not depend on the details of the world financial system but instead focus on productivity differences across countries We have already made some progress in understanding the potential sources of productivity differences across countries and as we make more progress we will start having better answers to the question of why capital does not flow from rich to poor countries in fact why it might sometimes flow from poor to rich countries 1922 Capital Flows under Imperfect International Financial Markets There are other reasons besides Proposition 194 why capital may not flow from rich to poor countries The rate of return to capital may be higher in poor countries but financial market frictions or issues of sovereign risk may prevent such flows For example lenders might worry that a country that has a negative asset position might declare international bankruptcy and not repay its debts Alternatively domestic financial problems in developing countries which are discussed in Chapter 21 may prevent or slow down the flows of capital from rich to poor countries For whatever reason if the international financial markets are not perfect and capital cannot flow freely from rich to poor countries we may expect large differences in the return to capital across countries1 Existing evidence on this topic is mixed Three different types of evidence are relevant First several studies including Treflers 1993 important work discussed in Chapter 3 and recent work by Caselli and Feyrer 2007 suggest that differences in the return to capital across countries are relatively limited These estimates are directly relevant to the question of whether there are significant differences in the returns to capital across countries but they are computed under assumptions that may not always hold in practice in Treflers case they rely on data on factor contents of trade and make a variety of assumptions on the impact of trade on factor prices as discussed in Chapter 3 Caselli and Feyrer on the other hand require comparable and accurate measures of qualityadjusted differences in capital stocks across countries and assume that there are no costs of adjustment Second and somewhat in contrast to the aggregate results some papers exploiting micro datafor example summarized in Banerjee and Duflo 2005suggest that the rate of return for additional investment in some firms in lessdeveloped countries could be as high as 100 Nonetheless this evidence even if taken at face value does not suggest that there are strong incentives for capital to flow from rich to poor countries since these higher rates of return may be generated by withincountry credit market imperfections In particular it may be that the rate of return is very high for a range of creditrationed firms but various incentive problems make it impossible for domestic or foreign financial institutions to lend to these firms on profitable 1 Limits on capital flows may also contribute to productivity differences eg by reducing productivity enhancing investments thus indirectly reducing the need for further capital flows 656 Chapter 19 Trade and Growth where Yjt is final output in country j at time t and XL j t and XK j t are respectively labor and capitalintensive intermediates inputs I use the letter X to denote these inputs since they refer to the amounts of these inputs used in production rather than the amount of inputs produced in country j In the presence of international trade these two quantities will typically differ In 197 F denotes a constant returns to scale production function and again satisfies Assumptions 1 and 2 from Chapter 2 except that it is defined over two intermediate inputs rather than labor and capital Notice that Assumption 2 also incorporates the Inada conditions The production of the final good is competitive The theory of international trade is a welldeveloped and rich area and provides useful results on the structure of production and trade Here my purpose is not to review these results but to illustrate the implications of HeckscherOhlin type international trade for economic growth Therefore I adopt the simplest possible setting which involves each intermediate input being produced by one factor In particular Y L j t AjLjt and 198 Y K j t Kjt 199 where the use of Y instead of X here emphasizes that these quantities refer to the local production not the use of these intermediates Also as usual Ljt is total labor input in country j at time t supplied inelastically and Kjt is the total capital stock of the country One feature about these intermediate production functions is worth noting there are potential productivity differences across countries in the production of the laborintensive good but not in the production of the capitalintensive good This is the same assumption as the one adopted in Ventura 1997 Exercise 1911 shows the implications of allowing differences in the productivity of the capitalintensive sector For now it suffices to note that this assumption makes it possible to derive a wellbehaved world equilibrium and it is in the spirit of allowing only laboraugmenting technological progress in the basic neoclassical model One may also presume that differences in Ajs reflect differences in the human capital embodied in labor Finally notice also that there is no technological progress This is again to simplify the exposition and Exercise 1913 extends the model in this section to incorporate laboraugmenting technological progress Throughout the rest of this chapter I assume that there is free international trade in commoditiesin intermediate goods This assumption is extreme since trading internation ally involves costs and many analyses of international trade incorporate the physical costs of transportation and tariffs Nevertheless this assumption is useful to simplify the analysis and to highlight how international trade affects crosscountry growth patterns The most important implication of this assumption is that the prices of traded commoditieshere the intermediate goodsin all countries are equal to their world prices determined by the world supply and demand for these commodities Let us denote the world prices of the laborintensive and the capitalintensive intermediates at time t by pLt and pKt respectively Both of these prices are in terms of the final good in the world market which is taken as numeraire with price normalized to 12 Given the production technologies in 198 and 199 competitive factor markets imply that the wage rate and the rental rate of capital in country j at time t are given by 2 In this model there is no loss of generality in assuming that the price of the final good is normalized to 1 in each country even if there is no trade in the final good This is because all goods are traded and there are no differences in costs of living purchasing power parity across countries This will no longer be the case in the models studied in the next section 193 Economic Growth in a HeckscherOhlin World 657 wjt AjpLt and Rjt pKt These two equations summarize the most important economic insights of the model studied here In closedeconomy models factor prices which shape the incentives to accumulate capital are determined by the capitallabor ratio in the economy recall Chapter 8 In contrast factor prices here are determined by world prices In particular since capital is used only in the production of the capitalintensive intermediate and there is free trade in intermediates the rental rate of capital in each country is given by the world price of the capitalintensive intermediate A similar reasoning applies to the wage rate with the only difference being that because of crosscountry differences in the productivity of labor wage rates are not equalized instead it is the effective wage rates wjtAj that are equalized Let us follow Trefler 1993 in referring to this pattern as conditional factor price equalization across countries meaning that once we take into account intrinsic productivity differences of factors there is equalization of effective factor prices across countries Conditional factor price equalization is weaker than the celebrated factor price equalization of international trade theory which would require wjts to be equalized across countries Instead here wjtAjs are equalized In this model conditional factor price equalization is a consequence of free and costless trade in goods since each factor is used only in the production of a single traded intermediate Nevertheless conditional factor price equalization results are more general than the specific structure here might suggest In the jargon of international trade theory with free trade of commodities there exists a cone of diversification such that when factor proportions of different countries are within this cone there will be conditional factor price equalization The assumptions here that labor is used in the production of the laborintensive intermediate and capital is used in the production of the capitalintensive intermediate and that international trade is costless ensure that the cone of diversification is large enough to include any possible configuration of the distribution of capital and labor stocks across countries Conditional factor price equalization is important because it implies that factor prices in each country are entirely independent of its capital stock and labor provided that the country in question is small relative to the rest of the world recall footnote 1 in the previous chapter The distinguishing feature of the model in this section is this independence of factor prices from accumulation decisions3 Because capital again depreciates at the rate δ the interest rate in country j at time t is rjt Rjt δ pKt δ 1910 Let us next specify the resource constraint While there is free international trade in com modities there is no intertemporal trade Thus we are abstracting from international lending and borrowing discussed in the previous two sections This enables us to isolate the effects of international trade in the simplest possible way Lack of international lending and borrow ing implies that at every date each country must run a balanced international trade Thus the following trade balance equation pKtXK j t Y K j t pLtXL j t Y L j t 0 1911 must hold for all j and all t This equation is intuitive it requires that for each country at each date the value of its net sales of the capitalintensive good should be made up by its net purchases of the laborintensive good For example if XK j t Y K j t 0 so that the country 3 This feature is common to many but not all HeckscherOhlin models of trade and conditional factor price equalization may also hold in other trade models 662 Chapter 19 Trade and Growth pattern of growth similar to that found in Chapter 8 with each country converging to a unique steady state There is one important difference however As in the model with international borrowing and lending in the previous section the nature of the transitional dynamics is very different from the closedeconomy neoclassical growth models Here despite the absence of international capital flows the rate of return to capital is equalized across countries Thus there are no transitional dynamics because a country with a higher rate of return to capital is accumulating capital faster than the rest This model therefore also emphasizes the potential pitfalls of using the closedeconomy growth model for the analysis of output and capital dynamics across countries and regions Exercise 1910 compares the equilibrium here to the closedeconomy equilibrium of the same model Nevertheless the results on transitional dynamics are perhaps the less interesting implica tions of the current model One of the main objectives of this chapter is to illustrate how the presence of international trade changes the conclusions of closedeconomy growth models The current framework already points out how this can happen Notice that while the world economy has a standard neoclassical technology satisfying Assumptions 1 and 2 each country faces an AK technology since it can accumulate as much capital as it wishes without running into diminishing returns as long as the country remains small In particular for every ad ditional unit of capital at time t a country receives a return of pKt which is independent of its own capital stock So how is it that the world does not generate endogenous growth The answer is that while each country faces an AK technology and thus can accumulate when the price of capitalintensive intermediates is high accumulation by all countries drives down the price of capitalintensive intermediate goods to a level that is consistent with steady state In other words the price of capitalintensive intermediates adjusts to ensure the steady state equilibrium where capital output and consumption per capita are constant see the proof of Proposition 198 While this process describes the longrun dynamics it also opens the door for a very different type of shortrun or mediumrun dynamics especially for countries that have different saving rates than others To illustrate this possibility in the simplest possible way consider the following thought experiment Let us start with the world economy in steady state and suppose that one of the countries experiences a decline in its discount rate from ρ to ρ ρ What happens The answer is provided in the next proposition Proposition 199 Consider the abovedescribed model Suppose that J is large and the world starts in steady state at time t 0 and then the discount rate of country 1 declines to ρ ρ After this change there exists some T 0 such that for all t 0 T country 1 grows at the rate g1 c1t c1t 1 θ ρ ρ Proof In steady state Proposition 198 and 1919 imply that pK ρ δ As long as country 1 is small which will be the case during some interval 0 T it faces this price as the return on capital Thus the countrys dynamics will be identical to those of the AK economy in Chapter 11 Section 111 with A ρ ρ The result that the growth rate is constant follows from the analysis there Given conditional factorprice equalization each country faces an AK technology and thus can accumulate capital and grow without experiencing diminishing returns The price of capitalintensive intermediates and thus the rate of return to capital is pinned down by the discount rate of other countries in the world so that country 1 with its lower discount rate has 194 Trade Specialization and the World Income Distribution 663 an incentive to save faster than the rest of the world and can achieve positive growth of income per capita while the rest of the world has constant income per capita Therefore the model of economic growth with HeckscherOhlin trade and with conditional factor price equalization can easily rationalize bouts of rapid growth growth miracles by the countries that change their policies or their saving rates or discount rates Ventura 1997 suggests this model as a potential explanation for why starting in the 1970s the East Asian tigers grew rapidly without experiencing diminishing returns Since in the 1970s and 1980s East Asian economies were indeed more open to international trade than many other developing economies and accumulated capital rapidly eg Young 1992 1995 Vogel 2006 this explanation is quite plausible It shows how international trade can temporarily prevent the diminishing returns to capital that would set in because of rapid accumulation and can enable sustained growth at higher rates Nevertheless such behavior cannot go on forever This follows from Assumption 2 which implies that world output cannot grow in the long run So how is Proposition 199 consistent with Assumption 2 The answer is that this proposition describes behavior in the medium run This is why the statement of the proposition is for t 0 T At some point country 1 becomes so large relative to the rest of the world that it will essentially own almost all capital in the world At that point or in fact before this point is reached country 1 can no longer be considered a small country its capital accumulation will have a major impact on the relative price of the capitalintensive intermediate Consequently the rate of return on capital will eventually fall so that accumulation by this country comes to an end Naturally an alternative path of adjustment could take place if at some future date the discount rate of country 1 increases back up to ρ so that the world economy again settles into a steady state The important lesson from this discussion is that while the current model can generate growth miracles these can only apply in the medium run This feature is related to the result highlighted in Exercise 199 that the current model does not admit a steadystate equilibrium or even a welldefined distribution of world income when discount rates differ across countries In other words the wellbehaved world equilibrium that emerges from this model relies on the knifeedge case in which all countries have the same discount rate and also the same productivity of the capitalintensive intermediates see Exercise 1911 This result is a consequence of the fact that in this HeckscherOhlin model each country is small and factor prices are independent of domestic factor proportions In the next section we will see how a simple Ricardian model without these features leads to different interactions between international trade and growth 194 Trade Specialization and the World Income Distribution In this section I present a model of the world economy in which countries trade intermediate goods because of Ricardian featuresproductivity or technology differences Each country will affect the prices of the goods that it supplies to the world This is a plausible feature While countries typically take the prices of the goods that they import as given they often influence the world prices of at least some of the goods that they export eg copper for Chile Microsoft Windowsfor the United States or Lamborghinis for Italy The key implication of this feature is that each countrys terms of trade the prices of its exports relative to its imports are endogenous and depend on the rate at which it accumulates capital Consequently domestic factor prices are also affected by capital accumulation We will see that such a model is more flexible than the one discussed in the previous section since it can allow for differences in discount rates and saving rates and also enables us to obtain a richer set of comparative static results The model economy presented here builds on Acemoglu and Ventura 2002 I start 670 Chapter 19 Trade and Growth rate of return on capital which from 1935 slows down capital accumulation This process ensures that the world economy and all economies move toward the unique steadystate world equilibrium Exercise 1917 asks you to provide a formal proof of stability The results summarized in this proposition are remarkable First despite the high degree of interaction among the various economies there exists a unique globally stable steadystate world equilibrium Second this equilibrium takes a relatively simple form Third and most important in this equilibrium all countries grow at the same rate g This third feature is quite surprising since each economy has access to a AK technology thus without any international trade eg when τ 0 or see Exercise 1918 each country would grow at a different rate eg those with lower ζjs or ρjs would have higher longrun growth rates The process of international trade acts as a powerful force keeping countries together ensuring that in the long run they all grow at the same rate In other words international trade together with termsof trade effects leads to a stable world income distribution Why The answer is related to the termsoftrade effects encapsulated in 1936 To understand the implications of this equation consider the special case where all countries have the same technology parameter that is μj μ for all j Suppose also that a particular country say country j has lower ζj and ρj than the rest of the world Then 1935 implies that this country accumulates more capital than others But 1936 makes it clear that this cannot go on forever and country j by virtue of being richer than the world average will also have a lower rate of return on capital This lower rate of return ultimately compensates the greater incentive to accumulate in country j so that capital accumulation in this country converges to the same rate as in the rest of the world Intuitively each country has market power in the goods that it supplies to the world when it exports more of a particular good the price of that good declines to ensure that world consumers purchase a greater amount of this good So when a country accumulates faster than the rest of the world and thus increases the supply of its exports relative to the supplies of other countries exports it will face worse terms of trades This negative termsoftrade effect reduces its income and its rate of return to capital recall 1923 and slows down capital accumulation This mechanism ensures that in the steadystate equilibrium all countries accumulate and grow at the same rate Therefore this model shows how pure trade linkages are sufficient to ensure that countries that would otherwise grow at different rates pull one another toward a common growth rate and the result is a stable world income distribution The role of trade can be seen most clearly by comparing the equilibrium here to closedeconomy equilibrium which is done in Exercise 1918 Naturally growth at a common rate does not imply that countries with different character istics have the same level of income Exactly as in models of technological interdependences in the previous chapter countries with better characteristics higher μj and lower ζj and ρj grow at the same rate as the rest of the world but will be richer than other countries This is most clearly shown by the following equation which summarizes the world income distribu tion Let y j YjtYt be the relative income of country j in steady state Then 1934 and 1939 yield y j μjζjρj g1ετ 1940 This equation shows that countries with better technology high μj lower distortions low ζj and lower discount rates low ρj will be relatively richer Equation 1940 also highlights that the elasticity of income with respect to ζj and ρj depends on the elasticity of substitution 195 Trade Technology Diffusion and the Product Cycle 677 0 LnLs LnLs NnNo wnws 1 E FIGURE 191 Determination of the relative wages in the North and the South in the international product cycle model crosscountry income differences and shows that even in the context of the current model it can sometimes lead to a larger gap between rich and poor countries 1952 Product Cycles and Technology Transfer The characterization of the equilibrium in Section 1951 was for a given number of new and old goods Our interest in this model is because its relative simplicity enables us to endogenize the number of new and old goods and it generates a pattern of product cycles across countries Here let us follow Krugman 1979 and endogenize the number of new and old goods using a model of exogenous technological change Exercise 1929 considers a version of this model with endogenous creation of new products In particular let us suppose that new goods are created in the North according to the following simple differential equation Nt ηNt with some initial condition N0 0 and innovation parameter η 0 Goods invented in the North can be imitated by the South As in the models of technology diffusion in the previous chapter this process is assumed to be slow and to follow the differential equation Not ιNnt where ι 0 is the imitation parameter This differential equation has a motivation similar to that of the technology diffusion equations in the previous chapter and captures the idea that the South can only imitate from the set of goods that have not so far been imitated of which 196 Trade and Endogenous Technological Change 679 of predicted trade for each country and use this as an instrument for actual trade openness Using this strategy they show that greater trade is associated with higher income per capita thus with greater longrun growth In addition recent microeconomic evidence from Bernard et al 2003 Bernard and Jensen 2004 and others show that firms that engage in exporting are typically more productive which might be partly due to learningbyexporting though at least some part of this correlation is likely due to selection Melitz 2003 Similarly firms in developing countries that import machinery from more advanced economies appear to be more productive and trade liberalization is associated with productivity increases both among continuing plants and due to reallocation eg Pavcnik 2002 Nevertheless some economists are skeptical of the growth effects of trade Rodriguez and Rodrik 2000 criticize the empirical evidence that trade promotes growth On the theoretical side several authors eg Matsuyama 1992 Young 1991 have presented models in which international trade can slow down growth in some countries In this and the next sections I investigate some of the simplest models that link trade to growth to investigate the potential impacts of international trade on economic growth I start with a model illustrating how trade opening may change the pace of endogenous technological change This model is inspired by Grossman and Helpman 1991b who investigate many dif ferent interactions between international trade and endogenous technological change Briefly the model consists of two independent economies that can be approximated by the baseline endogenous technological change model with expanding input varieties as in Chapter 13 In fact the model is identical to the labequipment specification in Section 131 The advantage of this model is that there are no knowledge spillovers thus we do not have to make assumptions about how knowledge spillovers change with trade opening5 I compare innovation and growth in these two economies in the equilibria without any international trade and with costless in ternational trade Naturally a smoother transition in which trade costs decline slowly is more realistic in practice but the sharp thought experiment of moving from autarky to full trade integration is sufficient for us to obtain the main insights concerning the effect of international trade on technological progress Given the analysis in Section 131 of Chapter 13 there is no need to repeat the same steps here It suffices to say that we consider two economies say 1 and 2 with identical technologies identical preferences and identical labor forces normalized to 1 and no population growth Preferences and technologies are also the same as those specified in Section 131 Consequently a slight variation on Proposition 131 in that section immediately implies the following result Proposition 1913 Suppose that the conditions ηβ ρ and 21 θηβ ρ 1955 hold Then in autarky there exists a unique equilibrium where starting from any level of technology both countries innovate and grow at the same rate gA 1 θ ηβ ρ 1956 Proof See Exercise 1931 5 If instead of the labequipment specification we were to use the specification with knowledge spillovers and the two countries produced different sets of inputs we would need to decide whether and how much the inputs produced in the foreign country increase the productivity of RD in the home country before and after trade opening Exercise 1933 shows that assumptions concerning how the extent of knowledge spillovers change with trade opening influence the conclusions regarding the effect of trade on growth 680 Chapter 19 Trade and Growth Next let us analyze what happens when these two economies start trading The exact im plications of trade depend on whether before trade opening the two countries were producing some of the same inputs recall that there is a continuum of available inputs that can be pro duced To the extent that they were producing some of the same inputs the static gains from trade will be limited If on the other hand the two countries were producing different inputs there will be larger static gains However our interest here is with the dynamic effects of trade opening that is with the effects of trade opening on economic growth The analysis from Chapter 13 again leads to the following result Proposition 1914 Suppose that condition 1955 holds Then after trade opening the world economy and both countries innovate and grow at the rate gT 1 θ 2ηβ ρ gA where gA is the autarky growth rate given by 1956 Proof See Exercise 1932 This proposition shows that opening to international trade encourages technological change and increases the growth rate of the world economy The reason is simple international trade enables each input producer to access a larger market and this makes inventing new inputs more profitable This greater profitability translates into a higher rate of innovation and more rapid growth The main effect captured in this simple model is reasonably robust Grossman and Helpman 1991b provide a number of extensions and also richer models of international trade eg with multiple factors The economic force a version of the market size effect that leads to the innovation gains from trade is also reasonably robust Nevertheless several caveats are necessary First as Exercise 1933 shows if the RD sector competes with production there are powerful offsetting effects because trade also increases the demand for production workers In this case the qualitative result in this sectionthat trade opening increases the rate of technological progressgenerally applies but it is also possible to construct versions of this baseline model in which this effect is entirely offset Exercise 1933 also provides an example of this type of extreme offset which should be borne in mind as a useful caveat Second Exercise 1934 shows that if the full scale effect is removed and we focus on an economy with semiendogenous growth as the model studied in Section 133 in Chapter 13 trade opening increases innovation temporarily but not in the long run 197 LearningbyDoing Trade and Growth The previous section showed how international trade can increase economic growth in all countries in the world by encouraging faster technological progress In addition to this effect of trade on growth working via technological change the static gains from trade are well rec ognized and understood By improving the allocation of resources in the world economy these static gains can also encourage economic growth Nevertheless as mentioned in Section 196 many commentators and some economists remain skeptical of the positive growth effects of international trade A popular argument often used to justify infant industry protection and industrial policy is that the static gains from trade come at the cost of dynamic gains because international trade induces some countries to specialize in industries with relatively low growth potential In this section I outline a simple model with this feature Richer models that also lead to similar conclusions have been presented by among others Young 1991 Matsuyama 197 LearningbyDoing Trade and Growth 683 Ant Ant η and Ast Ast 0 The world economy converges to a growth rate of g η in the long run The ratio of income in the North and the South is given by Ynt Yst Ant ε1 ε for all t Consequently if ε 1 then the North becomes progressively richer relative to the South so that limt YntYst If instead ε 1 then the relative incomes of the North and the South remain constant so that YntYst constant for all t Proof See Exercise 1936 This proposition contains the main result on how international trade can harm certain countries when there are learningbydoing externalities in some sectors The South has a slight comparative disadvantage in sector 1 Yet in the absence of trade it devotes enough of its resources to that sector and achieves the same growth rate as the North However if there is free trade the South specializes in sector 2 because of its slight comparative disadvantage in sector 1 and fails to benefit from the learningbydoing opportunities offered by sector 1 As a result the South becomes progressively poorer relative to the North This proposition therefore captures the main critique against international trade coming from models such as Young 1991 and proponents of the infant industry arguments However the proposition also shows some of the shortcomings of these arguments For example if ε 1or sufficiently close to 1 specialization in sector 2 does not hurt the South The reason is closely related to the effects highlighted in Section 194 the increase in the productivity of sector 1 in the North creates a negative termsoftrade effect against the North This effect is always present but when ε 1 it becomes sufficiently powerful to prevent the impoverishment of the South even though they have specialized in the sector with the low growth potential Another caveat is highlighted in Exercise 1936 in the world economy described here infant industry protection will not help the South Even if international trade is prevented in the South for a period of duration T 0 for protecting some infant industry the ultimate outcome is the same as in Proposition 1916 So what are we to make of the results in this section and the general issue of the impact of trade on growth An immediate answer is that the juxtaposition of the models of this and the previous sections suggest that the effect of trade on growth must be an empirical one Since there are models that highlight both the positive and the negative effects of trade on growth the debate can be resolved only by empirical work Nevertheless the theoretical perspectives are still useful A couple of issues are particularly worth noting First the effect of trade integration on the rate of endogenous technological progress may be limited because of the factors already discussed at the end of Section 196 For example significant effects are possible only when trade opening does not increase wages in the final good sector competing for workers against the RD sector which is the case when the RD sector does not compete for workers with the final good sector Moreover if the extreme scale effects are removed trade opening creates a temporary boost in innovation but does not necessarily change the longrun growth rate Nonetheless the benefits of the greater market size for firms involved in innovation must be to some degree present in any model of endogenous technological change Taking all these factors into account we should expect some inducement to innovation from trade opening Whether these effects are commensurate with or 684 Chapter 19 Trade and Growth even greater than the static gains of international trade is much harder to ascertain It may well be that the static gains from trade are more important than the subsequent innovation gains On the other side of the tradeoff are the potential costs of trade in terms of inducing specialization in the wrong sectors The model in this section illustrates this possibility Nevertheless I believe that the potential negative effects of trade on growth because of such incorrect specialization should not be exaggerated First there is no strong evidence that international trade leads to incorrect specialization in practice Second international flows of information which often increase with trade opening imply that improvements in productivity in some countries affect productivity in others that were not initially specializing in those sectors eg South Korea was initially an importer of cars and is now a net exporter its productivity in the automotive sector having increased with technology transfer Finally as the main result in this section showed termsoftrade effects ameliorate any negative impact of specialization 198 Taking Stock This chapter had three main objectives The first was to emphasize the shortcoming of using the closedeconomy models for the analysis of the economic growth patterns across countries or regions We have seen that both intertemporal trade and trade in commodities change the dynamics and also possibly the longrun implications of the closedeconomy neoclassical growth models For example international capital flows remove transitional dynamics because economies that are short of capital do not need to accumulate it slowly but can borrow it in international markets Naturally there are limits to how much international borrowing can take place Countries are sovereign entities and thus it is relatively easy for them to declare bankruptcy once they have borrowed a lot Consequently the sovereign borrowing risk might place limits on the ability of countries to use international markets to smooth consumption and rapidly increase their investments Even in this case some amount of international lending takes place and this has an important effect on the equilibrium dynamics of output and the capital stock Nevertheless the available evidence typically confirms the FeldsteinHorioka puzzle which states that changes in investment are correlated with changes in savings An investigation of why despite significant gross capital flows net international capital flows do not play a greater role in international consumption and investment smoothing and what the implications are for economic growth is an interesting area for future research We have further seen that international trade in commodities also changes the implications of the neoclassical growth model For example in the model of economic growth in Section 193 international trade in goods plays the same role as international lending and borrowing and it significantly changes crosscountry output dynamics Thus even in the absence of international lending and borrowing the implications of approaches that model the entire world equilibrium are significantly different from those focusing on closedeconomy dynamics The model of Ricardian trade and termsoftrade effects in Section 194 also illustrated the potential sharp implications of international trade for economic growth In that model there would be no convergence across countries without trade but international trade via the terms oftrade effects it induces creates a powerful force that links incomes around the world Consequently the longrun equilibrium involves a stable world income distribution and the shortrun dynamics are different from the closedeconomy models The second objective was to highlight how the nature of international trade interacts with the process of economic growth Sections 193 and 194 focused on this issue The model of economic growth with HeckscherOhlin trade showed how economic growth increases the effective elasticity of output with respect to capital for each country because of conditional 199 References and Literature 685 factor price equalization This model is useful for understanding how certain economies such as the East Asian tigers can grow rapidly for extended periods relying on capital accumulation without experiencing diminishing returns However our analysis also showed that conditional factor price equalization can lead to extreme results In contrast the model in Section 194 emphasized how a simple form of Ricardian trade based on technological comparative advantage creates a new source of diminishing returns to accumulation for each country based on termsoftrade effects As a country accumulates more capital it starts exporting more of the goods in which it specializes The result is a worsening of its terms of trade reducing the rate of return to further capital accumulation The analysis showed how this force leads to a stable world income distribution whereby rapidly growing economies pull up the laggards to grow at the same rate as themselves How are we to reconcile the different implications of the models in Sections 193 and 194 One possibility is to imagine a world that is a mixture of the models of these two sections It may be that some goods are standardized and can be produced in any country When producing these goods there are no termsoftrade effects So if a country can grow only by producing these goods it can escape the standard diminishing returns to capital thanks to international trade This might be a good approximation to the situation experienced by the East Asian tigers in the 1970s and 1980s when they specialized in mediumtech goods However as countries become richer they also produce more differentiated goods and they may encounter termsoftrade effects Consequently if a country is at the stage of development where it produces more of the differentiated goods further capital accumulation result in diminishing returns through the mechanism highlighted in Section 194 Regardless of how the forces emphasized in these two approaches are combined they both show the importance of modeling the world equilibrium and of viewing the changes in the rate of return to capital in the context of the international trading relations The third objective of this chapter was to investigate the effect of international trade on economic growth Sections 196 and 197 illustrated two different approaches one emphasizing the beneficial effects of trade on growth the other one the potential negative effects Both classes of models are useful to have in ones arsenal in the analysis of world equilibrium and economic growth The usefulness of these models notwithstanding the impact of international trade on economic growth is ultimately an empirical question though our theoretical analysis has already highlighted some important mechanisms and also suggested that the negative effects of trade on growth are unlikely to be important Whether the positive effects of trade on technological progress are quantitatively significant remains an open question It may well be that the static gains of trade are more important than the dynamic ones Nevertheless any analysis of international trade must take into account its implications on economic growth and technological change 199 References and Literature This chapter covered a variety of models Section 191 focused on the implications of inter national financial flows on economic growth This topic is discussed in detail in Barro and SalaiMartin 2004 Chapter 3 both with and without limits to financial flows Obstfeld and Rogoff 1996 Chapters 1 and 2 provide a more detailed analysis of international borrowing and lending Chapter 6 of Obstfeld and Rogoff provides an introduction to the implications of imperfections in international capital markets Work that models these imperfections and their implications includes Bulow and Rogoff 1989ab Atkeson 1991 Kehoe and Perri 2002 and Matsuyama 2004 The FeldsteinHorioka puzzle which was also discussed in Section 191 is still an active area of research Obstfeld and Taylor 2002 present a survey of much of 686 Chapter 19 Trade and Growth the research on this topic Taylor 1994 Baxter and Crucini 1993 and Kraay and Ventura 2007 propose potential resolutions for the FeldsteinHorioka puzzle Section 192 is motivated by Lucass classic 1990 article There is a large literature on why capital does not flow from rich to poor countries Obstfeld and Taylor 1994 contain a survey of the work in this area The work by Caselli and Feyrer 2007 discussed above provides a method for estimating crosscountry differences in the marginal productive capital and argues that differences in the return to capital are limited This work supports models that account for the lack of capital flows based on productivity differences such as the model presented in Section 192 Recent work by Chirinko and Mallick 2007 argues that the Caselli and Feyrer 2007 procedure may lead to misleading results because they do not incorporate adjustment costs in investment in their calculations and that once these costs are incorporated returns to capital differ significantly across countries See also recent work by Gourinchas and Jeanne 2006 on the lack of major investment or growth gains following financial integration and Alfaro KalemliOzcan and Volosovych 2005 on the links between institutional differences and capital flows The rest of the chapter relies on some basic knowledge of international trade theory Space restrictions preclude a detailed review The reader is referred to a standard text for example Dixit and Norman 1980 Section 193 provides a slight generalization of the model in Ventura 1997 it considers a general constant returns to scale production function rather than CES production function used in Ventura A similar but lessrich model was first analyzed by Stiglitz 1971 Stiglitz did not include laboraugmenting productivity differences across nations and assumed exogenous saving rates Other papers that combine HeckscherOhlin trade with models of economic growth include Atkeson and Kehoe 2000 and Cunat and Maffezoli 2001 Section 194 builds on Acemoglu and Ventura 2002 This model uses the structure of preferences first introduced by Armington 1969 but in the production of the final good rather than in preferences see also Ventura 2005 The model in Section 195 builds on Krugmans 1979 seminal article on the product cycle Grossman and Helpman 1991b provide richer models of the product cycle with endogenous technology similar to the economy discussed in Exercise 1929 Antras 2005 provides a new perspective on the international product cycle that relies on the importance of incomplete contracts In his model contractual problems between Northern producers and Southern subsidiaries constitute a barrier slowing down the transfer of goods to the South Only after goods become sufficiently standardized do the contracting problems become less severe and can the transfer of production to the South take place There is a large empirical literature on the impact of trade on growth Many of the bestknown papers in this literature were discussed at the beginning of Section 196 The rest of Section 196 builds on RiveraBatiz and Romer 1991 and Grossman and Helpman 1991b but uses the formulation from Section 131 in Chapter 13 Grossman and Help man 1991b assume that RD requires labor and introduce competition between the RD sector and the final good sector In this case the nature of the knowledge spillovers becomes important for the implications of trade on the pace of endogenous technological progress RiveraBatiz and Romer 1991 also discuss the implications of the form of the innovation possibilities frontier for the effects of trade on technological change This point which is developed in Exercise 1933 also features in recent work by Atkeson and Burstein 2007 Grossman and Helpman 1991b also present models with multiple sectors and factor propor tion differences across countries Another potential effect of international trade on technology works by influencing the direction of technological change This topic is analyzed in detail in Acemoglu 2003b where I show that trade opening with imperfect IPR can make new tech nologies more skillbiased than before trade opening Similar models are also analyzed in Thoenig and Verdier 2003 and Epifani and Gancia 2006 1910 Exercises 687 Section 197 presents a model inspired by Young 1991 and Matsuyama 1992 Lucas 1988 and Galor and Mountford 2008 also present similar models which feature interaction between specialization and learningbydoing Other models in which international trade may be costly for some countries rely on differences in the amount of rents generated by different sectors because of imperfections in the labor market or institutional problems Nunn 2006 and Levchenko 2007 present models in which trade leads to the transfer of rentcreating jobs from countries with weak institutions to those with better institutions and may be harmful to countries with weak institutions 1910 Exercises 191 Prove Propositions 191 and 192 Hint for Proposition 192 use 195 together with the fact that consumption and output grow at the same rate in each country to show that in the steady state it is optimal for each country or each household in each country to choose ajt 0 192 Consider the world economy with free flows of capital but assume that each country has a different discount factor ρj a Prove that Proposition 191 still holds b Show that there does not exist a steadystate equilibrium with ajt 0 for all j Explain the intuition for this result c Characterize the asymptotic equilibrium the equilibrium path as t Suppose that ρj ρj for all j j Show that the share of world net output that is consumed in country j tends to 1 What does this imply for the relationship between GDP and GNP across countries d How would you modify the model to make the asymptotic equilibrium in part c more realistic 193 This exercise asks you to prove Proposition 193 a Show that cjtcjt is constant for all j and j b Show that given the result in Proposition 191 the integrated world equilibrium can be represented by a single aggregate production function Hint use an argument similar to that leading to Proposition 196 c Relate this result and Proposition 196 to Theorem 54 in Chapter 5 Explain why these aggregation results would not hold without free capital flows d Given the result in parts a and b apply an analysis similar to that for the global stability of the equilibrium path in the basic neoclassical growth model to establish the global stability of the equilibrium path here Given global stability prove the uniqueness of the equilibrium path 194 Consider a world economy with international capital flows but suppose that because of sovereign default risk a country cannot borrow more than a fraction φ 0 of its capital stock Consequently in Section 191 we have the restriction that ajt φkjt a Characterize the steadystate equilibrium of the world economy and show that the steady state is not affected by this constraint Explain the intuition for this result carefully b Characterize the transitional dynamics of the world economy under this constraint Show that Corollary 191 no longer holds 195 Barro and SalaiMartin 1991 2004 use growth regressions to look at the patterns of convergence across US regions and states They find that there is a slow pattern of convergence across regions and states and they interpret this through the lenses of the neoclassical growth model Explain why Corollary 191 implies that this interpretation is not appropriate Suggest instead an alternative 1910 Exercises 689 b Explain the roles of the different parameters in determining crosscountry income dispersion Using reasonable parameter values discuss whether the model with international trade can generate larger differences in income per capita across countries than the neoclassical growth model 1920 Derive 1943 1921 Prove Proposition 1911 1922 Prove Proposition 1912 1923 Consider the steadystate world equilibrium in the model of Section 194 a Show that an increase in τ does not necessarily increase the steadystate world equilibrium growth rate g as given by 1938 Provide an intuition for this result b Show that even when τ does not increase growth it increases world welfare Hint to simplify the answer to this part of the question you can simply look at steadystate welfare c Interpret the finding in part b in light of the debate about the effect of trade on growth d Provide a sufficient condition for an increase in τ to increase the world growth rate and interpret this condition 1924 Consider the model of Section 194 except that instead of utility maximization by a representative household assume that each country saves a constant fraction sj of its income Characterize the equilibrium in this case and show that termsoftrade effects again lead to a stable world income distribution 1925 Consider the model of Section 194 but assume that ε 1 Characterize the equilibrium Show that in this case countries that have lower discount rates will be relatively poor Provide a precise intuition for this result Explain why the assumption that ε 1 may not be plausible 1926 Consider the baseline AK model in Section 194 Suppose that production and allocation decisions within each country are made by a countryspecific social planner who maximizes the utility of the representative household within the country a Show that the allocation in the text is no longer an equilibrium Explain b Characterize the equilibrium in this case and show that all of the qualitative results derived in the text apply Provide generalizations of Propositions 1911 and 1912 c Show that world welfare is lower in this case than in the equilibrium in the text d Do you find the equilibrium in this exercise or the one in the text more plausible Justify your answer 1927 Consider the model with labor in Section 194 Suppose that countries can invest to create new varieties of products Suppose that if a particular firm creates such a variety it becomes the monopolist and can charge a markup equal to the monopoly price to all consumers in the world until this variety is destroyed endogenously which happens at the exponential rate δ 0 a Show that the optimal monopoly price for a firm in country j at time t is given by pjt εrjtε 1 Interpret this equation b Suppose that a new variety can be created by using 1η units of labor Show how this changes the labor market clearing condition and specify the freeentry condition c Define a world BGP as an equilibrium in which all countries grow at the same rate Show that such an equilibrium exists and is uniquely defined Explain the economic forces that lead to the existence of such a stable equilibrium Hint show that in this BGP the number of varieties that each country produces is constant d What is the effect of an increase in the discount rate ρ on the number of varieties that a country produces Interpret this result 690 Chapter 19 Trade and Growth e Discuss informally how the analysis and the results would be modified if new products were produced using a combination of labor and capital 1928 Show that in the model of Section 195 an increase in ι always weakly closes the relative income gap between the North and the South Characterize the conditions under which an increase in ι makes the North worse off in terms of reducing its real income 1929 This exercise asks you to endogenize innovation decisions in the model of Section 195 Assume that new goods are created by technology firms in the North as in the model in Section 134 in Chapter 13 and these firms are monopolist suppliers until the good they have invented is copied by the South The technology of production is the same as before and assume that new goods can be produced by using final goods with the technology Nt ηZt where Zt is final good spending Imitation is still exogenous and takes place at the rate ι Once a good is imitated it can be produced competitively in the South a Show that for a good that is not copied by the South the equilibrium price is pt ν ε ε 1wnt b Characterize the static equilibrium for given levels of Nnt and Not c Compute the net present value of a new product for a Northern firm Why does it differ from 138 in Section 134 d Impose the freeentry condition and derive the equilibrium rate of technological change for the world economy Compute the world growth rate e What is the effect of an increase in ι on the equilibrium Can an increase in ι make the South worse off Explain the intuition for this result 1930 Consider a variation of the product cycle model in Section 195 Suppose there is no trade so that the number of goods consumed in each country differs a Show that wages and incomes in the North and the South at time t are wnt Nt 1 ε1 and wst Not 1 ε1 b Derive a condition for relative income differences to be smaller in this case than in the model with international trade Provide a precise intuition for why international trade may increase relative income differences c If trade increases the income differences between the North and the South does it mean that it reduces welfare in the South Hint if you wish you can again use the steadystate welfare levels 1931 Prove Proposition 1913 1932 Prove Proposition 1914 1933 Consider the model in Section 196 but assume that new products are created with the innovation possibilities frontier as in Section 132 in Chapter 13 Assume that before trade knowledge spillovers are created by the entire set of available inputs in the world economy that is the innovation possibilities frontier is Njt ηNtLj Rt for country j where Nt N1t N2t and Lj Rt is the number workers working in RD in country j Thus trade opening does not change knowledge spillovers a Show that in this model trade opening has no effect on the equilibrium growth rate Provide a precise intuition for this result 1910 Exercises 691 b Assume that before trade opening the innovation possibilities frontier takes the form Njt ηNjtLj Rt Show that in this case trade opening leads to an increase in the equilibrium growth rate as in Proposition 1914 Explain why the results are different c Which of the specifications in parts a and b is more plausible In light of your answer to this question how do you think trade opening should affect economic growth 1934 Consider the model in Section 196 with two differences First population grows at the rate n in both countries Second the innovation possibilities frontier is given by Njt ηNtφZjt for country j where Nt N1t N2t Show that at first trade opening leads to more innovation but the longrun growth rate of each country remains unchanged 1935 Prove Proposition 1915 1936 a Prove Proposition 1916 b Explain why when ε 1 specialization in the sector without learningbydoing does not have an adverse effect on the relative income of the South c What are the implications of trade opening on relative incomes if ε 1 d Characterize the equilibrium if all economies are closed until time t T and then open to international trade at time T What are the implications of this result for infant industry protection 1937 Consider the economy in Section 197 but suppose that the South is bigger than the North In particular assume that 1 δε LSLN ε1 1 δε 1961 a Show that in this case not all Southern workers work in sector 2 and there is some learning bydoing in the South Why is 1961 necessary for this result b How does this affect the longrun equilibrium Hint show that the limiting value of L1 s is equal to 0 Why is 1961 necessary for this result PART VII ECONOMIC DEVELOPMENT AND ECONOMIC GROWTH I n this part of the book I discuss the relationship between economic development and economic growth The first question that the reader may rightly ask is why there is or there should be a distinction between economic development and economic growth This question is particularly apt because I have argued in Chapter 1 that societies that are rich developedtoday are those that have grown steadily over the past 200 years and those that are poor or less developed are those that have not achieved this type of steady growth This perspective suggests that economic development and economic growth are essentially the same thing and should be studied together Nevertheless there are two reasons one good and one bad for drawing a distinction between development and growth The good reason is that even though economic development and growth are part of the same process models of growth emphasize different aspects of this process than models of economic development In particular the models studied so far focus on either balanced growth or transitional dynamics leading to balanced growth Even though these transitional dynamics have been analyzed in a number of contexts our main interest has been to ensure that they take us toward a BGP Behavior along or near the BGP of a neoclassical or endogenous growth economy provides a good approximation to the behavior of relatively developed societies But many salient features of economic growth at earlier stages of development are not easy to map to this orderly behavior of balanced growth In fact Kuznets and other economists have documented that even in moredeveloped economies many aspects of the process of economic growth are far from the balanced benchmark implied by the standard neoclassical growth model Motivated by these patterns in his classic book Modern Economic Growth Kuznets 1966 p 1 defines economic growth as follows We identify the economic growth of nations as a sustained increase in per capita or per worker product most often accompanied by an increase in population and usually by sweeping structural changes In modern times these were changes in the industrial structure within which product was turned out and resources employedaway from agriculture toward nonagricultural activities the process of industrialization in the distribution of population between the countryside and the cities the process of ur banization in the relative economic position of groups within the nation distinguished 694 Part VII Economic Development and Economic Growth by employment status attachment to various industries level of per capita income and the like in the distribution of product by useamong household consumption capital formation and the government consumption and within each of these major categories by further subdivisions in the allocation of product by its origin within the nations boundaries and elsewhere and so on Although one might debate whether this is the most functional definition of economic growth it does capture a range of important changes that accompany economic growth in most societies And yet the models of economic growth studied so far do not do justice to the complex process described by Kuznets They provide a framework for explaining the sustained increase in income per capita or output per worker But our models do not feature Kuznetss sweeping structural changes A complementary perspective to Kuznetss vision is provided by early development econo mists such as Hirschman Nurske and RosensteinRodan who emphasized the importance of potential market failures and poverty traps in the process of development If such market fail ures and poverty traps are an important determinant of economic performance then we may expect them to be more widespread in lessdeveloped poorer economies1 Thus one might expect Kuznetss structural change to be accompanied by a process that involves the organiza tion of production becoming more efficient and the economy moving from the interior of the aggregate production possibilities set toward its frontier Throughout I use the term struc tural change to describe changes in the composition of production and employment while structural transformations refers to changes in the organization and efficiency of production accompanying the process of development A useful theoretical perspective might therefore be to consider the early stages of economic development taking place in the midst ofor even viastructural changes and transforma tions We may then expect these changes to ultimately bring the economy to the neighborhood of balanced growth where our focus has so far been If this perspective is indeed useful then we would like to develop models that can account for both the structural changes and transfor mations at the early stages of development and the behavior approximated by balanced growth at the later stages We would also like to understand why some societies embark upon these transformations while others do not Some of the models presented so far take steps in this direction For example the model of takeoff in Section 176 of Chapter 17 captures a specific type of transformation from volatile lowproductivity growth into sustained stable growth In addition many of the models in Chapter 18 emphasize the difference between frontier economies and technological follow ers Nevertheless I have not offered a framework that can do justice to Kuznetss and other early development economists vision This is largely because the current growth literature is far from a satisfactory framework that can achieve this objective In this light the distinction between economic growth and economic development can be justified by arguing that in the absence of a unified framework or perhaps precisely as a prerequisite for developing a uni fied framework we need to study the two aspects of the longrun growth process separately Economic growth according to this division of labor focuses on balanced growth the growth behavior of the world economy and other aspects of the growth process approximating the be havior of relatively developed economies Economic development on the other hand becomes the study of structural changes and transformationsand the efficiency implications of these transformationsat the early stages of development Models of economic development would 1 In fact these theoretical perspectives may be the justification for referring to relatively poor economies as underdeveloped rather than as developing In what follows unless there is a special reason for using these terms I stick with the less tainted adjectives lessdeveloped or relatively poor Part VII Economic Development and Economic Growth 695 then focus on structural changes in production and consumption on urbanization on the size and the composition of the population on the occupational structure and on changes in living and social arrangements The study of economic development would thus seek to understand when why and how these processes take place and whether they contribute to a lessdeveloped economy moving toward the frontier of its production possibilities set Since as emphasized by Kuznets economic growth in relatively developed economies also incorporates important elements of structural change part of our analysis in the context of economic development also sheds light on the nature of economic growth in more advanced nations for example by helping us understand why and how relatively balanced growth can often go handinhand with major changes in the sectoral composition of output and employment The secondnotsosatisfactoryreason for the distinction between economic growth and economic development is that there are separate literatures on these two topics with very dif ferent emphases and often different questions The economic growth literature focuses on the theoretical and empirical questions we have so far addressed in this book The economic de velopment literature on the other hand focuses on empirical analyses of education poverty discrimination womens economic and social status child outcomes health lending relations and agriculture in lessdeveloped economies Much of this literature is nontheoretical It doc uments how economic relationships work in lessdeveloped economies or identifies specific market failures This literature has provided us with numerous facts that are helpful in under standing the economic relations in lessdeveloped economies and has sometimes acted as a conduit for micro reforms that have improved the lives of the citizens of these economies But this literature does not ask questions about the aspects of the process of economic development I have emphasized herethat is it does not pose the question of why some countries are less productive and poorer and how these lessdeveloped economies can undergo the process of structural transformation associated with and necessary for modern economic growth Thus though the reason for drawing a distinction between economic growth and economic devel opment might be literaturedriven it may still be useful Moreover based on this distinction one may attempt to bridge the gap between the development and growth literatures by com bining the theoretical tools developed in this book with the wealth of evidence collected by the empirical development literature Such a combination might ultimately lead to a more satisfac tory framework for understanding the process of economic development though unfortunately space restrictions preclude me from pursuing these issues in detail here These two reasons motivate my acceptance of the standard distinction between economic development and economic growth Although I go along with this standard distinction through out I emphasize how it is exactly the same tools that are useful for understanding the process of economic developmentthe structural changes and transformations emphasized by Kuznets Hirschman Nurske and RosensteinRodanas well as the more orderly process of economic growth My hope is that this approach will engender both greater efforts to develop a unified theoretical framework useful for understanding the process of development and theoretical ap proaches that can make contact with and benefit from the wealth of evidence collected by the empirical development literature I organize this part of the book into two chapters Chapter 20 focuses on models that make only a minimal departure from the balanced growth approaches studied so far while still shed ding some light on the structural changes emphasized by Kuznets The models in this chapter can thus be viewed as extensions of the neoclassical growth models in Chapters 8 and 11 de signed to confront various important empirical patterns that are salient in the development process However these models neither do full justice to the process of sweeping structural changes emphasized by Kuznets nor do they capture the complex aspects of the process of eco nomic development associated with the move from the interior of the production possibilities set toward the frontier Chapter 21 presents several models that investigate various facets of 696 Part VII Economic Development and Economic Growth this process including financial development the demographic transition urbanization and other social changes Furthermore they highlight the importance of potential market failures that may cause development traps These models present a range of exciting questions and different modeling approaches but at the expense of providing less unity Each model makes a different set of assumptions and the profession is far from having a unified framework for the analysis of the major structural transformations involved in the process of development The purpose of Chapter 21 is not to provide such a unified framework but to introduce the reader to these interesting and important questions It should also be noted that the division between the two chapters is not perfect Some of the models of structural transformation studied in Chapter 21 can be seen as closely related to the structural change models in Chapter 20 Moreover some topics such as the beginning of industrialization can be treated both as a process of structural change and as an outcome of a society solving certain market failures Thus the decision of whether a particular topic should be in Chapter 20 or Chapter 21 was somewhat arbitrary 20 Structural Change and Economic Growth S ections 201 and 202 focus on the shift of employment and production from agriculture to manufacturing and then from manufacturing to services This is a useful starting point both because changes in the composition of employment and production are an important part of the process of economic development and because as emphasized by Kuznets and others similar changes are present even beneath the facade of balanced modern growth Consequently these two sections focus on preferencerelated demandside and technology related supplyside reasons for why we may expect structural change as an economy becomes richer but they also emphasize how such structural changes can be reconciled with balanced growth Section 203 turns to a related theme and shows how preindustrial agricultural productivity may be a key determinant of the process of industrialization and takeoff 201 Nonbalanced Growth The Demand Side Figure 201 provides a summary of some of the major changes in the structure of production that the US economy has undergone over the past 150 years It shows that the share of employment in agriculture stood at about 90 of the labor force at the beginning of the nineteenth cen tury while only a small fraction of the US labor force worked in manufacturing and services By the second half of the nineteenth century both manufacturing and services had expanded to more than 20 of employment accompanied by a steep decline in the share of agriculture Over the past 150 years or so the share of employment in agriculture has continued to decline and now stands at less than 5 while more than 70 of US workers are now employed in service industries The share of manufacturing first increased when the share of agriculture started its decline but has been on a downward trend over the past 40 years or so and now stands at just over 20 When we look at consumption shares the general trends are similar though the share of consumption expenditures on agricultural products is still substantial because of changes in relative prices and relative productivities and also partly because of imports of agricultural goods The changes in the composition of employment in the British economy toward the end of the eighteenth century are also consistent with the US patterns shown in Figure 201 see eg Mokyr 1993 Similar patterns are present in all OECD economies as 697 201 Nonbalanced Growth The Demand Side 699 where cAt γ A denotes per capita agricultural consumption at time t cMt R is manufacturing consumption and cSt R is services consumption γ A γ S ηA ηM and ηS are positive constants and ηA ηM ηS 1 This StoneGeary form is a highly tractable way of introducing income elasticities that are different from one for different subcomponents of consumption and Engels Law In particular this aggregator implies that there is a minimum or subsistence level of agricultural food consumption equal to γ A The household must consume at least this much food to survive and in fact consumption and utility are not defined when the household does not consume the minimum amount of food After this level of food consumption has been achieved the household starts to demand other items in particular manufactured goods eg textiles and durables and services eg health entertainment wholesale and retail However as we will see shortly the presence of the γ S term in the aggregator implies that the household will spend a positive amount on services only after certain levels of agricultural and manufacturing consumption have been reached Suppose that the economy is closed thus agricultural manufacturing and services con sumption must be met by domestic production I follow Kongsamut Rebelo and Xie and assume the following production functions Y At BAFKAt XtLAt Y Mt BMFKMt XtLMt Y St BSFKSt XtLSt 203 where Y jt for j A M S denotes the output of agricultural manufacturing and services at time t Kjt and Ljt for j A M S are the levels of capital and labor respectively al located to the agricultural manufacturing and services sectors at time t Bj for j A M S is a Hicksneutral productivity term for the three sectors and finally Xt is a laboraugmenting Harrodneutral productivity term affecting all sectors I use the letter X instead of the stan dard A to distinguish this term from the agricultural good The function F satisfies the usual neoclassical assumptions Assumptions 1 and 2 from Chapter 2 Two other features in 203 are worth noting First the production functions for all three sectors are identical Sec ond the same laboraugmenting technology term affects all three sectors Both of these features are clearly unrealistic but they are useful for isolating the demandside sources of structural change and for contrasting them with the supplyside factors that will be discussed in the next section Furthermore Exercise 207 shows that they can be relaxed to some degree Let us take the initial population L0 0 and the initial capital stock K0 0 as given and also assume that there is a constant rate of growth of the laboraugmenting technology term that is Xt Xt g 204 for all t with initial condition X0 0 To ensure that the transversality condition of the representative household holds I impose the same assumption as in the basic neoclassical growth model of Chapter 8Assumption 4 which implies that ρ n 1 θ g Market clearing for labor and capital requires that KAt KMt KSt Kt 205 LAt LMt LSt Lt 206 where Kt and Lt are the total supplies of capital and labor respectively at time t 201 Nonbalanced Growth The Demand Side 701 An equilibrium is straightforward to characterize in this economy Because the production functions of all three sectors are identical the following result obtains immediately Proposition 201 Suppose 2013 holds Then in any equilibrium the following conditions are satisfied KAt XtLAt KMt XtLMt KSt XtLSt Kt XtLt kt 2014 for all t where the last equality defines kt as the aggregate effective capitallabor ratio of the economy and also pAt BM BA and pSt BM BS for all t 2015 Proof See Exercise 202 The results in this proposition are intuitive First the fact that the production functions are identical implies that the capitallabor ratios in the three sectors must be equalized Second given 2014 the equilibrium price relationships 2015 follow because the marginal products of capital and labor have to be equalized in all three sectors Proposition 201 does not make use of the preference side Incorporating utility maximiza tion on the side of the representative household in particular deriving the standard Euler equation for the representative household and then using 209 and 2010 we obtain the following additional equilibrium conditions Proposition 202 Suppose 2013 holds Then in any equilibrium we have cMt cMt 1 θ rt ρ 2016 for all t Moreover provided that Assumption 4 holds household utility is finite and the transversality condition is satisfied In addition we have BMcAt γ A BAηA cMt ηM and BMcSt γ S BSηS cMt ηM for all t 2017 Proof See Exercise 203 Proposition 203 Suppose that either γ A 0 andor γ S 0 Then there does not exist an equilibrium in which all sectors grow at the same rate Proof See Exercise 204 This result is not surprising Since the preferences of the representative household incorpo rate Engels Law the household always wants to change the composition of its consumption and this is reflected in a change in the composition of production Nevertheless a BGP in which consumption asymptotically grows at a constant rate still exists I refer to this as a constant growth path CGP to emphasize that this notion allows for nonbalanced sec toral growth In a CGP the consumption aggregate grows at a constant rate while output and employment in the three sectors grow at different rates Given the preferences in 201 the constant growth rate of consumption also implies that the interest rate must be constant 702 Chapter 20 Structural Change and Economic Growth Proposition 204 Suppose 2013 holds Then in the abovedescribed economy a unique CGP exists if and only if γ A BA γ S BS 2018 In a CGP kt k for all t and cAt cAt g cAt γ A cAt cMt cMt g cSt cSt g cSt γ S cSt 2019 LAt LAt n g γ ALtLAt BAXtF k 1 LMt LMt n and LSt LSt n g γ SLtLSt BSXtF k 1 for all t Moreover in a CGP the share of national income accruing to capital is constant Proof See Exercise 205 This model therefore delivers a tractable framework for the analysis of structural change that has potential relevance both for the experience of economies at the early stages of development and for understanding the patterns of growth of relatively advanced countries Engels Law augmented with the highly incomeelastic demand for services generates a demandside force for nonbalanced growth In particular as their incomes grow households wish to spend a greater fraction of their budgets on services and a smaller fraction on food agricultural goods This tendency makes an equilibrium with fully balanced growth impossible Instead different sectors grow at different rates and there is reallocation of labor and capital across sectors Nevertheless Proposition 204 shows that under condition 2018 a CGP exists and in this equilibrium structural change takes place even though the interest rate and the share of capital in national income are constant This model therefore delivers many of the features that are useful for thinking about longrun economic development the equilibrium path can be consistent with the Kaldor facts and there is a continuous process of structural change whereby the share of agriculture in production and employment declines over the development process and the share of services increases On the downside several potential shortcomings of the current model are worth noting First one may argue that the process of structural change in this model falls short of the sweeping changes discussed by Kuznets Although I focused on the CGP it is straightforward to incorpo rate transitional dynamics into the model Exercise 206 shows that if the effective capitallabor ratio starts out below its CGP value of k in Proposition 204 then there will be additional tran sitional dynamics in this model complementing the structural changes Nevertheless even these transitional dynamics fall short of the sweeping structural changes emphasized by Kuznets Second the assumption that all three sectors have the same production function appears restrictive This assumption can be relaxed to some degree Exercise 207 discusses how this can be done Perhaps more important is the assumption that investments for all three sectors use only the manufacturing good This assumption is similar in nature to the assumption that only capital is used to produce capital investment goods in Rebelos 1991 model recall Chapter 11 Exercise 2010 shows that if this assumption is relaxed it is no longer possible to reconcile the Kuznets and the Kaldor facts in the context of this model Third the model presented here is designed to generate a constant share of employment in manufacturing Although this pattern is broadly consistent with the US experience over the 202 Nonbalanced Growth The Supply Side 703 past 150 years when we look at even earlier stages of development almost all employment is in agriculture Thus the early stages of structural change must also involve an increase in the share of employment in manufacturing Several models in the literature generate this pattern by also introducing land as an additional factor of production Exercise 208 provides an example and Section 212 presents a model incorporating land as a major factor of production in the context of the study of population dynamics Finally and most importantly the condition necessary for a CGP 2018 is a rather knifeedge condition Why should this specific equality between technology and preference parameters hold In the final analysis there is no compelling argument that this condition should be satisfied see Exercise 209 202 Nonbalanced Growth The Supply Side The previous section showed how the process of structural change can be driven by a general ized form of Engels Law that is by the desires of the households to change the composition of their consumption as they become richer An alternative approach to why growth may be nonbalanced was first proposed by Baumols 1967 seminal work Baumol suggested that un even growth or what I am referring to here as nonbalanced growth will be a general feature of the growth process because different sectors grow at different rates owing to different rates of technological progress eg technological progress might be faster in manufacturing than in agriculture or services Although Baumols original article derived this result only under a variety of special assumptions the general insight that there might be technological supply side forces pushing the economy toward nonbalanced growth is considerably more general Here I review some ideas based on Acemoglu and Guerrieri 2008 who emphasize the tech nological causes of nonbalanced growth Ultimately both the rich patterns of structural change during the early stages of development and those we witness in more advanced economies to day require models that combine technological and preference factors Nevertheless isolating these factors in separate models is both more tractable and conceptually more transparent For this reason in this section I focus on technological causes of nonbalanced growth abstract ing from Engels Law throughout and only return to the combination of technological and preference factors in Exercise 2017 2021 General Insights At some level Baumols theory of nonbalanced growth can be viewed as selfevidentif some sectors have higher rates of technological progress there must be some nonbalanced elements in equilibrium My first purpose in this section is to show that there are more subtle and compelling reasons for supplyside nonbalanced growth than those originally emphasized by Baumol In particular most growth models like the Kongsamut Rebelo and Xie model presented in Section 201 assume that production functions in different sectors are identical In practice however industries differ considerably in terms of their capital intensity and the intensity with which they use other factors eg compare the retail sector to durables manufacturing or transport In short different industries have different factor proportions The main economic point I emphasize in this section is that factor proportion differences across sectors combined with capital deepening lead to nonbalanced economic growth I illustrate this point first using a simple but fairly general environment This environment consists of two sectors each with a constant returns to scale production function and arbitrary preferences over the goods that are produced in these two sectors Both sectors employ capital 704 Chapter 20 Structural Change and Economic Growth K and labor L To highlight that the exact nature of the accumulation process is not essential for the results I take the path of capital and labor supplies Kt Lt t0 as given and assume that Kt and Lt are differentiable functions of time Labor is supplied inelastically Preferences are defined over the final output or a consumption aggregator as in 202 in the Section 201 Whether we use the specification with a consumption aggregator or a formulation with intermediates used competitively in the production of a final good makes no difference to any of the results With this in mind let final output be denoted by Y and assume that it is produced as an aggregate of the output of two sectors Y1 and Y2 Yt FY1t Y2t which again satisfies Assumptions 1 and 2 see Chapter 2 Sectoral production functions are given by Y1t A1tG1K1t L1t and 2020 Y2t A2tG2K2t L2t 2021 where L1t L2t K1t and K2t denote the amount of labor and capital employed in the two sectors and the functions G1 and G2 are also assumed to satisfy the equivalents of Assumptions 1 and 2 The terms A1t and A2t are Hicksneutral technology terms Market clearing for capital and labor implies that K1t K2t Kt and L1t L2t Lt 2022 at each t Without loss of generality I ignore capital depreciation Let us take the final good as the numeraire in every period and denote the prices of Y1 and Y2 by p1 and p2 and wage and rental rate of capital interest rate by w and r respectively Product and factor markets are competitive so that product and factor prices satisfy p1t p2t FY1t Y2tY1 FY1t Y2tY2 2023 and wt p1tA1tG1K1t L1t L1 p2tA2tG2K2t L2t L2 rt p1tA1tG1K1t L1t K1 p2tA2tG2K2t L2t K2 2024 An equilibrium given factor supply paths Kt Lt t0 is a path of product and factor prices p1t p2t wt rt t0 and factor allocations K1t K2t L1t L2t t0 such that 2022 2023 and 2024 are satisfied Let the shares of capital in the two sectors be defined as σ1t rtK1t p1tY1t and σ2t rtK2t p2tY2t 2025 There is capital deepening at time t if KtKt LtLt There are factor proportion differences at time t if σ1t σ2t And finally technological progress is balanced at time t 706 Chapter 20 Structural Change and Economic Growth Since A1A1 A2A2 εg 1 k1 k1 εg 2 k2 k2 2030 Differentiating the wage condition 2029 with respect to time and using 2027 and some algebra gives A1 A1 σ1 1 σ1 εg 1 k1 k1 A2 A2 σ2 1 σ2 εg 2 k2 k2 Since A1A1 A2A2 and σ1 σ2 this equation is inconsistent with 2030 yielding a contradiction and proving the claim The intuition for this result is straightforward Suppose that there is capital deepening and that for concreteness sector 2 is more capital intensive σ1 σ2 If both capital and labor were allocated to the two sectors at constant proportions over time the more capitalintensive sector sector 2 would grow faster than sector 1 In equilibrium the faster growth in sector 2 would change equilibrium prices and the decline in the relative price of sector 2 would cause some of the labor and capital to be reallocated to sector 1 However this reallocation could not entirely offset the greater increase in the output of sector 2 since if it did the relative price change that stimulated the reallocation in the first place would not occur Consequently equilibrium growth must be nonbalanced Proposition 205 is related to the wellknown Rybczynskis Theorem in international trade Rybczynskis Theorem states that in an open economy within the cone of diversification where factor prices do not depend on factor endowments changes in factor endowments are absorbed by changes in the sectoral output mix Proposition 205 can be viewed both as a closedeconomy analogue and as a generalization of Rybczynskis Theorem it shows that changes in factor endowments capital deepening is absorbed by faster growth in one sector than the other even though relative prices of goods and factors change in response to the change in factor endowments 2022 Balanced Growth and Kuznets Facts The Section 2021 provided general insights about how technological factors can lead to nonbalanced growth To obtain a general result on the implications of capital deepening and factor proportion differences across sectors on nonbalanced growth Proposition 205 was stated for a given arbitrary path of capital and labor supplies Kt Lt t0 However without endogenizing the path of capital accumulation and specifying the pattern of population growth we cannot address whether a model relying on technological factors can also provide a useful framework for thinking about the Kuznets facts without significantly deviating from the balanced growth patterns exhibited by many relatively developed economies For this purpose I now specialize the environment of Section 2021 by incorporating specific preferences and production functions and then provide a full characterization of a simpler economy The economy is again in infinite horizon and population grows at the exogenous rate n 0 Let us also assume that the economy admits a representative household with standard preferences given by 201 that also supplies labor inelastically Proposition 205 emphasized the importance of capital deepening which now results from exogenous technological progress 714 Chapter 20 Structural Change and Economic Growth and also similar expressions for n 1 and n 2 Combining these equations implies that g 1 g 2 which contradicts the hypothesis g 1 g 2 0 The argument for ε 1 is analogous 2 Suppose g 2 g 1 and ε 1 Then the same steps as above imply that there is a unique solution to equilibrium conditions 2033 2057 and 2058 which is given by 2062 2065 But now 2062 directly contradicts g 1 0 Finally suppose g 2 g 1 and ε 1 Then the unique solution is given by the equations in part 1 above But in this case 2067 contradicts the hypothesis that g 1 0 completing the proof Several implications of this proposition are worth emphasizing First as long as a11 α1 a21 α2 growth is nonbalanced The intuition for this result is the same as that for Proposition 205 Suppose for concreteness that ε 1and a11 α1 a21 α2 which would be the case eg if a1 a2 Then differential capital intensities in the two sec tors combined with capital deepening in the economy which itself results from technological progress ensures faster growth in the more capitalintensive sector sector 2 Intuitively if capital were allocated proportionately to the two sectors sector 2 would grow faster Because of the changes in prices capital and labor would be reallocated in favor of the less capital intensive sector and relative employment in sector 1 would increase However crucially this reallocation would not be enough to fully offset the faster growth of real output in the more capitalintensive sector This result also highlights that the assumption of balanced techno logical progress in Proposition 205 which in this context corresponds to a1 a2 was not necessary for the result there but we simply needed to rule out the knifeedge case where the relative rates of technological progress between the two sectors were exactly in the right proportion to ensure balanced growth in this context a11 α1 a21 α2 Second the CGP growth rates are relatively simple especially because I restricted atten tion to the set of parameters that ensure that sector 1 is the asymptotically dominant sector see 2061 If in addition ε 1 the model leads to the richest set of dynamics where the more slowly growing sector determines the longrun growth rate of the economy while the more rapidly growing sector continually sheds capital and labor but does so at exactly the rate to ensure that it still grows faster than the rest of the economy Third in the CGP the share of capital and labor allocated to one of the sectors tends to 1 eg when sector 1 is the asymptotically dominant sector λ κ 1 Nevertheless at all points in time both sectors produce positive amounts so this limit point is never reached In fact at all times both sectors grow at rates greater than the rate of population growth in the economy Moreover when ε 1 the sector that is shrinking in terms of capital and labor share grows faster than the rest of the economy at all points in time even asymptotically Therefore the rate at which capital and labor are allocated away from this sector is determined in equilibrium to be exactly such that this sector still grows faster than the rest of the economy This is the sense in which nonbalanced growth is not a trivial outcome in this economy with one of the sectors shutting down but results from the positive but differential growth of the two sectors Finally it can be verified that the capital share in national income and the interest rate are constant in the CGP see Exercise 2015 For example when 2061 holds σ K α1 In contrast when this condition does not hold then σ K α2 Thus the asymptotic capital share in national income always reflects the capital share of the asymptotically dominant sector Therefore this model based on technological sources of nonbalanced growth is also broadly consistent with the Kaldor facts as well as the Kuznets facts though this model also generates significant deviations from the orderly behavior implied by the Kaldor facts when the economy is away from the CGP The analysis so far does not establish that the CGP is asymptotically stable This is done in Exercise 2016 which also provides an alternative proof of Proposition 2010 Consequently a model based on technological factors can also give useful insights about structural change Naturally to understand the sweeping longrun 716 Chapter 20 Structural Change and Economic Growth These preferences are similar to those in 201 cAt again denotes the consumption of the agricultural good cMt is the consumption of the manufacturing good at time t and the parameter γ A is the minimum subsistence food requirement In addition ρ is the discount factor and η 0 1 designates the importance of agricultural goods versus manufacturing goods in the utility function The representative household supplies labor inelastically Let us also focus on the closed economy in the text leaving the extension to an open economy to Exercise 2020 Output in the two sectors is produced with the following production functions Y Mt XtFLMt and Y At BAGLAt 2069 where as before Y M and Y A denote the total production of the manufacturing and the agricul tural goods respectively and LM and LA denote the total labor employed in the two sectors Both production functions F and G exhibit diminishing returns to labor More formally F and G are differentiable and strictly concave In particular F0 0 F 0 F 0 G0 0 G 0 and G 0 Diminishing returns to labor might arise because they both use land or some other factor of production as well as labor Nevertheless it is simpler to assume diminishing returns rather than introduce another factor of production Diminishing returns implies that when labor is priced competitively there are equilibrium profits and these are redistributed to households The key feature for this model of industrialization is that there is no technological progress in agriculture but the production function for the manufacturing good in 2069 includes the term Xt which allows for technological progress in manufacturing Although there is no technological progress in agriculture the productivity parameter BA potentially differs across countries reflecting either previous technological progress in terms of new agricultural meth ods or differences in land quality for simplicity I am focusing on a single country Existing evidence shows that there are very large differences in labor productivity and the TFP of agricul tural activities among countries even today thus allowing for potential productivity differences in agriculture is reasonable Current research also shows that the image of agriculture as a quasistagnant sector without technological progress is not accurate and in fact this sector ex periences both substantial capitallabor substitution and major technological change including the introduction of new varieties of seeds mechanization and organizational changes affecting productivity Nevertheless the current model provides a good starting point for our purposes Labor market clearing requires that LMt LAt 1 Let nt denote the fraction of labor employed in manufacturing as of time t Since there is full employment in this economy LMt nt and LAt 1 nt The key assumption is that manufacturing productivity Xt evolves over time as a result of learningbydoing externalities as in Romers 1986a model in Chapter 11 In particular suppose that the growth of the manufacturing technology Xt is proportional to the amount of current production in manufacturing Xt κY Mt 2070 where κ 0 measures the extent of these learningbydoing effects and the initial productivity level is X0 0 at time t 0 taken as given As in the Romer model learningbydoing is external to individual firms This type of external learningbydoing is too reducedform to generate insights about how productivity improvements take place in the industrial sector Nevertheless our analysis so far makes it clear that one can endogenize technology choices by 203 Agricultural Productivity and Industrialization 717 introducing monopolistic competition and under the standard assumptions made in Part IV this richer model of endogenous technological change generates a market size effect and leads to an equation similar to 2070 Exercise 2019 asks you to consider such a model In equilibrium each firm chooses its labor demand to equate the value of the marginal product to the wage rate wt Let us choose the agricultural good as the numeraire so that its price is normalized to 1 and also assume that the equilibrium is interior with both sectors being active Then in equilibrium we have wt BAG1 nt and wt ptXtF nt where pt is the relative price of the manufactured good in terms of the numeraire the agricultural good Market clearing then implies that BAG1 nt ptXtF nt 2071 The term γ A 0 in 2068 implies that as in Section 201 preferences in the economy are nonhomothetic and that the income elasticity of demand for agricultural goods is less than unity while that for manufacturing goods is greater than unity As we have already seen this is the simplest way of introducing Engels Law Suppose that productivity is high enough to meet the minimum agricultural consumption requirements of the entire population which here is normalized to 1 BAG1 γ A 0 2072 If 2072 were violated the economys agricultural sector would not be productive enough to provide the subsistence level of food to all households Finally the budget constraint of the representative household at each date t can be written as cAt ptcMt wt πt where πt is the profits per representative household resulting from the diminishing returns in the production technologies An equilibrium in this economy is defined in the standard way as paths of consumption levels in the two sectors and allocations of labor between the two sectors at all dates such that households maximize utility and firms maximize profits given prices and goods and factor prices are such that all markets clear Maximization of 2068 implies that cAt γ A ηptcMt 1 η 2073 Since the economy is closed production must equal consumption and thus cAt Y At BAG1 nt and cMt Y Mt XtFnt Now combining these equations with 2071 and 2073 yields φnt γ A BA 2074 where φn G1 n ηG1 nFn 1 η F n 204 Taking Stock 719 An important advantage of the current model is its tractability This enables us to adapt it easily to analyze related questions such as the impact of trade opening on industrialization This is done in Exercise 2020 which shows that the role of agricultural productivity in closed and open economies can be very different For example in an open economy greater agricultural productivity may delay or prevent industrialization rather than encouraging it as in the closed economy The reason for this is related to the forces highlighted in Section 197 of Chapter 19 specialization according to comparative advantage may have negative longrun consequences in the presence of sectorspecific externalities However as already discussed in that section the evidence for large externalities of this sort is not very strong Consequently the role of international trade in the process of industrialization is likely to be more complex than that suggested by Exercise 2020 Nevertheless this exercise illustrates how open economy models enrich the study of structural change 204 Taking Stock This chapter took a first step toward the analysis of structural changes involved in the process of economic development Our first step has been relatively modest The focus has been on the structural changes associated with 1 the shifts in output and employment away from agriculture to manufacturing and to services and 2 the changes between sectors of different capital intensities Section 201 focused on demandside reasons resulting from the structure of preferences for why growth may be nonbalanced It incorporated Engels Law into the basic neoclassical growth model so that households spend a smaller fraction of their budget on agricultural goods as they become richer This framework is ideally suited to the analysis of the structural changes across broad sectors such as agriculture manufacturing and services Section 202 on the other hand turned to supplyside technological reasons for nonbalanced growth which were first highlighted by Baumols 1967 classic paper However instead of assuming exogenously given different rates of technological progress across sectors this section emphasized how sectoral differences in capital intensity can lead to nonbalanced growth Capitalintensive sectors tend to grow more rapidly as a result of an equiproportionate increase in the capitallabor ratio This feature combined with capital deepening at the economy level naturally leads to a pattern of nonbalanced growth Such nonbalanced growth may contribute to structural change across agricultural manufacturing and service sectors but becomes particularly relevant when we look at sectors differentiated according to their capital intensity A particular focus of both Sections 201 and 202 was to reconcile nonbalanced growth at the sectoral level with the patterns of relatively balanced growth at the aggregate As already noted in Chapter 2 balanced growth need not be taken literally It is at best an approximation Nevertheless it seems to be a good approximation to many features of the growth process in advanced economies where interest rates and the share of capital income in GDP appear to have been relatively constant over the past 100 years or so It is therefore important to understand how significant reallocation of resources at the sectoral level can coexist with the more balanced behavior at the aggregate The models in Sections 201 and 202 suggested some clues about why this may be the case but the answers provided here should be viewed as preliminary rather than definitive I also discussed a simple model of the origins of industrialization This model showed how agricultural productivity might have a significant effect on the timing of industrialization The study of the process of industrialization is important in part because as discussed in Chapter 1 existing evidence suggests that the timing and nature of industrialization may have important implications for the crosscountry income differences we observe today The study of economic 720 Chapter 20 Structural Change and Economic Growth development may therefore necessitate an analysis of why some countries industrialized early while others were delayed or never started the process of industrialization Understanding the sources of structural changes and how they can be reconciled with the broad patterns of balanced growth in the aggregate sheds light on the process of economic growth and development In this sense the models in this section enrich our understanding of economic growth considerably And yet this is only a modest step toward the investigation of the sweeping structural changes emphasized by Kuznets because we have not departed from the neoclassical approach to economic growth In particular Sections 201 and 202 used generalized versions of the basic neoclassical growth model of Chapter 8 and Section 203 used a variant of the Romer 1986a model from Chapter 11 It should be emphasized again that the topics discussed in this chapter though closely related to the basic neoclassical growth model are areas of frontier research We are far from a satisfactory framework for understanding the process of reallocation of capital and labor across sectors how this changes at different stages of development and how it remains consistent with relatively balanced aggregate growth and the Kaldor facts I have therefore not attempted to provide a unified framework that combines the transition from agriculture to industrialization nonhomothetic preferences resulting from Engels Law and technological factors leading to nonbalanced growth The development of such unified models as well as richer models of nonbalanced growth are areas for future research 205 References and Literature The early development literature contains many important works documenting the major struc tural changes taking place in the process of development Kuznets 1957 1973 and Chenery 1960 provide some of the best overviews of the broad evidence and the literature though similar issues were discussed by even earlier development economists such as Rosenstein Rodan 1943 Nurske 1958 and Rostow 1960 Figure 201 which uses data from Historical Statistics of the United States Carter et al 2006 gives a summary of these broad changes The model of nonbalanced growth based on Engels Law presented in Section 201 is from Kongsamut Rebelo and Xie 2001 Previous works that have analyzed similar models include Murphy Shleifer and Vishny 1989 Echevarria 1997 and Laitner 2000 More recent work building on Kongsamut Rebelo and Xie 2001 includes Caselli and Coleman 2001b and Gollin Parente and Rogerson 2002 Some of these models include land as an additional factor of production necessary for agriculture Exercise 208 provides an example of such a model The recent literature also places greater emphasis on sources of agricultural productivity and emphasizes that differences in agricultural productivity across countries are often as large as or even larger than productivity differences in other sectors Gollin Parente and Rogerson 2002 is one of the first papers in this vein The works mentioned in the previous paragraph like the model I presented in Section 201 appeal to Engels Law and model the resulting nonhomothetic preferences by positing StoneGeary preferences as in 202 A more flexible and richer approach is to allow for hierarchies of needs in consumption whereby households consume different goods in a particular sequence eg food needs to be consumed before textiles and textiles need to be consumed before electronics and so on This and related approaches are used in Stokey 1988 Foellmi and Zweimuller 2002 Matsuyama 2002 and Buera and Kaboski 2006 to generate richer models of structural change Space restrictions preclude me from presenting these hierarchy of needs models even though they are both insightful and elegant alternatives to the standard approach of using StoneGeary preferences 206 Exercises 721 Section 202 builds on Acemoglu and Guerrieri 2008 The precursor to this work is Baumol 1967 which emphasizes the importance of differential productivity growth on nonbalanced growth However Baumol did not derive a pattern of nonbalanced growth including realloca tion of capital and labor across sectors and assumed differential rates of productivity growth to be exogenous Ngai and Pissarides 2006 and Zuleta and Young 2006 provide modern versions of Baumols hypothesis Instead the approach in Section 202 emphasizes how the combination of different capital intensities and capital deepening in the aggregate can endoge nously lead to this pattern The model in Section 203 is based on Matsuyama 1992 and is also closely related to the model presented in Section 197 in Chapter 19 The role of agriculture in industrialization is discussed in Mokyr 1993 Overton 1996 and Mundlak 2000 206 Exercises 201 Show that the consumption aggregator in 202 leads to Engels Law Suggest alternative con sumption aggregators that will generate similar patterns 202 Prove Proposition 201 203 a Set up the optimal control problem for a representative household in the model of Section 201 b From the Euler equations and the transversality condition verify 2016 in Proposition 202 c Use 209 and 2010 to derive 2017 of the proposition 204 Prove Proposition 203 Show that an equilibrium path always exists even though it never features equal rates of growth for all sectors 205 a Prove Proposition 204 In particular show that if 2018 is not satisfied a CGP cannot exist and that this condition is sufficient for a unique CGP to exist b Characterize the CGP effective capitallabor ratio k 206 In the model of Section 201 show that as long as condition 2018 is satisfied when the economy starts with an effective capitallabor ratio K0 X0L0 different from k the CGP is globally stable and the effective capitallabor ratio monotonically converges to k as t 207 Consider a generalization of the model of Section 201 in which the sectoral production functions are given by the following CobbDouglas forms Y At KAtαABAtLAt1αA Y Mt KMtαMBMtLMt1αM Y St KStαSBStLSt1αS and assume that BAt BMt and BSt grow at the rates gA gM and gS respectively a Derive the equivalents of Propositions 201 and 202 b Show that there exists a generalization of condition 2018 such that this model will have a CGP as defined in Section 201 Hint the generalization includes two separate conditions on technology growth rates and preferences 208 Consider a version of the model in Section 201 with only manufacturing and agricultural goods The consumption aggregator is ct cAt γ AηAcMtηM with γ A 0 Assume 722 Chapter 20 Structural Change and Economic Growth that the production functions for agricultural and manufacturing goods take the form Y At XtLAtζZ1ζ and Y Mt XtLMt respectively where Z is land There are no savings or capital a Characterize the competitive equilibrium in this economy b Show that this economy also exhibits structural change in particular show that the share of manufacturing sector grows over time c What happens to land rents along the equilibrium path 209 In the model of Section 201 suppose that condition 2018 is not satisfied Assume that the production function F is CobbDouglas Characterize the asymptotic growth path of the economy the growth path of the economy as t 2010 Consider the model of Section 201 but assume that there exists a final good produced with the technology Yt Y At γ AηAY MtηMY St γ SηS a Show that all the results in Section 201 hold without any change as long as capital goods are produced out of intermediate Y M as implied by 207 b Next assume that capital goods are produced out of the final good so that the resource constraint becomes Kt ctLt Yt where ct is the per capita consumption of the final good Show that in this model a CGP does not exist 2011 In the model of Section 2021 suppose that aggregate output is given by the constant returns to scale production function Y FY1t Y2t YNt Defining σjt as the capital share in sector j 1 N as in 2025 show that if at time t there are factor proportion differ ences among the N sectors in the sense that there exists i and j N such that σit σjt technological progress is balanced between i and j that is AitAit AjtAjt and there is capital deepening that is KtKt LtLt then growth is not balanced and YitYit YjtYjt 2012 Derive 2039 2040 2041 and 2042 2013 Prove Proposition 206 2014 a Complete the proof of Proposition 2010 by considering the case in which ε 1 and g 1 g 2 0 b State and prove the equivalent of Proposition 2010 when the converse of condition 2061 holds 2015 Show that in the allocation in Proposition 2010 the asymptotic interest rate is constant and derive a closedform expression for this interest rate 2016 In this exercise you are first asked to provide an alternative proof of Proposition 2010 and then characterize the local transitional dynamics in the neighborhood of the CGP Throughout suppose that either ε 1 and a11 α1 a21 α2 or that ε 1 and a11 α1 a21 α2 a Reexpress the equilibrium equations in terms of the following three variables ϕt ctLtA1t11α1 χt KtLtA1t11α1 and κt In particular show that the following three differential equations together with the appropriate transversality condi tion and initial values χ0 and κ0 characterize the dynamic equilibrium ϕt ϕt 1 θ α1γ ηt1ελt1α1κt1α1χt1α1 ρ n a1 1 α1 χt χt λt1α1κtα1χt1α1ηt χt1ϕt n a1 1 α1 κt κt 1 κtα2 α1 χt χt a2 1α2 1α1a1 1 ε1 α2 α1κt λt 2076 724 Chapter 20 Structural Change and Economic Growth for creating new machines as in Section 157 Characterize the equilibrium of this economy and show that the qualitative features are the same as the model in the text 2020 Consider an open economy version of the model of Section 203 In particular suppose that the economy trades with the rest of the world taking product prices as given The rest of the world is characterized by the same technology except that it has an initial level of productivity in the manufacturing sector equal to XF0 and an agricultural productivity given by BF Suppose that there are no spillovers in learningbydoing so that 2070 applies to the home economy and the law of motion of manufacturing productivity in the rest of the world is given by XFt κY MFt where Y MFt is total foreign manufacturing production at time t a Show that comparative advantage in this economy is determined by the comparison of X0BA to XF0BF Interpret this result b Suppose that X0BA XF0BF so that the home economy has a comparative advantage in agricultural production Show that the initial share of employment in manufacturing in the home economy n0 must satisfy X0F n0 BAG1 n0 XF0F nF0 BFG1 nF0 2078 where nF0 is the share of manufacturing employment in the rest of the world Show that n0 given by this equation is strictly less than n as given by 2075 c What happens to manufacturing employment in the home economy starting as in part b of this exercise Hint derive an equivalent of 2078 for any t differentiate this with respect to time and then use the laws of motion of X and XF d Explain why agricultural productivity which was conducive to faster industrialization in the closed economy may lead to delayed industrialization or to deindustrialization in the open economy e Consider an economy specializing in agriculture as in the earlier parts of this exercise Is welfare at time t 0 necessarily lower when this economy is open to trade than when it is closed to trade Relate your answer to the analysis in Section 197 of Chapter 19 21 Structural Transformations and Market Failures in Development A more complex transformation of the economy than the changes in the structure of production studied in the previous chapter takes place with the process of economic development Among other things this transformation involves major social changes and induces greater coordination of economic activities Loosely speaking we can think of a society that is relatively developed as functioning along or at any rate near the frontier of its production possibilities set while a lessdeveloped economy may be in the interior of its notional production possibilities set This may be because certain arrangements necessary for an economy to reach the frontier of its production possibility set require a large amount of capital or some specific technological advances in which case even though we may think of the society as functioning in the interior of its production possibility set this may not be the outcome of market failure thus the qualifier notional in the previous sentence Alternatively lessdeveloped economies may be in the interior of their production possibility set because these societies are subject to severe market failures In this chapter I discuss these approaches to economic development I first focus on various dimensions of structural transformations and how these may be limited by the amount of capital or technology available in a society I then discuss a number of approaches suggesting that lessdeveloped economies might be suffering disproportionately from market failures or may even be stuck in development traps In this context I also discuss differences between models with multiple equilibria and with multiple steady states The topics covered in this chapter are part of a large and diverse literature My purpose is not to do justice to this literature but to emphasize how certain major structural transformations take place as part of the process of economic development and also to highlight the potential importance of market failures in this process Given this objective and the large number of potential models my choice of models is selective and my treatment is more informal than in the rest of the book In addition I often make reducedform assumptions to keep the exposition brief and simple 725 726 Chapter 21 Structural Transformations and Market Failures in Development 211 Financial Development An important aspect of the structural transformation brought about by economic development is a change in financial relations and a deepening of financial markets Section 176 in Chapter 17 presented a model in which economic growth goes handinhand with financial deepening However the model in that section only focused on a specific aspect of the role of financial in stitutions In general financial development brings about a number of complementary changes in the economy First there is greater depth in the financial market allowing better diversi fication of aggregate risksa feature also emphasized in the model of Section 176 Second one of the key roles of financial markets is to allow risk sharing and consumption smoothing for individuals In line with this role financial development also allows better diversifica tion of idiosyncratic risks Section 176 showed that better diversification of aggregate risks leads to improved allocation of funds across sectors or projects Similarly better sharing of idiosyncratic risks leads to improved allocation of funds across individuals Third financial development might also reduce credit constraints on investors and thus may directly enable the transfer of funds to individuals with improvements in investment opportunities The second and the third channels not only affect the allocation of resources in the society but also the distri bution of income because diversification of idiosyncratic risks and relaxation of credit market constraints might lead to improvements in income and risk sharing On the other hand as the possibility of such risksharing arrangements reduces consumption risk individuals might take riskier actions also potentially affecting the distribution of income To provide a brief introduction to these issues I now present a simple model of financial development focusing on the diversification of idiosyncratic risks and complementing the analysis in Section 176 The model is inspired by the work of Townsend 1979 and Greenwood and Jovanovic 1990 It illustrates how financial development takes place endogenously and interacts with economic growth it also provides some simple insights into the implications of financial development for income distribution Given the similarity of the model to that in Section 176 my treatment here is relatively informal I consider an OLG economy in which each individual lives for two periods and has prefer ences given by EtUtct ct 1 log ct βEt log ct 1 211 where ct denotes the consumption of the unique final good of the economy and Et denotes the expectation operator given time t information There is no population growth and the total population of each generation is normalized to 1 Let us assume that each individual is born with some labor endowment l The distribution of endowments across agents is given by the distribution function Gl over some support l l This distribution of labor endowments is constant over time with mean L 1 and labor is supplied inelastically by all individuals in the first period of their lives In the second period of their lives individuals simply consume their capital income The aggregate production function of the economy is given by Yt KtαLt1α Ktα where α 0 1 and the second equality uses the fact that total labor supply equals to 1 at each date As in Section 176 the only risk is in transforming savings into capital thus the life cycle of an individual looks identical to that shown in Figure 173 Moreover suppose that agents can either save all of their labor earnings from the first period of their lives using a safe technology with rate of return q in terms of capital at the next date or invest all of their 212 Fertility Mortality and the Demographic Transition 729 start using their savings more efficiently Thus with a mechanism similar to that in Section 176 economic development improves the allocation of funds in the economy and increases productivity Consequently this model like the one in Section 176 implies that economic development and financial development go handinhand 2 However there is also a distinct sense in which the economy here allows for a potential causal effect of financial development on economic growth Imagine that societies differ according to their values of ξ which can be interpreted as a measure of the institutionally or technologically determined costs of monitoring or some other costs associated with financial transactions that may depend on the degree of investor protection Societies with lower ξ values have greater participation in financial markets and this endogenously increases their productivity Thus while the equilibrium behavior of financial and economic development are jointly determined differences in financial development driven by exogenous institutional factors related to ξ have a potential causal effect on economic growth 3 As noted above at any given point in time it is the richer agentsthose with greater labor endowmentthat join the financial market Therefore initially the financial market helps those who are already well off to increase the rate of return on their savings This can be thought of as the unequalizing effect of the financial market 4 The fact that participation in financial markets increases with Kt also implies that as the economy grows at least at the early stages of economic development the unequalizing effect of financial intermediation become stronger Therefore presuming that the economy starts with relatively few rich individuals the first expansion of the financial market increases the level of overall inequality in the economy as a greater fraction of the agents in the economy now enjoy the greater returns 5 As Kt increases even further eventually the equalizing effect of the financial market starts operating At this point the fraction of the population joining the financial mar ket and enjoying the greater returns is steadily increasing If the steadystate level of capital stock K is such that l W 1 αKα then eventually all individuals join the financial market and receive the same rate of return on their savings The last two observations are interesting in part because the relationship between growth and inequality is a topic of great interest to development economics one to which I return later in this chapter One of the most important ideas in this context is that of the Kuznets curve which claims that economic growth first increases and then reduces income inequality in the society Whether the Kuznets curve is a good description of the relationship between growth and inequality is a topic of current debate While many European societies seem to have gone through a phase of increasing and then decreasing inequality during the nineteenth century the evidence for the twentieth century is more mixed The last two observations show that a model with endogenous financial development based on risk sharing among individuals can generate a pattern consistent with the Kuznets curve Whether there is indeed a Kuznets curve in general and if so whether the mechanism highlighted here plays an important role in generating this pattern are questions for future theoretical and empirical work 212 Fertility Mortality and the Demographic Transition Chapter 1 highlighted the major questions related to growth of income per capita over time and its dispersion across countries today Our focus so far has been on these per capita income differences Equally striking differences exist in the level of population across countries and 730 Chapter 21 Structural Transformations and Market Failures in Development Asia Western offshoots Europe Africa Latin Americ a 10 50 500 2000 5000 Population millions 0 500 1000 1500 2000 FIGURE 211 Total population in different parts of the world over the past 2000 years over time Figure 211 uses data from Maddison 2002 and shows the levels and the evolution of population in different parts of the world over the past 2000 years The figure is in log scale so a linear curve indicates a constant rate of population growth The figure shows that starting about 250 years ago there is a significant increase in the population growth rate in many areas of the world This more rapid population growth continues in much of the world but importantly the rate of population growth slows down in Western Europe some time in the nineteenth century though thanks partly to immigration not so in the Western offshoots There is no similar slowdown of population growth in lessdeveloped parts of the world On the contrary in many lessdeveloped nations the rate of population growth seems to have increased over the past 50 years or so We have already discussed one of the reasons for this in Chapter 4the spread of antibiotics basic sanitation and other healthcare measures around the world that have reduced the high mortality rates in many countries However equally notable is the demographic transition which in the course of the nineteenth century reduced fertility in Western Europe Why population has grown slowly and then accelerated to reach a breakneck speed of growth over the past 150 years and why population growth rates differ across countries are major questions for economic development In this section I present the most basic approaches to population dynamics and fertility I first discuss a simple version of the famous Malthusian model and then use a variant of this model to investigate potential causes of the demographic transition Thomas Malthus was one of the most brilliant and influential economists of the nineteenth century and is responsible for one of the first general equilibrium growth models Section 2121 presents a version of this model The Malthusian model is responsible for earning the discipline of economics the name the dismal science because of its dire prediction that population will adjust up or down by births or deaths until all individuals are at the subsistence level of consumption Nevertheless this dire prediction is not the most important part of the Malthusian model At 212 Fertility Mortality and the Demographic Transition 731 the heart of this model is the negative relationship between income per capita and population which is itself endogenously determined In this sense it is closely related to the Solow and the neoclassical growth models augmented with a behavioral rule that determines the rate of population growth It is this lessextreme version of the Malthusian model that is presented next I then enrich this model by the important and influential idea due to Gary Becker that there is a tradeoff between the quantity and quality of children and that this tradeoff changes over the process of development I show how a simple model can incorporate the notion that over the course of development markets and parents may start valuing the quality human capital of their offspring more and how this shift in valuation may lead to a pattern reminiscent of the demographic transition 2121 A Simple Malthusian Model Consider the following nonOLG model that starts with a population of L0 0 at time t 0 A representative individual living at time t supplies one unit of labor inelastically and has utility ctβyt 1nt 1 1 2η0nt 12 216 where ct denotes the consumption of the unique final good of the economy by the individual himself nt 1 denotes the number of offspring the individual begets yt 1 is the income of each offspring and β 0 and η0 0 The last term in the square brackets represents the cost of child rearing and is assumed to be convex to reflect the fact that the costs of having more and more children will be higher eg because of time constraints of parents though one can also make arguments for why child rearing might exhibit increasing returns to scale over a certain range Clearly these preferences introduce a number of simplifying assumptions First each individual is allowed to have as many offspring as he likes which is unrealistic because it does not restrict the number of offspring to a natural number The technology also does not incorporate possible specialization in child rearing and market work within the family Second these preferences introduce the warm glow type of altruism we encountered in Chapter 9 so that parents receive utility not from the future utility of their offspring but from some characteristic of their offspring Here it is a transform of the total income of all the offspring that features in the utility function of the parent Third the costs of child rearing are in terms of utils rather than forgone income and current consumption multiplies both the benefits and the costs of having additional children This feature which is motivated by a balanced growth type of reasoning implies that the demand for children is independent of current income otherwise growth would automatically lead to greater demand for children All three of these assumptions are adopted for simplicity I have also written the number of offspring that an individual has at time t as nt 1 since this determines population at time t 1 Each individual has one unit of labor and there are no savings The production function for the unique good takes the form Yt ZαLt1α 217 where Z is the total amount of land available for production and Lt is total labor supply There is no capital and land is introduced to create diminishing returns to labor which is an important element of the Malthusian model Without loss of generality I normalize the total amount of land to Z 1 A key question in models of this sort is what happens to the returns to land The most satisfactory way of dealing with this problem would be to allocate the property rights to land among the individuals and let them bequeath land to their offspring This however introduces another layer of complication and since my purpose here is to illustrate 732 Chapter 21 Structural Transformations and Market Failures in Development the basic ideas I follow the unsatisfactory assumption often made in the literature that land is owned by another set of agents whose behavior is not analyzed here By definition population at time t 1 is given by Lt 1 nt 1Lt 218 which takes into account new births as well as the death of the parent Labor markets are competitive so the wage at time t 1 is given by wt 1 1 αLt 1α 219 Since there is no other source of income 219 is also equal to the income of each individual living at time t 1 yt 1 Thus an individual with income wt at time t solves the problem of maximizing 216 subject to the constraint that ct wt together with the equation yt 1 1 αLt 1α Naturally in equilibrium nt 1 must be consistent with Lt 1 according to 218 Individual maximization implies that nt 1 1 αη1 0 Lt 1α Now substituting for 218 and rearranging we obtain Lt 1 1 α 1 1α η 1 0 1α Lt 1 1α 2110 This equation implies that Lt 1 is an increasing concave function of Lt In fact the law of motion for population implied by 2110 resembles the dynamics of capitallabor ratio in the Solow growth model or the OLG model and is plotted in Figure 212 The figure makes it clear that starting with any L0 0 there exists a unique globally stable state L given by L 1 α1αη1α 0 2111 If the economy starts with L0 L then population slowly and monotonically adjusts toward this steadystate level Moreover 219 shows that as population increases wages fall If in contrast L0 L then the society experiences a decline in population and rising real wages It is straightforward to introduce shocks to population and show that in this case the economy fluctuates around the steadystate population level L with an invariant distribution depending on the distribution of the shocks and experience cycles reminiscent of the Malthusian cycles with periods of increasing population and decreasing wages followed by periods of decreasing population and increasing wages see Exercise 213 The main difference between this model and the simplest or crudest version of the Malthusian model is that there is no biologically determined subsistence level of consumption The steadystate level of consumption instead reflects technology and preferences and is given by c 1 αLα η0 2122 The Demographic Transition To study the demographic transition I now introduce a qualityquantity tradeoff along the lines of the ideas suggested by Becker Each parent can choose his offspring to be unskilled or skilled To make them skilled the parent has to exert the additional effort for child rearing denoted by et 0 1 If he chooses not to do this his offspring will be unskilled 734 Chapter 21 Structural Transformations and Market Failures in Development that η1 is sufficiently greater than η0 and in particular that X0η1 η0 so that even at the initial level of the modern technology rearing a skilled child is more costly than an unskilled child Finally I assume learningbydoing is external as in Romer 1986a so that Xt 1 Xt κSt 2114 which implies that the improvement in the technology of the modern sector is a function of the number of skilled workers employed in this sector This type of reducedform assumption is clearly unsatisfactory but as noted in Chapter 20 in particular recall Exercise 2019 one could obtain similar results with an endogenous technology model featuring the market size effect Another important feature of this production function is that it does not use land This assumption is consistent with the fact that most modern production processes make little use of land instead relying on technology physical capital and human capital The output of the traditional and the modern sectors are perfect substitutesthey both produce the same final good In view of the observation that all unskilled workers work in the traditional sector and all skilled workers work in the modern sector wages of skilled and unskilled workers at time t are wUt 1 αUtα and 2115 wSt Xt 2116 respectively where 2115 is identical to 219 except that it features only the unskilled workers instead of the entire labor force Let us next turn to the fertility and qualityquantity decisions of individuals As before current income has no effect on fertility and qualityquantity decisions Thus we do not need to distinguish between highskill and lowskill parents Using this observation let us simply look at the optimal number of offspring that an individual will have when he chooses et 0 This number is given by nUt 1 wUt 1η1 0 1 αη1 0 Ut 1α 2117 where the second equality uses 2115 If the parent instead decides to exert effort et 1 and invest in the skills of his offspring then he will choose a number of offspring equal to nSt 1 η1 1 wSt 1Xt 11 η1 1 2118 The comparison of 2117 and 2118 suggests that unless unskilled wages are very low an individual who decides to provide additional skills to his offspring will have fewer offspring This is because bringing up skilled children is more expensive ie because η1 is sufficiently larger than η0 Thus the comparison of these two equations captures the qualityquantity tradeoff Substituting these equations back into the utility function 2113 we obtain the utility from the two strategies normalized by consumption ie the utility divided by ctβ as V Ut 1 21 α2η1 0 Ut 12α and V St 1 2η1 1 Xt 1 Inspection of these two expressions shows that in equilibrium some workers must be unskilled since otherwise V U would become infinite Therefore in equilibrium we have V Ut V St for all t 2119 212 Fertility Mortality and the Demographic Transition 735 This equilibrium condition implies that there are two possible configurations First X0 can be so low that 2119 holds as a strict inequality at all t and all individuals remain unskilled at all dates The condition for inequality 2119 to be strict at time t 0 is η1 1 X0 1 α2η1 0 L12α which uses the fact that when there are no skilled workers there is no production in the modern sector and thus X1 X0 If this inequality were satisfied there would be no skilled children at date t 0 However as long as L1 is less than L as given in 2111 the population grows It is therefore possible that at some point 2119 holds with equality The condition ensuring that equality never happens is that η1 1 X0 1 α2η1 0 L2α 2120 In this case 2119 would hold as strict inequality at all dates there would be no investment in skills and the law of motion of population would be identical to that in Section 2121 We can think of this case as a pure Malthusian economy If on the other hand 2120 is not satisfied then at least at some point individuals start investing in the skills of their offspring From then on 2119 must hold as equality Let the fraction of parents having unskilled children at time t be denoted by ut 1 Then by definition it follows that Ut 1 ut 1nUt 1Lt 1 α21αη11α 0 ut 111αLt11α 2121 and St 1 1 ut 1nSt 1Lt η1 1 1 ut 1Lt 2122 Moreover to satisfy 2119 as equality we need 1 α2η1 0 Ut 12α η1 1 Xt 1 Rearranging this expression yields Xt 1 1 α21αη1α1α 0 η1ut 12α1αLt2α1α 2123 Equilibrium dynamics are then determined by 21212123 together with 2116 While the details of the behavior of this dynamical system are somewhat involved the general picture is clear Most interestingly if an economy has both a low level of X0 and a low level of L0 but does not satisfy condition 2120 then it starts in the Malthusian regime only making use of the traditional technology and not investing in skills As population increases wages fall and at that point parents start finding it beneficial to invest in the skills of their children and firms start using the modern technology Parents who invest in the skills of their children will typically have fewer children than parents rearing unskilled offspring because η1 is sufficiently larger than η0 2117 is greater than 2118 The aggregate rate of population growth and fertility are still high at first but as the modern technology improves and the demand for skills increases a larger fraction of parents start investing in the skills of their children and the rate of population growth declines Ultimately the rate of population growth approaches 736 Chapter 21 Structural Transformations and Market Failures in Development η1 1 This model thus gives a stylized representation of the demographic transition based on the qualityquantity tradeoff There exist substantially richer models of the demographic transition in the literature For example there are many ways of introducing qualityquantity tradeoffs and what spurs a change in this tradeoff may be an increase in capital intensity of production changes in the wages of workers or changes in the wages of women differentially affecting the desirability of market and home activities Nevertheless the general qualitative features are similar to those in the model presented here and in most of these approaches the qualityquantity tradeoff is the major reason for the demographic transition Despite this emphasis on the qualityquantity tradeoff there is relatively little direct evidence that this tradeoff is important in general or that it leads to the demographic transition Other social scientists have suggested social norms the large declines in mortality starting in the nineteenth century and the reduced need for child labor as potential factors contributing to the demographic transition As of yet there is no general consensus on the causes of the demographic transition or on the role of the quality quantity tradeoff in determining population dynamics The study of population growth and demographic transition is an exciting and important area and theoretical and empirical analyses of the factors affecting fertility decisions and how they interact with the reallocation of workers across different tasks sectors remain important and interesting questions to be explored 213 Migration Urbanization and the Dual Economy Another major structural transformation that occurs during the process of development relates to changes in social and living arrangements For example as an economy develops more individuals move from rural areas to cities and also undergo the social changes associated with separation from a small community and becoming part of a larger more anonymous envi ronment Other social changes might also be important For instance certain social scientists regard the replacement of collective responsibility systems by individual responsibility sys tems as an important social transformation This replacement is clearly related to changes in the living arrangements of individuals eg villages versus cities or extended versus nuclear families It is also linked to whether different types of contracts are being enforced by social norms and community enforcement and whether they are enforced by legal institutions There may also be a similar shift in the importance of the market as more activities are mediated by prices rather than taking place inside the home or using the resources of an extended family or broader community This process of social change is both complex and interesting to study though a detailed discussion of the literature and possible approaches to these issues is beyond the scope of this book Nevertheless a brief discussion of some of these social changes is useful to illustrate other more diverse facets of structural transformations associated with economic development I illustrate the main ideas by focusing on the process of migration from rural areas and on ur banization Another reason to study migration and urbanization is that the reallocation of labor from rural to urban areas is closely related to the popular concept of the dual economy which is an important theme of some of the older literature on development economics According to this notion lessdeveloped economies consist of a modern sector and a traditional sector but the connection between these two sectors is imperfect The model of industrialization in the previous chapter Section 203 featured a traditional and a modern sector but these sectors traded their outputs and competed for labor in competitive markets Dual economy approaches instead emphasize situations in which the traditional and the modern sectors function in par allel but with only limited interactions Moreover the traditional sector is often viewed as less 213 Migration Urbanization and the Dual Economy 737 efficient than the modern sector thus the lack of interaction may also be a way of shielding the traditional economy from its more efficient competitor A natural implication of this approach is then to view the process of development as one in which the lessefficient traditional sector is replaced by the moreefficient modern sector Lack of development may in turn correspond to the societys inability to generate such reallocation I first present a model of migration that builds on the work by Lewis 1954 A lessdeveloped economy is modeled as a dual economy with the traditional sector associated with villages and the modern sector with the cities I then present a model that builds on Banerjee and Newman 1998 and Acemoglu and Zilibotti 1999 in which the traditional sector and the rural economy have a comparative advantage in community enforcement even thoughin line with the other dual economy approachesthe modern economy the city enables the use of more efficient technologies This model also illustrates how certain aspects of the traditional sector can shield the lessproductive firms from moreproductive competitors and slow down the process of development Finally I show how the import of technologies from moredeveloped economies along the lines of the models discussed in Section 184 of Chapter 18 may also lead to dual economy features as a byproduct of the introduction of more skillintensive modern technologies into lessdeveloped economies 2131 Surplus Labor and the Dual Economy Lewis argued that lessdeveloped economies typically have surplus laborthat is unemployed or underemployed labor often in the villages The dual economy can then be viewed as the juxtaposition of the modern sector where workers are productively employed with the traditional sector where they are underemployed The general tendency of lessdeveloped economies to have lower levels of employmentpopulation ratios was one of the motivations for Lewiss model A key feature of the model is the presence of some barriers preventing or slowing down the allocation of workers away from the traditional sector toward urban areas and the modern sector I now present a reducedform model that formalizes these notions Consider a continuoustime infinitehorizon economy that consists of two sectors or re gions which I refer to as urban and rural Total population is normalized to 1 At time t 0 LU0 individuals are in the urban area and LR0 1 LU0 are in the rural area In the rural area the only economic activity is agriculture and for simplicity suppose that the production function for agriculture is linear Thus total agricultural output is Y At BALRt where BA 0 In the urban area the main economic activity is manufacturing Manufacturing can only employ workers in the urban area and employs all available workers The production function therefore takes the form Y Mt FKt LUt where Kt is the capital stock with initial condition K0 The function F is a standard neoclassical production function satisfying Assumptions 1 and 2 Chapter 2 Let us also assume for simplicity that the manufacturing and agricultural goods are perfect substitutes Labor markets in both the rural and urban area are competitive There is no technological change in either sector The key assumption is that because of barriers to mobility there is only a slow migration of workers from rural to urban areas even when manufacturing wages are greater than rural 738 Chapter 21 Structural Transformations and Market Failures in Development wages In particular let us capture the dynamics in this model in a reducedform way assuming that capital accumulates only out of the savings of individuals in the urban area and thus Kt sFKt LUt δKt 2124 where s is the exogenous saving rate and δ is the depreciation rate of capital The important feature implied by 2124 is that greater output in the modern sector leads to further accumu lation of capital for the modern sector An alternative adopted in Section 203 of the previous chapter and also used in Section 2132 is to allow the size of the modern sector to directly influence its productivity growth eg because of learningbydoing externalities as in Romer 1986a or because of endogenous technological change depending on the market size com manded by this sector see Exercise 2019 For the purposes of the model here which of these alternatives is adopted has no major consequences Given competitive labor markets the wage rates in the urban and rural areas are wUt FKt LUt L and wRt BA Let us assume that FK0 1 L BA 2125 so that even if all workers are employed in the manufacturing sector at the initial capital stock they will have a higher marginal product than working in agriculture Migration dynamics are assumed to take the following simple form LRt μLRt if wUt wRt μLRt 0 if wUt wRt 0 if wUt wRt 2126 Equation 2126 implies that as long as wages in the urban sector are greater than those in the rural sector there is a constant rate of migration The speed of migration does not depend on the wage gap which is an assumption adopted only to simplify the exposition We may want to think of μ as small so that there are barriers to migration and so even when there are substantial gains to migrating to the cities migration takes place slowly When there is no wage gain to migrating there will be no migration Now 2125 implies that at date t 0 there is migration from the rural areas to the cities Moreover assuming that K0LU0 is below the steadystate capitallabor ratio the wage remains high and continues to attract further workers To analyze this process in slightly greater detail let us define k0 K0 LU0 as the capitallabor ratio in manufacturing the modern sector As usual let us also define the per capita production function in manufacturing as f kt Clearly wUt f kt ktf kt Combining 2124 and 2126 we find that as long as f kt ktf kt BA the dynamics of this capitallabor ratio is given by kt sf kt δ μνtkt 2127 213 Migration Urbanization and the Dual Economy 739 where νt LRtLUt is the ratio of the rural to urban population Notice that when urban wages are greater than rural wages the rate of migration μ times the ratio νt plays the role of the rate of population growth in the basic Solow model In contrast when f kt ktf kt BA there is no migration and we have kt sf kt δkt 2128 Let us focus on the former case Let k be the level of capitallabor ratio such that urban and rural wages are equalized given by f k kf k BA 2129 Once this level is reached migration stops and νt remains constant After this level equilib rium dynamics are given by 2128 Therefore the steady state must involve sf ˆk ˆk δ 2130 For the analysis of transitional dynamics which are our primary interest here there are several cases to study Let us focus on the one that appears most relevant for the experiences of many lessdeveloped economies leaving the rest to Exercise 214 In particular suppose that the following conditions hold 1 k0 ˆk so that the economy starts with a lower capitallabor ratio in the urban sector than in the steadystate level This assumption also implies that sf k0 δk0 0 2 k0 k which implies that f k0 k0f k0 BA that is wages are initially higher in the urban sector than in the rural sector 3 sf k0 δ μν0k0 0 so that given the distribution of population between urban and rural areas the initial migration leads to a decline in the capitallabor ratio In this case the economy starts with rural to urban migration at date t 0 Since initially ν0 is high this migration reduces the capitallabor ratio in the urban area which evolves according to the differential equation 2127 There are then two possibilities In the first the capitallabor ratio never falls below k thus rural to urban migration takes place at the maximum possible rate of μ forever Nevertheless the effect of this migration on the urban capitallabor ratio is reduced over time as νt declines with migration Since we know that sf k0 δk0 0 at some point the urban capitallabor ratio will start increasing and it will eventually converge to the unique steadystate level ˆk This convergence can take a long time however and notably it is not necessarily monotone the capitallabor ratio and urban wages first fall and then increase The second possibility is that the initial surge in rural to urban migration reduces the capitallabor ratio to k at some point say at date t When this happens wages remain constant at BA in both sectors and the rate of migration LRtLRt adjusts exactly so that the capitallabor ratio remains at k for a while recall that when urban and rural wages are equal 2126 admits any level of migration between zero and the maximum rate μ In fact the urban capitallabor ratio can remain at this level for an extended period of time During this time wages in both sectors remain stagnant Ultimately however νt will again decline sufficiently that the capitallabor ratio in the urban sector must start increasing Once this happens urban wages also start increasing migration takes place at the maximal rate μ and the economy again slowly converges to the capitallabor ratio ˆk in the urban sector 740 Chapter 21 Structural Transformations and Market Failures in Development Therefore this discussion illustrates how a simple model of migration can generate rich population dynamics in rural and urban areas and also dynamics of wage difference between the modern and the traditional sectors The dynamics discussed above especially in the first case give the flavor of a dual economy Wages and the marginal product of labor are higher in the urban area than in the rural one If in addition μ is low the allocation of workers from the rural to the urban area is slow despite the higher wages Thus the pattern of dual economy may be pronounced and may persist for a long time It is also notable that rural to urban migration increases total output in the economy because it enables workers to be allocated to activities in which their marginal product is higher This process of migration increasing the output level in the economy also happens slowly because of the relatively slow process of migration The above discussion implies that for the parameter configurations on which we have focused the dual economy structure not only affects the social outlook of the society which remains rural and agricultural for an extended period of time especially when μ is small but also leads to lower output than the economy could have generated by allocating labor more rapidly to the manufacturing sector One should be cautious in referring to this as a market failure however since we did not specify the reason why migration is slow 2132 Community Enforcement Migration and Development I now present a model that builds on Banerjee and Newman 1998 and Acemoglu and Zilibotti 1999 Banerjee and Newman consider an economy in which the traditional sector has low productivity but is less affected by informational asymmetries Thus individuals can engage in borrowing and lending with limited monitoring and incentive costs In contrast the modern sector is more productive but informational asymmetries create more severe credit market problems Banerjee and Newman discuss how the process of development is associated with the reallocation of economic activity from the traditional to the modern sector and how this reallocation is slowed down by the informational advantage of the traditional sector Acemoglu and Zilibotti 1999 view the development process as one of information accumulation and argue that greater information enables individuals to write more sophisticated contracts and enter into more complex production relations This process is then associated with changes in technology changes in financial relations and social transformations since greater availability of information and better contracts enable individuals to abandon lessefficient and less informationdependent social and productive relationships The model in this subsection is simpler than those of both of these papers but features a similar economic mechanism Individuals who live in rural areas are subject to community enforcement Thus they can enter into economic and social relationships without being unduly affected by moral hazard problems When individuals move to cities they can take part in more productive activities but other enforcement systems are necessary to ensure compliance to social rules contracts and norms These systems are typically associated with certain costs As in the model of industrialization in Section 203 in the previous chapter I also assume that the modern sector is subject to learningbydoing externalities Thus the productivity advantage of the modern sector grows as more individuals migrate to cities and work there However the community enforcement advantage of villages slows down this process Both labor markets are competitive and total population is normalized to 1 There are three differences between this model and the one in Section 2131 First migration between the rural and urban areas is costless Thus at any point in time an individual can switch from one sector to another Second instead of capital accumulation there is an externality so that output in the modern sector is given by 742 Chapter 21 Structural Transformations and Market Failures in Development Time Fraction of population living in the city 1 FIGURE 213 Dynamic behavior of the population in rural and urban areas Several features of this law of motion are worth noting First the typical evolution of Xt is given as in Figure 213 with an Sshaped pattern This is because starting with a low initial value of X0 equilibrium urban employment φXtBA ξ is also low during the early stages of development Thus there is limited learningbydoing and the modern sector technology progresses only slowly However as Xt increases φXtBA ξ also increases raising the rate of technological change in the modern sector Ultimately however LUt cannot exceed 1 so φXtBA ξ tends to a constant and thus the rate of growth of X declines Therefore this reducedform model generates an Sshaped pattern of technological change in the modern sector and an associated pattern of migration of workers from rural to urban areas Second and more importantly the process of technological change in the modern sector and migration to the cities are slowed down by the comparative advantage of the rural areas in community enforcement In particular the greater is ξ the slower is technological change and migration into urban areas Since employment in the urban areas creates positive external ities the community enforcement system in rural areas slows down the process of economic development in the economy as a whole We may therefore conjecture that a higher ξ corre sponding to a greater community enforcement advantage of the traditional sector generally reduces growth and welfare in the economy Counteracting this effect however are the static gains created by the better community enforcement system in rural areas A high level of ξ increases the initial level of consumption in the economy Consequently there is a tradeoff between the dynamic and static welfare implications of different levels of ξ This tradeoff is investigated formally in Exercise 215 It is worth noting that unlike the model in Section 2131 there are no barriers to migration here workers in the villages and cities receive the same wage However the functioning of the economy and the structure of social relations are different in these two areas While villages and the rural economy rely on community enforcement the city uses the modern technology and impersonal institutional checks to enforce various economic and social arrangements Consequently the dual economy in this model manifests itself as much in the social as in the economic dimension 746 Chapter 21 Structural Transformations and Market Failures in Development Integrate 2138 over ν 0 1 use the fact that εν t has mean zero divide both sides by At and use 2136 to obtain a simple linear relationship between a countrys distance to frontier at at date t and its distance to the frontier at 1 at date t 1 at 1 1 g η γ at 1 2139 This equation is similar to the technological catchup equation 184 in Section 182 It shows how the dual process of imitation and innovation may lead to a process of convergence In particular as long as γ 1 g 2139 implies that at eventually converges to 1 This equation also shows that the relative importance of imitation and innovation depends on the distance to the frontier of the economy in question In particular when at is large meaning the country is close to the frontier innovation γ matters more for growth In contrast when at is small meaning the country is farther from the frontier imitation η is relatively more important To obtain further insights let us now endogenize η and γ using a reducedform approach Following the analysis in Acemoglu Aghion and Zilibotti 2006 I model the parameters η and γ as functions of the investments undertaken by the entrepreneurs and the contractual arrangement between firms and entrepreneurs The key idea is that there are two types of entrepreneurs highskill and lowskill When an entrepreneur starts a business his skill level is unknown and is revealed over time through his subsequent performance Thus two types of growth strategies are possible The first one emphasizes selection of highskill entrepreneurs and replaces any entrepreneur who is revealed to have low skill This growth strategy involves a high degree of churning creative destruction and a large number of young entrepreneurs as older unsuccessful entrepreneurs are replaced by new young entrepreneurs The second strategy maintains experienced entrepreneurs in place even when they have low skills This strategy therefore involves an organization of firms relying on longerterm relationships here between entrepreneurs and the credit market an emphasis on experience and cumulative earnings and less creative destruction While lowskill entrepreneurs are less productive than highskill ones there are potential reasons for preferring an experienced lowskill entrepreneur to a new young entrepreneur For example experience may increase productivity at least in certain tasks Alternatively Acemoglu Aghion and Zilibotti 2006 show that in the presence of credit market imperfections the retained earnings of an old entrepreneur may provide him with an advantage in the credit market because he can leverage his existing earnings to raise more money and undertake greater productivityenhancing investments I denote the strategy based on selection by R 0 while the strategy that maintains experienced entrepreneurs in place is denoted by R 1 The key reducedform assumption here is that experienced entrepreneurs either because of the value of experience or because of their retained earnings are better at increasing the productivity of their company when this involves the imitation of technologies from the world frontier which can be thought to correspond to relatively routine tasks Highskill entrepreneurs on the other hand are more innovative and generate higher growth through innovation Thus the tradeoff between R 1 and R 0 and the associated tradeoff between organizational forms boils down to the tradeoff between imitation of technologies from the world technology frontier and innovation For this reason I refer to the first strategy as an imitationbased growth strategy and to the second as an innovationbased growth strategy Motivated by these considerations let us assume that the equation for the law of motion of the distance to frontier 2139 takes the form 214 Distance to the Frontier and Changes in the Organization of Production 747 at 1 1gη γ at 1 if Rt 1 1 1gη γ at 1 if Rt 0 2140 Let us also impose the following conditions η η and γ γ 1 g 2141 The first part of this assumption follows immediately from the notion that highskill entre preneurs are better at innovation while the second part in particular that γ γ builds in the feature that experienced entrepreneurs are better at imitation When the imitationbased growth strategy is pursued experienced entrepreneurs are not replaced and consequently there is greater transfer of technology from the world technology frontier The final part of this assumption γ 1 g simply ensures that imitationbased growth does not lead to faster growth than the world technology frontier We can thus interpret assumption 2137 as stating that the world technology frontier advances due to innovationbased growth strategy which is natural since a country at the world technology frontier cannot imitate others Figure 214 plots 2140 and shows that the economy with longterm contracts R 1 achieves greater growth higher level of at for given at 1 through the imitation channel but lower growth through the innovation channel The figure also shows that which regime maximizes the growth rate of the economy depends on the level of at 1 that is on the distance of the economy to the world technology frontier In particular inspection of 2140 is sufficient to establish that there exists a threshold ˆa η η γ γ 0 1 2142 such that when at 1 ˆa the imitationbased strategy R 1 leads to greater growth and when at 1 ˆa the innovationbased strategy R 0 achieves higher growth Thus for the economy to follow a growthmaximizing sequence of strategies it should start with R 1 and then switch to an innovationbased strategy R 0 once it is sufficiently close to the world technology frontier In the imitationbased regime incumbent entrepreneurs are sheltered from the competition of younger entrepreneurs and this may enable the economy to make better use of the experience of older entrepreneurs or to finance greater investments out of their retained earnings In contrast the innovationbased regime is based on an organizational form relying on greater selection of entrepreneurs and places greater emphasis on maximizing innovation at the expense of experience imitation and investment Figure 214 describes the law of motion of technology in an economy as a function of the organization of firms markets as captured by R It does not specify what the equilibrium sequence Rt t0 is To determine this sequence we need to specify the equilibrium behavior which involves the selection of entrepreneurs as well as the functioning of credit markets Space restrictions preclude me from providing a full analysis of the equilibrium in such a model Instead I informally discuss some of the main insights of such an analysis Conceptually one might want to distinguish among four configurations which arise as equilibria under different institutional settings and parameter values 1 Growthmaximizing equilibrium the first and the most obvious possibility is an equi librium that is growth maximizing In particular if markets and entrepreneurs have growth maximization as their objective and are able to solve the agency problems have 750 Chapter 21 Structural Transformations and Market Failures in Development government achieve this Subsidies to investment would be one possibility Acemoglu Aghion and Zilibotti 2006 show that the degree of competition in the product market also has an indirect effect on the equilibrium as emphasized by the notation arδ In particular a higher level of δ which corresponds to lower competition in the product market higher χ increases arδ and thus may close the gap between arδ and ˆa Nevertheless it has to be noted that reducing competition creates other static distortions because of higher markups Moreover and more importantly we will see in the next two configurations that reducing competition can have much more detrimental effects on economic growth so any use of competition policy for this purpose must be subject to serious caveats 3 Sclerotic equilibrium the third possibility is a sclerotic equilibrium in which arδ ˆa so that lowproductivity incumbents survive even when they are potentially damaging to economic growth Acemoglu Aghion and Zilibotti 2006 show that this configuration can also arise in equilibrium because the retained earnings of incumbent entrepreneurs act as a shield protecting them against the forces of creative destruction brought about by new entrepreneurs Consequently the retained earnings or other advantages of experienced entrepreneurs both have social benefits and costs and which of these dominates depends on parameter values When the benefits dominate the equilibrium may feature too rapid a switch to the innovationbased strategy and when the costs dominate the economy may experience sclerosis in the imitation regime with excessive protection of incumbents The resulting pattern in this case is drawn in Figure 216 Now the economy fails to achieve the maximum growth rate for a range of values of a such that a ˆa arδ In this range the innovationbased regime would be growth maximizing but the economy is stuck with the imitationbased regime because the retained earnings and the power of the incumbents prevent the transition to the more efficient organizational forms An interesting feature is that as Figure 216 shows this economy also follows a pattern in line with Kuznetss vision it starts with a distinct set of organizations represented by R 1 and then switches to a different set of arrangements R 0 Like the previous two types of equilibria this case also features convergence to the world technology frontier that is to a 1 4 Nonconvergence trap equilibrium the fourth possibility is related to the third one and also involves arδ ˆa However now the gap between arδ and ˆa is larger as depicted in Figure 217 and includes the level of a atrap such that atrap η 1 g γ Inspection of 2140 immediately reveals that if at 1 atrap and Rt 1 the economy remains at atrap Therefore in this case the retained earnings or the experience of incumbent firms afford them so much protection that the economy never transitions to the innovationbased equilibrium This scenario not only retards growth for a temporary interval but also pushes the economy into a nonconvergence trap In particular this is the only equilibrium pattern in which the economy fails to converge to the frontier in the imitationbased regime R 1 the economy does not grow beyond atrap and at this distance from the frontier the equilibrium always involves R 1 This equilibrium therefore illustrates the most dangerous scenariothat of non convergence Encouraging imitationbased growth for example by supporting incum 758 Chapter 21 Structural Transformations and Market Failures in Development other potential forces leading to multiple equilibria are more important as sources of persis tence or as mechanisms generating multiple steady states while still maintaining a unique equilibrium path 216 Inequality Credit Market Imperfections and Human Capital The previous section illustrated how aggregate demand externalities can generate development traps Investment by different firms may require coordination leading to multiple equilibria Underdevelopment may be thought to correspond to a situation in which the coordination is on the bad equilibrium and the development process starts with the big push ensuring coordination to the highinvestment equilibrium Here I illustrate a related set of issues in the context of the impact of the distribution of income on human capital under imperfect credit markets In contrast to the previous section I emphasize the possibility of multiple steady states rather than multiple equilibria In addition while I focus on human capital investments inequality and credit market problems influence not only human capital investments but also business creation occupational choices and other aspects of the organization of production Nevertheless models focusing on the link between inequality and human capital are more tractable and constitute a natural continuation of the theories of human capital investments presented in Chapter 10 2161 A Simple Case with No Borrowing When credit markets are imperfect a major determinant of human capital investments is the distribution of income as well as the degree of imperfection in credit markets I start with a discussion of the simplest case in which there is no borrowing or lending which introduces an extreme form of credit market problems I then enrich this model by introducing imperfect credit markets where the cost of borrowing is greater than the interest rate received by households engaged in saving The economy consists of a continuum 1 of dynasties Each individual lives for two periods childhood and adulthood and begets an offspring in his adulthood There is consumption only at the end of adulthood Preferences are given by 1 δ log cit δ log eit where c is consumption at the end of the individuals life and e is the educational spending on the offspring of this individual The budget constraint is cit eit wit where w denotes the wage income of the individual Notice that preferences here have the warm glow type of altruism encountered in Chapter 9 and in Section 212 In particular parents do not care about the utility of their offspring but simply about what they bequeath to them here education As usual this assumption significantly simplifies the analysis The labor market is competitive and the wage income of each individual is simply a linear function of his human capital hit wit Ahit 216 Inequality Credit Market Imperfections and Human Capital 761 Thus nothing determines which equilibrium the economy will be in At best we can appeal to expectations arguing that the better equilibrium will emerge when everybody expects the better equilibrium to emerge One can informally appeal to the role of history for example suggesting that if an economy has been in the low investment equilibrium for a while it is likely to stay there but this argument is misleading First of all the model is a static one thus a discussion of an economy that has been in the low equilibrium for a while is not quite meaningful Second even if the model were turned into a dynamic one by repeating it over time the history of being in one equilibrium for a number of periods has no effect on the existence of multiple equilibria in the next period In particular each static equilibrium would still remain an equilibrium in the dynamic environment and the economy could suddenly jump from one equilibrium to another Thus models with multiple equilibria have a degree of indeterminacy that is both theoretically awkward and empirically difficult to map to reality Models with multiple steady states avoid these thorny issues The equilibrium is unique but the initial conditions determine where the dynamical system will eventually end up Because the equilibrium is unique there is no issue of indeterminacy or expectations affecting the path of the economy But because multiple steady states are possible the model can be useful for thinking about potential development traps This model also shows the importance of the distribution of income in an economy with imperfect credit markets here with no credit markets In particular the distribution of income affects which individuals are unable to invest in human capital accumulation and thus influences the longrun income level of the economy For this reason models of this sort are sometimes interpreted as implying that an unequal distribution of income leads to lower output and growth The above example with two classes seems to support this conclusion However it is not a general result and it is important to emphasize that this class of models does not make specific predictions about the relationship between inequality and growth To illustrate this consider again the same economy with two classes but now starting with h1 h2 δA1 In this case neither group accumulates human capital but redistributing resources away from group 1 to group 2 thus increasing inequality so that we push group 2 to h2 δA1 would increase human capital accumulation This feature is general in models with nonconvexities there are no unambiguous results about whether greater inequality is good or bad for economic growth it depends on whether greater inequality pushes more people below or above the critical thresholds Somewhat sharper results can be obtained about the effect of inequality on human capital accumulation and development under additional assumptions Exercise 2110 presents a parameterization of inequality in the model here that shows that greater inequality leads to lower human capital and lower output per capita in relatively rich economies but to greater investments in human capital in poorer economies 2162 Human Capital Investments with Imperfect Credit Markets I now enrich the environment in Section 2161 by introducing credit markets following Galor and Zeiras 1993 model Each individual still lives for two periods In his youth he can either work or acquire education The utility function of each individual is 1 δ log cit δ log bit where again c denotes consumption at the end of the life of the individual The budget constraint is cit bit yit 762 Chapter 21 Structural Transformations and Market Failures in Development where yit is individual is income at time t Note that preferences still take the warm glow form but the utility of the parent now depends on the monetary bequest to the offspring bit rather than on the level of education expenditures It is now the individuals themselves who use the monetary bequests to invest in education The logarithmic formulation once again ensures a constant saving rate equal to δ Education is a binary outcome and educated skilled workers earn wage ws while unedu cated workers earn wu The required education expenditure to become skilled is h and workers acquiring education do not earn the unskilled wage wu during the first period of their lives The fact that education is a binary decision introduces the aforementioned nonconvexity in human capital investment decisions3 Imperfect capital markets are modeled by assuming that there is some monitoring required for loans to be paid back The cost of monitoring creates a wedge between the borrowing and lending rates In particular assume that there is a linear savings technology open to all agents which fixes the lending rate at some constant r However the borrowing rate is i r because of costs of monitoring necessary to induce agents to pay back the loans see Exercise 2112 for a more microfounded version of these borrowing costs Also assume that ws 1 rh wu2 r 2155 which implies that investment in human capital is profitable when financed at the lending rate r Consider an individual with wealth x If x h 2155 implies that the individual invests in education If x h then whether it is profitable to invest in education depends on the wealth of the individual and on the borrowing interest rate i Let us now write the utility of this individual with x h in the two scenarios and also the bequest that he will leave to his offspring These are Usx logws 1 ix h log1 δ1δδδ bsx δws 1 ix h when he invests in education When he chooses not to invest then the equations become Uux log1 rwu x wu log1 δ1δδδ bux δ1 rwu x wu Comparing these expressions it is clear that an individual prefers to invest in education if and only if x f 2 rwu 1 ih ws i r The dynamics of individual wealth can then be obtained simply by using the bequests of unconstrained constrainedinvesting and constrainednoninvesting agents 3 An alternative to nonconvexities in human capital investments is presented in Galor and Moav 2004 who show that multiple steady states are possible when there are no nonconvexities credit markets are imperfect and the marginal propensity to save is higher for richer dynasties This assumption is motivated by Kaldors 1957 paper and was discussed in Exercise 212 in Chapter 2 764 Chapter 21 Structural Transformations and Market Failures in Development capital and low wealth Therefore this model extends the insights of the simple model with no borrowing from Section 2161 to a richer environment in which individuals make forward looking human capital investments The key is again the interaction between credit market imperfections which here make the interest rate for borrowing greater than the interest rate for saving and inequality As in the earlier model it is straightforward to construct examples where an increase in inequality can lead to either worse or better outcomes depending on whether the scenario pushes more individuals into the basin of attraction of the low steady state An important feature of the model here is that because it allows individuals to borrow and lend in financial markets it enables an investigation of the implications of financial development for human capital investments In an economy with better financial institutions the wedge between the borrowing rate and the lending rate is smaller that is i is smaller for a given level of r With a smaller i more agents escape the poverty trap and in fact the poverty trap may not exist at all there may not be an intersection between 2156 and the 45 line where 2156 is steeper Thus financial development not only improves risk sharing as demonstrated in Section 211 but by relaxing credit market constraints it also contributes to human capital accumulation Although the model in this section is considerably richer than that in Section 2161 it is still a partial equilibrium model Multiple steady states are possible for different individuals as a function of their initial level of human capital or wealth but individual dynamics are not affected by general equilibrium prices Galor and Zeira 1993 Banerjee and Newman 1993 Aghion and Bolton 1997 and Piketty 1997 consider richer environments in which income dynamics of each dynasty individual are affected by general equilibrium prices eg interest rate or wage rate which are themselves functions of the income inequality Exercise 2111 shows that the type of multiple steady states generated by the model presented here may not be robust to the addition of noise in income dynamicsinstead of multiple steady states the longrun equilibrium may generate a stationary distribution of human capital levels though this stationary distribution would exhibit considerable persistence4 In contrast models in which prices are determined in general equilibrium and affect wealth income dynamics can generate more robust multiplicity of steady states 217 Toward a Unified Theory of Development and Growth A unifying theme recurs in to the models discussed in this chapter They have either emphasized the transformation of the economy and the society during the process of development or potential reasons for the failure of such a transformation This transformation takes the form of the structure of production changing the process of industrialization getting underway a greater fraction of the population migrating from rural areas to cities financial markets becoming more developed mortality and fertility rates changing through health improvements and the demographic transition and the extent of inefficiencies and market failures becoming less pronounced over time In many instances the driving force for this process is reinforced by the structural transformation that it causes 4 Note that this is related to the Markovian nature of the model Markovian models can generate multiple steady states because the Markov chain or the Markov process implied by the model is not ergodic eg poor individuals can never accumulate enough to become rich A small amount of noise then ensures that different parts of the distribution communicate making the Markov process ergodic and thus removing the multiplicity of steady states 217 Toward a Unified Theory of Development and Growth 765 My purpose in this section is not to offer a unified model of structural transformations and market failures in development An attempt to pack many different aspects of development into a single model often leads to a framework that is complicated whereas I believe that relatively abstract representations of reality are more insightful Moreover the literature has not made sufficient progress for us to be able to develop a unified framework Instead I provide a reduced form model intended to bring out the salient common features of the models presented in this chapter In all of the models presented in this and the previous chapters economic development is associated with capital deepening that is with greater use of capital instead of human labor Thus we can also approximate the growth process with an increase in the capitallabor ratio of the economy kt This does not necessarily mean that capital accumulation is the engine of economic growth In fact previous chapters have emphasized how technological change is often at the root of the process of economic growth and economic development and thus capital deepening may be the result of technological change Moreover Section 214 shows how the crucial variable capturing the stage of development might be the distance of an economys technology from the world technology frontier Nevertheless even in these cases an increase in the capitallabor ratio takes place along the equilibrium path and can thus be used as a proxy for the stage of development though in this case one must be careful not to confuse increasing the capitallabor ratio with ensuring economic development With this caveat in mind in this section I take the capitallabor ratio as the proxy for the stage of development and for analytical convenience I use the Solow model to represent the dynamics of the capitallabor ratio In particular consider a continuoustime economy with output per capita given by yt f kt xt 2157 where kt is the capitallabor ratio and xt is some social variable such as financial development urbanization the structure of production or the structure of the family As usual f is assumed to be differentiable increasing and strictly concave in k The social variable x potentially affects the efficiency of the production process and thus is part of the per capita production function in 2157 As a convention suppose that an increase in x corresponds to structural change eg a move from the countryside to the cities Therefore f is also increasing in x and the partial derivative with respect to x is nonnegative that is fx 0 Naturally not all structural change is beneficial Nevertheless for simplicity I focus on the case in which f is increasing in x Suppose that structural change can be represented by the differential equation xt gkt xt 2158 where g is assumed to be twice differentiable Since x corresponds to structural change associated with development g should be increasing in k and in particular its partial derivative with respect to k is strictly positive that is gk 0 The standard meanreversion type of reasoning suggests that the case in which the derivative gx is negative is the most reasonable benchmark If x is above its natural level it should decline and if it is below its natural level it should increase Motivated by this reasoning let us also assume that gx 0 Capital accumulates according to the Solow growth model as in Chapter 2 so that kt sf kt xt δkt 2159 where I have suppressed population growth and there is no technological change for simplicity For a fixed x capital naturally accumulates in an identical fashion to that in the basic Solow model The structure of this economy is slightly more involved because xt also changes 219 References and Literature 769 orderly growth behavior captured by the neoclassical and endogenous technology models These models may also need to take market failures and how these market failures change over time more seriously This view stems from the recognition that the essence of economic development is the process of structural transformation including financial development the demographic transition migration urbanization organizational change and other social changes Another potentially important aspect of economic development is the possibility that the inefficiencies in the organization of production credit markets and product markets may culminate in development traps These inefficiencies may stem from lack of coordination in the presence of aggregate demand externalities or from the interaction between imperfect credit markets and human capital investments These topics not only highlight some of the questions that need to be addressed for understanding the process of economic development but also bring a range of issues that are often secondary in the standard growth literature to the forefront of analysis These include among other things the organization of financial markets the distribution of income and wealth and issues of incentives eg problems of moral hazard adverse selection and incomplete contracts in both credit markets and production relationships The recognition that the analysis of economic development necessitates a special focus on these topics also opens the way for a more constructive interaction between empirical devel opment studies and the theories of economic development surveyed in this chapter As already noted there is now a large literature on empirical development economics documenting the extent of credit market imperfections the impact of inequality on human capital investments and occupation choices the process of social change and various other market failures in less developed economies By and large this literature is about market failures in lessdeveloped economies and sometimes also focuses on how these market failures can be rectified The standard models of economic growth do not feature these market failures A fruitful area for future research is then the combination of theoretical models of economic growth and development that pay attention to market failures with the rich empirical evidence on the incidence characterization and costs of these market failures This combination has the advan tage of being theoretically rigorous and empirically grounded and perhaps most importantly it can focus on what I believe to be the essence of development economicsthe questions of why some countries are less developed how they can grow more rapidly and how they can jumpstart the process of structural transformation necessary for economic development 219 References and Literature By its nature this chapter has covered a large amount of material My selection of topics has reflected my own interests and was also motivated by a desire to keep this chapter from becoming even longer than it already is Section 211 scratches the surface of a rich literature on financial development and economic growth On the theoretical side Townsend 1979 Greenwood and Jovanovic 1990 and Bencivenga and Smith 1991 focus on the interaction between financial development on the one hand and risk sharing the allocation of funds across different tasks and individuals on the other Obstfeld 1994 and Acemoglu and Zilibotti 1997 focus on the relationship between financial development and the diversification of risks There is also a large empirical literature looking at the effect of financial development on economic growth An excellent survey of this literature is provided in Levine 2005 Some of the bestknown empirical papers include King and Levine 1993 which documents the crosscountry correlation between measures of financial development and economic growth Rajan and Zingales 1998 which shows that 770 Chapter 21 Structural Transformations and Market Failures in Development lack of financial development has particularly negative effects on sectors that have greater external borrowing needs and Jayaratne and Strahan 1996 which documents how banking deregulation that increased competition in US financial markets led to more rapid financial and economic growth in the United States In discussing financial development I also mentioned the literature on the Kuznets curve There is no consensus on whether there is a Kuznets curve Work that focuses on historical data such as Lindert and Williamson 1976 or Bourguignon and Morrison 2002 reports aggregate patterns consistent with a Kuznets curve while studies using panels of countries in the postwar era such as Fields 1980 do not find a consistent pattern resembling this curve The literature on fertility the demographic transition and growth is also vast The main trends in world population and crosscountry differences in population growth are summarized in LiviBacci 1997 and Maddison 2003 The idea that parents face a tradeoff between the numbers and the human capital of their childrenthe quality and quantity tradeoff was proposed by Becker 1981 The aggregate patterns in LiviBacci 1997 are consistent with this idea though there is little micro evidence supporting this tradeoff Recent work on micro data by Black Devereux and Salvanes 2005 Angrist Lavy and Schlosser 2006 and Qian 2007 looks at evidence from Norway Israel and China but does not find strong support for the qualityquantity tradeoff Fertility choices were first introduced into growth models by Becker and Barro 1988 and Barro and Becker 1989 Becker Murphy and Tamura 1990 provide the first endogenous growth model with fertility choice More recent work on the demographic transition and the transition from a Malthusian regime to one of sustained growth include Goodfriend and McDermott 1995 Galor and Weil 1996 2000 Hansen and Prescott 2002 and Doepke 2004 KalemliOzcan 2002 and FernandezVillaverde 2003 focus on the effect of declining mortality on fertility choices in a growth context A recent series of papers by Galor and Moav 2002 2004 combine fertility choice qualityquantity tradeoff and natural selection Galor 2005 provides an excellent overview of this literature The first model presented in Section 212 is a simplified version of Malthuss classic model in his 1798 book while the second model is a simplified version of Becker and Barro 1988 and Galor and Weil 2000 Urbanization is another major aspect of the process of economic development Bairoch 1988 provides an overview of the history of urbanization The first model in Section 213 builds on Arthur Lewiss 1954 classic which argued that early development can be viewed as a situation in which there is surplus labor available to the modern sector and thus growth is constrained by capital and technology but not by labor Harris and Todaros wellknown 1970 paper also emphasizes the importance of migration though their model features free migration between rural and urban areas and suggests that unemployment in urban areas is the key equilibriating variable The second model presented in Section 2132 is based on Banerjee and Newman 1998 and Acemoglu and Zilibotti 1999 Banerjee and Newman emphasize the advantage of smaller rural communities in reducing moral hazard problems in credit relations and show how this interacts with the process of urbanization which involves individuals migrating to areas where their marginal product is higher Acemoglu and Zilibotti argue that development leads to information accumulation In particular as more individuals perform similar tasks more so cially useful information is revealed which enables more complex contractual and production relations Section 2132 also touched on another important aspect of social and economic relations in lessdeveloped economies the importance of community enforcement Clifford Geertz 1963 emphasizes the importance of community enforcement mechanisms and how they may sometimes conflict with markets Section 214 builds on Acemoglu Aghion and Zilibotti 2006 2110 Exercises 771 Section 215 is based on Murphy Shleifer and Vishnys famous 1989 paper which for malized ideas first proposed by RosensteinRodan 1943 Other models that demonstrate the possibility of multiple equilibria in monopolistic competition models featuring nonconvexities include Kiyotaki 1988 who derives a similar result in a model with endogenous labor supply choices as well as investment decisions Matsuyama 1995 provides an excellent overview of these models and a clear discussion of why pecuniary externalities can lead to multiple equilibria in the presence of monopolistic competition The distinction between multiple equilibria and multiple steady states is discussed in Krugman 1991 and Matsuyama 1991 Both of these papers highlight the idea that in models with multiple equilibria expectations determine which equilibria emerge while with multiple steady states there can be or often is a unique equilibrium and initial conditions history determine where the economy will end up The model in Section 2162 is based on the first model in Galor and Zeira 1993 Similar ideas are investigated in Banerjee and Newman 1993 in the context of the effect of inequality on occupational choice and in Aghion and Bolton 1997 and Piketty 1997 in the context of the interaction between inequality and entrepreneurial investments Other work on the dy namics of inequality and its interactions with efficiency include Loury 1981 Tamura 1991 Benabou 1996 Durlauf 1996 Fernandez and Rogerson 1996 Glomm and Ravikumar 1992 and Acemoglu 1997b 2110 Exercises 211 Analyze the equilibrium of the economy in Section 211 relaxing the assumption that each individual has to invest either all or none of his wealth in the risky saving technology Does this generalization affect the qualitative results derived in the text 212 Consider the economy in Section 211 a Show that in 215 Kt 1 is everywhere increasing in Kt and that there exists some K such that the capital stock grows over time when Kt K b Can there be more than one steadystate level of capital stock in this economy If so provide an intuition for this type of multiplicity c Provide sufficient conditions for the steadystate level of capital stock K to be unique Show that in this case Kt 1 Kt when Kt K 213 In the model of Section 2121 suppose that the population growth equation takes the form Lt 1 εtnt 1 1Lt instead of 218 where εt is a random variable that takes one of two values 1 ε or 1 ε reflecting random factors affecting population growth Characterize the stochastic equilibrium In particular plot the stochastic correspondence representing the dynamic equilibrium behavior and analyze how shocks affect population growth and income dynamics 214 Characterize the full dynamics of migration urban capitallabor ratio and wages in the model of Section 2131 ie consider the cases in which conditions 1 2 and 3 in that section do not all hold simultaneously 215 Consider the model of Section 2132 and suppose that all individuals have time t 0 utility given by the standard CRRA preferences Taking the equilibrium path in that section as given find a level of community enforcement advantage ξ that would maximize time t 0 utility What happens if the actual comparative advantage of community enforcement of villages is greater than this level 216 Consider the maximization problem 2131 772 Chapter 21 Structural Transformations and Market Failures in Development a Explain why this maximization problem characterizes the equilibrium allocation of workers to tasks What kind of price system would support this allocation b Derive the firstorder conditions 2132 c Provide sufficient conditions such that the solution to this problem involves all skilled workers being employed at technology h d Provide an example in which no worker is employed at technology h even though Ah Ah for all h 0 h e Can there be a solution where more than two technologies are being used in equilibrium If so explain the conditions for such an equilibrium to arise 217 Consider a variant of the model in Section 214 in which firms have to make a decision on organizational form in particular they decide whether to vertically integrate For this purpose consider a slight modification of 2138 Aν t η At 1 γ ν tAt 1 with γ ν t γ θν t Suppose that entrepreneurial effort increases θν t and the internal organization of the firm affects how much effort the entrepreneur devotes to innovation activities In particular suppose that θν t 0 if there is vertical integration because the entrepreneur is overloaded and has limited time for innovation activities In contrast with outsourcing θν t θ 0 However when there is outsourcing the entrepreneur has to share a fraction β 0 of the profits with the manager owner of the firm to which certain tasks have been outsourced whereas in a vertically integrated structure he can keep the entire revenue a Determine the profitmaximizing outsourcing decision for an entrepreneur as a function of at In particular show that there exists a threshold a such that there is vertical integration for all at a and outsourcing for all at a b Contrast this equilibrium behavior with the growthmaximizing internal organization of the firm 218 Show that when multiple equilibria exist in the model of Section 215 the equilibrium with investment Pareto dominates the one without 219 Consider the model of Section 2161 and remove the nonconvexity in the accumulation equation 2152 so that the human capital of the offspring of individual i is given by hit 1 eitγ for any level of eit and γ 0 1 Show that there exists a unique level of human capital to which each dynasty converges Based on this result explain the role of nonconvexities in generating multiple steady states 2110 Consider the model of Section 2161 and suppose that the initial inequality is given by a uniform distribution with mean human capital of h0 and support over h0 λ h0 λ An increase in λ corresponds to greater inequality a Show that when h0 is sufficiently small an increase in λ increases longrun average human capital and income whereas when h0 is sufficiently large an increase in λ reduces them Hint use Figures 218 and 219 b What other types of distributions besides uniform would lead to the same result c Show that the same result generalizes to the model of Section 2162 d On the basis of this result discuss whether we should expect greater inequality to lead to higher income in poor societies and lower income in rich societies If your answer is no then sketch an environment in which this is not the case 2111 Consider the model presented in Section 2162 Make the following two modifications First the utility function is now 1 δ1δδδc1δbδ 2160 2110 Exercises 773 and second unskilled agents receive a wage of wu ε where ε is a random shock with mean zero a Suppose that ε is distributed with support λ λ Show that if λ is sufficiently close to zero then the multiple steady states characterized in Section 2162 survive in the sense that depending on their initial conditions some dynasties become highly skilled and others become low skilled b Why was it convenient to change the utility function from the log form used in the text to 2160 c Now suppose that ε is distributed with support λ where λ wu Show that in this case there is a unique ergodic distribution of wealth and no poverty trap Explain why the results here are different from those in part a d How would the results be different if in addition the skilled wage is equal to ws υ where υ is another random shock of mean zero Hint simply sketch the analysis and the structure of the equilibrium without repeating the full analysis of part c 2112 a In the model of Section 2162 suppose that each individual can run away without paying his debts and if he does so he is never caught However a bank can prevent this by paying a monitoring cost per unit of borrowing equal to m Suppose that there are many banks competing a la Bertrand for lending opportunities Under these assumptions show that all bank lending is accompanied with monitoring and the lending rate satisfies i r m Show that in this case all results in the text apply b Next suppose that the bank can prevent the individual from running away by paying a fixed monitoring cost of M Under the same assumptions as in part a show that in this case the interest rate charged to an individual who borrows an amount x h is i r Mx h Given this assumption characterize the equilibrium of the model in Section 2162 How do the conclusions change in this case c Next suppose that there is no way of preventing individuals from running away but if an individual runs he is caught with probability p and if caught a fraction λ 0 1 of his income is confiscated Given this assumption characterize the equilibrium dynamics of the model in Section 2162 How do the conclusions change d Now consider an increase in ws for a given level of wu so that the skill premium in the economy increases In which of the three scenarios outlined in parts ac does this have the largest effect on human capital investments 2113 In this exercise you are asked to study Banerjee and Newmans 1994 model of occupational choice The utility of each individual is again 1 δ1δδδc1δbδ z where z denotes whether the individual is exerting effort with cost of effort normalized to 1 Each agent chooses one of four possible occupations These are 1 subsistence and no work which leads to no labor income and has a rate of return on assets equal to ˆr 1δ 2 work for a wage v 3 selfemployment which requires investment I plus the labor of the individual and 4 entrepreneurship which requires investment μI plus the employment of μ workers and the individual becomes the boss monitoring the workers and does not take part in production All occupations other than subsistence involve effort Let us assume that both entrepreneurship and selfemployment generate a rate of return greater than subsistence ie the mean return for both activities is r ˆr a Derive the indirect utility function associated with the preferences above Show that no individual will work as a worker for a wage less than 1 b Assume that μIr ˆr 1 1 Ir ˆr 1 0 Interpret this assumption Hint it relates the profitabilities of entrepreneurship and selfemployment at the minimum possible wage of 1 774 Chapter 21 Structural Transformations and Market Failures in Development c Suppose that only agents who have wealth w w can borrow enough to become self employed and only agents who have wealth w w w can borrow μI to become an entrepreneur Provide an intuition for these borrowing constraints d Now compute the expected indirect utility from the four occupations Show that if v v μ 1r ˆrIμ then selfemployment is preferred to entrepreneurship e Suppose the wealth distribution at time t is given by Gtw On the basis of the results in part d show that the demand for labor in this economy is given by x 0 if v v x 0 μ1 Gtw if v v x μ1 Gtw if v v f Let v r ˆrI v Then show that the supply of labor is given by s 0 if v 1 s 0 Gtw if v 1 s Gtw if 1 v v s Gtw 1 if v v s 1 if v v g Show that if Gtw μ1 Gtw there is an excess supply of labor and the equilibrium wage rate is v 1 Show that if Gtw μ1 Gtw there is an excess demand for labor and the equilibrium wage rate is v v h Now derive the wealth bequest dynamics for a worker with wealth w as follows 1 sub sistence and no work bt δˆrw 2 worker bt δˆrw v 3 selfemployment bt δrI ˆrw I and 4 entrepreneurship bt δrμI ˆrw μI μv Explain the intuition for each of these expressions i Now using the wealth dynamics in part b show that multiple steady states with different wealth distributions and occupational choices are possible In particular show that the steady state wealth level of a worker when the wage rate is v is wwv δv1 δˆr while the steadystate wealth level of a selfemployed individual is wse δr ˆrI1 δˆr and the wealth level of an entrepreneur is wev δrμI ˆrμI μv1 δˆr Now show that when wwv 1 w and wev v w a steady state in which the equilibrium wage rate is equal to v 1 would involve workers not accumulating sufficient wealth to become selfemployed while entrepreneurs accumulate enough wealth to remain entrepreneurs Explain why Hint it depends on the equilibrium wage rate j Given the result in part i show that if we start with a wealth distribution such that μ1 Gw Gw the steady state involves an equilibrium wage v 1 and no self employment whereas for μ1 Gw Gw the equilibrium wage is v v and there is selfemployment Contrast the level of output in these two steady states k Is the comparison of the steady states in terms of output in this model plausible Is it consistent with historical evidence What are the pros and cons of this model relative to the GalorZeira model discussed in Section 2162 2114 This exercise asks you to analyze the dynamics of the reducedform model in Section 217 more formally than done in the text 2110 Exercises 775 1 Show that when fx 0 the locus for kk 0 implied by 2158 is an upwardsloping curve 2 Consider the differential equations 2158 and 2159 and a steady state k x By linearizing the two differential equations around k x show that if fxk x is sufficiently small the steady state is locally stable 3 Provide a uniform bound on fxk x so that there exists a unique steady state Show that when this bound applies the unique steady state is globally stable 4 Construct a parameterized example where there are multiple steady states Interpret the conditions necessary for this example Do you find them economically likely PART VIII THE POLITICAL ECONOMY OF GROWTH I n this part of the book I turn from the mechanics of economic growth to an investigation of potential causes of economic growth Almost all models studied so far take economic institutions eg property rights and types of written contracts policies eg tax rates dis tortions and subsidies and often the market structure as given They then derive implications for economic growth and crosscountry income differences While these models constitute the core of growth theory they leave unanswered some of the central questions raised in Chapters 1 and 4 why do some societies choose institutions and policies that discourage growth while others choose growthenhancing social arrangements In this part of the book I make a first attempt to provide some answers to these questions based on political economythat is on differences in institutions and policies arising from different ways of aggregating individual preferences across societies and on differences in the type and nature of social conflict In par ticular I emphasize a number of key themes and attempt to provide a tractable and informative formalization of these issues The main themes are as follows 1 Different institutions policies generate different economic allocations In the context of growth models this may correspond to distinct growth rates or steadystate levels of output These institutions also generate different winners and losers however Con sequently there will be social conflict concerning the types of policies and institutions that a society should adopt 2 Two interrelated factors are central in shaping collective equilibrium choices in the presence of social conflict the form of political institutions and the political power of different groups Individuals and groups with significant political power are more likely to be influential and sway policies toward their preferences Exactly how political power is distributed within the society and how individuals can exer cise their political power resulting from their votes connections or brute force de pends on political institutions For example a dictatorship that concentrates political power in the hands of a small group implies a different distribution of political power 778 Part VIII The Political Economy of Growth in the society than a democracy which corresponds to a society with a greater degree of political equality We expect that these various political regimes induce different sets of economic institutions and policies and thus lead to different economic outcomes The purpose of the next two chapters is to investigate this process of collective deci sion making and the implications of different choices of institutions and policies on economic growth 3 The technology the nature of the endowments and the distribution of income and endowments in the economy influence both preferences and the distribution of political power For example the nature of political conflict and the resulting political economy equilibrium is likely to be different in a society where much of the land and the capital stock is concentrated in the hands of a few individuals and families than one in which there is a more equitable distribution of resources We would also expect politics to function differently in a society where the major assets are in the form of human capital vested in individuals than in one where natural resources such as diamonds or oil are the major assets The issues raised and addressed in this part of the book are central to the field of political economy Since this is a book on economic growth not on political economy I do not try to do justice to the large and growing literature in this area Instead I focus on topics and models that I deem to be most important for the questions posed above I also save space by focusing when possible on the neoclassical growth model in discrete time rather than some of the richer models that have been presented in this book This might at first appear an odd choice Why focus on the neoclassical growth model which does not generate growth other than by exogenous technological change to study the political economy of growth Yet the neoclassical growth model offers two significant advantages First it provides the most tractable framework to analyze the main political economy conflicts Second because competitive equilibria in this model are Pareto optimal the role of political economy distortions become more transparent Naturally once the basic forces are understood it is relatively straightforward to incorporate them into endogenous growth models or other richer structures Some of the exercises consider these extensions Finally throughout I focus on discretetime models because this makes gametheoretic interactions easier to study I have organized the material on the political economy of growth into two chapters Chapter 22 takes political institutions as given and focuses on the implications of distributional conflict under different scenarios In this chapter I highlight why and when distributional conflict can lead to distortionary policies that retard growth I also offer various complementary frameworks for the analysis of these questions Chapter 23 then turns to the implications of different political regimes for economic growth and includes a brief discussion of how political institutions themselves are determined endogenously Before presenting this material it is useful to start with an abstract discussion of the relationship between economic institutions political institutions and economic outcomes and of how individual preferences over economic and political institutions are formed Much of the political science literature posits that individuals have direct preferences over political institutions and perhaps also over economic institutions For example individuals might derive utility from living under a democratic system While this assertion is plausible the approach developed so far emphasizes another potentially equally important reason for individuals to have preferences over political institutions Economic institutions and policies have a direct effect on economic outcomes eg the ef fects of tax policies regulation and contracting institutions described in previous chapters Thus a major determinant of individual preferences over economic institutions and policies ought to be the allocations that result from these arrangements Based on this viewpoint Part VIII The Political Economy of Growth 779 throughout I focus on these induced preferences over economic institutions The same reason ing applies to political institutions These determine the political rules under which individuals interact In direct democracy for example key decisions are made by majoritarian voting In representative democracy majorities choose representatives who then make the policy choices and face the risk of being removed from office if they pursue policies that are not in line with the preferences of the electorate In contrast in nondemocratic regimes such as dictatorships or autocracies a small clique such as an oligarchy of rich individuals or a junta of generals make the key decisions As a result different policies and economic institutions are likely to emerge in different political systems and individuals should have induced preferences over political institutions To emphasize this point let us represent the chain of causation described above by a set of mappings Let P denote the set of political regimes or institutions R be the set of feasible policies or economic institutions and X denote the set of feasible allocations which include different levels of consumption of all goods and services by all individuals in the society Ignoring any stochasticity in outcomes for simplicity we can think of each political institution in the set P leading to some specific set of economic institutions in the set R Let this be represented by the mapping π Similarly different policies lead to different allocations ignoring again stochastic elements and multiple equilibria let this be represented by the mapping ρ Schematically we can write P π R ρ X Now suppose that each individual i has a utility function ui X R representing his prefer ences over possible allocations in X Suppose also that individuals are consequentialist in the sense that they do not care about economic or political institutions beyond these institutions influences on allocations Then their preferences over some economic institution R R are simply given by ui ρ R ui ρ R R This mapping therefore captures their induced preferences over economic institutions as a function of the economic allocations that these institutions induce Preferences over political institutions are also induced in the same manner The utility that individual i derives from some political institution P P is ui ρ π P ui ρ π P R Induced preferences over institutions are important since an equilibrium framework ought to explain the emergence and change of political institutions as a function of these preferences This brief introduction has therefore laid two types of foundations for the next two chap ters First as taken up in Chapter 22 we must understand how different types of economic institutions and policies affect economic outcomes including economic performance and the distribution of resourcesthe mapping ρ Based on this understanding we will analyze the preferences of different groups over these economic institutions policies and determine the conditions under which different groups will have a preference for distortionary nongrowth enhancing economic arrangements Second to understand political change and how it interacts with economic decisions and economic growth we need to study induced preferences over political institutionsthe mapping π This is the topic of Chapter 23 22 Institutions Political Economy and Growth T his chapter makes a first attempt at answering the following question that has been in the background of much of what we have done so far why do similar societies choose different institutions and policies leading to very different economic growth outcomes The analysis so far has highlighted the role of capital accumulation human capital and technology in economic growth Throughout I have stressed that the level of physical capi tal the extent of human capital and even the technology of societies should be thought of as endogenous that is as responding to incentives This brings us to the fundamental ques tion why do different societies provide different incentives to firms and workers Chapter 4 suggested that differences in institutions are important determinants of these incentives and of crosscountry variations in investments in physical capital human capital and technology The purpose of this and the next chapter is to provide models that can help us understand why institutions might have such an effect and why institutions themselves differ across societies 221 The Impact of Institutions on LongRun Development As already emphasized in Chapter 4 institutions matterat least when we look at clusters of economic and political institutions over long horizons Most of the models in the book incor porate this feature since they highlight various effects of economic institutions and policies on economic allocations For example tax and subsidy policies and market structures may af fect physical capital accumulation human capital investments and technological progress and contracting institutions and the structure of the credit markets influence technology choices and the efficiency of production Perhaps even more important all models studied so far assume a relatively orderly working of the market economy Add to these models some degree of inse curity of property rights or entry barriers preventing activities by the more productive firms and they imply major inefficiencies Both theory and casual empiricism suggest that these fac tors are important We must thus recognize that doing business is very different in the United States than in subSaharan Africa Entrepreneurs and businessmen in the United States or 781 782 Chapter 22 Institutions Political Economy and Growth pretty much in any OECD country face relatively secure property rights and a stable orderly environment Individuals or corporations that wish to create new businesses face relatively few barriers The situation is starkly different in much of the rest of the world for example in sub Saharan Africa the Caribbean and large parts of Central America and Asia Similarly the lives of the majority of the population are radically different across these societies most citizens have access to a wide variety of public goods and the ability to invest in their human capital in most OECD countries but not in many lessdeveloped economies Economists often summarize these variations across societies as institutional differences or differences in institutions and policies The term is slightly unfortunate but is one that is widely used and accepted in the literature Institutions mean different things in different contexts and none of these exactly corresponds to the meaning intended here As already emphasized in Chapter 4 by institutional differences we are referring to differences in a broad cluster of social arrangements including security of property rights for businesses as well as for regular citizens and the ability of firms and individuals to write contracts to facilitate their economic transactions contracting institutions the entry barriers faced by new firms the socially imposed costs and barriers facing individual investments in human capital and incentives of politicians to provide public goods This definition of institutions is quite encompassing To make theoretical and empirical progress one typically needs a narrower definition Toward this goal I have already distinguished between economic institutions and policies which correspond to taxes the security of property rights contracting institutions entry barriers and other economic arrangements and political institutions which correspond to the rules and regulations affecting political decision making including checks and balances against presidents prime ministers or dictators as well as methods of aggregating the different opinions of individuals in the society eg electoral laws In terms of the notation introduced in the introduction to this part the effect of economic institutions on economic outcomes is summarized by the mapping ρ while the implications of political institutions for the types of economic institutions and policies is captured by the mapping π It is also useful to note that the difference between economic institutions and policies is not always clear so it is often their combination not one or the other that is important For example we can refer to the security of property rights as economic institutions but we would not typically refer to tax rates as institutions Yet entirely insecure property rights and 100 taxation of income have much in common One difference might be that institutions are more durable than policies1Thus in what follows I make a distinction between economic institutions and policies economic institutions provide a framework in which policies are set However when the distinction between economic institutions and policies is unimportant I typically use economic institutions as a standin for both The evidence presented in Chapter 4 suggests that institutional differences do matter for economic growth The focus of this section is not to review this evidence but to build on it and ask the next question if economic institutions are so important for economic growth why do some societies choose institutions that do not encourage growth In fact based on available historical evidence we can go further why do some societies choose institutions and policies that specifically block technological and economic progress The rest of this chapter and much of the next one provide a framework for answering these questions I start with an informal discussion of the main building blocks for constructing an answer The first important element of the political economy approach is social conflict There are few if any economic changes that would benefit all agents in the society Thus every change in institutions and policies creates winners and losers relative to the status quo Take the 1 In Section 229 I discuss another potential reason that taxation and security of property rights might be different which relates to how the proceeds are used 221 The Impact of Institutions on LongRun Development 783 simplest example removing entry barriers so that a previously monopolized market becomes competitive While consumers benefit from this policy because of lower prices the monopolist who was previously enjoying a privileged position and high profits will be a loser The effect on workers depends on the exact market structure If the labor market is competitive workers also benefit since the demand for labor increases with the entry of new firms But if there are labor market imperfections so that the employees of the monopolist were previously sharing some of the rents accruing to this firm they will also be potential losers from the reform Consequently there will not be unanimous support for removing entry barriers even when removal increases growth and output in the economy This example highlights a general principle because of the different allocations that they induce individuals have different conflicting preferences over economic institutions So if there are conflicting preferences over collective choices in general and over institutions and policies in particular how do societies make decisions Political economy is the formal analysis of this process of collective decision making If there is social conflict between a monopolist who wishes to retain entry barriers and consumers who wish to dismantle them the equilibrium of a political process decides the outcome This process may be orderly in democracies or disorderly or even chaotic in other political regimes as illustrated by the alltoo frequent civil wars throughout human history Whether it is a democratic or a nondemocratic process that leads to the equilibrium policy the political power of the parties with conflicting interests plays a central role Put simply if two individuals disagree over a particular choice eg about how to divide a dollar their relative powers determine the ultimate choice In the political arena this corresponds to the political power of different individuals and groups For example in the monopoly example we may expect the monopolist to have political power because it has already amassed income and wealth and may be able to lobby politicians In a nondemocratic society where the rule of law is tenuous we can even imagine the monopolist utilizing thugs and paramilitaries to quash the opposition On the other hand in a democracy consumers may have sufficient political power to overcome the interests and wishes of the monopolist through the ballot box or by forming their own lobbying groups The second key element of the political economy approach is commitment problems which act as a source of inefficiency and augment the distortions created by social conflict Political decisions at each date are made by the political process at that date eg by those holding political power at that time commitment to future sequences of political and economic decisions are not possible unless they happen to be equilibrium commitments we will see that whether we use the concept of Subgame Perfect Equilibrium SPE or Markov Perfect Equilibrium MPE plays an important role At this point it is important to distinguish between nongrowthenhancing policies or distortionary policies and Pareto inefficiency Many political economy models do not lead to Pareto inefficiency This is because their equilibria can be represented as solutions to weighted social welfare functions see Section 227 The resulting allocation is then a point along the constrained Pareto frontier of the economy given the set of available instruments Nevertheless many such allocations involve distortionary and nongrowthenhancing policies2 In addition when commitment problems are present the political equilibrium can involve constrained Pareto inefficiencies as well because there may exist future policy sequences that can make all parties better off but they may not be implemented in equilibrium Consider a situation in which political power is in the hands of a specific group or an individualthe political elite To simplify the thought experiment let us ignore any constraints 2 Consider for example an allocation in which a dictator such as Mobutu Sese Seko in Zaire expropriates all the investors in the country it is possible to change policies to increase investment and growth but this will typically imply taking resources and power away from Mobutu and making him worse off 784 Chapter 22 Institutions Political Economy and Growth on the exercise of this political power Then the elite can set policies to induce allocations that are most beneficial for themselves and thus the political equilibrium can be thought of as the solution to the maximization of a social welfare function giving all the weight to the elite Even though the resulting equilibrium may not be Pareto inefficient it typically involves non growthenhancing policies The key question is under what circumstances does the exercise of political power by the elite lead to such distortionary policies I argue that there are two broad reasons for why those with political power choose dis tortionary policies The first is revenue extraction that is the attempt by the elite to extract resources from other members of the society Central to this source of distortionary policies are two aspects of the society 1 a decoupling between political power which is in the hands of the elite and economic opportunities which lie with the entrepreneurs and the workers and 2 a limited set of fiscal instruments These two aspects combined imply that the elite will use the available distortionary fiscal instruments to transfer resources from the rest of the so ciety to themselves We will also see that the same type of distortionary policies emerge even when there is no political elite but decisions are made democratically see Section 228 The restriction to a limited set of fiscal instruments such as distortionary linear taxes is important here Had there been nondistortionary taxes such as lumpsum taxes the elite could extract resources from the rest of the society without discouraging economic growth But lumpsum taxes are often not feasible and more generally most forms of redistribution create distortions by reducing incentives for work or effort or by discouraging investment Second the elite may choose distortionary policies because they are in competition with other social groups This competition may be economic For example the elite may be engaged in production and recognize that taxes on other producers will reduce the demand for factors thus increasing the elites profits indirectly I refer to this as the factor price manipulation motive for distortionary policies The competition between the elite and other social groups may also be political Enrichment by other groups might pose a threat to the elites ability to use and benefit from their political power in the future distortionary taxes are then beneficial for the elite as a way of impoverishing their political competitors I refer to this as the political replacement motive for distortionary policies The rest of the chapter illustrates these various mechanisms An important implication of the models I present is that factor price manipulation and political replacement motives often lead to greater distortions and are more damaging to the growth potential of a society than the revenue extraction motive This basic framework also clarifies the additional inefficiencies created by commitment problems Because the elite cannot commit to future policies there may be a holdup problem whereby investments once undertaken are expropriated or taxed at prohibitively high rates Holdup problems are likely to be important in a wide variety of circumstances for example when the relevant investments are in longterm projects so that a range of policies is decided after these investments are undertaken I also use this framework to illustrate how and under what conditions economic institutions can constrain equilibrium policies In Sections 227 and 228 I show how political economy equilibria can be studied in models with greater heterogeneity and how distributional conflicts in such societies also lead to distortionary policies Finally I end this chapter by emphasizing the role of public goods provision by the government and how political economy considerations affect equilibrium investment by the state and the politically powerful groups controlling it in public goods 222 Distributional Conflict and Economic Growth in a Simple Society In this and the next four sections I discuss the implications of distributional conflict for economic growth in a simple societyIn a simple society individuals are permanently allocated 222 Distributional Conflict and Economic Growth in a Simple Society 791 This tax rate ˆτ maximizes tax revenues from middleclass entrepreneurs and puts the elite at the peak of the Laffer curve Substituting for ˆkτ from 2212 we obtain the following expression for ˆτ f ˆkˆτ ˆτ 1 ˆτ f ˆkˆτ2 f ˆkˆτ 0 2216 Intuitively the utilitymaximizing tax rate for the elite trades off the increase in revenues resulting from a small increase in the tax rate f ˆkˆτ against the loss in revenues that results because the increase in the tax rate reduces the equilibrium capitallabor ratio ˆτf ˆkˆτˆkτ This tax rate ˆτ is always between 0 and 1 see Exercise 221 though the maximization problem of the elite is not necessarily concave and 2216 may have more than one solution If this is the case ˆτ always corresponds to the global maximum for the elite8 This analysis so far establishes the following result Proposition 222 Suppose that 226 holds Then for any initial distribution of capital stocks among entrepreneurs Ki0iSm there exists a unique MPE where at each t 0 1 the elite set the tax ˆτ 0 1 as given in 2216 all entrepreneurs choose the capitallabor ratio ˆkˆτ as given by 2211 and the equilibrium wage rate is ˆwˆτ as given by 2213 We have that ˆkˆτ k where k is given by 224 and ˆwˆτ w where w is given by 227 This proposition shows that the unique political equilibrium involves positive taxation of entrepreneurs by the elite Consequently the capitallabor ratio the output level and the wage rate are strictly lower than they would be in an economy without taxation Exercise 222 shows how this framework can be extended so that policies also affect the equilibrium growth rate Let us now return to the fundamental question raised at the beginning of this chapter why would a society impose distortionary taxes on businessesentrepreneurs The model in this section gives a simple answer political power is in the hands of the elite who would like to extract revenues from the entrepreneurs Given the available tax instruments the only way they can achieve this is by imposing distortionary taxes Thus the source of inefficiencies in this economy is the combination of revenue extraction motive by the politically powerful combined with a limited menu of fiscal instruments While the analysis so far shows how distortionary policies can emerge and reduce the level of investment and output below the firstbest level it is important to emphasize that the equilibrium here is not Pareto inefficient In fact given the set of fiscal instruments the equilibrium allocation is the solution to the maximization of a social welfare function that puts all the weight on the elite Pareto inefficiency requires that given the set of instruments and informational constraints there should exist an alternative feasible allocation that would make each agent either better off or at least as well off as they were in the initial allocation Given the restriction to linear taxes there is no way of improving the utility of the middle class entrepreneurs and the workers without making the elite worse off9 This observation implies that when we explicitly incorporate political economy aspects into the analysis there 8 Here I ignore the cases in which there might be multiple global maxima 9 In a slightly modified environment there exist mechanisms that would lead to Pareto improvements though these mechanisms could not be supported as MPE but could be supported as SPE For example with a finite number of entrepreneurs there exist SPEs in which each entrepreneur makes voluntary donations to the elite and chooses the firstbest capitallabor ratio and the elite refrain from distortionary taxation see Exercise 224 This example shows that the MPE could easily lead to Pareto inefficient equilibria even though this is not the case in our baseline economy It also highlights why models with a continuum of agents where such mechanisms are not possible are often more intuitive 792 Chapter 22 Institutions Political Economy and Growth are typically no free lunchesthat is there is often no easy way of making all agents better off Thus political economy considerations typically involve tradeoffs between winners and losers Since the allocation in Proposition 222 involves distortionary policies and reduces output relative to the firstbest allocation we might want to refer to this outcome as inefficient even though it is not Pareto inefficient In fact this label is often used for such allocations in the literature and I follow this practice But it is important to remember that inefficiencies here do not mean Pareto inefficiencies As a preliminary answer to our motivating question Proposition 222 is a useful starting point However it leaves a number of important questions unanswered First it does not provide useful comparative statics regarding when we should expect more distortionary policies Second it takes the distribution of political power as given If political power were in the hands of the middleclass entrepreneurs rather than the nonproductive elite the choice of fiscal instruments would be very different Third this analysis takes the menu of available fiscal instruments as given If the elite had access to lumpsum taxes they could extract revenues from the entrepreneurs without creating distortions I extend the current framework to provide answers to these questions in this and the next chapters Before doing this let us first consider a more specific version of the economy analyzed so far where the production function is Cobb Douglas This CobbDouglas economy by virtue of its tractability is a workhorse model for Sections 224226 Exercise 2217 briefly discusses how the approach here can be generalized when individuals have concave preferences 223 The Canonical CobbDouglas Model of Distributional Conflict Consider a specialized version of the economy analyzed in the previous section with two differences First the production function of each entrepreneur takes the form Yit 1 α KitαAitLit1α 2217 where Ait is a laboraugmenting groupspecific or individualspecific productivity term For now we can set Ait Am for all i Sm The term 1α in the front is included as a convenient normalization This CobbDouglas form enables an explicitform characterization of the political equilibrium and also links equilibrium taxes to the elasticity of output with respect to capital Second the analysis so far has shown that with linear preferences incomplete depreciation of capital plays no qualitative role so I also simplify the notation by assuming full depreciation of capital that is δ 1 Given 2217 the per capita production function is f ki 1 α Am1αkα i Combining this production function with the assumption that δ 1 2210 implies that at date t 1 each entrepreneur chooses a capitallabor ratio kt 1 such that kt 1 β1 τt 111αAm 2218 The utilitymaximizing tax policy of the elite is still given by 2216 which combined with 2217 implies that the utilitymaximizing tax for the elite at each date is ˆτ 1 α 224 Distributional Conflict and Competition 795 τ mt τ RE 1 α and τ et T mt T wt 0 2223 for all t and T et is then determined from 2219 holding as equality Proof See Exercise 225 The equilibrium is therefore similar to that in Section 222 Notice however that this proposition is stated under the assumption that Condition 221 fails to holdso that the equilibrium wage rate is wt 0 for all t If this were not the case the elite would also recognize the effect of their taxation policy on equilibrium wages This would introduce the competition motive in the choice of policies which is our next focus An extreme form of this factor price manipulation effect is shown in the next proposition Proposition 225 Suppose that Condition 221 holds and φ 0 Then the unique MPE features τ mt τ FP M 1 and τ et T mt T wt 0 for all t Proof See Exercise 226 In this proposition φ is set equal to 0 so that there is no revenue extraction motive Instead the only motive for taxation is to affect the equilibrium wage rate in 2222 Clearly for this we need Condition 221 to hold otherwise the wage rate would be equal to zero and the elite would not have the ability or the desire to manipulate factor prices Proposition 225 implies that the equilibrium tax rate in this case τ FPM is greater than the tax rate when the only motive for taxation was revenue extraction τ RE This might at first appear paradoxical but is in fact quite intuitive With the factor price manipulation mechanism the objective of the elite is to reduce the profitability of the middle class whereas for revenue extraction the elite would like the middle class to invest and generate revenues Consequently τ RE puts the elite at the top of the Laffer curve while τ FPM tries to harm middleclass entrepreneurs as much as possible so as to reduce their labor demand and thus equilibrium wages It is also worth noting that unlike the pure revenue extraction case the tax policy of the elite is indirectly extracting resources from the workers whose wages are being reduced The role of the assumption that φ 0 in this context also needs to be emphasized Taxing the middle class at the highest rate is clearly inefficient Why is there not a more efficient way of transferring resources to the elite The answer again relates to the limited fiscal instruments available to the elite In particular φ 0 implies that they cannot use taxes to extract revenues from the middle class so they are forced to use inefficient means of increasing their consumptionby directly impoverishing the middle class The absence of any means of transferring resources from the middle class to the elite is not essential for the factor price manipulation mechanism however This is illustrated next by combining the factor price manipulation motive with revenue extraction though the absence of nondistortionary lump sum taxes is naturally important The next proposition derives the equilibrium when Condition 221 holds and φ 0 so that both the factor price manipulation and the revenue extraction motives are present In Proposition 225 the factor price manipulation motive by itself leads to the extreme result that the tax on the middle class should be as high as possible Revenue extraction though typically another motive for imposing taxes on the middle class serves to reduce the power of the factor price manipulation effect The reason is that high taxes also reduce the revenues extracted by the elite moving the economy beyond the peak of the Laffer curve To derive the political equilibrium in this case first note that the elite will again neither tax themselves nor 224 Distributional Conflict and Competition 797 Second since Proposition 226 incorporates both the revenue extraction and the factor price manipulation motives it contains the main comparative static results of interest One result is that the equilibrium tax rate is decreasing in φ because as φ increases revenue extraction becomes more efficient which has a moderating effect on the tax preferences of the elite In tuitively this shows the positive side of state capacity with greater state capacity the elite can raise revenues through taxation and thus their motives to impoverish competing groups become weaker we will see a potentially negative side of state capacity below Another comparative static result is that the equilibrium tax rate is increasing in θe The reason for this is again the interplay between the revenue extraction and factor price manipulation mechanisms When there are more elite producers reducing factor prices becomes more important relative to rais ing tax revenues This comparative static thus reiterates that when the factor price manipulation effect is more important there are typically greater distortions A third result is that a decline in α raises equilibrium taxes for the same reason as in the pure revenue extraction case taxes create fewer distortions and this increases the revenuemaximizing tax rate Finally for future reference note that rents from natural resources RN have no effect on equilibrium policies 2242 Political Competition The Political Replacement Effect Section 2241 illustrated how competition in the factor market induces the elite to choose distortionary policies to reduce the labor demand from the middle class In this subsection I discuss the implications of competition in the political arena The main difference is that I now allow for switches of political power In particular let us denote the probability that in period t political power permanently shifts from the elite to the middle class by ηt Once they come to power the middle class will pursue the policies that maximize their own utility We can easily derive what these policies are using the same analysis as in Section 2241 see Exercise 228 Denote the utility of the elite when they are in control of politics and when the middle class are in control of politics by V eE and V eM respectively When the probability of the elite losing power to the middle class η is exogenous the analysis in Section 2241 applies without any significant change New political economy ef fects arise when the probability that the elite will lose power is endogenous To save space while communicating the main ideas I use a reducedform model and assume that the proba bility that the elite will lose power to the middle class is a function of the net income level of the middle class ηt ηθmCmt 0 1 2230 where Cmt is the net income of a representative middleclass entrepreneur which is also equal to his consumption I assume that η is differentiable and strictly increasing with derivative η 0 This assumption implies that when the middle class are richer they are more likely to gain power eg with greater resources they may be more successful in solving their collective action problems or they may increase their military power To simplify the discussion let us focus on the case in which Condition 221 does not hold so that equilibrium wage is equal to 0 and there is no factor price manipulation motive Thus in the absence of the political replacement motive the only reason for taxation is revenue extraction resulting in an equilibrium tax rate of τ RE Given these assumptions and the definitions of V eE and V eM we can write the maximization problem of the elite when choosing the tax rate τ mt at t 1 as V eE max τ m βα1αAe Lα φβα1ατ m1 τ mα1αAmθm Lα RNθe β1 ητ mV eE ητ mV eM 225 Subgame Perfect versus Markov Perfect Equilibria 799 225 Subgame Perfect versus Markov Perfect Equilibria The concept of equilibrium so far has been MPE A natural question is whether the results are different when we turn to the concept of SPE In general the set of SPEs in dynamic games is larger than the set of MPEs and some SPEs can lead to more efficient allocation of resources see Appendix C I first show that in the setup analyzed so far the SPEs and MPEs coincide I then turn to two modified versions of the environment studied so far where there are holdup problems resulting from the timing of taxation or from ex ante technology adoption decisions In these environments commitment problems lead to greater inefficiencies and SPE may be more efficient than MPE because it allows for greater equilibrium commitment on the part of the elite 2251 SPE versus MPE without Holdup The MPEs are a subset of the SPEs because the latter include equilibria supported by history dependent punishment strategies If there is no room for such history dependence SPEs coin cide with the MPEs In the models analyzed so far such punishment strategies are not possible Intuitively in the economic sphere each individual is infinitesimal and acts competitively tak ing prices as given Therefore 2220 and 2221 determine factor demands uniquely in any equilibrium Given the factor demands the payoffs from various policy sequences are also uniquely pinned down Thus payoffs to the elite from different strategies are independent of history and there cannot be any SPEs other than the MPE characterized above Proposition 227 The MPEs characterized in Propositions 224226 are the unique SPEs Proof See Exercise 2210 Exercise 2211 shows that the MPE in the model of Section 2242 is also the unique SPE This last result however depends on the assumption that there is only one possible power switch from the elite to the middle class If there were multiple power switches potential punishment strategies could be constructed and the set of SPEs could include nonMarkovian equilibria 2252 Lack of CommitmentHoldup The models discussed so far feature full commitment to oneperiodahead taxes by the elite In particular at the end of period t the elite can commit to the tax rate on output that applies at time t 1 Using a term from organizational economics this corresponds to a situation without any holdup Holdup on the other hand corresponds to a situation without commitment to taxes or policies so that after entrepreneurs have undertaken their investments they can be held up by higher rates of taxation or by expropriation These types of holdup problems are endemic in political economy since binding commitments to future policies are difficult or impossible Those who have political power at a certain point in time make the relevant decisions at that point Moreover when the key investments are long term so that once an investment is made it is irreversible there is a holdup problem even if there is a oneperiod commitment since there will be taxes on the revenue stream of this investment after the investment decisions are sunk The problem with holdup is that the elite are unable to commit to a particular tax rate before middleclass producers undertake their investments because taxes are set after investments This lack of commitment generally increases the amount of taxation and distortion Moreover 800 Chapter 22 Institutions Political Economy and Growth in contrast to the allocations so far which featured distortions but were Pareto efficient the presence of commitment problems leads to Pareto inefficiency To illustrate the main issues that arise in the presence of commitment problems I consider the same model as above but change the timing of events such that taxes on output at time t are decided in period t that is after the capital investments for this period have already been made The economic equilibrium is essentially unchanged and in particular 2220 and 2221 still determine factor demands with the only difference being that τ m and τ e now refer to expected taxes Naturally in equilibrium expected and actual taxes coincide What is different is the calculus of the elite in setting taxes Previously they took into account that higher taxes on output at date t would discourage investment for production at date t Since taxes are now set after investment decisions are sunk this effect is absent As a result in the MPE the elite always tax at the maximum rate so in all cases there is a unique MPE where τ mt 1 for all t Proposition 228 With holdup there is a unique MPE with τ mt τ HP 1 for all t Clearly this holdup equilibrium is more inefficient than the equilibria characterized above For example consider a situation in which Condition 221 does not hold so that with the original timing of events without holdup the equilibrium tax rate is τ mt 1 α But with holdup the equilibrium tax is τ mt 1 and the middle class stop producing This policy is not only costly for the middleclass entrepreneurs but also for the elite since they lose all their tax revenues In this model the unique MPE is no longer the only SPE since there is room for an implicit agreement between different groups whereby the elite credibly promise a different tax rate than τ HP 1 The MPE is now Pareto inefficient and a social planner with access to the same fiscal instruments can improve the utility of all agents in the economy To illustrate the difference between the MPE and the SPE and the associated Pareto inefficiency of the MPE consider the example where Condition 221 fails to hold In the MPE the elite raise no tax revenue because the middle class produce zero output Recall that the history of the game is the complete set of actions taken up to that point Then consider the following triggerstrategy profile the elite set τ mt 1 α for all t and the middleclass producers invest according to 2220 with τ mt 1 α as long as the history consists of τ ms 1 α and investments have been consistent with 2220 for all s t If there is any other action in the history then the elite set τ m 1and the middleclass producers invest zero Does this strategy constitute an SPE First it is clear that the middle class have no profitable deviation since at each t they are choosing their best response to taxes along the equilibrium path implied by 2220 To check whether the elite have a profitable deviation note that with this strategy profile they are raising a tax revenue of φ1 ααα1αβα1αAmθm Lα in every period thus receiving transfers worth φ 1 β1 αα12α1αβα1αAmθm L 2231 If in contrast they deviate at any point the most profitable deviation for them is to set τ m 1 and they will raise a tax revenue of φα12α1αβα1αAmθm L 2232 in that period Following such a deviation consider a continuation equilibrium that switches to the unique MPE which is the worst possible continuation SPE in this model and yields zero continuation utility to the elite see Appendix C Therefore the abovedescribed trigger 226 Inefficient Economic Institutions A First Pass 803 2 Regulation of technology these institutions concern direct or indirect factors affecting the productivity of firms and individuals The analysis of factor price manipulation in Section 2253 provides a partial answer to one of the questions raised above why would the political system use inefficient instruments A full analysis of this question requires a setup with a richer menu of fiscal instruments such as lumpsum taxes A glimpse of such an analysis is provided in Exercise 2216 Propositions 225 and 226 provide the beginning of an answer since they show that the equilibrium tax rate would be strictly above the revenuemaximizing level Our first task is to derive some implications from these observations about constitutional limits on taxation by the elite 2261 Emergence of Secure Property Rights The environment is the same as in the previous section with the only difference being that at time t 0 before any decisions are taken the elite can choose some τ in the interval 0 1as the constitutionally mandated maximum tax rate Thus future taxes must be less than τ A lower τ provides greater security of property rights to the middle class Naturally a key question is how a constitution that imposes τ 1 would be made credible I do not address this question here and take it as given that such a constitutional limit on future taxes can be imposed though this assumption to some degree goes against the presumption that commitment to future policies is not possible My objective is to investigate whether when such constitutional guarantees are feasible the elite would like to institute themthat is whether they prefer τ 1 or τ 1 Proposition 2211 Without holdup and technology adoption the elite weakly prefer τ 1 The proof is immediate without holdup or technology adoption putting further restrictions on the taxes can only reduce the elites utility This proposition implies that when economic institutions are decided by the elite who will also hold political power in the future and there are no holdup issues then the elite derive no benefits from introducing constitutional limits on their future taxes and will not introduce further security of property rights The results are different when there are holdup problems To illustrate this let us first go back to the environment with holdup where taxes for time t are decided after the capital stock for time t is determined Let us focus on the MPE and on the general case where both the revenue extraction and factor price manipulation motives are present Proposition 2212 Consider the game with holdup and suppose that Condition 221 holds and φ 0 Then the unique MPE involves τ mt τ for all t The elite prefer to set τ τ COM 1 at t 0 Proof See Exercise 2213 The intuition for this proposition is simple in the presence of holdup problems Proposition 228 shows that the unique MPE involves τ m 1 However this is Pareto inefficient in fact if the elite could commit to a tax rate of τ τ COM they would increase their consumption and also the consumption levels of the middle class and the workers If the elite could use economic institutions to regulate future taxes for example by setting constitutional limits then they may wish to use these to encourage investment By manipulating economic institutions the elite may approach their desired policy indeed in this simple economy they can commit to the tax rate that maximizes their utility This result shows that under certain circumstances the elite may wish to change economic institutions to provide additional property rights protection to producers Note however that the 804 Chapter 22 Institutions Political Economy and Growth restriction to MPE is important in this proposition If we allow historydependent punishment strategies and look at the SPE then the elite would be able to improve over the MPE allocation in Proposition 229 and depending on parameters they may even be able to implicitly and credibly commit to an equilibrium in which the tax rate at each date is equal to τ RE If this were the case there would be less need for changing economic institutions to place limits on future taxes Whether the MPE or the SPE is more relevant in such a situation depends on what the expectations of the different parties are and on the degree of coordination among the players which is typically determined by historical or other institutional factors When the source of additional inefficiency is technology adoption rather than the holdup problem resulting from the timing of taxes there is a greater need for a change in economic institutionseven if we focus on the SPE This result is stated in the next proposition Proposition 2213 Consider the game with technology adoption and suppose that Con dition 221 does not hold and φ 0 Then the unique MPE and the unique SPE involve τ mt τ RE 1 α given by 2228 At t 0 the elite prefer to set τ τ T A 1 α as defined in Proposition 2210 Proof See Exercise 2214 This proposition highlights that in environments where longterm investments or technology adoption decisions are important implicit promises as in Proposition 229 are of limited use Instead explicit credible guarantees through economic institutions are necessary to provide incentives and security to middleclass entrepreneurs so that they undertake the appropriate technology investments Thus while implicit promises and other informal arrangements could play the same role as economic institutions under some circumstances there are often limits to how well they can perform this role Consequently constitutional limits on distortionary policies and expropriation if feasible may emerge endogenously in the political equilibrium as a substitute for andor an improvement over such implicit promises 2262 Blocking Economic Development The focus in Section 2261 was on choosing economic institutions at t 0 to provide more secure property rights and better investment incentives to middleclass entrepreneurs These types of economic institutions play an important role in practice and variation in the security of property rights for businesses across societies likely explains part of the variation in economic performance we observe Nevertheless security of property rights and limits on taxes are only one aspect of economic institutions In many societies rather than encouraging economic activity the elite actively try to block economic development Why would the elite choose specifically inefficient policies to reduce the productivity of entrepreneurs and block economic development To provide the basic ideas in the simplest possible way I extend the basic framework in this section in one direction at time t 0 the government thus the elite controlling political power chooses a policy affecting the technology choices of producers denoted by g 0 1 This choice can be thought of as investment in infrastructure or the provision of law and order with g 1corresponding to creating a better business environment Alternatively g 0 may directly correspond to actions taken by the elite to block technology adoption by middleclass entrepreneurs Let us assume that g 0 1 affects the productivity of middleclass producers in all future periods and in particular Am Amg with Am1 Am0 To simplify the discussion suppose further that g 1 is costless and has no effect on the productivity of the elite The key question is whether the elite will choose g 1 increasing the middleclass entrepreneurs productivity or choose to block technology adoption 227 Heterogeneous Preferences Social Choice and the Median Voter 805 When the only mechanism at work is revenue extraction the answer is that the elite would like the middle class to have the best technology Proposition2214 Suppose that Condition 221 fails to hold and φ 0 Then the economic equilibrium always involves wt 0 and in the unique MPE the elite choose g 1 This proposition delineates a range of situations in which the elite would not block the technology adoption decisions of middleclass entrepreneurs This result follows immediately since g 1 increases the tax revenues and has no other effect on the elites consumption Consequently in this case the elite benefit from the increase in the output of the middleclass entrepreneurs and thus would like them to be as productive as possible Intuitively there is no competition between the elite and the middle class either in factor markets or in the political arena and when the middleclass entrepreneurs are more productive they generate greater tax revenues for the elite The situation is different when the elite wish to manipulate factor prices To illustrate this possibility suppose that there exists an upper bound on taxes equal to τ 1 Proposition 2215 Suppose Condition 221 holds φ 0 τ 1 and 1 τ11α AeAm Then in any MPE or SPE the elite choose g 0 Proof See Exercise 2215 Intuitively with τ 1 labor demand from the middle class is high enough to generate positive equilibrium wages even at the maximum tax rate Since φ 0 taxes raise no revenues for the elite and their only objective is to reduce the labor demand from the middle class and thus wages as much as possible This makes g 0 their preferred policy Consequently the factor price manipulation mechanism suggests that when it is within their power to do so the elite will choose economic institutions to reduce the productivity of competing middleclass producers Proposition 2215 shows how the elite may take actions to directly reduce the productivity of other competing entrepreneurs thus retarding or blocking economic development A similar effect applies when the political power of the elite is contested see Exercise 2216 This section has demonstrated how the elites preferences over policies translate into prefer ences over economic institutions When the elite prefer to commit to lower taxes this can lead to the emergence of economic institutions that provide greater security of property rights On the other hand the factor price manipulation or the political replacement effects may also in duce the elite to choose arrangements that block technology adoption or more generally reduce the productivity of competing groups 227 HeterogeneousPreferencesSocialChoiceandtheMedianVoter My next objective is to relax the focus on simple societies and investigate how a richer and more realistic form of heterogeneity among the members of the society influences policy choices I do this in two steps In this section I provide a brief overview of how political economy decisions are made in a society with heterogeneous agents The main tool in this context is the Median Voter Theorem and its cousin the Downsian Policy Convergence Theorem I show that these two theorems together provide a useful characterization of democratic politics under limited heterogeneity among agents In Section 228 I then use these results to show that the qualitative results derived in Section 222 generalize to a model with heterogeneity among entrepreneurs The bottom line of the analysis in Section 228 is that the source of distortionary inefficient 806 Chapter 22 Institutions Political Economy and Growth policies that arise from the desire of the political system to extract revenues from a subset of the population holds more generally than in the simple society investigated in Section 222 The Median Voter Theorem MVT has a long pedigree in economics and has been applied in many different contexts Given its wide use in political economy models I start with a sec tion stating and outlining this theorem Despite its simplicity and elegance the MVT is not applicable to situations in which the menu of policies cannot be reduced to a onedimensional policy choice I end this section by outlining some alternative ways of aggregating heteroge neous preferences when there are multipledimensional decisions This analysis also illustrates why in many circumstances the determination of political equilibria can be represented as the maximization of a weighted social welfare function 2271 Basics Let us consider an abstract economy consisting of a set of individuals H Throughout this section I take H to be a finite set and denote the number of individuals by H though the results here can be extended to the case in which H consists of a continuum of individuals Individual i H has a utility function uxi Yp p αi Here xi is his action with a set of feasible actions denoted by Xi p denotes the vector of political choices eg institutions policies or other collective choices with the menu of policies denoted by R since P was used for the set of political institutions at the beginning of this part and Yp is a vector of general equilibrium variables such as prices or externalities that result from all agents actions as well as policies Instead of writing a different utility function ui for each agent I have parameterized the differences in preferences by the variable αi This is without loss of generality simply define ui ui αi and is convenient for some of the analysis that follows Clearly the equilibrium variables such as prices represented by Yp here need not be uniquely defined for a given set of policies p Nevertheless since multiple equilibria are not the focus here I ignore these complications and assume that Yp is uniquely defined I also assume that individual objective functions are strictly quasiconcave so that each individual has a unique optimal action xip Yp αi arg max xiXi uxi Yp p αi Substituting this maximizing choice of individual i into his utility function we obtain individ ual is indirect utility function Up αi which summarizes his ranking of the policies p R It is also sometimes convenient to write p i p when individual i weakly prefers p to p according to Up αi and p i p when he has a strict preference 2272 Voting and the Condorcet Paradox Aggregating the preferences of heterogeneous agents through voting or other mechanisms is not always easy or feasible Arrows Impossibility Theorem in social choice theory highlights this issue from a normative perspective The same problem arises in the context of voting and is most clearly illustrated by the wellknown Condorcet paradox example which I present next Imagine a society consisting of three individuals 1 2 and 3 and three choices The individuals preferences are as follows 227 Heterogeneous Preferences Social Choice and the Median Voter 807 1 a c b 2 b a c 3 c b a Moreover let us make the political mechanism somewhat more specific and assume that it satisfies the following three requirements which together make up the open agenda direct democracy system A1Direct democracy The citizens make the policy choices by majoritarian voting A2Sincere voting In every vote each citizen votes for the alternative that gives her the highest utility according to her policy preferences Up αi Strategic voting where each individual chooses a utilitymaximizing vote is discussed below A3Open agenda Citizens vote over pairs of policy alternatives such that the winning policy in one round is posed against a new alternative in the next round and the set of alternatives includes all feasible policies Later I replace the open agenda assumption with parties offering policy alternatives thus moving from direct democracy some way toward indirect or representative democracy Consider a contest between policies a and b Agents 2 and 3 vote for b over a so b is the majority winner Next by the open agenda assumption policy alternative c is run against b Now agents 1 and 3 prefer c to b which is the new majority winner Then c runs against a but now agents 1 and 2 prefer a so a is the majority winner Therefore in this case we have cycling over the various alternatives or put differently there is no equilibrium of the voting process that selects a unique policy outcome For future reference let us now define a Condorcet winner as a policy choice that does not lead to such cycling Definition 221 A Condorcet winner is a policy p that beats any other feasible policy in a pairwise vote Clearly in the example of the Condorcet paradox there is no Condorcet winner 2273 SinglePeaked Preferences Suppose that the policy space is unidimensional so that p is a real number that is R R In this case a simple way to rule out the Condorcet paradox is to assume that preferences are single peaked for all voters We will see below that the restriction that R is unidimensional is essential and singlepeaked preferences are generally not well defined when there are multiple policy dimensions Let us first define the preferred policyor the political bliss point of voter i To simplify notation suppose that this preference is uniquely defined and denote it by pαi arg max pR Up αi We say that voter i has singlepeaked preferences if his preference ordering for alternative policies is dictated by their relative distance from his bliss point pαi More generally we have the following definition Definition 222 Let pαi R be individual is unique bliss point over R Then the pol icy preferences of citizen i are single peaked if and only if for all p p R such that p p pαi or p p pαi we have Up αi Up αi 808 Chapter 22 Institutions Political Economy and Growth When R R singlepeaked preferences are equivalent to the strict quasiconcavity of Up αi We can easily verify that in the Condorcet paradox not all agents possess single peaked preferences For example taking the ordering to be a b c agent 1 who has preferences a c b does not have singlepeaked preferences if we took a different ordering of the alternatives then the preferences of one of the other two agents would violate the single peakedness assumption see Exercise 2218 The next theorem shows that with singlepeaked preferences there always exists a Con dorcet winner Before stating this theorem let us define the median voter of the society Given the assumption that each individual has a unique bliss point over R we can rank individuals according to their bliss points the pαis Also to remove uninteresting ambiguities let us imagine that H is an odd number Then the median voter is the individual who has exactly H 12 bliss points to his left and H 12 bliss points to his right Put differently his bliss point is exactly in the middle of the distribution of bliss points We denote this individual by αM and his bliss point ideal policy by pM Theorem 221 Median Voter Theorem Suppose that H is an odd number that A1 and A2 from Section 2272 hold and that all voters have singlepeaked policy preferences over a given ordering of policy alternatives R Then a Condorcet winner always exists and coincides with the medianranked bliss point pM Moreover pM is the unique equilibrium policy stable point under the open agenda majoritarian rule that is under A1A3 Proof The proof is by a separation argument Order the individuals according to their bliss points pαi and label the medianranked bliss point by pM By the assumption that H is an odd number pM is uniquely defined though αM may not be Suppose that there is a vote between pM and some other policy p pM By definition of singlepeaked preferences for every individual with pM pαi we have UpM αi Up αi By A2 these individuals vote sincerely and thus in favor of pM The coalition voting for supporting pM thus constitutes a majority The argument for the case where p pM is identical The assumption that the society consists of an odd number of individuals was made only to shorten the statement of the theorem and the proof Exercise 2219 asks you to generalize the theorem and its proof to the case in which H is an even number More important than whether there is an odd or even number of individuals in the society is the assumption of sincere voting Clearly rational agents could deviate from truthful reporting of their preferences and thus from sincere voting when this is beneficial for them So an obvious question is whether the MVT generalizes to the case in which individuals do not vote sincerely The answer is yes To see this let us modify the sincere voting assumption to strategic voting A2 Strategic voting Define a vote function of individual i in a pairwise contest between p and p by vip p p p Let a voting counting rule in a society with H citizens be V p pH p p for any p p R eg the majoritarian vot ing rule V maj picks p over p when this policy receives more votes than p Let V vip p vip p be the policy outcome from voting rule V applied to the pairwise contest p p when the remaining individuals cast their votes according to the vector vip p and individual i votes vip p Strategic voting requires that the voting behavior of each individual is a best response to those of others that is vip p arg max vipp UV vip p vip p αi 227 Heterogeneous Preferences Social Choice and the Median Voter 809 In other words strategic voting implies that each individual chooses the voting strategy that maximizes his utility given the voting strategies of other agents Finally recall that a weaklydominant strategy for individual i is a strategy that gives weakly higher payoff to individual i than any of his other strategies regardless of the strategy profile of other players Theorem 222 Median Voter Theorem with Strategic Voting Suppose that H is an odd number A1 and A2 hold and all voters have singlepeaked policy preferences over a given ordering of policy alternatives R Then sincere voting is a weaklydominant strategy for each player and there exists a unique weaklydominant equilibrium This equilibrium features the medianranked bliss point pM as the Condorcet winner Proof The voting rule the political system in this case is majoritarian denoted by V maj Consider two policies p p R and fix an individual i H Assume without loss of general ity that Up αi Up αi Suppose first that for any vi p p V majvi vip p p or V majvi vip p p that is i is not pivotal Thus vip p p is a best response for individual i Suppose next that i is pivotal that is V majvip p vip p p if vip p p and V majvip p vip p p otherwise In this case the action vip p p is clearly a best response for i Since this argument applies for each i H it establishes that voting sincerely is a weaklydominant strategy and the conclusion of the theorem follows from Theorem 221 Notice that the second part of the Theorem 221 which applied to open agenda elections is absent in Theorem 222 This is because the open agenda assumption does not lead to a well defined game so a gametheoretic analysis of strategic voting is not possible In fact there is no guarantee that sincere voting is optimal in dynamic situations even with singlepeaked preferences see Exercise 2220 2274 Party Competition and the Downsian Policy Convergence Theorem The focus so far has been on voting between two alternative policies or on open agenda voting which can be viewed as an extreme form of direct democracy The MVT becomes potentially more relevant and more powerful when applied in the context of indirect democracy that is when combined with a simple model of party competition I now give a brief overview of this situation and derive the Downsian Policy Convergence Theorem which is the basis of much applied work in political economy Suppose that there is a Condorcet winner and there are two parties A and B competing for political office Assume that the parties do not have an ideological bias and would like to come to power In particular they both maximize the probability of coming to power for example because they receive a rent or utility of Q 0 when they are in power Assume also that parties simultaneously announce their respective policies and are com mitted to these policies Then the behavior of the two parties can be represented by the Nash Equilibrium corresponding to the following pair of maximization problems Party A max pA PpA pBQ and Party B max pB 1 PpA pBQ 810 Chapter 22 Institutions Political Economy and Growth where Q 0 denotes the rents of being in power and PpA pB is the probability that party A comes to power when the two parties platforms are pA and pB respectively Let the bliss point of the median voter be pM When the MVT applies we have PpA pB pM 0 PpA pM pB 1 and PpA pM pB pM 0 1 2235 The last equation in 2235 follows since when both parties offer exactly the same policy it is a best response for all citizens to vote for either party However the literature typically assumes randomization A4RandomizationPpA pM pB pM 12 This assumption can be rationalized by arguing that when they are indifferent individuals randomize between the two parties with equal probabilities Theorem 223 Downsian Policy Convergence Theorem Suppose that there are two parties competing for office A4 holds and all voters have singlepeaked policy preferences over a given ordering of policy alternatives Then both parties choose the medianranked bliss point pM as their policy platform Proof Suppose this is not the case Then there is a profitable deviation for one of the parties For example if pA pB pM one of the parties can announce pM and is sure to win the election When pA pM and pB pM party A can also announce pM and so increase its chance of winning to 12 Exercise 2221 provides a generalization of this theorem without assumption A4 This theorem demonstrates that policy converges between the two parties and that party competition implements the Condorcet winner Therefore in situations in which the MVT applies the democratic process of decision making with competition between two parties leads to a situation in which both parties choose their policy platform to coincide with the bliss point of the median voter Thus the MVT and the Downsian Policy Convergence Theorem together enable us to simplify the process of aggregating the heterogeneous preferences of individuals over policies and assert that under the appropriate assumptions democratic decision making leads to the preferred policy of the median voter The Downsian Policy Convergence Theorem is useful in this context since it gives a better approximation to democratic policy making than do open agenda elections There is a sense in which Theorem 223 is slightly misleading however While the theorem is correct for a society with two parties it gives the impression of a general tendency toward policy convergence in all democratic societies Many democratic societies have more than two parties A natural generalization of this theorem would be to consider three or more parties Unfortunately as Exercise 2222 shows these results do not generalize to three parties Thus some care is necessary in applying the Downsian Policy Convergence Theorem in the context of different political institutions Theorem 223 also does not apply when there is no Condorcet winner In particular if we take a situation in which there is cycling as in the Condorcet paradox example of Section 2272 there is typically no purestrategy equilibrium in the political competition game This is further discussed in Exercise 2222 2275 Beyond SinglePeaked Preferences Singlepeaked preferences play a very important role in the results of Theorem 221 by ensuring the existence of a Condorcet winner However single peakedness is a very strong assumption and does not have a natural analogue in situations in which voting concerns more than one policy choice see Exercise 2225 When there are multiple policy choices or when voting 814 Chapter 22 Institutions Political Economy and Growth Theorem 226 Probabilistic Voting Theorem Consider a set of policy choices R RK let p R be a policy vector and let preferences be given by the 2237 with the distri bution function of σ g i as H g If a purestrategy symmetric equilibrium exists then equilibrium policy is given by the p that maximizes 2241 The important point to note about this result is its seeming generality as long as a pure strategy symmetric equilibrium in the party competition game exists it corresponds to a maximum of some weighted social welfare function This generality is somewhat exaggerated however because such a symmetric equilibrium does not always exist The sufficient conditions to guarantee the existence of such an equilibrium are rather restrictive and are discussed in Exercise 2226 228 Distributional Conflict and Economic Growth Heterogeneity and the Median Voter I now return to the model of Section 222 and relax the assumption that political power is in the hands of the elite Instead I now introduce heterogeneity among the agents and then apply the tools from Section 227 in particular the MVT and Downsian Policy Convergence Theorem Theorems 221225 to analyze the political economy equilibrium of this model Recall that these theorems show that if there is a onedimensional policy choice and individuals have singlepeaked preferences or preferences over the menu of policies that satisfy the single crossing property then the political equilibrium coincides with the most preferred policy of the median voter To simplify the analysis I modify the environment in Section 222 slightly First there are no longer any elites Instead economic decisions are made by majoritarian voting among all agents Second to abstract from political conflict between entrepreneurs and workers I also assume that there are no workers Instead the economy consists of a continuum 1 of yeoman entrepreneurs each denoted by i 0 1and with access to a neoclassical production function Yit FKit AiLit where Ai is a timeinvariant laboraugmenting productivity measure and is the only source of heterogeneity among the yeomanentrepreneurs In particular F satisfies Assumptions 1 and 2 from Chapter 2 I assume that Ai has a distribution given by μA among the entrepreneurs The yeomanentrepreneur assumption means that each entrepreneur can only employ himself as the worker so Lit 1 for all i 0 1 and for all t I also set the depreciation rate of capital δ equal to 1 to simplify notation All individuals have linear preferences given by 221 As in Section 222 the investment decisions at time t 1depend only on the tax rate announced for time t 1 This latter feature is particularly important here since we know from Section 227 that the MVT does not generally apply with multidimensional policy choices The fact that at each point in time all actions depend on a single policy variable enables us to use the MVT The timing of events is similar to that in Section 222 At each date t there is voting over a linear tax rate on output τt 1 0 1 that will apply to all entrepreneurs in the next period at t 1 Voting is between two parties so that Theorems 221225 apply The proceeds of taxation are redistributed as a lumpsum transfer T t 1 0 to all agents Let us focus on MPE and first check that the conditions of the MVT are satisfied Let us define kit KitAi as the effective capitallabor ratio the ratio of capital to effective labor of entrepreneur i and recall that pt includes the sequence of taxes starting 229 The Provision of Public Goods Weak versus Strong States 817 the cost to the median voter is related to his productivity If the median entrepreneur is more productive than the average there are two forces making him oppose redistributive taxation he is effectively redistributing away from himself and there is also the distortionary effect of taxation captured by the second term in 2247 Third and most important in the case in which the productivity of the median voter is below average the political equilibrium involves positive distortionary taxation on all entrepreneurs To obtain the intuition for this result recall that tax revenues are equal to zero at τ 0 A small increase in taxes starting at τ 0 induces a secondorder loss for each entrepreneur and when AM A a firstorder redistributive gain for the median voter This result is important in part because most realworld wealth and income distributions appear to be skewed to the left with the median lower than the mean thus this configuration is more likely in practice Furthermore this result is most interesting in comparison with those in previous sections which also led to positive distortionary taxation but in environments where the nonproductive elite were in power Proposition 2218 shows that the same qualitative result generalizes to the case in which there is democratic politics and the median voter is an entrepreneur himself but is less productive than the average Finally Proposition 2218 gives a new comparative static result It shows that holding average productivity constant a decline in the productivity of the median entrepreneur voter leads to greater distortionary taxation Since higher taxes correspond to lower output and the larger gap between the mean and the median of the productivity distribution can be viewed as a measure of inequality this result suggests a political mechanism by which greater inequality may translate into higher distortions and lower output Nevertheless some care is necessary in interpreting this last result since the gap between the mean and the median is not an unambiguous measure of inequality Exercise 2230 gives an example in which a mean preserving spread of the distribution leads to a smaller gap between the mean and the median This caveat notwithstanding the literature often interprets this last result as providing a link between inequality and distortionary taxation Exercise 2231 presents a version of this model in which taxes affect the equilibrium growth rate 229 The Provision of Public Goods Weak versus Strong States The analysis so far has emphasized the distortionary effects of taxation and expropriation This paints a picture in which the major political economy determinant of poor economic performance is the extent of taxation and expropriation While the disincentive effects of taxation are undoubtedly important whether taxes are high is only one of the dimensions of policy that might affect economic growth For example in many endogenous growth models subsidies to RD also encourage faster growth even if this policy involves some taxation of capital and labor More generally public goods provision investment in infrastructure and provision of law and order are important functions of a government and the failure to perform these functions may have significantly negative consequences for economic performance In fact existing evidence does not support the view that growth or high levels of output are strongly associated with official taxation On the contrary poor economies typically have lower levels of tax revenues and government spending This is most stark if we compare OECD countries to subSaharan Africa Consequently the political economy of growth must also pay attention to whether governments perform the roles that they are supposed to The standard nonpolitical economy approach to this question starts by positing the existence of a benevolent government and looks for policy combinations that would maximize social welfare Once we incorporate political economy considerations we must also recognize that the government may 229 The Provision of Public Goods Weak versus Strong States 821 investment that is more important for economic development ie α is low a higher τ is required justified10 The main conclusion from this analysis is that when both the state and the citizens make productive investments it is no longer true that limiting the rents that accrue to the state is always good for economic performance Instead there needs to be a certain degree of balance of powers between the state and its citizens When the political elite controlling the power of the state expect too few rents in the future they have no incentive to invest in public goods Consequently excessively weak states may be as damaging for economic development as the unchecked power and expropriation of excessively strong states A number of shortcomings of the analysis in this section should be noted The first is that it relies on economic exit options of the citizens in the informal sector as the source of their con trol over the state whereas in practice political controls may be more important The second is that it focuses on the MPE without any possibility of an implicit agreement between the state and the citizens In Acemoglu 2005 I generalize the results presented here in these directions I show that similar results can be obtained when the constraints on the power of the state are not economic but political In particular we can envisage a situation in which citizens can stochas tically replace the government if taxes are too high In this case when citizens are politically powerful the extent of taxation and the amount of public goods provision are again limited In addition using a model with variable political checks on the state one can analyze the SPE where there might be an implicit agreement between the state and the citizens to allow for some amount of taxation and correspondingly high levels of public goods provision This equilibrium configuration can be viewed as an example of a consensually strong statesince the citizens al low the economic power of the state to be high partly because they believe they can control the state and the political elites by using elections or other means The configuration with the con sensually strong state might provide a potential explanation for the higher tax rates and higher levels of public goods provision in OECD countries than in many lessdeveloped economies This perspective also suggests a useful distinction between taxation and expropriation High taxes appear to have similar effects on investment and economic performance as does expropriation One difference between expropriation and taxes might be uncertainty It can be argued that producers know exactly at what rate they will be taxed while expropriation is inherently risky In the presence of risk aversion expropriation could be more costly than taxation The analysis here suggests another useful distinction which comes not from the revenue side but from the expenditure side Expropriation might correspond to the government taking a share of the output of the producers for its own consumption while in an equilibrium with a consensually strong state some of the revenues from taxation are spent on public goods which are useful for the producers If this distinction is important one of the reasons why taxation is viewed as fundamentally different from expropriation may be because taxation is often associated with some of the proceeds being given back to the citizens in the form of public goods Perhaps the most important aspect of the analysis in this section is the emphasis on different facets of growthenhancing institutions Economic growth not only requires secure property rights and low taxes but also complementary investments often most efficiently undertaken by the government Provision of law and order investment in infrastructure and public goods are obvious examples Thus growthpromoting institutions should not only provide some degree of security of property rights to individuals but also incentivize the government to undertake the appropriate public goods investments In this light excessively weak governments might be as costly to economic performance as the unchecked power of excessively strong governments 10 This discussion focuses on the outputmaximizing value of the parameter τ Exercise 2232 discusses how different taxes affect welfare of the elite and the citizens 822 Chapter 22 Institutions Political Economy and Growth 2210 Taking Stock To understand why some countries are poor and others are rich we need to understand why some countries choose growthenhancing policies while others choose policies that block eco nomic development This chapter emphasized a number of key themes in developing answers to these questions First the sources of institutional differences and nongrowthenhancing institutions must be sought in social conflict among different individuals and groups Social conflict implies that there is no guarantee that the society will adopt economic institutions and policies that encourage economic growth Such social arrangements benefit many individuals in the society but they also create losersindividuals and groups whose rents are destroyed or eroded by the introduction of new technologies When individuals in the society have conflict ing preferences over institutions and policies the distribution of political power in the society plays an important role in determining which institutions and policies are chosen and whether nongrowthenhancing institutions will be reformed In this chapter I emphasized that nongrowthenhancing policies can emerge without any significant Pareto inefficiencies I illustrated this point first by focusing on a simple society in which individuals belong to a social group the conflict of interest is among social groups and all political power rests in the hands of the political elite I showed that this environment combined with linear preferences implies that even the restrictive MPE concept leads to constrained Pareto efficient allocations Despite their Pareto efficiency equilibrium allocations may involve significant distortions suggesting as a byproduct that Pareto efficiency may not be the right concept to focus on in the analysis of the political economy of growth In addition to providing a simple useful framework for the analysis of policy the model with political power vested in the hands of the elite also leads to a range of comparative static results that shed light on what types of societies adopt policies that encourage growth and which societies are likely to block economic development The following are some of the main comparative static results 1 taxes are likely to be higher when the demand for capital by entrepreneurs is inelastic because in this case the revenuemaximizing tax rate for the elite is higher 2 taxes are higher when the factor price manipulation effect is more important relative to the revenue extraction effect 3 taxes are higher when the political power of the elite is contested and reducing the income level of the competing groups will lead to political consolidation for the elite 4 taxes are higher and more distortionary when there are significant holdup problems because investments are longterm or entrepreneurs have ex ante technology adoption decisions 5 in the absence of the political replacement effect greater state capacity leads to lower taxes and 6 when the political replacement effect is important both greater state capacity and greater rents from natural resources may lead to more distortionary policies because they increase the political stakes the value of holding on to political power This chapter has further illustrated that the revenue extraction mechanism emphasized in the context of elitedominated politics is also present in more complex societies If political decisions with heterogeneous productivity or preferences are made democratically then they often reflect the policy preferences of the median voter When the median voter is poorer than the average individual entrepreneur in the society she may want to use distortionary policies to transfer resources to herself This type of distortionary revenue extraction by the median voter is qualitatively similar to revenue extraction from middleclass entrepreneurs by the elite though it is in the context of a more general environment with heterogeneity among the entrepreneurs The analysis also leads to a new comparative static result when the gap between the mean and the median of the productivity distribution is greater the incentives to extract revenues are stronger and policies are more likely to be distortionary 2211 References and Literature 823 Finally I emphasized that taxation is not the only relevant policy affecting economic growth The provision of public goods in the form of securing law and order investments in infrastructure or even appropriate regulation might also be important for inducing a high rate of economic growth Will the state provide the appropriate amounts and types of public goods In the context of a political economy model the answer depends on whether the politically powerful groups controlling the state have the incentives to provide such goods The economic or the political elite only invest in public goods if they expect to reap the benefits of these investments in the future This raises the issue of weak versus strong states While an emphasis on taxes suggests that checks on the economic or political power of the state should be conducive to more growthenhancing policies weak states are unwilling to invest in public goods because those controlling the state realize that they will not be able to tax future revenues created by these public goods investments Consequently an intermediate strength of the state might be most conducive to growthenhancing policies The more important point here is that an analysis of the effect of economic institutions and policies on growth should take into account both individual incentives for investment and the government incentives for public goods provision The material in this chapter is no more than an introduction to the exciting and important field of the political economy of growth Many issues have not been addressed Among those omitted the following appear most important First in addition to taxes expropriation and public goods whether the society provides a level playing field to a broad cross section of society is important For example broadbased human capital investments which are important for modern economic growth require the provision of incentives and the ability to invest not only for a few businesses but for the entire population Similarly security of property rights for existing businesses must be balanced against the ease of entry for new firms Second the entire analysis in this chapter takes as given the distribution of political power in the society It is clear however that different distributions of power in the society lead to different policies and thus to distinct growth trajectories Consequently it seems important to understand how the distribution of political power and equilibrium political institutions might evolve endogenously and how this distribution interacts with the economic equilibrium Some of these issues are discussed in the next chapter 2211 References and Literature The material in this chapter draws on the large political economy literature and also on some of the recent work on the political economy of growth My purpose has not been to provide a balanced survey of these literatures but to emphasize the most important features pertaining to the sources of differences in economic institutions and policies across societies with the hope of shedding some light on differential crosscountry growth performances I focused throughout on the neoclassical growth model and its variants to isolate the contribution of political economy mechanisms and to keep the exposition manageable Persson and Tabellini 2000 and Drazen 2001 provide introductions to political economy Eggertsson 2005 provides an informal discussion of institutions The material in Sections 222226 and the discussion of revenue extraction and factor price manipulation effects draw on Acemoglu 2007b but the setup has been modified to be more consistent with the neoclassical growth model The factor price manipulation effect fea tures in Acemoglu 2007b 2008a The political replacement effect is introduced in Acemoglu and Robinson 2000b and is further discussed in Acemoglu 2007b A detailed analysis of why the political elite may block technological innovations to increase the likelihood of their 824 Chapter 22 Institutions Political Economy and Growth survival is presented in Acemoglu and Robinson 2006b That paper also shows how both relatively secure elites and those in competitive political environments do not have incentives to block technological change but those with intermediate levels of security that might be challenged by new technologies may try to block economic development Models with com petitive economic behavior by pricetaking agents and strategic political decisions were first developed by Chari and Kehoe 1990 for the analysis of the timeconsistency of the behavior of a benevolent government The material in Section 227 is standard See for example Arrow 1951 and Austen Smith and Banks 1999 for Arrows Impossibility Theorem Singlepeaked preferences are first introduced in Black 1948 The singlecrossing property is introduced in Roberts 1977 and further developed by Gans and Smart 1996 The notion of intermediate preferences introduced in Exercise 2224 is due to Grandmont 1978 The Downsian model of political competition is introduced in Downs 1957 and builds heavily on Hotellings seminal 1929 paper AustenSmith and Banks 1999 discuss the Downsian party competition model in detail The probabilistic voting model is due to Lindbeck and Weibull 1987 and Coughlin 1992 My exposition here was simplified by the assumption that parties care about their vote share not the probability of coming to power The Median Voter Theorem presented in Section 228 was first applied to an economy with linear redistributive taxes by Romer 1975 and Roberts 1977 Meltzer and Richard 1981 used the RobertsRomer model to relate taxation to inequality and to the extent of the voting franchise Several authors have since applied the RobertsRomer model in growth settings The most notable examples are Alesina and Rodrik 1994 Persson and Tabellini 1994 SaintPaul and Verdier 1993 and Benabou 2000 The models in Alesina and Rodrik 1994 and Persson and Tabellini 1994 are similar to the one I developed in Section 228 except that they do not characterize a welldefined MPE Instead they assume that either 1 voting takes place at the beginning of time at t 0 and over a single tax rate that will apply at all future dates or 2 agents are myopic and do not take into account future votes though they do take into account their own future economic decisions In addition these papers focus on an economy with endogenous growth so that differences in taxes lead to differences in equilibrium growth rates see Exercise 2231 Both Alesina and Rodrik 1994 and Persson and Tabellini 1994 emphasize the negative effects of inequality on economic growth interpreting the gap between the mean and the median as a measure of inequality They also present cross country evidence suggesting that inequality is negatively correlated with economic growth This crosscountry growth evidence is difficult to interpret however both because there are many omitted variables in such growth regressions and also because other researchers find very different associations between inequality and growth see eg Forbes 2000 Banerjee and Duflo 2003 SaintPaul and Verdier 1993 on the other hand show that higher inequality can lead to greater growth when tax revenues are invested in human capital accumulation Benabou 2000 shows how a negative relationship between inequality and growth is consistent with higher inequality leading to less redistribution in a world in which greater redistribution may be growthenhancing again because tax revenues are invested in education None of these papers characterize the MPE of a dynamic economy instead assuming that voting is either myopic or takes place only once at the beginning of time Krusell and RıosRull 1996 and Hassler et al 2005 provide characterizations of MPEs in related political environments Section 229 builds on Acemoglu 2005 The idea that weak states may be an important impediment to economic growth is popular among political scientists and political sociologists and is most famously articulated in Migdal 1988 Wade 1990 Evans 1995 and Herbst 2000 These approaches do not analyze the incentives of the politicians or the government Acemoglu 2005 provides the first formal framework to analyze these issues The material in Section 229 embeds the baseline model in that paper into a neoclassical growth model 2212 Exercises 827 1 a b c 2 b c a 3 c b a Suppose the following dynamic voting protocol is in effect first there is a vote between a and b then the winner goes against c and the winner of this contest will be implemented Focus on SPE where voters do not use weakly dominated strategies at any stage a Show that these preferences are single peaked but sincere voting is not equilibrium behavior Hint suppose that players 1 and 2 are voting sincerely and show that player 3 prefers not to vote sincerely b Characterize the SPE of this game under strategic voting by all players c Consider a generalization in which the society H consists of H individuals and there are finite number of policies R p1 p2 pM For simplicity suppose that H is an odd number Voting takes M 1 stages In the first stage there is a vote between p1 and p2 In the second stage there is a vote between the winner of the first stage and p3 until we have a final vote against pM The winner of the final vote is the policy choice of the society Prove that if preferences of all agents are single peaked then the unique SPE implements the bliss point of the median voter 2221 Modify and prove Theorem 223 without using assumption A4 2222 This exercise reviews Downsian party competition and then shows that Theorem 223 does not apply if there are three parties competing In particular consider Downsian party competition in a society consisting of a continuum 1 of individuals with singlepeaked preferences The policy space R is the 0 1 interval and assume that the bliss points of the individuals are uniformly distributed over this space a To start with suppose that there are two parties A and B They both would like to max imize the probability of coming to power The game involves both parties simultaneously announcing pA 0 1 and pB 0 1 and then voters voting for one of the two parties The platform of the party with most votes gets implemented Determine the equilibrium of this game How would the result be different if the parties maximized their vote share rather than the probability of coming to power b Now assume that there are three parties simultaneously announcing their policies pA 0 1 pB 0 1 and pC 0 1 and the platform of the party with most votes is implemented Assume that parties maximize the probability of coming to power Characterize all pure strategy equilibria c Now assume that the three parties maximize their vote shares Prove that there exists no purestrategy equilibrium d In part c characterize the mixedstrategy equilibrium Hint assume the same symmetric probability distribution for two parties and make sure that given these distributions the third party is indifferent over all policies in the support of the distribution 2223 Prove Theorem 225 2224 This exercise involves generalizing the idea of singlecrossing property used in Theorem 224 to multidimensional policy spaces The appropriate notion of preferences of individuals turns out to be intermediate preferences Let R RK where K N and policies p belong to R We say that voters have intermediate preferences if their indirect utility function Up αi can be written as Up αi G1p BαiG2p where Bαi is monotone monotonically increasing or monotonically decreasing in αi and the functions G1p and G2p are common to all voters Suppose that A2 holds and voters have intermediate preferences The bliss point vector of individual i is pαi R that maximizes individual is utility Prove that when preferences are intermediate a Condorcet winner always exists and coincides with bliss point of the voter with the median value of αi that is pM pαM 830 Chapter 22 Institutions Political Economy and Growth individual with the median capital holdings ωM will be implemented Show that as this median capital holdings falls the rate of capital taxation increases What is the effect of this on economic growth d Show that the equilibrium characterized in part c is not an MPE Explain why not How would you set up the problem to characterize such an equilibrium Hint just describe how you would set up the problem no need to solve for the equilibrium 2232 a Prove Proposition 2219 b Derive the outputmaximizing tax rate as in 2260 c Let τ τ wm τ e and τ c be the values of τ that respectively maximize output social welfare the elites utility and citizens utility for all t 0 Show that 0 τ c τ τ e 1 and 0 τ c τ wm τ e 1 23 Political Institutions and Economic Growth T he previous chapter investigated why some societies choose inefficient economic in stitutions and policies It emphasized the importance of social conflict among different groups and the lack of commitment to future policies as major sources of nongrowth enhancing policies Much of the discussion was in the context of a given set of political institutions which shaped both the extent and kind of social conflict among different individ uals and groups and what types of policies were possible or could be committed to A natural conjecture in this context is that political institutions influence a societys choices of economic institutions and policies and thus its growth trajectory This conjecture leads to the following two questions Do certain political institutions mediate social conflict more successfully thus potentially avoiding nongrowthenhancing policies Why do different societies choose or end up with different political institutions This chapter provides some preliminary answers to these two questions I start with a brief summary of the empirical evidence on the effect of different political regimes on economic growth Section 232 then uses the baseline model in Section 222 from the previous chapter to illustrate that once we take the existence of conflicting preferences into account no political regime is perfect and each creates different types of costs and benefits associated with different losers and winners in the society Whether a particular set of political institutions leads to growthenhancing policies then depends on the details of how it functions on the technology and the factor endowments of the society and on which groups benefit from these institutions Section 233 then turns to the dynamic tradeoffs between different regimes emphasizing how democratic regimes might compensate for the shortrun distortions that they create by generating longrun benefits both by avoiding sclerotic outcomes and by creating greater flexibility This section also emphasizes how different political regimes deal with the process of creative destruction which as we saw in Chapter 14 is one of the engines of modern economic growth The arguments in Section 233 suggest that democracies may be better at taking advantage of the forces of creative destruction How political institutions themselves emerge and change is discussed briefly in Section 234 831 832 Chapter 23 Political Institutions and Economic Growth 231 Political Regimes and Economic Growth In thinking about the impact of political institutions on economic outcomes and growth most scholars would probably start with the contrast between democratic and nondemocratic regimes But there are many different types and shades of democracy Democracy is typically defined by a set of procedural rules for instance by whether there are free and fair elections in which most adults can participate and whether there is free entry of parties into politics But this definition of democracy leaves many distinctive institutional features of democracies unspecified Democracies can be parliamentary or presidential They can use different electoral rules giving varying degrees of voice to minorities Perhaps more importantly there are different degrees of free and fair and most adults Most elections even those in Europe or the United States involve some degree of fraud and some restrictions on the entry of parties or candidates Moreover many individuals are effectively or sometimes explicitly disenfranchised Similarly political scientists consider Britain and the United States in the late nineteenth century to have been democratic though only men had the right to vote Few people would consider the United States in the 1960s to have been a nondemocracy but many blacks were disenfranchised These specifics create various shades of democracy that may affect the economic outcome The differences between nondemocratic societies are probably even more pronounced China under the rule of the Communist Party since 1948 is an undisputed case of a nondemo cratic regime but it is very different in nature from the oligarchic regime in place in Britain before the process of democratization started with the First Reform Act of 1832 In Britain before 1832 there were prime ministers and parliaments though they were elected by a small minority of the populationthose with wealth education and privilege who made up less than 10 of the adult population Furthermore the powers of the state never rivaled those of the Communist Party in China The Chinese example is also different from Augusto Pinochets military dictatorship in Chile or that of Park Chung Hee in South Korea Once we consider regimes based on personal rule such as that of Mobutu Sese Seko in Zaire and monarchies such as the rule of the Saud family in Saudi Arabia the contrast is even more marked Nevertheless there is an important commonality among these nondemocracies and an important contrast between nondemocratic and democratic regimes making these categories still useful for conceptual and empirical analysis Despite all their imperfections and different shades democratic regimes at least when they have a certain minimal degree of functionality provide greater political equality than nondemocratic ones The free entry of parties and the practice of oneperson onevote in a democracy are the foundations of this and ensure some voice for each individual When democracies function well majorities have some often a significant influence on policiesthough they themselves may be constrained by certain constitutional restrictions In contrast nondemocracies rather than representing the wishes of the population at large represent the preferences of a subgroup of the population which I have so far referred to as the elite The identity of the elite differs across nondemocratic societies In China it is mainly the wishes of the leaders of Communist Party that matter In Chile under Pinochet most decisions were taken by a military junta and it was their preferences and perhaps those of certain affluent segments of the society supporting the dictatorship that counted In Britain before the First Reform Act of 1832 it was the small wealthy minority that was politically influential With this cautionary introduction on the distinctions between democracies and nondemoc racies what are the major differences between these political regimes First one might imagine that democracies and nondemocracies have different growth performances The first place to look for such differences is the postwar era for which there are better data on economic growth Using crosscountry regression evidence Przeworski and Limongi 1993 and Barro 1999 231 Political Regimes and Economic Growth 833 conclude that democracies do not perform better than nondemocracies However there is no universal consensus on this matter For example Minier 1998 reports results showing both positive effects of democratizations and negative effects of transitions to nondemocracy on growth Nevertheless the bulk of the available evidence suggests that on average democracies do not grow much faster than nondemocracies at least once one controls for other potential determinants of economic growth This result is surprising and even perhaps disturbing One might have expected significantly worse growth performances among nondemocracies since this group includes highly unsuccessful countries such as Iraq under Saddam Hussein Zaire under Mobutu and Haiti under the Duvaliers Counteracting this group however are plenty of unsuccessful democracies including India until the 1990s and many newly independent former colonies that started their independence as electoral democracies though often quickly falling prey to coups or the personal rule of some strongman There are also many successful non democracies including Singapore under Lee Kwan Yew South Korea under General Park or more recently China Thus to understand how different political institutions affect economic decisions and economic growth we need to go beyond the distinction between democracy and nondemocracy If there are no marked growth differences between democracies and nondemocracies are there instead other significant policy or distributional differences Rodrik 1999 documents that democracies have higher labor shares and interprets this as the outcome of greater redis tribution in democracies Acemoglu and Robinson 2006a summarize a range of case studies showing how democracies pursue more redistributive policies In contrast Gil Mulligan and SalaiMartin 2004 use crosssectional regressions to show that many policies in particular overall government spending and spending on social security do not differ between democra cies and dictatorships Therefore there is no consensus in the literature on whether democracies pursue different fiscal policies and whether this has a major impact on the distribution of re sources in the society But the evidence in Rodrik 1999 and some of the evidence summarized in Acemoglu and Robinson 2006a indicate that at least in some cases democracies pursue significantly more redistributive policies than do nondemocracies and we can take these dif ferences as our starting point or at least as a working hypothesis But it is useful to bear in mind that the differences in policy between democracies and nondemocracies even if present appear to be much less pronounced than one might have expected on the basis of theory alone It should also be noted at this point that the comparison of democracies to nondemocracies over the postwar era might be overly restrictive When we look at a longer time horizon it appears that democracies experience better economic performance Most of the countries that industrialized rapidly during the nineteenth century were more democratic than those that failed to do so The comparisons of the United States to South American countries or of Britain and France to Russia and AustriaHungary are particularly informative in this context For example the United States which was one of the most democratic societies at the time was not any richer and may have been significantly poorer than the highly nondemocratic and repressive Caribbean colonies at the end of the eighteenth century However the nineteenth century and early twentieth century witnessed rapid growth and industrialization in the United States and stagnation in the entire Caribbean area and in much of the rest of South America This historical episode therefore suggests that the more democratic societies may have been better at taking advantage of the new investment and growth opportunities that came with the age of industrialization in the nineteenth century The contrast of Britain and France to Russia and AustriaHungary is similar Even though the former two countries were already richer at the beginning of the nineteenth century than the latter two the income differences were small Differences in political institutions were much more marked however Britain was already on its way to becoming a parliamentary democracy and France had already undergone the Revolution of 1789 Britain and France adopted progrowth policies throughout much of the 834 Chapter 23 Political Institutions and Economic Growth nineteenth century even when this was costly to their existing landowning elites whereas Russia and AustriaHungary explicitly blocked industrialization to protect the economic and political interests of their landowning aristocracies Longrun regressions such as those discussed in Chapter 4 are also consistent with this pattern and show a significant effect of a broad cluster of institutions on economic growth While we cannot confidently say that this effect represents the impact of political institutions on growth this cluster of institutions comprises both political and economic elements and it is likely that the growthenhancing cluster of institutions could not exist without the political institutions supporting the economic policies that encouraged investment and free entry I next turn to a theoretical investigation of how we might expect different political institu tions to affect economic policies and economic outcomes 232 Political Institutions and GrowthEnhancing Policies Consider the canonical CobbDouglas model analyzed in Section 223 in the previous chapter The model was analyzed under the assumption that a subset of the producers the elite was in power I now briefly discuss the equilibrium in the same environment when the middle class or the workers are in power and then contrast the resulting allocations 2321 The Dictatorship of the Middle Class versus the Dictatorship of the Elite First let us suppose that the middle class hold political power so that we have the dictatorship of the middle class instead of the dictatorship of the elite in the previous chapter The situation is symmetric to that in the previous chapter with the middle class and the elite having exchanged places In particular the analysis leading to Proposition 226 immediately yields the following result Proposition 231 Consider the environment of Section 223 but the middle class instead of the elite holding political power Suppose that Condition 221 holds φ 0 and Am φαα1αAe θe θm 231 Then the unique MPE features τ mt 0 and τ et τ COM κ L θm α φ 1 κ L θm α φ for all t where κ L θe α φ is defined in 2229 Proof See Exercise 231 This proposition shows that political equilibria under elite control and middleclass control are identical except that the two groups have switched places Political institutions therefore influence policies and the resulting equilibrium allocation of resources In particular in the elitecontrolled society the middle class are taxed both to create revenues for the elite and to reduce their labor demand In the middleclassdominated society the competing group of producers that are out of political power are the elite even though the name elite has the connotation of political power So now the elite are taxed to generate tax revenues and create more favorable labor market conditions for the middle class The contrast between 232 Political Institutions and GrowthEnhancing Policies 835 the elitedominated and the middleclassdominated politics approximates certain wellknown historical episodes For example in the context of the historical development of European societies political power was first in the hands of landowners who exercised it to keep labor tied to the land and reduce the power and profitability of merchants and early industrialists With the economic and constitutional changes of the late medieval period power shifted away from landowning aristocracies toward the merchants and industrialists ie the middle class in terms of the model here and it was their turn to adopt policies favorable to their own economic interests and costly for landowners So which one of these two sets of political institutionsthe dictatorship of the middle class or that of the eliteis better The answer is that they cannot be compared easily First as already emphasized in the previous chapter the equilibrium in Section 224 is Pareto optimal given the set of fiscal instruments it is not possible to make any other member of the society better off without making the elite worse off In the same way the current allocation of resources is Pareto optimal but it picks a different point along the Pareto frontiera point that favors the middle class instead of the elite What about the level of output Even here there is no straightforward ranking Either of these two societies may achieve a higher level of income per capita depending on which group has more productive investment opportunities When the middle class are more productive a society in which the elite are in power creates significant distortions In contrast if the elite have more profitable and socially beneficial production opportunities then having political power vested with the elite is more beneficial for economic performance than the dictatorship of the middle class The following proposition illustrates a particularly simple version of this result Proposition232 Consider the environment of Section 223 with CobbDouglas technology Suppose that Condition 221 and the inequalities in 2227 and 231 hold θe θm and φ 0 Then the dictatorship of the middle class generates higher income per capita when Am Ae and the dictatorship of the elite generates higher income per capita when Ae Am Proof See Exercise 232 This proposition gives a simple example of a situation in which the political institutions that lead to better economic performance in terms of income per capita depend on whether more productive group also holds political power When political and economic power are decoupled there is greater inefficiency An immediate implication of this result is that it is difficult to think about efficient political institutions without considering the selfinterested objectives of those who hold and wield political power and without fully analyzing how their productivity and economic activities compare to those of others Naturally one can think of political institutions that will outperform both the elitedominated politics of the previous chapter and the middle classdominated politics of this section In this case the key question is whether such political institutions are feasible once more realistic political economy and economic interactions are introduced The analysis of the design of feasible political institutions in the presence of political economy constraints is an interesting area but very much in its infancy For now we can simply note that under most circumstances the choice of political institutions in practice is among arrangements that create different types of distortions and different winners and losers 2322 Democracy or Dictatorship of the Workers The Section 2321 contrasted the dictatorship of the middle class to that of the elite A third possibility is to have a more democratic political system in which the majority decides policies Since in realistic scenarios the workers outnumber both the elite and the middle class entrepreneurs policies that favor the economic interests of the workers who have so far 836 Chapter 23 Political Institutions and Economic Growth been passive in this model simply supplying labor at the equilibrium wage rate will then be implemented While such a system resembles democracy in some ways especially since it implies greater political equality than the dictatorship of the elite or the middle class it can also be viewed as the dictatorship of the workers it is now the workers who dictate policies in the same way that the elite or the middle class did under their own dictatorships1 This again emphasizes that different political institutions create different winners and losers depending on which group has more political power As before the analysis is straightforward though the nature of the political equilibrium depends even more strongly on whether Condition 221 holds Proposition 233 Consider the environment of Section 223 and suppose that workers hold political power 1 Suppose that Condition 221 fails to hold so that there is excess labor supply Then the unique MPE features τ mt τ et τ RE 1 α 2 Suppose that Condition 221 holds so that there is no excess labor supply and that θe θm θ If in addition Am Ae then in the unique MPE τ et 0 and also τ mt τ Dm where 1 τ Dm11αAm Ae or τ Dm 1 α and α11αAm Ae If Am Ae then in the unique MPE τ mt 0 and also τ et τ De where 1 τ De11αAe Am or τ De 1 α and α11αAe Am Proof See Exercise 233 The most interesting implication of this proposition comes from the comparison of the cases with and without excess supply When Condition 221 fails to hold there is excess labor supply and taxes have no effect on wages Anticipating this workers favor taxes on both groups of producers to raise revenues to be redistributed to themselves Democracy then generates this outcome as the political equilibrium Clearly this result is more distortionary than either the dictatorship of the elite or that of the middle class because in the latter two political scenarios one of the producer groups was not taxed The situation is very different when Condition 221 holds In that case recall that both the dictatorships of the elite and of the middle class generated significant distortions owing to the factor price manipulation effectin particular they imposed taxes on competing producers to keep wages low In contrast workers dislike taxes precisely because of their effect on wages Consequently in this case workers have more moderate preferences regarding taxation and democracy generates lower taxes than both the dictatorships of the elite and of the middle class This proposition therefore again highlights that which set of political institutions generates a greater level of income per capita or higher economic growth depends on investment opportunities and market structure When workers or a subgroup that is influential in democracy can tax entrepreneurs without suffering the consequences democracy generates high levels of redistributive taxation and can lead to a lower income per capita than elite or middle classdominated politics However when workers recognize the impact of taxes on their own wages democracy generates more moderate political outcomes 1 Distinguishing the dictatorship of workers or poor segments of the society from a true democracy is an important issue but falls beyond the scope of my focus here 233 Dynamic Tradeoffs 837 The simple analysis in this section therefore already gives us some clues about why there are no clearcut relationships between political regimes and economic growth When the equivalent of Condition 221 holds so that distortionary policies reduce wages democracy is likely to generate higher aggregate output and growth than nondemocratic regimes In contrast democracy leads to worse economic performance by pursuing populist policies and imposing high taxes when the equivalent of Condition 221 fails to hold Naturally the model presented here is very simple in many ways and Condition 221 or its close cousins may not be appropriate for evaluating whether democracy or other regimes are more growthenhancing Nevertheless this analysis emphasizes that democracies like other regimes look after the interests of the groups that have political power and the resulting allocations often involve different types of distortions Whether these distortions are more or less severe than those generated by alternative political regimes depends on technology factor endowments and the types of policies available to the political system In light of the analysis so far this result is not surprising but its implications are nonetheless important In particular it highlights that there are no a priori theoretical reasons to expect that there should be a simple empirical relationship between democracy and growth 233 Dynamic Tradeoffs The previous section contrasted economic allocations under different political regimes Al though the underlying economic environment was a simplified version of the infinitehorizon neoclassical growth model the tradeoffs among the regimes were static In this section I examine an environment that also incorporates entry into entrepreneurship social mobility and a simple form of creative destruction Using this environment I contrast democracy to oligarchy The emphasis is on the dynamic tradeoffs between the two regimes 2331 The Baseline Model The model economy is populated by a continuum of measure 1 of infinitelylived agents each with preferences given by 221 as in the previous chapter In addition for reasons that will soon become clear I assume that each individual dies with a small probability ε 0 in every period and a measure ε of new individuals are born with the convention that after death there is zero utility and β 0 1 is the discount factor inclusive of the probability of death I consider the limit of this economy with ε 0 There are two occupations production workers and entrepreneurs This introduces the possibility of social mobility In particular each agent can either be employed as a worker or can become an entrepreneur I assume that all agents have the same productivity as workers but their productivity in entrepreneurship differs In particular agent i at time t has entrepreneurial talentskills ait AL AH with AL AH To become an entrepreneur an agent needs to set up a firm if he does not have an active firm already Setting up a new firm may be costly because of entry barriers created by existing entrepreneurs Each agent therefore starts period t with skill level ait AH AL some amount of capital kit invested from the previous date recall that capital investments are again made one period in advance and another state variable denoting whether he already possesses a firm I denote this variable by eit 0 1 with eit 1 corresponding to the individual having chosen entrepreneurship at date t 1 for date t The individual who is already an incumbent entrepreneur at t ie eit 1 may find it cheaper to become an entrepreneur at t 1 because potential entry barriers into entrepreneurship do not apply to incumbents I 838 Chapter 23 Political Institutions and Economic Growth refer to an agent with eit 1as a member of the elite at t both because he avoids the entry costs and because in an oligarchy he will be a member of the political elite making the policy choices In summary at date t each agent chooses eit 1 0 1 and if eit 1 1 he becomes an entrepreneur and also makes an investment decision for next period kit 1 R at date t 1 he decides how much labor lit 1 R to hire Agents also make the policy choices in this society How the preferences of various agents map into policies differs depending on the political regime and is discussed below There are three policy choices Two of those are similar to the policies we have seen so far a tax rate τt 0 τ on output and a lumpsum transfer distributed to all agents denoted by T t 0 Notice that I have already imposed an upper bound on taxes τ 1 This bound may result from the ability of individuals to hide their output in the informal sector or from the distortionary effects of taxation it is taken as given here The new policy instrument is a cost Bt 0 imposed on new entrepreneurs when they set up a firm I assume that the entry barrier Bt is pure waste for example corresponding to the bureaucratic procedures that individuals have to go through to open a new business Thus lumpsum transfers are financed only from taxes An entrepreneur with skill level ait capital level kit and labor lit produces yit 1 α kitαaitlit1α 232 units of the final good As in Section 223 I assume that there is full depreciation of capital so kit is also the level of investment of entrepreneur i at time t 1 in terms of the unique final good I further simplify the analysis by assuming that all firms have to operate at the same size L so lit L see Exercise 235 for the implications of relaxing this assumption Finally I adopt the convention that the entrepreneur himself can work in his firm as one of the workers which implies that the opportunity cost of becoming an entrepreneur is 0 The most important assumption here is that each entrepreneur has to run the firm himself so it is his productivity ait that matters for output An alternative would be to allow costly delegation of managerial positions to other more productive agents In this case low productivity entrepreneurs may prefer to hire more productive managers Throughout I assume that delegation is prohibitively costly To simplify expressions I also define bt Btβ L which corresponds to discounted per worker entry cost and is the relevant object when we look at the profitability of different occupational choices Profits the returns to entrepreneur i gross of the cost of entry barriers at time t are then equal to πit 1 τtyit wtlit 1 β kit which takes into account that the investment cost kit was incurred in the previous period and thus the opportunity cost of investment which is forgone consumption is multiplied by the inverse of the discount factor This expression for profits takes into account that the entrepreneur produces output yit pays a fraction τt of this output in taxes and also pays a total wage bill of wtlit Given a tax rate τt and a wage rate wt 0 and using the fact that lit L the net profits of an entrepreneur with talent ait at time t are πkit ait wt τt 1 α 1 τtkitαait L1α wt L 1 β kit 233 840 Chapter 23 Political Institutions and Economic Growth This Markov chain also implies that the fraction of agents with high skill in the stationary distribution is see Exercise 236 M σ L 1 σ H σ L 0 1 237 Since there is a large number continuum of agents the fraction of agents with high skill at any point is M I also assume that M L 1 so that without entry barriers highskill entrepreneurs generate more than sufficient demand to employ the entire labor supply Moreover suppose that M is small and L is large in particular L 2 Then the workers are always in the majority which simplifies the political economy discussion below The timing of events is as follows At the beginning of time t ait eit and kit are given for all individuals as a result of their decision at date t 1 and the realization of uncertainty regarding ability Then the following sequence of moves takes place 1 Entrepreneurs demand labor the labor market clearing wage rate wt is determined and production takes place 2 The tax rate on entrepreneurs τt 0 τ is set 3 The skill level of each agent for the next period ait 1 is realized 4 The entry barrier for new entrepreneurs bt 1 is set 5 All agents make occupational choices eit 1 and entrepreneurs make investment decisions kit 1 for the next period Entry barriers and taxes are set by different agents in the various political regimes as specified below Notice that taxes are set after the investment decisions This raises the holdup problems discussed in the previous chapter and acts as an additional source of inefficiency The fact that τt τ 1 puts a limit on these holdup problems Individuals make their occupational choices and investment decisions knowing their ability level that is ait 1 is realized before the decisions about eit 1 and kit 1 Notice also that if an individual does not operate his firm he loses the license so next time he wants to set up a firm he needs to incur the entry cost and the assumption that lit L rules out the possibility of operating the firm at a much smaller scale Finally we need to specify the initial conditions I assume that the distribution of talent in the society is given by the stationary distribution and nobody starts out as an entrepreneur so that ei1 0 for all i Given linear preferences the initial level of capital holdings is not important Let us again focus on MPE where strategies are a function only of the payoffrelevant states For individual i the payoffrelevant state at time t includes his own state eit ait kit ait 1 and potentially the fraction of entrepreneurs that are high skill2 denoted by μt and defined as μt Prait AH eit 1 Prait AH i SE t 2 Here eit kit and ait are part of the individuals state at time t because they influence an entrepreneurs labor demand In addition ait 1 is revealed at time t and influences his occupational choice and investment decisions eit 1 and kit 1 for t 1 and so is also part of his state 846 Chapter 23 Political Institutions and Economic Growth in the future except through its impact on payoffrelevant state variables Therefore given τt τ the utility of agent i with eit 1 0 and ait AL depends on bt only through the equilibrium wage wEt and the transfer T Et Highproductivity workers those with eit 1 0 and ait AH may become entrepreneurs but as the above analysis shows in this case NV qt ait AH eit 1 0 0 and W H W L so their utility is also identical to those of lowskill workers Consequently all workers prefer a level of bt that maximizes wEt T Et Since the preferences of all workers are the same and they are in the majority the democratic equilibrium maximizes these preferences A democratic equilibrium starting at time t is therefore given by policy wage and economic decision sequences ˆpt ˆwt and ˆxt respectively such that ˆwt and ˆxt constitute an economic equilibrium given ˆpt and ˆpt τ bt 1 is such that bt 1 arg max bt10wEt 1 T Et 1 Inspection of 2318 and 2320 shows that wages and tax revenues are both maximized when bt 1 0 for all t so the democratic equilibrium will not impose any entry barriers This result is intuitive workers do not wish to protect incumbents because such protection reduces labor demand and wages Since there are no entry barriers only highskill agents become entrepreneursthat is eit 1 only if ait AH at all t Given this stationary sequence of MPE policies we can use the value functions 2311 and 2313 to obtain V H W H W L W wD T D 1 β where wD is the equilibrium wage in a democracy and T D is the level of transfers when τt τ and bt 0 for all t Equation 2315 implies that wD 1 αβ1 τα1αAHα The following proposition therefore follows immediately Proposition 234 There exists a unique democratic equilibrium In this equilibrium τt τ and bt 0 for all t Moreover eit 1 if and only if ait AH so μt 1 The equilibrium wage rate is given by wt wD 1 α α βα1α1 τ11αAH and the aggregate output is Y Dt Y D 1 α β1 τα1αAH Note that aggregate output is constant over time and there is perfect equality in equilibrium because the excess supply of highskill entrepreneurs ensures that they receive no rents These features will contrast with the oligarchic equilibrium 2333 Oligarchy In oligarchy policies are determined by voting among the elite4 At the time of voting over the entry barriers bt the elite consist of those with eit 1 1 and at the time of voting over 4 Notice that this assumption means political power rests with current entrepreneurs As discussed in the previous chapter there may often be a decoupling between economic and political power so that key decisions 850 Chapter 23 Political Institutions and Economic Growth a different dimension of the tradeoff between different regimesthat related to the dynamics they imply While democracy may create shortrun distortions it can lead to better longrun performance because it avoids political sclerosisthat is incumbents becoming politically powerful and erecting entry barriers against new and better entrepreneurs This model also suggests the type of patterns we already discussed in Section 231 lack of a clear relationship between democracy and growth over the past 50 years combined with the examples of democ racies that have been able to achieve industrialization during critical periods in the nineteenth century In fact a simple extension of the framework here provides additional insights that are useful for thinking about why democracies may be successful in preventing political sclerosis the forces highlighted here also suggest that democracies are more flexible than oligarchies For example Exercise 2310 shows that democracies are typically better able to adapt to the arrival of new technologies because there are no incumbents with rents to protect who can successfully block or slow down the introduction of new technology This type of flexibility might be one of the more important advantages of democratic regimes Even though the model presented in this section provides ideas and results that are useful for understanding the comparative development experiences of democratic and nondemocratic regimes like the model discussed in the previous section it focuses on the costs of democ racy resulting from its more redistributive nature In particular it emphasizes that democratic regimes redistribute income away from the rich and the entrepreneurs toward the poorer seg ments of the society and this leads to distortions that reduce income per capita An alternative source of distortions in democracy is that democratic regimes may become dysfunctional be cause the elite still exercise power through corruption or other means despite the existence of democratic institutions It is possible that when the society becomes democratic but the elite maintain significant political power they may try to control the democratic agenda in poten tially more inefficient ways than in nondemocracyfor example by corruption rather than by direct decree In this case democracy can lead to worse economic outcomes not because it is pursuing populist redistributive policies as in the models presented here but because of political inefficiencies resulting from the continuing power of the elite 234 Understanding Endogenous Political Change 2341 General Insights The analysis so far has focused on the implications of different political institutions on eco nomic growth and how their economic consequences shape the preferences of different agents over these political institutions How do equilibrium political institutions emerge And why do institutions change Returning to the model of Section 233 we can imagine that democracy emerges because oligarchs voluntarily give up power and institute a democracy While this might be in their interest under some circumstances it will generally be costly for them to give up their monopoly of political power and the economic rents that this monopoly brings Not surprisingly most institutional changes in practice do not happen voluntarily but result from social conflict Consider for example the democratization of most Western European nations during the nineteenth century and early twentieth century or the democratization experience in Latin America during the twentieth century In these cases democracy was not voluntarily granted by the existing elites but resulted from the process of social conflict in which those previously disenfranchised demanded political rights and in some cases were able to secure them But how does this happen A nondemocratic regime by its nature vests political power with a narrow group Those who are excluded from this group the nonelites do not have the right 234 Understanding Endogenous Political Change 851 to vote nor do they have any voice in collective decisions So how can they influence the political equilibrium and induce equilibrium political change The answer to this question lies in drawing a distinction between de jure formal and de facto political power De jure political power refers to power that originates from the political institutions in society and has been the form of political power on which we have so far focused Political institutions determine who gets to vote how representatives make choices and the general rules of collective decision making in society However there is another equally important type of political power that features in equilibrium political changes The political power of protesters that marched against the existing regime before the First Reform Act in Britain in 1832 was not of the de jure kind The law of the land did not empower them to influence the political course of actionsin fact they were quite explicitly disenfranchised But they had a different kind of power emanating from their majority in the population and their ability to solve the collective action problem and to organize protests This type of political power which lives outside the political institutions is de facto political power De facto political power is important for political change since de jure political power itself acts as a source of persistencenot of change For example consider the model of the previous section The elite are typically content with the oligarchic regime If de jure power is the only source of power only the elite have decisionmaking powers in the society and they are unlikely to change the political regime from oligarchy toward democracy However if the nonelites citizens or workers had some source of powerwhich by its nature has to be de facto powerthen political change becomes a possibility Perhaps in some periods the nonelites will be able to solve their collective action problem and thus exercise enough pressure on the system to force some changes In the extreme they can induce the elite to disband the oligarchy and transition to democracy or they can themselves topple the oligarchic regime The interaction between de jure and de facto political power is the most promising way to approach the analysis of equilibrium political change Moreover this interaction becomes particularly interesting when studied in a dynamic framework This is for at least two reasons First most of the issues we are discussing such as commitment problems and institutional change are dynamic in nature Second whether the distribution of de facto political power is permanent or changing stochastically over time has major consequences for the structure of political equilibrium When a particular disenfranchised group has permanent de facto political power it can use this power at each date to demand concessions from those holding de jure political power Such a situation leads to a redistribution of resources in favor of this group but not necessarily to institutional change because the requisite redistribution can take place within the context of the existing political regime Next consider a situation in which the de facto political power of the disenfranchised group is highly transientin the sense that they have been able to solve their collective action problem and can exercise de facto political power today but it is unlikely that they will have the same type of power tomorrow Then the disenfranchised group cannot rely on the use of their de facto political power in the future to receive concessions To obtain redistribution of resources favorable to themselves in the future they have to use their current power This scenario generally involves a change in political institutions as a way of changing the future distribution of de jure power More explicitly consider a situation in which a particular group of disenfranchised individuals currently have the de facto political power to change the distribution of resources in their favor but they also understand that this de facto power will be gone tomorrow But any limited transfer of resources or other concessions made to this disenfranchised group today is likely to be reversed in the future Therefore the transient nature of their de facto political power encourages this disenfranchised group to take actions to change political institutions to cement their power more firmlyso that they can change their transient de facto political power into more durable de jure political power This informal 852 Chapter 23 Political Institutions and Economic Growth discussion suggests a particular channel through which the interplay between de facto and de jure political power can lead to equilibrium changes in political institutions 2342 A Framework for the Study of the Dynamics of Political Institutions The discussion so far illustrates how we can use the interaction between de facto and de jure political power to study equilibrium political changes While the discussion has given some clues about what the incentives of different parties with and without de jure political power will be in a dynamic game it is so far unclear how one would construct models to analyze these forces and generate useful comparative statics I now suggest a general framework that is useful for thinking about the dynamic interactions between de facto and de jure political power Imagine a dynamic model in which there are two state variables political institutions and the distribution of resources For example Pt P denotes a specific set of political institutions at time t such as democracy or nondemocracy parliamentary versus presidential system or different types of nondemocratic institutions The set P denotes the set of feasible political institutions Similarly let Wt W denote a variable encoding the distribution of resources at time t For example in a society consisting of two groups the rich and the poor Wt could be the relative incomes of the two groups In a society with many individuals it could be the distribution function of income or wealth Again W is the set of all possible distributions of resources It is useful to think of both Pt and Wt as state variables for three reasons First they change relatively slowly thus corresponding to the loose notion of a state variable Second they are typically the payoffrelevant Markovian states Third these two variables determine the two sources of political power essential for understanding equilibrium political change Pt determines the distribution of de jure political power Jt J which for example determines who has the right to vote or the constraints on politicians The distribution of resources on the other hand affects the distribution of de facto political power De facto political power is typically the result of the ability of certain groups to solve their collective action problem or it emerges when certain groups have the resources to hire their own armies paramilitaries and supporters or simply use the money for lobbying and bribing Let the distribution of de facto political power in the society at time t be Ft F As in the beginning of Part VIII let us also denote economic institutions by Rt R and let Yt Y be a measure of economic performance such as income per capita or growth A dynamic framework that is useful for thinking about political change and its implications for economic growth consists of a mapping ϕ P Z J which determines the distribution of de jure power at time t as a function of political institutions at time t Pt P as well as some potential stochastic elements captured by zt Z It also comprises a mapping that de termines the equilibrium distribution of de facto power in a similar manner φ W Z F Then given the realization of Jt J and Ft F another mapping ι J F R P determines both economic institutions today Rt R and political institutions tomorrow Pt 1 P Put differently the distributions of de facto and de jure political power regu late what types of economic institutions emerge in equilibrium which thus corresponds to the mapping π introduced at the beginning of Part VIII and they also determine whether there will be political reform leading to changes in future de jure power eg example a switch from nondemocracy to democracy to increase the future de jure power of the citizens who hold significant de facto power today Finally an economic equilibrium mapping ρ R Y W determines both economic performance and the distribution of economic re sources For example if economic institutions involve competitive markets and secure property rights they lead to high aggregate output whereas insecure property rights and entry barriers 854 Chapter 23 Political Institutions and Economic Growth expanding the franchise and giving political power to the previously disenfranchised which created the precedents for modern democracy In the context of European political history the first important move toward democracy came in Britain with the First Reform Act of 1832 This act removed many of the worst inequities under the old electoral system and established the right to vote based strictly on property and income The reform was passed in the context of rising popular unrest and discontent at the political status quo in Britain By the 1820s the Industrial Revolution was well under way and the decade prior to 1832 saw continual rioting and popular unrest Notable were the Luddite Riots from 1811 to 1816 the Spa Fields Riots of 1816 the Peterloo Massacre in 1819 and the Swing Riots of 1830 Another catalyst for the reforms was the July revolution of 1830 in Paris There is general agreement among historians that the motive for the 1832 Reform was to avoid social disturbances The 1832 Reform Act increased the total electorate from 492700 to 806000 which represented about 145 of the adult male population Yet the majority of British population still could not vote There is also evidence of continued corruption and intimidation of voters until the Ballot Act of 1872 and the Corrupt and Illegal Practices Act of 1883 The Reform Act therefore did not create mass democracy it was instead designed as a strategic concession As a result parliamentary reform was still on the agenda in the middle of the century Following the sharp business cycle downturn in the second half of the nineteenth century the founding of the National Reform Union in 1864 the Reform League in 1865 and the Hyde Park Riots of July 1866 major electoral reforms were again instigated The Second Reform Act in 1867 increased the total electorate from 136 million to 248 million and made workingclass voters the majority in all urban constituencies The electorate was doubled again by the Third Reform Act of 1884 which extended the same voting regulations that already existed in the boroughs urban constituencies to the counties rural constituencies The Redistribution Act of 1885 removed many remaining inequalities in the distribution of seats and from this point on Britain had only singlemember electoral constituencies After 1884 about 60 of adult males were enfranchised Once again social disorder appears to have been an important factor behind the 1884 Act The Reform Acts of 186784 were a turning point in the history of the British state Eco nomic institutions also began to change Liberal and Conservative governments introduced a considerable amount of labor market legislation fundamentally changing the nature of indus trial relations in favor of workers During 190614 the Liberal Party under the leadership of Henry Asquith and David Lloyd George introduced the modern redistributive state into Britain including health and unemployment insurance government financed pensions min imum wages and a commitment to redistributive taxation As a result of the fiscal changes taxes as a proportion of GNP more than doubled in the 30 years following 1870 and then doubled again and taxes became more progressive Finally the Education Act of 1870 com mitted the government to the systematic provision of universal education and the educational attainment of the population increased dramatically at this time As a result of these changes inequality in Britain appears to have fallen sharply in the second half of the nineteenth century Overall the picture that emerges from British political history is clear Beginning in 1832 when Britain was governed by the relatively rich primarily rural aristocracy a series of strategic concessions were made over an 86year period These concessions were aimed at incorporating the previously disenfranchised into politics since the alternative was seen to be social unrest chaos and possibly revolution However when faced with the threat of revolt and social chaos the elite may also attempt to avoid giving away political power They may instead make economic concessions such as in come redistribution or other policies that favor nonelites and the disenfranchised Nevertheless 234 Understanding Endogenous Political Change 855 because the promise of concessions is typically not credible when threats are transient such promises are often insufficient to defuse social unrest Democratization can then be viewed as a credible commitment to future redistribution It is credible because it redistributes de jure political power away from the elite to the masses In democracy the poorer segments of the society become more powerful and can use their de jure political power to implement eco nomic institutions and policies consistent with their interests Thus democratization was a way of transforming the transient de facto power of the disenfranchised poor into more durable de jure political power The above account of events makes it clear that democracy in many Western societies and particularly in Britain did not emerge from the voluntary acts of an enlightened elite Democ racy was in many ways forced on the elite because of the threat of revolution Nevertheless many other countries faced the same pressures and political elites decided to repress the disen franchised rather than make concessions to them This happened with regularity in Europe in the nineteenth century though by the turn of the twentieth century most West European nations had accepted that democracy was inevitable Repression lasted much longer in much of South America and is still the preferred option for current political elites in China or Burma And yet repression is costly not only for the repressed but also for the elite for example because it destroys assets disrupts production and requires investments in repressive technologies Therefore faced with demands for democracy political elites face a tradeoff In the urban ized environment of nineteenth century Europe Britain was 70 urbanized at the time of the Second Reform Act the disenfranchised masses were relatively well organized and therefore difficult to repress Moreover industrialization had led to an economy based on physical and in creasingly human capital Such assets are easily destroyed by repression and conflict making repression an increasingly costly option for the elite In contrast in predominantly agrarian so cieties as in many parts of Latin America earlier in the century or currentday Burma physical and human capital are relatively unimportant and repression is easier and cheaper Moreover not only is repression cheaper in such environments democracy is potentially much worse for the elite because of the prospect of radical land reform Since physical capital is much harder to redistribute the elites in Western Europe found the prospect of democracy much less threatening 2344 Modeling Democratization So far I have offered a verbal account of how one might develop a model of the democratization process in line with the abstract framework of Section 2342 Once the main ideas are understood a formal framework is not difficult to construct The following is a simplified version of framework in Acemoglu and Robinson 2006a See also Exercise 2312 The society consists of two groups the elite and the masses the poor or the citizens Political power is initially in the hands of the elite but the masses are more numerous Thus if there is democratization the masses become politically more powerful and dictate the policies All individuals are infinitely lived and the elite are richer than the masses Because the society starts as a nondemocracy de jure power is in the hands of the elite Let us suppose that the only policy choice is a redistributive tax τ the proceeds of which are distributed lumpsum The elite prefer zero taxation τ 0 since they are richer and any taxation redistributes income away from them to the poorer masses Let us imagine that while de jure power in the nondemocracy lies with the elite the poor sometimes have de facto political power In particular suppose that with probability q in each period the masses are able to solve their collective action problem and can threaten to undertake 856 Chapter 23 Political Institutions and Economic Growth a revolution A revolution is very costly for the elite but generates only limited gains for the masses These limited gains may nonetheless be better than living in a nondemocracy and the inequitable distribution of resources that it involves So when they are able to solve their collective action problem the revolution constraint of the masses becomes binding In this case the rich need to make concessions to avoid a revolution As in the historical account above the elite have three options to defuse the revolutionary threat The first is to make concessions through redistributive policies today which will work if q is high In the limit where q 1 a revolution is possible at each date thus the elite can credibly commit to making redistribution toward the masses at each date because if they fail to do so the masses can immediately undertake a revolution However the same strategy does not work when q is small Consider the polar case where q 0 In this case the masses expect never to have the same type of de facto political power in the future Presuming that the amount to redistribution that the elite can give to the masses during a particular period is limited they will not be satisfied by temporary concessions In this case the elite may prefer to use repression Repression will be successful if the revolutionary threat is not well organized and it will be profitable for the elite if they have a great deal to lose from democratization Thus repression will be the action of choice for elites who fear major redistribution under democracy such as the landbased elites in Central America and Burma But in a highly urbanized and industrialized society like Britain where the costs of repression are likely to be substantial and the elite have less to fear from democratization the third option enfranchisement becomes an attractive choice This option involves the elite changing the political system and initiating a transition to democracy to alter the distribution of de jure power in favor of the masses With their newly gained decisionmaking power the masses know that they can choose policies in the future that will create a more equitable distribution of resources for themselves and will typically be happy to accept democratic institutions instead of a revolution that is costly for themselves as well as for the elite Compared to the abstract framework in Section 2342 the model sketched here is stripped down and to save space I have not even provided the equations to establish the main claims First the distribution of resources is no longer a state variable it is constant and does not affect transitions or the distribution of political power Second de jure political power is simply a nonstochastic outcome of political institutions in a nondemocracy the elite make the decisions and in a democracy there is a oneperson onevote policy and the masses thanks to their majority become the decisive voters Finally there are limited economic decisions Thus in its current form this model is not satisfactory for analyzing the impact of political institutions on economic institutions or the relationship between political regimes and economic growth Some of the extensions of this approach presented in Acemoglu and Robinson 2006a 2008 go some way toward incorporating economic institutions and decisions Nevertheless much work still remains to be done on the dynamic interactions between political institutions and economic growth 235 Taking Stock This chapter provided a brief overview of some of the issues related to the effects of political institutions on economic growth Based on the ideas presented in Chapter 22 we may expect differences in economic institutions to be related to political institutions For example if political power is in the hands of an elite that is opposed to growth growthenhancing policies are less likely to emerge The empirical evidence in Chapter 4 also provides support for such 236 References and Literature 857 a view because the cluster of economic institutions that provide secure property rights to a broad cross section of societytogether with political institutions that place constraints on the elite and on politiciansappear to be conducive to economic growth Nevertheless the relationship between political regimes and growth is more complicated for a number of reasons First the empirical evidence is less clearcut than we may have originally presumedwhile there are historical examples of the positive effects of democratic institutions on economic growth the postwar evidence does not provide strong support for the view that democracies and political institutions that constrain rulers and politicians always generate more economic growth Second political institutions themselves are endogenous and change dynamically These two factors imply that we need to study how political institutions affect economic outcomes more carefully and we should also consider the modeling of equilibrium political institutions Both of these areas are at the forefront of research in political economy and are likely to play a more important role in the research on economic growth in the coming years I also presented a number of model approaches that can shed light on the relationship between political institutions and economic growth I emphasized that ideal or perfect po litical institutions are unlikely to exist because different political institutions create different sets of winners and losers and distinct distortions Oligarchies for example favor the already rich and create distortions by protecting these established interests Democracies on the other hand typically involve higher taxes on the rich and on businesses to generate income to re distribute to the less well off In general it is impossible to unambiguously conclude whether democracies or oligarchies or yet other political systems favoring other groups will be more growth enhancing However certain ideas seem both plausible and consistent with the data One aspect I tried to emphasize is that the dynamic tradeoffs between democracies and other regimes may be different than the static tradeoffs While democracies may create static dis tortions because of their greater redistributive tendencies they may outperform oligarchies in the long run because they avoid political sclerosis that results when incumbents are able to dominate the political system and erect entry barriers to protect their businesses even when efficiency dictates that new individuals and businesses should enter and replace theirs Thus democracy may be more conducive than other political regimes to the process of creative de struction that is part of modern capitalist growth Democracy may also be more flexible and adaptable to the arrival of new technologies Finally I also gave a very brief overview of some of the issues that arise when we wish to model the dynamics of political institutions Section 234 provided both a general discussion of the types of models that would be useful for such an analysis and examples of how these models can be developed Once again this is an area of active current research and the material presented here is no more than the tip of the iceberg It is meant to encourage the reader to think more about various aspects of the relationship between political institutions and economic growth 236 References and Literature This chapter relates to a large literature in political economy and political science Because of space constraints I do not provide a comprehensive literature review The key references on the effect of political regimes on economic growth are provided in Section 231 Section 232 built on the models presented in the previous chapter Section 233 is directly based on Acemoglu 2008a Other models that discuss the functioning of oligarchic societies include Leamer 1998 Bourguignon and Verdier 2000 Robinson and Nugent 2001 Sonin 237 Exercises 859 a Show that there exists ψ such that if ψ ψ all existing entrepreneurs raise the entry barriers and switch to the new technology b Show that if ψ ψ then again the entry barriers will be raised but now only entrepreneurs who have low skills with the old technology switch to the new technology c Analyze the response of a democracy to the arrival of the same technology d Compare output per capita in a democracy and an oligarchy after the arrival of new tech nology and explain why democracy is more flexible in dealing with the arrival of new technologies 2311 This exercise shows that entry barriers typically lead to multiple equilibrium wages in dynamic models Consider the following twoperiod model The production function is given by 232 and the distribution of entrepreneurial talent is given by a continuous cumulative distribution function Ga There is an entry cost into entrepreneurship equal to b at each date and each entrepreneur hires one worker and does not work as a worker himself Total population is equal to 1 a Ignore the second period and characterize the equilibrium wage and determine which indi viduals become entrepreneurs Show that the equilibrium is unique b Now introduce the second period and suppose that all agents discount the future at the rate β Show that there are multiple equilibrium wages in the second period and as a result multiple equilibrium wages in the initial period c Suppose that a fraction ε of all agents die in the second period and are replaced by new agents New agents have to pay the entry cost into entrepreneurship if they want to become entrepreneurs Suppose that their talent distribution is also given by Ga Characterize the equilibrium in this case and show that it is unique d Consider the limiting equilibrium in part c with ε 0 Explain why this limit leads to a unique equilibrium while there are multiple equilibria at ε 0 2312 Consider an economy populated by λ rich agents who initially hold power and 1 λ poor agents who are excluded from power with λ 12 All agents are infinitely lived and discount the future at the rate β 0 1 Each rich agent has income θλ while each poor agent has income 1 θ1 λ where θ λ The political system determines a linear tax rate τ the proceeds of which are redistributed lumpsum Each agent can hide her money in an alternative nontaxable production technology and in the process she loses a fraction φ of her income There are no other costs of taxation The poor can undertake a revolution and if they do so in all future periods they obtain a fraction μt of the total income of the society ie an income of μt1 λ per poor agent The poor cannot revolt against democracy The rich receive zero payoff after a revolution At the beginning of every period the rich also decide whether to extend the franchise If the franchise is extended the poor decide the tax rate in all future periods a Define the MPE in this game b First suppose that μt μl at all times Also assume that 0 μl 1 θ Show that in the MPE there is no taxation when the rich are in power and the tax rate is τ φ when the poor are in power Show that along the equilibrium path there is no extension of the franchise and no taxation c Suppose that μl 1 θ 1 φ1 θ φ1 λ Characterize the MPE in this case Why is the restriction μl 1 φ1 θ φ1 λ necessary d Now consider the SPE of this game when μl 1 θ Construct an equilibrium where there is extension of the franchise along the equilibrium path Hint To simplify take β 1 and then consider a strategy profile where the rich are always expected to set τ 0 in the future Then show that in this case the poor would undertake a revolution Also explain why the continuation strategy of τ 0 by the rich in all future periods could be part of an SPE Why is there extension of the franchise now Can you construct a similar nonMarkovian equilibrium when μl 1 θ 860 Chapter 23 Political Institutions and Economic Growth e Explain why the MPE led to different predictions than the nonMarkovian equilibria Which one is more satisfactory f Now suppose that μt μl with probability 1 q and μt μh with probability q where μh 1 θ μl Construct an MPE where the rich extend the franchise and from then on a poor agent sets that tax rate Determine the parameter values that are necessary for such an equilibrium to exist Explain why an extension of the franchise is useful for rich agents g Now consider nonMarkovian equilibria again Suppose that the unique MPE results in franchise extension Can you construct an SPE equilibrium as β 1 where there is no franchise extension Epilogue Mechanics and Causes of Economic Growth I nstead of summarizing the models and ideas presented so far I end with a brief discussion of what we have learned from the models in this book and how they offer a useful perspective on world growth and crosscountry income differences I then provide a quick overview of some of the many remaining questions which are important to emphasize both as a measure of our ignorance and as potential topics for future research What Have We Learned Let us first summarize the most important aspects and takeaway lessons of our analysis Growth as the source of current income differences At an empirical level the investigation of economic growth is important not only for understanding the growth process but also because the analysis of the sources of crosscountry income differences today requires us to understand why some countries have grown rapidly over the past 200 years while others have not Chapter 1 The role of physical capital human capital and technology Crosscountry differences in economic performance and growth over time are related to physical capital human capital and technology Part of our analysis has focused on the contributions of these factors to production and growth Chapters 2 and 3 One conclusion that has emerged concerns the importance of technology in understanding both crosscountry and overtime differences in economic performance Here technology refers to advances in techniques of production advances in knowledge and the general efficiency of the organization of production Endogenous investment decisions While we can make empirical progress by taking cross country differences in physical and human capital as given we also need to endogenize these investment decisions to develop a more satisfactory understanding of the mechanics and the causes of income and growth differences across countries A large part of the book has focused on understanding physical and human capital accumulation Chapters 811 Investments in physical and human capital are forwardlooking and depend on the rewards that individuals expect from their investments Understanding differences in these investments is therefore intimately linked to understanding how reward structuresthat is the pecuniary 861 862 Epilogue Mechanics and Causes of Economic Growth and nonpecuniary rewards and incentives for different activitiesdiffer across societies and how individuals respond to differences in reward structures Endogenous technology I have also emphasized throughout that technology should be thought of as endogenous not as manna from heaven There are good empirical and theoretical reasons for thinking that new technologies are created by profitseeking individuals and firms through research development and tinkering In addition decisions to adopt new technologies are likely to be highly responsive to profit incentives Since technology appears to be a prime driver of economic growth over time and a major factor in crosscountry differences in economic performance we must understand how technology responds to factor endowments market structures and rewards Developing a conceptual framework that emphasizes the endogeneity of technology has been one of the major objectives of this book The modeling of endogenous technology necessitates ideas and tools that are somewhat different from those involved in the modeling of physical and human capital investments Three factors are particularly important First the fixed costs of creating new technologies combined with the nonrival nature of technology necessitates the use of models in which innovators have ex post after innovation monopoly power The same might apply though perhaps to a lesser degree to firms that adopt new technologies The presence of monopoly power changes the welfare properties of decentralized equilibria and creates a range of new interactions and externalities Chapters 12 and 13 and Section 215 in Chapter 21 Second the process of innovation is implicitly one of competition and creative destruction The modeling of endogenous technology necessitates more detailed models of the industrial organization of innovation These models shed light on the impact of market structure competition regulation and IPR protection on innovation and technology adoption Chapters 12 and 14 Third endogenous technology implies that not only the aggregate rate of technological change but also the types of technologies that are developed will be responsive to rewards Key factors influencing the types of technologies that societies develop are again reward structures and factor endowments For example changes in relative supplies of different factors are likely to affect which types of technologies will be developed and adopted Chapter 15 Linkages across societies and balanced growth at the world level While endogenous technology and endogenous growth are major ingredients in our thinking about the process of economic growth in general and the history of world economic growth in particular it is also important to recognize that most economies do not invent their own technologies but adopt them from the world technology frontier or adapt them from existing technologies Chapter 18 In fact the process of technology transfer across nations might be one of the reasons why after the initial phase of industrialization countries that have been part of the global economy have grown at broadly similar rates Chapter 1 Therefore the modeling of crosscountry income differences and the process of economic growth for a large part of the world requires a detailed analysis of technology diffusion and international economic linkages Two topics deserve special attention in this context The first is the contracting institutions supporting contracts between upstream and downstream firms between firms and workers and between firms and financial institutions These institutional arrangements affect the amount of investment the selection of entrepreneurs and firms and the efficiency with which different tasks are allocated across firms and workers There are marked differences in contracting institutions across societies and these differences appear to be a major factor influencing technology adoption and diffusion in the world economy Contracting institutions not only have a direct effect on technology and prosperity but they also shape the internal organization of firms which contributes to the efficiency of production and influences how innovative firms will be Section 185 in Chapter 18 The second is international trading relationships International trade not only generates static gains familiar to economists but also influences the innovation and growth process The international division of labor and the product cycle are examples of What Have We Learned 863 how international trading relationships help the process of technology diffusion and enhance the specialization of production Chapter 19 Takeoffs and failures The past 200 years of world economic growth stand in stark contrast to the thousands of years before Despite intermittent growth in some parts of the world during certain epochs the world economy was largely stagnant until the end of the eighteenth century This stagnation had multiple aspects These included low productivity high volatility in aggregate and individual outcomes a largely rural and agricultural economy and a Malthusian configuration in which increases in output were often accompanied by increases in population thus having only a limited effect on per capita income Another major aspect of stagnation has been the failed growth attempts many societies grew for certain periods of time and then lapsed back into depressions and stagnation This cycle changed at the end of the eighteenth century We owe our prosperity today to the takeoff in economic activity and especially in industrial activity that started in Britain and Western Europe and spread to certain other parts of the world most notably to Western European offshoots such as the United States and Canada The nations that are rich today are precisely those where this process of takeoff originated or those that were able to rapidly adopt and build on the technologies underlying this takeoff Chapter 1 A study of current income differences across countries requires understanding why some countries failed to take advantage of the new technologies and production opportunities Structural changes and transformations Modern economic growth and development are accompanied by a set of sweeping structural changes and transformations These include changes in the composition of production and consumption the shift from agriculture to industry and from industry to services urbanization financial development changes in inequality of income and inequality of opportunity the transformation of social and living arrangements changes in the internal organization of firms and the demographic transition While the process of economic development is multifaceted much of its essence lies in the structural transformation of the economy and the society at large Section 176 in Chapter 17 and Chapters 20 and 21 Many of these transformations are interesting to study for their own sake They are also important ingredients for sustained growth Lack of structural transformation is not only a symptom of stagnation but is also often one of its causes Societies may fail to take off and benefit from the available technology and investment opportunities partly because they have not managed to undergo the requisite structural transformations and thus lack the type of financial relations the appropriate skills or the types of firms that are conducive to the adoption of new technologies Policy institutions and political economy The reward structures faced by firms and individ uals play a central role in shaping whether they undertake the investments in new technology and in human capital necessary for takeoff industrialization and economic growth These reward structures are determined by policies and institutions Policies and institutions also directly affect whether a society can embark on modern economic growth for a variety of interrelated reasons Chapter 4 First they directly determine the societys reward structure thus shaping whether investments in physical and human capital and technological innovations are profitable Second they determine whether the infrastructure and contracting arrangements necessary for modern economic relations are present For example modern economic growth would be impossible in the absence of some degree of contract enforcement the maintenance of law and order and at least a minimum amount of investment in public infrastructure Third they influence and regulate the market structure thus determining whether the forces of creative destruction are operational so that new and more efficient firms can replace less efficient incum bents Finally institutions and policies may sometimes or perhaps often block the adoption and use of new technologies to protect politically powerful incumbent producers or stabilize the established political regime Thus to understand the process of modern economic growth we need to study the institutional and policy choices that societies make We then need to 864 Epilogue Mechanics and Causes of Economic Growth investigate the political economy of growth paying special attention to which individuals and groups will be the winners from economic growth and which the losers When losers cannot be compensated and have sufficient political power we may expect the political economy equi librium to lead to policies and institutions that are not growth enhancing The basic analysis of the political economy of growth generates insights about what types of distortionary policies may block growth when these distortionary policies will be adopted and how technology market structure and factor endowments interact with the incentives of the social groups in power to encourage or discourage economic growth Chapter 22 Endogenous political institutions Policies and institutions are central to understanding the growth process over time and crosscountry differences in economic performance These social choices are in turn determined in the context of a societys political institutions Democracies and dictatorships are likely to make different policy choices and create distinct types of reward structures But political institutions themselves are not exogenous They can change along the equilibrium path as a result of their own dynamics and of stimuli coming from changes in technology trading opportunities and factor endowments Chapter 23 For a more complete understanding of world economic growth and the income differences today we therefore need to study 1 how political institutions affect policies and economic institutions thus shaping incentives for firms and workers 2 how political institutions themselves change especially when interacting with economic outcomes and technology and 3 why political institutions and the associated economic institutions did not lead to sustained economic growth throughout history why they enabled economic takeoff 200 years ago and why in some countries they blocked the adoption and use of superior technologies and derailed the process of economic growth In this summary I have focused on the ideas most relevant for examining the process of world economic growth and crosscountry income differences we observe today The focus in the book has been not only on ideas but also on careful mathematical modeling of these ideas to develop coherent and rigorous theoretical approaches I do not repeat here the theoretical foundations of these ideas which range from basic consumer producer and general equilib rium theory to dynamic models of accumulation models of monopolistic competition models of world equilibria and dynamic models of political economy But I emphasize again that a thorough study of the theoretical foundations of these ideas is necessary both to develop a sat isfactory understanding of the main issues and to find the best way of making them empirically operational A Possible Perspective on Growth and Stagnation over the Past 200 Years The previous section summarized the most important ideas highlighted in this book I now discuss how some of these ideas might be useful in shedding light on the process of world economic growth and crosscountry divergence that have motivated our investigation from the start The central questions are 1 Why did the world economy not experience sustained growth before 1800 2 Why did economic takeoff start around 1800 and in Western Europe 3 Why did some societies manage to benefit from the new technologies and organizational forms that emerged starting in 1800 while others steadfastly refused or failed to do so A Possible Perspective on Growth and Stagnation over the Past 200 Years 865 I now offer a narrative that provides some tentative answers to these three questions While certain parts of the mechanisms I propose here have been investigated econometrically and other parts are supported by historical evidence the reader should view this narrative as a first attempt at providing coherent answers to these central questions Two aspects of these answers are noteworthy First they build on the theoretical insights that the models presented so far generate Second in the spirit of the discussion in Chapter 4 they link the proximate causes of economic phenomena to fundamental causes and in particular to institutions And here I take a shortcut Although I emphasized in Chapter 23 that there are no perfect political institutions and that each set of different political arrangements is likely to favor some groups at the ex pense of others I simplify the discussion in this part by making a core distinction between two sets of institutional arrangements one less conducive to growth than the other one The first which I refer to as authoritarian political systems encompasses absolutist monarchies dicta torships autocracies and various types of oligarchies that concentrate power in the hands of a small minority and pursue economic policies that are favorable to the interests of this minority Authoritarian systems often rely on some amount of repression because they seek to maintain an unequal distribution of political power and economic benefits They also adopt economic institutions and policies that protect incumbents and create rents for those who hold political power The second set of institutions are participatory regimesThese regimes place constraints on rulers and politicians thus preventing the absolutist tendencies in political systems and give voice to new economic interests so that a strict decoupling between political and economic power is avoided Such regimes include constitutional monarchies where broader sections of the society take part in economic and political decision making and democracies where political participation is greater than in nondemocratic regimes The distinguishing feature of participatory regimes is that they provide voice and economic and political security to a broader cross section of society than do authoritarian regimes As a result they are more open to entry by new businesses and provide a more level playing field and better security of prop erty rights to a relatively broad section of the society Thus in some ways the contrast between authoritarian political systems and participatory regimes is related to the contrast between the growthpromoting cluster of institutions and the growthblocking extractive institutions em phasized and illustrated in Chapter 4 The reader should note that many different terms could have been used instead of authoritarian and participatory and some details of the dis tinction may be arbitrary More importantly it should be borne in mind that even the most participatory regime involves an unequal distribution of political power and those who have more political power can use the fiscal and political instruments of the state for their own benefits and for the detriment of the society at large Why this type of behavior is sometimes successfully curtailed or limited is a question at the forefront of current research and I do not dwell on it here Why Did the World Not Experience Sustained Growth before 1800 While sustained growth is a recent phenomenon growth and improvements in living standards certainly have occurred many times in the past Human history is also full of major techno logical breakthroughs Even before the Neolithic Revolution many technological innovations increased the productivity of huntergatherers The transition to farming after about 9000 bc is perhaps the most significant technological revolution of all times it led to increased agri cultural productivity and the development of socially and politically more complex societies Archaeologists have also documented various instances of economic growth in premodern periods Historians estimate that consumption per capita doubled during the great flowering of 866 Epilogue Mechanics and Causes of Economic Growth ancient Greek society from 800 bc to 50 bc Morris 2004 Similar improvements in living standards were experienced by the Roman republic and empire after 400 bc Hopkins 1980 and also appear to have been experienced by preColumbian civilizations in South America es pecially by the Olmec the Maya the Aztec and even perhaps the Inca Webster 2002 Mann 2004 Although data on these ancient growth experiences are limited the available evidence suggests that the basic neoclassical model in which growth relies mostly on physical capital accumulation provides a good description of the developments in these ancient economies see eg Morris 2004 However these growth experiences were qualitatively different from those that the world experienced after its economic takeoff starting in the late eighteenth and early nineteenth centuries Four factors appear to have been particularly important and set these growth episodes apart from modern economic growth First earlier episodes were relatively shortlived or took place at a relatively slow pace1 In most cases the initial spurt of growth soon crumbled for one reason or another somehow reminiscent of the failed takeoffs in the model of Section 176 in Chapter 17 Secondly and relatedly growth was never based on continuous technological innovations thus it never resembled the technologybased growth described in Chapters 1315 Third in most cases economic institutions that would be necessary to support sustained growth did not develop Financial relations were generally primitive contracting institutions remained informal markets were heavily regulated with various internal tariffs and incomes and savings did not reach the levels necessary for the mass market and simultaneous investments in a range of activities to become profitable Put differently the structural transformations accompanying development discussed in Chapter 21 did not take place Fourth and arguably most important and the cause of the first three all these episodes took place within the context of authoritarian political regimes They were not broadbased growth experiences Instead this was elitedriven growth for the benefit of the elite that largely exploited existing comparative advantages Thus it is not surprising that the improvements in living standards did not affect the entire society but only a minority Why did these growth episodes not turn into a process of takeoff ultimately leading to sustained growth My main answer is related to that offered in Section 233 in Chapter 23 Growth under authoritarian regimes is possible Entrepreneurs and workers can become better at what they do achieve a better division of labor and improve the technologies they work with by tinkering and learning by doing Moreover those with political power and their allies do have the necessary security of property rights to undertake investments And some technological breakthroughs can happen by chance Nevertheless a distinguishing feature of growth under authoritarian institutions is that it protects the interests of the current elite So in the final analysis growth must always rely on existing techniques and production relationships It will not unleash the process of creative destruction and the entry of new talent and new businesses necessary to carry a nation to the state of sustained growth In addition technological constraints may have also played a role For example the relatively rapid growth in the nineteenth century required skilled workers and before the printing press was invented it would have been prohibitively costly for a critical mass of workers to acquire the necessary skills Although the progress of technological knowledge is not monotonic and useful production techniques are sometimes forgotten the technological knowhow available to potential entrepreneurs at the end of the eighteenth century was undoubtedly greater than that available to potential entrepreneurs in Rome or ancient Greece 1 For example in ancient Greece Morris 2004 estimates that income per capita doubled or at most tripled in the 500 years between 800 bc and 300 bc and this was largely caused by catchup growth starting from unusually low levels in 800 bc A Possible Perspective on Growth and Stagnation over the Past 200 Years 867 Let me next elaborate on the aspects of political economy that appear to be critical and pro vide a few examples to illustrate the limits to growth under authoritarian regimes The available evidence shows that the Chinese empire was technologically innovative during many distinct phases of its history Productivity in the Chinese economy especially in the Yangtze Delta and other fertile lands was high enough to support a high density of population But the Chinese economy never came close to sustained growth Authoritarian political institutions have regu lated economic activity tightly for most of Chinese history The society was hierarchical with a clear distinction between the elite and the masses This system did not allow free entry into business by new entrepreneurs who would adopt and exploit new technologies and unleash the powers of creative destruction When prospects for economic growth conflicted with political stability the elite opted for maintaining stability even if this came at the expense of potential economic growth Thus China tightly controlled overseas and internal trade did not develop the broadbased property rights and contracting institutions necessary for modern economic growth and did not allow an autonomous middle class to emerge as an economic and political force Elvin 1973 Mokyr 1990 Wong 1997 The ancient Greek and Roman civilizations are often viewed as the first democratic societies One might therefore be tempted to count them as participatory regimes that should have achieved sustained economic growth But this is not necessarily the case First as noted above participatory regimes do not guarantee sustained economic growth when other preconditions have not been met But more importantly these societies were democratic only in comparison to others at the time Both societies were representative only for a small fraction of the population Production relied on slavery and coercion Moreover despite certain democratic practices there was a clear distinction between a small elite which monopolized economic and political power and the masses which consisted of both free plebs and slaves Economic growth in both ancient Greece and Rome did not rely on continuous innovation Both societies managed to achieve high levels of productivity in agriculture but without changing the organization of production in a radical manner Both societies benefited from their military superiority for a while and challenges to their military power were also important factors in their decline The Ottoman Empire provides another example of a society that was successful for an extended period of time but without ever transitioning to sustained growth The Ottoman Empire especially during the fourteenth fifteenth and sixteenth centuries achieved relative prosperity and military strength Agricultural productivity was high in many parts of the empire and military tribute contributed to state coffers and generated revenues to be distributed to parts of the population But the state elite who controlled decision making within the empire never encouraged broadbased economic growth There was no private property in land trade was permitted as long as it was consistent with the states objectives but was always tightly controlled and any new technology that could destabilize the power of the state was blocked Like China Greece and Rome the Ottoman growth first tapered off and then turned into decline Pamuk 2004 The final example I mention is the Spanish monarchy By the beginning of the sixteenth century the Spanish crown had achieved both political dominance over its own lands under Ferdinand and Isabella and control of a large overseas empire through its colonial enterprises Many parts of greater Spain including the lands of Aragon and the south that had been recently reconquered from the Moors were already prosperous in the fifteenth century The whole of Spain became much wealthier with the transfer of gold silver and other resources from the colonies in the sixteenth century But this wealth did not translate into sustained growth The colonial experiment was managed under a highly authoritarian regime set up by Ferdinand and Isabella and the most lucrative businesses were allocated to the allies of the crown The greater revenues generated from the colonies only helped to tighten the grip of the crown on the rest of the society and the economy Instead of abating absolutism increased Trade and 868 Epilogue Mechanics and Causes of Economic Growth industry remained highly regulated and groups not directly allied to the crown were viewed suspiciously and discriminated against The most extreme example of this the persecution of Jews that had started under the Inquisition continued and spilled over to other independent merchants Subsequent to the transfer of wealth from the colonies Spain experienced a very lengthy period of stagnation with economic and political decline Elliott 1963 It is also remarkable that in none of these cases did complementary economic institutions develop Financial institutions remained rudimentary The Roman Republic developed a pre cursor to the modern corporation and allowed some contracts between free citizens but by and large economic prosperity was built on traditional economic activities that did not neces sitate complex relationships among producers and between firms and workers Consequently the structural transformations that accompany economic growth never took place in these so cieties Life was largely rural and social relations were dominated by the state and community enforcement Perhaps more important there was little investment in human capital except for the elite for whom education was seldom a means to higher productivity Without broadbased human capital and political rights creative destruction becomes even more difficult as a large fraction of the population is barred from entrepreneurial activities All of the cases discussed here confirm this expectation Overall these cases illustrate that societies that encourage increases in the productivity of the elite in traditional activities can secure growth for a while But they are unlikely to engender creative destruction Growth goes handinhand with the political domination of the elite and thus with entry barriers protecting the status and the power of the elite In this light the answer to the question of why not before 1800 is twofold First no society before 1800 invested in human capital allowed new firms to bring new technology and generally unleashed the powers of creative destruction This failure might have been partly due to the difficulty of undertaking broadbased human capital investments in societies without the printing press and with only limited communication technologies But it was also related to the reward structures for and constraints on workers and firms An important consequence of this pattern of growth is that no society experienced the sweeping structural transformations that are an essential part of modern economic growth Chapter 21 Second no society took steps toward sustained growth because all these societies lived under authoritarian political regimes Why Did Economic Takeoff Start around 1800 and in Western Europe The division of labor emphasized by Adam Smith and capital accumulation always present growth opportunities to societies Furthermore human ingenuity is strong enough to create room for major technological breakthroughs in almost any environment Thus there is always a growth impetus in human societies Jones 1988 Nevertheless this impetus may only be latent because it exists in the context of a set of political and economic institutions When these institutions do not encourage growthwhen they do not provide the right kind of reward structure and so punish rather than reward innovationswe do not expect the growth impetus to lead to sustained growth Even in such environments economic growth is possible and this is why China Greece Rome and other empires experienced growth for part of their history But this prosperity did not exploit the full growth impetus instead it took place in the context of political regimes that by their nature had to control the growth impetus because this impetus would ultimately bring these regimes down West European growth starting in the late eighteenth century was different because Western Europe underwent three important structural transformations starting in the late Middle Ages These structural transformations created an environment in which the latent growth impetus could turn into an engine of sustained growth A Possible Perspective on Growth and Stagnation over the Past 200 Years 869 The first was the collapse of one of the pillars of the ancient regime the decline of feudal relations in Western Europe Starting in the thirteenth century and especially after the Black Death during the midfourteenth century feudal economic relations crumbled in many parts of Western Europe Serfs were freed from their feudal dues either by default because the relationship collapsed or by fleeing to the expanding city centers Postan 1966 This emancipation heralded the beginning of an important social transformation urbanization and changes in social relations But perhaps more importantly it created a labor force ready to work at cheap wages in industrial and commercial activities It also removed one of the greatest sources of conflict between existing elites and new entrepreneurscompetition in the labor market Chapter 22 The decline of the feudal order further weakened the power base of the European authoritarian regimes Pirenne 1937 The second structural transformation was related With the decline in population in the four teenth century real incomes increased in much of Europe and many cities created sufficiently large markets for merchants to seek new imports and for industrialists to seek new products During the Middle Ages a range of important technologies in metallurgy armaments agricul ture and basic industry eg textiles were already perfected White 1964 Mokyr 1990 Thus the European economy had reached the technological maturity to act as a platform for entrepreneurial activity in a range of areas and income levels were sufficient to encourage investment in physical capital and technology to spearhead new production relations The third and most important change was political The late Middle Ages also witnessed the start of a political process that inexorably led to the collapse of absolutist monarchies and to the rise of constitutional regimes The constitutional regimes that emerged in the sixteenth and sev enteenth centuries in Western Europe were the first examples of participatory regimes because they shifted political power to a large group of individuals that were previously outsiders to po litical power This group included the gentry small merchants protoindustrialists as well as overseas traders and financiers These regimes then provided secure property rights and growth enhancing institutions for a broad cross section of society These institutional changes created the requisite environment for new investments and technological changes and the beginning of sustained growth which would culminate in the Commercial Revolution in the Netherlands and Britain during the seventeenth century and in the British Industrial Revolution at the end of the eighteenth century By the nineteenth century industry and commerce had spread to much of Western Europe see Chapter 4 North and Thomas 1973 It is noteworthy that constitutional monarchies were not democracies as we understand them today There was no oneperson onevote principle and the distinction between the rich and the poor was quite palpable Nevertheless these regimes emerged as responses to the demands by the merchants and industrialists More importantly these constitutional regimes not only reformed the political institutions of Western Europe but undertook a series of economic reforms facilitating modern capitalist growth Internal tariffs and regulations were lifted Entry into domestic businesses and foreign trade was greatly facilitated For example the process of financial development in Britain began with the founding of the Bank of England and other financial reforms These constitutional regimes which emerged first in Britain and the Netherlands and then in France and other parts of Western Europe paved the way for sustained economic growth based on property rights for a broad cross section of society contract enforcement the rule of law and free entry into existing and new business lines According to the theoretical perspective developed in earlier chapters these improved conditions should have led to greater investments in physical capital human capital and technology This is indeed what happened and the process of modern economic growth was launched Economic relations now relied on new businesses investing in industry commerce and the formation of complex organizational forms and production relations Growth did not immediately accelerate Economic growth 870 Epilogue Mechanics and Causes of Economic Growth was present but modest during the seventeenth and eighteenth centuries Maddison 2001 But these institutional changes laid the foundations for the more rapid growth that was soon to come Financial institutions developed the urban areas expanded further new technologies were invented and markets became the primary arena for transactions and competition North and Thomas 1973 By 1800 the process of technological change and investment had progressed so much as to be dubbed the Industrial Revolution Ashton 1969 Mokyr 1993 The first phase of the Industrial Revolution was followed by the production of yet newer technologies more complex organizations greater reliance on skills and human capital in the production process and increasing globalization of the world economy By the second half of the nineteenth century Western Europe had reached unprecedented growth levels Naturally a complete answer to the question in the title of this subsection requires an explanation for why the constitutional regimes that were so important for modern economic growth emerged in Western Europe starting in the late sixteenth century and seventeenth century These institutions had their roots in the late medieval aristocratic parliaments in Europe but more importantly they were the outcome of radical reform resulting from the change in the political balance of power in Europe starting in the sixteenth century Ertman 1997 The sixteenth century witnessed a major economic transformation of Europe following the increase in international trade due to the discovery of the New World and the rounding of the Cape of Good Hope Davis 1973 Acemoglu Johnson and Robinson 2005a Together with increased overseas trade came greater commercial activity within Europe These changes led to a modest increase in living standards and more importantly to greater economic and political power for a new group of merchants traders and industrialists These new men were not the traditional allies of the European monarchies They therefore demanded and often were powerful enough to obtain changes in political institutions that provided them with greater security of property rights and government action to help them in their economic endeavors By this time with the collapse of the feudal order the foundations of the authoritarian regimes that were in place in the Middle Ages were already weak Nevertheless the changes leading to the constitutional regimes did not come easy The Dutch had to fight the Hapsburg monarchy to gain their independence as a republic Britain had to endure its civil war and the Glorious Revolution France had to go through the Revolution of 1789 But in all cases the ancien regime gave way to more representative institutions with greater checks on absolute power and greater participation by merchants industrialists and entrepreneurs It was important that the social changes led to a new set of political institutions and not simply to concessions This distinction is related to the theoretical ideas emphasized in Section 233 of Chapter 23 the nascent groups demanded longterm guarantees for the protection of their property rights and their participation in economic life Such guarantees were most easily delivered by changes in political institutions not by shortterm concessions These changes created the set of political institutions that would then enable the emergence of the economic institutions mentioned above The collapse of the authoritarian political regimes and the rise of the first participatory regimes then opened the way for modern economic growth Why Did Some Societies Manage to Benefit from New Technologies While Others Failed to Do So The economic takeoff started in Western Europe but quickly spread to certain other parts of the world The chief importer of economic institutions and economic growth was the United States The United States founded by settler colonists who had just defeated the British A Possible Perspective on Growth and Stagnation over the Past 200 Years 871 crown to gain their independence and set up a smallholder society already had participatory political institutions This was a society built by the people who would live in it and they were particularly willing to create checks and balances to prevent the subsequent emergence of a strong political or economic elite This environment turned out to be a perfect conduit for modern economic growth The lack of a strong political and economic elite meant that a broad cross section of society could take part in economic activity import technologies from Western Europe and then build their own technologies to quickly become the major industrial power in the world Galenson 1996 Engerman and Sokoloff 1997 Keyssar 2000 Acemoglu Johnson and Robinson 2002 In the context of this example the importance of technology adoption from the world technology frontier is in line with the emphasis in Chapter 18 while the growthpromoting effects of a lack of elite creating entry barriers is consistent with the approach in Section 233 in Chapter 23 Similar processes took place in other West European offshoots for example in Canada Yet in other parts of the world adoption of new technologies and the process of economic growth came as part of a movement toward defensive modernization Japan started its economic and political modernization with the Meiji restoration or perhaps even before and a central element of this modernization effort was the importation of new technologies However these attitudes to new technologies were by no means universal New technologies were not adopted but resisted in many parts of the world This included most of Eastern Europefor example Russia and AustriaHungarywhere the existing landbased elites saw new technologies as a threat both to their economic interests because they would lead to the end of the feudal relations that still continued in this part of Europe and to their political interests which relied on limiting the power of new merchants and slowing down the process of peasants migrating to cities to become the new working class see Freudenberger 1967 and Mosse 1992 for evidence and Chapter 22 for a theoretical perspective Similarly the previously prosperous plantation economies in the Caribbean had no interest in introducing new technologies and allowing free entry by entrepreneurs These societies continued to rely on their agricultural staples Industrialization competition in free labor markets and workers investing in their human capital were seen as potential threats to the economic and political powers of the elite The newly independent nations in Latin America were also dominated by a political elite which continued the tradition of the colonial elite and showed little interest in industrialization Much of Southeast Asia the Indian subcontinent and almost all of subSaharan Africa were still West European colonies and were governed under authoritarian and repressive regimes often as producers of raw materials for the rapidly industrializing Western European nations or as sources of tribute Free labor markets factor mobility creative destruction and new technologies did not feature in the colonial political trajectories of these countries Chapter 4 Thus the nineteenth century was only to see the industrialization of a few select places By the twentieth century however more and more nations started importing the technologies that had been developed and used in Western Europe This process of technology transfer pulled the countries integrated into in the global economy toward higher income levels Chapter 19 But this growth episode did not benefit all countries Many had to wait for their independence from their colonial masters and even then the end of colonialism led to a period of instability and infighting among wouldbe elites Once some degree of political stability was achieved and economic institutions that encourage growth were put in place growth started For ex ample growth in Australia and New Zealand was followed by that in Hong Kong by that in South Korea then by the rest of Southeast Asia and finally by India In each of these cases as emphasized in Chapters 20 and 21 growth went handinhand with structural transformations Once the structural transformations were under way they facilitated further growth Consistent with the picture in Chapters 18 and 19 societies integrated into the global economy started 872 Epilogue Mechanics and Causes of Economic Growth importing technologies and achieved growth rates in line with the growth of the world technol ogy frontier and often exceeding those during their initial phase of catchup In most cases this process meant growth for the new members of the global economy but not necessarily the disappearance of the income gap between these new members and the earlier industrializers Meanwhile many parts of the world continued to suffer political instability that discouraged investment in capital and new technology or even exhibited overt hostility to new technologies These included parts of subSaharan Africa and until recently much of Central America Returning to some of the examples discussed in Chapter 1 Nigeria and Guatemala failed to create incentives for their entrepreneurs or workers both during their colonial periods and after independence Both these countries also experienced significant political instability and economically disastrous civil wars in the postwar era Brazil managed to achieve some degree of growth but it was mostly based on investment by large heavily protected corporations and not on a sustained process of technological change and creative destruction thus it was more similar to the oligarchic growth in terms of the model of Section 233 in Chapter 23 In these and other cases policies that failed to provide secure property rights to new entrepreneurs and those that blocked the adoption of new technologiesas well as political instability and infighting among the elitesseem to have played an important role in the failure to join the world economy and its growth process Overall these areas fell behind the world average in the nineteenth century and continued to trail for most of the twentieth century Many nations in sub Saharan Africa such as Congo Sudan and Zimbabwe are still experiencing political turmoil and fail to offer even the most basic rights to their entrepreneurs and citizens Consequently many are falling still further behind the world average Many Remaining Questions The previous section provided a narrative emphasizing how technological changes transformed the world economy starting in the eighteenth century and how certain societies took advantage of these changes while others failed to do so Parts of the story receive support from the data The importance of industrialization to the initial takeoff is now well documented There is a broad consensus that economic institutions protecting property rights and allowing for free entry and introduction of new technologies were important in the nineteenth century and continue to be important today in securing economic growth There is also a general consensus that political instability weak property rights and lack of infrastructure are major impediments to growth in subSaharan Africa Nevertheless the narrative here is speculative These factors might be important but they may not be the main explaination of the evolution of the world income distribution over the past 200 years And as yet there is no consensus on the role of political institutions in this process Thus what I have presented here should be taken for what it is a speculative answer that needs to be further investigated My purpose in outlining it was not only that I suspect this answer has much truth to it but also to show how the various models developed in this book can help us better frame answers to fundamental questions of economic growth and of economics and social sciences in general I should add that further investigation of the causes of the worlds takeoff into sustained growth and the failure of some nations to take advantage of this process is only one of the many remaining challenges The political economy of growth is important because it enables us to ask and answer questions about the fundamental causes of economic growth But many other aspects of the process of growth require further study In some sense the field of economic growth is one of the more mature areas in economics and certainly within macroeconomics it is the area where there is broadest agreement on what types Many Remaining Questions 873 of models are useful for the study of economic dynamics and for empirical analysis And yet there is so much that we still do not know I now end by mentioning a few areas with great potential for further theoretical and empirical advances First although here I have largely focused on factors facilitating or preventing the adoption of technologies in lessdeveloped nations there is still much to be done to understand the pace of technological progress in frontier economies Our models of endogenous technological change give us the basic framework for thinking about how profit incentives shape investments in new technologies But we still know relatively little about the industrial organization of innovation for example on how market structure affects economic growth Chapters 12 and 14 highlighted how different market structures may create different incentives for technological change But most of our understanding of these issues is qualitative For example in the context of the economics of innovation we lack a frameworksimilar to that used for the analysis of the effects of capital and labor income taxes and indirect taxes in public financewhich could be used to analyze the effects of various regulations IPR policies and anticompetitive laws on innovation and economic growth Since the pace at which the world technology frontier progresses has a direct effect on the growth of many nations even small improvements in the environment for innovation in advanced economies could have important dividends for the rest of the world In addition to the industrial organization of innovation the contractual structure of inno vation needs further study We live in a complex society in which most firms are linked to others as suppliers or downstream customers and most firms are connected to the rest of the economy indirectly through their relationship with financial markets These relationships are mediated by various explicit and implicit contracts Similarly the employment relationship that underlies the productivity of most firms relies on contractual relations between employers and employees We know that moral hazard and holdup problems occur in these contractual relationships But how important are they for the process of economic growth Can improve ments in contracting institutions improve innovation and technological upgrading in frontier economies Can they also facilitate technology transfer These are basic but as yet un answered questions The contractual foundations of economic growth are still in their infancy and require much work The previous section emphasized how several economies started the growth process by importing technologies and thus integrating into the global economy Today we live in an increasingly globalized and globalizing economy But there is still much to understand about how technology is transferred from some firms to others and from advanced economies to less developed ones The models I presented in Chapter 19 emphasized the importance of human capital barriers to technology adoption issues of appropriate technology and contracting problems Nevertheless most of the models are still at the qualitative level and we lack a framework that can make quantitative predictions about the pace of technology diffusion We have also not yet incorporated many important notions related to technology transfer into our basic frameworks These include among others ideas related to tacit knowledge appropriate technology the workings of the international division of labor the role of international IPR protection and the interaction between trade and technology diffusion The reader will have also noticed that the material presented in Chapter 21 is much less unified and perhaps more speculative than that in the rest of the book Although some of this reflects the fact that I had to simplify a variety of models to be able to present them in a limited space much of it is because we are far from a satisfactory framework for understanding the process of economic development and the structural transformations that it involves Some aspects of these structural transformations such as the increased importance of manufacturing and then services relative to agriculture can be viewed as a byproduct of economic growth But other aspects of this process including financial development changes in contract enforcement 874 Epilogue Mechanics and Causes of Economic Growth regimes urbanization and the amount and composition of human capital investments may be facilitators or even preconditions for economic growth and development Thus the lack of significant structural transformation might be an important factor in delaying or preventing economic growth To understand these questions we require models with stronger theoretical foundations a systematic approach to these related issues and a greater effort to link the models of economic development to the wealth of empirical evidence that the profession has now accumulated on economic behavior in lessdeveloped economies Last but not least given the narrative in the last section and the discussion in Chapters 4 22 and 23 it comes as no surprise that I think many important insights about economic growth lie in political economy But understanding politics is in many ways harder than understanding economics because political relations are even more multifaceted Although I believe that the political economy and growth literatures have made important advances in this area over the past decade or so much remains to be done The political economy of growth is in its infancy and as we further investigate why societies make different collective choices we will gain a better understanding of the process of economic growth A Odds and Ends in Real Analysis and Applications to Optimization T his appendix reviews basic material from real analysis Its main purpose is to make the book selfcontained and to include explicit statements of some of the theorems that are used in the text The material here is not meant to be a comprehensive treatment of real analysis Accordingly many results are stated without proof and other important results are omitted as long as they are not referred to in the text and are not necessary for the results presented here I state some useful results as Facts often leaving their proofs as exercises These results are typically used or referred to in the text The more important results are stated as Theorems The material here is not a substitute for a basic Mathematics for Economists review or textbook An excellent book of this sort is Simon and Blume 1994 and I presume that the reader is familiar with most of the material in this or a similar book In particular I assume that the reader is comfortable with linear algebra functions relations set theoretic language multivariate calculus and basic proof techniques To gain a deeper understanding and appreciation of the material here the reader is encour aged to consult one of many excellent books on real analysis functional analysis and general topology Some of the material here is simply a review of introductory real analysis more or less at the level of the classic books by Apostol 1975 or Rudin 1976 Some of the mate rial particularly that concerning topology and infinitedimensional analysis is more advanced and can be found in Kelley 1955 Kolmogorov and Fomin 1970 Conway 1990 Royden 1994 and Aliprantis and Border 1999 Excellent references for applications of these ideas to optimization problems include Berge 1963 and Luenberger 1969 A recent treatment of some of these topics with economic applications is presented in Ok 2007 877 886 Appendix A Odds and Ends in Real Analysis and Applications to Optimization a topology τp and ℓ τp is a topological space which is sometimes denoted by the same symbol as the corresponding metric space ℓp As suggested by this example many topological spaces of interest are derived from a metric space In this case we say that they are metrizable and for all practical purposes we can treat metrizable topological spaces as metric spaces DefinitionA17 A topological space X τ is metrizable if there exists a metric d on X such that when V τ then V is also open in the metric space X d according to Definition A3 Fact A11 If a topological space X τ is metrizable with some metric d then it defines the same notions of convergence and continuity as the metric space X d Proof This follows immediately from the fact that X τ and X d have the same open sets Not all general topological spaces have the nice properties of metric spaces Fortunately this is not an issue for the properties of topological spaces that are related to continuity and compactness which we focus on here Nevertheless it is useful to note that a particularly relevant property of general topological spaces is the Hausdorff property which requires that any distinct points x and y of a topological space X τ should be separated that is in a topological space with the Hausdorff property there exist Vx Vy τ such that x Vx y Vy and Vx Vy It is clear that every metric space has the Hausdorff property see Exercise A14 For our purposes the Hausdorff property is not necessary Returning to general topological spaces the notions of convergence of sequences subse quences nets and subnets can be stated for general topological spaces Here I only give the definitions for convergence of sequences and nets those for subsequences and subnets are defined similarly Definition A18 Let X τ be a topological space A sequence xn n1 a net xααA in X is convergent and has limit point x X if for each V τ with x V there exists N N there exists some α A such that xn V for all n N xα V for all α α We write this as limn xn lim xn x or as xn n1 x Continuity is defined in a similar manner Definition A19 Let X τX and Y τY be topological spaces and consider the mapping φ X Y φ is continuous at x X if for every U τY with φx U there exists V τX with x V such that φV U φ is continuous on X if it is continuous at each x X The parallel between this definition and the equivalent characterization of continuity in metric spaces in Definition A14 is evident This leads to the following theorem Theorem A4 Open Sets and Continuity II Let X τX and Y τY be topological spaces and consider the mapping φ X Y φ is continuous if and only if for every Y Y that is open in Y φ1Y is open in X The proof of this theorem is identical to that of Theorem A2 and is thus omitted In general topological spaces convergence in terms of sequences is not sufficient to char acterize continuity but convergence in terms of nets is Theorem A5 Continuity and Convergence of Nets Let X τX and Y τY be topo logical spaces The mapping φ X Y is continuous at x X if and only if φxααA φx for any net xααA x A6 Correspondences and Berges Maximum Theorem 895 Finally correspondences are also useful in expressing the properties of maximizers in Berges Maximum Theorem Theorem A16 As with functions for a correspondence F X Y I use the notation FX to de note the image of the set X under the correspondence F so that FX is defined as FX y Y x X with y Fx DefinitionA31 Let X dX and Y dY be metric spaces and consider the correspondence F X Y Let Nεx refer to neighborhoods in X dX Then 1 F is upper hemicontinuous at x X if for every open subset Y of Y with Fx Y there exists ε 0 such that FNεx Y F is upper hemicontinuous on X if it is upper hemicontinuous at each x X 2 F is lower hemicontinuous at x X if for every open subset Y of Y for which Fx Y there exists ε 0 such that Fx Y for all x Nεx F is lower hemicontinuous on X if it is lower hemicontinuous at each x X and 3 F is continuous at x X if and only if it is both upper and lower hemicontinuous at x X F is continuous on X if and only if it is continuous at each x X These notions are slightly easier to understand if we specialize them to Euclidean spaces First we say that a correspondence F X Y is closedvalued compactvalued if Fx is closed compact in Y for each x For Euclidean spaces the following definition is equivalent to Definition A31 and more generally it implies Definition A31 see Exercise A18 and Fact A18 Definition A32 Let X RKX and Y RKY where KX KY N and consider a compact valued correspondence F X Y Then 1 F is upper hemicontinuous at x X if for every sequence xn n1 x and every sequence yn ynk of ynn1 with yn Fxn for each n there exists a convergent subsequence n1 such that ynk y Fx and 2 F is lower hemicontinuous at x X if Fx is nonemptyvalued and for every y Fx and every sequence xn n1 x there exists some N N and a sequence yn n1 with yn Fxn for all n N and yn n1 y Figure A1 illustrates these notions diagrammatically In this figure the correspondence Fx is upper and lower hemicontinuous and thus continuous at x1 it is upper hemicontinuous but not lower hemicontinuous at x2 and it is lower hemicontinuous but not upper hemicontin uous at x3 Upper hemicontinuity and lower hemicontinuity according to Definition A32 imply the corresponding concepts in Definition A31 for general metric spaces Fact A18 Let X dX and Y dY be metric spaces and consider the correspondence F X Y If F is upper hemicontinuous lower hemicontinuous at x X according to Definition A32 then it is upper hemicontinuous lower hemicontinuous at x X according to Definition A31 Proof Suppose to obtain a contradiction that part 1 of Definition A32 holds at x but F is not upper hemicontinuous at x Then there exists an open set Y Y such that Fx Y but for any ε 0 FNεx is not a subset of Y Then for any ε 0 there exists xε Nεx and yε Fxε such that yε Y Construct the sequence xn yn n1 such that each xn yn satisfies this property for ε 1n Clearly xn n1 x Therefore by hypothesis there exists a convergent subsequence ynk y Fx Since Y is open Y Y is closed and since 896 Appendix A Odds and Ends in Real Analysis and Applications to Optimization 0 x1 x2 x3 x Fx FIGURE A1 Upper and lower hemicontinuity ynk Y Y for each nk the limit point y must also be in the closed set Y Y But y Y Y together with y Fx yields a contradiction because Fx Y proving the first part of the Fact Suppose to obtain a contradiction that part 2 of Definition A32 holds at x but F is not lower hemicontinuous at x Then there exists an open set Y Y such that Fx Y but for any ε 0 there exists xε FNεx such that Fxϵ Y Consider the sequence xn n1 with xn x let ε 1n and suppose that this sequence satisfies the property just stated ie for any ε 0 there exists xε FNεx such that Fxε Y Also let y Fx Y By part 2 of Definition A32 there exists a sequence yn n1 and some N 1such that yn Fxn for all n N and yn n1 y However by the construction of the sequence xn n1 yn Y Once again since Y Y is closed it must be the case that the limit point y also lies in the closed set Y Y This contradicts y Fx Y and establishes the second part of the Fact Definition A33 Let X dX and Y dY be metric spaces and consider the correspon dence F X Y Then F has a closed graph is closed at x X if for every sequence xn yn n1 x y such that yn Fxn for each n we also have y Fx In addition F has a closed graph on the set X if it is closed at each x X The following fact is a simple consequence of Definition A32 FactA19 Let X RKX and Y RKY where KX KY N and consider the correspondence F X Y that is upper hemicontinuous If Fx is a closed set in Y ie if F is closedvalued for each x X then F has a closed graph Proof See Exercise A20 For finitedimensional spaces correspondences with closed graphs are also upper hemi continuous provided that they satisfy a simple boundedness hypothesis Fact A20 Let X RKX and Y RKY where KX KY N and consider a correspondence F X Y Suppose that F has a closed graph at x X and that there exists a neighborhood Vx of x such that FVx is bounded Then F is upper hemicontinuous at x 900 Appendix A Odds and Ends in Real Analysis and Applications to Optimization Proof The result follows immediately from Theorem A18 using Part 3 of Theorem A17 which shows that a continuous map is a nonemptyvalued convexvalued and upper hemi continuous correspondence A8 Differentiation Taylor Series and the Mean Value Theorem In this and the next sections I briefly discuss differentiation and some important results related to differentiation that are useful for the analysis in the text The material in this section should be more familiar thus I am somewhat more brief in my treatment than in other sections of this appendix In this section the focus is on a realvalued function of one variable f R R Functions of several variables and vectorvalued functions are discussed in the next section The reader will recall that the derivative function for f R R has a simple definition Take a point x in an open set X on which the function f is defined Then when the limit exists and is finite the derivative of f at x is defined as f x lim h0 f x h f x h A5 Clearly the term f x h is well defined for h sufficiently small since x is in the open set X Moreover this limit exists at point x only if f is continuous at x X This property is more general differentiability implies continuity see Fact A22 Using the elementary properties of limits A5 can be rearranged as lim h0 f x h f x Lxh h 0 A6 where Lx f x This expression emphasizes that we can think of the derivative of the function f x f x as a linear operatorIn fact one might want to define f x precisely as the linear operator Lx that satisfies equation A6 Note that f x is linear in hnot in x It is generally a nonlinear function of x but it defines a linear function from X the open subset of X where f is defined to R that assigns the value f xh to each h such that x h X This perspective is particularly useful in the next section Definition A37 When f x exists at x f is differentiable at x If f x exists at all x in some subset X X then f is differentiable on the entire X If in addition f is a continuous function of x on X f is continuously differentiable When X is a closed set then f being differentiable or continuously differentiable on X is equivalent to f being differentiable or continuously differentiable in the interior of X and then also having an extension or a continuous extension of its derivative to the boundary of X A slightly stronger requirement which also guarantees continuous differentiability on X is that there exists an open set X X such that f is continuously differentiable on X When f is not differentiable at x ie f x does not exist it may still have directional derivativesin particular left and right derivatives These derivatives are defined by f x lim h0f x h f xh and f x lim h0f x h f xh which can be well defined even when A5 is not Directional derivatives are used in the second version of the proof of Theorem 66 in Chapter 6 The next example illustrates how simple functions may have left and right derivatives but may fail to be differentiable A8 Differentiation Taylor Series and the Mean Value Theorem 901 Example A13 Let f be defined as f x x for x 0 and f x x for x 0 f has left and right derivatives at 0 but is not differentiable according to A5 since a unique f x does not exist Differentiability is a stronger requirement than continuity Fact A22 Let X R and f X R be a realvalued function If f is differentiable at x X then it is also continuous at x Proof See Exercise A24 It is also useful to note that differentiability over some set X does not imply continuous differentiability The following example illustrates this point Example A14 Consider the function f such that f x x2 sin1x for all x 0 and f 0 0 It can be verified that f is continuous and differentiable with derivative f x 2x sin1x cos1x and f 0 0 But clearly limx0 f x 0 Higherorder derivatives are defined in a similar manner Again starting with a realvalued function f suppose that this function has a continuous derivative f x Taking x in some open set X where f X is well defined the second derivative of f denoted f x is f x lim h0 f x h f x h Higherorder derivatives are defined similarly If a realvalued function f has continuous derivatives up to order n on some set X then it is said to be Cn on X If f is a C1 function we also say that it is continuously differentiable A C function has continuous derivatives of any order which may be constant after some level as is the case with polynomials The following simple fact shows how first and secondorder derivatives relate to concavity equivalent results naturally hold for convexity Fact A23 Suppose that X R and that f X R is differentiable Then 1 f is concave on X if and only if for all x y X f y f x f xy x A7 2 f is concave on X if and only if f x is nonincreasing in x for all x X 3 If in addition f is twice differentiable then f is concave on X if and only if f x 0 for all x X Proof Part 1 Suppose first that f is concave and take without loss of generality y x Then f λy 1 λx λf y 1 λf x for all λ 0 1 Rearranging this expression yields f y f x f x λy x f x λy x y x Let ε λy x and note that this inequality is true for any λ 0 1 and thus for any ε 0 in the neighborhood of 0 Therefore we have f y f x f x ε f x ε y x f xy x 902 Appendix A Odds and Ends in Real Analysis and Applications to Optimization where the second line follows by taking the limit ε 0 and using the fact that by the differen tiability of f this limit uniquely defines f x Conversely suppose that A7 holds Then for any λ 0 1 it follows that f y f λy 1 λx 1 λf λy 1 λxy x and f x f λy 1 λx λf λy 1 λxy x Multiplying the first inequality by λ and the second by 1 λ and summing the two we obtain that for all λ 0 1 f λy 1 λx λf y 1 λf x Part 2 Suppose f is concave or equivalently A7 holds Then for y x we have f x f y f x y x f x f y x y f y where the last inequality uses the fact that x y 0 Conversely if y x and f x f y then the previous string of inequalities imply that either f xy x f y f x or f yx y f x f y thus violating A7 Part 3 This part follows immediately from Part 2 when f is twice differentiable The next three results are often very useful in applications The first one is a generalization of the Intermediate Value Theorem Theorem A3 to derivatives Theorem A20 Mean Value Theorem Suppose that f a b R is continuously differentiable on a b with b a Then there exists x a b such that f x f b f a b a Moreover if f a f b then for any c intermediate between f a and f b there exists x a b such that f x c Proof See Exercise A25 A particular difficulty often encountered in evaluating limits of the form limxx f xgx where f and g are continuous realvalued functions is that we may have both f x 0 and gx 0 The following result known as lHˆopitals Rule provides one way of evaluating these types of limits Theorem A21 lHˆopitals Rule Suppose that f a b R and g a b R are differentiable functions on a b suppose that gx 0 for x a b and let c a b If lim xc f x gx exists and either lim xc f x lim xc gx 0 or lim xc f x lim xc gx 904 Appendix A Odds and Ends in Real Analysis and Applications to Optimization CorollaryA4 Suppose that f a b R is twice continuously differentiable and concave Then for any x y a b we have f y f x f xy x Proof By Theorem A22 f y f x f xy x f zy x22 for some z between x and y From Fact A23 f z 0 for a concave function and thus the conclusion follows A9 Functions of Several Variables and the Inverse and Implicit Function Theorems Throughout this section I limit myself to differentiation in Euclidean spaces that is our interests are with mappings φ X Y where X RKX and Y RKY with KX KY N In the text when mappings of this form arise and emphasis is needed I refer to φ as a vector function or vectorvalued function since φx RKY for x X The theory of differentiation and the types of results that I present here can be developed in more general spaces than Euclidean spaces For example Luenbergers 1969 classic treat ment of general optimization problems considers X and Y to be Banach spaces complete normed vector spaces which allow for a convenient definition of linear operators see Section A10 Nevertheless for the results presented here restricting attention to Euclidean spaces is without loss of generality and enables me to reduce notation and avoid unnecessary complex ities The case Kx KY 1 was treated in the previous section Building on the results and the intuitions of that section let us now move to more general mappings For φ X Y where X RKX and Y RKY the equivalent of the derivative is the linear operator Jx X Y In particular in analogy to A6 we have the following definition of differentiability4 Let h X be a vector and let h denote its Euclidean norm Then for x X where X is an open set with φX Y well defined φ is differentiable if the limit lim h0 φx h φx Jxh h 0 A8 at x exists and defines a unique linear operator Jx mapping from RKX onto RKY In this case the derivative of φx is denoted by Jx The derivative is again a linear operator because it assigns the value Jxh to any vector h such that x h X We refer to Jx as the Jacobian matrix or as simply the Jacobian of φ at x and often denote it by Dφx The latter is a more convenient notation than Jx since it indicates which function we are referring to We will see below that the Jacobian when it exists is also the matrix of partial derivatives of φ We can also denote the matrix of partial derivatives by Dx1φx1 x2 for x1 RK1 x2 RK2 and K1 K2 N FactA24 Let X RKX Y RKY where KX KY N and φ X Y If φ is differentiable at x X then it is also continuous at x 4 More precisely this is the definition of Frechet differentiability The alternative weaker notion of Gateaux differentiability is also useful in many instances see eg Luenberger 1969 For our purposes there is no need to distinguish between these two notions since in finitedimensional spaces they are equivalent A9 Several Variables and the Inverse and Implicit Function Theorems 905 Let us next take X RKX and consider the mapping φ X R also referred to as a function of several variables Its partial derivatives with respect to each component of X are defined identically to the derivative of a realvalued function of one variable holding all the other variables constant Let x x1 xKX and assume that φ is differentiable with respect to its kth component Then the kth partial derivative of φ is φx1 xKX xk φkx where φkx lim h0 φx1 xk1 xk h xk1 xKXφx1 xk1 xk xk1 xKX h Now assuming that φ has partial derivatives with respect to each xk for k 1 KX the Jacobian in this case is simply a row vector Jx φ1x φKXx A general mapping φ X Y where Y is a subset of RKY can then be thought of as consisting of KY realvalued functions of several variables φ1x φKYx We can define the partial derivatives of each of these functions in a similar fashion and denote them by φj kx The Jacobian can then be written as Jx φ1 1x φ1 KXx φKY 1 x φKY KXx Higherorder derivatives can be defined in a similar fashion When φ X X Jx is a KX KX matrix and in this case we can investigate whether it is invertible ie whether the inverse J 1x at x exists This property plays an important role in the Inverse Function and Implicit Function Theorems below When the matrix of partial derivatives exists we refer to it as the Jacobian but this does not guarantee that the mapping in question is differentiable The following example illustrates the problem Example A15 Consider the function of several variables φx1 x2 over the entire R2 such that φx1 x2 0 if x1 x2 0 and φx1 x2 x2 1x2 2 x1 x2 otherwise The partial derivatives of this function are φx1 x2 x1 x2 1x2 2 2x1x3 2 x1 x22 and φx1 x2 x2 x2 1x2 2 2x3 1x2 x1 x22 906 Appendix A Odds and Ends in Real Analysis and Applications to Optimization It can be verified that these partial derivatives exist everywhere in R2 and in particu lar φ0 0x1 φ0 0x2 0 However it is also clear that φ is not continuous at x1 x2 0 let x x1 x2 and evaluate the limit x 0 using LHospitals Rule as limx0 φx x 2 Thus in view of Fact A24 φ is not differentiable The fact that φ is not differentiable can also be established using directly the definition of differentiability provided above The situation illustrated in Example A15 is important to bear in mind and it implies that a welldefined matrix of partial derivatives does not guarantee differentiability Thus one may wish to distinguish between the linear operator Jx defined above and the Jacobian consisting of the partial derivatives Dφx Nevertheless in this book there is no need to draw this distinction and throughout Dφx refers to the Jacobian ie to the matrix of partial derivatives Continuous differentiability is defined analogously to the onedimensional case Definition A38 A mapping φ is of class Cn ntimes continuously differentiable on some set X if it has continuous derivatives up to the nth order Fact A25 A mapping φ X Y with X RKX Y RKY where KX KY N and X open is of class C1 on X if its partial derivatives φj kx for k 1 KX and j 1 KY exist and are continuous functions of x for each x X When there is no need for further generality I require that the relevant utility or production functions are continuously differentiable of class C1 or the stronger requirement that they are twice differentiable Taylors Theorem and its corollaries can be generalized to mappings discussed here I state this result for φ X R with X RKX Let Dφ and D2φ denote the vector of first derivatives and the Jacobian of φ respectively Let y x be the Euclidean norm of the KXdimensional vector y x and zT be the transpose of vector z The following is a simpler version of the equivalent form of Taylors Theorem in Corollary A3 Its proof is similar to that of Theorem A22 and is omitted Theorem A23 Taylors Theorem II Suppose that φ X R is a C1 function and its second derivative D2φx exists for all x X Then for any x and y x in X φy φx DφxT y x oy x If in addition φ X R is a C2 function with third derivative D3φx for all x X then for any x and y x in X φy φx DφxT y x y xT D2φxy x oy x2 The following two theorems are the basis of much of the comparative static results in economics They are therefore among the most important mathematical results for economic analysis Consider a mapping φ X X for X RKX A key question is whether this mapping will have an inverse φ1 X X If for some subset X of X φ is singlevalued and has an inverse φ1 which is also a singlevalued then we say that it is onetoone Theorem A24 Inverse Function Theorem Consider a C1 mapping φ X X for X RKX Suppose that the Jacobian of φ Jx evaluated at some interior point x of X is invertible Then there exist open sets X and X in X such that x X φx X and φ is onetoone on X with φX X Moreover φ1φx x for all x in X and φ1 is also a C1 mapping A10 Separation Theorems 907 The proof of this theorem can be found in any real analysis book and is omitted Theorem A25 Implicit Function Theorem Consider a C1 mapping φ X Y Y with X RKX and Y RKY Suppose that x y X Y φx y 0 all the entries of the Jacobian of φ with respect to x y Dxyφx y are finite and Dyφx y is invertible Then there exists an open set X containing x and a unique C1 mapping γ X Y such that γ x y and φx γ x 0 A9 for all x X This theorem is called the Implicit Function Theorem because the mapping γ is defined implicitly Exercise 65 in Chapter 6 provided the proof of a special case of this theorem The more general case can also be proved with the same methods as in that exercise An alternative proof uses the Inverse Function Theorem Since the former proof has already been discussed and the latter one is contained in most real analysis books I omit the proof The main utility of this theorem comes from the fact that since φ and γ are C1and A9 holds for an open set around x A9 can be differentiated with respect to x to obtain an expression for how the solution y to the set of equations φx y 0 behaves as a function of x If we think of x as representing a set of parameters and y as the endogenous variables determined by some economic relationship summarized by A9 then this procedure can tell us how the endogenous variables change in response to the changes in the environment captured by the parameter x I make repeated use of this approach throughout the book A10 Separation Theorems In this section I briefly discuss the separation of convex disjoint sets using linear functionals or hyperplanes These results form the basis of the Second Welfare Theorem Theorem 57 They also provide the basis of many important results in constrained optimization see Section A11 For this section I take X to be a vector space linear space Recall that linearity implies that if x y X and λ is a real number then x y X and λx X see Section A7 The element of X with the property that x λx for all λ R is denoted by θ Definition A39 The realvalued nonnegative function X R is taken to be a norm on X which implies that for any x y X and any λ R 1 Properness x 0 and x 0 if and only if x θ 2 Linearity λx λ x and 3 Triangle Inequality x y x y A vector space equipped with a norm is a normed vector space A complete normed vector space is a Banach space If a function p X R satisfies properness and the triangle inequality but not necessarily the linearity condition then it is a seminorm Many of the metric spaces given in Example A1 are also normed vector spaces with the appropriate norm In fact a simple way of obtaining the norm in many cases is to take the distance function d and try the norm x dx θ Notice however that this method will not always work since metrics do not need to satisfy the linearity condition in Definition A39 A11 Constrained Optimization 911 the separation theorems of the previous section Let me illustrate this by focusing on finite dimensional optimization problems Consider the maximization problem sup xX f x A10 subject to gx 0 where X is an open subset of RK f X R g X RN and N K N The constrained maximization problem A10 satisfies the Slater condition if there exists x X such that gx 0 meaning that each component of the mapping g takes a negative value This condition is equivalent to the set G xgx 0 having an interior point We say that g is convex if each component function of g is convex Thus the set G is also convex but the converse is not necessarily true see Exercise A32 As usual we define the Lagrangian function as Lx λ f x λ gx for λ RN The vector λ is the Lagrange multiplier and λ gx denotes the inner product of the two vectors here λ and the vectorvalued function g evaluated at x thus it is equal to a real number A central theorem in constrained maximization is the following Theorem A29 SaddlePoint Theorem Suppose that in A10 f is a concave function g is convex and the Slater condition is satisfied 1 If x is a solution to A10 then there exists λ RN such that Lx λ Lx λ Lx λ for all x X and λ RN A11 In this case x λ satisfies the complementary slackness condition λ gx 0 A12 2 If x λ X RN satisfies gx 0 and A11 then x is a solution to A10 Proof Part 1 Consider the space Y RN1 with subsets Y 1 a b Y a f x and b 0 and Y 2 a b Y x X with a f x and b gx where a R b RN and b 0 means that each element of the Ndimensional vector b is negative Y 1 is clearly convex Moreover the concavity of f and the convexity of g ensure that Y 2 is also convex By the hypothesis that x is a solution to A10 the two sets are disjoint Then Theorem A28 implies that there exists a hyperplane separating these two sets In other words there exists a nonzero vector η RN1 such that η y1 c η y2 for all y1 Y 1 and y2 Y 2 Moreover the same conclusion holds for all y1 Y 1 and y2 Y 2 Then let η ρ λ with ρ R and λ RN so that ρa1 λ b1 ρa2 λ b2 for all a1 b1 Y 1 a2 b2 Y 2 A13 912 Appendix A Odds and Ends in Real Analysis and Applications to Optimization For f x 0 Y 2 we have ρa1 λ b1 ρf x A14 for all a1 b1 Y 1 Now taking a1 f x and b1 0 implies λ 0 suppose instead that one component of the vector λ is negative then take b1to have zeros everywhere except for that component yielding a contradiction to A14 Similarly setting b1 0 and a1 f x we obtain ρ 0 Moreover by the definition of a hyperplane either ρ is negative or a component of λ must be strictly positive Next the optimality of x implies that for any x X we have f x gx Y 2 Since f x 0 Y 1 A13 implies that ρf x ρf x λ gx A15 for all x X Now to obtain a contradiction suppose that ρ 0 Then by the Slater condition there exists x X such that gx 0 so that λ gx 0 for any nonzero vector λ violating A15 Therefore λ 0 However this contradicts the fact that the separating hyperplane is nonzero so that we cannot have both ρ 0 and λ 0 Therefore ρ 0 Now define λ λ ρ 0 The complementary slackness condition then follows immediately from A15 In particular evaluate the righthand side at x X which implies that λ gx 0 Since λ 0 and gx 0 we must have λ gx ρλ gx 0 Now using the complementary slackness condition and A15 together with ρ 0 yields Lx λ f x λ gx f x Lx λ for all x X which establishes the first inequality in A11 To establish the second inequality again use the complementary slackness condition and the fact that gx 0 to obtain Lx λ f x f x λ gx Lx λ for all λ RN which completes the proof of the first part Part 2 Suppose to obtain a contradiction that A11 holds but x is not a solution to A10 Thus there exists x X with gx 0 and f x f x Then f x λ gx f x λ gx which exploits the facts that λ gx 0 and λ gx 0 since λ 0 and gx 0 But this expression contradicts A11 and so establishes the desired result We often refer to maximization problems where as in Theorem A29 F is concave and g is convex as concave problems Exercise A33 shows that the Slater condition cannot be dispensed with in Theorem A29 Despite their importance constraint qualification conditions such as the Slater condition or the linear independence condition in the next theorem are often not stated explicitly in economic applications because in most problems they are naturally satisfied Nevertheless it is important to be aware that these conditions are necessary and that ignoring them can sometimes lead to misleading results A11 Constrained Optimization 913 An immediate corollary of the first inequality in A11 is that if x Int X and if f and g are differentiable then Dxf x λ Dxgx A16 where as usual Dxf and Dxg denote the Jacobians of f and g Equation A16 is the usual firstorder necessary condition for interior constrained maximum In this case because the maximization problem is concave A16 together with gx 0 is also sufficient for a maximum The next result is the famous KuhnTucker Theorem which shows that A16 is necessary for an interior maximum provided that f and g are differentiable even when the concavity convexity assumptions do not hold Theorem A30 KuhnTucker Theorem Consider the constrained maximization problem sup xRK f x subject to gx 0 and hx 0 where f x X R g x X RN and h x X RM for some K N M N Let x Int X be a solution to this maximization problem and suppose that N1 N of the inequality constraints are active in the sense that they hold as equality at x Define h X RMN1 to be the mapping of these N1 active constraints stacked with hx so that hx 0 Suppose that the following constraint qualification condition is satisfied the Jacobian matrix Dxhx has rank N1 M Then the following KuhnTucker condition is satisfied there exist Lagrange multipliers λ RN and μ RM such that Dxf x λ Dxgx μ Dxhx 0 A17 and the complementary slackness condition λ gx 0 holds Proof Sketch The constraint qualification condition ensures that there exists an N1 M dimensional manifold at x defined by the equality and active inequality constraints Since g and h are differentiable this manifold is differentiable at x Let vεx denote a feasible direction along this manifold for small ε RK in particular such that x εvεx ε remains along this manifold and thus satisfies Dx hx εvεx ε 0 For ε sufficiently small the N N1 nonactive constraints are still satisfied thus x εvεx ε is feasible If Dxf x εvεx ε 0 then f x εvεx ε f x or f x εvεx ε f x implying that x cannot be a local and thus global maximum As a next step consider the M N1 1 Kdimensional matrix A where the first row is Dxf xT and the rest is given by Dxhx The preceding argument implies that for all nonzero ε RK such that Dx hx εvεx ε 0 we also have A ε vεx ε 0 Therefore both Dx hx and A have the same rank which by the constraint qualification condition is equal to M N1 914 Appendix A Odds and Ends in Real Analysis and Applications to Optimization Since A has M N1 1rows the first row of A must be a linear combination of its remaining M N1 rows which equivalently implies that there exists an M N1dimensional vector μ such that Dxf x μDx hx Assigning zero multipliers to all nonactive constraints this result is equivalent to A17 The complementary slackness condition then follows immedi ately since we have zero multipliers for the nonactive constraints and gjx 0 for the active constraints The constraint qualification condition which required that the active constraints should be linearly independent plays a similar role to the Slater condition in Theorem A29 Exercise A34 shows that this constraint qualification condition cannot be dispensed with though somewhat weaker conditions can be used instead of this full rank condition The complementary slackness condition in Theorem A30 is a central result and has been used repeatedly in the text as a necessary condition for a maximum Let us end this Appendix with the famous and eminently useful Envelope Theorem Theorem A31 Envelope Theorem Consider the constrained maximization problem vp max xX f x p subject to gx p 0 and hx p 0 where X RK p R and f X R R g X R RN and h X R RM are differentiable K N M N Let xp Int X be a solution to this maximization problem Denote the Lagrange multipliers associated with the inequality and equality constraints by λ RN and μ RM Suppose also that v is differentiable at p Then we have v p p f x p p p λ Dpgx p p μ Dpx p p A18 Proof Since xp is the solution to the maximization problem we have v p f x p p A19 By hypothesis v is differentiable at p so v pp exists Moreover applying the Implicit Function Theorem to the necessary conditions for a maximum given in Theorem A30 x is also differentiable at p Therefore from A19 we can write v p p f x p p p Dxf x p p Dpx p A20 where once again Dxx p p Dpx p is the inner product and thus is a real number Let g X R RN1 denote the N1 N active inequality constraints Differentiating the active inequality constraints and the equality constraints with respect to p we also have Dp gx p p Dx gx p p Dpx p and Dphx p p Dxhx p p Dpx p The equivalent of A17 for this problem recall Theorem A30 implies that Dxf x p p λ Dxgx p p μ Dxx p p 0 B Review of Ordinary Differential Equations I n this appendix I give a brief overview of some basic results on differential equations and include a few results on difference equations I limit myself to results that are useful for the material covered in the body of the text In particular I provide the background for the major theorems on stability Theorems 22 23 24 25 718 and 719 which are presented and then extensively used in the text I also provide some basic theorems on existence uniqueness and continuity of solutions to differential equations Most of the material here can be found in basic differential equation textbooks such as Boyce and DiPrima 1977 Luenberger 1979 is an excellent reference since it gives a symmetric treatment of differential and difference equations The results on existence uniqueness and continuity of solutions can be found in more advanced books such as Walter 1991 or Perko 2001 Before presenting the results on differential equations I also provide a brief overview of eigenvalues and eigenvectors and some basic results on integrals Throughout I continue to assume basic familiarity with matrix algebra and calculus B1 Eigenvalues and Eigenvectors Let A be an n n square real matrixmeaning that all of its entries are real numbers The n n matrix D is diagonal if all of its nondiagonal elements are equal to zero that is D d1 0 0 0 d2 0 0 0 dn The n n identity matrix I is the diagonal matrix with 1s on the diagonal I 1 0 0 0 1 1 0 0 0 1 917 B11 Exercises 933 B4 Show that B12 is the general solution to the firstorder differential equation B4 B5 Verify that the system of linear differential equations B13 satisfies the conditions of Theorem B10 B6 Verify B18 B7 Prove B19 B8 Show that if g R R R satisfies the Lipschitz condition in Definition B1 then gx t is continuous in x B9 This exercise asks you to use the techniques for solving separable differential equations to characterize the family of utility functions with a constant coefficient of relative risk aversion In particular recall that the ArrowPratt measure of relative risk aversion of a twice differentiable utility function u is given by Ruc ucc uc Suppose that Ruc r 0 and let vc uc Then we obtain vc vc r c Using this equation characterize the family of utility functions that have a constant coefficient of relative risk aversion B10 Prove Theorem B15 B11 Consider the nthorder difference equation xt n Hxt n 1 xt t where H Rn R Prove that if the initial values x0 x1 xn 1 are specified this equation has a unique solution for any t D List of Theorems I n this appendix I list the theorems presented in various chapters for reference Many of these theorems refer to mathematical results used in different parts of the book Some of them are economic results that are more general and more widely applicable than the results I label as propositions To conserve space I do not list additional mathematical results given in lemmas corollaries and facts Chapter 2 21 Eulers Theorem 22 Stability for Systems of Linear Difference Equations 23 Local Stability for Systems of Nonlinear Difference Equations 24 Stability of Linear Differential Equations 25 Local Stability of Nonlinear Differential Equations 26 Uzawas Theorem I 27 Uzawas Theorem II Chapter 5 51 DebreuMantelSonnenschein Theorem 52 Gormans Aggregation Theorem 53 Existence of a Normative Representative Household 54 Representative Firm Theorem 55 First Welfare Theorem I Economies with Finite Number of Households 56 First Welfare Theorem II Economies with Infinite Number of Households 57 Second Welfare Theorem 58 Equivalence of Sequential and Nonsequential Trading with Arrow Securities Chapter 6 61 Equivalence of Sequential and Recursive Formulations 62 Principle of Optimality in Dynamic Programming 944 Chapter 16 945 63 Existence of Solutions in Dynamic Programming 64 Concavity of the Value Function 65 Monotonicity of the Value Function 66 Differentiability of the Value Function 67 Contraction Mapping Theorem 68 Applications of Contraction Mappings 69 Blackwells Sufficient Conditions for a Contraction 610 Euler Equations and the Transversality Condition 611 Existence of Solutions in Nonstationary Problems 612 Euler Equations and the Transversality Condition in Nonstationary Problems Chapter 7 71 Necessary Conditions for an Interior Optimum with Free End Points 72 Necessary Conditions II for Interior Optimum with Fixed End Points 73 Necessary Conditions III for Interior Optimum with InequalityConstrained End Points 74 Simplified Version of Pontryagins Maximum Principle 75 Mangasarians Sufficiency Conditions for an Optimum 76 Arrows Sufficiency Conditions for an Optimum 77 Pontyagins Maximum Principle for Multivariate Problems 78 Sufficiency Conditions for Multivariate Problems 79 Pontyagins InfiniteHorizon Maximum Principle 710 HamiltonJacobiBellman Equation 711 Sufficiency Conditions for InfiniteHorizon Optimal Control 712 Transversality Condition for InfiniteHorizon Problems 713 Maximum Principle for Discounted InfiniteHorizon Problems 714 Sufficiency Conditions for Discounted InfiniteHorizon Problems 715 Existence of Solutions in Optimal Control 716 Concavity of the Value Function in Optimal Control 717 Differentiability of the Value Function in Optimal Control 718 SaddlePath Stability in Systems of Linear Differential Equations 719 SaddlePath Stability in Systems of Nonlinear Differential Equations Chapter 10 101 Separation Theorem for Investment in Human Capital Chapter 16 161 Equivalence of Sequential and Recursive Formulations 162 Principle of Optimality in Stochastic Dynamic Programming 163 Existence of Solutions in Stochastic Dynamic Programming 164 Concavity of the Value Function 165 Monotonicity of the Value Function in State Variables 166 Differentiability of the Value Function 167 Monotonicity of the Value Function in Stochastic Variables 946 Appendix D List of Theorems 168 Euler Equations and the Transversality Condition 169 Existence of Solutions with Markov Processes 1610 Continuity of Value Functions with Markov Processes 1611 Concavity of Value Functions with Markov Processes 1612 Monotonicity of Value Functions with Markov Processes 1613 Differentiability of Value Functions with Markov Processes Chapter 22 221 Median Voter Theorem 222 Median Voter Theorem with Strategic Voting 223 Downsian Policy Convergence Theorem 224 Extended Median Voter Theorem 225 Extended Downsian Policy Convergence Theorem 226 Probabilistic Voting Theorem Appendix A A1 Properties of Open and Closed Sets in Metric Spaces A2 Open Sets and Continuity in Metric Spaces A3 Intermediate Value Theorem A4 Open Sets and Continuity in Topological Spaces A5 Continuity and Convergence of Nets in Topological Spaces A6 HeineBorel Theorem A7 BolzanoWeierstrass Theorem A8 Continuity and Compact Images in Topological Spaces A9 Weierstrasss Theorem A10 Uniform Continuity over Compact Sets A11 Projection Maps and the Product Topology A12 Continuity of Discounted Utilities in the Product Topology A13 Tychonoffs Theorem A14 Totally Bounded and Compact Spaces A15 ArzelaAscoli Theorem A16 Berges Maximum Theorem A17 Properties of Maximizers under QuasiConcavity A18 Kakutanis Fixed Point Theorem A19 Brouwers Fixed Point Theorem A20 Mean Value Theorem A21 lHˆopitals Rule A22 Taylors Theorem I A23 Taylors Theorem II Functions of Several Variables A24 Inverse Function Theorem A25 Implicit Function Theorem A26 Continuity of Linear Functionals in Normed Vector Spaces A27 Geometric HahnBanach Theorem A28 Separating Hyperplane Theorem A29 SaddlePoint Theorem Appendix C 947 A30 KuhnTucker Theorem A31 Envelope Theorem Appendix B B1 Fundamental Theorem of Calculus I B2 Fundamental Theorem of Calculus II B3 Integration by Parts B4 Leibnizs Rule B5 Solutions to Systems of Linear Differential Equations with Constant Coefficients B6 Solutions to General Systems of Linear Differential Equations B7 GrobmanHartman Theorem Stability of Nonlinear Systems of Differential Equations B8 Picards Theorem I Existence and Uniqueness for Differential Equations B9 Existence and Uniqueness of Differential Equations on Compact Sets I B10 Picards Theorem II on Existence and Uniqueness for Systems of 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Zuleta Hernando and Andrew Young 2006 Labors SharesAggregate and Industry Accounting for Both in a Model with Induced Innovation University of Mississippi mimeo Name Index Page numbers for entries occurring in notes are followed by an n Abernathy William J 479 Abraham Kathrine G 581n Abramowitz Moses 79 Abreu Dilip 941 Acemoglu Daron 24 25 129 130 133 135 136 137 138 139 141 142 143 319 380 384 416 427 430 489 490 522 523 526 527 528 529 588 598 605 626 630 631 640 644 663 686 703 715 721 737 740 744 746 749 750 769 770 771 818 821 823 824 833 853 855 857 858 870 871 Aczel J 223 Aghion Philippe 23 355 384 430 453 458 459 468 469n 470 480 490 491 643 744 746 749 750 764 770 771 Aiken Howard 415 Aiyagari S Rao 583 604 Akcigit Ufuk 489 490 Alesina Alberto 142 824 Alfaro Laura 686 Aliprantis Charalambos 877 Allais Maurice 354 Allen Franklin 605 Allen Robert C 24 Anant T C A 453 490 Andreoni James 355 Angrist Joshua D 380 384 528 770 Antras Pol 631 640 644 686 Apostol Tom M 877 Araujo A 277 Armington Paul S 686 Arrow Kenneth J 31 54 55 70 73 171 237 277 318 418 421 430 824 Ashton Thomas Southcliffe 24 870 Aten Bettina 3n 23 Atkeson Andrew 685 686 Atkinson Anthony 625 644 Aumann Robert J 640 AustenSmith David 824 Autor David 384 528 Azariadis Costas 354 379 384 Backus David 453 Baily Martin N 612 Bairoch Paul 143 512 770 Banerjee Abhijit V 355 654 737 740 764 770 771 773 824 Banfield Edward C 122 142 Banks Jeffrey S 824 Barro Robert J 15 17 23 24 25 69 70 80 82 83 106 142 318 319 404 685 687 770 83233 Barsky Robert 581n Bartelsman Eric J 612 Basu Susanto 625 644 Batou Jean 143 Baum R F 276 Baumol William J 25 82 106 703 719 721 Baxter Marianne 686 Becker Gary S 85 359 380 384 618 732 770 Becker Robert 178 Bellman Richard 185 222 Benabou Roland 527 771 824 Bencivenga Valerie 769 Benhabib Jess 382 384 643 BenPorath Yoram 384 Benveniste Lawrence M 222 277 Berge Claude 877 Bernard Andrew 679 Bewley Truman F 177 355 566 583 604 Billingsley Patrick 525 Bils Mark J 581n Black Duncan 824 Black Sandra E 612 770 Blackwell David 222 561 Blanchard Olivier J 277 318 327 346 348 35253 354 355 499 528 531 604 Bloom David E 119 141 Blume Lawrence 69 70 877 Blundell Richard 491 Boldrin Michele 418 430 Bolton Patrick 355 764 771 Border Kim 877 Boserup Ester 141 Boulton Matthew 414 415 Bourguignon Francois 142 770 857 Boyce William E 70 917 Boyd John Harvey 178 Braudel Fernand 588 Broadberry Stephen 24 Brock William A 566 604 Browning Martin 562 Bryant Victor 223 Buera Francisco 720 Bulow Jeremy I 685 Burstein Ariel 686 Caballero Ricardo J 277 528 556 Campbell David 612 Caputo Michael 277 Card David 384 Carter Susan B 720 971 972 Name Index Caselli Francesco 25 107 176 384 527 612 643 654 686 720 Cass David 318 355 Ceruzzi Paul E 415 Cesari Lamberto 276 Chamberlain Gary 562 Chamberlin Edward 422 426 430 Chandler Tertius 143 Chari V V 319 824 Chenery Hollis 720 Chevre Pierre 143 Chiang Alpha C 276 Chirinko Robert S 686 Ciccone Antonio 380 384 Coatsworth John H 858 Coleman Wilbur John 527 612 720 Collier Ruth B 858 Conrad Jon M 277 Conway John B 877 910 Cooley Thomas F 604 Coughlin Peter J 824 Crossley Thomas F 562 Crucini Mario J 686 Cunat Alejandro 686 Curtin Philip D 143 Dasgupta Partha 277 430 David Paul A 644 Davis Ralph 870 Davis Steven 612 Davis Y Donald 107 Deaton Angus S 25 153n 176 177 562 Debreu Gerard 150 177 De La Croix David 354 Denardo Eric V 222 223 Devereux Paul J 770 Diamond Jared M 25 118 129 14142 Diamond Peter 327 328 353 354 355 Dinopoulos Elias 453 490 DiPrima Richard C 70 917 Dixit Avinash K 142 277 422 430 686 Doepke Matthias 770 Dollar David 678 Domar Evsey D 26 75 Doms Mark 612 Dorfman Robert 277 Downs Anthony 824 Drandakis E 528 Drazen Allan 379 384 823 Duflo Esther 380 384 654 824 Dunne Timothy 612 Duranton Gilles 527 Durlauf Steven N 25 106 142 771 Echevarria Cristina 720 Eggertsson Thrainn 823 Eggimann Gilbert 143 Ekeland Ivar 223 Elliott John H 868 Eltis David 858 Elvin Mark 867 Engel Ernst 698 Engerman Stanley L 858 871 Epifani Paolo 686 Epstein Larry G 178 Ertman Thomas 870 Esquivel Gerard 25 Ethier Stewart 562 604 Evans Eric J 858 Evans Peter 824 Fafchamps Marcel 142 Feinstein Charles 25 Feldstein Martin 649 655 Fernandez Raquel 771 FernandezVillaverde Jesus 770 Feyrer James 654 686 Fields Gary 770 Finkelstein Amy 416 430 Fischer Stanley 277 318 354 604 Fisher Irving 554 Fleming Wendell H 276 277 Foellmi Reto 720 Fomin Sergei V 177 223 27576 877 910 Forbes Kristen J 824 Foster Andrew 382 384 Foster Lucia 612 Francois Patrick 490 Frankel Jeffrey 67879 Fraumeni Barbara 80 106 Freeman Christopher 430 Freudenberger Herman 871 Friedman Milton 554 Fudenberg Drew 431 490 934 941 Funk Peter 529 Futia Carl A 562 604 Gabaix Xavier 107 Gale Douglas 605 Galenson David W 871 Gallup John Luke 119 Galor Oded 70 141 354 355 384 643 681 687 761 762n 763 764 770 771 858 Gancia Gino 453 527 528 686 Gans Joshua S 824 Geary Robert C 319 Geertz Clifford 770 Gelfand I M 27576 Gerschenkron Alexander 615 643 Gikhman I I 562 604 Gil Richard 833 Gilles Christian 355 Glomm Gerhard 771 Goldin Claudia 528 Gollin Douglas 720 Gollop F M 80 106 GomezGalvarriato Aurora 25 Goodfriend Marvin 770 Gordon Robert J 76 Gorman W M 151 176 Gourinchas PierreOlivier 686 Grandmont JeanMichel 824 Green Jerry R 69 150 176 177 Greenwood Jeremy 76 384 643 726 769 Greif Avner 142 Griffith Rachel 491 Griliches Zvi 384 415 613 Grossman Gene M 433 453 458 490 679 680 686 Grossman Herschel 142 Guerrieri Veronica 703 721 Guiso Luigi 142 Gupta Bishnupriya 24 Gutierrez Hector 143 Guvenen Fatih 384 Habakkuk H J 512 528 Hakenes Hendrik 38n 7071 Halkin Hubert 277 Hall Robert E 96 97 1067 142 554 556 Haltiwanger John C 612 Haltiwanger Jon 581n Hammermesh Daniel 528 Hammour Mohammad 528 Hansen Gary D 141 770 Harris Christopher 490 Harris John 770 Harrison Lawrence E 142 Harrod Roy 26 59 75 Hart Oliver D 605 Hassler John 824 Hayashi Fumia 277 Name Index 973 Heal Geoffrey 277 Heckman James 384 Hellwig Martin 418 Helpman Elhanan 23 433 453 458 490 631 640 644 679 680 686 Herbst Jeffery I 824 Hercowitz Zvi 76 Heston Allen 3n 12 23 25 91 97 Hicks John 58 527 Hildenbrand Werner 176 177 Hirschman Albert 694 695 752 Hirshleifer Jack 142 Homer Sydney 57 70 Hopkins Keith 866 Horioka Charles 649 655 Hotelling Harold 277 824 Houthakker Hendrik S 523 524 526 529 Howard Ronald A 561 Howitt Peter 23 384 430 453 458 459 468 469n 470 490 64344 Hsieh ChangTai 25 319 Hulten Charles 612 Huntington Samuel P 142 Imbs Jean 588 605 Inada KenIchi 70 Irmen Andreas 38n 7071 418 Jacobs Jane 379 384 Jaffee Adam 415 430 James John A 528 Jayaratne Jay 770 Jeanne Olivier 686 Jensen Bradford 679 Johnson Paul A 106 Johnson Simon 24 25 129 130 133 135 136 137 138 139 141 142 143 858 870 871 Jones Benjamin F 117 143 Jones Charles I 23 25 69 70 96 97 1067 142 319 446 453 497 523 526 527 529 Jones Eric 868 Jones Larry 404 Jones Richard 141 142 Jorgensen Dale 80 106 Jovanovic Boyan 726 769 Judd Kenneth 221 453 Kaboski Joseph 720 Kaldor Nicholas 57 70 72 762n KalemliOzcan Sebnem 686 770 Kamien Morton 277 Kamihigashi Takashi 223 562 Karlin Samuel 222 Katz Lawrence 384 528 Katz Lawrence F 528 Keefer Philip 142 Kehoe Patrick J 319 453 685 686 824 Kehoe Timothy J 453 Kelley John 877 Kennedy Charles 52728 Keyssar Alexander 871 Kiley Michael 527 Kim Minseong 142 King Robert G 604 769 Kirman Alan 176 177 Kiyotaki Nobuhiro 771 Klenow Peter J 25 96 1067 319 Klepper Steven 490 Klette Tor Jacob 490 Knack Stephen 142 Knight Frank 535n Kolmogorov Andrei 177 223 877 910 Kongsamut Piyabha 698 699 720 Koopmans Tjalling C 318 Koren Miklos 588 605 Kortum Samuel 490 562 Kraay Aart 686 Kremer Michael 113 141 453 Kreps David 178 Kreyszig Erwin 177 223 Krizan Cornell J 612 Krueger Alan 24 528 Krugman Paul 674 677 686 771 Krusell Per 76 142 528 583 604 824 Kupperman Karen O 142 Kurtz Thomas 562 604 Kuruscu Burhanettin 384 Kurz Mordecai 277 318 Kuznets Simon 8 69394 695 697 702 720 750 751 Kydland Finn E 579 604 Lagos Ricardo 529 Laitner John 720 Landes David S 142 Lane Julia I 612 Lang Sean 858 Lavy Victor 770 Leamer Edward 857 Lee JongWha 23 Lefort Fernando 25 Leonard Daniel 277 LeRoy Stephen F 355 Levchenko Andrei 687 Levine David K 418 430 Levine Ross 769 Lewis William Arthur 142 737 770 Limongi Fernando 83233 Lindahl Mikael 24 Lindbeck Assar 824 Lindert Peter H 770 858 Linn Joshua 416 430 LiviBacci Massimo 770 Ljungqvist Lars 221 318 562 604 Lochner Lance 384 Locke John 142 Long John B 604 LopezAlonso Moramay 25 Loury Glenn 771 Lucas Robert E 177 22223 379 384 399 404 407 553 560 561 604 649 654 686 687 Luenberger David 70 177 276 877 904 904n 910 917 Lynch Lisa 612 Machiavelli Niccolo 141 Maddison Angus 12 13 24 25 126 730 770 870 Maffezoli Marco 686 Magill Michael J P 276 Makowski Louis 605 Mallick Debdulal 686 Malthus Thomas R 730 770 Mangasarian O O 236 277 Mankiw N Gregory 90 9193 94 9596 106 Mann Charles C 866 Mantel Rolf R 177 Manuelli Rodolfo 384 404 Marris Robin 25 Marshall Alfred 118 141 384 Martimort David 490 MasColell Andreu 69 150 176 177 Matsuyama Kiminori 430 453 490 679 68081 685 687 715 720 721 771 Mauro Paolo 142 McCall John 561 562 McCandless George T 354 McCleary Rachel 142 McDermott John 770 McEvedy Colin 141 142 McGrattan Ellen 319 Melitz Mark 679 974 Name Index Meltzer Allan H 824 Michel Philippe 277 354 Migdal Joel 824 Mill John Stuart 142 415 Mincer Jacob 85 359 361 380 384 618 Minier Jenny A 833 Mirman Leonard J 222 566 604 Mitch David 858 Moav Omer 141 384 643 762n 763 770 858 Mokyr Joel 24 25 412 415 430 499 528 697 715 721 867 869 870 Montesquieu Charles de Secondat 11718 122 124 Moretti Enrico 380n 384 Morgenstern Oskar 156 177 Morris Ian 866n Morrison Christian 770 Mosse W E 871 Mountford Andrew 681 687 Muellbauer John 153n 176 177 Mulligan Casey 833 Mundlak Yair 721 Murphy Kevin M 528 720 752 756 757 770 771 Myerson Rogerson 431 934 Myrdal Gunnar 141 Nelson Richard R 380 382 384 61819 620 643 Newell Richard 415 430 Newman Andrew 355 737 740 764 770 771 773 Ngai Rachel 721 Nickell Stephen 491 Norman Victor 686 North Douglass C 25 119 141 142 870 Nugent Jeffrey 857 Nunn Nathan 687 Nurske Ragnar 694 695 715 720 752 757 Obstfeld Maurice 68586 769 Ok Efe 877 Olken Benjamin A 117 143 Osborne Martin 431 934 Overton Mark 715 721 Pamuk Sevket 867 Parente Stephen 720 Parente Stephen L 319 619 620 Park Chung Hee 832 Parker Jonathan A 581n Pavcnik Nina 679 Peri Giovanni 380 384 Perko Lawrence 917 Perri Fabrizio 685 Persson Torsten 142 823 824 Phelps Edmund 528 Phelps Edmund S 70 380 382 384 61819 620 643 Piketty Thomas 70 355 764 771 Pindyck Robert S 277 Pinochet Augusto 832 Pirenne Henri 869 Pissarides Christopher 562 721 Plosser Charles I 604 Pollak Richard 176 Pomeranz Kenneth 24 Pontryagin Lev S 227 236 248 275 276 Popp David 416 430 Porras Condey Raul 25 Postan M M 869 Prescott Edward C 141 177 222 23 319 553 561 579 604 619 620 770 Pritchett Lant 25 Przeworski Adam 83233 Puterman Martin L 222 561 Putnam Robert 122 142 Qian Nancy 770 Quah Danny 25 106 Ragot Xavier 527 Rajan Raghuram 76970 Ramey Garey 605 Ramey Valerie 605 Ramsey Frank 250 318 Rauch James E 379 380 384 Ravikumar B 771 Rebelo Sergio 387 395 404 444 604 671 698 699 700 702 720 Reinganum Jennifer 430 Richard Scott 824 Ringer Fritz 858 RıosRull JoseVıctor 142 824 Rishel Raymond W 276 277 RiveraBatiz Luis A 453 686 Roberts Joanne 490 Roberts Kevin W S 824 Robinson James A 24 25 129 130 133 135 141 142 143 823 824 833 853 855 857 858 870 871 Rockefeller Tyrell R 276 Rodriguez Andres 25 96 1067 Rodriguez Francisco 679 Rodrik Dani 142 679 824 833 Rogerson Richard 562 720 Rogerson Roger 771 Rogoff Kenneth 685 Romer David 90 9193 94 9596 106 277 318 67879 Romer Paul M 276 38788 398 399 404 41314 418 430 433 439 453 681 686 720 734 738 Romer Thomas 824 Rosenberg Nathan 415 RosensteinRodan Paul 694 695 715 720 752 757 771 Rosenzweig Mark 382 384 Rostow Walt Whitman 13 24 598 715 720 Royden Halsey 561 877 Rubinstein Ariel 431 934 Rudin Walter 561 877 Ryder Harl E 354 Sachs Jeffrey 118 141 678 Sachs Jeffrey D 119 141 Saez Emmanuel 70 SaintPaul Gilles 142 824 SalaiMartin Xavier 15 17 23 24 25 69 70 82 83 106 142 318 404 685 687 833 Salop Steven 430 432 Salter W E G 644 Salvanes Kjell 770 Samuelson Paul A 327 328 353 354 355 511 528 Sapienza Paola 142 Sargent Thomas J 221 318 562 604 Scheinkman Jose A 222 223 277 Scherer Frederick M 414 Schlicht Ekkehart 60 70 Schlosser Analia 770 Schmookler Jacob 415 430 Schultz Theodore 85 380 384 61819 643 Schumpeter Joseph A 8 417 418 421 430 Schwartz Nancy 277 Scotchmer Suzanne 430 Scrimgeour Dean 70 Segerstrom Paul S 453 490 Seierstad Atle 277 Name Index 975 Seshadri Anant 384 Shapley Lloyd S 222 640 Shell Karl 328 35455 Shimer Robert 562 Shleifer Andrei 720 752 756 757 771 Simon Carl 69 70 877 Simon Julian 113 141 Skaperdas Stergios 142 Skinner Jonathan S 528 Skorohod A V 562 604 Smart Michael 824 Smith Adam 142 167 868 Smith Anthony 583 604 Smith Bruce 769 Sokoloff Kenneth 858 871 Solon Gary 581n Solow Robert M 26 69 77 79 106 404 Sonin Konstantin 842 857 Sonnenschein Hugo 177 Spence Michael 422 430 Spiegel Mark M 382 384 643 Spletzer James R 612 Stavins Robert 415 430 Stewart Frances 644 Stiglitz Joseph E 355 422 430 625 644 686 Stokey Nancy 177 22223 553 561 604 720 Stone Richard 319 Strahan Philip 770 Summerhill William 142 Summers Lawrence H 604 Summers Robert 3n 12 23 25 91 97 Sundaram Rangarajan 223 Swan Trevor W 26 69 Sydsaeter Knut 277 Sylla Richard 57 70 Tabellini Guido 142 823 824 Taber Christopher 384 Tamura Robert 770 771 Taylor Alan M 68586 Temple Jonathan R W 106 Tenreyro Silvana 588 605 Thoenig Matthias 527 528 686 Thomas Robert 25 141 142 870 Thompson Peter 453 Tirole Jean 355 430 431 490 934 941 Tobin James 274 277 Todaro Michael 770 Tornell Aaron 142 Townsend Robert 726 727 769 Trefler Daniel 1015 107 654 657 Troske Kenneth 612 Tsiddon Daniel 384 Uhlig Harald 462n Uzawa Hirofumi 60 70 404 407 Van Long Ngo 277 Van Reenen Jon 491 Velasco Andes 142 Veliz Claudio 142 Ventura Jaume 176 319 355 656 663 686 Verdier Thierry 142 490 527 528 686 824 857 Vernon Raymond 674 Vickers John 490 Violante Gianluca 384 643 Vishny Robert W 720 752 756 757 771 Vogel Ezra 663 674 Vollrath Dietrich 858 Volosovych Vadym 686 von Neumann John 156 177 318 404 Wacziarg Romain 588 605 Wade Robert 824 Wallace Neil 354 355 Walter Wolfgang 917 930 Wan Henry Jr 70 177 Warner Andrew 678 Watt James 414 415 Weber Max 122 142 Webster David L 866 Weibull Jorgen 824 Weil David N 23 90 9193 94 95 96 106 137 141 143 625 644 770 Weil Philippe 355 Weingast Barry R 142 Weinstein David E 107 Weitzman Martin L 277 Whinston Michael D 69 150 176 177 White Lynn T 869 Wiarda Howard J 142 Williams David 561 Williamson Jeffrey 770 Wilson Charles A 562 Wilson Francis 25 Wong R Bin 867 Wooldridge Jeffery M 106 Wright Randall 562 Xie Danyang 698 699 720 Xu Bin 527 528 Yaari Menahem E 327 346 348 354 355 Yorukoglu Mehmet 384 643 Young Alwyn 25 143 453 663 679 680 681 687 Young Andrew 721 Zeira Joseph 355 761 764 771 Zeldes Stephen P 556 Zilcha Itzak 222 562 Zilibotti Fabrizio 427 453 527 588 598 605 626 630 644 715 737 740 744 746 749 750 769 770 Zin Stanley E 178 Zingales Luigi 142 76970 Zuleta Hernando 721 Zweimuller Josef 720 Subject Index Page numbers for entries occurring in figures are followed by an f those for entries occurring in notes by an n and those for entries occurring in tables by a t admissible pairs 228 228n 23839 241 advanced countries international division of labor 67477 86263 sectoral employment shares in 69798 tax rates in 821 technologies optimized for conditions in 624 625 643 See also crosscountry income differences Africa disease burden in 119 133 European colonies in 135 871 See also lessdeveloped countries agents See households aggregate production function with health capital 137 with human capital 85 in Solow model 26 2829 77 aggregate production possibilities set 15859 AghionHowitt model 46870 agricultural productivity crosscountry differences in 716 employment shifts 715 geographic factors in 11819 industrialization and 71519 in open economies 719 technological change and 865 agriculture consumption expenditures on products of 697 698 699 employment in 697 history of 865 technological change in 716 865 AK model competitive equilibrium of 38992 39394 environment of 38889 with international trade 665 71 neoclassical version of 387 38892 with physical and human capital 39394 policy differences and 392 sustained growth in 5556 56f twosector 39598 altruism impure 34245 353 intergenerational 15758 pure 342 warm glow 34245 731 antitrust policies 44243 appropriability effect 420 429 465 749 appropriate technology 62630 643 ArrowDebreu equilibrium 171 173 602 Arrow securities definition of 172n sequential trading with 17172 173 57779 symmetric 594 Arrows Impossibility Theorem 806 Arrows replacement effect 421 429 Arrows sufficiency conditions 23738 ArzelaAscoli Theorem 89294 Asia economic growth miracles in 2021 117 123 126 663 674 685 European colonies in 13536 871 See also lessdeveloped countries assets bubbles on 342 pricing 560 See also investment securities asymptotic stability 44 augmented Solow model 8589 87f 9293 authoritarian political systems 865 86668 870 See also nondemocratic regimes autocracies 865 balanced growth definition of 57 Harrodneutral technological change and 64 models with 58 in neoclassical growth model with technological change 307 world 862 balanced growth path BGP 65 66 67 balanced portfolios 42829 594 601 Barro growth regression 1516 83 basin of attraction 75960 Bellman equation 185 BenPorath model 36366 365f 366f 383 bequests 34445 See also altruism Berges Maximum Theorem 198 199 213 894 89798 Bernoulli utility functions 149 Bewley model 58385 604 BGP See balanced growth path biased technological change capitalbiased 499 51920 difference from factoraugmenting technological change 500502 importance of 498500 skillbiased 49899 501 501f 512 strong equilibrium relative bias 500 503 510 51718 522 527 unskillbiased 499 weak equilibrium relative bias 500 5023 510 517 522 527 big push model 752 977 978 Subject Index Blackwells sufficient conditions for contraction 19394 block recursiveness 615 BolzanoWeierstrass Theorem 888 borrowing endogenous constraints on 554 See also debt Britain agricultural productivity in 715 democratization in 832 83334 85354 economic growth in 9 economic takeoff in 863 financial development in 869 First Reform Act of 1832 832 854 former colonies of 134 35 Industrial Revolution in 9 715 854 869 sectoral employment shares in 697 BrockMirman model 566 56771 Brouwers Fixed Point Theorem 899900 bubbles 342 business stealing effect 42122 429 465 calculus fundamental theorems of 91920 canonical overlapping generations model 33334 33536 335f capital accumulation of 59899 depreciation of 3132 97 diminishing returns to 29 47 expenditures on 79 health 137 measurement issues 79 overaccumulation in 585 rental rates of 32 share in US GDP 5758 57f in Solow model 31 stock of 596 59798 See also financial capital flows human capital physical capital capitalaugmenting technological change See Solowneutral technology capitalbiased technological change 499 51920 capital deepening 46 398 403 519 704 706 765 capitallabor ratios capital flows and 65354 crosscountry differences 100 effective 65 elasticity of substitution 519 710 711 equalization across countries 651 653 659 factor prices and 101 102 657 inappropriate technologies and 62526 increases in 765 in Solow model 36 38f 4041 capital markets imperfect 762 international 65354 capitalskill complementarity 37174 CassKoopmans model 318 See also neoclassical growth model CES See constant elasticity of substitution CES preferences DixitStiglitz preferences 15253 425 CGP See constant growth path children qualityquantity tradeoff of parents 73233 73436 cities human capital externalities and 379 lack of community enforcement in 741 See also urbanization climate 118 124 See also geography hypothesis CobbDouglas production function 3637 5254 8182 colonies European contracting institutions in 13637 cultural influences of colonizing power 13436 disease environments in 13233 growth takeoff in former 13 14 14f 863 87071 indigenous institutions in 129n 130 institutional differences among 12627 129n 13034 131f institutions imposed by colonizers of 128 29 latitudes of 134 legal systems of 136 property rights institutions in 130 131f 133 134f 135f 136 37 reversals of fortune in 127 12832 129f settler mortality in 13233 134f 135f technological change in 12930 commitment problems 783 784 799 See also holdup problems commodities sequential trading of 17174 See also markets community enforcement 74042 comparative advantage 655 671 685 719 See also Ricardian model of international trade comparative dynamics with basic Solow model 67 68 with standard neoclassical growth model 31315 314f competition among political parties 80910 81213 competition policies 44243 competitive equilibria definition of 16263 in optimal growth problem 21921 Pareto optimal 161 176 in stochastic growth models 57179 symmetric 21921 under uncertainty 57179 welfare theorems 16367 competitive markets 30 162 complete markets 162 566 57172 composition effect 485 487 489 computational tools 221 concave problems 256 25859 27677 912 concavity of functions 898 901 of Hamiltonian 239 of instantaneous payoff function 188 of value function 189 199200 26667 543 553 conditional convergence 1517 83 conditional factor price equalization 101 102 657 660 Condorcet paradox 8067 808 Condorcet winners 807 808 cone of diversification 657 constant elasticity of substitution CES aggregator 423 constant elasticity of substitution CES preferences 15253 425 constant elasticity of substitution CES production function 5455 constant growth path CGP 701 712 constant relative risk aversion CRRA utility function 3089 constant returns to scale 29 constitutional monarchies 865 869 870 constrained optimization 91015 consumption constant growth path of 701 712 Engel curves 151 152 698 699 701 702 715 hierarchies of needs 720 intertemporal elasticity of substitution and 297 love for variety 423 425 nonbalanced sectoral growth 701 702 of nonrenewable resources 25253 optimal plans 209 permanent income hypothesis 554 56 561 relationship to income per capita 78 7f in Solow model 42 consumption Euler equation 209 consumption set 161 Subject Index 979 contingent claims insuring against risk with 566 pricing of 57172 sequential trading with 57779 continuoustime models advantages of 48 perpetual youth model 34753 354 Solow model 4755 6467 stochastic growth 535 continuoustime neoclassical growth model See neoclassical growth model continuoustime optimal growth problem 26869 continuoustime optimization problems 22728 applications of 23335 26974 275 approach 275 existence of solutions 25966 finitehorizon 228 29 infinitehorizon control 24050 Maximum Principle 23539 transversality condition of 232 variational approach 22935 contracting institutions 782 as barrier to technology transfer 686 effects of differences on technology adoption 63041 862 emergence of 869 in former European colonies 13637 future research on 873 influence on economic outcomes 13637 862 in lessdeveloped countries 13637 741 Contraction Mapping Theorem 19094 control variables 183 537 convergence CobbDouglas production function and 81 82 conditional 1517 83 global 82 in optimal growth model 218 219 of policies 805 80910 811 speed of 81 convexity 898 costate variables 23031 236 creative destruction in democracies 857 economic growth and 489 economic institutions and 12021 labor market implications of 47172 losers from 421 productivity growth resulting from 47677 social and political tensions from 8 467 48990 source of 460 uneven growth resulting from 47071 See also innovations new entrants Schumpeterian growth models credit market imperfections 746 758 76164 See also debt crosscountry income differences absolute gap between rich and poor countries 4 5f conditional convergence 1517 83 distribution of GDP per capita 36 4f 5f growth rate differences and 911 growth regressions 8085 human and physical capital investment decisions and 93 86162 human capital differences and 37071 378 380 inappropriate technologies and 62526 630 increasing inequality 46 5f with international trade 67071 67374 in nineteenth and twentieth centuries 1214 13f 14f 15f origins of 1114 per capita 36 910 11f 14 15f persistence of 13940 139f possible perspective on 86472 productivity differences and 96 100 98f 99f proximate causes of 31213 regression analysis using augmented Solow model 9096 92t 93t stability of 11 685 technology differences and 9096 105 timing of growth takeoffs and 603 welfare impact of 79 CRRA See constant relative risk aversion cultural differences hypothesis 20 21 12223 arguments against 130 136 channels affecting economic growth 111 122 distinction from institutional differences 112 evidence in European colonies 13436 culture definition of 111 112 influences on economic behavior 111 122 institutions and 112 measurement issues 12223 religion and 122 135 currentvalue Hamiltonian 254 255 DebreuMantelSonnenschein Theorem 150 debt consumptiondenominated loans 39697 international borrowing and lending 317 international financial capital flows 64853 natural limit 208 29091 noPonzi condition 207 sovereign 654 684 demandside sources of structural change 697703 719 democracy advantages of 857 contrast with nondemocratic regimes 832 definition of 832 dictatorship of workers 83537 direct 779 807 dynamic tradeoffs with oligarchy 83750 857 dysfunctional 850 economic growth in 83234 850 electoral rules of 832 elite political power in 850 emergence of 85051 85356 equilibrium 84546 84850 flexibility of 850 857 indirect 809 industrialization in 83334 Montesquieu on geography and 124 open agenda 807 809 party competition in 80910 81213 political economy model of 80514 political equality in 832 836 political participation in 865 redistributive policies in 833 836 849 850 85456 857 representative 779 demographics See migration population growth urbanization demographic transition 730 73236 764 developing countries See lessdeveloped countries development See economic development development poverty traps 757 760 764 769 dictatorial allocations 162 dictatorships 832 83435 865 See also authoritarian political systems nondemocratic regimes difference equations 93032 linear 44 51 nonlinear 4445 51 differentiability 900907 Frechet 904n Gateaux 904n of instantaneous payoff function 188 of solutions 930 of value function 190 200201 267 543 553 differential equations 92021 continuity of solutions 930 differentiability of solutions 930 linear firstorder 921 24 nonlinear 926 927f separable and exact 92728 systems of linear 92426 systems of nonlinear 926 927f directed technological change factor prices and 50911 Harrodneutral purely laboraugmenting 499 profit incentives and 499500 skill premium and 498 510 51113 511f 513f 514f 517 wage structure and 498 51113 See also biased technological change 980 Subject Index directed technological change models advantages of 497 52627 applications of 52223 baseline 50314 511f with knowledge spillovers 51418 520 521 without scale effects 51819 See also endogenous technology models discounted infinitehorizon optimization problems 25359 275 discounting 256 See also exponential discounting discretetime infinitehorizon optimization 18285 discretetime models neoclassical growth 3056 overlapping generations 32934 perpetual youth 34547 stochastic growth 535 See also Solow model disease burden in European colonies 13233 134f 135f influence on economic outcomes 119 13740 influence on institutional development 13233 134f 135f labor productivity effects of 137 distance to world technology frontier 615 616 74546 747 748f 749f 751f distortionary policies 468 78384 793 8025 822 864 distributional conflicts CobbDouglas model of 79298 political power and 822 in simple society 78492 See also social conflict division of labor economic growth and 868 international 67477 86263 DixitStiglitz aggregator 423 DixitStiglitz model with continuum of products 42526 with finite number of products 42225 limitations of 428 limit prices 42728 loveforvariety feature 423 425 DixitStiglitz preferences See CES preferences Downsian Policy Convergence Theorem 805 80910 811 814 dual economy community enforcement in 74042 modern sector 73637 surplus labor in 737 technologies in 74344 traditional sector 73637 urbanization rates in 736 739 742 742f wages in 73738 740 dynamic general equilibrium models 161 176 dynamic inefficiency in overlapping generations model 33839 35354 in Solow model 4243 dynamic infinitehorizon games 93442 dynamic programming computational tools 221 contraction mapping theorem 19094 importance of 22122 Principle of Optimality 186 189 19798 542 43 54748 sequence problem and 21011 See also stationary dynamic programming stochastic dynamic programming dynastic preferences 158 Eastern Europe 871 economic development big push model 752 capital deepening 765 766f 767f 768f distinction from growth 69395 future research on 87374 institutional influences on longterm 78184 models of 69495 76869 policies blocking 8045 822 871 structural transformations in 694 76468 766f 767f 768f 863 87374 traps 757 economic growth Asian miracles 2021 117 123 126 663 685 balanced 57 58 correlates of 1819 definition of 69394 distinction from development 69395 future research on 87274 links to economic development 76468 in premodern periods 86566 867 868 prenineteenth century 13 588 863 86568 proximate causes of 1920 106 109 31213 regression analysis of determinants of 8384 sustained 5556 56f 863 uneven 47071 winners and losers from 89 See also fundamental causes takeoff growth economic growth rates crosscountry differences in 911 distribution of 9 10f GDP per worker and 1617 16f 17f geometric averages 24 human capital investments and 1819 investment levels and 1819 18f 86162 in nineteenth and twentieth centuries 1214 13f 14f 15f 11214 regression analysis of 8085 technological diffusion and 862 variability of 588 economic institutions distinction from policies 782 distinction from political institutions 782 distortionary policies and 8025 incentives provided by 12021 political institutions and 779 782 85253 853f 85657 preferences over 778 779 783 relationship to economic outcomes 778 782 See also contracting institutions distortionary policies entry barriers institutions property rights institutions tax policies economies of scale 11314 education See human capital investments schooling eigenvalues 91718 eigenvectors 91718 elasticity of substitution 519 710 711 See also constant elasticity of substitution elections See voters electoral laws 782 832 See also political institutions voting elites in democracies 850 economic development blocked by 8045 871 with political power 78384 789 93 79597 832 866 871 property rights protection provided by 8034 reactions to social conflict 85455 See also oligarchy employment sectoral shifts in 715 in Solow model 30 structural change in United States 697 698f See also labor markets endogenous borrowing constraints 554 endogenous growth models 38788 AK model 387 388 92 application to data 403 Romer model 398402 technological diffusion 62123 endogenous political change 85056 endogenous technology models appropriate technology 62630 differences from Romer model 452 with expanding input variety 43346 45152 458 with expanding product variety 44852 generalizations of 522 importance of 45253 862 Joness model 523 Subject Index 981 26 with knowledge spillovers 44448 labequipment model with input varieties 43344 laboraugmenting technological change 52326 limitations of 452 458 497 linearity of 402 policies in 44244 process innovation 433 product innovation 433 44852 Romer model 398402 scale effect in 439 446 technological diffusion 61921 technology adoption with contractual differences 63141 trade liberalization effects on 679 80 uses of 409 See also directed technological change Romer model Schumpeterian growth models Engel curves 151 152 Engels Law 698 699 701 702 715 entrepreneurs distortionary taxes on 791 economic institutions and 804 highskill 746 747 innovations by 747 lowskill 746 747 with political power 814 17 retained earnings of 750 search for ideas 55660 561 social mobility and 83738 technology adoption by 747 8012 in Western Europe 869 See also new entrants entry barriers 479 782 838 Envelope Theorem 190 91415 equilibrium ArrowDebreu 171 173 602 democratic 84546 84850 dynamic 707 entry 84244 844f equalization 675 678 meaning of 43 multiple equilibria models 11415 116 75258 76061 Nash 416 430 939 nonconvergence trap 75051 752f oligarchic 847 49 sclerotic 750 751f 843 844 845f specialization 675 678 static 59394 707 stationary 583 584 585 underinvestment 74850 749f world 616 651 659 66768 See also competitive equilibria Equivalence of Values Theorem 542 Euler equations 2025 212 consumption 209 stochastic 54952 Eulers theorem 2930 Europe See Eastern Europe Western Europe excess sensitivity tests 556 existence theorems 92930 exogenous growth model 61317 expanding variety models 433 input variety 43346 45152 458 product variety 44852 expected utility functions 149 expected utility theory 15657 exponential discounting 148 16061 25359 expropriation distinction from taxation 821 holdup problems 784 799 protection against risk of 123 124f 130 131f 133 134f 135f 802 804 Extended Downsian Policy Convergence Theorem 811 Extended Median Voter Theorem 811 externalities aggregate demand 425 426 75258 human capital 94 37880 383 learningbydoing 681 683 715 716 738 pecuniary 33839 378 383 442 physical capital 399 technological 398 399400 44448 51418 520 679n factoraugmenting technological change 500502 factor price equalization 101 102 657 660 factor price manipulation effect 784 79497 803 805 822 factor proportion differences 704 705 factors of production See capital labor feasible variations 230 FeldsteinHorioka puzzle 653 655 684 68586 felicity function 148 fertility 764 See also population growth fiat money 342 financial capital flows growth and 64853 under imperfect international capital markets 65455 under perfect international capital markets 65354 to poor countries 65355 financial development effects on economic growth 729 model of 72629 risk sharing through 588 599 in Western Europe 869 870 financial intermediaries 592 599 6023 firms optimization problem of 3234 production functions of 158 profit maximization problem of 3233 representative 2728 15859 in Solow model 2728 value of investment to 274 First Welfare Theorem 16366 167 importance of 176 with infinite number of households 16465 nonapplicability to OLG models 32829 339 fixed point theorems 884 Frechet distribution 52526 frontier technologies 609 642 643 skill requirements of 62630 See also technological diffusion world technology frontier functional equations 18586 functions absolute continuity of 892 definition of 881 of several variables 9056 vector 904 fundamental causes of economic growth 1921 analysis with neoclassical growth model 31213 cultural differences 20 21 111 112 12223 130 13436 distinction from proximate causes 106 109 geographic differences 2021 111 11719 12324 importance of investigating 110 luck 20 11011 11417 603 See also institutional differences hypothesis game theory dynamic infinitehorizon games 93442 general equilibrium models ArrowDebreu equilibrium of 171 assumptions in 176 competitive equilibria in 163 dynamic 161 176 economic growth theory and 16167 infinite number of commodities in 31 geography hypothesis 2021 111 11719 arguments against 129 136 disease burden 119 13740 empirical support for 12324 125f latitude and income relationship 12324 125f 134 sophisticated 12930 Geometric HahnBanach Theorem 167 90910 globalization 652 87172 See also international trade golden rule saving rate 42 70 Gorman preferences 151 152 154 308 982 Subject Index Gormans Aggregation Theorem 151 governments See policies political institutions public goods state government spending 34n 317 GrobmanHartman Theorem 926 gross domestic product GDP distribution of per capita 36 4f 5f per capita increases in 910 11f 57 per worker 11 12f 1617 16f 17f See also crosscountry income differences growth See economic growth takeoff growth accounting 78 Habakkuk hypothesis 522 528 HahnBanach Theorem 910 Geometric 167 90910 Hamiltonian 235 concavity of 239 currentvalue 254 255 maximized 237 notation of 235n Hamiltonian dynamical system 236n HamiltonJacobiBellman HJB equation 24344 economic intuitions from 24748 heuristic derivation of 24446 stationary version of 24446 248 HarrodDomar model 26 27 29 Harrodneutral purely laboraugmenting technology 59 59f 60 61 62 64 499 519 health improvements in 13839 730 764 life expectancies at birth 78 8f 13839 138f productivity and 119 relationship to economic growth 13940 See also disease burden health capital 137 HeckscherOhlin international trade theory 101 65563 68485 HeineBorel Theorem 887 hemicontinuity 89597 896f Hicksneutral technology 40 58 59f HJB See HamiltonJacobiBellman equation holdup problems 784 799801 803 822 households budget constraints of 208 29092 296 554 infinite planning horizons of 15658 life cycles of typical 591 591f lifetime budget constraint of 554 local nonsatiation of 16364 maximization problem of 29092 29497 30911 normative representative 150 15355 ownership of factors of production 3031 permanent income hypothesis 55456 561 representative 27 14952 in Solow model 27 strong representative 152 153 154 See also consumption preferences human capital AK model with 39394 in augmented Solow model 8589 87f 9293 crosscountry income differences and 37071 378 380 definition of 85 359 depreciation of 363 firmspecific skills 472 imbalance between physical capital and 367 36970 371 374 37778 383 in imperfect labor markets 37479 in neoclassical growth model 36771 quality of 383 role in technology diffusion 612 61819 62630 stocks of 9697 See also capital schooling human capital externalities 94 37880 383 human capital investment models BenPorath 36366 365f 366f 383 NelsonPhelps 38082 383 human capital investments barriers to 370 782 dynamics of individual decisions 75960 760f estimating 94 in imperfect credit markets 76164 763f income distribution and 75861 onthejob training 366 383 in premodern periods 868 productivity increases from 367 383 618 rates of 91 relationship to economic growth 1819 19f 9293 38283 86162 returns to education 9495 9697 36163 382 498 498f 512 13 schooling decisions 35963 separation theorem 35961 technological change and 38082 383 training 366 383 See also schooling human capital theory 85 359 ideas nonrivalry of 41314 search for 55660 561 See also innovations Implicit Function Theorem 41 907 Inada conditions 33 34f inappropriate technology 62426 630 643 74344 income differences See crosscountry income differences income distribution human capital investments and 75861 world 46 5f 403 615 income inequality crosscountry 46 5f distortionary taxation and 817 Kuznets curve 729 relationship to economic growth 729 wages 528 income per capita consumption per capita and 78 7f crosscountry differences in 36 910 11f 14 15f life expectancy and 78 8f population density and 12728 129f population growth rates and 73032 urbanization rates and 12728 127f 128f incomes demand and 15152 Engel curves 151 152 698 699 701 702 715 See also wages incomplete markets 33839 566 58385 604 individuals See households voters induced innovation 52728 industrialization 1314 bigpush type of 757 in Britain 9 715 854 869 in democracies 83334 distinction from takeoff 24 in nineteenth century 132 715 855 870 political effects of 855 Protestantism and 122 relationship to agricultural productivity 71519 timing of 71920 trade liberalization effects on 719 See also dual economy takeoff growth industrial organization of innovation 472 490 862 873 inefficiency See dynamic inefficiency Pareto inefficiency inequality See income inequality infant industry protection 680 683 InfiniteHorizon Maximum Principle 243 24850 infinitehorizon optimization continuoustime 24050 discounted 25359 discretetime 18285 economic intuitions from 24648 necessary and sufficient conditions 24044 246 nonstationary 21115 transversality condition 246 25053 infinite planning horizons 15658 initial value problems 921 92930 Subject Index 983 inner product 158 innovation possibilities frontier 413 433 43435 444 52728 620 innovations appropriability effect of 420 429 749 cumulative 47980 drastic 41819 excessive 422 429 incremental 473 induced 52728 industrial organization of 472 490 862 873 limit pricing and 419 macro 41213 41415 micro 41213 by new entrants 42122 747 862 nonexcludability 414 417 policies affecting 620 process 41112 433 458 459 product 411 433 44852 profit incentives for 440 452 quality improvements 459 473 479 replacement effect of 42021 429 search for ideas 55660 561 social value of 41920 stepbystep 47989 value in partial equilibrium 41622 See also creative destruction DixitStiglitz model technological change inputs expanding variety models 43346 45152 458 institutional differences hypothesis 20 21 11112 analysis with neoclassical growth model 31517 empirical support for 12325 13334 as factor in takeoff to modern economic growth 86364 86970 importance of investigating 14041 influence on investment decisions 862 influence on technology adoption 862 meaning of term 782 natural experiments 12537 reversals of fortune in former colonies and 13032 131f role of incentives 11921 863 sources of differences 822 tax policy differences 31517 institutions as constraints on individuals 11920 culture and 112 definition of 111 11920 782 endogeneity of 121 extractive 132 865 growthpromoting 865 869 incentives provided by 119 120 313 863 long run development and 78184 political leaders and 117 reforms 112 121 relationship to preferences 77879 resource allocation and 30 societal choices of 782 See also contracting institutions economic institutions policies political institutions property rights institutions integrated world economy 652 87172 intellectual property rights IPR protection composition effect of changes 485 487 489 disincentive effect of changes 485 patents 414 435 443 485 relationship to growth 489 relationship to NorthSouth income gap 678 weak enforcement as barrier to technology transfer 643 interest rates on consumptiondenominated loans 39697 in Solow model 31 32 Intermediate Value Theorem 39 884 international division of labor 67477 86263 international financial capital flows See financial capital flows international product cycle model division of labor 67477 86263 equilibrium in 67576 677f with incomplete contracts 686 technology transfer in 67778 international trade comparative advantage in 655 671 685 719 cone of diversification 657 economic growth with 65563 670 67885 HeckscherOhlin model of 101 65563 68485 income differences with 670 71 infant industry protection 680 683 liberalization of 67980 68384 719 negative growth effects of 679 68083 productivity differences and 1015 103f Ricardian model of 66374 684 685 Rybczynskis Theorem 706 technological diffusion and 67478 862 63 termsoftrade effects 670 67374 684 685 world income distribution and 67071 67374 intertemporal elasticity of substitution 297 intertemporal utility maximization problem 2079 invariant limiting distribution 570 Inverse Function Theorem 9067 investment balanced portfolios 601 endogenous decisions on 86162 financial intermediaries 592 599 602 3 institutional differences and 862 minimum size requirements 58990 590f Pareto efficient portfolio allocations 600602 601f in public goods 81721 q theory of 26974 riskreturn relationship 590 saving rates and 655 subsidies to 750 taxes on returns 313 31516 392 under uncertainty 56061 value to firm 274 See also assets capital human capital investments securities investment goods prices of 316 IPR See intellectual property rights Jacobian matrix 904 Joness model 52326 Kakutanis Fixed Point Theorem 899 Kaldor facts 57 698 702 714 knowledge accumulation of 398 as nonrival and nonexcludable good 398 knowledge spillovers in directed technological change models 51418 520 521 in endogenous technology models 44448 international trade and 679n reduced effect of 44648 KongsamutRebeloXie model 698703 KuhnTucker Theorem 91314 Kuznets curve 729 Kuznets facts 698 702 714 labequipment model with input varieties balanced growth path in 43839 environment of 43336 equilibrium characterization in 43638 innovation possibilities frontier 43435 Pareto optimal allocations in 44042 policy effects in 44244 sources of inefficiency in 442 443 transitional dynamics of 43940 labor diminishing returns to 29 elasticity of substitution between skilled and unskilled 51718 household ownership of 3031 inelastic supply of 3031 measurement issues 79 share in US GDP 5758 57f See also capitallabor ratios wages laboraugmenting technological change 59 62 51922 52326 See also Harrodneutral technology 984 Subject Index labor markets imperfect 37479 implications of creative destruction 47172 population growth and 4851 relationship to technological change 499 search model 561 supply choices 57982 Latin America culture 122 democratization in 850 growth rates in 10 political institutions in 832 871 872 preColumbian civilizations in 127 129n 130 866 repression of social conflict in 855 See also lessdeveloped countries leaders political 117 learningbydoing externalities 681 683 715 716 738 Lebesgue integral 152n LeChatelier principle 511 Leibnizs Rule 920 Leontief production function 5455 lessdeveloped countries appropriate technology for 626 30 643 contracting institutions in 13637 741 debt 654 development traps 757 760 764 769 former European colonies 135 871 inappropriate technologies for 62426 630 643 74344 integration into global economy 87172 international division of labor 67477 86263 lack of capital flows to 65355 market failures in 725 population growth rates of 730 skills available in 62630 variable growth rates of 588 See also cross country income differences dual economy technological diffusion lHˆopitals Rule 39 9023 life expectancies at birth 78 8f 13839 138f limit prices 419 427 744 linear difference equations stability for systems of 44 51 linear differential equations firstorder 92124 systems of 92426 local nonsatiation 16364 loveforvariety feature 423 425 luck hypothesis 20 11011 11417 drawbacks of 116 17 formalization of 603 multiple equilibria and 11415 116 Malthusian model 73032 733f Mangasarians sufficiency conditions 23637 manufacturing sector 697 699 716 See also industrialization market failures 725 752 markets competitive 30 162 complete 162 566 571 72 credit 746 758 76164 financial development 588 599 72629 869 870 incomplete 33839 566 583 85 604 sequential trading 17174 57779 See also commodities labor markets stock market market size effect direction of technological change and 500 508 510 51314 51819 distinction from scale effect 518 innovation and 414 415 416 on technology adoption 634 Markov chains 538 Markovian models 760 764n Markov Perfect Equilibrium MPE comparison to Subgame Perfect Equilibria 936 937 definition of 937 existence of 93839 in political economy model 790 in repeated games 941 in stepbystep innovation model 482 versus Subgame Perfect Equilibria 799802 939 Markov processes 538n 55253 martingales 556 maximized Hamiltonian 237 Maximum Principle for discounted infinitehorizon problems 25456 economic intuitions from 24648 275 infinitehorizon 243 24850 for multivariate problems 23940 simplified 23536 terminal value constraint on 276 McCall labor market search model 561 Mean Value Theorem 902 Median Voter Theorem MVT 805 8069 810 814 extended 811 with strategic voting 809 metric spaces 191 87880 88183 middle class emergence of 869 870 with political power 79798 83435 migration during economic development 736 model of 73740 See also dual economy urbanization Mincer equation 94 96 36263 minimum size requirements 58990 590f monarchies absolutist 865 constitutional 865 869 870 Spanish 86768 monopolistic firms antitrust policies 44243 political power of 468 profit maximization objective of 42627 452 monopoly power of innovating firm 41822 427 monotonicity of instantaneous payoff function 188 of value function 190 200 544 553 moral hazard 585 740 770 873 mortality rates 133 730 764 See also disease burden MPE See Markov Perfect Equilibrium multifactor productivity See total factor productivity multiple equilibria models aggregate demand externalities 75258 differences from multiple steadystate models 76061 luck hypothesis and 11415 Paretoranked equilibria 116 multiple steady state models 116 117 75861 764 764n multivariate problems Maximum Principle for 23940 sufficiency conditions for 240 MVT See Median Voter Theorem Nash equilibria 416 430 939 natural debt limit 208 29091 298 natural resources 21n 111 NelsonPhelps model of human capital 38082 383 neoclassical growth model advantages of 311 318 AK model 387 38892 applications of 317 canonical 30911 comparative dynamics with 31315 314f comparative static results of 301 comparison to Solow model 27 318 competitive equilibrium of 293 299 Subject Index 985 300 consumption behavior in 29798 in continuous time 287 discounting assumption in 288 discount rate and saving rate 301 in discrete time 3056 environment of 28789 equilibrium characterization in 29398 explanations of crosscountry income differences 403 extensions of 317 household maximization problem in 29092 29497 30911 infinite planning horizons of households in 15658 with labor supply 57982 linearity of 402 normative representative household in 15355 optimal growth problem and 298304 with physical and human capital 36771 preference orderings of 14749 preferences in 287 problem formulation in 16061 proximate and fundamental causes of growth 31213 quantitative evaluation of 31517 Ramsey model 318 representative firm assumption in 158 59 representative household assumption in 14952 sequential trading in 17174 steadystate equilibrium in 300301 30910 with technological change 30612 transitional dynamics of 3024 303f with uncertainty BrockMirman model 566 56771 uniqueness of equilibrium in 3024 311 use of 287 welfare theorems in 16171 nets 255 new entrants aggregate demand externalities and 426 barriers to 479 782 838 business stealing effect 421 22 429 465 free entry by 475 823 fringe of potential competitors 419 427 44243 highskill entrepreneurs 746 innovation by 42122 747 862 productivity growth by 47279 research and development by 46061 See also creative destruction entrepreneurs noarbitrage conditions 575 nonbalanced sectoral growth See structural change nonconvergence trap 75051 752f nonconvexities 589 590 nondemocratic regimes authoritarian 865 86668 870 contrast with democratic regimes 832 economic growth in 83234 elite rule in 832 Montesquieu on geography and 124 variations 832 See also dictatorships oligarchy nonexcludability 28 398 414 417 nongrowthenhancing policies See distortionary policies nonlinear difference equations local stability for systems of 4445 51 nonlinear differential equations 926 927f nonrenewable resources 25253 nonrival goods 28 398 nonrivalry of ideas 41314 nonstationary infinitehorizon optimization 21115 noPonzi condition 207 29192 296 31819 normative representative households 150 15355 normed vector spaces 90710 OLG models See overlapping generations models oligarchy 865 British 832 dynamic tradeoffs with democracy 83750 857 equilibrium 84749 longrun inefficiency of 849 policy decisions in 84649 See also authoritarian political systems omitted variable bias 93 optimal control theory 227 23839 optimal growth model 21819 optimal growth paths Second Welfare Theorem and 167 optimal growth problem 17475 application of stationary dynamic programming 2067 competitive equilibrium in 21921 in continuous time 26869 in discrete time 21519 3056 existence of solutions 25966 of neoclassical economy 21519 optimality principle of 186 189 19798 24243 54243 54748 optimal plans 18384 186 189 209 optimal stopping problems 561 output per worker 6 6f See also gross domestic product overlapping generations OLG models advantages of 327 applications of 354 baseline 32934 35354 canonical 33334 33536 335f capitalskill complementarity in 37174 competitive equilibrium of 331 33639 consumption 33031 in continuous time 34753 352f 354 in discrete time 32934 dynamic inefficiency in 33839 35354 financial development model 72629 with impure altruism 34245 353 37174 non applicability of First Welfare Theorem to 32829 339 overaccumulation in 33639 353 Pareto optimality of competitive equilibrium in 33639 restrictions on utility and production functions of 33234 savings 33031 with social security 33942 354 steadystate equilibria of 33134 332f stochastic 56667 58688 587f with warm glow preferences 34245 See also perpetual youth model Pareto distribution 524 Pareto inefficiency distinction from nongrowthenhancing policies 783 in political economy models 800 801 Pareto optimal allocations decentralization as competitive equilibria 16671 176 definition of 153 163 normative representative household and 15354 in Romer model 4012 See also optimal growth problem Pareto optimal equilibria 161 176 participatory regimes 865 869 870 871 See also democracy party competition 80910 81213 patents 414 435 443 485 See also intellectual property rights protection payasyougo social security system 339 34042 pecuniary externalities 33839 378 383 442 perfect monitoring games 934 permanent income hypothesis 55456 561 perpetual inventory method 97 perpetual youth model 156 327 in continuous time 34753 352f 354 in discrete time 34547 986 Subject Index physical capital AK model with 39394 depreciation of 97 imbalance between human capital and 367 36970 371 374 37778 383 investments and economic growth rates 1819 18f 9293 86162 in neoclassical growth model 36771 See also capital physical capital externalities 399 Picards Theorem 929 plans 187 541 feasible 542 See also optimal plans Poisson death model See perpetual youth model policies child labor laws 380 competition 44243 convergence of 805 80910 811 distinction from economic institutions 782 distortionary 468 783 84 793 8025 822 864 economic development blocked by 8045 822 871 in endogenous technology models 44244 as factor in takeoff to modern economic growth 86364 growthenhancing 823 86364 holdup problems of 784 799801 infant industry protection 680 683 investment subsidies 750 mappings to allocations 779 political conflicts over 468 48990 preferred 807 public goods provision 81721 research subsidies 442 478 620 817 in Schumpeterian growth models 46768 47879 48990 in stationary dynamic programming 185 technology adoption barriers 872 See also intellectual property rights protection tax policies policy correspondences 186 policy functions 186 190 political economy analysis with neoclassical growth model 778 collective decision making 783 commitment problems 783 784 799 conflicts among societal interests 121 140 777 78283 822 future research on 874 growthenhancing policies 823 86364 leaders influence on economic growth 117 models of 140 tensions from economic growth 89 421 winners and losers 792 See also institutions policies political economy models CobbDouglas 79298 83437 dynamics of political institutions 85253 853f 855 56 dynamic tradeoffs between regimes 83750 857 with heterogeneous preferences 80514 probabilistic voting model 81214 public goods provision 817 21 of simple society 78492 tax policy decisions with heterogeneous voters 81417 political institutions distinction from economic institutions 782 dynamic model of 85253 853f 85556 dynamic tradeoffs between 83750 857 endogenous change in 85056 864 geographic differences and 124 impact on economic growth 83237 857 influence of social conflict 822 850 853 855 mapping to economic institutions 779 782 85253 853f 85657 participatory 865 869 870 871 power distribution and 77778 870 preferences over 778 See also democracy institutions nondemocratic regimes political party competition 80910 81213 political power de facto 85152 de jure 85152 855 distributional conflicts and 78492 822 distribution of 77778 822 823 870 of elites 78384 789 93 79597 832 866 871 of entrepreneurs 81417 factors influencing distribution of 778 of middle class 79798 83435 of monopolistic firms 468 support of nongrowthenhancing policies 78384 822 political replacement effect 784 79798 822 political stakes 798 Ponzi games 207 292 34142 See also noPonzi condition poor countries See lessdeveloped countries population density economic institutions and 130 32 131f relationship to income per capita 12728 129f population growth demographic transition 730 73236 764 differences in rates of 72930 730f economic growth and 113 health improvements as cause of 139 Malthusian model of 73032 733f relationship to technological change 11314 in Solow model 4851 See also scale effects poverty traps See development traps preferences CES DixitStiglitz 15253 425 dynastic 158 Gorman 151 152 154 308 induced 77879 787 orderings 14749 over economic institutions 778 779 783 over political institutions 778 relationship to institutions 77879 singlecrossing property of 811 of voters 80710 822 warm glow 34245 price effect direction of technological change and 500 508 510 price index ideal 42324 prices asset 560 limit 419 427 744 Principle of Optimality 186 189 19798 24243 54243 54748 Prisoners Dilemma 942 private return to schooling 37980 probabilistic voting model 81214 Probabilistic Voting Theorem 814 process innovations 41112 433 458 459 See also innovations product cycles international 67478 86263 product innovations 411 433 44852 See also innovations production functions CobbDouglas 3637 5254 81 82 constant elasticity of substitution 5455 with health capital 137 with human capital 85 Leontief 5455 meta 413 in Solow model 26 2829 77 technology 413 production structure change in 74451 764 productivity crosscountry differences in 1015 103f 624 differences within countries 61113 642 effects of disease burden 137 Hicksneutral 40 human capital investments and 367 383 618 in manufacturing sector 716 naıve estimation approach 100101 1024 103f 104f relationship to earnings 95 trade liberalization and 679 Trefler estimation approach 1015 103f 104f See also agricultural productivity total factor productivity Subject Index 987 productivity growth creative destruction and 47677 models of 47279 role of innovation 41213 433 products expanding variety models 44852 product topology 88991 profit motives technological change and 41416 property rights institutions 782 emergence of 8034 869 in former colonies 130 131f 133 134f 135f 136 37 importance of 120 limits on policy choices 802 protection against expropriation risk 123 124f 130 131f 133 134f 135f relationship to economic growth 123 124f 13637 See also intellectual property rights protection proximate causes of economic growth 1920 106 109 31213 public goods economic growth and 817 821 823 nonrival and nonexcludable 28 398 414 provision of 81721 823 pure 414 qtheory of investment 26974 RD See research and development Real Business Cycle RBC models 566 57982 Rebelo model 39598 444 671 702 repeated games 94142 replacement effect 42021 429 representative firm 2728 15859 representative household assumption of 27 14952 159 176 maximization problem of 220 normative 150 15355 strong 152 153 154 research and development RD cumulative 460 employment 444 investors in firms 42829 knowledge spillovers from past 44448 51418 520 679n subsidies to 442 478 620 817 taxes on spending 467 68 47879 uncertainty in 42829 See also innovations technological change resource allocations 162 dictatorial 162 institutional structures and 30 See also optimal growth problem revenue extraction effect 784 791 79597 798 803 805 822 reverse causality 93 Ricardian model of international trade economic growth implications of trade 684 685 environment of 66465 general 67174 simplified 66371 Riemann integral 919 risk aggregate 726 diversification of 566 595 59899 726 727 idiosyncratic 566 599 726 727 relationship to returns 590 sovereign 654 655 684 riskless arbitrage 575 Romer model 398402 competitive equilibrium of 400 401 environment of 399400 knowledge accumulation in 452 learningbydoing externalities 681 716 738 parallels to endogenous technology models 452 Pareto optimal allocations in 4012 scale effect in 401 439 Roys identity 151 rural areas community enforcement in 74042 See also agriculture dual economy Rybczynskis Theorem 706 saddlepath stability 269 27172 302 SaddlePoint Theorem 91112 saving rates correlation with investment rates 655 golden rule 42 70 relationship to discount rate 301 in Solow model 27 35 301 scale effects 401 439 direction of technological change and 51314 518 distinction from market size effect 518 growth without 44648 in technology adoption 414 schooling college premium 498 498f 51213 effects of child labor laws 380 external return to 380 measurement issues 24 private return to 37980 relationship to earnings 9495 96 relationship to economic growth 18 19f 2425 returns to 9495 96 97 36163 382 universal 854 See also human capital investments Schumpeterian growth models 45859 advantages of 468 AghionHowitt model 46870 applications of 490 balanced growth path in 46365 baseline 459 68 489 490 equilibrium in 46163 extensions of 490 limitations of 472 490 onesector 46872 Pareto optimal allocations in 46567 policies in 46768 478 79 productivity growth by incumbents and entrants 47279 stepbystep innovation 47989 See also creative destruction sclerotic equilibrium 750 751f 843 844 845f search for ideas 55660 561 Second Welfare Theorem 163 16667 907 application to optimal growth problem 175 importance of 176 proof of 16871 securities balanced portfolios of 42829 594 complex 602 prices 31 See also Arrow securities assets interest rates investment semiendogenous growth models 448 Separating Hyperplane Theorem 910 separation theorems 35961 90710 sequence problem 21011 sequences 88183 sequential trading 17174 57779 services sector consumption spending in 69899 employment in 697 sets See metric spaces Shapley value 635 636 64041 Shell model 32829 SimonKremer model 11314 simple society definition of 78485 model of 78492 Simplified Maximum Principle 23536 sincere voting 807 808 singlecrossing property 811 skillbiased technological change 49899 501 501f 512 988 Subject Index skill premium 498 510 51113 511f 513f 514f 517 skills relative supply of 49899 498f See also human capital Slater condition 911 social capital 122 social conflict 121 140 777 78283 CobbDouglas model of 79298 elite reactions to 85455 influence on institutions 822 850 853 855 repression of 855 856 simple society model of 78492 urbanization rates and 855 social mobility 83738 social planner See optimal growth problem social security fully funded 33940 in overlapping generations model 33942 354 unfunded 339 34042 social welfare function weighted 812 societies dysfunctional 122 heterogeneity of 80514 mobility in 83738 simple society model 78492 structural transformations in 736 See also culture political economy Solow model aggregate production function in 26 28 29 77 application to data 79 9096 105 augmented version with human capital 8589 87f 9293 capital labor ratio in 36 38f 4041 comparative dynamics with 6768 comparison to neoclassical model 27 318 in continuous time 4755 6467 in discrete time 27 34 47 5664 economic development 76568 endowments in 3032 environment of 2734 equilibrium difference equation of 37 equilibrium in 3234 3543 firm optimization problem in 3234 fundamental law of motion of 3435 growth accounting framework 7780 growth regressions with 8085 growth sources in 81 saving rate in 27 35 301 simplicity of 2627 steady state equilibrium in 3743 38f 39f 40f 47 stochastic form of 588 strengths and weaknesses of 105 sustained growth in 5556 technological diffusion in 61319 with technological progress 5667 78 81 technology in 28 transitional dynamics of 4347 47f 5155 value of 6869 Solowneutral technology 58 59f 62 sovereign risk 654 655 684 spaces dual 909 metric 191 87880 88183 normed vector 90710 topological 88589 vector 898 SPE See Subgame Perfect Equilibrium specialization in international trade 665 675 678 stability asymptotic 44 global 45 46 47 local 4445 51 saddlepath 269 27172 302 stable arm 271 3024 303f standards of living crosscountry differences in 79 state balance of powers with citizens 821 capacity of 797 798 consensually strong 821 strong 818 82021 823 weak 818 82021 See also governments political institutions state dependence 51415 517 520 state variables 183 537 stationary dynamic programming applications of 20111 assumptions in 18788 basic equations of 2029 Euler equations 2025 functional equations 18586 optimal growth problem 216 policy functions 186 recursive formulation 18586 190 221 theorems of 18790 194201 transversality condition 2035 stationary problems 18485 stepbystep innovations 47989 stochastic correspondence 587 587f 596 597f stochastic dynamic programming applications of 55461 with expectations 53744 general Markov processes 55253 proofs of theorems 54449 theorems of 54244 transversality condition 550 stochastic Euler equations 54952 stochastic growth models applications of 535 57982 Bewley model 58385 604 BrockMirman 566 567 71 in continuous time 535 in discrete time 535 equilibrium growth under uncertainty 57179 of long run growth 588603 overlapping generations models 56667 58688 587f Solow model 588 stochastic permanent income hypothesis model 55456 561 stock market 42829 594 See also capital markets securities StoneGeary preferences 319 strategic voting 807 structural change agricultural productivity and 71519 definition of 694 demandside sources of 697703 719 in economic growth 695 863 KongsamutRebeloXie model 698703 supplyside sources of 70315 719 technological causes of 70315 See also industrialization structural transformations 725 definition of 694 demographic transition 730 73236 economic takeoff and 86870 factors slowing 742 financial development 588 599 72629 869 870 future research on 873 74 migration 736 in organizations 751 to production structure 74451 764 social and living arrangements 736 social tensions caused by 89 sustained growth and 863 87172 See also economic development urbanization Subgame Perfect Equilibrium SPE 416 comparison to Markov Perfect Equilibria 936 937 definition of 936 existence of 939 versus Markov Perfect Equilibria 799 802 939 payoffs in 93941 symmetric 635 636 637 in technology adoption model 633 subsidies to investment 750 to research 442 478 620 817 subsistence level of agricultural consumption 699 supplyside sources of structural change 70315 719 takeoff growth causes of 11214 explanation in stochastic growth models 588 59899 603 institutional and policy choices allowing 86364 population growth and 113 14 structural change model and 71519 structural Subject Index 989 transformations and 86870 timing of 603 in Western Europe 12 1314 14f 588 603 863 86870 in West European offshoots 1314 14f 863 87071 See also economic growth tax policies AK model and 392 analysis with neoclassical growth model 31315 314f capital returns taxes 313 31516 392 chosen by elites 79091 79596 decision models 81417 distinction from expropriation 821 distortionary 787 798 817 effects on crosscountry income differences 31517 as entry barriers 838 human capital investment taxes 370 limits on policy choices 802 preferred rates 81516 redistributive 313 784 793 814 81617 833 836 849 85456 taxes on RD spending 46768 47879 Taylors Theorem 903 906 technological change in agriculture 716 865 balanced 7045 capitalaugmenting 58 62 factoraugmenting 500502 future research on 873 Harrodneutral purely laboraugmenting 59 59f 60 61 62 64 499 519 Hicksneutral 58 59f history of 869 imitation and innovation levels 74547 75051 laboraugmenting 59 62 51922 52326 labor markets and 499 learningby doing externalities 681 683 local innovations 615 in manufacturing 716 monopoly power of innovating firm 41822 427 neutral 5859 59f in nineteenth century 512 population growth and 11314 production costs reduced by 41112 458 profit motives and 41416 quality improvements 412 458 scientific breakthroughs 41416 in Solow model 5667 78 81 Solowneutral 58 59f 62 in standard neoclassical growth model 306 12 supplyside sources of structural change 70315 trade liberalization effects on 67980 68384 types of 41112 Uzawas Theorem 5964 value of innovation to firm 41622 See also biased technological change creative destruction directed technological change DixitStiglitz model endogenous technology models industrialization innovations technological diffusion advantages for backward economies 642 balanced world growth and 862 barriers to 617 619 63031 63940 benchmark model of 61319 distance to world technology frontier and 615 616 74546 em pirical data on 61113 endogenous growth and 61923 to European colonies 12930 explanations of cross country differences 62324 future research on 873 human capital role in 61819 62630 international prod uct cycle model 67478 international trade and 67478 86263 level differences 61617 models of 60910 611 speed of transfer process 615 64243 Sshape of 613 in twentieth century 871 from world technology frontier 615 862 See also technology adoption technological spillovers 398 399400 44448 51418 520 679n technology appropriate 62630 643 crosscountry income differences and 9096 105 crosscountry variations in 19 861 862 differences within countries 61113 642 inappropriate 62426 630 643 74344 increasing returns to scale 414 meaning of 19 nonrivalry of ideas 41314 orthogonal 91 9394 technology adoption contracting institutions and 63041 862 costs of 620 determinants of decisions 612 862 effects of economic institutions 803 804 entrepreneurs decisions 8012 human capital and 38082 383 612 model of 63141 policies blocking 872 relationship to economic growth 861 862 See also technological diffusion technology transfer See technological diffusion terminal value constraint 276 29192 termsoftrade effects 663 670 67374 684 685 TFP See total factor productivity time consistency 14849 timeseparable utility 148 Tobins q 274 topological spaces 88589 topology continuity and compactness 88589 product 88991 total factor productivity TFP calibrating differences across countries 96100 98f 99f causes of differences in 1056 differences within countries 612 expected 596 97 growth of 78 measurement issues 403 procyclical nature of 581 trade See international trade training 366 383 See also human capital investments transitional dynamics of equilibrium difference equation 4344 of labequipment model with input varieties 439 40 in qtheory of investment 272 of Solow model 4347 47f 5155 of standard neoclassical growth model 3024 303f of world economy 65253 transversality condition of continuoustime optimization problem 232 for discounted infinitehorizon problems 255 256 Euler equations and 212 in infinite dimensional problems 2035 for infinitehorizon optimization problems 246 25053 market value version of 296 noPonzi condition and 296 sequence problem and 21011 for stochastic dynamic programming 550 stronger 256 weaker 255 Turnpike Theorems 219 twosector AK model 39598 Tychonoffs Theorem 198 213 891 uncertainty aggregate shocks as source of 566 571 investment under 56061 in research and development 42829 See also risk unit cost functions 66667 United States democracy in 849 871 economic growth in 9 economic institutions in 87071 income per capita in 3 relative labor scarcity in nineteenth century 522 sectoral employment shares in 697 698f settlers of 87071 990 Subject Index urbanization barriers to mobility and 73738 during economic development 736 764 in Western Europe 869 870 See also cities urbanization rates in dual economy 736 739 742 742f economic institutions and 13032 131f relationship to income per capita 12728 127f 128f social conflict and 855 utility functions Bernoulli 149 constant relative risk aversion CRRA 3089 expected 149 exponential discounting in 148 16061 indirect 806 instantaneous 148 time separability in 148 von Neumann Morgenstern 149 Uzawas Theorem 5964 value function concavity of 189 199200 26667 543 553 differentiability of 190 200201 267 543 553 monotonicity of 190 200 544 553 uniqueness of 189 19899 value of marginal product 509 variational approach to continuoustime optimization problems 22935 variations 230 vector functions 904 vector spaces 898 von NeumannMorgenstern utility functions 149 voters aggregating preferences of 806 810 median 808 810 822 singlepeaked preferences of 80710 weakly dominant strategies 809 See also democracy Median Voter Theorem voting Condorcet paradox 8067 808 electoral laws 782 832 probabilistic model of 81214 sincere 807 808 strategic 807 8089 wages in dual economy 73738 740 inequality of 528 in international product cycle model 676 677f 678 procyclical nature of 581n relationship to years of schooling 94 skill premium 498 510 51113 511f 513f 514f 517 in Solow model 31 See also incomes Walrass Law 31 weak states 818 82021 823 Weierstrasss Theorem 198 213 88889 welfare theorems See First Welfare Theorem Second Welfare Theorem Western Europe democratization in 850 85354 demographic transition in 730 feudal relations in 869 870 growth takeoff in 12 1314 14f 588 603 863 86870 industrialization in 122 offshoots of 12 13f Protestantism in 122 social conflict in 855 urbanization in 869 870 See also advanced countries colonies European world equilibrium 616 651 659 66768 world income distribution See crosscountry income differences world technology frontier 381 614 745 862 distance to 615 616 74546 747 748f 749f 751f