Recursive Methods in Economic Dynamics - Stokey Lucas Prescott - Livro Completo

Recursive Methods in Economic Dynamics NANCY L STOKEY AND ROBERT E LUCAS JR with Edward C Prescott Harvard University Press Cambridge Massachusetts and London England Copyright 1989 by the President and Fellows of Harvard College All rights reserved Printed in the United States of America Fifth printing 1999 This book is printed on acidfree paper and its binding materials have been chosen for strength and durability Library of Congress CataloginginPublication Data Stokey Nancy L Recursive methods in economic dynamics Nancy L Stokey and Robert E Lucas Jr with the collaboration of Edward C Prescott p cm Includes index ISBN 0674750969 alk paper 1 Economics Mathematical 2 Dynamic programming I Lucas Robert E Jr II Prescott Edward C III Title HB1355745 1989 3380151dc19 8837681 CIP To our parents Universita Karlova V Praze CEP RGE J1ZIVROHa Pollickyor vezni 7 Preface This book was motivated by our conviction that recursive methods should be part of every economists set of analytical tools Applications of these methods appear in almost every substatice area of economics the theory of investment the theory of the consumer search theory public finance growth theory and so on but neither the methods nor the applications have ever been drawn together and presented in a systematic way Our goal has been to do precisely this We have attempted to develop the basic tools of recursive analysis in a mathematically rigorous way while at the same time stressing the wide applicability of recursive methods and suggesting new areas where they might usefully be exploited Our first outlines for the book included a few chapters devoted to mathematical preliminaries followed by numerous chapters treating the various substatice areas to which mathematical recursive methods have been applied We hoped to keep the technical material to a minimum by simply citing the existing literature for most of the required mathematical results and to focus on substantive issues This plan failed rather quickly as it soon became apparent that the reader would be required either to take most of the important results on faith or else to keep a dozen mathematics books close at hand and refer to them constantly Neither apparent nor the overall structure of the book The methods became the organizing principle and we were led to make major alterations in the overall structure of the book The methods became the organizing principle and we began to focus on providing a fairly comprehensive rigorous and selfcontained treatment of the tools and basic techniques in recursive analysis We then found it natural to group applications by the nature of the technical tools involved rather than by their economic substance Thus Parts IIIV of the book deal with deterministic models stochastic models and equilib Contents Symbols Used I THE RECURSIVE APPROACH 1 Introduction 2 An Overview 21 A Deterministic Model of Optimal Growth 22 A Stochastic Model of Optimal Growth 23 Competitive Equilibrium Growth 24 Conclusions and Plans II DETERMINISTIC MODELS 3 Mathematical Preliminaries 31 Metric Spaces and Normed Vector Spaces 32 The Contraction Mapping Theorem 33 The Theorem of the Maximum 4 Dynamic Programming under Certainty 41 The Principle of Optimality 42 Bounded Returns 43 Constant Returns to Scale 44 Unbounded Returns 45 Euler Equations III STOCHASTIC MODELS 7 Measure Theory and Integration 71 Measurable Spaces 72 Measures 73 Measurable Functions 74 Integration 75 Product Spaces 76 The Monotone Class Lemma 77 Conditional Expectation 8 Markov Processes 81 Transition Functions 82 Probability Measures on Spaces of Sequences 83 Iterated Integrals 84 Transitions Defined by Stochastic Difference Equations 9 Stochastic Dynamic Programming 91 The Principle of Optimality 92 Bounded Returns 93 Constant Returns 94 Unbounded Returns 95 Stochastic Euler Equations 96 Policy Functions and Transition Functions 10 Applications of Stochastic Dynamic Programming 101 The OneSector Model of Optimal Growth 102 Optimal Growth with Two Capital Goods 103 Optimal Growth with Many Goods 104 Industry Investment under Uncertainty 105 Production Inventory Accumulation 106 Asset Prices in an Exchange Economy 107 A Model of Search Unemployment 108 The Dynamic Unemployment Model 109 Variations on the Search Model 1010 A Model of Job Matching 1011 Job Matching and Unemployment 11 Strong Convergence of Markov Processes 111 Markov Chains 112 Convergence Concepts for Measures 113 Characterizations of Strong Convergence 114 Sufficient Conditions for Strong Convergence 12 Weak Convergence of Markov Processes 121 Characterizations of Weak Convergence 122 Distribution Functions 123 Weak Convergence of Distribution Functions content and the style of the final product We are also indebted to Ricard Torres whose comments on the manuscript led to many improvements and in several places to major revisions along lines he proposed We owe special thanks to Michael Aronson whose patience and enthusiasm have supported this project from its beginningmore years ago than any of us cares to remember We are grateful too to Jodi Simpson whose editing led to many refinements of style and logic her skillful work is much valued June Nason began typing our early drafts on an IBM Selectric and stayed to finish the job on a LaserJet printer We appreciate her cheerful assistance and the fact she showed by never asking how a job could remain urgent for six years Finally we would like to thank Mary Ellen Geer for helping us see the book through to its completion viii Preface rium theory respectively with substantive applications appearing in all three places Indeed many of the applications appear more than once with different aspects of the same problem treated as the appropriate tools are developed Once we decided to write the book focused on analytical tools rather than economic models the choice of techniques evolved become more important than ever We wanted the book to be rigorous enough to be useful to researchers and at the same time to be accessible to as wide an audience as possible In pursuing these twin goals we aimed for a rigorous and fairly selfcontained treatment of analytical tools but one that requires relatively little by way of mathematical analysis and should have had a course in advanced calculus or real analysis and some probability theory also deltaepsilon arguments A little background should be comfortable with although not at all essential The treated in a largely selfcontained way Our first introduction was as level at which to restrict names that ariseand there are a wide varietyare The majority clearly decisions fully maintained way choosing the appropriate to The of The at slight We end we number measurement sure in theory and our treatment yields This means elsewhere well The reader will econometrics uncertaintybecome measure repeatedly in standard language of The term recursive methods is broad enough to include a variety of interesting tools that might have been included in the book but are not There is a large literature on recursive utility for example but we treat that topic only in Chapter 9 except for examples discussed briefly in Chapters 4 and 8 we ignore that There is also a growing body of expertise on methods for the numerical solution of recursive models that we have not attempted to incorporate into volume Although models of equilibrium and related recursive methods can be analyzed by recursive methods our examples of equilibrium are almost exclusively competitive We have included a large collection of applications but we certainly have not exhausted the many applied literatures where recursive methods are being used Yet these omissions are not we feel cause for apology The book is longer enough as it is and we will certainly not be disappointed if one of the functions is to stimulate the reader to a more serious exploration of some of these closely related areas We have tried to write this book in a way that make full for several different types of readers Those who are familiar with dynamic economic models and dynamic optimization and control theory are invited simply to consult the table of contents and proceed to the particular topics that interest them We have tried to make chapters and sections sufficiently selfcontained for the book can be used in this way We turn next to dynamic models The manuscript has at a variety of stages been used for graduatelevel courses at Chicago Minnesota Northwestern and elsewhere We have taught and will teach it to midlevel from students and The book is about right length and level for a yearlong course for secondyear students but can easily be adapted for shorter courses as well After the introductory material in Chapters 1 and 2 it is probably advisable to omit Chapters 6 and 7 Section 41 on general discuss an general adviser and then choose a few applications from Chapter 5 For a onequarter course there are then several possibilities One could skip to Chapters 15 and 16 and if time permits go on to 17 and 18 Also to be covered with measured is then to applications Chapter 3 Section 31 then 81 then 92 and Application from Section 10 Covering the required measure theory Sections 7073 takes about three weeks and could be done in a onesemester book course A consistent thread throughout the book technically selfcontained is that completing it involved a much higher ratio of exposition to new results than any of us had anticipated Ed Prescott found he did not wish to spend so much of his time away from the research frontier and so proposed what we called a cooperation with the involvement correspondence expresses the phrase with the collaboration of However there is no part of the book that has not benefited from his ideas and contributions We are also to many friends and colleagues for their comments and criticisms Hugo Hoppenhayn Larry Jones Lars Ljungquist Rodolfo Lars Hansen Masao Ogaki Jose Victor RiosRull and Jose Scheinkman for Manuell Arthur Kupfierman read large portions of the manuscript at an early stage and his detailed comments enhanced both the Symbols Used xX element AB AB subset strict subset AB AB superset strict superset empty set intersection AB difference defined only if AB Ac complement closure Int interior boundary indicator function Cartesian product real numbers extended real numbers ddimensional Euclidean space R R subspace of R containing nonnegative vectors strictly positive vectors R R a b a b open interval closed interval B B Borel intervals Bx Borel subsets of R of R Borel subsets of X defined for XB ρx y distance norm x norm CX space of bounded continuous functions on X f f positive and negative parts of the function f f f finite sequence fi1 fi infinite sequence xii1 xii1 7 Symbols Used converges converges from below converges from above measurable space space of measurable realvalued functions subset of MX B containing nonnegative functions space of bounded measurable realvalued functions on X B LX B μ space of μintegrable functions on X B ΛX B space of probability measures on X B μae except on a set A with μA 0 mutually singular absolutely continuous with respect to converges in the total variation norm converges weakly product σalgebra PART I The Recursive Approach 8 Contents 13 Applications of Convergence Results for Markov Processes 375 124 Monotone Markov Processes 383 125 Dependence of the Invariant Measure on a Parameter 386 126 A Loose End 389 131 A DiscreteSpace s s Inventory Problem 389 132 A ContinuousState s S Process 390 133 The OneSector Optimal Growth 391 134 Industry Investment under Uncertainty 395 135 Equilibrium in a Pure Currency Economy 397 136 A Pure Currency Economy with Linear Utility 401 137 Credit Economy with Linear Utility 402 138 An Equilibrium Search Economy 404 14 Laws of Large Numbers 414 141 Definitions and Preliminaries 416 142 A Strong Law for Markov Processes 425 IV COMPETITIVE EQUILIBRIUM 15 Pareto Optima and Competitive Equilibria 441 151 Dual Spaces 445 152 The First and Second Welfare Theorems 451 153 Issues in the Choice of a Commodity Space 458 154 Inner Product Representations of Prices 463 16 Applications of Equilibrium Theory 475 161 A OneSector Model of Growth under Certainty 476 162 A ManySector Model of Stochastic Growth 481 163 An Economy with Sustained Growth 485 164 Industry Investment under Uncertainty 491 165 Truncation of a General Uncertainty 493 166 A Peculiar Example 495 167 An Economy with Many Consumers 501 17 FixedPoint Arguments 502 171 An OverlappingGenerations Model 508 172 An Application of the Contraction Mapping Theorem 516 173 The Brouwer FixedPoint Theorem 519 174 The Schauder FixedPoint Theorem 525 175 Fixed Points of Monotone Operators 531 176 Partially Observed Shocks 542 18 Equilibria in Systems with Distortions 543 181 An Indirect Approach 547 182 A Local Approach Based on FirstOrder Conditions 554 183 A Global Approach Based on FirstOrder Conditions 563 References 574 Index of Theorems 579 General Index 579 Introduction Research in economic dynamics has undergone a remarkable transformation in recent decades A generation ago empirical researchers were typically obliged to add ad hoc dynamic and stochastic elements after thoughts to predictions derived from static as a first approximation economic models Today in every field of application we have theories that deal explicitly with rational economic agents operating through time in an environment ruled by the kind of economic equilibrium has made dynamic stochastic general equilibrium of a system at rest similar methods are now available for analyzing theoretical models with equilibrium outcomes described by the same kinds of compact equilibrium outcomes that we use to describe observed economic behavior These theoretical developments are based on a wide variety of results in economics mathematics and statistics the contingentclaim view of economic applications of the calculus of variations pioneered long ago by Ramsey 1928 and Hotelling 1931 the theory of dynamic programming of Bellman 1957 and Blackwell 1956 Our goal in this book is to provide a selfcontained treatment of these theoretical ideas that form the basis of modern dynamic dynamics Our approach is distinguished by its systematic use of recursive methods that make it possible to treat a wide variety of dynamic economic problemsboth deterministic and stochasticfrom what we mean by a recursive approach of view To illustrate these methods we begin with a list of concrete examples drawn from the much longer list of applications to be treated in detail in later chapters These examples also serve to illustrate the kinds of substantive economic questions that can be studied by the analytical methods in this book 3 This consumptionsavings decision is the only allocation decision the economy must make Capital is assumed to depreciate at a constant rate 1 ct it Fkt nt In this section we study the problem of optimal growth when there is no uncertainty Assume that the production function is yt Fkt nt where F R2 R is continuously differentiable strictly increasing homogeneous of degree one and strictly quasiconcave with limk0 Fk1 0 limk Fkk1 0 21 A Deterministic Model of Optimal Growth In this section we study the problem of optimal growth in an economy composed of many identical households In each period t there is a resource allocation using two inputs capital kt in place at the beginning of the period and labor nt a product of labor is always positive it is clear that an optimum requires nt all t Hence nt represented both by capital and output per worker define fk Fk1 1 δk to be the total supply of goods available per worker including undepreciated capital when beginningofperiod capital is kt Introduction First consider an economy that produces a single good that can be either consumed or invested The expected discounted present value of profits alternatively suppose there are many firms in this industry In competitive equilibrium what investment strategies for all of these firms and what do they imply for the behavior of industry production and prices These problems evidently have much in common In each case a decisionmakera social planner a worker a manager an entire market a firm or collection of firmsmust choose a sequence of actions through time In the first example there is no uncertainty so the entire sequence environment as well as that of the other firms is determined by the that the best future actions depend on the outside shocks and it is clear that we might formulate each of these problems mathematically and consider how we might formulate each of these problems mathematically and consider how we might formulate each of these problems mathematically and consider how we might formulate each of these problems mathematically and consider how we might formulate each of these problems mathematically and consider how we might formulate each of these problems mathematically and consider how we might formulate each of these problems mathematically and consider how we might formulate each of these problems mathematically and consider how we might formulate each of these problems mathematically and consider how we might formulate each of these problems mathematically and consider how we might formulate each of these problems mathematically and consider how we might formulate each of these problems mathematically and consider how we might formulate each of these problems mathematically and consider the problem of optimal savings that Frank Ramsey formulated and solved in 1928 Ramsey viewed the problem as one of maximizing a function total utility of an infinity of variables consumption and stock subject to a set of differential equations imposed by the technology He set up the problem in continuous time and thus proposed by the calculus of variations to obtain a very sharp characterization of the utilitymaximizing dynamics the capital stock should converge monotonically to the level that if sustained maximizes consumption per unit of time In the Ramsey problem the feature of the production possibility set that changes over time is the current stock of capital This observation suggests that an alternative way to describe the problem is in terms of a function that gives societys optimal current investment as a function of its current capital stock Thus an alternative mathematical strategy can seek the homogeneous function defined the optimal investment as a function to compute the optimal sequence by direct use of the dynamic programming This way of looking at the problemdecide on the immediate action to take as a function of the current situationis called a recursive formulation because it exploits the recursive structure that a decision problem of the same general structure recurs each period To illustrate a concrete list of examples drawn from this longer list of applications to be treated in detail in later chapters These examples also serve to illustrate the kinds of substantive economic questions that can be studied by the analytical methods in this book 5 Introduction First considered an economy that produces a single good that can be either consumed or invested The expected discounted present value is immediate utility to the single decisionmaker thereby making increased production possible in the future capital stock hereby making increased production possible in the future capital stock hereby making increased production possible in the future capital stock hereby making increased production possible in the future capital stock hereby making increased production possible in the future capital stock hereby making increased production possible in the future capital stock hereby making increased production possible in the future capital stock hereby making increased production possible in the Thus an economys investment decisionmakers problem of choosing an investment plan to maximize the expected present value of his utility What is the consumerinvestment policy that maximizes utility over an infinite horizon Next consider an economy that is otherwise similar to the one just described but that is subject to random shocks affecting the amount of output that can be produced with a given stock of capital How should the consumption decision be made if the objective is to maximize the expected sum of utilities Suppose a worker wishes to maximize the present value of his earnings In any period he is presented with a wage offer at which he can work or not work If he works he earns the wage but the search and retains the same job next period If he does not work he continues searching and activity that yields him a new wage offer from a known probability distribution What decision rule should he adopt if his goal is to maximize the expected present value of his lifetime earnings A store manager has a given number of units of a specific type Demand is stochastic so in any period he may either stock out and forgo the sales he would have made with a larger inventory or incur the costs of carrying over unsold items The cost of this action includes an fixed delivery charge plus a charge per item ordered The order must be placed before the manager knows the current period demand If the goal is to maximize the expected discounted present value of profits when should he place an order and when an order is placed how large should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be An economy is endowed with a fixed number of productive assets that have previously given yields described by a stochastic process These assets are owned privately by a number of agents who make their portfolio choices to maximize asset returns and minimize risk How are the competitive equilibrium prices determined in this market How are the consumer preferences over equilibrium prices and the current product process How is the answer to this question affected in a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shifting demand curve for his product His current production capacity is determined by his past investments but he has the option to invest in additions to capacity additional investment will be available for production in the future What investment should it be A monopolist faces a stochastic shiftin concave with f0 0 fk 0 limk0 fk limk fk 1 δ The planning problem can then be written as maxkt10 t0 βt Ufkt kt1 3 st 0 kt1 fkt t 0 k0 0 given 4 Although ultimately we are interested in the case where the planning horizon is infinite it is instructive to begin with the much easier problem of a finite horizon If the horizon in 3 were a finite value T instead of infinity then 34 would be an entirely standard concave programming problem with a finite horizon the sequence ktt0T of states and the objective function 4 is closed bounded and convex subsets of RT1 and the objective one solution and it is completely characterized by the KuhnTucker conditions To obtain these conditions note that since U0 0 and U0 it is clear that the inequality constraints in 4 do not bind except for kT1 and it is also clear that kT1 0 Hence the solution satisfies the firstorder and boundary conditions β fkt Ufkt kt1 kt 1 Ufkt1 kt t 1 2 T 5 k0 0 given kT1 0 6 given Equation 5 is a secondorder difference equation in kt hence it has a twoparameter family of solutions The unique optimum for the maximization problem of interest is the one solution in this family that in addition satisfies the two boundary conditions in 6 The following exercise illustrates how 56 can be used to solve for the optimum in a particular example Exercise 21 Show that the assumptions on F above imply that f R R is continuously differentiable strictly increasing and strictly Exercise 22 Let f β α such that 0 α β 1 and let Uc lnc No this does not fit all the assumptions we placed on f and U above but go ahead anyway a Write 5 for this case and use the change of variable zt kt1α kt to convert the result in the firstorder difference equation in zt Plot zt1 against zt and plot the 45 line on the same diagram b The boundary condition 6 implies that zT1 0 Using this condition show that the unique solution is ct T 1 c Check that the path for capital satisfies 7 bt1 αβ 1 αβTt 1 αβT1t ktα t 0 1 T given k0 satisfies 56 Now consider the infinitehorizon version of the planning problem in Exercise 22 Note that if T is large then the coefficient of ktα in essentially constant αβ for a long time For the solution to the infinitehorizon problem we not take the limit of the solutions in 7 as T approaches infinity After all we are discussing households that discount the future at a geometric rate Taking the limit in 7 we find that kt1 αβktα t 0 1 8 maxct Uc0 β vk1 st c0 k1 fk0 k0 0 k0 0 given 10 where g R R is a fixed savings function Our intuition suggests that this must be so since the planning problem takes the same form every period only the beginningofperiod capital stock changing from one period to the next what else but kt could influence the choice of kt1 Unfortunately Exercise 22 does not exactly help in pursuing this conjecture The range of variability of k is obviously specific to each problem so the strategy we used in the finitehorizon problem and choosing infinite sequences ct kt1inftyt0 for consumption and altogether and for the kt1s to push the idea involves ignoring 5 and 6 Although we state that the problem is that the planner in period t 0 is that of choosing todays consumption c0 and capital tomorrow k1 for a period and another cannot wait out in the period and later The rest cannot be considered as we knew the planers preferences over these two goods we could simply maximize the appropriate function of c0 k1 over the opportunity set defined by 1b But what are the planners preferences over current consumption and k1 Suppose that 914 had already been solved for all possible values of k0 Then we could define a function v R R by taking vk0 to be the value of the maximized objective function 3 for each k0 A function v thus related to 914 is called a value function Our job would be to give the value of the utility from period 1 on that could be obtained with a beginningofperiod capital stock k1 and β vk1 would be the value of this utility discounted back to period 0 Then in terms of the value function v the planners problem in period 0 would be maxc0k1 Uc0 β vk1 st c0 k1 fk0 k0 0 given 10 If the function v were known we could use 10 to define a function g R R as follows for each kt let kt1 gkt be the maximum in 10 with g so defined 9 be the values that attain the maximum in 10 With g defined this way would completely describe the dynamics of capital accumulation from any given initial stock k0 We do not at this point know v but we have defined it as the maxi b Verify that this function v satisfies 11 Suppose we have established the existence of an optimal savings policy g either by analyzing conditions 56 or by analyzing the functional equation 11 What can we do with this information For the particular parametric example in Exercises 2123 we can solve the functional equation recursively by paperandpencil methods We can then use the resulting difference equation 8 to compute the optimal sequence of capital stocks kt This example is not a direct solution except for most other parametric savings functions it is not possible to obtain an explicit solution for the sequence kt In such cases a numerical approach can be used to compute explicit solutions When all parameters are specified numerically it is possible to use an algorithm based on 11 to obtain an approximate solution to g Then kt can be computed using 9 given any initial value k0 In addition there are often qualitative features of the savings function g and hence of the capital paths generated by 9 that hold under a very wide range of assumptions and are specifically of interest For example 56 or the firstorder and envelope conditions for 11 together with assumptions on U and f to characterize the optimal savings function g We can then in turn use the properties of g so established to characterize solutions kt to 9 The following exercise illustrates the second of these steps Exercise 24 a Let the as specified in Exercise 21 and suppose that the optimal savings function g is characterized by a constant savings rate gk s fk all k where s 0 Plot g and on the same diagram plot the 45 line The points at which gk k are called the stationary points steadystates or fixed points of g Prove that there is exactly one positive stationary point k with k 0 b Use the diagram to show that 0 then the sequence kt given by 9 converges to k That is let k0 be a sequence satisfying 9 given some k0 0 Prove that lim kt k for any k0 0 Show that this convergence is monotonic Can it occur in a finite number of periods This exercise contains most of the information that can be established 2 An Overview mixed objective function for the problem in 34 Thus if solving 10 provides the solution for that problem then vk0 must be the maximal objective function for 10 as well That is v must satisfy vk max0 k fk Ufk k β vk where as before we have used the fact that goods will not be wasted Notice that when the problem is looked at in this recursive way the time subscripts have become a nuisance we do not care what the date is We can rewrite the problem facing a planner with current capital stock k as 11 vk max0 k fk Ufk k β vk This one equation in the unknown function v is called a functional equation and we will see later that it is a very tractable mathematical object The study of functional equations is called dynamic programming the analysis of such functional equations is called the dynamic programming and might establish the existence of the function Example 4 To study this problem we try solving the associated functional equation 11 for the example 3 and 4 by arriving at the associated policy function Several steps are involved in trying out this analysis First we need to be sure that the solutions to a problem posed in terms of an infinite sequence kt s to the related functional equations are also the solutions that by the functional recursions kt solve the problem This shows that we must establish the existence and uniqueness of solutions like 11 We must establish the existence and uniqueness of a value differential equation and where possible a value function satisfying the functional equation and to establish properties of the associated properties of v Finally we must show how qualitative properties of g are translated into properties of the sequences generated by g Since a wide range of problems from very different substantive areas of economics all have this same mathematical structure we want to develop these results in a way that is widely applicable Doing this is the task of Part II 22 A Stochastic Model of Optimal Growth The deterministic model of optimal growth discussed above has a variety of stochastic counterparts corresponding to different assumptions about the nature of the uncertainty In this section we consider a model whichthe uncertainty affects the technology only and does so in a specific way Assume that output is given by yt zt fkt where zt independent and identically distributed iid random variables and f is defined as it was in the last section The shocks may be thought of as arising from crop failures technological breakthroughs and so on The feasibility constraints for the economy are then 1 kt1 ct zt fkt ct kt1 0 all t Assume that the households in this economy rank stochastic consumption sequences according to the expected utility they deliver where their underlying common utility function takes the same additively separable form as before Euc0 c1 Ei0 βi uci 2 Here E denotes expected value with respect to the probability distribution of the random variables ci i01 Now consider the problem of choosing a benevolent social planner in this stochastic environment As before his objective is to maximize the objective function in 2 subject to the constraints in 1 Before proceeding we need to be clear about the timing of information actions and decisions about the objects of choice for the planner and about the distribution of the random variables zii0 Assume that the timing of information and actions in each period follows as At the beginning of period t the value zt of the exogenous shock is realized Thus the pair kt zt and hence value of total output yt are known when consumption ct takes place and endofperiod capital kt1 is called the state of the economy at date t As we did in the deterministic case we can think of the planner in period 0 as choosing in addition to the pair c0 k1 an infinite sequence ct kt1t1 describing all future consumption and capital pairs In the stochastic case this sequence of numbers is called a stochastic sequence of contingency plans Specifically consumption at t is one for each period kt1 in each period t12 are functions of the realizations of the shocks z1 z2 This sequence contingent on the realization that is available when the decision is being carried out but is unknown in period 0 when the decision is being made 22 Stochastic Growth More generally given a stochastic difference equation of the form in 4 and a transition function G for the exogenous shocks we can define a transition function H as we did in 6 Then for any initial value k0 0 the sequence ψt of distribution functions for the ks is given by Exercise 27 suggests that if g and G are in some suitable families then H is such that this sequence converges in some sense to a limiting distribution function ψ satisfying ψk Hk k dψk 8 A distribution function ψ satisfying 8 is called an invariant distribution for the transition function H The idea is that if the distribution ψ gives a probabilistic description of the capital stock in any period t then also describes the distribution of the capital stock in period t 1 A distribution is thus a stochastic analogue to a stationary point of a deterministic system Now suppose that g and G are given and that the associated transition function H has a unique invariant distribution ψ Suppose further that for any k0 0 the sequence ψt defined by 7 converges to ψ Let φ be a continuous function and consider the sample average 1Tφkt for some sample path One might expect that this sample average is for long horizons approximately equal to the mathematical expectation of φ taken with respect to the limiting distribution ψ That is one might expect that limT 1T t1T φkt φk dψk at least along most sample paths A statement of this sort is called a law of large numbers Later we will specify precisely what is meant by most sample paths and what conditions under which 9 holds When 9 holds we sample average on the left in 9 from observed time series calculate the integral on the right in 9 from the theory and use a comparison of the two as a test of the theory The first calculation this book is concerned with methods As for the discussion above suggests the techniques of dynamic programming are if anything even more useful for analyzing stochastic models 22 Stochastic Growth The methods used to characterize the optimal policy in the stochastic case are completely analogous to those used for the deterministic case If we assume differentiability and an interior optimum the firstorder condition for 3 is Uck z βEz ugk z z This condition implicitly defines a policy function g that has as its arguments the two state variables k and z Then the optimal capital path is given by the stochastic difference equation 4 kt1 gkt zt where zt is an iid sequence of random shocks The following exercise looks at 34 for the special case of log utility and CobbDouglas technology studied in the last section Exercise 26 Let Uc lnc and fk kα 0 α 1 as we did in Exercises 2224 Conjecture that an optimal policy given in 5 kt1 αβktα all t Calculate the value of the objective function 2 under this policy given k0 and z0 and call this value vk z Verify that the function v so defined satisfies 3 Working out the dynamics of the state variable kt implied by the policy function g is quite different in the stochastic case Equation 4 and its specialization 5 for stochastic growth mean the state in terms of the random variables kt generated by such difference equations are called a firstorder Markov process It is useful to recall the results about Markov processes for the stochastic case In Exercise 24 we think about the determinacy and stability of numerical calculation in the sequence kt described by 5 is not going to converge to any single value in the presence of the recurring shocks zt Can anything be said about its behavior Taking logs in 5 we obtain lnkt1 lnαβ α lnkt lnzt 2 An Overview Since the shocks zt are iid random variables so are the logs lnzt Now suppose that the latter are normally distributed with common mean μ and variance σ2 Exercise 27 Given k0 lnztt1 is a sequence of normally distributed random variables with means μt and variances for t 1 2 find these means and variances and calculate their limiting values as t 2 An Overview than they are for looking at deterministic problems Exercise 25 illustrates the complexity of looking at stochastic problems in terms of sequences even when the horizon is finite On the other hand the functional equations like 3 are a cornerstone of deterministic dynamic programming and stochastic shocks when talking about distributions like the ones in 3 A few basic results about expectation operators like 3 involves an optimal solution to a functional equation and hence we are interested in policy functions for the problem 4 and hence we can try applying the product rules of standard analysis is significantly harder than the analysis of solutions to deterministic difference equations but it is not unmanageable Clearly a stability theory for stochastic systems requires several things First we must define precisely in mathematical terms the sequence of distribution functions Then we need to develop sufficient conditions of transition functions like the function H above to ensure that H has a unique invariant distribution and that the sequence of distribution functions given by 7 converges to a distribution that corresponds to the given distribution Finally connect the theory to observed behavior we must develop conditions under which a law of large numbers holds The reader should not be surprised that carrying out this agenda requires a fairly large body of groundwork Some definitions notation and basic results from modern probability theory are needed as well as some basic information about Markov processes This preliminary material as well as the analysis of stochastic recursive models is the content of Part III 23 Competitive Equilibrium Growth In the last two sections we were concerned exclusively with the allocation problem faced by a hypothetical social planner In this section we show that solutions to planning problems of this type can under appropriate conditions be interpreted as competitive equilibria that give behavior of market economies The argument establishing this is based of course on the classical connection between competitive equilibria and Pareto optima These connections hold under fairly broad assumptions and in later chapters we will establish them in a very general setting At that time we will also show that in situations where the connection between competi 23 Equilibrium Growth tive equilibria and Pareto optima breaks down as it does in the presence of taxes or other distortions the study of competitive equilibria can be carried out by a direct analysis of the appropriate firstorder conditions Recall that in the models discussed above there many identical households and we looked the common preferences of these households were to be the many identical firms all with the same constantreturnstoscale technology so the technology available to the economy was the same as that available to each firm Thus the planning problem considered in Sections 21 and 22 can be viewed as problems of maximizing a weighted average of households utilities specialized to a case where all households had identical tastes were given equal weight and hence received identical treatment by a social planner A special but very important type of equilibrium and solution concept is the one we considered were Paretooptimal allocations In this section we show that these allocations are exactly the ones that correspond to competitive equilibria For simplicity let be a finite time horizon and consider the case of a competitive equilibrium Suppose that we have solved the finitehorizon optimal growth problem of Section 21 and that ct kt1t0 is the solution Our goal is to item of prices that support these quantities as a competitive equilibrium However we must first specify the households the firms and how prices must be set in these markets It is crucial to be specific on these matters Assume that there are two types of productive units in firms and households that own all factors of production and all shares in goods produced by firms consumers are equally distributed across households Each household sells factor services to firms and buys consumers goods on a rental basis to produce output Households want to maximize utility Besides labor is produced by firms consuming some and accumulated capital labor on a rental basis to produce output of quality for hire capital and households and return any profits that result to shareholders Finally assume that all transactions take place in a single marketthe market meets in period t All trading takes place at that time so all prices and quantities are simultaneously determined After this market has closed in periods t 0 1 agents simply deliver the quantities of factors and goods they have contracted to sell and receive those they have contracted to buy Assume that the convention for prices in this one big market is as allocation and that ct yields highest total utility in the objective function 11 Then this allocation must violate 12 or the household would have chosen it But if 12 is violated then 16 implies that π t0T βtFk t 1 rk t h t wtu t 0 π contradicting the hypothesis that k t n t t0 was a profitmaximizing choice of inputs This result is a version of the first fundamental theorem of welfare economics Conversely suppose that c k u t0 is a solution to the planners problem in Section 21 Then k t h t u t0 is the unique sequence satisfying the firstorder and boundary conditions βt fk tUfk t1 k t1 kt k t t1 2 T 17 βt Ufk tUfk t1 k t1 Ufk t1 k t1 k0 x0 18 and ct is given by ct fk t kt1 t0 1 T 19 where the function fk t Fk t 1 1 δk t is as defined in Section 21 To construct a competitive equilibrium with these quantities we must find supporting prices p t r t w tt0 that solve 9 and 15 together suggest that goods prices must satisfy To do this note that 9 and 15 together suggest that goods prices must satisfy p t p t fk t t1 2 T 20 where h0 0 is arbitrary and 9 and 10 imply that real wage and rental rates must satisfy u t k t rk t fk t t1 2 T 21 w t fk t rk t fk t t1 2 T 22 It is not difficult to verify that these prices together with the quantities in 1719 constitute a competitive equilibrium and we leave the proof as 2 An Overview 26 available n t 1 and k t x t all t Using these facts and substitu ing from 5 to eliminate i t we can write the households problem as max βt Uc t t0T st t0T pdfc t k t1 r t 1 δk t w t 0 11 12 c t 0 k t1 0 t0 1 T 13 given k0 x0 Since limc0 Uc the nonnegativity constraints on the c ts in 13 are never binding Hence the firstorder conditions for the household are βt1 Uc t1 Uc t 1 fk t1 t0 1 T 1 14 βUc t λ1 δpk t1 μ t with equality if k t1 0 15 where λ is the multiplier associated with the budget constraint 12 Therefore a competitive equilibrium is characterized by quantities and prices c t k t h t p t r t w tt0 with all goods and factor prices strictly positive h0 0 that solve 8 at the given prices such that c t k t h tt0 solves 1113 at the given prices k0 x0 kT1 0 and in addition 16 Fk t 1 c t k t1 1 δk t all t Now that we have defined and partially characterized a competitive equilibrium for the economy of Section 21 we can be more specific about the connection between the planners and equilibriums solution First note that if c t k t p t w t r tt0 is an equilibrium then c t k t h tt0 is a solution to the planning problem discussed in Section 21 To prove this we need only show that c t k t h tt0 is a feasible is Pareto optimal Suppose to the contrary that c t k t h tt0 23 Equilibrium Growth 27 2 An Overview follows Let p be the price of a unit of output delivered in period t for t 0 1 T expressed in an abstract unitsofaccount Let wt be the price of a unit of labor delivered in period t expressed in units of goods in period t so that wt is the real wage Similarly let rt be real rental price of capital in period t Given the prices pt wt rtt0T the problem faced by the representative firm is to choose input demands and output supplies ct xt nt ytt0T that maximize net discounted profits Thus its decision problem is max π t0T pt Fkt nt rt kt wt nt st yt Fkt nt t 0 1 T Given the same price sequence the typical household must choose demand for consumption and investment and supplies of current capital and labor ct xt kt1 xt1 nt t0T In making these choices the household faces several constraints First the total value of goods purchased cannot exceed the total value of goods sold plus transfers that it receives from the government Second the households holdings of real capital in each period t1 are equal to its holdings in period t net of depreciation plus any new investment Third the maturity of each factor supplied by the household in each period must be nonnegative Finally since the quantity available to period in that period must be nonnegative Thus its decision problem is max Z t0T Bt Uct st t0T pt ct xt1 rt kt wt nt w ut t 0 1 T xt1 1δ kt 0 kt xt 1 x1 x0 t 0 1 T ct 0 xt1 0 t 0 1 T Note that capital stocks owned xt1 and capital supplied to firms kt are required to be nonnegative However gross investment it may be negative This assumption is the one that was made implicitly in Section 21 A competitive equilibrium is a set of prices pt wt rt w utt0T an allocation ct xt kt1 xt1 nt t0T for the typical household and an allocation kt nt yt Tt0 for the typical firm such that a kt nt yt rt wt solves 2927 at the stated prices b ct xt kt1 xt1 nt solves 2324 at the stated prices c markets clear that is for all t To find a competitive equilibrium we begin by conjecturing that by certain features Later we will verify that these conjectures are correct First since the representative households preferences are strictly monotone we conjecture that goods both factors are strictly positive at each period t at bt 0 Also since the price of investment goods are strictly positive for all t we conjecture that both factor prices are strictly positive for all periods t rt 0 wt 0 and all r and w must be positive quantities of goods wt and rt are nonnegative since in equilibrium quantities of goods 23 Equilibrium Growth are nonnegative and capital stocks are strictly positive Now consider the typical firm If the price of goods is strictly positive in each period then the firm supplies to the market all of the output that it produces and labor demand simply hires capital 2 MISSING capital produces that is 2 holds with equality for all t Also this problem is equivalent to a series of one period maximization problems Hence its input demands solve maxkt nt pt Fkt nt rt kt wt nt 8 It then follows that real factor prices must be equal to marginal products rt Fkkt nt t 0 1 T wt Fnkt nt t 0 1 T Since F is homogeneous of degree one when we substitute from 8 we find that π 0 Note too that rT1 0 Next consider the typical household Since supplying available factors causes no disutility to the household in every period it supplies all that it PART II Deterministic Models 25 Bibliographic Notes Modern growth theory began with Frank Ramseys 1928 classic paper and then lay dormant for almost 30 years Although a substantial body of literature on growth developed during the 1930s and 1940s this work is quite different from the neoclassical theory of growth both in mot 24 Conclusions and Plans We began this chapter with a deterministic model of optimal growth and then explored a number of variations of it In the course of the discussion we have raised a variety of substantial issues technical and economic past and present and have also posed a number of questions in both categories lightly with promises of better treatments to come It is time to spell out these promises in more detail We will do this by describing briefly the plan for the rest of the sub Department of Economics Princeton University Deterministic Models Introduction In such cases establishing the existence and qualitative properties of a competitive equilibrium requires looking directly at the equilibrium in the appropriate function spaces In the text above V H such that V and H are the value and policy functions for the households dynamic programming problem given the economywide law of motion for the stochastic variables the analogues of 27 and 28 require that we look directly at the equilibrium in functional spaces Establishing the existence of a competitive equilibrium involves satisfying those equations Given h the functions ϕk V1k h and H1k h satisfy the methods for studying re turn In this section we have focused on solving Parts II and III can be used to study problems that arise in more general settings Both the finite and infinite horizon problem hold equivalent recursive functional equations where the problem is analyzed as an infinite dimensional dynamic programming problem However the methods of analysis need to be established differently for these two approaches to the study of competitive equilibrium These two approaches to the study of com petitive equilibria are the subject of Part IV det ruministic dynamic programming in a deterministic 1 Remark 21 is a typical example 15 141 the inter applications est zlebased These economic models that are drawn straight to give be the applicability of these methods characterize recursive systems over time the theory of stability for auto mistic recursive methods for characterizing the behavior of deter from a variety of probability structures and for a variety of mixtures of broad applicability These applications include global stability and some of the review results on global stability and Chapter 6 treats methods applicable to the kinds of infinite dimensional economics that arise in dynamic applications We also treat the issue of constructing prices for problems involving infinite time horizons andor uncertainty Chapter 16 contains a number of applications designed to illustrate how a variety of planning problems can be inter III stochastic systems In general these we saw in Section 22 are treated in Part includes chapters 1315 in the analysis of Chapters 46 to chosen to take a modern attack one that allows us to deal with very general classes of stochastic shocks The methods of dynamic programming problems of stochastic differential equations and that yields a definitive look at the recursive program matic counterpoint of stability theory To take this approach we the stochastic counter part of stability theory Robinson 1971 This must first develop some of the basic tools of the theory of measure and integration This background is presented in Chapters 7 and 8 Chapter 7 is a self contained treatment of the definitions and results from measure theory that are needed in later chapters and Chapter 8 contains an introduc tion to Markov processes the theorems the processes the stochastic difference equations the natural generalization of the stochastic With these equations discussed above electron Theorems Chapter 9 deals with stochastic dynamic programming paralleling Chapter 4 as closely as possible With the rewards from Chapters 7 and 8 we are now in a position to hope that imperfectionsthat cannot be analyzed in this way In many such cases it is still possible to construct recursive equilibrium models using these methods Chapters 8 and 9 illustrate the preliminary The existence of market equilibrium is also the solution to a benevolent social planners problema fact vastly simplifies the analysis However which markets subject to distortions due to taxes external effects or various 33 Conclusion 24 Conclusions and Plans and illustrates the necessary ingredients of such a treatment First we must show that for stability of such systems problems like the social security problem the finitehorizon obtain some features of a Markov process the general Markov chain and a functional equivalent in some sense to the sequence of decisions is equal in the form of a functional equation With this established we can study functional equations for bounded constant behavior possible in these methods Analyzing economic models that are amena ble to analysis using these tools These applications which are drawn from a variety of substantive areas of economic theory are intended to give us some idea of the broad applicability of these methods Chapter 6 treats methods for characterizing the behavior of determi nistic recursive systems over time the theory of stability for au tonomous recursive systems and some of the review results on global stability and some methods that apply to nonautonomous systems Chapter 5 we turn to substantive economic models that are amenable to analysis using these tools These economic models that are drawn and illustrates necessary ingredients Unlike the finite horizon problem constant be timeinvariant The sequence of decisions is equivalent to the strategy functions used in Chapter 1 and the treatment of such problems 32 21 An Overview tic analogues to models discussed in Chapter 5 others are entirely new Chapters 11 and 12 survey results on convergence in various senses for Markov processes of the sorts of stochastic systems for which much work has been done This material is the body of theory suited to characterizing the dynamics for state variables generated by optimal policy functions for stochastic dynamic programming Some of these are continuations are discussed in Chap ter 10 others are new Chapter 14 provides a law of large numbers for Markov processes The use of recursive systems within a general equilibrium framework as illustrated in Sections 23 above is the subject of Part IV Chapters 15 18 Chapter 15 returns at a more abstract level to the connections be tween Paretooptimal and competitive equilibrium allocations In par ticular we there review two foundations for theorems of welfare economics that apply to the kinds of infinite dimensional commodity spaces that arise in dynamic applications We also treat the local stability of competitive equilibria with some applications of economic theory to local methods and with some examples that illustrate the types of behavior possible in unstable systems 33 and growth and utility theory and optimal growth Paisley and Brock All connections between the dynamic programming approach and economic equilibrium exist only at the level of theorems the central issue is the existence and construction of solutions of Bellman equations One way to look at the growth model studied in Section 22 with the theory of dynamic programming Arrow and Enthoven 1961 is as an equilibrium à la Debreu 1959 the results of that model show the existence of competitive equilibria Assar Lindbeck 1960 Blackorby and Shoven 1985 The first part of the chapter broadens the view showing many cases where the dynamic programming formulation is equivalent to an economic general equilibrium modeloften of a form which can be studied using constructive methods To introduce the idea behind that consider an equilibrium problem which is linear in the measure where the theoretical difficulties discussed in chapter 3 can no longer be applied 34 23 Deterministic Models Growth models are among the earliest deterministic models in economic theory and they have a long history now reviewed in some detail The text of the early literature especially of the optimal growth literature cannot be given here very carefullyit is scattered and apart from the collection in Bourgin 1966 only a few of the original references are easily accessible The classical growth models discussed in the text have probably been known since Ramsey 1928 Others in the literature were developed over the variety of decades that followed into the 1930s and 1940s Weibull 1997 More recently explicit formulations as mathematical problems with wellposed solutions and unique equilibrium growth paths have appeared Dixit 1987 Brock 1980 1982 Brock and Mirman 1972 Judd 1985 See for example Judd 1985 Chapter 2 and Ljungqvist and Sargent 2000 The canonical onesector economy is a very special case of this literature which we have chosen to use to introduce a number of ideas that are technical theoretical and economic A general equilibrium view of this onesector economy and a modern treatment in other growth models as well can be found in Uzawa 1961 Early papers on optimal growth include Ramsey 1928 Cass 1965 Koopmans 1965 and Debreu 1962 Ramseys original presentation is often hard to read the problem is solved by variational methods his inter pretation is that the representative agent is not selfish but a benevolent social planner Cass 1965 and Koopmans 1965 formulate the problem rigorously in the dynamic programming framework Debreu 1962 follows with a modern treatment of the existence of equilibria and optimality properties of the solution The key feature of such growth models is a state variable that accumulates the capital stock Capital accumulation is described by an equation of the form Kt1 fKt Ct The utility of consumption in each period is given by UCt continuous concave and strictly increasing The goal is then to maximize discounted utility βt UCt subject to feasibility constraints and the initial capital stock The solution involves finding an optimal consumption path and the resulting capital accumulation path Such models are solved using methods like dynamic programming and Euler equations 35 25 Bibliographic Notes Modern growth theory began with Frank Ramseys 1928 classic paper and then lay dormant for almost 30 years Although a substantial body of literature on growth developed during the 1930s and 1940s this work is quite different from the neoclassical theory of growth both in mot 3 Mathematical Preliminaries In Chapter 2 the optimal growth problem max ⁸ᵢ₀ βᶦUcᵢ cₖ kₖ₁ₖ₀ fkₖ st cₖ kₖ₁ fkₖ cₖ kₖ₁ 0 t 0 1 given k₀ was seen to lead to the functional equation vk maxUc βvyₛₜ c y fk y c 0 The purpose of this chapter and the next is to show precisely the relationship between these two problems and others like them and to develop the mathematical methods that have proved useful in studying the latter In Section 21 we argued in an informal way that the solutions to the two problems should be closely connected and this argument will be made rigorous later In the rest of this introduction we consider alternative methods for finding solutions to 1 outline the one to be pursued and describe the mathematical issues it raises In the remaining sections of the chapter we deal with these issues in turn We draw upon this 39 3 Mathematical Preliminaries initial capital stock k₀ is w₀k₀ ᵢ₀ βᶦUfkᵢ g₀kᵢ kₖ₁ g₀kₖ t 0 1 2 where w₀k Ufk y βw₀g₀k that is used later Exercise 31 Show that If the utility from the policy g₀ is used as the initial guess for a value function that is if v₀ w₀then 2 is the problem facing a planner who can choose capital in all subsequent periods but must follow the policy w₀ in all subsequent periods v₁k is the level of lifetime utility attained and the maximizing value of ycall it g₁kis the optimal level for endofperiod capital Both v₁ and g₁ are functions of beginningofperiod capital Now notice that since v₁ is the value function for the first period the planner is choosing a feasible choice in the first period the policy g₀ from the beginning and do no worse than he would by following the policy g₀ for any feasible policy g₀ and associated initial value function w₀ is v₁k maxUfk y βw₀yₛₜ y fk Ufk g₀k βw₀g₀k v₀k vk 4 where the last line follows from Exercise 31 Now suppose the planner has the option of choosing capital accumulation optimally for two periods but must follow the policy g₀ thereafter If y is his choice for endofperiod capital in the first period then from the second period on he can do is to choose g₁ for endofperiod 41 3 Mathematical Preliminaries capital and enjoy total utility v₁y His problem in the first period is thus maxUc βv₁y subject to the constraints in 1 The maximized value of this objective function was defined in 3 as v₂k Hence it follows from 4 that v₂k v₁k u₁k Continuing in this way one establishes by induction that vₙ₁k vₙk for all k n 0 1 2 reflecting the fact that planning flexibility over longer and longer finite horizons offers new options without taking any other options away Consequently it seems reasonable to suppose that the sequence of functions vₙ might converge to a solution to 1 That is the method of successive approximations proceeds in a more systematic way Suppose as is usually the case that v₀ then clearly v₀ is a reasonable way to locate and characterize solutions This method can be described in a somewhat different and much more convenient language As we showed in the discussion above for any function w R R we can define a new functioncall it Tw R R by Twk maxUfk y βwyₛₜ y fk When we use this notation the method of successive approximations amounts to choosing vₙ₁ by choosing vₙ and studying the sequence vₙₙ defined by vₙ₁ Tvₙ n 0 1 2 The hope behind this iterative process is that as n increases the successive approximations vₙ get closer to a function u that actually satisfies 1 Moreover the hope is that the limit of the sequence vₙₙ is a solution v Why If can be shown that lim vₙ v and v satisfies 1 then it will follow that this limit is the only function satisfying 1 Is there any reason to hope for success in this analytical strategy Recall that our reason for being optimistic is that the sequence vₙ defined by vₙ₁ Tvₙ n 0 1 2 The goal then is to show that this sequence converges and that the limit function v satisfies 1 Alternatively we can simply view the operator T as a mapping from some set C of functions into itself C In this setting solving 1 is equivalent to locating a fixed point of the operator T that is a function v C satisfying v Tv and the method of successive approximations is viewed as a way to construct this fixed point To study operators T like the one defined in 5 we need to draw on several basic mathematical results To show that T maps an appropriate 42 43 31 Normed Vector Spaces 31 Metric Spaces and Normed Vector Spaces The preceding section motivates the study of certain functional equations as a means of finding solutions to problems posed in terms of infinite sequences To pursue the study of these problems we will in Chapter 4 look at infinite sequences of states w and candidates for the value function v and think of o as a sequence of sequences and vector spaces of functions as elements of infinitedimensional normed vector spaces and we begin here with the definitions of vector spaces metric spaces and normed vector spaces We then discuss the notions of convergence and Cauchy convergence and define completeness for a metric space Theorem 31 then establishes the existence of a fixed point for a complete metric space Recall we begin with the definition of a vector space Definition A real vector space X is a set of elements vectors together with two operations addition and scalar multiplication For any two vectors x y X and scalar scalar multiplication gives a vector αx X If these operations obey the usual algebraic laws that is for all x y z X and α β R x y z x y z 3 Theorem of the Maximum We will want to apply the Contraction Mapping Theorem to analyze dynamic programming problems that are much more general than the examples that have been discussed to this point If x is the beginningofperiod state variable an element of this set X and y is the endofperiod state to be chosen we would like to let the currentperiodof Fx y and state be chosen by us given x be specified as generally as possible On the other hand we want the operator T defined by Tυx sup Fx y βυy st y feasible given x to take the space CX of bounded continuous functions of the state into itself We would also like to be able to characterize the set of maximizing values of y given x To describe the feasible set we use the idea of a correspondence from a set X into a set Y Γ assigns a set Γx Υ to each x X In the case of Γ X X hence weekly restrictions on the correspondance of Γ X describing the feasibility constraints on the currentperiod return function F which together ensure that if f CX and Tυx 3 Mathematical Preliminaries We have demonstrated that both terms in the last expression converge to zero as η hence ρTυ v 0 or Tυ v Finally we must show that there is no other function g S satisfying Tυ υ Suppose to the contrary that υ v is another solution Then 0 α ρυ v ρTυ Tv βρTυ v which cannot hold since β 1 This proves part a To prove part b observe that for any η 1 ρTnυ v ρTTn1υ Tv βρTn1υ v so that b follows by induction Recall from Exercise 36b that if S ρ is a complete metric space and S is a closed subset of S then S ρ is also a complete metric space Now suppose that T S S is a contraction mapping and the supnorm that T maps S into itself where S is a closed subset of S Then T is also a contraction mapping on S Hence the unique fixed point of T on S lies in S This observation is often useful for establishing qualitative properties of a fixed point Specifically in some situations we want to apply the Contraction Mapping Theorem twice once on a large space to establish the existence of a fixed point and again on a smaller space to characterize the fixed point more precisely The following corollary formalizes this argument COROLLARY 1 Let S ρ be a complete metric space and let T S S be a contraction mapping with fixed point v S If S is a closed subset of S and TS S then v S Proof Choose v0 S and note that Tnυ0 is a sequence in S converging to v Since S is closed it follows that v v S If in addition TS S then it follows that v v S Part b of the Contraction Mapping Theorem bounds the distance ρTnυ v between the nth approximation and the fixed point in terms of the distance ρυ0 v between the initial approximation and the fixed point However if υ0 is not known as is the case if one is computing the point neither is the magnitude of the bound Exercise 39 gives a computationally useful inequality 32 Contraction Mapping Theorem Figure 31 Continuing by induction we get ρvn1 vn βnρv1 v0 n 1 2 1 Hence for any m n ρvm vn ρvm vm1 ρvn1 vn βm1ρv1 v0 βnρv1 v0 βn βm1 ρv1 v0 βn 1 β ρv1 v0 2 where the first line uses the triangle inequality and the second follows from 1 It is clear from 1 the clear that vn is a Cauchy sequence Since S is complete it follows that v converges to some v S To show that Tυ υ note that for all n and all v0 S ρTυ v ρTυ ΤΝν βρυ v β ρv2 v1 where 0 β 1 The second follows from the contraction property of T Hence from the same argument as in the proof of Theorem 32 The first exercise shows how the Contraction Mapping Theorem is used to prove existence and uniqueness of a solution to a differential equation 33 Theorem of the Maximum Figure 32 Γx x2 x1 x with satisfy this requirement the restriction will not be binding A definition of lhc for all correspondences is available but it is stated in terms of images of open sets For our purposes this definition is much less convenient and its wider scope is never useful DEFINITION A correspondence Γ X Y is continuous at x X if it is both whc and lhc at x A correspondence Γ X Y is called lhc or continuous if it has that property at every point x X The following exercises highlight some important facts about upper and lower hemicontinuity Note that if Γ X Y then for any set X X we define ΓX y Y y Γx for some x X Exercise 311 a Show that if Γ is singlevalued and uhc then it is continuous b Let Γ Rk Rm and define φ Rk Rl by φx y1 y2 EΓx for some y2 Rm 3 Mathematical Preliminaries supFx y βvy y CX Moreover we wish to determine the implied properties of the correspondence Gx containing the maximizing values of y for each x The main result in this section is the Theorem of the Maximum which Γ accomplishes both tasks Let Rl Rk Y be a metric space Y and R be a singlevalued function and let Γ X Y be a possibly multivalued correspondence Our interest is in problems of the form supFx y for each x fx y is continuous in y and the set Γy is nonempty and c compact then for each x the maximum is attained In this case the function 1 hx max fx y y Γx is well defined as is the nonempty set 2 Gx y Γx fx y hx of y values that attain the maximum In this section further restrictions on Γ will be added to ensure that the function h and the set G vary in a continuous way with x There are a variety of continuity for correspondences and each can be characterized in terms of sequences For our purposes it is convenient to use definitions stated in terms of sequences DEFINITION A correspondence Γ X Y is lower hemicontinuous lhc at x if Γx is nonempty and if for every y Γx and every sequence xn x there exists N 1 and a sequence yn Γxn such that yn y and yn Γxn all N fΓx is nonempty for all x X then it is always possible to take N 1 DEFINITION A correspondence Γ X Y is upper hemicontinuous uhc at x if Γx is nonempty and if for every sequence xn x and every sequence yn such that yn Γxn all n there exists a convergent subsequence ynj such that ynj Γxnj all nj DEFINITION A compactvalued correspondence Γ X Y is upper hemi continuous uhc at x if Γx is nonempty and if for every sequence xn x and every sequence yn such that yn Γxn all n there exists a convergent subsequence ynj such that ynj Γxnj all nj Figure 32 displays a correspondence that is lhc but not uhc and uhc but not lhc at all other points Note that our definition of uhc applies only to correspondences that are compactvalued Since all of the correspondences we will be dealing 33 Theorem of the Maximum Show that Γ X Y be lhc and suppose that d Let Φ X Y ψ lhc Γx y Φx ψx ψx N ψx all x X Show by example that Γ need not be lhc Show that if d and ψ are both convexvalued and if int Φx int ψx then Γ is lhc at x e Show that if Φ X Y and ψ X Y are lhc then the correspondence φ and ψ are lhc defined by Γx Z z Ψx for some y φx is also lhc f Let Γ X Y be lhc Show that Γ X Y Y1 Yk defined by Γx y1 yk where yi Γix i 1 k is also compactvalued and uhc e Show that if Γ X Y are compactvalued and uhc then the correspondence φ X Y defined by the correspondence φx Γx Z defined by Γx z ψx for some y φx is also compactvalued and uhc f Let Γ X Y Y1 yk where yi Γix i 1 k that Γx y Y is compactvalued and uhc The next two exercises show some of the relationships between constraints stated in terms of inequalities involving continuous functions and those stated in terms of continuous correspondences These relationships are extremely important for many problems in economics where constraints are often stated in terms of production functions budget constraints and so on Exercise 313 Show that Γ is continuous b Let Γ Left Rk R and define Γ by Γx y Rk y Leftx be a continuous function and define the correspondence Γ Rk Rk by Γx 0 fx x Γx z y Rk 0 yi fx z i 1 l and Σi xi x Show that Γ is continuous To see why a must be an interior point consider the case where X is a disk and A is any set with x on its boundary so the tip is directly above the boundary of X Let x be the point below the tip of the cone and take a sequence xn along the boundary of the disk Then each set Γxn contains a convergent subsequence We now prove the answers provided in the following theorem THEOREM 34 Let Γ X Y be a nonemptyvalued correspondence and let A be the graph of Γ Suppose that A is closed and that for any bounded set X the set ΓX is bounded Then Γ is uhc Proof For each x X Γx is closed since A is closed and bounded by hypothesis Hence Γ is compactvalued Let xn x and let xn yn A all n Since xn yn lies in a and Γ is convex it follows that x y A all y Moreover since 0 and all of the y lies in A and converges to x y it follows that it as was to be shown THEOREM 36 Theorem of the Maximum Let X Rᵏ and Y Rᵐ uhc continuous function gₙ X Y be a compactvalued correspondence and the function f X Y R defined in 1 is continuous and correspondence G X Y defined in 2 is nonempty compactvalued and uhc Fix x X The set Fx is nonempty and compact and fx is continuous hence the maximum in 1 is attained and the set Aₑ y Y y Fx gₙx gₙy δₑ is nonempty Moreover since Gx Fx and the set Gx is compact it follows that Aₑ is bounded Suppose yₙ y and yₙ Aₑ Also sup fxₙ yₙ fx y Hence for each n all n gₙ is compact and continuous and yₙ is contained in the compact set A₂ Thus gₙxₙ gₙyₙ all n so Gx is closed Next we will show that Gx is uhc Fix x and let xₙ xₙ be any sequence converging to x Choose yₙ Gxₙ for each n and since there exists a subsequence yₙₖ converging to y with y Gx it follows that fxₙ yₙ fxₙₖ yₙₖ Since F is uhc there exists a subsequence yₙₖ converging to y with y Fx since fxₙₖ yₙₖ fx y Hence Γxₙₖ yₙₖ holds sup norm Call xₙ yₙ analogous argument finitely many times to show the convergence THEOREM 38 Let X Y Γ and A be as defined in Lemma 37 Let fₙ be a sequence of continuous realvalued functions that for each n and each x X fₙx is strictly concave in its second argument Assume that f has the same properties and that fₙ f uniformly in the sup norm Define the functions gₙ and g by gₙx argmaxy Fx fₙx y gx argmaxy Fx fx y 41 The Principle of Optimality In this section we study the relationship between solutions to the problems SP and FE Note that sup has been used instead of max in both so that we can ignoring the momentthe question of whether the optimum is attained or notspeak of course is that the SP when the optimum is attained count as the value of the supremum in SP in v to FE evaluated at x gives the value of the supremum in the supremum in SP if and only if it satisfies vᵤxₜ Fxₜ xₜ₁ βᵤxₜ₁ t 0 1 2 1 Richard Bellman called these ideas the Principle of Optimality Intuitive as it is the Principle requires proof Seeing our beat precisely the conditions under which holds are Theorem 42 establishes the supremum function vₜ for the main results a Show that the onesector growth model is well b Show that the manysector growth model satisfies the supremum function vₜ for the sequence problems SP satisfies the supremum function vₜ Theorems 43 and 46 contain the Theorems 44 and 45 bound that deal with the nature of the optimize problem function Theorems 44 and 45 then The characterization of optimal policies Theorem 44 shows that fxₜ₁ₜ₀ is 41 Dynamic Programming under Certainty a sequence attaining the supremum in SP then it satisfies 1 for v conversely Theorem 45 establishes that any sequence xttl that satisfies 1 for v v and also satisfies a boundedness condition attains the supremum under SP taken together thus establish conditions under which solutions to SP and to FE coincide exactly To begin we must establish 1 Let X be the set of possible values for the state v ie the notation we will need to impose any restrictions on the set X It may be a subset of a Euclidean space a set of functions a set of probability distributions or any other set Let TX X be the correspondence describing the feasibility constraints That is for each x E X Tx is the set of feasible values for the state variable next period if the current state is x Let A be the graph of T Axy E X x Xy E Tx Let the realvalued function A R be the oneperiod return function and let β 0 be the stationary discount factor Thus the given for the problem are XFTF and β First we must find conditions under which the problem SP is well defined That is we must find a function is well defined for every x0 E X and any both 2 feasible set Call any feasible set denote natural for any x let Πx0x0x1x2 be the set of plans that are feasible from x0 That is Πx0 is the set of all sequences xt satisfying the constraints in SP Let x0 be a typical element of Πx0 The following assumption ensures that Πx0 is nonempty for all x0 ε X ASSUMPTION 41 For each n ε 01 define u nΠ1 R by unx0 Σ βnFxnxnl Then unΠx0 R by Then unx is the partial sum of the discounted returns in periods 0 through n from the feasible plan x underst x Under Assumption 42 for any x0 ε X and any x ε Πx0 where x x1x2 Proof Under Assumption 42 for any x0 ε X and any x ε Πx0 unx lim n Σn βnRFnxn xn1 Fx0x1 βux1x2 Fx0x1 βux1x2 THEOREM 42 Let XFF and β satsify Assumptions 4142 Then the function v satisfies FE Proof If β 0 the result is trivial Suppose that β 0 and choose ξ0 E X Suppose vx0 is finite Then 2 and 3 hold and it is sufficient to show that this implies 4 and 5 hold To establish 4 let x1 ε Γx0 and ε 0 be given Then by 3 there exists x n x0x1x2 E Πx0 and that ux2 zx1 Note too that x x0x1x2 E Πx0 Hence ASSUMPTION 42 For all x ε X and all y ε Πx0 lim n 0β1 θl 0 β 1 there exist θ θlβ such that y E Tx implies θFx0 F is increasing in its last t arguments F is concave in its first t arguments and decreasing in its last t arguments all k Show that Assumption 42 is satisfied if X R and Ψlθ00 is concave in its first t arguments and decreasing in its last t arguments Show that Assumption 42 is satisfied if X R 0 β 1 there exists 0 θβ such that y E Tx implies θFx0 F is increasing in its last t arguments and decreasing in its last t arguments and 0 E Tx all x Then unx is the partial sum of the discounted returns in periods 0 41 Dynamic Programming under Certainty define u nΠx0 R by ux lim n sup unx vx0 sup ux x ε Πx0 Thus v is the supremum in SP Note that it follows from the uniqueness of the function satisfying the following three conditions a if vx0 then v is continuous b if vx0 then there exists a sequence yk in Γx0 such that lim k Fx0 yk βv yk c if vx0 βvy Fx0y βux 7 Fx0 βux βux1 βux2 lim supn uxn it follows that 6 holds for the sequence x in Πx0 where x x0 x1 in Πx0 Our interest is in the connections between the supermum function v and solutions to the functional equation FE In interpreting the next result it is important to remember that v is always uniquely defined provided Assumption 4142 hold whereas FE may for all we know so far have zero one or many solutions We will say that v satisfies the functional eqaution if three conditions hold a If v then v satisfies FE b if vx0 all x ε Γx c if vx0 all x1x2 ε Πx0 follows immediately The next theorem provides a partial converse to Theorem 42 It shows that v is the only solution to the functional equation that satisfies a certain boundedness condition THEOREM 43 Let XFF and B satisfy Assumptions 4142 If v is a solution to FE and satisfies c αx y αx αy d αβx αβx e αx θ is a zero vector θ X that has the following properties f x θ x and g x x θ Finally lx x The adjective real simply indicates that scalar multiplication is defined taking the real R of elements of the complex plane or some other set as scalars All of the vector spaces used in this book are real and the adjective will not be repeated Important features of a vector space are that it has a zero element and that it is closed under addition and scalar multiplication Vector spaces are also called linear spaces Exercise 32 Show that the following are vector spaces a any finitedimensional Euclidean space Rn where x Rn b the set X xα α R some α on R c the set R all consisting of all infinite sequences x0 x1 x2 d the set of all continuous functions on the interval a b Show that the unit circle in R2 e the set of all integers I 1 0 1 f the set of all nonnegative functions on a b To discuss convergence in a vector space or in any other space we need to have the notion of distance The notion of distance in Euclidean space is generalized in the abstract notion of a metric a function defined on any two elements of a set the value of which has an interpretation as the distance between them DEFINITION A metric space is a set S together with a metric distance function ρ S S R such that for all x y z S a ρx y 0 with ρx y 0 if and only if x y b ρx y ρy x and c ρx z ρx y ρy z 41 Principle of Optimality There are a variety of ways of ensuring that Assumption 42 holds Clearly it is satisfied if the function F is bounded and 0 β 1 Alterna tively for any xy E A let Fxy max0 Fxy max0 Fxy Then Assumption 42 holds if for each x E Πx0 either lim n Σn 0 βnFxn xnl or lim n Σn 0 βnFxn xnl or both Thus a sufficient condition for Assumptions 4142 is that F be bounded above or below and 0 β 1 Another sufficient condition is that for each x0 X and y Π𝜔 there exist θ 0 β and 0 θ 8 such that Fxn xnl cθ l all l The following exercise provides a way of verifying that the latter holds Exercise 42 a Show that Assumption 42 is satisfied if X R 0 β 1 there exists θ θlβ such that y Tx implies θFx0 F is increasing in its last t arguments F is concave in its first t arguments and decreasing in its last t arguments all k Show that Assumption 42 is satisfied if X R 0 β 1 there exists 0 θβ such that y Tx implies θFx0 F is increasing in its last t arguments and decreasing in its last t arguments and 0 Tx all x The definition of a metric thus abstracts the four basic properties of Euclidean distance the distance between distinct points is strictly positive the distance from a point to itself is zero distance is symmetric and the triangle inequality holds Show that the following are metric spaces Exercise 33 a Let S be the set of integers with ρx y x y b Let S be the set of integers with ρx y 0 if x y 1 if x y c Let S be the set of all continuous strictly increasing functions on a b with ρx y maxxt yt d Let S be the set of all continuous strictly increasing functions on a b with ρx y ab xt yt dt e Let S be the set of all rational numbers with ρ R R with ρx y x y f Let S R with ρx y x y 1 x y For vector spaces metrics are usually defined in such a way that the distance between any two points is equal to the distance of their difference from the zero point That is since for any points x and y in a vector space S the point x y is also in S the metric on a vector space is usually defined in such a way that ρx y ρx y 0 To define such a metric we need the concept of a norm DEFINITION A normed vector space is a vector space S R such that that for all x y S and α R a x 0 with x 0 if and only if x θ b αx α x and c x y x y the triangle inequality Show that the following are normed vector spaces Exercise 34 a Let S R1 with x i11 xi1212 Euclidean space b Let S R1 with x x c Let S be the set of all bounded sequences x1 x2 xj R all k with x max xk d Let S be R1 with x i xi e Let S be the set of all bounded continuous functions on a b with x supasb xt f Let S be the set of all continuous functions on a b with x ab xt dt It is standard to view any normed vector space S as a metric space where the metric is taken to be ρx y x y all x y S The notion of convergence of a sequence of real numbers carries over without change to any metric space DEFINITION A sequence xnn0 in S converges to x S if for each ε 0 there exists Nε such that ρxn x ε all n Nε Thus a sequence xn in a metric space S ρ converges to x S if and only if the sequence of distances ρxn x as a sequence in R converges to zero In this case we write fxn as a sequence in R converges to zero In this case we write fxn f Verifying convergence usually involves having a candidate for the limit point x so that the inequality 1 can be checked When a candidate is not immediately available the following alternative criterion is often useful DEFINITION A sequence xnn0 in S is a Cauchy sequence satisfies the Cauchy criterion if for each ε 0 there exists Nε such that ρxn xm ε all m n Nε Thus a sequence is Cauchy if the points get closer and closer to each other The following exercise illustrates some of those in Exercise 31 The next example is no more difficult than some of those clearly each of 36 but since it is important in what follows and illustrates and illustrates the proof here the steps involved in verifying completeness we present the proof here 41 Dynamic Programming under Certainty A xx Ex Fx be the graph of F let F A R be the return function and let β 0 be the discount factor Throughout this section we will impose the following two assumptions on X Γ F and β useful however we must work with space where it implies the existence of a limit point DEFINITION A metric space S ρ is complete if every Cauchy sequence in S converges to an element in S In complete metric spaces then verifying that a sequence satisfies the Cauchy criterion is a way of verifying the existence of a limit point in S Verifying completeness of particular spaces can take some work We take as given the following FACT The set of real numbers R with the metric ρx y x y is complete Exercise 36 a Show that the metric spaces in Exercises 33ab and 34 are and 34 are complete and that those in Exercises 33ce and 34f are not Show that the spaces in 33c is complete if it strictly increasing is replaced with monotone increasing b Show what the subsets of S then S ρ is a complete metric space and S is a closed subset of S then S ρ is a complete metric space A complete normed vector space is called a Banach space THEOREM 31 Let X R1 and let CX be the set of bounded continuous functions f X R with the sup norm f supxX fx Then CX is a not complete That is give an example of a sequence of functions that is Cauchy in the given norm that does not converge to a function in the set Is this sequence Cauchy in the set of all k part a c Let Cka b be the set of continuously differentiable functions on a b with the norm f k0 maxxab fαx Show that this space is complete if and only if αi 0 Proof That CX is a normed vector space follows from Exercise 34e Hence it suffices to show that if fn is a Cauchy sequence there exists f CX such that for any ε 0 there exists Nε such that fn f ε all n Nε 42 Bounded Returns In this section we study functional equations of the form Three steps are involved to find a candidate function f to show that fn converges to f in the sup norm and to show that f is bounded and continuous Each step involves its own entirely distinct logic Fix x X then the sequence of real numbers fnx satisfies fnx fmx supxX fnx fmx fn fm Therefore it satisfies the Cauchy criterion and by the completeness of the real numbers it converges to a limit pointcall it fx The completeness of X R that we take to be 0 0 be our candidate function f It shows that if fn f then fn f ε2 Since fn satisfies the Cauchy criterion this can be done Now for any fixed x X and all m Nε fnx fx fnx fmx fmx fx fn fm fmx fx ε2 fmx fx Since fnx converges to fx we can choose m separately for each fixed x X so that fmx fx ε2 Since the choice of x was arbitrary it follows that fn f ε2 for n Nε Since ε 0 was arbitrary the desired result follows Finally we must show that f is bounded and continuous Boundedness is obvious To prove that f is continuous we must show that for every ε 0 there exists δ 0 such that fx fyE δ where E is the Euclidean norm on R1 Let ε 0 be given Choose δ so that fE δ since the Euclidean norm on R1 Let ε and x be given Choose k so possible Then choose δ so that fn f in the sup norm such a choice is x yE δ implies fnx fny ε3 1 is the supremum function for the associated sequence problem That is the supermum together establish that under Assumptions 43 44 and 48 bounded continuation values vn converge to v Moreover it then follows from Theorems 45 and 46 there exists at least one optimal plan any plan generated by the nonempty correspondence G To characterize v and G more sharply we need more information about F and T The next two results show how Corollary 1 to the Contraction Mapping Theorem can be used to obtain more precise characterizations of v and G ASSUMPTION 45 For each y f Fxy is strictly increasing in each of its first arguments ASSUMPTION 46 Γ is monotone in the sense that x x implies Γx Γx THEOREM 47 Let X Γ F and β satisfy Assumptions 4346 and let v be unique solution to 1 Then v is strictly increasing Proof Let CX CX be the set of bounded continuous nondecreasing functions on X and let CX CX be the set of strictly increasing functions on X and let CX be the set of strictly concave functions on X Since CX is a closed subset of the complete metric space CX by Theorem 326 and Corollary 1 to the Contraction Mapping Theorem Theorem 329 it is sufficient to show that Γf CX and let To verify that this is so let f E CX and let x0 x1 θ 01 and x x1 θ 1 θ x1 Let yi E Γxi attain Tfxi for i 01 Then by Assumption 48 θ θ 1θ yi θ0 1θ1 Γx0 It follows that TΓfx0 1θ1 βfw0 1θFx11 βfy1 TΓfx Fx y0 βFw0 TΓπx0 1 θTx1 θTΓx0 1 θTΓx1 where the first line uses c and the fact that y Γx the second uses the hypothesis is concavity restriction on F in Assumption 47 and the last follows from concavidity and y0 and y1 were selected Since x0 and x1 were arbitrary it follows that Γf is strictly concave Hence Gx argmaxTΓx is strictly concave Since F is also concave Assumption 47 it follows that TΓx is convex Γx is convex Assump tion 48 it follows that the maximum in 3 is attained at a unique y value Hence G is a singleton The continuity of C then follows from 48c Exercise 311 Theorem 47 and 48 characterize the value function by using the fact that the operator Τ preserves certain properties Thus if γ has property 49 P and if P is preserved by T then we can conclude that each function in the sequence Tn v0 has property P Then if P is preserved under uni form convergence T also preserves property P Theorem 44 about the property of u and u For whatever differentiability assumptions Ui j and β do require can be identified by analyzing the properties of Γ and F whatever differentiability assumptions we choose next to what is known about this issue It has been shown by Benveniste and Scheinkman 1979 that under general conditions value function V is once differentiable That is 5 is valid under quite broad conditions However known conditions 5 ensuring that V is twice differentiable and hence that g is once differen tiable are extremely seldom see Araujo and Scheinkman 1981 Thus differentiability is elusive whereas monotone it is usually possible to estab lish that fact by a direct argument involving a firstorder condition like 5 We begin with the theorem proved by Benveniste and Scheinkman THEOREM 410 Benveniste and Scheinkman Let X Rl be a convex set let V X R be concave let b ℝk and let D be a neighborhood of x0 If there is a concave continuously differentiable function W D ℝ with Wx0 Vx0 and with Wx Vx for all x D then V is differentiable at x0 and with Wx0 Vx0 Vx0 Wx0 and any subgradient p of V at x0 must satisfy p x x0 Vx Wx0 all x D where the first inequality uses the definition of a subgradient and the second uses the fact that Wx Vx with equality at x0 Since W is concave function with a unique subgradient at an interior point x0 is differentiable at x0 cf Rockafellar 1970 Theorem 251 p 242 Figure 41 illustrates the idea behind this result Applying this result to dynamic programs is straightforward given the following additional restriction ASSUMPTION 49 F is continuously differentiable on the interior of A Moreover if we knew that v was twice differentiable the monotonicity of g could be established by differentiating 5 with respect to x and exam 50 THEOREM 411 Differentiability of the value function Let X Γ F and β satisfy Assumptions 4344 and 4749 let v and let u satisfy x0 int X and gx0 int Γx0 then v is continuously differentiable at x0 with derivatives given by vix0 Fix0 gx0 for i 12 l Wx F x gx0 βvgx0 Proof Since gx0 int Γx0 and F is continuous it follows that W is concave and differentiable Assumption 49 If it follows that W is concave and differentiable Moreover since gx0 Γx for all x D it follows that Wx max Fxy βvy y Γx vx all x D with equality at x0 and the desired results follow immediately Note that the proof requires only that F be differentiable in its first argument Since fn is continuous such a choice is possible Then fx fy fx fnx fnx fny fny fy f fn fnx fny f fn Although we have organized these component arguments into a theorem about a function space each should be familiar to students of calculus Convergence in the sup norm is simply uniform convergence The proof above is then just a manual of the standard proofs that a sequence of functions that satisfies the Cauchy criterion uniformly converges uniformly and that uniform convergence preserves preserves continuity Exercise 37 a Let C1a b be the set of all continuously differentiable functions on a b X R with the norm f supxX fx fx Show that Ca b is a Banach space Hint Notice that f maxsupxX fx supxX fx b Show that this set of functions with the norm f supxXfx is not complete That is give an example of a sequence of functions that is Cauchy in the given norm that does not converge to a function in the set Is this sequence Cauchy in the set of all k part a c Let Cka b be the set of continuously differentiable functions on a b with the norm f k0 maxxab fαx Show that this space is complete if and only if αi 0 32 The Contraction Mapping Theorem In this section we prove two main results The first is the Contraction Mapping Theorem an extremely simple and powerful fixed point theorem The second is a set of sufficient conditions due to Blackwell for establishing that certain operators are contraction mappings The With differentiability of the value function established it is often straightforward to show that the optimal policy function g is monotone and to bound its slope Exercise 45 Consider the firstorder condition 5 Assume that U f and v are strictly increasing strictly concave and continuously differentiable and that 0 β g x f x Use 5 to show that g is strictly increasing and has slope less than the slope of f That is 0 g x g x f x f x if x x Hint Refer to Figure 42 In specific applications it is often possible to obtain much sharper characterizations of σ or of G or of both than those provided by the theorems above It is useful to keep in mind that once the existence and uniqueness of the solution to 1 has been established the right side of that equation can be treated as an ordinary maximization problem Thus whatever tools can be brought to bear on that problem should be exploited But such arguments usually rely on properties of F or of G both that are specific to the application at hand The problems in Chapter 5 provide a variety of illustrations of specific arguments of this type The functional equation and every optimal sequence xt if any exist is the associated policy correspondence a of the exercise over has the property corresponded in part of the exercise Our next task is to choose an appropriate space of functions within which to look for solutions to the functional equation and then to define an appropriate operator on that space in view of the results in Exercise 46c It is natural to seek solutions f X R that are continuous and homogeneous of degree one and bounded in the sense that fxxr all x X To capture the latter fact it is useful to use the norm f max xX fx 1 Let HX be the space of functions f X R that are continuous and homogeneous of degree one and bounded in the norm in 1 Define the operator T on HX by T fx sup yΓx Fx y β fy 2 Exercise 47 a Show that HX with the norm in 1 is a complete normed vector space b Show that under Assumptions 410 and 411 T HX HX It follows directly from Exercise 46a that for any f HX all x0 X all xk Γx0 fx αfEx all x0 X all xk Γx0 Since αβ 1 it then follows that the hypotheses of Theorems 43 and 45 hold Therefore the only solution in HX to the functional equation and the associated policy solution and value function generated by the associated policy correspondence g are unique and given by the unique fixed point of the operator T Thus Principle of Optimality applies to this type of constantreturnstoscale problem The constantreturnstoscale property of the operator T can be verified by using a modification of Blackwells contraction mapping theorem Theorem 39 For any function f that is homogeneous of degree one and for any α R we will in this context define the function f dα f αx fx 1 θ x θ fx 1 θ fx all 0 1 where η α is also homogeneous of degree one THEOREM 412 Let X Rk be a convex cone and let HX be as above with the norm η Let T HX satisfies T Tβ a monotonicity f g implies T f T g b discounting Tf γ xα T f γ βαxα for all f H and all α 0 Then T is a contraction with modulus γ Proof By homogeneity of degree one fx xr fxxr all x 0 Choose any f g HX Then fx gx fx gxxr gx fxxr gxxrβx f g all x 0 That is f g f g Hence monotonicity and discounting respectively imply that T f T g f g and Tg f g T g f g Reversing the roles of f and g and combining the two results we find that T f T g f g as was to be shown Our next result uses this theorem to establish that the operator T defined in 2 is a contraction with modulus αβ THEOREM 413 Let X F E and β satisfy Assumptions 410 and 411 and let HX be as above Then the operator T defined in 2 has a unique fixed point v HX In addition T w0 v αβnv0 v n 0 1 2 all v0 HX Proof As shown in Exercise 47 HX is a complete metric space and T HX satisfies the monotonicity condition of Theorem 47 and α 0 Then Tf αx sup yΓx Fx y βf αy sup yΓx Fx y βfy βαyr sup yΓx Fx y βfy βαxr T fx αβαx where the third line uses Assumption 410 Since x X was arbitrary it follows that Tf α T f βαT α satisfies the discounting condition and by Theorem 412 T is a contraction of modulus αβ Then follows from the Contraction Mapping Theorem Theorem 32 that T has a unique fixed pointvalue v HX and that v satisfies vx sup yΓx Fx y βvy Finally suppose that y Gx Then the Maximum Theorem 36 inequality of the Theorem of the Maximum Theorem 36 and homogeneity of F and v implies that βvλy is the homogeneity of F and v implies that βvλy Hence y Gx implies λy Gλx Exercise 48 Call a function f X R quasiconcave if x fx fy implies fθx 1 θy fx Call f strictly quasiconcave if f is strictly quasiconcave and f X R is homogeneous of degree one and quasiconcave then f is strictly quasiconcave homogeneous of degree one in part a that f is strictly quasiconcave Show that if x x X and x αx for any α R 1 fθx θfx 1 θfx all θ 0 1 and the associated policy correspondence G X X is compactvalued and uhc c Under what conditions is the fixed point v of the operator T defined in 2 strictly quasiconcave Hint Look at the proof of Theorem 48 and apply parts a and b of this exercise d Under what conditions is v differentiable 44 Unbounded Returns In this section we present a theorem that is useful when Assumptions 4142 hold so that the supremum function v satisfies the functional equation Theorem 42 but the boundedness hypotheses needed for Theorem 48 do not hold In such cases the functional equation may have other solutions as well The main result of this section is Theorem 414 which gives sufficient conditions for a solution to the functional equation to be the supremum function We then show how this result can be applied to two economic models with specific functional forms The first is a onesector model of optimal growth with a logarithmic utility function and a CobbDouglas production function the second is a standard investment model with a quadratic objective function and linear constraints The proof of Theorem 414 exploits only the monotonicity of the operator T defined on the set of all functions fXR by Tfx sup yΓx Fxy β fy The idea behind the proof is to start with a function ϑ that is an upper bound for v and then to apply the operator T to ϑ iterating down to a fixed point THEOREM 414 Let X Γ F and β satisfy Assumptions 4142 and let ϑ v and v be defined as they were in Section 41 Suppose there is a function ϑ X R such that 1 Tϑ ϑ 2 lim βn vxn 0 all x0 X all x Πx0 3 vx0 ϑx0 all x0 X If the function vXR defined by vx lim Tn ϑx converges then v v Proof First we will show that v is well defined and that v z Since the operator T is monotone 1 implies that Tn1 ϑ Tn ϑ a all n Hence for each x X Tn ϑx is a decreasing sequence of real numbers Furthermore if the sequence converges then vx is the limiting value if the sequence diverges then vx z Thus v is well defined and v z Theorem 43 Assumption 4142 hold Theorem 49 Hypothesis H implies that v satisfies the functional equation v T v Moreover 3 implies that v ϑ Hence by monotonicity of T Tn v Tn ϑ v establishing the desired result This theorem is particularly useful in the study of the unit elasticity and linearquadratic models described above For these cases it is easy to see that a solution v of the functional equation cf Exercise 293 Theorem 414 then ensures that this guess does indeed provide a solution to the problem at hand Moreover as will be seen below in these examples the value function and policy function have convenient closed forms that involve only a number of parameters This fact makes these two parametric structures especially useful for computing examples for computational purposes and for econometric estimation We will apply Theorem 414 first to the unit elastic form of the onesector optimal growth model max βt lnkt1α kt1 st 0 kt1 kα t t012 where α β 01 and the set X is the open interval 0 The return function is unbounded above and below on this interval Note that even if we were to restrict attention to the set X 01 of maintainable capital stocks the return would be unbounded below Since Γk 0 βk Tnvk 1βn 1β ln1 αβ αβ 1 αβ lnαβ α 1 αβ lnk n 12 To apply Theorem 44 clearly Assumption 41 holds for all k X To apply tools to this problem we must also show that Assumption 42 holds To do this note that the technology constraint implies that lnk t1 α lnk t and Given k0 it then follows that any feasible path lnk t αt lnk0 all t Hence for any k0 and any feasible path kt Πk0 the sequence of oneperiod returns satisfies 4 Fkt kt1 Fk t αlnk t all t Therefore lim βtFkt k t1 lim n Πk0 n β Fkt kt1 β 1αβ lnk0 all k0 0 t0 t0 where F is as defined in Section 41 Hence Assumption 42 holds Next we need a function ϑ that is an upper bound for the supremum function v and for the supremum function v Since 4 implies that v α lnk1αβ all k 0 we may take ϑk α lnk1 αβ With ϑ so defined clearly 13 hold Moreover with T defined by T defined by v we can always check to see if this is the case After obtaining a solution we can always check to see if this is the case Moreover reasonable restrictions on the optimal sequence x t are satisfied Hence 45 Euler Equations Theorem 414 is also useful in dealing with manydimensional quadratic problems An upper bound satisfying 13 is easy to calculate since any concave quadratic is bounded above by 0 The iterates Tnv are readily computed since they are defined by a finite number of parameters If the sequence converges Theorem 414 implies that the function is the supremum function and Theorem 45 implies that the linear policy that attains it is optimal If the problem is strictly concave there are no other optimal policies Hence Theorem 414 a CobbDouglas growth with a logarithmic utility function and a onesector model of optimal growth with linear state constraints Fxy αx 1 2 y2 1 2 x y2 abc 0 5 Think of the term αx bx2 as describing a firms net revenue when its capital stock is x and the term cy x22 as the cost of changing the capital stock from x to y Then given a constant interest rate r 0 the problem facing the firm is max wt1 βt I xt1 xt 2 t0 1 2 αx t 1 2 b x t2 1 2 c x t1 x t 2 where δ 1l1 γ To apply Theorem 414 to this problem first note that the return function F in 5 is bounded above by α 2 2b Hence the return we can verify by direct calculation that Tnvk 1β n 1β ln1 αβ αβ 1 αβ lnαβ α 1 αβ lnk n 12 This sequence converges to νk 1β 1β ln1 αβ αβ 1 αβ lnαβ α 1 αβ lnk Recall from Exercise 23 that this function is a fixed point of T Hence by Theorem 414 moreover since Theorem 43 applies the associated policy function constant saving rate policy gk αβk generates the optimal sequence of capital stocks Theorem 414 is also applicable to problems with quadratic functional forms There are many interesting new linear quadratic dynamic economic example suffices to illustrate the main ideas Let X R and let Tx R all x R Consider the return function fined by vx α 2 β1 δ all x R satisfied 13 Moreover it follows that β n β β where β β It is a simple exercise to verify from 6 that β n β γ β β b c and then from 7 and 8 that α α and γ γ The limit function ϑx clearly satisfies the functional equation and hence Theorem 41 implies it is the supremum function associated policy function is gx δx c and it follows from Theorem 414 that any sequence x t generated from it is optimal In this particular case the sequence never hits the restriction x t to the interior of domain 0 αb since negative capital has no interpretation and accumulating more capital than αb is costly and decreases revenue Section 42 would apply But the constraints of the quadratic problem vanish from the marginal advantage of the quadratic variables This fact that marginal returns are linear in the state variables This is also described by firstorder conditions the optimal policy function is also linear in the variables Hence the conventional form is realized only if maximal solutions attained at interior points of the feasible set Setting X R and x R and Tν we may take νk α lnk1 αβ After obtaining a solution we can always check to see if this is the case With β so defined clearly 13 hold Moreover max sup Tfk sup lnαk y βfy yαk 0 k y k t1 k α0 Theorem 414 There is a classical eighteenthcentury mode of attack on the sequence problem sup β t x t x t1 st x t1 Γ x t t0 1 2 SP that involves treating it as straightforward programming problem in decision variables x ti 0n Necessary conditions for an optimal program can be developed from the observation that if x tii0 solves the problem SP given x0 then for t 0 1 x t1 must solve max Fx t y βFy x t2 st y Γ x t1 That is a feasible variation on the sequence x tii0 leads to an improvement on an optimal policy A derivation of necessary conditions by this kind of argument is called a variational approach In the present context the conditions so derived are called Euler equations since Euler first obtained them from the continuoustime analogue to this problem Theorem 414 is useful for an Euler equation solution Let Assumptions 4345 47 and 49 hold Let Ft denote the lvector consisting of the partial derivatives F1 of F with respect to its first l arguments and let F denote the vector Fst1xt1 Since F is continuously differentiable and strictly concave if xt1 is in the interior of the set Γxt for all t the firstorder conditions for 1 are 0 Fxxtxt1 βFxxt1xt2 t 0 1 2 2 This is a system of l secondorder difference equations in the vector xt of state variables With vvector x given its solutions form an lparameter family and additional boundary conditions are needed to single out the one solution that is in fact optimal These additional boundary conditions are supplied by the transversality condition lim t βt Fx xtxt1 xt 0 3 This condition has the following interpretation Since the vector of derivatives Fx is the vector of marginal returns from increases in the current state variables the product Fx is a kind of statesecutor price payout at the start of the period of a claim to capital goods prices In this case β times differentiable and each gn is p1 times differentiable and 4 0 Fxx gx The envelope condition for this same maximum problem is 5 vx Fxx gx and set x x and gx gx1 Now set x xt and gxt xt1 Then 4 0 Fxxtxt1 βvxt1 6 Between these two equations then gives the Euler equations FE is always attained in the interior of Γx and let vy denote the vector νy vjy of partial derivatives of v Then the firstorder conditions for the maximum problem FE are vx max Fx y βvy yΓx FE The Euler equations can also be derived directly from the functional b Use Theorem 415 to obtain an alternative proof that the policy function gx 6x c β c is optimal for the quadratic investment model of Section 44 Strauch 1966 where a more complete treatment of the unboundedloss case can be found Our proofs of the necessity of the Euler equations and the sufficiency of the conditions are standard The necessity of the transversality condition is a difficult issue and resolving it involves conditions beyond those imposed here Peleg and Ryder 1972 and Weitzman 1973 both deal with this issue See Ekeland and Schenkman 1986 for a recent treatment For deterministic problems there is a close connection between maximization of indeed the term Euler equation and those formulated in continuous time Others are more straightforward and include many applications we discuss in Chapter 5 were originally formulated and studied in continuous time There are many good texts discussing the mathematical techniques used for such problems the calculus of variations and the closely related Principle of Pontryagin et al 1962 Arrow and Kurz 1970 and Kamien and Schwartz 1981 are excellent examples 5 Applications of Dynamic Programming under Certainty This chapter contains some economic problems that illustrate how the methods developed in the last chapter can be applied Some of the problems are straightforward and many exercises are solved and presented Others are more openended and in these cases specific results can be obtained only if additional assumptions are imposed The problems are not ordered in terms of difficulty 51 The OneSector Model of Optimal Growth In Chapter 2 we introduced the problem of optimal growth in a onegood economy max βt Ufxt xt1 st 0 xt1 fxt t 0 1 given x0 0 1 This problem is defined by the parameter β the functions U R R and f R R and the initial capital stock x0 The assumptions we will U1 0 β 1 U2 U is continuous U3 U is strictly increasing U4 U is strictly concave U5 U is continuously differentiable Since F is continuous concave and differentiable Assumptions 44 47 and 49 D lim T t0T βt Fxxtxt1 Fxtxt1 lim T t0T βt Fxxtxt1 xt xt Fxtxt1 Fxtxt1 Since x0 x0 0 rearranging terms gives D lim T t0T βt Fxxtxt1 βFxxt1xt2 xt xt Since xt satisfies 2 the terms in the summation are all zero Therefore substituting from 2 into the last term as well and then using 3 gives D lim T βT FxxTxT1 xT xT lim T βT FxxTxT1 xT where the last line uses the fact that Fx 0 Assumption 45 and xt 0 all t It then follows from 3 that D 0 establishing the desired result Note that Theorem 415 does not require any restrictions on Γ or β because the theorem applies only if a sequence satisfying 2 and 3 has already been found Restrictions on Γ and β are needed to ensure that such a sequence can be located Exercise 49 a Use Theorem 415 to obtain an alternative proof that the policy function gk αβkα is optimal for the unitelastic optimal growth model of Section 44 The next exercise gives two variations on Theorem 43 that are sometimes useful when 8 does not hold The next theorem provides a partial converse to Theorem 44 It shows that any sequence satisfying 9 and a boundedness condition is an optimal plan Now consider plans that attain the optimum Given any x 0 the set of feasible plans Πx0 consists of the sequences