Recursive Macroeconomic Theory - Ljungqvist e Sargent - 4a Edição

recursive macroeconomic theory LARS LJUNGQVIST THOMAS J SARGENT F OU RTH E D I TI ON COVER DESIGNED BY NANNA REYKDAL LARS LJUNGQVIST THOMAS J SARGENT F O UR T H EDIT ION ECONOMICS COVER DESIGNED BY NANNA REYKDAL ECONOMICS LARS LJUNGQVIST THOMAS J SARGENT 9780262038669 Recursive methods provide powerful ways to pose and solve problems in dynamic macroeconomics Recursive Macroeconomic Theory offers both an introduction to recursive methods and more advanced material Only practice in solving diverse problems fully conveys the advantages of the recursive approach so the book provides many applications This fourth edition features two new chapters and substantial revisions to other chapters that demonstrate the power of recursive methods One new chapter applies the recursive approach to Ramsey taxation and sharply characterizes the time inconsistency of optimal policies These insights are used in other chapters to simplify recursive formulations of Ramsey plans and credible government policies The second new chapter explores the mechanics of matching models and identifies a common channel through which productivity shocks are magnified Other chapters have been refined and extended including new material on heterogeneous beliefs in models of both complete and incomplete markets and a deeper account of forces that shape aggregate labor supply elasticities in life cycle models The book is suitable for first and secondyear graduate courses in macroeconomics Most chapters conclude with exercises many exercises and examples use Matlab or Python Lars Ljungqvist is Professor of Economics at the Stockholm School of Economics Thomas J Sargent is Berkley Professor of Economics and Business at New York University and Senior Fellow at the Hoover Institution He was a recipient of the 2011 Nobel Prize in Economic Sciences Through a lucid exposition a rigorous development of the tools and a thorough analysis of each topic this extraordinary textbook offers a stateoftheart representation of the field A must for every student with the ambition of becoming a scholar of macroeconomics and for every teacher with the passion to guide students through this journey This book makes you proud to be a macroeconomist Gianluca Violante Professor of Economics Princeton University This classic textbook has taught generations of students the tools of dynamic economics At the same time it has been continually updated to serve as a key reference manual for researchers Once you have been Bellmanized by this book your life as an economist will never be the same Monika Piazzesi Joan Kenney Professor of Economics Stanford University Recursive Macroeconomic Theory thoroughly works through a wide variety of applications of recursive methods to the analysis of central themes in macroeconomics As Alexander the Great is said to have always kept a copy of the Iliad under his pillow I think the modern macroeconomist would do well to keep a copy of this excellent work close at hand Fernando Alvarez William C Norby Professor of Economics University of Chicago The MIT Press Massachusetts Institute of Technology Cambridge Massachusetts 02142 httpmitpressmitedu FO UR T H ED IT ION 9 780262 038669 9 0 0 0 0 recursive macroeconomic theory recursive macroeconomic theory Recursive Macroeconomic Theory Fourth edition To our parents Zabrina and Carolyn Recursive Macroeconomic Theory Fourth edition Lars Ljungqvist Stockholm School of Economics Thomas J Sargent New York University and Hoover Institution The MIT Press Cambridge Massachusetts London England c 2018 Massachusetts Institute of Technology All rights reserved No part of this book may be reproduced in any form by any electronic or mechanical means including photocopying recording or information storage and retrieval without permission in writing from the publisher Printed and bound in the United States of America Library of Congress CataloginginPublication Data Names Ljungqvist Lars author Sargent Thomas J author Title Recursive macroeconomic theory Lars Ljungqvist and Thomas J Sargent Description Fourth Edition Cambridge MA MIT Press 2018 Revised edition of the authors Recursive macroeconomic theory c2012 Includes bibliographical references and index Identifiers LCCN 2018002500 ISBN 9780262038669 hardcover alk paper Subjects LCSH Macroeconomics Recursive functions Statics and dynamics Social sciences Classification LCC HB1725 L59 2018 DDC 3390151135dc23 LC record available at httpslccnlocgov2018002500 10 9 8 7 6 5 4 3 2 1 Contents Acknowledgments xxi Preface to the fourth edition xxiii Part I Imperialism of Recursive Methods 1 Overview 3 11 Warning 12 A common ancestor 13 The savings problem 131 Linear quadratic permanent income theory 132 Precaution ary saving 133 Complete markets insurance and the distribution of wealth 134 Bewley models 135 History dependence in stan dard consumption models 136 Growth theory 137 Limiting results from dynamic optimal taxation 138 Asset pricing 139 Multiple assets 14 Recursive methods 141 Dynamic programming and the Lucas Critique 142 Dynamic programming challenged 143 Impe rialistic response of dynamic programming 144 History dependence and dynamic programming squared 145 Dynamic principalagent problems 146 More applications v vi Contents Part II Tools 2 Time Series 29 21 Two workhorses 22 Markov chains 221 Stationary distri butions 222 Asymptotic stationarity 223 Forecasting the state 224 Forecasting functions of the state 225 Forecasting functions 226 Enough onestepahead forecasts determine P 227 Invariant functions and ergodicity 228 Simulating a Markov chain 229 The likelihood function 23 Continuousstate Markov chain 24 Stochas tic linear difference equations 241 First and second moments 242 Summary of moment formulas 243 Impulse response function 244 Prediction and discounting 245 Geometric sums of quadratic forms 25 Population regression 251 Multiple regressors 26 Estimation of model parameters 27 The Kalman filter 28 Estimation again 29 Vector autoregressions and the Kalman filter 291 Conditioning on the semiinfinite past of y 292 A timeinvariant VAR 293 Inter preting VARs 210 Applications of the Kalman filter 2101 Muths reverse engineering exercise 2102 Jovanovics application 211 The spectrum 2111 Examples 212 Example the LQ permanent in come model 2121 Another representation 2122 Debt dynamics 2123 Two classic examples 2124 Spreading consumption cross sec tion 2125 Invariant subspace approach 213 Concluding remarks A Linear difference equations 2A1 A firstorder difference equation 2A2 A secondorder difference equation 215 Exercises 3 Dynamic Programming 105 31 Sequential problems 311 Three computational methods 312 CobbDouglas transition logarithmic preferences 313 Euler equa tions 314 A sample Euler equation 32 Stochastic control problems 33 Concluding remarks 34 Exercise 4 Practical Dynamic Programming 115 41 The curse of dimensionality 42 Discretestate dynamic program ming 43 Bookkeeping 44 Application of Howard improvement algo rithm 45 Numerical implementation 451 Modified policy iteration 46 Sample Bellman equations 461 Example 1 calculating expected Contents vii utility 462 Example 2 risksensitive preferences 463 Example 3 costs of business cycles 47 Polynomial approximations 471 Recom mended computational strategy 472 Chebyshev polynomials 473 Algorithm summary 474 Shapepreserving splines 48 Concluding remarks 5 Linear Quadratic Dynamic Programming 129 51 Introduction 52 The optimal linear regulator problem 521 Value function iteration 522 Discounted linear regulator problem 523 Policy improvement algorithm 53 The stochastic optimal lin ear regulator problem 531 Discussion of certainty equivalence 54 Shadow prices in the linear regulator 541 Stability 55 A Lagrangian formulation 56 The Kalman filter again 57 Concluding remarks A Matrix formulas 59 Exercises 6 Search and Unemployment 157 61 Introduction 62 Preliminaries 621 Nonnegative random vari ables 622 Meanpreserving spreads 63 McCalls model of intertem poral job search 631 Characterizing reservation wage 632 Effects of meanpreserving spreads 633 Allowing quits 634 Waiting times 635 Firing 64 A lake model 65 A model of career choice 66 Offer distribution unknown 67 An equilibrium price distribution 671 A BurdettJudd setup 672 Consumer problem with noisy search 673 Firms 674 Equilibrium 675 Special cases 68 Jovanovics match ing model 681 Recursive formulation and solution 682 Endogenous statistics 69 A longer horizon version of Jovanovics model 691 The Bellman equations 610 Concluding remarks A More numerical dy namic programming 6A1 Example 4 search 6A2 Example 5 a Jovanovic model 612 Exercises Part III Competitive Equilibria and Applications 7 Recursive Competitive Equilibrium I 225 71 An equilibrium concept 72 Example adjustment costs 721 A planning problem 73 Recursive competitive equilibrium 74 Equi librium human capital accumulation 741 Planning problem 742 viii Contents Decentralization 75 Equilibrium occupational choice 751 A plan ning problem 752 Decentralization 76 Markov perfect equilibrium 761 Computation 77 Linear Markov perfect equilibria 771 An example 78 Concluding remarks 79 Exercises 8 Equilibrium with Complete Markets 249 81 Time 0 versus sequential trading 82 The physical setting pref erences and endowments 83 Alternative trading arrangements 831 History dependence 84 Pareto problem 841 Time invariance of Pareto weights 85 Time 0 trading ArrowDebreu securities 851 Equilibrium pricing function 852 Optimality of equilibrium alloca tion 853 Interpretation of trading arrangement 854 Equilibrium computation 86 Simpler computational algorithm 861 Example 1 risk sharing 862 Implications for equilibrium computation 863 Ex ample 2 no aggregate uncertainty 864 Example 3 periodic endow ment processes 865 Example 4 87 Primer on asset pricing 871 Pricing redundant assets 872 Riskless consol 873 Riskless strips 874 Tail assets 875 Oneperiod returns 88 Sequential trading 881 Arrow securities 882 Financial wealth as an endogenous state variable 883 Reopening markets 884 Debt limits 885 Sequential trading 886 Equivalence of allocations 89 Recursive competitive equilibrium 891 Endowments governed by a Markov process 892 Equilibrium outcomes inherit the Markov property 893 Recursive formulation of optimization and equilibrium 894 Computing an equi librium with sequential trading of Arrowsecurities 810 j step pricing kernel 8101 Arbitragefree pricing 811 Term structure of yields on riskfree claims 8111 Constructing yields 812 Recursive version of Pareto problem 813 Concluding remarks Appendices Departures from key assumptions A Heterogenous discounting B Heterogenous beliefs 8B1 Example one types beliefs are closer to the truth 8B2 Equilibrium prices reflect beliefs 8B3 Mispricing 8B4 Learning 8B5 Role of complete markets C Incomplete markets 8C1 An example economy 8C2 Asset payoff correlated with iid aggregate endowment 8C3 Beneficial market incompleteness 818 Exercises 9 Overlapping Generations 331 91 Endowments and preferences 92 Time 0 trading 921 Example equilibria 922 Relation to welfare theorems 923 Nonstationary equilibria 924 Computing equilibria 93 Sequential trading 94 Money 941 Computing more equilibria with valued fiat currency Contents ix 942 Equivalence of equilibria 95 Deficit finance 951 Steady states and the Laffer curve 96 Equivalent setups 961 The economy 962 Growth 97 Optimality and the existence of monetary equilib ria 971 BalaskoShell criterion for optimality 98 Withingeneration heterogeneity 981 Nonmonetary equilibrium 982 Monetary equi librium 983 Nonstationary equilibria 984 The real bills doctrine 99 Giftgiving equilibrium 910 Concluding remarks 911 Exercises 10 Ricardian Equivalence 379 101 Borrowing limits and Ricardian equivalence 102 Infinitely lived agent economy 1021 Optimal consumptionsavings decision when bt1 0 1022 Optimal consumptionsavings decision when bt1 bt1 103 Government finance 1031 Effect on household 104 Linked generations interpretation 105 Concluding remarks 11 Fiscal Policies in a Growth Model 391 111 Introduction 112 Economy 1121 Preferences technology in formation 1122 Components of a competitive equilibrium 113 The term structure of interest rates 114 Digression sequential version of government budget constraint 1141 Irrelevance of maturity structure of government debt 115 Competitive equilibria with distorting taxes 1151 The household noarbitrage and assetpricing formulas 1152 User cost of capital formula 1153 Household firstorder conditions 1154 A theory of the term structure of interest rates 1155 Firm 116 Computing equilibria 1161 Inelastic labor supply 1162 The equilibrium steady state 1163 Computing the equilibrium path with the shooting algorithm 1164 Other equilibrium quantities 1165 Steadystate R 1166 Lumpsum taxes available 1167 No lump sum taxes available 117 A digression on backsolving 118 Effects of taxes on equilibrium allocations and prices 119 Transition experi ments with inelastic labor supply 1110 Linear approximation 11101 Relationship between the λi s 11102 Conditions for existence and uniqueness 11103 Onceandforall jumps 11104 Simplification of formulas 11105 A onetime pulse 11106 Convergence rates and anticipation rates 11107 A remark about accuracy Euler equation errors 1111 Growth 1112 Elastic labor supply 11121 Steady state calculations 11122 Some experiments 1113 A twocountry model 11131 Initial conditions 11132 Equilibrium steady state values 11133 Initial equilibrium values 11134 Shooting algorithm x Contents 11135 Transition exercises 1114 Concluding remarks A Log linear approximations 1116 Exercises 12 Recursive Competitive Equilibrium II 471 121 Endogenous aggregate state variable 122 The stochastic growth model 123 Lagrangian formulation of the planning problem 124 Time 0 trading ArrowDebreu securities 1241 Household 1242 Firm of type I 1243 Firm of type II 1244 Equilibrium prices and quantities 1245 Implied wealth dynamics 125 Sequential trading Arrow securities 1251 Household 1252 Firm of type I 1253 Firm of type II 1254 Equilibrium prices and quantities 1255 Financing a type II firm 126 Recursive formulation 1261 Technology is gov erned by a Markov process 1262 Aggregate state of the economy 127 Recursive formulation of the planning problem 128 Recursive formulation of sequential trading 1281 A Big K little k device 1282 Price system 1283 Household problem 1284 Firm of type I 1285 Firm of type II 129 Recursive competitive equilibrium 1291 Equilibrium restrictions across decision rules 1292 Using the plan ning problem 1210 Concluding remarks A The permanent income model revisited 12A1 Reinterpreting the singleagent model 12A2 Decentralization and scaled prices 12A3 Matching equilibrium and planning allocations 12A4 Interpretation 13 Asset Pricing Theory 503 131 Introduction 132 Euler equations 133 Martingale theories of consumption and stock prices 134 Equivalent martingale mea sure 135 Equilibrium asset pricing 136 Stock prices without bub bles 137 Computing asset prices 1371 Example 1 logarithmic preferences 1372 Example 2 finitestate version 1373 Exam ple 3 growth 138 Term structure of interest rates 139 State contingent prices 1391 Insurance premium 1392 Manmade un certainty 1393 The ModiglianiMiller theorem 1310 Government debt 13101 The Ricardian proposition 13102 No Ponzi schemes A HarrisonKreps 1978 heterogeneous beliefs 13A1 Optimism and Pessimism 13A2 Equilibrium price function 13A3 Comparisons of equilibrium price functions 13A4 Single belief prices 13A5 Pric ing under heterogeneous beliefs 13A6 Insufficient funds B Gaussian assetpricing model 1313 Exercises Contents xi 14 Asset Pricing Empirics 549 141 Introduction 142 Interpretation of riskaversion parameter 143 The equity premium puzzle 144 Market price of risk 145 Hansen Jagannathan bounds 1451 Law of one price implies that EmR 1 1452 Inner product representation of price functional 1453 Admis sible stochastic discount factors 146 Failure of CRRA to attain HJ bound 147 Nonexpected utility 1471 Another representation of the utility recursion 1472 Stochastic discount factor 1473 Twisted probability distributions 148 Reinterpretation of the utility recursion 1481 Risk aversion versus model misspecification aversion 1482 Re cursive representation of probability distortions 1483 Entropy 1484 Expressing ambiguity aversion 1485 Ambiguity averse preferences 1486 Market price of model uncertainty 1487 Measuring model uncertainty 149 Costs of aggregate fluctuations 1410 Reverse engi neered consumption heterogeneity 1411 Affine risk prices 14111 An application 14112 Affine term structure of yields 1412 Riskneutral probabilities 14121 Asset pricing in a nutshell 1413 Distorted be liefs 1414 Concluding remarks A Riesz representation theorem B Computing stochastic discount factors C A log normal bond pricing model 14C1 Slope of yield curve 14C2 Backus and Zins stochas tic discount factor 14C3 Reverse engineering a stochastic discount factor 1418 Exercises 15 Economic Growth 631 151 Introduction 152 The economy 1521 Balanced growth path 153 Exogenous growth 154 Externality from spillovers 155 All fac tors reproducible 1551 Onesector model 1552 Twosector model 156 Research and monopolistic competition 1561 Monopolistic competition outcome 1562 Planner solution 157 Growth in spite of nonreproducible factors 1571 Core of capital goods produced without nonreproducible inputs 1572 Research labor enjoying an ex ternality 158 Concluding remarks 159 Exercises 16 Optimal Taxation with Commitment 661 161 Introduction 162 A nonstochastic economy 1621 Govern ment 1622 Household 1623 Firms 163 The Ramsey problem 164 Zero capital tax 165 Primal approach to the Ramsey problem 1651 Constructing the Ramsey plan 1652 Revisiting a zero capital tax 166 Taxation of initial capital 167 Nonzero capital tax due to incomplete taxation 168 A stochastic economy 1681 Government xii Contents 1682 Household 1683 Firms 169 Indeterminacy of debt and cap ital taxes 1610 A Ramsey plan under uncertainty 1611 Ex ante capital tax varies around zero 16111 Sketch of the proof of Proposi tion 2 1612 A stochastic economy without capital 16121 Computa tional strategy 16122 More specialized computations 16123 Time consistency 1613 Examples of labor tax smoothing 16131 Example 1 gt g for all t 0 16132 Example 2 gt 0 for t T and nonstochastic gT 0 16133 Example 3 gt 0 for t T and gT is stochastic 16134 Time 0 is special with b0 0 1614 Lessons for optimal debt policy 1615 Taxation without statecontingent debt 16151 Future values of gt become deterministic 16152 Stochastic gt but special preferences 16153 Example 3 revisited gt 0 for t T and gT is stochastic 1616 Nominal debt as statecontingent real debt 16161 Setup and main ideas 16162 Optimal taxation in a nonmonetary economy 16163 Optimal policy in a corresponding monetary economy 16164 Sticky prices 1617 Relation to fiscal the ories of the price level 16171 Budget constraint versus asset pricing equation 16172 Disappearance of quantity theory 16173 Price level indeterminacy under interest rate peg 16174 Monetary or fis cal theory of the price level 1618 Zero tax on human capital 1619 Should all taxes be zero 1620 Concluding remarks 1621 Exercises Part IV Savings Problems and Bewley Models 17 SelfInsurance 759 171 Introduction 172 The consumers environment 173 Non stochastic endowment 1731 An ad hoc borrowing constraint non negative assets 1732 Example periodic endowment process 174 Quadratic preferences 175 Stochastic endowment process iid case 176 Stochastic endowment process general case 177 Intuition 178 Endogenous labor supply 179 Concluding remarks A Supermartin gale convergence theorem 1711 Exercises 18 Incomplete Markets Models 785 181 Introduction 182 A savings problem 1821 Wealthemployment distributions 1822 Reinterpretation of the distribution λ 1823 Ex ample 1 a pure credit model 1824 Equilibrium computation 1825 Example 2 a model with capital 1826 Computation of equilibrium Contents xiii 183 Unification and further analysis 184 The nonstochastic sav ings problem when β1 r 1 185 Borrowing limits natural and ad hoc 1851 A candidate for a single state variable 1852 Su permartingale convergence again 186 Average assets as a function of r 187 Computed examples 188 Several Bewley models 1881 Optimal stationary allocation 189 A model with capital and private IOUs 1810 Private IOUs only 18101 Limitation of what credit can achieve 18102 Proximity of r to ρ 18103 Inside money or free banking interpretation 18104 Bewleys basic model of fiat money 1811 A model of seigniorage 1812 Exchange rate indeterminacy 1813 Interest on currency 18131 Explicit interest 18132 The upper bound on M p 18133 A very special case 18134 Implicit in terest through deflation 1814 Precautionary savings 1815 Models with fluctuating aggregate variables 18151 Aiyagaris model again 18152 Krusell and Smiths extension 1816 Concluding remarks 1817 Exercises Part V Recursive Contracts 19 Dynamic Stackelberg Problems 839 191 History dependence 192 The Stackelberg problem 193 Timing protocol 194 Recursive formulation 1941 Two Bellman equations 1942 Subproblem 1 1943 Subproblem 2 1944 Timing protocol 1945 Time inconsistency 195 Large firm facing a competitive fringe 1951 The competitive fringe 1952 The large firms problem 1953 Numerical example 196 Concluding remarks 197 Exercises 20 Two Ramsey Problems Revisited 857 201 Introduction 202 The LucasStokey economy 2021 Find ing the state is an art 2022 Intertemporal delegation 2023 Bell man equations 2024 Subproblem 1 Continuation Ramsey problem 2025 Subproblem 2 Ramsey problem 2026 Firstorder conditions 2027 State variable degeneracy 2028 Symptom and source of time inconsistency 203 Recursive formulation of AMSS model 2031 Re casting state variables 2032 Measurability constraints 2033 Bell man equations 2034 Martingale replaces statevariable degeneracy 204 Concluding remarks xiv Contents 21 Incentives and Insurance 871 211 Insurance with recursive contracts 212 Basic environment 213 Onesided no commitment 2131 Selfenforcing contract 2132 Recursive formulation and solution 2133 Recursive computation of contract 2134 Profits 2135 Pv is strictly concave and contin uously differentiable 2136 Many households 2137 An example 214 A Lagrangian method 215 Insurance with asymmetric infor mation 2151 Efficiency implies bs1 bs ws1 ws 2152 Local upward and downward constraints are enough 2153 Concavity of P 2154 Local downward constraints always bind 2155 Coinsurance 2156 P v is a martingale 2157 Comparison to model with com mitment problem 2158 Spreading continuation values 2159 Mar tingale convergence and poverty 21510 Extension to general equilib rium 21511 Comparison with selfinsurance 216 Insurance with unobservable storage 2161 Feasibility 2162 Incentive compatibil ity 2163 Efficient allocation 2164 The twoperiod case 2165 Role of the planner 2166 Decentralization in a closed economy 217 Concluding remarks A Historical development 21A1 Spear and Sri vastava 21A2 Timing 21A3 Use of lotteries 219 Exercises 22 Equilibrium without Commitment 933 221 Twosided lack of commitment 222 A closed system 223 Recursive formulation 224 Equilibrium consumption 2241 Con sumption dynamics 2242 Consumption intervals cannot contain each other 2243 Endowments are contained in the consumption intervals 2244 All consumption intervals are nondegenerate unless autarky is the only sustainable allocation 225 Pareto frontier and ex ante divi sion of the gains 226 Consumption distribution 2261 Asymptotic distribution 2262 Temporary imperfect risk sharing 2263 Per manent imperfect risk sharing 227 Alternative recursive formulation 228 Pareto frontier revisited 2281 Values are continuous in implicit consumption 2282 Differentiability of the Pareto frontier 229 Con tinuation values a la Kocherlakota 2291 Asymptotic distribution is nondegenerate for imperfect risk sharing except when S 2 2292 Continuation values do not always respond to binding participation con straints 2210 A twostate example amnesia overwhelms memory 22101 Pareto frontier 22102 Interpretation 2211 A threestate example 22111 Perturbation of parameter values 22112 Pareto frontier 2212 Empirical motivation 2213 Generalization 2214 De centralization 2215 Endogenous borrowing constraints 2216 Con cluding remarks 2217 Exercises Contents xv 23 Optimal Unemployment Insurance 987 231 Historydependent unemployment insurance 232 A onespell model 2321 The autarky problem 2322 Unemployment insurance with full information 2323 The incentive problem 2324 Unem ployment insurance with asymmetric information 2325 Computed example 2326 Computational details 2327 Interpretations 2328 Extension an onthejob tax 2329 Extension intermittent unem ployment spells 233 A multiplespell model with lifetime contracts 2331 The setup 2332 A recursive lifetime contract 2333 Com pensation dynamics when unemployed 2334 Compensation dynamics while employed 2335 Summary 234 Concluding remarks 235 Ex ercises 24 Credible Government Policies I 1011 241 Introduction 2411 Diverse sources of history dependence 242 Oneperiod economy 2421 Competitive equilibrium 2422 Ram sey problem 2423 Nash equilibrium 243 Nash and Ramsey out comes 2431 Taxation example 2432 Blackbox example with dis crete choice sets 244 Reputational mechanisms general idea 2441 Dynamic programming squared 2442 Etymology of dynamic pro gramming squared 245 The infinitely repeated economy 2451 A strategy profile implies a history and a value 2452 Recursive formu lation 246 Subgame perfect equilibrium SPE 247 Examples of SPE 2471 Infinite repetition of oneperiod Nash equilibrium 2472 Supporting better outcomes with trigger strategies 2473 When rever sion to Nash is not bad enough 248 Values of all SPEs 2481 Basic idea of dynamic programming squared 249 APS machinery 2410 Selfenforcing SPE 24101 The quest for something worse than rep etition of Nash outcome 2411 Recursive strategies 2412 Examples of SPE with recursive strategies 24121 Infinite repetition of Nash outcome 24122 Infinite repetition of a betterthanNash outcome 24123 Something worse a stickandcarrot strategy 2413 Best and worst SPE values 24131 When v1 is outside the candidate set 2414 Examples alternative ways to achieve the worst 24141 Attaining the worst method 1 24142 Attaining the worst method 2 24143 Attaining the worst method 3 24144 Numerical example 2415 In terpretations 2416 Extensions 2417 Exercises xvi Contents 25 Credible Government Policies II 1059 251 Historydependent government policies 252 The setting 2521 Household problem 2522 Government 2523 Analysis of house holds problem 2524 θt1 as intermediating variable 253 Recur sive approach to Ramsey problem 2531 Subproblem 1 Continua tion Ramsey problem 2532 Subproblem 2 Ramsey problem 2533 Finding set Ω 2534 An example 254 Changs formulation 2541 Competitive equilibrium 255 Inventory of key objects 256 Analy sis 2561 Notation 2562 An operator 257 Sustainable plans 258 Concluding remarks 26 Two Topics in International Trade 1083 261 Two dynamic contracting problems 262 Moral hazard and dif ficult enforcement 2621 Autarky 2622 Investment with full insur ance 2623 Limited commitment and unobserved investment 2624 Optimal capital outflows under distress 263 Gradualism in trade pol icy 2631 Closedeconomy model 2632 A Ricardian model of two countries under free trade 2633 Trade with a tariff 2634 Wel fare and Nash tariff 2635 Trade concessions 2636 A repeated tariff game 2637 Timeinvariant transfers 2638 Gradualism time varying trade policies 2639 Baseline policies 26310 Multiplicity of payoffs and continuation values 264 Another model 265 Concluding remarks A Computations for Atkesons model 267 Exercises Part VI Classical Monetary and Labor Economics 27 FiscalMonetary Theories of Inflation 1123 271 The issues 272 A shopping time monetary economy 2721 Household 2722 Government 2723 Equilibrium 2724 Short run versus long run 2725 Stationary equilibrium 2726 Initial date time 0 2727 Equilibrium determination 273 Ten mone tary doctrines 2731 Quantity theory of money 2732 Sustained deficits cause inflation 2733 Fiscal prerequisites of zero inflation policy 2734 Unpleasant monetarist arithmetic 2735 An open market operation delivering neutrality 2736 The optimum quan tity of money 2737 Legal restrictions to boost demand for currency 2738 One big open market operation 2739 A fiscal theory of the Contents xvii price level 27310 Exchange rate indeterminacy 27311 Determi nacy of the exchange rate retrieved 274 An example of exchange rate indeterminacy 2741 Trading before sunspot realization 2742 Fis cal theory of the price level 2743 A game theoretic view of the fiscal theory of the price level 275 Optimal inflation tax the Friedman rule 2751 Economic environment 2752 Households optimization prob lem 2753 Ramsey plan 276 Time consistency of monetary policy 2761 Model with monopolistically competitive wage setting 2762 Perfect foresight equilibrium 2763 Ramsey plan 2764 Credibility of the Friedman rule 277 Concluding remarks 278 Exercises 28 Credit and Currency 1171 281 Credit and currency with longlived agents 282 Preferences and endowments 283 Complete markets 2831 A Pareto problem 2832 A complete markets equilibrium 2833 Ricardian proposition 2834 Loan market interpretation 284 A monetary economy 285 Townsends turnpike interpretation 286 The Friedman rule 2861 Welfare 287 Inflationary finance 288 Legal restrictions 289 A twomoney model 2810 A model of commodity money 28101 Equi librium 28102 Virtue of fiat money 2811 Concluding remarks 2812 Exercises 29 Equilibrium Search Matching and Lotteries 1207 291 Introduction 292 An island model 2921 A single market island 2922 The aggregate economy 293 A matching model 2931 A steady state 2932 Welfare analysis 2933 Size of the match surplus 294 Matching model with heterogeneous jobs 2941 A steady state 2942 Welfare analysis 2943 The allocating role of wages I separate markets 2944 The allocating role of wages II wage announcements 295 Employment lotteries 296 Lotteries for households versus lotteries for firms 2961 An aggregate production function 2962 Timevarying capacity utilization 297 Employment effects of layoff taxes 2971 A model of employment lotteries with lay off taxes 2972 An island model with layoff taxes 2973 A matching model with layoff taxes 298 KiyotakiWright search model of money 2981 Monetary equilibria 2982 Welfare 299 Concluding remarks 2910 Exercises xviii Contents 30 Matching Models Mechanics 1269 301 Introduction 302 Fundamental surplus 3021 Sensitivity of unemployment to market tightness 3022 Nash bargaining model 3023 Shimers critique 3024 Relationship to workers outside value 3025 Relationship to match surplus 3026 Fixed matching cost 3027 Sticky wages 3028 Alternatingoffer wage bargaining 303 Business cycle simulations 3031 Halls sticky wage 3032 Hage dorn and Manovskiis high value of leisure 3033 Hall and Milgroms alternatingoffer bargaining 3034 Matching and bargaining proto cols in a DSGE model 304 Overlapping generations in one match ing function 3041 A steady state 3042 Reservation productivity is increasing in age 3043 Wage rate is decreasing in age 3044 Welfare analysis 3045 The optimal policy 305 Directed search agespecific matching functions 3051 Value functions and market tightness 3052 Job finding rate is decreasing in age 3053 Block recursive equilibrium computation 3054 Welfare analysis 306 Con cluding remarks 31 Foundations of Aggregate Labor Supply 1315 311 Introduction 312 Equivalent allocations 3121 Choosing ca reer length 3122 Employment lotteries 313 Taxation and social security 3131 Taxation 3132 Social security 314 Earnings experience profiles 3141 Time averaging 3142 Employment lot teries 3143 Prescott tax and transfer scheme 3144 No discounting now matters 315 Intensive margin 3151 Employment lotteries 3152 Time averaging 3153 Prescott taxation 316 BenPorath human capital 3161 Time averaging 3162 Employment lotteries 3163 Prescott taxation 317 Earnings shocks 3171 Interpretation of wealth and substitution effects 318 Time averaging in a Bewley model 3181 Incomplete markets 3182 Complete markets 3183 Simulations of Prescott taxation 319 L and S equivalence meets C and Ks agents 3191 Guess the value function 3192 Verify optimality of time averaging 3193 Equivalence of time averaging and lotteries 3110 Two pillars for high elasticity at extensive margin 3111 No pillars at intensive margin 31111 Special example of high elasticity at intensive margin 31112 Fragility of the special example 3112 Concluding remarks Contents xix Part VII Technical Appendices A Functional Analysis 1373 A1 Metric spaces and operators A2 Discounted dynamic program ming A21 Policy improvement algorithm A22 A search problem B Linear Projections and Hidden Markov Models 1385 B1 Linear projections B2 Hidden Markov models B3 Nonlinear filtering 1 References 1391 2 Subject Index 1425 3 Author Index 1431 4 Matlab Index 1437 To our parents Zabrina and Carolyn Acknowledgments We wrote this book during the 1990s 2000s and 2010s while teaching graduate courses in macro and monetary economics We owe a substantial debt to the students in these classes for learning with us We would especially like to thank Marco Bassetto Victor Chernozhukov Riccardo Colacito Mariacristina De Nardi William Dupor William Fuchs George Hall Sagiri Kitao Hanno Lustig Monika Piazzesi Navin Kartik Martin Schneider Yongseok Shin Christopher Sleet Stijn Van Nieuwerburgh Laura Veldkamp Neng Wang Chao Wei Mark Wright and Sevin Yeltekin for commenting on drafts of earlier editions when they were graduate students In prefaces to earlier editions we forecast that they would soon be prominent economists and we are happy that has come true We also thank Isaac Baley Anmol Bhandari Saki Bigio Jaroslav Borovicka David Evans Sebastian Graves Christopher Huckfeldt Anna Orlik Ignacio Presno Cecilia Parlatore Siritto Balint Szoke and Hakon Travoll for helpful comments on earlier drafts of this edition Each of these people made substantial sugges tions for improving this book We expect much from members of this group as we did from an earlier group of students that Sargent 1987b thanked We received useful criticisms from Roberto Chang Gary Hansen Jonathan Heathcote Berthold Herrendorf Mark Huggett Charles Jones Narayana Kocher lakota Dirk Krueger Per Krusell Francesco Lippi Rodolfo Manuelli Beatrix Paal Adina Popescu Jonathan Thomas Nicola Tosini and Jesus Fernandez Villaverde Rodolfo Manuelli and Pierre Olivier Weill kindly allowed us to reproduce some of their exercises We indicate the exercises that they donated Some of the exercises in chapters 6 9 and 28 are versions of ones in Sargent 1987b Francois Velde provided substantial help with the TEX and Unix macros that produced this book Maria Bharwada helped typeset the first edition For providing good environments to work on this book Ljungqvist thanks the Stockholm School of Economics and New York University and Sargent thanks the Hoover Institution and the departments of economics at the Uni versity of Chicago Stanford University Princeton University and New York University xxi To our parents Zabrina and Carolyn Preface to the fourth edition Recursive Methods Much of this book is about how to use recursive methods to study dynamic macroeconomic models Recursive methods are very important in the analysis of dynamic systems in economics and other sciences They originated after World War II in diverse literatures promoted by Wald sequential analysis Bellman dynamic programming and Kalman Kalman filtering Dynamics Dynamics studies sequences of vectors of random variables indexed by time called time series Time series are immense objects with as many components as the number of variables times the number of time periods A dynamic economic model characterizes and interprets covariations among all of these components in terms of the purposes and opportunities of economic agents Agents choose components of the time series in light of their opinions about other components Recursive methods break a dynamic problem into pieces by forming a se quence of problems each one being a constrained choice between utility today and utility tomorrow The idea is to find a way to describe the position of the system now where it might be tomorrow and how agents care now about where it is tomorrow Thus recursive methods study dynamics indirectly by characterizing a pair of functions a transition function mapping the state today into the state tomorrow and another function mapping the state today into the other endogenous variables of the model today The state is a vector of variables that characterizes the systems current position Time series are generated from these objects by iterating transition laws Recursive methods focus on a tradeoff between the current periods utility and a continuation value for utility in all future periods and the evolution of state variables that capture all consequences of todays actions and events Half xxiii xxiv Preface to the fourth edition of the job is accomplished once we choose and understand the roles of suitable state variables Another reason for learning about recursive methods is the increased im portance of numerical simulations in macroeconomics Many computational algorithms use recursive methods When such numerical simulations are called for in this book we give some suggestions for how to proceed but rely on other sources to provide important details1 Philosophy We think that only experience from solving practical problems fully conveys the power of the recursive approach Therefore this book provides many applica tions The book mixes tools and applications We present the tools with just enough technical sophistication for our applications but little more We aim to give readers a taste of the power of the methods and to direct them to sources where they can learn more Macroeconomic dynamics is now an immense field with diverse applications We do not pretend to survey the field only to sample it We intend our sample to equip the reader to approach much of the field with confidence Fortunately for us good books cover parts of the field that we neglect for example Adda and Cooper 2003 Aghion and Howitt 1998 Altug and Labadie 1994 Azariadis 1993 Barro and SalaiMartin 1995 Benassy 2011 Blanchard and Fischer 1989 Christensen and Kiefer 2009 Canova 2007 Cooley 1995 Cooper 1999 DeJong and Dave 2011 Farmer 1993 Gali 2008 Hansen and Sar gent 2013 Majumdar 2009 Pissarides 1990 Romer 1996 Shimer 2010 Stachurski 2009 Walsh 1998 and Woodford 2000 Bertsekas 1976 and Stokey and Lucas with Prescott 1989 remain standard references for recursive methods in macroeconomics Technical Appendix A in this book revises mate rial from chapter 2 of Sargent 1987b 1 Judd 1998 and Miranda and Fackler 2002 provide good treatments of numerical methods in economics Preface to the fourth edition xxv Changes in the fourth edition This edition contains two new chapters and substantial revisions of many other chapters from earlier editions New to this edition are chapter 20 on recursive formulations of optimal taxation problems and chapter 30 about the structure underlying models with matching functions that map unemployment and va cancies into jobfinding and jobfilling probabilities Chapter 19 has been exten sively revised and simplified in ways that closely link its formulation of Stackel berg plans to one in our new chapter 20 The new chapters and revisions cover topics that widen and deepen the message that recursive methods are pervasive and powerful New chapters Chapter 20 applies dynamic programming squared to two of the optimal tax ation models studied in chapter 16 namely Lucas and Stokeys 1983 model of optimal taxation and borrowing in an economy with complete markets and an incomplete markets version of that model Among other insights that recursive formulations bring to these models is a sharp characterization of the time in consistency of optimal plans that emerges in the form of two value functions for each optimal taxation problem one for time t 0 another for all times t 1 Distinct value functions state vectors decision rules and Bellman equations at times t 0 and t 1 are telltale signs of timeinconsistency Chapter 30 studies the mechanics of matching models and sheds light on them by exploring two substantive issues the Shimer puzzle and the role of heterogeneity The Shimer puzzle is the finding that common calibrations of the standard matching model do not generate fluctuations in unemployment rates nearly as large as those observed during business cycles2 The chapter looks under the hoods of various matching models that have been reconfigured to generate big responses in unemployment to movements in productivity An outcome of this crossmodel investigation is the discovery of a single channel that we call fundamental surplus that is common across all of the models Turning to heterogeneity we study a class of models with multiple typespecific matching 2 A puzzle is always relative to a model A puzzle is a prediction of a model that is contradicted by data xxvi Preface to the fourth edition functions whose equilibria have a block recursive structure in which value func tions and market tightnesses are independent of distributions of agents Such models deliver significant analytical tractability but the directed search that prevails within them also attenuates the congestion externalities that are at the center of other kinds of matching models Ideas Beyond emphasizing recursive methods the economics of this book revolves around several main ideas 1 The competitive equilibrium model of a dynamic stochastic economy This model contains complete markets meaning that all commodities at different dates that are contingent on random events can be traded in a market with a centralized clearing arrangement In one version of the model all trades occur at the beginning of time In another trading in oneperiod claims occurs sequentially The model is a foundation for assetpricing theory growth theory real business cycle theory and normative public finance There is no room for fiat money in the standard competitive equilibrium model so we shall have to alter the model to make room for fiat money 2 A class of incomplete markets models with heterogeneous agents These models arbitrarily restrict the types of assets that can be traded thereby possibly igniting a precautionary motive for agents to hold those assets Such models have been used to study the distribution of wealth and the evolution of an individual or familys wealth over time One model in this class lets money in 3 Several models of fiat money We add a shopping time specification to a competitive equilibrium model to get a simple vehicle for explaining ten doctrines of monetary economics These doctrines depend on the govern ments intertemporal budget constraint and the demand for fiat money aspects that transcend many models We also use Samuelsons overlapping generations model Bewleys incomplete markets model and Townsends turnpike model to perform a variety of policy experiments 4 Restrictions on government policy implied by the arithmetic of budget sets Most of the ten monetary doctrines reflect properties of the governments Preface to the fourth edition xxvii budget constraint Other important doctrines do too These doctrines known as ModiglianiMiller and Ricardian equivalence theorems have a common structure that come from identifying an equivalence class of gov ernment policies that produce the same allocations We display the struc ture of such theorems with an eye to finding features whose absence causes them to fail letting particular policies matter 5 Ramsey taxation problems What is the optimal tax structure when only distorting taxes are available The primal approach to taxation recasts this question as a problem in which a government chooses allocations directly and tax rates only indirectly Permissible allocations are those that satisfy resource constraints and implementability constraints where the latter are budget constraints in which the consumer and firm firstorder conditions are used to eliminate prices and tax rates We study labor and capital taxation and examine the optimality of the inflation tax prescribed by the Friedman rule 6 Social insurance with private information and enforcement problems We use the recursive contracts approach to study a variety of problems in which a benevolent social insurer balances providing insurance against providing incentives Applications include the provision of unemployment insurance and the design of loan contracts when a lender has an imperfect capacity to monitor a borrower 7 Reputation models in macroeconomics We study how far reputation can go to overcome a governments inability to commit to a policy The theory describes multiple systems of expectations about its behavior to which a government wants to conform The theory has many applications including implementing optimal taxation policies and making monetary policy in the presence of a temptation to inflate offered by a Phillips curve 8 Search models Search theory makes assumptions different from ones under lying a complete markets competitive equilibrium model It imagines that there is no centralized place where exchanges can be made or that there are not standardized commodities Buyers andor sellers have to devote effort to search for opportunities to buy or sell goods or factors of production opportunities that might arrive randomly We describe the basic McCall search model and various applications We also describe some equilibrium xxviii Preface to the fourth edition versions of the McCall model and compare them with models of another type that postulate matching functions 9 Matching models A matching function accepts measures of job seekers and vacancies as inputs and maps them into opportunities to form matches Models with matching functions build in congestion externalities that job searchers impose on other job searchers and that vacancy posters impose on other job searchers The models study how these externalities contend with each other and how they shape jobfinding rates jobfilling rates and unemployment rates In the last fifteen years matching models have been revised in ways intended to make them fit business cycle facts and welfare state outcomes 10 Employment lotteries versus career timeaveraging A model that was pop ular until recently interpreted the aggregate labor supply as the fraction of people that a planner assigns to work by using a lottery in which the losers must work and the winners enjoy leisure An alternative model instead fo cuses on an individual worker who chooses the fraction of his or her life to work within a lifecycle model The two frameworks have strikingly similar implications about some aggregate outcomes but not about others 11 Heterogeneous beliefs While for very good reasons most applied macroeco nomic models continue to assume rational expectations it is useful to study frameworks in which there are multiple beliefs either across people or in models of ambiguity and robustness within the mind of one decision maker Parts of chapters 8 and 14 study such models Theory and evidence Though this book aims to give the reader the tools to read about applications we spend little time on empirical applications However the empirical failures of one class of models have been a main force prompting development of another class of models Thus the perceived empirical failures of the standard complete markets general equilibrium model stimulated the development of the incomplete markets and recursive contracts models For example the complete markets model forms a standard benchmark model or point of departure for theories and empirical work on consumption and asset pricing The complete markets Preface to the fourth edition xxix model has these empirical problems 1 there is too much correlation between individual income and consumption growth in micro data eg Cochrane 1991 and Attanasio and Davis 1995 2 the equity premium is larger in the data than is implied by a representative agent assetpricing model with reasonable riskaversion parameter eg Mehra and Prescott 1985 and 3 the riskfree interest rate is too low relative to the observed aggregate rate of consumption growth Weil 1989 While there have been numerous attempts to explain these puzzles by altering the preferences in the standard complete markets model there has also been work that abandons the complete markets assumption and replaces it with some version of either exogenously or endogenously incomplete markets The Bewley models of chapters 17 and 18 are examples of exogenously incomplete markets By ruling out complete markets this model structure helps with empirical problems 1 and 3 above eg see Huggett 1993 but not much with problem 2 In chapter 21 we study some models that can be thought of as having endogenously incomplete markets They can also explain puzzle 1 mentioned earlier in this paragraph at this time it is not really known how far they take us toward solving problem 2 though Alvarez and Jermann 1999 report promise Micro foundations This book is about micro foundations for macroeconomics Browning Hansen and Heckman 1999 describe two justifications for putting microfoundations underneath macroeconomic models The first is aesthetic and preempirical models with micro foundations are by construction coherent and explicit And because they contain descriptions of agents purposes they allow us to analyze policy interventions using standard methods of welfare economics Lucas 1987 gives a distinct second reason a model with micro foundations broadens the sources of empirical evidence that can be used to assign numerical values to the models parameters Lucas endorses Kydland and Prescotts 1982 pro cedure of borrowing parameter values from micro studies Browning Hansen and Heckman 1999 challenge Lucass recommended empirical strategy Most seriously they point out that in many contexts the specifications underlying the microeconomic studies cited by a calibrator conflict with those of the macroe conomic model being calibrated It is typically not obvious how to transport xxx Preface to the fourth edition parameters from one data set and model specification to another data set and model specification Although we take seriously the doubts about Lucass justification for mi croeconomic foundations that Browning Hansen and Heckman raise we remain strongly attached to micro foundations For us it remains enough to appeal to the first justification namely the coherence provided by micro foundations and the virtues that come from having the ability to see the agents in an artificial economy We see Browning Hansen and Heckman as raising many legitimate questions about empirical strategies for implementing macro models with micro foundations We dont think that the clock will soon be turned back to a time when macroeconomics was done without micro foundations Road map Chapter 1 is either a preview or review or both It is either a readers guide to what is to come or a concise review of main themes that have been studied There is a case for reading it quickly before diving into the other chapters while not expecting fully to understand everything that is written there After many of the other chapters have been mastered it could be useful to read it again Chapter 2 describes two basic models of a time series a Markov chain and a linear firstorder difference equation In different ways these models use the algebra of firstorder difference equations to form tractable models of time series Each model has its own notion of the state of a system These time series models define essential objects in terms of which the choice problems of later chapters are formed and their solutions are represented Chapters 3 4 and 5 introduce aspects of dynamic programming includ ing numerical dynamic programming Chapter 3 describes the basic functional equation of dynamic programming the Bellman equation and several of its properties Chapter 4 describes some numerical ways for solving dynamic pro grams based on Markov chains Chapter 5 describes linear quadratic dynamic programming and some uses and extensions of it including how to use it to approximate solutions of problems that are not linear quadratic This chapter also tells how the Kalman filter from chapter 2 is mathematically equivalent to Preface to the fourth edition xxxi the linear quadratic dynamic programming problem from chapter 53 Chapter 6 describes a classic twoaction dynamic programming problem the McCall search model as well as Jovanovics extension of it a good application of the Kalman filter While single agents appear in chapters 3 through 6 systems with multiple agents whose environments and choices must be reconciled through markets appear for the first time in chapters 7 and 8 Chapter 7 uses linear quadratic dynamic programming to introduce two important and related equilibrium con cepts rational expectations equilibrium and Markov perfect equilibrium Each of these equilibrium concepts can be viewed as a fixed point in a space of beliefs about what other agents intend to do and each is formulated using recursive methods Chapter 8 introduces two notions of competitive equilibrium in dy namic stochastic pure exchange economies then applies them to pricing various consumption streams Chapter 9 interprets an overlapping generations model as a version of the general competitive model with a peculiar preference pattern It then goes on to use a sequential formulation of equilibria to display how the overlapping generations model can be used to study issues in monetary and fiscal economics including Social Security Chapter 10 compares an important aspect of an overlapping generations model with an infinitely lived agent model with a particular kind of incomplete market structure This chapter is thus our first encounter with an incomplete markets model The chapter analyzes the Ricardian equivalence theorem in two distinct but isomorphic settings one a model with infinitely lived agents who face borrowing constraints another with overlapping generations of twoperiod lived agents with a bequest motive We describe situations in which the timing of taxes does or does not matter and explain how binding borrowing constraints in the infinitelived model correspond to nonoperational bequest motives in the overlapping generations model Chapter 11 studies fiscal policy within a nonstochastic growth model with distorting taxes This chapter studies how foresight about future policies and transient responses to past ones contribute to current outcomes In particular this chapter describes feedforward and feedback components of mathematical 3 The equivalence is through duality in the sense of mathematical programming xxxii Preface to the fourth edition formulas for equilibrium outcomes Chapter 12 describes the recursive com petitive equilibrium concept and applies it within the context of the stochastic growth model Chapter 13 studies asset pricing and a host of practical doctrines associated with asset pricing including Ricardian equivalence again and ModiglianiMiller theorems for private and government finance Chapter 14 studies empirical strategies for implementing asset pricing models Building on work by Darrell Duffie Lars Peter Hansen and their coauthors chapter 14 discusses ways of characterizing asset pricing puzzles associated with the preference specifications and market structures commonly used in other parts of macroeconomics It then describes alterations of those structures that hold promise for resolving some of those puzzles Chapter 15 is about economic growth It describes the basic growth model and analyzes the key features of the specification of the technology that allows the model to exhibit balanced growth Chapter 16 studies competitive equilibria distorted by taxes and our first mechanism design problems namely ones that seek to find the optimal temporal pattern of distorting taxes In a nonstochastic economy a striking finding is that the optimal tax rate on capital is zero in the long run Chapter 17 is about selfinsurance We study a single agent whose limited menu of assets gives him an incentive to selfinsure by accumulating assets We study a special case of what has sometimes been called the savings problem and analyze in detail the motive for selfinsurance and the surprising implications it has for the agents ultimate consumption and asset holdings The type of agent studied in this chapter will be a component of the incomplete markets models to be studied in chapter 18 Chapter 18 studies incomplete markets economies with heterogeneous agents and imperfect markets for sharing risks The models of market incompleteness in this chapter come from simply ruling out markets in many assets without motivating the absence of those asset markets from the physical structure of the economy We wait until chapter 21 to study reasons that such markets may not exist The next chapters describe recursive contracts Chapter 19 describes what we call dynamic programming squared and uses linear quadratic dynamic pro gramming to explain it in a context in which key objects can be computed easily A tell tale sign of a dynamic programming squared problem is that there is a Bellman equation inside another Bellman equation Chapter 20 uses dynamic Preface to the fourth edition xxxiii programming squared to reformulate two optimal taxation models from chapter 16 recursively Chapter 21 describes models in the mechanism design tradi tion work that starts to provide a foundation for incomplete assets markets and that recovers specifications resembling models of chapter 18 Chapter 21 is about the optimal provision of social insurance in the presence of information and enforcement problems Relative to earlier chapters chapter 21 escalates the sophistication with which recursive methods are applied by utilizing promised values as state variables Chapter 22 extends the analysis to a general equi librium setting and draws out some implications for asset prices among other things Chapter 23 uses recursive contracts to design optimal unemployment insurance and worker compensation schemes Chapters 24 and 25 apply some of the same ideas to problems in reputa tional macroeconomics using promised values to formulate a notion of credi bility We study how a reputational mechanism can make policies sustainable even when a government cant commit meaning choose a plan for all t 0 onceandforall at time 0 in the way assumed in the analysis of chapter 16 We use this reputational approach in chapter 27 to assess whether the Friedman rule is sustainable Chapter 26 describes a model of gradualism in trade policy that has features in common with the first model of chapter 21 Chapter 27 switches gears by adding money to a very simple competitive equilibrium model in a superficial way the excuse for that superficial device is that it permits us to present and unify ten wellknown monetary doctrines Chapter 28 presents a less superficial model of money the turnpike model of Townsend which is basically a special nonstochastic version of one of the models of chapter 18 The specialization allows us to focus on a variety of monetary doctrines Chapter 29 describes multiple agent models of search and matching Except for a section on money in a search model we focus on applications to labor To bring out the economic forces at work in different frameworks we examine the general equilibrium effects of layoff taxes Chapter 30 investigates some fundamental forces common to a variety of otherwise quite disparate matching models Chapter 31 compares forces in an employment lotteries model with those operating in a timeaveraging model of aggregate labor supply Two appendixes collect various technical results on functional analysis and linear projections and hidden Markov models xxxiv Preface to the fourth edition Alternative uses of the book We have used parts of this book to teach both first and secondyear gradu ate courses in macroeconomics and monetary economics at the University of Chicago Stanford University New York University Princeton University and the Stockholm School of Economics Here are some alternative plans for courses 1 A onesemester firstyear course chapters 26 8 9 10 and either chapter 13 15 or 16 2 A secondsemester firstyear course add chapters 8 12 13 14 15 16 parts of 17 and 18 and all of 21 3 A first course in monetary economics chapters 9 24 25 26 27 28 and the last section of 29 4 A secondyear macroeconomics course select from chapters 1331 5 A selfcontained course about recursive contracts chapters 1926 As an example Sargent used the following structure for a onequarter first year course at the University of Chicago for the first and last weeks of the quarter students were asked to read the monograph by Lucas 1987 Students were prohibited from reading the monograph in the intervening weeks During the middle eight weeks of the quarter students read material from chapters 6 about search theory chapter 8 about complete markets chapters 9 27 and 28 about models of money and a little bit of chapters 21 22 and 23 on social insurance with incentive constraints The substantive theme of the course was the issues set out in a nontechnical way by Lucas 1987 However to understand Lucass arguments it helps to know the tools and models studied in the middle weeks of the course Those weeks also exposed students to a range of alternative models that could be used to measure Lucass arguments against some of the criticisms made for example by Manuelli and Sargent 1988 Another onequarter course would assign Lucass 1992 article on efficiency and distribution in the first and last weeks In the intervening weeks of the course assign chapters 17 18 and 21 As another example Ljungqvist used the following material in a fourweek segment on employmentunemployment in firstyear macroeconomics at the Stockholm School of Economics Labor market issues command a strong in terest especially in Europe Those issues help motivate studying the tools in Preface to the fourth edition xxxv chapters 6 and 29 about search and matching models and parts of 23 on the optimal provision of unemployment compensation On one level both chap ters 6 and 29 focus on labor markets as a central application of the theories presented but on another level the skills and understanding acquired in these chapters transcend the specific topic of labor market dynamics For example the thorough practice on formulating and solving dynamic programming prob lems in chapter 6 is generally useful to any student of economics and the models of chapter 29 are an entrypass to other heterogeneousagent models like those in chapter 18 Further an excellent way to motivate the study of recursive con tracts in chapter 23 is to ask how unemployment compensation should optimally be provided in the presence of incentive problems As a final example Sargent used versions of the material in 6 11 and 14 to teach undergraduate classes at Princeton and NYU Computer programs Various exercises and examples use Matlab programs These programs are referred to in a special index at the end of the book They can be down loaded from wwwtomsargentcomsource codemitbookzip Python and Julia programs for some of the models studied in this book are described at httpslecturesquanteconorg Notation We use the symbol to denote the conclusion of a proof The editors of this book requested that where possible brackets and braces be used in place of multiple parentheses to denote composite functions Thus the reader will often encounter fuc to express the composite function f u xxxvi Preface to the fourth edition Brief history of the notion of the state This book reflects progress economists have made in refining the notion of state so that more and more problems can be formulated recursively The art in ap plying recursive methods is to find a convenient definition of the state It is often not obvious what the state is or even whether a finitedimensional state exists eg maybe the entire infinite history of the system is needed to characterize its current position Extending the range of problems susceptible to recursive methods has been one of the major accomplishments of macroeconomic theory since 1970 In diverse contexts this enterprise has been about discovering a convenient state and constructing a firstorder difference equation to describe its motion In models equivalent to singleagent control problems state vari ables are either capital stocks or information variables that help predict the future4 In singleagent models of optimization in the presence of measurement errors the true state vector is latent or hidden from the optimizer and the economist and needs to be estimated Here beliefs come to serve as the patent state For example in a Gaussian setting the mathematical expectation and covariance matrix of the latent state vector conditioned on the available history of observations serves as the state In authoring his celebrated filter Kalman 1960 showed how an estimator of the hidden state could be constructed re cursively by means of a difference equation that uses the current observables to update the estimator of last periods hidden state5 Muth 1960 Lucas 1972 Kareken Muench and Wallace 1973 Jovanovic 1979 and Jovanovic and Nyarko 1996 all used versions of the Kalman filter to study systems in which agents make decisions with imperfect observations about the state For a while it seemed that some very important problems in macroeco nomics could not be formulated recursively Kydland and Prescott 1977 ar gued that it would be difficult to apply recursive methods to macroeconomic 4 Any available variables that Granger cause variables impinging on the optimizers ob jective function or constraints enter the state as information variables See CWJ Granger 1969 5 In competitive multipleagent models in the presence of measurement errors the dimen sion of the hidden state threatens to explode because beliefs about beliefs about naturally appear a problem studied by Townsend 1983 This threat has been overcome through thoughtful and economical definitions of the state For example one way is to give up on seeking a purely autoregressive recursive structure and to include a moving average piece in the descriptor of beliefs See Sargent 1991 Townsends equilibria have the property that prices fully reveal the private information of diversely informed agents Preface to the fourth edition xxxvii policy design problems including two examples about taxation and a Phillips curve As Kydland and Prescott formulated them the problems were not re cursive the fact that the publics forecasts of the governments future decisions influence the publics current decisions made the governments problem simul taneous not sequential But soon Kydland and Prescott 1980 and Hansen Epple and Roberds 1985 proposed a recursive formulation of such problems by expanding the state of the economy to include a Lagrange multiplier or costate variable associated with the governments budget constraint The costate vari able acts as the marginal cost of keeping a promise made earlier by the govern ment Marcet and Marimon 1999 extended and formalized a recursive version of such problems A significant breakthrough in the application of recursive methods was achieved by several researchers including Spear and Srivastava 1987 Thomas and Worrall 1988 and Abreu Pearce and Stacchetti 1990 They discovered a state variable for recursively formulating an infinitely repeated moral hazard problem That problem requires the principal to track a history of outcomes and to use it to construct statistics for drawing inferences about the agents actions Problems involving selfenforcement of contracts and a governments reputation share this feature A continuation value promised by the principal to the agent can summarize the history Making the promised valued a state variable allows a recursive solution in terms of a function mapping the inherited promised value and random variables realized today into an action or allocation today and a promised value for tomorrow The sequential nature of the solu tion allows us to recover historydependent strategies just as we use a stochastic difference equation to find a moving average representation6 It is now standard to use a continuation value as a state variable in models of credibility and dynamic incentives We shall study several such models in this book including ones for optimal unemployment insurance and for designing loan contracts that must overcome information and enforcement problems 6 Related ideas are used by Shavell and Weiss 1979 Abreu Pearce and Stacchetti 1986 1990 in repeated games and Green 1987 and Phelan and Townsend 1991 in dynamic mechanism design Andrew Atkeson 1991 extended these ideas to study loans made by borrowers who cannot tell whether they are making consumption loans or investment loans Part I Imperialism of Recursive Methods Chapter 1 Overview 11 Warning This chapter provides a nontechnical summary of some themes of this book We debated whether to put this chapter first or last A way to use this chapter is to read it twice once before reading anything else in the book then again after having mastered the techniques presented in the rest of the book That second time this chapter will be easy and enjoyable reading and it will remind you of connections that transcend a variety of apparently disparate topics But on first reading this chapter will be difficult partly because the discussion is mainly literary and therefore incomplete Measure what you have learned by comparing your understandings after those first and second readings Or just skip this chapter and read it after the others 12 A common ancestor Clues in our mitochondrial DNA tell biologists that we humans share a com mon ancestor called Eve who lived 100000 years ago All of macroeconomics too seems to have descended from a common source Irving Fishers and Mil ton Friedmans consumption Euler equation the cornerstone of the permanent income theory of consumption Modern macroeconomics records the fruits and frustrations of a long lovehate affair with the permanent income mechanism As a way of summarizing some important themes in our book we briefly chronicle some of the high and low points of this long affair 3 13 The savings problem A consumer wants to maximize E0 t0 βt uct 131 where β 01 u is a twice continuously differentiable increasing strictly concave utility function and E0 denotes a mathematical expectation conditioned on time 0 information The consumer faces a sequence of budget constraints1 At1 Rt1At yt ct 132 for t 0 where At1 A is the consumers holdings of an asset at the beginning of period t1 A is a lower bound on asset holdings yt is a random endowment sequence ct is consumption of a single good and Rt1 is the gross rate of return on the asset between t and t1 In the general version of the problem both Rt1 and yt can be random though special cases of the problem restrict Rt1 further A firstorder necessary condition for this problem is βEt Rt1 uct1uct 1 if At1 A 133 This Euler inequality recurs as either the cornerstone or the straw man in many theories contained in this book Different modeling choices put 133 to work in different ways One can restrict u β the return process Rt1 the lower bound on assets A the income process yt and the consumption process ct in various ways By making alternative choices about restrictions to impose on subsets of these objects macroeconomists have constructed theories about consumption asset prices and the distribution of wealth Alternative versions of equation 133 also underlie Chamleys 1986 and Judds 1985b striking results about eventually not taxing capital 1 We use a different notation in chapter 17 At here conforms to bt in chapter 17 131 Linear quadratic permanent income theory To obtain a version of the permanent income theory of Friedman 1955 and Hall 1978 set Rt1 R impose R β1 assume the quadratic utility function uct ct γ2 and allow consumption ct to be negative We also allow yt to be an arbitrary stationary process and dispense with the lower bound A The Euler inequality 133 then implies that consumption is a martingale Etct1 ct 134 Subject to a boundary condition that2 E0t0 βtA2t equation 134 and the budget constraints 132 can be solved to yield ct r1rEtj011rjytj At 135 where 1 r R Equation 135 expresses consumption as a fixed marginal propensity to consume r1r that is applied to the sum of human wealth namely Etj0 11r j ytj and financial wealth At This equation has the following notable features 1 consumption is smoothed on average across time current consumption depends only on the expected present value of nonfinancial income 2 feature 1 opens the way to Ricardian equivalence redistributions of lumpsum taxes over time that leave the expected present value of nonfinancial income unaltered do not affect consumption 3 there is certainty equivalence increases in the conditional variances of future incomes about their forecast values do not affect consumption though they do diminish the consumers utility 4 a byproduct of certainty equivalence is that the marginal propensities to consume out of financial and nonfinancial wealth are equal This theory continues to be a workhorse in much good applied work see Ligon 1998 and Blundell and Preston 1999 for creative applications Chapter 5 describes conditions under which certainty equivalence prevails while chapters 2 and 5 also describe the structure of the crossequation restrictions that the 2 The motivation for using this boundary condition instead of a lower bound A on asset holdings is that there is no natural lower bound on asset holdings when consumption is permitted to be negative Chapters 8 and 18 discuss what are called natural borrowing limits the lowest possible appropriate values of A in the case that c is nonnegative 6 Overview hypothesis of rational expectations imposes and that empirical studies heavily exploit 132 Precautionary saving A literature on the savings problem or precautionary saving investigates the consequences of altering the assumption in the linear quadratic permanent income theory that u is quadratic an assumption that makes the marginal util ity of consumption become negative for large enough c Rather than assuming that u is quadratic the literature on the savings problem assumes that u is increasing and strictly concave This assumption keeps the marginal utility of consumption above zero We retain other features of the linear quadratic model βR 1 yt is a stationary process but now impose a borrowing limit At a With these assumptions something amazing occurs Euler inequality 133 implies that the marginal utility of consumption is a nonnegative supermartin gale3 That gives the model the striking implication that ct as and At as where as means almost sure convergence Consumption and wealth will fluctuate randomly in response to income fluctuations but so long as randomness in income continues they will drift upward over time without bound If randomness eventually expires in the tail of the income process then both consumption and income converge But even small perpetual random fluctuations in income are enough to cause both consumption and assets to di verge to This response of the optimal consumption plan to randomness is required by the Euler equation 133 and is called precautionary savings By keeping the marginal utility of consumption positive precautionary savings models arrest the certainty equivalence that prevails in the linear quadratic per manent income model Chapter 17 studies the savings problem in depth and struggles to understand the workings of the powerful martingale convergence 3 See chapter 17 The situation is simplest in the case that the yt process is iid so that the value function can be expressed as a function of level yt At alone V A y Applying the BenvenisteScheinkman formula from chapter 3 shows that V A y uc which implies that when βR 1 133 becomes EtV At1 yt1 V At yt which states that the derivative of the value function is a nonnegative supermartingale That in turn implies that A almost surely diverges to theorem The supermartingale convergence theorem also plays an important role in the model insurance with private information in chapter 21 133 Complete markets insurance and the distribution of wealth To build a model of the distribution of wealth we consider a setting with many consumers To start imagine a large number of ex ante identical consumers with preferences 131 who are allowed to share their income risk by trading oneperiod contingent claims For simplicity assume that the saving possibility represented by the budget constraint 132 is no longer available4 but that it is replaced by access to an extensive set of insurance markets Assume that household i has an income process yit gist where st is a state vector governed by a Markov process with transition density πss where s and s are elements of a common state space S See chapters 2 and 8 for material about Markov chains and their uses in equilibrium models Each period every household can trade oneperiod statecontingent claims to consumption next period Let Qss be the price of one unit of consumption next period in state s when the state this period is s When household i has the opportunity to trade such statecontingent securities its firstorder conditions for maximizing 131 are Qst1st β ucit1 st1ucit st πst1st 136 Notice that st1 Qst1stdst1 is the price of a riskfree claim on consumption one period ahead it is thus the reciprocal of the gross riskfree interest rate R Therefore if we sum both sides of 136 over st1 we obtain our standard consumption Euler condition 133 at equality5 Thus the complete markets equation 136 is consistent with our complete markets Euler equation 133 but 136 imposes more We will exploit this fact extensively in chapter 16 In a widely studied special case there is no aggregate risk so that i yit di i gistdi constant In that case it can be shown that the competitive equilibrium statecontingent prices become Qst1st βπ st1st 137 4 It can be shown that even if it were available people would not want to use it 5 That the asset is riskfree becomes manifested in Rt1 being a function of st so that it is known at t This in turn implies that the riskfree gross rate of return R is β16 If we substitute 137 into 136 we discover that cit1st1 citst for all st1 st Thus the consumption of consumer i is constant across time and across states of nature s so that in equilibrium all idiosyncratic risk is insured away Higher presentvalueofendowment consumers will have permanently higher consumption than lower presentvalueofendowment consumers so that there is a nondegenerate crosssection distribution of wealth and consumption In this model the crosssection distributions of wealth and consumption replicate themselves over time and furthermore each individual forever occupies the same position in that distribution A model that has the crosssection distribution of wealth and consumption being time invariant is not a bad approximation to the data But there is ample evidence that individual households positions within the distribution of wealth move over time7 Several models described in this book alter consumers trading opportunities in ways designed to frustrate risk sharing enough to cause individuals position in the distribution of wealth to change with luck and enterprise One class that emphasizes luck is the set of incomplete markets models started by Truman Bewley It eliminates the households access to almost all markets and returns it to the environment of the precautionary savings model 134 Bewley models At first glance the precautionary savings model with βR 1 seems like a bad starting point for building a theory that aspires to explain a situation in which crosssection distributions of consumption and wealth are constant over time even as individuals experience random fluctuations within that distribution A panel of households described by the precautionary savings model with βR 1 would have crosssection distributions of wealth and consumption that march upward and never settle down What have come to be called Bewley models are 6 This follows because the price of a riskfree claim to consumption tomorrow at date t in state st is st1 Qst1st βst1 πst1st β 7 See DíazGiménez Quadrini and RíosRull 1997 Krueger and Perri 2004 2006 Rodriguez DíazGiménez Quadrini and RíosRull 2002 and Davies and Shorrocks 2000 The savings problem 9 constructed by lowering the interest rate R to allow those crosssection distri butions to settle down8 Bewley models are arranged so that the cross section distributions of consumption wealth and income are constant over time and so that the asymptotic stationary distributions of consumption wealth and income for an individual consumer across time equal the corresponding cross section distributions across people A Bewley model can thus be thought of as starting with a continuum of consumers operating according to the precaution ary savings model with βR 1 and its diverging individual asset process We then lower the interest rate enough to make assets converge to a distribution whose crosssection average clears a market for a riskfree asset Different ver sions of Bewley models are distinguished by what the riskfree asset is In some versions it is a consumption loan from one consumer to another in others it is fiat money in others it can be either consumption loans or fiat money and in yet others it is claims on physical capital Chapter 18 studies these alternative interpretations of the riskfree asset As a function of a constant gross interest rate R Figure 131 plots the time series average of asset holdings for an individual consumer At R β1 the time series mean of the individuals assets diverges so that EaR is infinite For R β1 the mean exists We require that a continuum of ex ante identical but ex post different consumers share the same time series average EaR and also that the distribution of a over time for a given agent equals the distribution of At1 at a point in time across agents If the asset in question is a pure consumption loan we require as an equilibrium condition that EaR 0 so that borrowing equals lending If the asset is fiat money then we require that EaR M p where M is a fixed stock of fiat money and p is the price level Thus a Bewley model lowers the interest rate R enough to offset the pre cautionary savings force that with βR 1 propels assets upward in the savings problem Precautionary saving remains an important force in Bewley models 8 It is worth thinking about the sources of the following differences In the complete markets model sketched in subsection 133 an equilibrium riskfree gross interest rate R satisfies Rβ 1 and each consumer completely smooths consumption across both states and time so that the distribution of consumption trivially converges The precautionary savings model of section 132 assumes that Rβ 1 and derives the outcome that each consumers consumption and financial wealth both diverge toward Why can βR 1 be compatible with nonexploding individual consumption and wealth levels in the complete markets model of subsection 133 but not in the precautionary savings model of subsection 132 10 Overview 0 EaR β1 EaR R Figure 131 Mean of time series average of household con sumption as function of riskfree gross interest rate R an increase in the volatility of income generally pushes the EaR curve to the right driving the equilibrium R downward 135 History dependence in standard consumption models Individuals positions in the wealth distribution are frozen in the complete mar kets model but not in the Bewley model reflecting the absence or presence re spectively of history dependence in equilibrium allocation rules for consumption The preceding version of the complete markets model erases history dependence while the savings problem model and the Bewley model do not History dependence is present in these models in an easy to handle recur sive way because the households asset level completely encodes the history of endowment realizations that it has experienced We want a way of represent ing history dependence more generally in contexts where a stock of assets does not suffice to summarize history History dependence can be troublesome be cause without a convenient lowdimensional state variable to encode history it requires that there be a separate decision rule for each date that expresses the time t decision as a function of the history at time t an object with a number of arguments that grows exponentially with t As analysts we have a strong The savings problem 11 incentive to find a lowdimensional state variable Fortunately economists have made tremendous strides in handling history dependence with recursive meth ods that summarize a history with a single number and that permit compact timeinvariant expressions for decision rules We shall discuss history depen dence later in this chapter and will encounter many such examples in chapters 19 through 26 136 Growth theory Equation 133 is also a key ingredient of growth theory see chapters 11 and 15 In the onesector growth model a representative household solves a version of the savings problem in which the single asset is interpreted as a claim on the return from a physical capital stock K that enters a constant returns to scale production function FK L where L is labor input When returns to capital are tax free the theory equates the gross rate of return Rt1 to the gross marginal product of capital net of depreciation namely Fkt1 1 δ where Fkk t 1 is the marginal product of capital and δ is a depreciation rate Suppose that we add leisure to the utility function so that we replace uc with the more general oneperiod utility function Uc ℓ where ℓ is the households leisure Then the appropriate version of the consumption Euler condition 133 at equality becomes Uc t βUc t 1 Fk t 1 1 δ 138 The constant returns to scale property implies that FkK N f k where k KN and FK N NfKN If there exists a steady state in which k and c are constant over time then equation 138 implies that it must satisfy ρ δ f k 139 where β1 1 ρ The value of k that solves this equation is called the augmented Golden rule steadystate level of the capitallabor ratio This celebrated equation shows how technology in the form of f and δ and time preference in the form of β are the determinants of the steadystate level of capital when income from capital is not taxed However if income from capital is taxed at the flat rate marginal rate τkt1 then the Euler equation 138 becomes modified Uc t βUc t 1 Fk t 1 1 τkt1 1 δ 1310 12 Overview If the flat rate tax on capital is constant and if a steadystate k exists it must satisfy ρ δ 1 τk f k 1311 This equation shows how taxing capital diminishes the steadystate capital labor ratio See chapter 11 for an extensive analysis of the onesector growth model when the government levies timevarying flat rate taxes on consumption capital and labor as well as offering an investment tax credit 137 Limiting results from dynamic optimal taxation Equations 139 and 1311 are central to the dynamic theory of optimal taxes Chamley 1986 and Judd 1985b forced the government to finance an exogenous stream of government purchases gave it the capacity to levy timevarying flat rate taxes on labor and capital at different rates formulated an optimal taxation problem a socalled Ramsey problem and studied the possible limiting behavior of the optimal taxes Two Euler equations play a decisive role in determining the limiting tax rate on capital in a nonstochastic economy the households Euler equation 1310 and a similar consumption Euler equation for the Ramsey planner that takes the form Wc t βWc t 1 Fk t 1 1 δ 1312 where W ct ℓt U ct ℓt Φ Uc t ct Uℓ t 1 ℓt 1313 and where Φ is a Lagrange multiplier on the governments intertemporal budget constraint As Jones Manuelli and Rossi 1997 emphasize if the function Wc ℓ is simply viewed as a peculiar utility function then what is called the primal version of the Ramsey problem can be viewed as an ordinary optimal growth problem with period utility function W instead of U 9 In a Ramsey allocation taxes must be such that both 138 and 1312 always hold among other equations Judd and Chamley note the following 9 Notice that so long as Φ 0 which occurs whenever taxes are necessary the objective in the primal version of the Ramsey problem disagrees with the preferences of the household over c ℓ allocations This conflict is the source of a timeinconsistency problem in the Ramsey problem with capital implication of the two Euler equations 138 and 1312 If the government expenditure sequence converges and if a steady state exists in which ct ℓt kt τkt all converge then it must be true that 139 holds in addition to 1311 But both of these conditions can prevail only if τk 0 Thus the steadystate properties of two versions of our consumption Euler equation 133 underlie Chamley and Judds remarkable result that asymptotically it is optimal not to tax capital In stochastic versions of dynamic optimal taxation problems we shall glean additional insights from 133 as embedded in the assetpricing equations 1316 and 1318 In optimal taxation problems the government has the ability to manipulate asset prices through its influence on the equilibrium consumption allocation that contributes to the stochastic discount factor mt1t defined in equation 1316 below The Ramsey government seeks a way wisely to use its power to revalue its existing debt by altering statehistory prices To appreciate what the Ramsey government is doing it helps to know the theory of asset pricing 138 Asset pricing The dynamic asset pricing theory of Breeden 1979 and Lucas 1978 also starts with 133 but alters what is fixed and what is free The BreedenLucas theory is silent about the endowment process yt and sweeps it into the background It fixes a function u and a discount factor β and takes a consumption process ct as given In particular assume that ct gXt where Xt is a Markov process with transition cdf FXX Given these inputs the theory is assigned the task of restricting the rate of return on an asset defined by Lucas as a claim on the consumption endowment Rt1 pt1 ct1 pt where pt is the price of the asset The Euler inequality 133 becomes Et β uct1 uct pt1 ct1 pt 1 This equation can be solved for a pricing function pt pXt In particular if we substitute pXt into 1314 we get Lucass functional equation for pX 14 Overview 139 Multiple assets If the consumer has access to several assets a version of 133 holds for each asset Etβ u ct1 u ct Rjt1 1 1315 where Rjt1 is the gross rate of return on asset j Given a utility function u a discount factor β and the hypothesis of rational expectations which allows the researcher to use empirical projections as counterparts of the theoretical projec tions Et equations 1315 put extensive restrictions across the moments of a vector time series for ct R1t1 RJt1 A key finding of the literature eg Hansen and Singleton 1983 is that for us with plausible curvature10 consumption is too smooth for ct Rjt1 to satisfy equation 1315 where ct is measured as aggregate consumption Lars Hansen and others have elegantly organized this evidence as follows Define the stochastic discount factor mt1t β u ct1 u ct 1316 and write 1315 as Etmt1tRjt1 1 1317 Represent the gross rate of return as Rjt1 ot1 qt where ot1 is a oneperiod payout on the asset and qt is the price of the asset at time t Then 1317 can be expressed as qt Etmt1tot1 1318 The structure of 1318 justifies calling mt1t a stochastic discount factor to determine the price of an asset multiply the random payoff for each state by the discount factor for that state then add over states by taking a conditional expectation Applying the definition of a conditional covariance and a Cauchy Schwartz inequality to this equation implies qt Etmt1t Etot1 σt mt1t Etmt1t σt ot1 1319 10 Chapter 14 describes Pratts 1964 mental experiment for deducing plausible curvature The savings problem 15 where σtyt1 denotes the conditional standard deviation of yt1 Setting ot1 1 in 1318 shows that Etmt1t must be the time t price of a riskfree oneperiod security Inequality 1319 bounds the ratio of the price of a risky security qt to the price of a riskfree security Etmt11 by the right side which equals the expected payout on that risky asset minus its conditional standard deviation σtot1 times a market price of risk σtmt1tEtmt1t By using data only on payouts ot1 and prices qt inequality 1319 has been used to estimate the market price of risk without restricting how mt1t relates to consumption If we take these atheoretical estimates of σtmt1tEtmt1t and compare them with the theoretical values of σtmt1tEtmt1t that we get with a plausible curvature for u and by imposing ˆmt1t β uct1 uct for aggregate consumption we find that the theoretical ˆm has far too little volatility to account for the atheoretical estimates of the conditional coefficient of variation of mt1t As we discuss extensively in chapter 14 this outcome reflects the fact that aggregate consumption is too smooth to account for atheoretical estimates of the market price of risk There have been two broad types of response to the empirical challenge The first retains 1317 but abandons 1316 and instead adopts a statistical model for mt1t Even without the link that equation 1316 provides to consumption equation 1317 imposes restrictions across asset returns and mt1t that can be used to identify the mt1t process Equation 1317 contains noarbitrage conditions that restrict the joint behavior of returns This has been a fruitful approach in the affine term structure literature see Backus and Zin 1993 Piazzesi 2000 and Ang and Piazzesi 200311 Another approach has been to disaggregate and to write the householdi version of 133 βEtRt1 u cit1 u cit 1 if Ait1 Ai 1320 If at time t a subset of households are on the corner 1320 will hold with equality only for another subset of households Households in the second set price assets12 11 Affine term structure models generalize earlier models that implemented rational ex pectations versions of the expectations theory of the term structure of interest rates See Campbell and Shiller 1991 Hansen and Sargent 1991 and Sargent 1979 12 David Runkle 1991 and Gregory Mankiw and Steven Zeldes 1991 checked 1320 for subsets of agents 16 Overview Chapter 22 describes a model of Harald Zhang 1997 and Alvarez and Jer mann 2000 2001 The model introduces participation collateral constraints and shocks in a way that makes a changing subset of agents i satisfy 1320 Zhang and Alvarez and Jermann formulate these models by adding participa tion constraints to the recursive formulation of the consumption problem based on 147 Next we briefly describe the structure of these models and their attitude toward our theme equation the consumption Euler equation 133 The idea of Zhang and Alvarez and Jermann was to meet the empirical asset pricing challenges by disrupting 133 As we shall see that requires eliminat ing some of the assets that some of the households can trade These advanced models exploit a convenient method for representing and manipulating history dependence 14 Recursive methods The pervasiveness of the consumption Euler inequality will be a significant sub stantive theme of this book We now turn to a methodological theme the imperialism of the recursive method called dynamic programming The notion that underlies dynamic programming is a finitedimensional object called the state that from the point of view of current and future payoffs completely summarizes the current situation of a decision maker If an optimum problem has a lowdimensional state vector immense simplifications follow A recurring theme of modern macroeconomics and of this book is that finding an appropriate state vector is an art To illustrate the idea of the state in a simple setting return to the savings problem and assume that the consumers endowment process is a timeinvariant function of a state st that follows a Markov process with timeinvariant one period transition density πss and initial density π0s so that yt yst To begin recall the description 135 of consumption that prevails in the special linear quadratic version of the savings problem Under our present assumption that yt is a timeinvariant function of the Markov state 135 and the house holds budget constraint imply the following representation of the households decision rule ct f At st 141a At1 gAt st Equation 141a represents consumption as a timeinvariant function of a state vector At st The Markov component st appears in 141a because it contains all of the information that is useful in forecasting future endowments for the linear quadratic model 135 reveals the households incentive to forecast future incomes and the asset level At summarizes the individuals current financial wealth The s component is assumed to be exogenous to the households decisions and has a stochastic motion governed by πss But the future path of A is chosen by the household and is described by 141b The system formed by 141 and the Markov transition density πss is said to be recursive because it expresses a current decision ct as a function of the state and tells how to update the state By iterating 141b notice that At1 can be expressed as a function of the history st st1 s0 and A0 The endogenous state variable financial wealth thus encodes all payoffrelevant aspects of the history of the exogenous component of the state st Define the value function VA0 s0 as the optimum value of the savings problem starting from initial state A0 s0 The value function V satisfies the following functional equation known as a Bellman equation V As maxcA uc βE V A s s where the maximization is subject to A RA y c and y ys Associated with a solution VAs of the Bellman equation is the pair of policy functions c fAs A gAs from 141 The ex ante value ie the value of 131 before s0 is drawn of the savings problem is then vA0 s VA0s π0s We shall make ample use of the ex ante value function 18 Overview 141 Dynamic programming and the Lucas Critique Dynamic programming is now recognized as a powerful method for studying private agents decisions and also the decisions of a government that wants to design an optimal policy in the face of constraints imposed on it by private agents best responses to that government policy But it has taken a long time for the power of dynamic programming to be realized for government policy design problems Dynamic programming had been applied since the late 1950s to design gov ernment decision rules to control an economy whose transition laws included rules that described the decisions of private agents In 1976 Robert E Lucas Jr published his now famous critique of dynamicprogrammingbased econo metric policy evaluation procedures The heart of Lucass critique was the implication for government policy evaluation of a basic property that pertains to any optimal decision rule for private agents with a form 143 that attains a Bellman equation like 142 The property is that the optimal decision rules f g depend on the transition density πss for the exogenous component of the state s As a consequence any widely understood government policy that alters the law of motion for a state variable like s that appears in private agents decision rules should alter those private decision rules In the applications that Lucas had in mind the s in private agents decision problems included variables useful for predicting tax rates the money supply and the aggregate price level Therefore Lucas asserted that econometric policy evaluation procedures that assumed that private agents decision rules are fixed in the face of alterations in government policy are flawed13 Most econometric policy evaluation procedures at the time were vulnerable to Lucass criticism To construct valid policy eval uation procedures Lucas advocated building new models that would attribute rational expectations to decision makers14 Lucass discussant Robert Gordon predicted that after that ambitious task had been accomplished we could then use dynamic programming to compute optimal policies ie to solve Ramsey problems 13 They were flawed because they assumed no response when they should have assumed best response of private agents decision rules to government decision rules 14 That is he wanted private decision rules to solve dynamic programming problems with the correct transition density π for s Recursive methods 19 142 Dynamic programming challenged But Edward C Prescotts 1977 paper entitled Should Control Theory Be Used for Economic Stabilization asserted that Gordon was too optimistic Prescott claimed that in his 1977 JPE paper with Kydland he had proved that it was log ically impossible to use dynamic programming to find optimal government poli cies in settings where private traders face genuinely dynamic problems Prescott said that dynamic programming was inapplicable to government policy design problems because the structure of best responses of current private decisions to future government policies prevents the government policy design problem from being recursive a manifestation of the time inconsistency of optimal govern ment plans The optimal government plan would therefore require a govern ment commitment technology and the government policy must take the form of a sequence of historydependent decision rules that could not be expressed as a function of natural state variables 143 Imperialistic response of dynamic programming Much of the subsequent history of macroeconomics belies Prescotts claim of logical impossibility More and more problems that smart people like Prescott in 1977 thought could not be attacked with dynamic programming can now be solved with dynamic programming Prescott didnt put it this way in 1977 but today we would in 1977 we lacked a way to handle history dependence within a dynamic programming framework Finding a recursive way to handle history dependence is a major achievement of the past 35 years and an important methodological theme of this book that opens the way to a variety of important applications We shall encounter important traces of the fascinating history of this topic in various chapters Important contributors to the task of overcoming Prescotts challenge seemed to work in isolation from one another being unaware of the complementary approaches being followed elsewhere Important contributors included Shavell and Weiss 1979 Kydland and Prescott 1980 Miller and Salmon 1985 Pearlman Currie and Levine 1985 Pearlman 1992 and Hansen Epple and Roberds 1985 These researchers achieved truly indepen dent discoveries of the same important idea 20 Overview As we discuss in detail in chapter 19 one important approach amounted to putting a government costate vector on the costate equations of the private decision makers then proceeding as usual to use optimal control for the govern ments problem A costate equation is a version of an Euler equation Solved forward the costate equation depicts the dependence of private decisions on forecasts of future government policies that Prescott was worried about The key idea in this approach was to formulate the governments problem by taking the costate equations of the private sector as additional constraints on the gov ernments problem These amount to promisekeeping constraints they are cast in terms of derivatives of value functions not value functions themselves be cause costate vectors are gradients of value functions After adding the costate equations of the private sector the followers to the transition law of the gov ernment the leader one could then solve the governments problem by using dynamic programming as usual One simply writes down a Bellman equation for the government planner taking the private sector costate variables as pseudo state variables Then it is almost business as usual Gordon was correct We say almost because after the Bellman equation is solved there is one more step to pick the initial value of the private sectors costate To maximize the governments criterion this initial condition should be set to zero because ini tially there are no promises to keep The governments optimal decision is a function of the natural state variable and the costate variables The date t costate variables encode history and record the cost to the government at t of confirming the private sectors prior expectations about the governments time t decisions expectations that were embedded in the private sectors decisions before t The solution is time inconsistent the government would always like to reinitialize the time t multiplier to zero and thereby discard past promises but that is ruled out by the assumption that the government is committed to follow the optimal plan See chapter 19 for many technical details computer programs and an application 144 History dependence and dynamic programming squared Rather than pursue the costate on the costate approach further we now turn to a closely related approach that we illustrate in a dynamic contract design problem While superficially different from the government policy design problem the contract problem has many features in common with it What is again needed is a recursive way to encode history dependence Rather than use costate variables we move up a derivative and work with promised values This leads to value functions appearing inside value functions or dynamic programming squared Define the history st of the Markov state by st st st1 s0 and let πtst be the density over histories induced by π π0 Define a consumption allocation rule as a sequence of functions the time component of which maps st into a choice of time t consumption ct σtst for t 0 Let c σtstt0 Define the ex ante value associated with an allocation rule as vc t0 st βt uσtst πtst For each possible realization of the period zero state s0 there is a continuation history st s0 The observation that a continuation history is itself a complete history is our first hint that a recursive formulation is possible For each possible realization of the first period s0 a consumption allocation rule implies a oneperiod continuation consumption rule cs0 A continuation consumption rule is itself a consumption rule that maps histories into time series of consumption The oneperiod continuation history treats the time t 1 component of the original history evaluated at s0 as the time t component of the continuation history The period t consumption of the oneperiod continuation consumption allocation conforms to the time t 1 component of original consumption allocation evaluated at s0 The time and state separability of 145 then allow us to represent vc recursively as vc s0 uc0s0 βvcs0 π0 s0 where vcs0 is the value of the continuation allocation We call vcs0 the continuation value In a special case that successive components of st are iid and have a discrete distribution we can write 146 as v s ucs βws Πs 147 where Πs Probyt ys and y1 y2 yS is a grid on which the endowment resides cs is consumption in state s given v and ws is the continuation value in state s given v Here we use v in 147 to denote what was vc in 146 and ws to denote what was vcs in 146 So far this has all been for an arbitrary consumption plan Evidently the ex ante value v attained by an optimal consumption program must satisfy v maxcswss1S s ucs βws Πs 148 where the maximization is subject to constraints that summarize the individuals opportunities to trade current statecontingent consumption cs against future statecontingent continuation values ws In these problems the value of v is an outcome that depends in the savings problem for example on the households initial level of assets In fact for the savings problem with iid endowment shocks the outcome is that v is a monotone function of A This monotonicity allows the following remarkable representation After solving for the optimal plan use the monotone transformation to let v replace A as a state variable and represent the optimal decision rule in the form cs fv s 149a ws gv s 149b The promised value v a forwardlooking variable if there ever was one is also the variable that functions as an index of history in 149 Equation 149b reminds us that v is a backward looking variable that registers the cumulative impact of past states st The definition of v as a promised value for example in 148 tells us that v is also a forwardlooking variable that encodes expectations promises about future consumption Recursive methods 23 145 Dynamic principalagent problems The right side of 148 tells the terms on which the household is willing to trade current utility for continuation utility Models that confront enforcement and information problems use the tradeoff identified by 148 to design intertem poral consumption plans that optimally balance risk sharing and intertemporal consumption smoothing against the need to offer correct incentives Next we turn to such models We remove the household from the market and hand it over to a planner or principal who offers the household a contract that the planner designs to deliver an ex ante promised value v subject to enforcement or information con straints16 Now v becomes a state variable that occurs in the planners value function We assume that the only way the household can transfer his endow ment over time is to deal with the planner The saving or borrowing technology 132 is no longer available to the agent though it might be to the planner We continue to consider the iid case mentioned above Let Pv be the ex ante optimal value of the planners problem The presence of a value function for the agents as an argument of the value function of the principal causes us sometimes to speak of dynamic programming squared The planner earns yt ct from the agent at time t by commandeering the agents endowment but returning consumption ct The value function Pv for a planner who must deliver promised value v satisfies P v max cswsS s1 ys cs βP ws Πs 1410 where the maximization is subject to the promisekeeping constraint 147 and some other constraints that depend on details of the problem as we indicate shortly The other constraints are contextspecific incentivecompatibility con straints and describe the best response of the agent to the arrangement offered by the principal Condition 147 is a promisekeeping constraint The planner is constrained to provide a vector of cs wsS s1 that delivers the value v We briefly describe two types of contract design problems and the con straints that confront the planner because of the opportunities that the envi ronment grants the agent To model the problem of enforcement without an information problem as sume that while the planner can observe yt each period the household always 16 Here we are sticking close to two models of Thomas and Worrall 1988 1990 28 Estimation again The innovations representation that emerges from the Kalman filter is xt1 Axt Kt at 281a yt Gxt at 281b where for t 1 xt Extyt1 and E at at GΣt G R Ωt Evidently for t 1 Eytyt1 Gxt and the distribution of yt conditional on yt1 is NGxt Ωt The objects Gxt Ωt emerging from the Kalman filter are thus sufficient statistics for the distribution of yt conditioned on yt1 for t 1 The sufficient statistics and also the innovation at yt Gxt can be calculated recursively from 2714 The unconditional distribution of y0 is evidently NGx0 Ω0 As a counterpart to 262 we can factor the likelihood function for a sample yT yT1 y0 as fyT y0 fyTyT1 fyT1yT2 fy1y0 fy0 282 The log of the conditional density of the m x 1 vector yt is log fytyt1 5 m log 2π 5 log det Ωt 5 at Ωt1 at 283 We can use 283 and 2714 to evaluate the likelihood function 282 recursively for a given set of parameter values θ that underlie the matrices A G C R Such calculations are at the heart of efficient strategies for computing maximumlikelihood estimators19 The likelihood function is also an essential object for a Bayesian statistician20 It completely summarizes how the data influence the Bayesian posterior via the following application of Bayes law Where θ is our parameter vector y0T our data record and pθ a probability density that summarizes our prior 19 See Hansen 1982 Eichenbaum 1991 Christiano and Eichenbaum 1992 Burnside Eichenbaum and Rebelo 1993 and Burnside and Eichenbaum 1996a 1996b for alternative estimation strategies 20 See Canova 2007 Christensen and Kiefer 2009 and DeJong and Dave 2011 for extensive descriptions of how Bayesian and maximum likelihood methods can be applied to macroeconomic and other dynamic models 24 Overview has the option of consuming its endowment yt and receiving an ex ante con tinuation value vaut with which to enter the next period where vaut is the ex ante value the consumer receives by always consuming his endowment The consumers freedom to walk away induces the planner to structure the insurance contract so that it is never in the households interest to defect from the contract the contract must be selfenforcing A selfenforcing contract requires that the following participation constraints be satisfied u cs βws u ys βvaut s 1411 A selfenforcing contract provides imperfect insurance when occasionally some of these participation constraints are binding When they are binding the planner sacrifices consumption smoothing in the interest of providing incentives for the contract to be selfenforcing An alternative specification eliminates the enforcement problem by assum ing that once the household enters the contract it does not have the option to walk away A planner wants to supply insurance to the household in the most efficient way but now the planner cannot observe the households endowment The planner must trust the household to report its endowment It is assumed that the household will truthfully report its endowment only if it wants to This leads the planner to add to the promisekeeping constraint 147 the following truthtelling constraints u cs βws u ys yτ cτ βwτ s τ 1412 where constraint 1412 pertains to a situation when the households true en dowment is ys but the household considers to falsely report that the endowment instead is yτ The left and right sides of 1412 are the utility of telling the truth and lying respectively If the household falsely reports yτ the planner awards the household a net transfer cτ yτ and a continuation value wτ If 1412 holds for all τ the household will always choose to report the true state s As we shall see in chapters 21 and 22 the planner elicits truthful reporting by manipulating how continuation values vary with the reported state House holds that report a low income today might receive a transfer today but they suffer an adverse consequence by getting a diminished continuation value start ing tomorrow The planner structures this menu of choices so that only low endowment households those that badly want a transfer today are willing to views or information about θ before seeing y0T our views about θ after seeing y0T are described by a posterior probability pθy0T that is constructed from Bayess law via pθy0T fy0T θ pθ fy0T θ pθ dθ where the denominator is the marginal joint density fy0T of y0T Recursive methods 25 accept the diminished continuation value that is the consequence of reporting that low income today At this point a supermartingale convergence theorem raises its ugly head again But this time it propels consumption and continuation utility downward The super martingale result leads to what some people have termed the im miseration property of models in which dynamic contracts are used to deliver incentives to reveal information To enhance our appreciation for the immiseration result we now touch on another aspect of macroeconomics lovehate affair with the Euler inequality 133 In both of the incentive models just described one with an enforce ment problem the other with an information problem it is important that the household not have access to a good riskfree investment technology like that represented in the constraint 132 that makes 133 the appropriate first order condition in the savings problem Indeed especially in the model with limited information the planner makes ample use of his ability to reallocate consumption intertemporally in ways that can violate 132 in order to elicit accurate information from the household In chapter 21 we shall follow Cole and Kocherlakota 2001 by allowing the household to save but not to dissave a riskfree asset that bears fixed gross interest rate R β1 The Euler inequal ity comes back into play and alters the character of the insurance arrangement so that outcomes resemble ones that occur in a Bewley model provided that the debt limit in the Bewley model is chosen appropriately 146 More applications We shall study many more applications of dynamic programming and dynamic programming squared including models of search in labor markets reputation and credible public policy gradualism in trade policy unemployment insurance and monetary economies It is time to get to work seriously studying the math ematical and economic tools that we need to approach these exciting topics Let us begin unity22 we can solve the preceding equation to get xt1 j0 A KGj K ytj 292 Then solving 291b for yt gives the vector autoregression yt G j0 A KGj K ytj1 at 293 where by construction E at ytj1 0 j 0 294 The orthogonality conditions 294 identify 293 as a vector autoregression Part II Tools This page is blank This page is blank Chapter 2 Time Series 21 Two workhorses This chapter describes two tractable models of time series finite state Markov chains and firstorder stochastic linear difference equations These models are organizing devices that put restrictions on a sequence of random vectors They are useful because they describe a time series with parsimony In later chapters we shall make two uses each of Markov chains and stochastic linear difference equations 1 to represent the exogenous information flows impinging on an agent or an economy and 2 to represent an optimum or equilibrium outcome of agents decision making The Markov chain and the firstorder stochastic linear difference both use a sharp notion of a state vector A state vector sum marizes the information about the current position of a system that is relevant for determining its future The Markov chain and the stochastic linear difference equation will be useful tools for studying dynamic optimization problems 22 Markov chains A stochastic process is a sequence of random vectors For us the sequence will be ordered by a time index taken to be the integers in this book So we study discrete time models We study a discretestate stochastic process with the following property Markov Property A stochastic process xt is said to have the Markov property if for all k 1 and all t Probxt1xt xt1 xtk Prob xt1xt We assume the Markov property and characterize the process by a Markov chain A timeinvariant Markov chain is defined by a triple of objects namely 29 an ndimensional state space consisting of vectors ei i 1 n where ei is an n 1 unit vector whose ith entry is 1 and all other entries are zero an n n transition matrix P which records the probabilities of moving from one value of the state to another in one period and an n 1 vector π0 whose ith element is the probability of being in state i at time 0 π0i Probx0 ei The elements of matrix P are Pij Probxt1 ejxt ei For these interpretations to be valid the matrix P and the vector π0 must satisfy the following assumption Assumption M a For i 1 n the matrix P satisfies j1n Pij 1 b The vector π0 satisfies i1n π0i 1 A matrix P that satisfies property 221 is called a stochastic matrix A stochastic matrix defines the probabilities of moving from one value of the state to another in one period The probability of moving from one value of the state to another in two periods is determined by P2 because Probxt2 ejxt ei h1n Probxt2 ejxt1 eh Probxt1 ehxt ei h1n Pih Phj Pij2 where Pij2 is the i j element of P2 Let Pijk denote the i j element of Pk By iterating on the preceding equation we discover that Probxtk ejxt ei Pijk The unconditional probability distributions of xt are determined by π1 Probx1 π0 P π2 Probx2 π0 P2 πk Probxk π0 Pk where πt Probxt is the 1 n vector whose ith element is Probxt ei 221 Stationary distributions Unconditional probability distributions evolve according to πt1 πt P An unconditional distribution is called stationary or invariant if it satisfies πt1 πt that is if the unconditional distribution remains unaltered with the passage of time From the law of motion 222 for unconditional distributions a stationary distribution must satisfy π π P or π I P 0 Transposing both sides of this equation gives I P π 0 which determines π as an eigenvector normalized to satisfy i1n πi 1 associated with a unit eigenvalue of P We say that P π is a stationary Markov chain if the initial distribution π is such that 223 holds The fact that P is a stochastic matrix ie it has nonnegative elements and satisfies j Pij 1 for all i guarantees that P has at least one unit eigenvalue and that there is at least one eigenvector π that satisfies equation 32 Time Series 224 This stationary distribution may not be unique because P can have a repeated unit eigenvalue Example 1 A Markov chain P 1 0 0 2 5 3 0 0 1 has two unit eigenvalues with associated stationary distributions π 1 0 0 and π 0 0 1 Here states 1 and 3 are both absorbing states Further more any initial distribution that puts zero probability on state 2 is a stationary distribution See exercises 210 and 211 Example 2 A Markov chain P 7 3 0 0 5 5 0 9 1 stationary distribution π 0 6429 3571 associated with its single unit eigenvalue Here states 2 and 3 form an absorbing subset of the state space 222 Asymptotic stationarity We often ask the following question about a Markov process for an arbitrary initial distribution π0 do the unconditional distributions πt approach a sta tionary distribution lim t πt π where π solves equation 224 If the answer is yes then does the limit distribution π depend on the initial distribution π0 If the limit π is inde pendent of the initial distribution π0 we say that the process is asymptotically stationary with a unique invariant distribution We call a solution π a sta tionary distribution or an invariant distribution of P We state these concepts formally in the following definition Definition 221 Let π be a unique vector that satisfies I P π 0 If for all initial distributions π0 it is true that P tπ0 converges to the same Markov chains 33 π we say that the Markov chain is asymptotically stationary with a unique invariant distribution The following theorems describe conditions under which a Markov chain is asymptotically stationary Theorem 221 Let P be a stochastic matrix with Pij 0 i j Then P has a unique stationary distribution and the process is asymptotically station ary Theorem 222 Let P be a stochastic matrix for which P n ij 0 i j for some value of n 1 Then P has a unique stationary distribution and the process is asymptotically stationary The conditions of Theorem 221 and Theorem 222 state that from any state there is a positive probability of moving to any other state in one or n steps Please note that some of the examples below will violate the conditions of The orem 222 for any n 223 Forecasting the state The minimum mean squared error forecast of the state next period is the con ditional mathematical expectation E xt1xt ei Pi1 Pi2 Pin P ei P i 225 where P i denotes the transpose of the ith row of the matrix P In section B2 of this books appendix B we use this equation to motivate the following firstorder stochastic difference equation for the state xt1 P xt vt1 226 where vt1 is a random disturbance that evidently satisfies Evt1xt 0 Now let y be an n 1 vector of real numbers and define yt yxt so that yt yi if xt ei Evidently we can write yt1 yP xt yvt1 227 The pair of equations 226 227 becomes a simple example of a hidden Markov model when the observation yt is too coarse to reveal the state See section B2 of technical appendix B for a discussion of such models 224 Forecasting functions of the state From the conditional and unconditional probability distributions that we have listed it follows that the unconditional expectations of yt for t 0 are determined by Eyt π0Pt y Conditional expectations are determined by E yt1xt ei j Pij yj Pyi 228 E yt2xt ei k P2ik yk P2yi 229 and so on where P2ik denotes the i k element of P2 and i denotes the ith row of the matrix An equivalent formula from 226 227 is Eyt1xt yPxt xtP y which equals Pyi when xt ei Notice that E E yt2xt1 ej xt ei j Pij k Pjk yk k j PijPjk yk k P2ik yk E yt2xt ei Equation the first and last terms yields EEyt2xt1xt Eyt2xt This is an example of the law of iterated expectations The law of iterated expectations states that for any random variable z and two information sets J I with J I EEzIJ EzJ As another example of the law of iterated expectations notice that Ey1 j π1j yj π1 y π0P y π0 Py and that E E y1x0 ei i π0i j Pij yj j i π0iPij yj π1 y Ey1 225 Forecasting functions There are powerful formulas for forecasting functions of a Markov state Again let y be an n 1 vector and consider the random variable yt yxt Then E ytkxt ei Pk yi where Pk yi denotes the ith row of Pk y Stacking all n rows together we express this as E ytkxt Pk y 2210 We also have k0 βk E ytkxt ei I βP1 yi where β 01 guarantees existence of I βP1 I βP β2P2 The matrix I βP1 is called a resolvent operator 226 Enough onestepahead forecasts determine P Onestepahead forecasts of a sufficiently rich set of random variables characterize a Markov chain In particular onestepahead conditional expectations of n independent functions ie n linearly independent vectors h1 hn uniquely determine the transition matrix P Thus let Ehkt1xt ei Phki We can collect the conditional expectations of hk for all initial states i in an n 1 vector Ehkt1xt Phk We can then collect conditional expectations for the n independent vectors h1 hn as Ph J where h h1 h2 hn and J is the n n matrix consisting of all conditional expectations of all n vectors h1 hn If we know h and J we can determine P from P Jh1 227 Invariant functions and ergodicity Let P π be a stationary nstate Markov chain with the state space X ei i 1 n An n 1 vector y defines a random variable yt y xt Let Eyx0 be the expectation of ys for s very large conditional on the initial state The following is a useful precursor to a law of large numbers Theorem 223 Let y define a random variable as a function of an underlying state x where x is governed by a stationary Markov chain P π Then 1T t1T yt E yx0 2211 with probability 1 To illustrate Theorem 223 consider the following example Example Consider the Markov chain P 1 0 0 1 π0 p 1 p for p 01 Consider the random variable yt y xt where y 10 0 The chain has two possible sample paths yt 10 t 0 which occurs with probability p and yt 0 t 0 which occurs with probability 1 p Thus 1T T t1 yt 10 with probability p and 1T T t1 yt 0 with probability 1 p The outcomes in this example indicate why we might want something more than 2211 In particular we would like to be free to replace Eyx0 with the constant unconditional mean Eyt Ey0 associated with the stationary distribution π To get this outcome we must strengthen what we assume about P by using the following concepts Suppose that P π is a stationary Markov chain Imagine repeatedly drawing x0 from π and then generating xt t 1 by successively drawing from transition densities given by the matrix P We use Definition 222 A random variable yt y xt is said to be invariant if yt y0 t 0 for all realizations of xt t 0 that occur with positive probability under P π Thus a random variable yt is invariant or an invariant function of the state if it remains constant at y0 while the underlying state xt moves through the state space X Notice how the definition leaves open the possibility that y0 itself might differ across sample paths indexed by different draws of the initial condition x0 from the initial and stationary density π The stationary Markov chain Pπ induces a joint density fxt1 xt over xt1 xt that is independent of calendar time t Pπ and the definition yt yxt also induce a joint density fyyt1 yt that is independent of calendar time In what follows we compute mathematical expectations with respect to the joint density fyyt1 yt For a finitestate Markov chain the following theorem gives a convenient way to characterize invariant functions of the state Theorem 224 Let P π be a stationary Markov chain If E yt1xt yt 2212 then the random variable yt yxt is invariant Proof By using the law of iterated expectations notice that E yt1 yt2 E E y2t1 2yt1yt y2t xt E Ey2t1xt 2E yt1xt yt Ey2t xt Ey2t1 2E y2t Ey2t 0 where the middle term on the right side of the second line uses that Eytxt yt the middle term on the right side of the third line uses hypothesis 2212 and the third line uses the hypothesis that π is a stationary distribution In a finite Markov chain if Eyt1 yt2 0 then yt1 yt for all yt1 yt that occur with positive probability under the stationary distribution As we shall have reason to study in chapters 17 and 18 any not necessarily stationary stochastic process yt that satisfies 2212 is said to be a martingale Theorem 224 tells us that a martingale that is a function of a finitestate stationary Markov state xt must be constant over time This result is a special case of the martingale convergence theorem that underlies some remarkable results about savings to be studied in chapter 171 1 Theorem 224 tells us that a stationary martingale process has so little freedom to move that it has to be constant forever not just eventually as asserted by the martingale convergence theorem Equation 2212 can be expressed as P y y or P I y 0 2213 which states that an invariant function of the state is a right eigenvector of P associated with a unit eigenvalue Thus associated with unit eigenvalues of P are 1 left eigenvectors that are stationary distributions of the chain recall equation 224 and 2 right eigenvectors that are invariant functions of the chain from equation 2213 Definition 223 Let Pπ be a stationary Markov chain The chain is said to be ergodic if the only invariant functions y are constant with probability 1 under the stationary unconditional probability distribution π ie yi yj for all i j with πi 0 πj 0 Remark Let π1 π2 πm be m distinct basis stationary distributions for an n state Markov chain with transition matrix P Each πk is an n 1 left eigenvector of P associated with a distinct unit eigenvalue Each πj is scaled to be a probability vector ie its components are nonnegative and sum to unity The set S of all stationary distributions is convex An element πb S can be represented as πb b1π1 b2π2 bmπm where bj 0 j bj 1 is a probability vector Remark A stationary density πb for which the pair P πb is an ergodic Markov chain is an extreme point of the convex set S meaning that it can be represented as πb πj for one of the basis stationary densities A law of large numbers for Markov chains is Theorem 225 Let y define a random variable on a stationary and ergodic Markov chain Pπ Then 1T t1 to T yt E y0 2214 with probability 1 This theorem tells us that the time series average converges to the population mean of the stationary distribution Three examples illustrate these concepts Example 1 A chain with transition matrix P 0 1 1 0 has a unique stationary distribution π 5 5 and the invariant functions are α α for any scalar α Therefore the process is ergodic and Theorem 225 applies Example 2 A chain with transition matrix P 1 0 0 1 has a continuum of stationary distributions γ 1 0 1 γ 0 1 for any γ 0 1 and invariant functions 0 α1 and α2 0 for any scalars α1 α2 Therefore the process is not ergodic when γ 0 1 for note that neither invariant function is constant across states that receive positive probability according to a stationary distribution associated with γ 0 1 Therefore the conclusion 2214 of Theorem 225 does not hold for an initial stationary distribution associated with γ 0 1 although the weaker result Theorem 223 does hold When γ 0 1 nature chooses state i 1 or i 2 with probabilities γ 1 γ respectively at time 0 Thereafter the chain remains stuck in the realized time 0 state Its failure ever to visit the unrealized state prevents the sample average from converging to the population mean of an arbitrary function y of the state Notice that conclusion 2214 of Theorem 225 does hold for the stationary distributions associated with γ 0 and γ 1 Example 3 A chain with transition matrix P 8 2 0 1 9 0 0 0 1 has a continuum of stationary distributions γ 13 23 0 1 γ 0 0 1 for γ 0 1 and invariant functions α1 1 1 0 and α2 0 0 1 for any scalars α1 α2 The conclusion 2214 of Theorem 225 does not hold for the stationary distributions associated with γ 0 1 but Theorem 223 does hold But again conclusion 2214 does hold for the stationary distributions associated with γ 0 and γ 1 228 Simulating a Markov chain It is easy to simulate a Markov chain using a random number generator The Matlab program markovm does the job Well use this program in some later chapters2 229 The likelihood function Let P be an n n stochastic matrix with states 12 n Let π0 be an n 1 vector with nonnegative elements summing to 1 with π0i being the probability that the state is i at time 0 Let it index the state at time t The Markov property implies that the probability of drawing the path x0 x1 xT1 xT ei0 ei1 eiT1 eiT is L Prob xiT xiT1 xi1 xi0 PiT1iT PiT2iT1 Pi0i1 π0i0 2215 The probability L is called the likelihood It is a function of both the sample realization x0 xT and the parameters of the stochastic matrix P For a sample x0 x1 xT let nij be the number of times that there occurs a one period transition from state i to state j Then the likelihood function can be written L π0i0 i j Pijnij a multinomial distribution Formula 2215 has two uses A first which we shall encounter often is to describe the probability of alternative histories of a Markov chain In chapter 8 we shall use this formula to study prices and allocations in competitive equilibria A second use is for estimating the parameters of a model whose solution is a Markov chain Maximum likelihood estimation for free parameters θ of a Markov process works as follows Let the transition matrix P and the initial distribution π0 be functions Pθ π0θ of a vector of free parameters θ Given a sample xtTt0 regard the likelihood function as a function of the parameters θ As the estimator of θ choose the value that maximizes the likelihood function L 2 An index in the back of the book lists Matlab programs 23 Continuousstate Markov chain In chapter 8 we shall use a somewhat different notation to express the same ideas This alternative notation can accommodate either discrete or continuous state Markov chains We shall let S denote the state space with typical element s S Let state transitions be described by the cumulative distribution function Πss Probst1 sst s and let the initial state s0 be described by the cumulative distribution function Π0s Probs0 s The transition density is πss dds Πss and the initial density is π0s dds Π0s For all s S πss 0 and s πssds 1 also s π0sds 13 Corresponding to 2215 the density over history st st st1 s0 is π st π stst1 π s1s0 π0 s0 231 For t 1 the time t unconditional distributions evolve according to πt st st1 π stst1 πt1 st1 d st1 A stationary or invariant distribution satisfies π s s π ss π s d s which is the counterpart to 223 Definition A Markov chain πss π0s is said to be stationary if π0 satisfies π0 s s π ss π0 s d s Definition Paralleling our discussion of finitestate Markov chains we can say that the function φs is invariant if φ s π ss ds φ s A stationary continuousstate Markov process is said to be ergodic if the only invariant functions φs are constant with probability 1 under the stationary distribution π 3 Thus when S is discrete πsjsi corresponds to Pij in our earlier notation A law of large numbers for Markov processes states Theorem 231 Let ys be a random variable a measurable function of s and let πss π0s be a stationary and ergodic continuousstate Markov process Assume that Ey Then 1T t1T yt Ey y s π0 s d s with probability 1 with respect to the distribution π0 24 Stochastic linear difference equations The firstorder linear vector stochastic difference equation is a useful example of a continuousstate Markov process Here we use xt IRn rather than st to denote the time t state and specify that the initial distribution π0x0 is Gaussian with mean μ0 and covariance matrix Σ0 and that the transition density πxx is Gaussian with mean Ax and covariance CC4 This specification pins down the joint distribution of the stochastic process xtt0 via formula 231 The joint distribution determines all moments of the process This specification can be represented in terms of the firstorder stochastic linear difference equation xt1 Axt Cwt1 241 for t 01 where xt is an n1 state vector x0 is a random initial condition drawn from a probability distribution with mean E x0 μ0 and covariance matrix Ex0 μ0x0 μ0 Σ0 A is an n n matrix C is an n m matrix and wt1 is an m 1 vector satisfying the following Assumption A1 wt1 is an iid process satisfying wt1 N 0 I 4 An n 1 vector z that is multivariate normal has the density function f z 2π5n Σ5 exp 5 z μ Σ1 z μ where μ Ez and Σ Ez μz μ We can weaken the Gaussian assumption A1 To focus only on first and second moments of the x process it is sufficient to make the weaker assumption ASSUMPTION A2 wt1 is an m 1 random vector satisfying Ewt1Jt 0 242a Ewt1wt1Jt I 242b where Jt wt wt1 w1 x0 is the information set at t and E Jt denotes the conditional expectation We impose no distributional assumptions beyond 242 A sequence wt1 satisfying equation 242a is said to be a martingale difference sequence adapted to Jt An even weaker assumption is ASSUMPTION A3 wt1 is a process satisfying Ewt1 0 for all t and Ewtwtj I if j 0 0 if j 0 A process satisfying assumption A3 is said to be a vector white noise 5 Assumption A1 or A2 implies assumption A3 but not vice versa Assumption A1 implies assumption A2 but not vice versa Assumption A3 is sufficient to justify the formulas that we report below for second moments We shall often append an observation equation yt Gxt to equation 241 and deal with the augmented system xt1 Axt Cwt1 243a yt Gxt 243b Here yt is a vector of variables observed at t which may include only some linear combinations of xt The system 243 is often called a linear statespace system 5 Note that 242a by itself allows the distribution of wt1 conditional on Jt to be heteroskedastic Example 1 Scalar secondorder autoregression Assume that zt and wt are scalar processes and that zt1 α ρ1zt ρ2zt1 wt1 Represent this relationship as the system zt1 zt 1 ρ1 ρ2 α 1 0 0 0 0 1 zt zt1 1 1 0 0 wt1 zt 1 0 0 zt zt1 1 which has form 243 Example 2 Firstorder scalar mixed moving average and autoregression Let zt1 ρzt wt1 γwt Express this relationship as zt1 wt1 ρ γ 0 0 zt wt 1 1 wt1 zt 1 0 zt wt Example 3 Vector autoregression Let zt be an n 1 vector of random variables We define a vector autoregression by a stochastic difference equation zt1 from j1 to 4 Ajzt1j Cywt1 244 where wt1 is an n1 martingale difference sequence satisfying equation 242 with x0 z0 z1 z2 z3 and Aj is an n n matrix for each j We can map equation 244 into equation 241 as follows zt1 zt zt1 zt2 A1 A2 A3 A4 I 0 0 0 0 I 0 0 0 0 I 0 zt zt1 zt2 zt3 Cy 0 0 0 wt1 245 Define A as the state transition matrix in equation 245 Assume that A has all of its eigenvalues bounded in modulus below unity Then equation 244 can be initialized so that zt is covariance stationary a term we define soon 241 First and second moments We can use equation 241 to deduce the first and second moments of the sequence of random vectors xtt0 A sequence of random vectors is called a stochastic process Definition 241 A stochastic process xt is said to be covariance stationary if it satisfies the following two properties a the mean is independent of time Ext Ex0 for all t and b the sequence of autocovariance matrices Extj Extjxt Ext depends on the separation between dates j 0 1 2 but not on t We use Definition 242 A square real valued matrix A is said to be stable if all of its eigenvalues modulus are strictly less than unity We shall often find it useful to assume that 243 takes the special form x1t1 x2t1 1 0 0 à x1t x2t 0 Ĉ wt1 246 where à is a stable matrix That à is a stable matrix implies that the only solution of à Iμ2 0 is μ2 0 ie 1 is not an eigenvalue of à It follows that the matrix A 1 0 0 à on the right side of 246 has one eigenvector associated with a single unit eigenvalue A I μ1 μ2 0 implies μ1 is an arbitrary scalar and μ2 0 The first equation of 246 implies that x1t1 x10 for all t 0 Picking the initial condition x10 pins down a particular eigenvector x10 0 of A As we shall see soon this eigenvector is our candidate for the unconditional mean of x that makes the process covariance stationary We will make an assumption that guarantees that there exists an initial condition μ0 Σ0 Ex0 Ex Ex0x Ex0 that makes the xt process covariance stationary Either of the following conditions works CONDITION A1 All of the eigenvalues of A in 243 are strictly less than 1 in modulus CONDITION A2 The statespace representation takes the special form 246 and all of the eigenvalues of à are strictly less than 1 in modulus 46 Time Series To discover the first and second moments of the xt process we regard the initial condition x0 as being drawn from a distribution with mean µ0 Ex0 and covariance Σ0 ExEx0xEx0 We shall deduce starting values for the mean and covariance that make the process covariance stationary though our formulas are also useful for describing what happens when we start from other initial conditions that generate transient behavior that stops the process from being covariance stationary Taking mathematical expectations on both sides of equation 241 gives µt1 Aµt 247 where µt Ext We will assume that all of the eigenvalues of A are strictly less than unity in modulus except possibly for one that is affiliated with the constant terms in the various equations Then xt possesses a stationary mean defined to satisfy µt1 µt which from equation 247 evidently satisfies I A µ 0 248 which characterizes the mean µ as an eigenvector associated with the single unit eigenvalue of A The condition that the remaining eigenvalues of A are less than unity in modulus implies that starting from any µ0 µt µ6 Notice that xt1 µt1 A xt µt Cwt1 249 From equation 249 we can compute that the law of motion of the covariance matrices Σt Ext µtxt µt Thus E xt1 µt1 xt1 µt1 AE xt µt xt µt A CC or Σt1 AΣtA CC 6 To understand this assume that the eigenvalues of A are distinct and use the repre sentation A P ΛP 1 where Λ is a diagonal matrix of the eigenvalues of A arranged in descending order of magnitude and P is a matrix composed of the corresponding eigenvec tors Then equation 247 can be represented as µ t1 Λµ t where µ t P 1µt which implies that µ t Λtµ 0 When all eigenvalues but the first are less than unity Λt converges to a matrix of zeros except for the 1 1 element and µ t converges to a vector of zeros except for the first element which stays at µ 01 its initial value which we are free to set equal to 1 to capture the constant Then µt P µ t converges to P1µ 01 P1 where P1 is the eigenvector corresponding to the unit eigenvalue Stochastic linear difference equations 47 A fixed point of this matrix difference equation evidently satisfies Σ AΣA CC 2410 A fixed point Σ is the covariance matrix Ext µxtµ under a stationary distribution of x Equation 2410 is a discrete Lyapunov equation in the nn matrix Σ It can be solved with the Matlab program doublejm By virtue of 241 and 247 note that for j 0 xtj µtj Aj xt µt Cwtj Aj1Cwt1 Postmultiplying both sides by xt µt and taking expectations shows that the autocovariance sequence satisfies Σtjt E xtj µtj xt µt AjΣt 2411 Note that Σtjt depends on both j the gap between dates and t the earlier date In the special case that Σt Σ that solves the discrete Lyapunov equa tion 2410 Σtjt Aj 0Σ and so depends only on the gap j between time periods In this case an autocovariance matrix sequence Σtjt j0 is often also called an autocovariogram Suppose that yt Gxt Then µyt Eyt Gµt and E ytj µytj yt µyt GΣtjtG 2412 for j 0 1 Equations 2412 show that the autocovariogram for a stochastic process governed by a stochastic linear difference equation obeys the nonstochastic version of that difference equation 242 Summary of moment formulas The accompanying table summarizes some formulas for various conditional and unconditional first and second moments of the state xt governed by our linear stochastic state space system ACG In section 25 we select some moments and use them to form population linear regressions 243 Impulse response function Suppose that the eigenvalues of A not associated with the constant are bounded above in modulus by unity Using the lag operator L defined by Lxt1 xt express equation 241 as I ALxt1 Cwt1 2413 Iterate equation 241 forward from t 0 to get xt Atx0 t1 j0 AjCwtj 2414 Evidently yt GAtx0 G 2415 t1 j0 AjCwtj and Eytx0 GAtx0 Equations 2414 and 2415 are examples of a moving average representation Viewed as a function of lag j hj AjC or hj GAjC is called the impulse response function The moving average representation and the associated impulse response function show how xtj or ytj is affected by lagged values of the shocks the wt1s Thus the contribution of a shock wtj to xt is AjC7 Equation 2415 implies that the tstep ahead conditional covariance matrices are given by E yt Eytx0yt Eytx0 G t1 h0 AhCCAh G 2416 244 Prediction and discounting From equation 241 we can compute the useful prediction formulas Etxtj Ajxt 2417 for j 1 where Et denotes the mathematical expectation conditioned on xt xt xt1 x0 Let yt Gxt and suppose that we want to compute Et j0 βjytj Evidently Et j0 βjytj G I βA1 xt 2418 provided that the eigenvalues of βA are less than unity in modulus Equation 2418 tells us how to compute an expected discounted sum where the discount factor β is constant 7 The Matlab programs dimpulsem and impulsem compute impulse response functions 245 Geometric sums of quadratic forms In some applications we want to calculate αt Et j0 βjxtjY xtj where xt obeys the stochastic difference equation 241 and Y is an n n matrix To get a formula for αt we use a guessandverify method We guess that αt can be written in the form αt xtνxt σ 2419 where ν is an n n matrix and σ is a scalar The definition of αt and the guess 2419 imply8 αt xtY xt βEt xt1νxt1 σ xtY xt βEt Axt Cwt1 ν Axt Cwt1 σ xt Y βAνA xt β traceνCC βσ It follows that ν and σ satisfy ν Y βAνA σ βσ β trace νCC 2420 The first equation of 2420 is a discrete Lyapunov equation in the square matrix ν and can be solved by using one of several algorithms9 After ν has been computed the second equation can be solved for the scalar σ We mention two important applications of formulas 2419 and 2420 8 Here we use the fact that for two conformable matrices A B traceAB traceBA to deduce Ewt1CνCwt1 EtraceνCwt1wt1C traceνCEwt1wt1C traceνCC 9 The Matlab control toolkit has a program called dlyapm that works when all of the eigenvalues of A are strictly less than unity the program called doublejm works even when there is a unit eigenvalue associated with the constant 2451 Asset pricing Let yt be governed by the statespace system 243 In addition assume that there is a scalar random process zt given by zt Hxt Regard the process yt as a payout or dividend from an asset and regard βtzt as a stochastic discount factor The price of a perpetual claim on the stream of payouts is αt Et j0 βj ztj ytj 2421 To compute αt we simply set Y HG in 2419 and 2420 In this application the term σ functions as a risk premium it is zero when C 0 2452 Evaluation of dynamic criterion Let a state xt be governed by xt1 Axt But Cwt1 2422 where ut is a control vector that is set by a decision maker according to a fixed rule ut F0xt 2423 Substituting 2423 into 2422 gives xt1 Aoxt Cwt1 where Ao A BF0 We want to compute the value function v x0 E0 t0 βt xtRxt utQut for fixed positive definite matrices R and Q fixed decision rule F0 in 2423 and arbitrary initial condition x0 Formulas 2419 and 2420 apply with Y R F0QF0 and A being replaced by Ao A BF0 Express the solution as v x0 x0 P0 x0 σ 2424 52 Time Series where by applying formulas 2419 and 2420 P0 satisfies the following formula P0 R F 0QF0 β A BF0 P0 A BF0 2425 which can be recognized to be a discrete Lyapunov equation of the form of the first equation 2420 Given F0 formula 2425 determines the matrix P0 in the value function that describes the expected discounted value of the sum of payoffs from sticking forever with this decision rule Now consider the following oneperiod problem Suppose that we must use decision rule F0 from time 1 onward so that the value at time 1 on starting from state x1 is v x1 x 1P0x1 σ 2426 Taking ut F0xt as given for t 1 what is the best choice of u0 This leads to the optimum problem max u0 x 0Rx0u 0Qu0βE Ax0 Bu0 Cw1 P0 Ax0 Bu0 Cw1βσ 2427 The firstorder conditions for this problem can be rearranged to attain u0 F1x0 2428 where F1 β Q βBP0B1 BP0A 2429 Given P0 formula 2429 gives the best decision rule u0 F1x0 if at t 0 you are permitted only a oneperiod deviation from the rule ut F0xt that has to be used for t 1 If F1 F0 we say that the decision maker would accept the opportunity to deviate from F0 for one period It is tempting to iterate on 2429 and 2425 as follows to seek a decision rule from which a decision maker would not want to deviate for one period 1 given an F0 find P0 2 reset F equal to the F1 found in step 1 then to substitute it for F0 in 2425 to compute a new P call it P1 3 return to step 1 and iterate to convergence This leads to the two equations Pj R F jQFj β A BFj Pj A BFj Fj1 β Q βBPjB1 BPjA 2430 which are to be initialized from an arbitrary F0 that ensures that βA BF0 is a stable matrix After this process has converged one cannot find a valueincreasing oneperiod deviation from the limiting decision rule ut Fxt 10 As we shall see in chapter 4 this is an excellent algorithm for solving a dynamic programming problem It is an example of the Howard policy improvement algorithm In chapter 5 we describe an alternative algorithm that iterates on the following equations Pj1 R FjQFj β A BFj Pj A BFj Fj β Q βBPj B1 BPj A 2431 that is to be initialized from an arbitrary positive semidefinite matrix P0 11 25 Population regression This section explains the notion of a population regression equation Suppose that we have a statespace system 243 with initial conditions that make it covariance stationary We can use the preceding formulas to compute the second moments of any pair of random variables These moments let us compute a linear regression Thus let X be a p 1 vector of random variables somehow selected from the stochastic process yt governed by the system 243 For example let p 2m where yt is an m 1 vector and take X yt yt1 for any t 1 Let Y be any scalar random variable selected from the m 1 stochastic process yt For example take Y yt11 for the same t used to define X where yt11 is the first component of yt1 We consider the following leastsquares approximation problem find a 1p vector of real numbers β that attain min β E Y βX2 251 Here βX is being used to estimate Y and we want the value of β that minimizes the expected squared error The firstorder necessary condition for minimizing EY βX2 with respect to β is E Y βX X 0 252 which can be rearranged as12 β EY X E XX 1 253 By using the formulas 248 2410 2411 and 2412 we can compute EXX and EY X for whatever selection of X and Y we choose The condition 252 is called the leastsquares normal equation It states that the projection error Y βX is orthogonal to X Therefore we can represent Y as Y βX ϵ 254 where EϵX 0 Equation 254 is called a population regression equation and βX is called the leastsquares projection of Y on X or the leastsquares regression of Y on X The vector β is called the population leastsquares regression vector The law of large numbers for continuousstate Markov processes Theorem 231 states conditions that guarantee that sample moments converge to population moments that is 1 S S s1 Xs Xs EXX and 1 S S s1 Ys Xs EY X Under those conditions sample leastsquares estimates converge to β There are as many such regressions as there are ways of selecting Y X We have shown how a model eg a triple ACG together with an initial distribution for x0 restricts a regression Going backward that is telling what a given regression tells about a model is more difficult Many regressions tell little about the model and what little they have to say can be difficult to decode As we indicate in sections 26 and 28 the likelihood function completely describes what a given data set says about a model in a way that is straightforward to decode 12 That EXX is nonnegative definite implies that the secondorder conditions for a minimum of condition 251 are satisfied Estimation of model parameters 55 251 Multiple regressors Now let Y be an n1 vector of random variables and think of regression solving the least squares problem for each of them to attain a representation Y βX ǫ 255 where β is now n p and ǫ is now an n 1 vector of least squares residuals The population regression coefficients are again given by β E Y X E XX1 256 We will use this formula repeatedly in section 27 to derive the Kalman filter 26 Estimation of model parameters We have shown how to map the matrices A C into all of the second moments of the stationary distribution of the stochastic process xt Linear economic models typically give A C as functions of a set of deeper parameters θ We shall give examples of such models in chapters 4 and 5 Such a model and the formulas of this chapter give us a mapping from θ to these theoretical moments of the xt process That mapping is an important ingredient of econometric methods designed to estimate a wide class of linear rational expectations models see Hansen and Sargent 1980 1981 Briefly these methods use the following procedures to match theory to data To simplify we shall assume that at time t observations are available on the entire state xt As discussed in section 28 the details are more complicated if only a subset of the state vector or a noisy signal of the state is observed though the basic principles remain the same Given a sample of observations for xtT t0 xt t 0 T the likelihood function is defined as the joint probability distribution fxT xT 1 x0 The likelihood function can be factored using f xT x0 f xT xT 1 x0 f xT 1xT 2 x0 f x1x0 f x0 261 where in each case f denotes an appropriate probability distribution For sys tem 241 fxt1xt x0 fxt1xt which follows from the Markov 56 Time Series property possessed by equation 241 Then the likelihood function has the recursive form f xT x0 f xT xT 1 f xT 1xT 2 f x1x0 f x0 262 If we assume that the wt s are Gaussian then the conditional distribution fxt1xt is Gaussian with mean Axt and covariance matrix CC Thus under the Gaussian distribution the log of the conditional density of the n dimensional vector xt1 becomes log f xt1xt 5n log 2π 5 log det CC 5 xt1 Axt CC1 xt1 Axt 263 Given an assumption about the distribution of the initial condition x0 equations 262 and 263 can be used to form the likelihood function of a sample of observations on xtT t0 One computes maximum likelihood estimates by using a hillclimbing algorithm to maximize the likelihood function with respect to free parameters that determine A C 13 When the state xt is not observed we need to go beyond the likelihood function for xt One approach uses filtering methods to build up the likeli hood function for the subset of observed variables14 In section 27 we derive the Kalman filter as an application of the population regression formulas of sec tion 25 Then in section 28 we use the Kalman filter as a device that tells us how to find state variables that allow us recursively to form a likelihood function for observations of variables that are not themselves Markov 13 For example putting those free parameters into a vector θ think of A C as being the matrix functions Aθ Cθ 14 See Hamilton 1994 Canova 2007 DeJong and Dave 2011 and section 28 below The Kalman filter 57 27 The Kalman filter As a fruitful application of the population regression formula 256 we derive the celebrated Kalman filter for the state space system for t 015 xt1 Axt Cwt1 271 yt Gxt vt 272 where xt is an n 1 state vector and yt is an m 1 vector of signals on the hidden state wt1 is a p1 vector iid sequence of normal random variables with mean 0 and identity covariance matrix and vt is another iid vector sequence of normal random variables with mean zero and covariance matrix R We assume that wt1 and vs are orthogonal ie Ewt1v s 0 for all t 1 and s greater than or equal to 0 We assume that x0 N ˆx0 Σ0 273 We assume that we observe yt y0 but not xt x0 at time t We know all first and second moments implied by the structure 271 272 273 We work forward in time starting at time t 0 before we observe y0 Specification 272 273 implies that the conditional distribution of y0 is y0 N Gˆx0 GΣ0G R 274 For t 0 let yt yt yt1 y0 We want seek an expression for the probability distribution of yt conditional on history yt1 that has a convenient recursive representation The Kalman filter attains that by constructing recur sive formulas for objects ˆxt Σt that appear in the following generalization of 274 yt N Gˆxt GΣtG R 275 The objects ˆxt Σt characterize the population regression ˆxt Extyt1 y0 and the covariance matrix Σt Ext ˆxtxt ˆxt At each date our approach is to regresse what we dont know on what we know Lets start at date t 0 We arrive at date 0 knowing ˆx0 Σ0 Then we 15 In exercise 222 we ask you to derive the Kalman filter for a state space system that uses a different timing convention and that allows the state and measurement noises to be correlated 58 Time Series observe y0 and make inferences It will turn out that among the objects with which we leave time t 0 will be ˆx1 Σ1 This gives a perspective from which we are in the same situation at the start of period 1 that we were at the start of period 0 an insight that activates a recursion We use the insight that the information in y0 that is new relative to the information ˆx0 Σ0 that we knew before observing y0 is a0 y0 Gˆx0 Thus before we observe y0 we regard x0 as a random vector with mean ˆx0 and covariance matrix Σ0 Then we observe the random vector y0 linked to x0 by the time 0 version of equation 272 We form revised beliefs about the mean of x0 after observing y0 by computing the distribution of x0 conditional on y0 The conditional mean Ex0y0 ˆx0 L0y0 Gˆx0 satisfies the appropriate version of the population regression formula 256 namely x0 ˆx0 L0 y0 Gˆx0 η 276 where η is a matrix of least squares residuals whose orthogonality to y0 Gˆx0 characterizes L0 as population least squares regression coefficients The least squares orthogonality conditions are E x0 ˆx0 y0 Gˆx0 L0E y0 Gˆx0 y0 Gˆx0 Evaluating the moment matrices and solving for L0 gives the formula L0 Σ0G GΣ0G R1 277 Having constructed Ex0y0 we can construct ˆx1 Ex1y0 as follows16 Equation 271 implies that Ex1ˆx0 Aˆx0 and that x1 Aˆx0 A x0 ˆx0 Cw1 278 Furthermore applying 276 shows that Ex1y0 Aˆx0 AL0y0 Gˆx0 which we express as ˆx1 Aˆx0 K0 y0 Gˆx0 279 where K0 AΣ0G GΣ0G R1 2710 16 It is understood that we know ˆx0 Instead of writing Ex1y0 ˆx0 we choose simply to write Ex1y0 but we intend the meaning to be the same More generally when we write Extyt1 it is understood that the mathematical expectation is also conditioned on ˆx0 The Kalman filter 59 Subtract 279 from 278 to get x1 ˆx1 A x0 ˆx0 Cw1 K0 y0 Gˆx0 2711 Use this equation and y0 Gx0 v0 to compute the following formula for the conditional variance Ex1 ˆx1x1 ˆx1 Σ1 Σ1 A K0G Σ0 A K0G CC K0RK 0 2712 Thus we have deduced the conditional distribution x1y0 Nˆx1 Σ1 Col lecting equations we can write a0 y0 Gˆx0 2713a K0 AΣ0G GΣ0G R1 2713b ˆx1 Aˆx0 K0a0 2713c Σ1 CC K0RK 0 A K0G Σ0 A K0G 2713d Among the outcomes of system 2713 is a conditional mean covariance pair ˆx1 Σ1 It is appropriate to view system 2713 as a mapping a mean co variance pair ˆx0 Σ0 into a mean covariance pair ˆx1 Σ1 with auxiliary intermediate outputs a0 K0 The Kalman filter iterates on this mapping to arrive at the following recursions for t 0 at yt Gˆxt 2714a Kt AΣtG GΣtG R1 2714b ˆxt1 Aˆxt Ktat 2714c Σt1 CC KtRK t A KtG Σt A KtG 2714d System 2714 is the celebrated Kalman filter and Kt is called the Kalman gain Substituting for Kt from 2714b allows us to rewrite 2714d as Σt1 AΣtA CC AΣtG GΣtG R1 GΣtA 2715 Equation 2715 is known as a matrix Riccati difference equation that restricts a sequence of covariance matrices Σt t017 17 In a different context we shall encounter equations that will remind us of 2714b 2715 See chapter 5 page 142 60 Time Series 2701 GramSchmidt process The Kalman filter in effect uses a sequence of least squares projections called a GramSchmidt process to construct an orthogonal basis for the information set yt1 yt2 y0 Instead of computing Extyt1 yt2 y0 as one big least squares regression the GramSchmidt process computes a sequence of much smaller regressions on successive components as of an orthogonal basis at1 at2 a0 for the linear space spanned by yt1 yt2 y0 The random vector at yt Eytyt1 y0 is called the innovation for yt with respect to the information set yt1 It is the part of yt that cannot be predicted from past values of y Note that Eata t GΣtG R the moment matrix whose inverse appears on the right side of the least squares regression formula 2714b A direct calculation that uses at Gxt ˆxt vt and at1 Gxt1 ˆxt1 vt1 to compute expected values of products shows that Eata t1 0 and more generally that Eatat1 a0 018 Sometimes 2714 is called a whitening filter that takes a process yt of signals as an input and produces a process at of innovations as an output The linear space Hat is an orthogonal basis for the linear space Hyt 2702 Hidden Markov model System 271 272 273 is an example of a hidden Markov model The stochastic process yt t0 of observables is not Markov but the hid den process xt t0 is Markov and so is the process ˆxt Σt that consti tutes sufficient statistics for the probability distributions of yt conditional on yt1 yt2 y0 18 An alternative argument based on first principles proceeds as follows Let Hyt denote the linear space of all linear combinations of yt Note that at1 yt1 Eyt1yt that at Hyt that by virtue of being a leastsquares error at1 Hyt and that therefore at1 at and more generally at1 at Thus at is a white noise process of innovations to the yt process 271272 to the one associated with the vector autoregression 293 the Kalman filter is a very useful tool for interpreting vector autoregressions 210 Applications of the Kalman filter 2101 Muths reverse engineering exercise Phillip Cagan 1956 and Milton Friedman 1957 posited that to form expectations of future values of a scalar yt people use the following adaptive expectations scheme yt1 K Σj0 1 Kj ytj 2101a or yt1 1 Kyt Kyt 2101b where yt1 is peoples expectation23 Friedman used this scheme to describe peoples forecasts of future income Cagan used it to model their forecasts of inflation during hyperinflations Cagan and Friedman did not assert that the scheme is an optimal one and so did not fully defend it Muth 1960 wanted to understand the circumstances under which this forecasting scheme would be optimal Therefore he sought a stochastic process for yt such that equation 2101 would be optimal In effect he posed and solved an inverse optimal prediction problem of the form You give me the forecasting scheme I have to find the stochastic process that makes the scheme optimal Muth solved the problem using classical nonrecursive methods The Kalman filter was first described in print in the same year as Muths solution of this problem Kalman 1960 The Kalman filter lets us solve Muths problem quickly Muth studied the model xt1 xt wt1 2102a yt xt vt 2102b 23 See Hamilton 1994 and Kim and Nelson 1999 for diverse applications of the Kalman filter Appendix B see Technical Appendixes briefly describes a discretestate nonlinear filtering problem Applications of the Kalman filter 65 where yt xt are scalar random processes and wt1 vt are mutually independent iid Gaussian random processes with means of zero and variances Ew2 t1 Q Ev2 t R and Evswt1 0 for all t s The initial condition is that x0 is Gaussian with mean ˆx0 and variance Σ0 Muth sought formulas for ˆxt1 Ext1yt where yt yt y0 0 05 1 15 2 25 0 05 1 15 2 25 Figure 2101 Graph of fΣ ΣRQQR ΣR Q R 1 against the 45degree line Iterations on the Riccati equation for Σt converge to the fixed point For this problem A 1 CC Q G 1 making the Kalman filtering equations become Kt Σt Σt R 2103a Σt1 Σt Q Σ2 t Σt R 2103b The second equation can be rewritten Σt1 Σt R Q QR Σt R 2104 For Q R 1 Figure 2101 plots the function fΣ ΣRQQR ΣR appearing on the right side of equation 2104 for values Σ 0 against the 45degree line Note that f0 Q This graph identifies the fixed point of iterations on fΣ as the intersection of f and the 45degree line That the slope of f is less than unity at the intersection assures us that the iterations on f will converge as t starting from any Σ0 0 Muth studied the solution of this problem as t Evidently Σt Σ Σ is the fixed point of a graph like Figure 2101 Then Kt K and the formula for xt1 becomes xt1 1 K xt Kyt 2105 where K Σ ΣR 01 This is a version of Cagans adaptive expectations formula It can be shown that K 0 1 is an increasing function of QR Thus K is the fraction of the innovation at that should be regarded as permanent and 1 K is the fraction that is purely transitory Iterating backward on equation 2105 gives xt1 K Σt j0 1 Kjytj 1 Kt1x0 which is a version of Cagan and Friedmans geometric distributed lag formula Using equations 2102 we find that E ytj yt Extj yt xt1 for all j 1 This result in conjunction with equation 2105 establishes that the adaptive expectation formula 2105 gives the optimal forecast of ytj for all horizons j 1 This finding is remarkable because for most processes the optimal forecast will depend on the horizon That there is a single optimal forecast for all horizons justifies the term permanent income that Milton Friedman 1955 chose to describe the forecast of income The dependence of the forecast on horizon can be studied using the formulas E xtjyt1 Aj xt 2106a E ytjyt1 GAj xt 2106b In the case of Muths example E ytjyt1 yt xt j 0 For Muths model the innovations representation is xt1 xt Kat yt xt at Applications of the Kalman filter 67 where at yt Eytyt1 yt2 The innovations representation implies that yt1 yt at1 K 1 at 2107 Equation 2107 represents yt as a process whose first difference is a first order moving average process Notice how Friedmans adaptive expectations coefficient K appears in this representation 2102 Jovanovics application In chapter 6 we will describe a version of Jovanovics 1979 matching model at the core of which is a signalextraction problem that simplifies Muths problem Let xt yt be scalars with A 1 C 0 G 1 R 0 Let x0 be Gaussian with mean µ and variance Σ0 Interpret xt which is evidently constant with this specification as the hidden value of θ a match parameter Let yt denote the history of ys from s 0 to s t Define mt ˆxt1 Eθyt and Σt1 Eθ mt2 Then the Kalman filter becomes mt 1 Kt mt1 Ktyt 2108a Kt Σt Σt R 2108b Σt1 ΣtR Σt R 2108c The recursions are to be initiated from m1 Σ0 a pair that embodies all prior knowledge about the position of the system It is easy to see from Figure 2101 that when CC Q 0 Σ 0 is the limit point of iterations on equation 2108c starting from any Σ0 0 Thus the value of the match parameter is eventually learned It is instructive to write equation 2108c as 1 Σt1 1 Σt 1 R 2109 The reciprocal of the variance is often called the precision of the estimate According to equation 2109 the precision increases without bound as t grows and Σt1 024 24 As a further special case consider when there is zero precision initially Σ0 Then solving the difference equation 2109 gives 1 Σt tR Substituting this into equations We can represent the Kalman filter in the form mt1 mt Kt1 at1 which implies that E mt1 mt2 Kt1 2 σat1 2 where at1 yt1 mt and the variance of at is equal to σat1 2 Σt1 R from equation 565 This implies E mt1 mt2 Σt1 2 Σt1 R For the purposes of our discretetime counterpart of the Jovanovic model in chapter 6 it will be convenient to represent the motion of mt1 by means of the equation mt1 mt gt1 ut1 where gt1 Σt1 2 Σt1R 5 and ut1 is a standardized iid normalized and standardized with mean zero and variance 1 constructed to obey gt1 ut1 Kt1 at1 211 The spectrum For a covariance stationary stochastic process all second moments can be encoded in a complexvalued matrix called the spectral density matrix The autocovariance sequence for the process determines the spectral density Conversely the spectral density can be used to determine the autocovariance sequence Under the assumption that A is a stable matrix25 the state xt converges to a unique covariance stationary probability distribution as t approaches infinity 2108 gives Kt t 11 so that the Kalman filter becomes m0 y0 and mt 1 t 11 mt1 t 11 yt which implies that mt t 11 Σt s0 ys the sample mean and Σt Rt 25 It is sufficient that the only eigenvalue of A not strictly less than unity in modulus is that associated with the constant which implies that A and C fit together in a way that validates 2112 212 Example the LQ permanent income model To illustrate several of the key ideas of this chapter this section describes the linear quadratic savings problem whose solution is a rational expectations version of the permanent income model of Friedman 1956 and Hall 1978 We use this model as a vehicle for illustrating impulse response functions alternative notions of the state the idea of cointegration and an invariant subspace method The LQ permanent income model is a modification and not quite a special case for reasons that will be apparent later of the following savings problem to be studied in chapter 17 A consumer has preferences over consumption streams that are ordered by the utility functional E0 t0 βt uct 2121 where Et is the mathematical expectation conditioned on the consumers time t information ct is time t consumption uc is a strictly concave oneperiod utility function and β 01 is a discount factor The consumer maximizes 2121 by choosing a consumption borrowing plan ct bt1t0 subject to the sequence of budget constraints ct bt R1 bt1 yt 2122 where yt is an exogenous stationary endowment process R is a constant gross riskfree interest rate bt is oneperiod riskfree debt maturing at t and b0 is a given initial condition We shall assume that R1 β For example we might assume that the endowment process has the statespace representation zt1 A22 zt C2 wt1 2123a yt Uy zt 2123b where wt1 is an iid process with mean zero and identity contemporaneous covariance matrix A22 is a stable matrix its eigenvalues being strictly below unity in modulus and Uy is a selection vector that identifies y with a particular linear combination of the zt We impose the following condition on the consumption borrowing plan E0 t0 βt bt2 2124 This condition suffices to rule out Ponzi schemes The state vector confronting the household at t is bt zt where bt is its oneperiod debt falling due at the beginning of period t and zt contains all variables useful for forecasting its future endowment We impose this condition to rule out an alwaysborrow scheme that would allow the household to enjoy bliss consumption forever The rationale for imposing this condition is to make the solution resemble the solution of problems to be studied in chapter 17 that impose nonnegativity on the consumption path Firstorder conditions for maximizing 2121 subject to 2122 are29 Et uct1 uct t 0 2125 For the rest of this section we assume the quadratic utility function uct 5ct γ2 where γ is a bliss level of consumption Then 2125 implies30 Et ct1 ct 2126 29 We shall study how to derive this firstorder condition in detail in later chapters 30 A linear marginal utility is essential for deriving 2126 from 2125 Suppose instead that we had imposed the following more standard assumptions on the utility function uc 70 Time Series 0 10 20 30 1 05 0 05 1 15 impulse response 0 1 2 3 10 0 10 1 spectrum 15 10 5 0 5 10 15 1 0 1 2 3 4 covariogram 20 40 60 80 4 2 0 2 4 sample path Figure 2111 Impulse response spectrum covariogram and sample path of process 1 13L 7L2yt wt 0 10 20 30 0 02 04 06 08 1 impulse response 0 1 2 3 10 0 10 1 10 2 spectrum 15 10 5 0 5 10 15 15 2 25 3 35 4 45 5 covariogram 20 40 60 80 4 2 0 2 4 6 sample path Figure 2112 Impulse response spectrum covariogram and sample path of process 1 9Lyt wt 2111 Examples To give some practice in reading spectral densities we used the Matlab program bigshow3m to generate Figures 2112 2113 2111 and 2114 The program The spectrum 71 0 10 20 30 0 02 04 06 08 1 impulse response 0 1 2 3 10 0 10 1 spectrum 15 10 5 0 5 10 15 0 05 1 15 2 25 covariogram 20 40 60 80 4 3 2 1 0 1 2 3 sample path Figure 2113 Impulse response spectrum covariogram and sample path of process 1 8L4yt wt 0 10 20 30 0 02 04 06 08 1 impulse response 0 1 2 3 10 0 10 1 10 2 spectrum 15 10 5 0 5 10 15 22 24 26 28 3 32 covariogram 20 40 60 80 3 25 2 15 1 05 0 05 sample path Figure 2114 Impulse response spectrum covariogram and sample path of process 1 98Lyt 1 7Lwt takes as an input a univariate process of the form a L yt b L wt The spectral density matrix of this covariance stationary distribution Sxω is defined to be the Fourier transform of the covariogram of xt Sxω τ Cxτeiωτ 2111 For the system 241 the spectral density of the stationary distribution is given by the formula Sxω I Aeiω1 CCI Aeiω1 ω π π 2112 The spectral density summarizes all covariances They can be recovered from Sxω by the Fourier inversion formula26 Cxτ 12πππ Sxωeiωτ dω Setting τ 0 in the inversion formula gives Cx0 12πππ Sxωdω which shows that the spectral density decomposes covariance across frequencies27 A formula used in the process of generalized method of moments GMM estimation emerges by setting ω 0 in equation 2111 which gives Sx0 τ Cxτ 26 Spectral densities for continuoustime systems are discussed by Kwakernaak and Sivan 1972 For an elementary discussion of discretetime systems see Sargent 1987a Also see Sargent 1987a chap 11 for definitions of the spectral density function and methods of evaluating this integral 27 More interestingly the spectral density achieves a decomposition of covariance into components that are orthogonal across frequencies where wt is a univariate martingale difference sequence with unit variance where aL 1 a2 L a3 L2 an Ln1 and bL b1 b2 L bn Ln1 and where we require that az 0 imply that z 1 The program computes and displays a realization of the process the impulse response function from w to y and the spectrum of y By using this program a reader can teach himself to read spectra and impulse response functions Figure 2112 is for the pure autoregressive process with aL 1 9L b 1 The spectrum sweeps downward in what CWJ Granger 1966 called the typical spectral shape for an economic time series Figure 2113 sets a 1 8L4 b 1 This is a process with a strong seasonal component That the spectrum peaks at π and π2 is a telltale sign of a strong seasonal component Figure 2111 sets a 1 13L 7L2 b 1 This is a process that has a spectral peak in the interior of 0 π and cycles in its covariogram28 Figure 2114 sets a 1 98L b 1 7L This is a version of a process studied by Muth 1960 After the first lag the impulse response declines as 99j where j is the lag length Along with the quadratic utility specification we allow consumption ct to be negative To deduce the optimal decision rule we have to solve the system of difference equations formed by 2122 and 2126 subject to the boundary condition 2124 To accomplish this solve 2122 forward and impose limT βTbt1 0 to get bt j0 βj ytj ctj 2127 Imposing limT βTbt1 0 suffices to impose 2124 on the debt path Take conditional expectations on both sides of 2127 and use 2126 and the law of iterated expectations to deduce bt j0 βj Etytj 11β ct 2128 or ct 1β j0 βj Etytj bt 2129 If we define the net rate of interest r by β 11r we can also express this equation as ct r1r j0 βj Etytj bt 21210 Equation 2129 or 21210 expresses consumption as equaling economic income namely a constant marginal propensity to consume or interest factor r1r times the sum of nonfinancial wealth j0 βj Etytj and financial wealth bt Notice that 2129 or 21210 represents ct as a function of the state bt zt0 uc 0 uc 0 and required that c 0 The Euler equation remains 2125 But the fact that u 0 implies via Jensens inequality that Etuct1 uEtct1 This inequality together with 2125 implies that Etct1 ct consumption is said to be a submartingale so that consumption stochastically diverges to The consumers savings also diverge to Chapter 17 discusses this precautionary savings divergence result in depth That ct can be negative explains why we impose condition 2124 instead of an upper bound on the level of borrowing such as the natural borrowing limit of chapters 8 17 and 18 confronting the household where from 2123 zt contains the information useful for forecasting the endowment process 2121 Another representation Pulling together our preceding results we can regard zt bt as the time t state where zt is an exogenous component of the state and bt is an endogenous component of the state vector The system can be represented as zt1 A22zt C2wt1 bt1 bt Uy IβA221 A22 I zt yt Uyzt ct 1β Uy IβA221 zt bt Another way to understand the solution is to show that after the optimal decision rule has been obtained there is a point of view that allows us to regard the state as being ct together with zt and to regard bt as an outcome Following Hall 1978 this is a sharp way to summarize the implication of the LQ permanent income theory We now proceed to transform the state vector in this way To represent the solution for bt substitute 2129 into 2122 and after rearranging obtain bt1 bt β1 1 j0 βj Etytj β1 yt 21211 Next shift 2129 forward one period and eliminate bt1 by using 2122 to obtain ct1 1β j0 Et1βj ytj1 1β β1ct bt yt If we add and subtract β11β j0 βj Etytj from the right side of the preceding equation and rearrange we obtain ct1 ct 1β j0 βjEt1ytj1 Etytj1 21212 The right side is the time t 1 innovation to the expected present value of the endowment process y It is useful to express this innovation in terms of a moving average representation for income yt Suppose that the endowment process has the moving average representation32 yt1 dL wt1 21213 where wt1 is an iid vector process with Ewt1 0 and contemporaneous covariance matrix Ewt1 wt1 I dL j0 dj Lj where L is the lag operator and the household has an information set33 wt wt wt1 at time t Then notice that ytj Etytj d0 wtj d1 wtj1 dj1 wt1 It follows that Et1ytj Etytj dj1 wt1 21214 Using 21214 in 21212 gives ct1 ct 1β dβ wt1 21215 The object dβ is the present value of the moving average coefficients in the representation for the endowment process yt After all of this work we can represent the optimal decision rule for ct bt1 in the form of the two equations 21212 and 2128 which we repeat here for convenience ct1 ct 1β j0 βj Et1ytj1 Etytj1 21216 bt j0 βj Etytj 11β ct 21217 Equation 21217 asserts that the households debt due at t equals the expected present value of its endowment minus the expected present value of its 32 Representation 2123 implies that dL Uy I A22L1 C2 33 A moving average representation for a process yt is said to be fundamental if the linear space spanned by yt is equal to the linear space spanned by wt A timeinvariant innovations representation attained via the Kalman filter as in section 27 is by construction fundamental consumption stream A high debt thus indicates a large expected present value of surpluses yt ct Recalling the form of the endowment process 2123 we can compute Et βj ztj I βA221 zt Et1 βj ztj1 I βA221 zt1 Et βj ztj1 I βA221 A22zt Substituting these formulas into 21216 and 21217 and using 2123a gives the following representation for the consumers optimum decision rule ct1 ct 1 β Uy I βA221 C2wt1 bt Uy I βA221 zt 11β ct yt Uyz t zt1 A22zt C2wt1 Representation 21218 reveals several things about the optimal decision rule 1 The state consists of the endogenous part ct and the exogenous part zt These contain all of the relevant information for forecasting future c y b Notice that financial assets bt have disappeared as a component of the state because they are properly encoded in ct 2 According to 21218 consumption is a random walk with innovation 1 βdβwt1 as implied also by 21215 This outcome confirms that the Euler equation 2126 is built into the solution That consumption is a random walk of course implies that it does not possess an asymptotic stationary distribution at least so long as zt exhibits perpetual random fluctuations as it will generally under 2123 This feature is inherited partly from the assumption that βR 1 3 The impulse 34 See appendix A of chapter 17 for a reinterpretation of precisely these outcomes in terms of a competitive equilibrium of a model with a complete set of markets in history and datecontingent claims to consumption 35 The failure of consumption to converge will occur again in chapter 17 when we drop quadratic utility and assume that consumption must be nonnegative response function of ct is a box for all j 1 the response of ctj to an increase in the innovation wt1 is 1 βdβ 1 βUy I βA221 C2 4 Solution 21218 reveals that the joint process ct bt possesses the property that Granger and Engle 1987 called cointegration In particular both ct and bt are nonstationary because they have unit roots see representation 21211 for bt but there is a linear combination of ct bt that is stationary provided that zt is stationary From 21217 the linear combination is 1 βbt ct Accordingly Granger and Engle would call 1 β 1 a cointegrating vector that when applied to the nonstationary vector process bt ct yields a process that is asymptotically stationary Equation 2128 can be arranged to take the form 1 β bt ct 1 β Et βj ytj which asserts that the cointegrating residual on the left side equals the conditional expectation of the geometric sum of future incomes on the right 36 2122 Debt dynamics If we subtract equation 21218b evaluated at time t from equation 21218b evaluated at time t 1 we obtain bt1 bt Uy I βA221 zt1 zt 11β ct1 ct Substituting zt1 zt A22 Izt C2wt1 and equation 21218a into the above equation and rearranging gives bt1 bt Uy I βA221 A22 I zt 36 See Campbell and Shiller 1988 and Lettau and Ludvigson 2001 2004 for interesting applications of related ideas 2123 Two classic examples We illustrate formulas 21218 with the following two examples In both examples the endowment follows the process yt z1t z2t where z1t1 z2t1 1 0 0 0 z1t z2t σ1 0 0 σ2 w1t1 w2t1 where wt1 is an iid 2 x 1 process distributed as N 0 I Here z1t is a permanent component of yt while z2t is a purely transitory component Example 1 Assume that the consumer observes the state zt at time t This implies that the consumer can construct wt1 from observations of zt1 and zt Application of formulas 21218 implies that ct1 ct σ1 w1t1 1 β σ2 w2t1 Since 1 β r1r where R 1r formula 21221 shows how an increment σ1 w1t1 to the permanent component of income z1t1 leads to a permanent oneforone increase in consumption and no increase in savings bt1 but how the purely transitory component of income σ2 w2t1 leads to a permanent increment in consumption by a fraction 1 β of transitory income while the remaining fraction β is saved leading to a permanent increment in b Application of formula 21220 to this example shows that bt1 bt z2t σ2 w2t which confirms that none of σ1 w1t is saved while all of σ2 w2t is saved Example 2 Assume that the consumer observes yt and its history up to t but not zt at time t Under this assumption it is appropriate to use an innovation representation to form A22 C2 Uy in formulas 21218 In particular using our results from section 2101 the pertinent state space representation for yt is yt1 at1 1 1K 0 0 yt at 1 1 at1 yt 1 0 yt at where K is the Kalman gain and at yt Eytyt1 From subsection 2101 we know that K 01 and that K increases as σ12σ22 increases ie as the ratio 80 Time Series of the variance of the permanent shock to the variance of the transitory shock to income increases Applying formulas 21218 implies ct1 ct 1 β 1 K at1 21223 where the endowment process can now be represented in terms of the univariate innovation to yt as yt1 yt at1 1 K at 21224 Equation 21224 indicates that the consumer regards a fraction K of an innovation at1 to yt1 as permanent and a fraction 1K as purely transitory He permanently increases his consumption by the full amount of his estimate of the permanent part of at1 but by only 1 β times his estimate of the purely transitory part of at1 Therefore in total he permanently increments his consumption by a fraction K 1 β1 K 1 β1 K of at1 and saves the remaining fraction β1 K of at1 According to equation 21224 the first difference of income is a firstorder moving average while 21223 asserts that the first difference of consumption is iid Application of formula 21220 to this example shows that bt1 bt K 1 at 21225 which indicates how the fraction K of the innovation to yt that is regarded as permanent influences the fraction of the innovation that is saved 2124 Spreading consumption cross section Starting from an arbitrary initial distribution for c0 and say the asymptotic stationary distribution for z0 if we were to apply formulas 2411 and 2412 to the state space system 21218 the common unit root affecting ct bt would cause the time t variance of ct to grow linearly with t If we think of the initial distribution as describing the joint distribution of c0 b0 for a cross section of ex ante identical households born at time 0 then these formulas would describe the evolution of the crosssection for bt ct as the population of households ages The distribution would spread out37 37 See Deaton and Paxton 1994 and Storesletten Telmer and Yaron 2004 for evidence that cross section distributions of consumption spread out with age 2125 Invariant subspace approach We can glean additional insights about the structure of the optimal decision rule by solving the decision problem in a mechanical but quite revealing way that easily generalizes to a host of problems as we shall see later in chapter 5 We can represent the system consisting of the Euler equation 2126 the budget constraint 2122 and the description of the endowment process 2123 as β 0 0 0 I 0 0 0 1 bt1 zt1 ct1 1 Uy 1 0 A22 0 0 0 1 bt zt ct 0 C2 C1 wt1 21226 where C1 is an undetermined coefficient Premultiply both sides by the inverse of the matrix on the left and write bt1 zt1 ct1 à bt zt ct Cwt1 21227 We want to find solutions of 21227 that satisfy the noexplosion condition 2124 We can do this by using machinery to be introduced in chapter 5 The key idea is to discover what part of the vector bt zt ct is truly a state from the view of the decision maker being inherited from the past and what part is a costate or jump variable that can adjust at t For our problem bt zt are truly components of the state but ct is free to adjust The theory determines ct at t as a function of the true state variables bt zt A powerful approach to determining this function is the following socalled invariant subspace method of chapter 5 Obtain the eigenvector decomposition of à à VΛV1 where Λ is a diagonal matrix consisting of the eigenvalues of à and V is a matrix of the associated eigenvectors Let V1 V11 V12 V21 V22 Then applying formula 5512 of chapter 5 implies that if 2124 is to hold the jump variable ct must satisfy ct V221 V21 bt zt 21228 Formula 21228 gives the unique value of ct that ensures that 2124 is satisfied or in other words that the state remains in the stabilizing subspace 82 Time Series Notice that the variables on the right side of 21228 conform with those called for by 21210 bt is there as a measure of financial wealth and zt is there because it includes all variables that are useful for forecasting the future endowments that appear in 21210 213 Concluding remarks In addition to giving us tools for thinking about time series the Markov chain and the stochastic linear difference equation have each introduced us to the notion of the state vector as a description of the present position of a system38 Subsequent chapters use both Markov chains and stochastic linear difference equations In the next chapter we study decision problems in which the goal is optimally to manage the evolution of a state vector that can be partially controlled A Linear difference equations 2A1 A firstorder difference equation This section describes the solution of a linear firstorder scalar difference equa tion First let λ 1 and let ut t be a bounded sequence of scalar real numbers Let L be the lag operator defined by Lxt xt1 and let L1 be the forward shift operator defined by L1xt xt1 Then 1 λL yt ut t 2A1 has the solution yt 1 λL1 ut kλt 2A2 38 See Quah 1990 and Blundell and Preston 1998 for applications of some of the tools of this chapter and of chapter 5 to studying some puzzles associated with a permanent income model for any real number k You can verify this fact by applying 1λL to both sides of equation 2A2 and noting that 1λLλt0 To pin down k we need one condition imposed from outside eg an initial or terminal condition on the path of y Now let λ 1 Rewrite equation 2A1 as yt1 λ1 yt λ1 ut t 2A3 or 1λ1 L1 yt λ1 ut1 2A4 A solution is yt λ1 1 1λ1L1 ut1 k λt 2A5 for any k To verify that this is a solution check the consequences of operating on both sides of equation 2A5 by 1λL and compare to 2A1 Solution 2A2 exists for λ 1 because the distributed lag in u converges Solution 2A5 exists when λ 1 because the distributed lead in u converges When λ 1 the distributed lag in u in 2A2 may diverge so that a solution of this form does not exist The distributed lead in u in 2A5 need not converge when λ 1 2A2 A secondorder difference equation Now consider the second order difference equation 1λ1 L1λ2 L yt1 ut 2A6 where ut is a bounded sequence y0 is an initial condition λ1 1 and λ2 1 We seek a bounded sequence ytt0 that satisfies 2A6 Using insights from the previous subsection operate on both sides of 2A6 by the forward inverse of 1λ2 L to rewrite equation 2A6 as 1λ1 L yt1 λ211λ21L1 ut1 or yt1 λ1 yt λ21 j0 λ21j utj1 2A7 84 Time Series Thus we obtained equation 2A7 by solving stable roots in this case λ1 backward and unstable roots in this case λ2 forward Equation 2A7 has a form that we shall encounter often λ1yt is called the feedback part and λ1 2 1λ1 2 L1 ut1 is called the feedforward part of the solution We have al ready encountered solutions of this form Thus notice that equation 21220 from subsection 2122 is almost of this form almost because in equation 21220 λ1 1 In section 55 of chapter 5 we return to these ideas in a more general setting Exercises Exercise 21 Consider the Markov chain P π0 9 1 3 7 5 5 and a random variable yt y xt where y 1 5 Compute the likelihood of the following three histories for yt for t014 a 15151 b 11111 c 55555 Exercise 22 Consider a twostate Markov chain Consider a random variable yt y xt where y 1 5 It is known that Eyt1xt 18 34 and that Eyt12xt 58 154 Find a transition matrix consistent with these conditional expectations Is this transition matrix unique ie can you find another one that is consistent with these conditional expectations Exercise 23 Consumption is governed by an nstate Markov chain P π0 where P is a stochastic matrix and π0 is an initial probability distribution Consumption takes one of the values in the n 1 vector c A consumer ranks stochastic processes of consumption t01 according to E t0 βt uct where E is the mathematical expectation and uc c1γ 1γ for some parameter γ 1 Let ui uci Let vi Et0 βt uct x0 ei and V Ev where β 0 1 is a discount factor a Let u and v be the n 1 vectors whose ith components are ui and vi respectively Verify the following formulas for v and V v IβP1 u and V i π0i vi b Consider the following two Markov processes Process 1 π0 5 5 P 1 0 0 1 Process 2 π0 5 5 P 5 5 5 5 For both Markov processes c 15 Assume that γ 25 β 95 Compute the unconditional discounted expected utility V for each of these processes Which of the two processes does the consumer prefer Redo the calculations for γ 4 Now which process does the consumer prefer c An econometrician observes a sample of 10 observations of consumption rates for our consumer He knows that one of the two preceding Markov processes generates the data but he does not know which one He assigns equal prior probability to the two chains Suppose that the 10 successive observations on consumption are as follows 1111111111 Compute the likelihood of this sample under process 1 and under process 2 Denote the likelihood function ProbdataModeli i1 2 d Suppose that the econometrician uses Bayes law to revise his initial probability estimates for the two models where in this context Bayes law states ProbMi data ProbdataMi ProbMi Σj ProbdataMj ProbMj where Mi denotes model i The denominator of this expression is the unconditional probability of the data After observing the data sample what probabilities does the econometrician place on the two possible models e Repeat the calculation in part d but now assume that the data sample is 1551551515 Exercise 24 Consider the univariate stochastic process yt1 α Σj14 ρj yt1j cwt1 where wt1 is a scalar martingale difference sequence adapted to Jt wt w1 y0 y1 y2 y3 α μ1 Σj ρj and the ρjs are such that the matrix A ρ1 ρ2 ρ3 ρ4 α 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 has all of its eigenvalues in modulus bounded below unity a Show how to map this process into a firstorder linear stochastic difference equation b For each of the following examples if possible assume that the initial conditions are such that yt is covariance stationary For each case state the appropriate initial conditions Then compute the covariance stationary mean and variance of yt assuming the following parameter sets of parameter values i ρ 12 3 0 0 μ 10 c 1 ii ρ 12 3 0 0 μ 10 c 2 iii ρ 9 0 0 0 μ 5 c 1 iv ρ 2 0 0 5 μ 5 c 1 v ρ 8 3 0 0 μ 5 c 1 Hint 1 The Matlab command XdoublejACC computes the solution of the matrix equation AXA CC X 39 Hint 2 The mean vector is the eigenvector of A associated with a unit eigenvalue scaled so that the mean of unity in the state vector is unity c For each case in part b compute the hjs in Et yt5 γ0 Σj03 hj ytj d For each case in part b compute the ĥjs in Et Σk0 95k ytk Σj03 ĥj ytj e For each case in part b compute the autocovariance Eyt μyytk μy for the three values k 1 5 10 Exercise 25 A consumers rate of consumption follows the stochastic process ct1 αc Σj12 ρj ctj1 Σj12 δj zt1j ψ1 w1t1 zt1 Σj12 γj ctj1 Σj12 φj ztj1 ψ2 w2t1 39 Matlab code for this book is at wwwtomsargentcomsourcecodemitbookzip where wt1 is a 2 x 1 martingale difference sequence adapted to Jt wt w1 c0 c1 z0 z1 with contemporaneous covariance matrix E wt1 wt1 Jt I and the coefficients ρj δj γj φj are such that the matrix A ρ1 ρ2 δ1 δ2 αc 1 0 0 0 0 γ1 γ2 φ1 φ2 0 0 0 1 0 0 0 0 0 0 1 has eigenvalues bounded strictly below unity in modulus The consumer evaluates consumption streams according to V0 E0 Σt0 95t uct where the oneperiod utility function is uct 5 ct 602 a Find a formula for V0 in terms of the parameters of the oneperiod utility function 3 and the stochastic process for consumption b Compute V0 for the following two sets of parameter values i ρ 8 3 αc 1 δ 2 0 γ 0 0 φ 7 2 ψ1 ψ2 1 ii Same as for part i except now ψ1 2 ψ2 1 Hint Remember doublejm Exercise 26 Consider the stochastic process ct zt defined by equations 1 in exercise 25 Assume the parameter values described in part b item i If possible assume the initial conditions are such that ct zt is covariance stationary a Compute the initial mean and covariance matrix that make the process covariance stationary b For the initial conditions in part a compute numerical values of the following population linear regression ct2 α0 α1 zt α2 zt4 wt Exercises 89 where Ewt 1 zt zt4 0 0 0 Exercise 27 Get the Matlab programs bigshow3m and freqm from wwwtomsargentcomsource codemitbookzip Use bigshow3 to compute and display a simulation of length 80 an impulse response function and a spectrum for each of the following scalar stochastic processes yt In each of the following wt is a scalar martingale difference sequence adapted to its own history and the initial values of lagged ys a yt wt b yt 1 5Lwt c yt 1 5L 4L2wt d 1 999Lyt 1 4Lwt e 1 8Lyt 1 5L 4L2wt f 1 8Lyt wt g yt 1 6Lwt Study the output and look for patterns When you are done you will be well on your way to knowing how to read spectral densities Exercise 28 This exercise deals with Cagans money demand under rational expectations A version of Cagans 1956 demand function for money is 1 mt pt α pt1 pt α 0 t 0 where mt is the log of the nominal money supply and pt is the price level at t Equation 1 states that the demand for real balances varies inversely with the expected rate of inflation pt1 pt There is no uncertainty so the expected inflation rate equals the actual one The money supply obeys the difference equation 2 1 L 1 ρL ms t 0 subject to initial condition for ms 1 ms 2 In equilibrium 3 mt ms t t 0 ie the demand for money equals the supply For now assume that ρ α 1 α 1 An equilibrium is a ptt0 that satisfies equations 1 2 and 3 for all t a Find an expression for an equilibrium pt of the form pt Σj0n wj mtj ft Please tell how to get formulas for the wj for all j and the ft for all t b How many equilibria are there c Is there an equilibrium with ft 0 for all t d Briefly tell where if anywhere condition 4 plays a role in your answer to part a e For the parameter values α 1 ρ 1 compute and display all the equilibria Exercise 29 The n 1 state vector of an economy is governed by the linear stochastic difference equation xt1 A xt Ct wt1 where Ct is a possibly timevarying matrix known at t and wt1 is an m 1 martingale difference sequence adapted to its own history with E wt1 wt1Jt I where Jt wt w1 x0 A scalar oneperiod payoff pt1 is given by pt1 P xt1 The stochastic discount factor for this economy is a scalar mt1 that obeys mt1 M xt1 M xt Finally the price at time t of the oneperiod payoff is given by qt ftxt where ft is some possibly timevarying function of the state That mt1 is a stochastic discount factor means that E mt1 pt1Jt qt a Compute ftxt describing in detail how it depends on A and Ct b Suppose that an econometrician has a time series data set Xt zt mt1 pt1 qt for t 1 T where zt is a strict subset of the variables in the state xt Assume that investors in the economy see xt even though the econometrician sees only a subset zt of xt Briefly describe a way to use these data to test implication 4 Possibly but perhaps not useful hint recall the law of iterated expectations Exercise 210 Let P be a transition matrix for a Markov chain Suppose that P has two distinct eigenvectors π1 π2 corresponding to unit eigenvalues of P Scale π1 and π2 so that they are vectors of probabilities ie elements are nonnegative and sum to unity Prove for any α 0 1 that α π1 1 α π2 is an invariant distribution of P Exercise 211 Consider a Markov chain with transition matrix P 1 0 0 2 5 3 0 0 1 with initial distribution π0 π10 π20 π30 Let πt π1t π2t π3t be the distribution over states at time t Prove that for t 0 π1t π10 2 1 5t1 5 π20 π2t 5t π20 π3t π30 3 1 5t1 5 π20 Exercise 212 Let P be a transition matrix for a Markov chain For t 1 2 prove that the j th column of Pt is the distribution across states at t when the initial distribution is πj0 1 πi0 0 i j Exercise 213 A household has preferences over consumption processes ctt0 that are ordered by 5 Σt0 βt ct 302 000001 bt2 where β 95 The household chooses a consumption borrowing plan to maximize 1 subject to the sequence of budget constraints ct bt β bt1 yt for t 0 where b0 is an initial condition β1 is the oneperiod gross riskfree interest rate bt is the households oneperiod debt that is due in period t and yt is its labor income which obeys the secondorder autoregressive process 1 ρ1 L ρ2 L2 yt1 1 ρ1 ρ2 5 05 wt1 where ρ1 13 ρ2 4 a Define the state of the household at t as xt 1 bt yt yt1 and the control as ut ct 30 Then express the transition law facing the household in the form 2422 Compute the eigenvalues of A Compute the zeros of the characteristic polynomial 1 ρ1 z ρ2 z2 and compare them with the eigenvalues of A Hint To compute the zeros in Matlab set a 4 13 1 and call rootsa The zeros of 1 ρ1 z ρ2 z2 equal the reciprocals of the eigenvalues of the associated A b Write a Matlab program that uses the Howard improvement algorithm 2430 to compute the households optimal decision rule for ut ct 30 Tell how many iterations it takes for this to converge also tell your convergence criterion c Use the households optimal decision rule to compute the law of motion for xt under the optimal decision rule in the form xt1 A B F xt C wt1 where ut F xt is the optimal decision rule Using Matlab compute the impulse response function of ct bt to wt1 Compare these with the theoretical expressions 21218 Exercise 214 Consider a Markov chain with transition matrix P 5 5 0 0 1 9 0 0 0 0 9 1 0 0 0 1 Exercises 95 f Compute maximum likelihood estimators of λ and δ g Compare the estimators you derived in parts e and f h Extra credit Compute the asymptotic covariance matrix of the maximum likelihood estimators of λ and δ Exercise 218 Random walk A Markov chain has state space X ei i 1 4 where ei is the unit vector and transition matrix P 1 0 0 0 5 0 5 0 0 5 0 5 0 0 0 1 A random variable yt yxt is defined by y 1 2 3 4 a Find all stationary distributions of this Markov chain b Under what stationary distributions if any is this chain ergodic Compute invariant functions of P c Compute Eyt1xt for xt ei i 1 4 d Compare your answer to part c with 2212 Is yt yxt invariant If not what hypothesis of Theorem 224 is violated e The stochastic process yt yxt is evidently a bounded martingale Verify that yt converges almost surely to a constant To what constants does it converge Exercise 219 IQ An infinitely lived persons true intelligence θ N100 100 ie mean 100 variance 100 For each date t 0 the person takes a test with the outcome being a univariate random variable yt θ vt where vt is an iid process with distribution N0 100 The persons initial IQ is IQ0 100 and at date t 1 before the date t test is taken it is IQt Eθyt1 where yt1 is the history of test scores from date 0 until date t 1 a Give a recursive formula for IQt and for EIQt θ2 Exercises 97 a Show how to select wt1 C and D so that Cwt1 and Dwt1 are mutually uncorrelated processes Also give an example in which Cwt1 and Dwt1 are correlated b Construct a recursive representation for ˆxt of the form ˆxt1 Aˆxt Ktat1 yt1 Gˆxt at1 where at1 yt1 Eyt1yt for t 0 and verify that Kt CD AΣtG DD GΣtG1 Σt1 A KtG Σt A KtG C KtD C KtD and Eat1a t1 GΣtGDD Hint apply the population regression formula Exercise 223 A monopolist learning and ergodicity A monopolist produces a quantity Qt of a single good in every period t 0 at zero cost At the beginning of each period t 0 before output price pt is observed the monopolist sets quantity Qt to maximize 1 Et1ptQt where pt satisfies the linear inverse demand curve 2 pt a bQt σpǫt where b 0 is a constant known to the firm ǫt is an iid scalar with distribu tion ǫt N0 1 and the constant in the inverse demand curve a is a scalar random variable unknown to the firm and whose unconditional distribution is a Nµa σ2 a where µa 0 is large relative to σa 0 Assume that the random variable a is independent of ǫt for all t Before the firm chooses Q0 it knows the unconditional distribution of a but not the realized value of a For each t 0 the firm wants to estimate a because it wants to make a good decision about output Qt At the end of each period t when it must set Qt1 the firm observes pt and also of course knows the value of Qt that it had set In 1 for t 1 Et1 denotes the mathematical expectation conditional on Exercises 101 a Find optimal decision rules for consumption for both consumers Prove that the consumers optimal decisions imply the following laws of motion for b1 t b2 t b1 t1 st 0 b1 t 25 b1 t1 st 1 b1 t 25 b2 t1 st 0 b2 t 25 b2 t1 st 1 b2 t 25 b Show that for each consumer ci t bi t are cointegrated c Verify that bi t1 is riskfree in the sense that conditional on information available at time t it is independent of news arriving at time t 1 d Verify that with the initial conditions b1 0 b2 0 0 the following two equalities obtain b1 t b2 t 0 t 1 c1 t c2 t 15 t 1 Use these conditions to interpret the decision rules that you have computed as describing a closed pure consumption loans economy in which consumers 1 and 2 borrow and lend with each other and in which the riskfree asset is a oneperiod IOU from one of the consumers to the other e Define the stochastic discount factor of consumer i as mi t1 βuci t1 uci t Show that the stochastic discount factors of consumer 1 and 2 are m1 t1 β 25 β1β γc1 t if st1 0 β 25 β1β γc1 t if st1 1 m2 t1 β 25 β1β γc2 t if st1 0 β 25 β1β γc2 t if st1 1 Are the stochastic discount factors of the two consumers equal f Verify that Etm1 t1 Etm2 t1 β 104 Time Series Please interpret V x ie please complete the sentence V x is the value of b For a given matrix F please guess a functional form for V x then describe an algorithm for solving the functional equation 1 for V x Please get as far as you can in computing V x c Consider the functional equation 2 U x max u r x u δβV Ax Bu where V x satisfies the Bellman equation 1 Further let Ux be attained by u Gx so that 2 U x r x Gx δβV A BG x Please interpret Ux as a value function d Please define a Markov perfect equilibrium for the sequence of problems solved by the sequence of decision makers who choose ut t0 e Please describe how to compute a Markov perfect equilibrium in this setting f Please compare your algorithm for computing a Markov perfect equilibrium with the Howard policy improvement algorithm g Let a0 at t0 Define a1 at t1 as the continuation of the sequence a0 Is a continuation of a Markov perfect equilibrium a Markov perfect equilib rium h Suppose instead that there is a dictator who at time 0 chooses ut t0 to maximize the time t 0 value of the criterion 0 Please write Bellman equations and tell how to solve them for an optimal plan for the time 0 dictator i Given x1 a time 1 dictator chooses ut t1 to maximize utility function 0 for time t Is a continuation of the time 0 dictators plan the time 1 dictators plan j Can you restrict δ 0 1 so that the time 0 dictators plan equals the outcome of the Markov perfect equilibrium that you described above 106 Dynamic Programming where again the maximization is subject to xt1 gxt ut with x0 given Of course we cannot possibly expect to know V x0 until after we have solved the problem but lets proceed on faith If we knew V x0 then the policy function h could be computed by solving for each x X the problem max u r x u βV x 314 where the maximization is subject to x gx u with x given and x denotes the state next period Thus we have exchanged the original problem of finding an infinite sequence of controls that maximizes expression 311 for the prob lem of finding the optimal value function V x and a function h that solves the continuum of maximum problems 314one maximum problem for each value of x This exchange doesnt look like progress but we shall see that it often is Our task has become jointly to solve for V x hx which are linked by the Bellman equation V x max u r x u βV g x u 315 The maximizer of the right side of equation 315 is a policy function hx that satisfies V x r x h x βV g x h x 316 Equation 315 or 316 is a functional equation to be solved for the pair of unknown functions V x hx Methods for solving the Bellman equation are based on mathematical struc tures that vary in their details depending on the precise nature of the functions r and g2 All of these structures contain versions of the following four findings Under various particular assumptions about r and g it turns out that 2 There are alternative sets of conditions that make the maximization 314 well behaved One set of conditions is as follows 1 r is concave and bounded and 2 the constraint set generated by g is convex and compact that is the set of xt1 xt xt1 gxt ut for admissible ut is convex and compact See Stokey Lucas and Prescott 1989 and Bertsekas 1976 for further details of convergence results See Benveniste and Scheinkman 1979 and Stokey Lucas and Prescott 1989 for the results on differentiability of the value function In Appendix A see Technical Appendixes we describe the mathematics for one standard set of assumptions about r g In chapter 5 we describe it for another set of assumptions about r g 110 Dynamic Programming This problem can be solved by hand using any of our three methods We begin with iteration on the Bellman equation Start with v0k 0 and solve the oneperiod problem choose c to maximize lnc subject to c k Akα The solution is evidently to set c Akα k 0 which produces an optimized value v1k ln A α ln k At the second step we find c 1 1βαAkα k βα 1βαAkα v2k ln A 1αβ β ln A αβ ln αβA 1αβ α1 αβ ln k Continuing and using the algebra of geometric series gives the limiting policy functions c 1βαAkα k βαAkα and the value function vk 1β1lnA1 βα βα 1βα lnAβα α 1βα ln k Here is how the guessandverify method applies to this problem Since we already know the answer well guess a function of the correct form but leave its coefficients undetermined8 Thus we make the guess v k E F ln k 3112 where E and F are undetermined constants The left and right sides of equation 3112 must agree for all values of k For this guess the firstorder necessary condition for the maximum problem on the right side of equation 3110 implies the following formula for the optimal policy k hk where k is next periods value and k is this periods value of the capital stock k βF 1 βF Akα 3113 Substitute equation 3113 into the Bellman equation and equate the result to the right side of equation 3112 Solving the resulting equation for E and F gives F α1 αβ and E 1 β1ln A1 αβ βα 1αβ ln Aβα It follows that k βαAkα 3114 Note that the term F α1 αβ can be interpreted as a geometric sum α1 αβ αβ2 Equation 3114 shows that the optimal policy is to have capital move according to the difference equation kt1 Aβαkα t or ln kt1 ln Aβα α ln kt That α is less than 1 implies that kt converges as t approaches infinity for any positive initial value k0 The stationary point is given by the solution of k Aβαkα or kα1 Aβα1 8 This is called the method of undetermined coefficients Chapter 4 Practical Dynamic Programming 41 The curse of dimensionality We often encounter problems where it is impossible to attain closed forms for iterating on the Bellman equation Then we have to adopt numerical approxi mations This chapter describes two popular methods for obtaining numerical approximations The first method replaces the original problem with another problem that forces the state vector to live on a finite and discrete grid of points then applies discretestate dynamic programming to this problem The curse of dimensionality impels us to keep the number of points in the discrete state space small The second approach uses polynomials to approximate the value function Judd 1998 is a comprehensive reference about numerical analysis of dynamic economic models and contains many insights about ways to compute dynamic models 42 Discretestate dynamic programming We introduce the method of discretization of the state space in the context of a particular discretestate version of an optimal savings problem An infinitely lived household likes to consume one good that it can acquire by spending labor income or accumulated savings The household has an endowment of labor at time t st that evolves according to an mstate Markov chain with transition matrix P and state space s1 s2 sm If the realization of the process at t is si then at time t the household receives labor income of amount wsi The wage w is fixed over time We shall sometimes assume that m is 2 and that st takes on value 0 in an unemployed state and 1 in an employed state In this case w has the interpretation of being the wage of employed workers The household can choose to hold a single asset in discrete amounts at A where A is a grid a1 a2 an How the model builder chooses the 115 Application of Howard improvement algorithm 119 The policy improvement algorithm consists of iterations on the following two steps 1 For fixed Pn solve I β Pn vPn cPn 443 for vPn 2 Find Pn1 such that cPn1 βPn1 I vPn BvPn 444 Step 1 is accomplished by setting vPn I βPn1 cPn 445 Step 2 amounts to finding a policy function ie a stochastic matrix Pn1 M that solves a twoperiod problem with vPn as the terminal value function Following Putterman and Brumelle the policy improvement algorithm can be interpreted as a version of Newtons method for finding the zero of Bv v Using equation 443 for n 1 to eliminate cPn1 from equation 444 gives I βPn1 vPn1 βPn1 I vPn BvPn which implies vPn1 vPn I βPn11 BvPn 446 From equation 444 βPn1 I can be regarded as the gradient of BvPn which supports the interpretation of equation 446 as implementing Newtons method4 4 Newtons method for finding the solution of Gz 0 is to iterate on zn1 zn Gzn1 Gzn 124 Practical Dynamic Programming assuming various values of γ that he judged to be within a reasonable range6 Lucas found that for reasonable values of γ it takes a very small adjustment in the trend rate of growth µ to compensate for even a substantial increase in the cyclical noise σz which meant to him that the costs of business cycle fluctuations are small Subsequent researchers have studied how other preference specifications would affect the calculated costs Tallarini 1996 2000 used a version of the preferences described in example 2 and found larger costs of business cycles when parameters are calibrated to match data on asset prices Hansen Sargent and Tallarini 1999 and Alvarez and Jermann 1999 considered local measures of the cost of business cycles and provided ways to link them to the equity premium puzzle to be studied in chapter 14 47 Polynomial approximations Judd 1998 describes a method for iterating on the Bellman equation using a polynomial to approximate the value function and a numerical optimizer to perform the optimization at each iteration We describe this method in the context of the Bellman equation for a particular problem that we shall encounter later In chapter 21 we shall study Hopenhayn and Nicolinis 1997 model of optimal unemployment insurance A planner wants to provide incentives to an unemployed worker to search for a new job while also partially insuring the worker against bad luck in the search process The planner seeks to deliver discounted expected utility V to an unemployed worker at minimum cost while providing proper incentives to search for work Hopenhayn and Nicolini show that the minimum cost CV satisfies the Bellman equation C V min V u c β 1 p a C V u 471 where c a are given by c u1 max 0 V a βp a V e 1 p a V u 472 6 See chapter 14 for a discussion of reasonable values of γ See Table 1 of Manuelli and Sargent 1988 for a correction to Lucass calculations Polynomial approximations 127 473 Algorithm summary In summary applied to the HopenhaynNicolini model the numerical procedure consists of the following steps 1 Choose upper and lower bounds for V u so that V and V u will be under stood to reside in the interval V u V u In particular set V u V e 1 βp0 the bound required to assure positive search effort computed in chapter 21 Set V u Vrmaut 2 Choose a degree n for the approximator a Chebyshev polynomial and a number m n 1 of nodes or grid points 3 Generate the m zeros of the Chebyshev polynomial on the set 1 1 given by 476 4 By a change of scale transform the zis to corresponding points V u ℓ in V u V u 5 Choose initial values of the n 1 coefficients in the Chebyshev polynomial for example cj 0 n Use these coefficients to define the function CiV u for iteration number i 0 6 Compute the function CiV c β1 paCiV u where c a are de termined as functions of V V u from equations 472 and 473 This computation builds in the functional forms and parameters of uc and pa as well as β 7 For each point V u ℓ use a numerical minimization program to find Ci1V u ℓ minV u CiVu 8 Using these m values of Cj1V u ℓ compute new values of the coefficients in the Chebyshev polynomials by using least squares formula 477 Return to step 5 and iterate to convergence 128 Practical Dynamic Programming 474 Shapepreserving splines Judd 1998 points out that because they do not preserve concavity using Chebyshev polynomials to approximate value functions can cause problems He recommends the Schumaker quadratic shapepreserving spline It ensures that the objective in the maximization step of iterating on a Bellman equation will be concave and differentiable Judd 1998 p 441 Using Schumaker splines avoids the type of internodal oscillations associated with other polynomial ap proximation methods The exact interpolation procedure is described in Judd 1998 p 233 A relatively small number of nodes usually is sufficient Judd and Solnick 1994 find that this approach outperforms linear interpolation and discretestate approximation methods in a deterministic optimal growth prob lem7 48 Concluding remarks This chapter has described two of three standard methods for approximating so lutions of dynamic programs numerically discretizing the state space and using polynomials to approximate the value function The next chapter describes the third method making the problem have a quadratic return function and linear transition law A benefit of making the restrictive linearquadratic assumptions is that they make solving a dynamic program easy by exploiting the ease with which stochastic linear difference equations can be manipulated 7 The Matlab program schumakerm written by Leonardo Rezende of the University of Illinois can be used to compute the spline Use the Matlab command ppval to evaluate the spline Chapter 5 Linear Quadratic Dynamic Programming 51 Introduction This chapter describes the class of dynamic programming problems in which the return function is quadratic and the transition function is linear This specification leads to the widely used optimal linear regulator problem for which the Bellman equation can be solved quickly using linear algebra We consider the special case in which the return function and transition function are both time invariant though the mathematics is almost identical when they are permitted to be deterministic functions of time After studying a recursive formulation and the associated Bellman equa tion in section 55 we analyze a Lagrangian formulation that provides useful insights about how Lagrange multipliers on transition laws relate to gradients of value functions These insights help us in chapter 19 when we study how the methods of this chapter apply to problems in which a Stackelberg leader chooses a sequence of actions to manipulate future decisions of a collection agents whose decisions depend on forecasts of the leaders decisions In that chapter we shall get a sharp characterization of the time inconsistency of a Stackelberg plan In section 56 we tell how the Kalman filtering problem from chapter 2 relates to the linearquadratic dynamic programming problem Suitably rein terpreted formulas that solve the optimal linear regulator are the Kalman filter 129 134 Linear Quadratic Dynamic Programming where ǫ is the realization of ǫt1 when xt x and where Eǫx 0 The preceding equation implies v x max u xRx uQu βE xAPAx xAPBu xAPCǫ uBPAx uBPBu uBPCǫ ǫCPAx ǫCPBu ǫCPCǫ βd Evaluating the expectations inside the braces and using Eǫx 0 gives v x max u xRx uQu βxAPAx β2xAPBu βuBPBu βEǫCPCǫ βd The firstorder condition for u is Q βBPB u βBPAx which implies equation 536 Using EǫCPCǫ tracePCC substituting equation 536 into the preceding expression for vx and using equation 534 gives P R βAPA β2APB Q βBPB1 BPA and d β 1 β1 trace PCC 531 Discussion of certainty equivalence The remarkable thing is that although through d the objective function 533 depends on CC the optimal decision rule ut Fxt is independent of CC This is the message of equation 536 and the discounted algebraic Riccati equation for P which are identical with the formulas derived earlier under certainty In other words the optimal decision rule ut hxt is indepen dent of the problems noise statistics4 The certainty equivalence principle is 4 Therefore in linear quadratic versions of the optimum savings problem there are no precautionary savings Compare outcomes from section 212 of chapter 2 and chapters 17 and 18 136 Linear Quadratic Dynamic Programming 541 Stability After substituting the optimal control ut Fxt into the law of motion xt1 Axt But we obtain the optimal closedloop system xt1 A BFxt This difference equation governs the evolution of xt under the optimal control The system is said to be stable if limt xt 0 starting from any initial x0 Rn Assume that the eigenvalues of A BF are distinct and use the eigenvalue decomposition A BF DΛD1 where the columns of D are the eigenvectors of A BF and Λ is a diagonal matrix of eigenvalues of ABF Write the closedloop equation as xt1 DΛD1xt The solution of this difference equation for t 0 is readily verified by repeated substitution to be xt DΛtD1x0 Evidently the system is stable for all x0 Rn if and only if the eigenvalues of A BF are all strictly less than unity in absolute value When this condition is met A BF is said to be a stable matrix6 A vast literature is devoted to characterizing the conditions on A B R and Q that imply that F is such that the optimal closedloop system matrix A BF is stable These conditions are surveyed by Anderson Hansen McGrattan and Sargent 1996 and can be briefly described here for the undiscounted case β 1 Roughly speaking the conditions on A B R and Q are as follows First A and B must be such that it is possible to pick a control law ut Fxt that drives xt to zero eventually starting from any x0 Rn the pair A B must be stabilizable Second the matrix R must be such that it is desirable to drive xt to zero as t It would take us too far afield to go deeply into this body of theory but we can give a flavor of the results by considering the following special assump tions and their implications Similar results can obtain under weaker conditions relevant for economic problems7 Assumption A1 The matrix R is positive definite There immediately follows Proposition 1 Under assumption A1 if a solution to the undiscounted reg ulator exists it satisfies limt xt 0 6 It is possible to amend the statements about stability in this section to permit A BF to have a single unit eigenvalue associated with a constant in the state vector See chapter 2 for examples 7 See Kwakernaak and Sivan 1972 and Anderson Hansen McGrattan and Sargent 1996 for much weaker conditions 142 Linear Quadratic Dynamic Programming split half inside and half outside the unit circle13 Systems in which eigenvalues properly adjusted for discounting fail to occur in reciprocal pairs arise when the system being solved is an equilibrium of a model in which there are distortions that prevent there being any optimum problem that the equilibrium solves See Woodford 1999 for an application of such methods to solve for linear approximations of equilibria of a monetary model with distortions See chapter 11 for some applications to an economy with distorting taxes 56 The Kalman filter again Suitably reinterpreted the same recursion 527 that solves the optimal linear regulator also determines the celebrated Kalman filter that we derived in section 27 of chapter 2 Recall that the Kalman filter is a recursive algorithm for computing the mathematical expectation Extyt1 y0 of a hidden state vector xt conditional on observing a history yt y0 of a vector of noisy signals on the hidden state The Kalman filter can be used to formulate or simplify a variety of signalextraction and prediction problems in economics We briefly remind the reader that the setting for the Kalman filter is the following linear statespace system14 Given x0 Nˆx0 Σ0 let xt1 Axt Cwt1 561a yt Gxt vt 561b where xt is an n 1 state vector wt is an iid sequence Gaussian vector with Ewtw t I and vt is an iid Gaussian vector orthogonal to ws for all t s with Evtv t R and A C and G are matrices conformable to the vectors they multiply Assume that the initial condition x0 is unobserved but is known to have a Gaussian distribution with mean ˆx0 and covariance matrix Σ0 At time t the history of observations yt yt y0 is available to estimate the location of xt and the location of xt1 The Kalman filter is a recursive algorithm for computing ˆxt1 Ext1yt The algorithm is ˆxt1 A KtG ˆxt Ktyt 562 13 See Whiteman 1983 Blanchard and Kahn 1980 and Anderson Hansen McGrattan and Sargent 1996 for applications and developments of these methods 14 We derived the Kalman filter as a recursive application of population regression in chapter 2 page 57 The Kalman filter again 143 where Kt AΣtG GΣtG R1 563a Σt1 AΣtA CC AΣtG GΣtG R1 GΣtA 563b Here Σt Ext ˆxtxt ˆxt and Kt is called the Kalman gain Sometimes the Kalman filter is written in terms of the innovation representation ˆxt1 Aˆxt Ktat 564a yt Gˆxt at 564b where at yt Gˆxt yt Eytyt1 The random vector at is called the innovation in yt being the part of yt that cannot be forecast linearly from its own past Subtracting equation 564b from 561b gives at Gxtˆxtvt multiplying each side by its own transpose and taking expectations gives the following formula for the innovation covariance matrix Eata t GΣtG R 565 Equations 563 display extensive similarities to equations 527 the recursions for the optimal linear regulator Indeed the mathematical structures are identical when viewed properly Note that equation 563b is a Riccati equation With the judicious use of matrix transposition and reversal of time the two systems of equations 563 and 527 can be made to match15 See chapter 2 especially section 210 for some applications of the Kalman filter16 15 See Hansen and Sargent ch 4 2008 for an account of how the LQ dynamic programming problem and the Kalman filter are connected through duality That chapter formulates the Kalman filtering problem in terms of a Lagrangian then judiciously transforms the firstorder conditions into an associated optimal linear regulator 16 The Matlab program kfilterm computes the Kalman filter Matlab has several pro grams that compute the Kalman filter for discrete time and continuous time models 144 Linear Quadratic Dynamic Programming 57 Concluding remarks In exchange for their restrictions the linear quadratic dynamic optimization problems of this chapter acquire tractability The Bellman equation leads to Riccati difference equations that are so easy to solve numerically that the curse of dimensionality loses most of its force It is easy to solve linear quadratic control or filtering with many state variables That it is difficult to solve those problems otherwise is why linear quadratic approximations are widely used In chapter 7 we go beyond the singleagent optimization problems of this chapter to study systems with multiple agents who simultaneously solve lin ear quadratic dynamic programming problems with the decision rules of some agents influencing transition laws of variables appearing in other agents decision problems We introduce two related equilibrium concepts to reconcile different agents decisions A Matrix formulas Let z x a each be n 1 vectors A C D and V each be n n matrices B an m n matrix and y an m 1 vector Then ax x a xAx x A Ax 2xAx xx A A xAx A xx yBz y Bz yBz z By yBz B yz The equation AV A C V to be solved for V is called a discrete Lyapunov equation and its generalization AV D C V is called the discrete Sylvester equation The discrete Sylvester equation has a unique solution if and only if the eigenvalues λi of A and δj of D satisfy the condition λiδj 1 i j 148 Linear Quadratic Dynamic Programming demand function for currency expressing real balances as an inverse function of the expected gross rate of inflation Speaking of Cagan the government is running a permanent real deficit of g per period measured in goods all of which it finances by currency creation The governments budget constraint at t is 2 Mt Mt1 pt g where the left side is the real value of the new currency printed at time t The economy starts at time t 0 with the initial level of nominal currency stock M1 100 being given For this model define an equilibrium as a pair of positive sequences pt 0 Mt 0 t0 that satisfy equations 1 and 2 portfolio balance and the government budget constraint respectively for t 0 and the initial condition assigned for M1 a Let γ1 100 γ2 50 g 05 Write a computer program to compute equilibria for this economy Describe your approach and display the program b Argue that there exists a continuum of equilibria Find the lowest value of the initial price level p0 for which there exists an equilibrium Hint 1 Notice the positivity condition that is part of the definition of equilibrium Hint 2 Try using the general approach to solving difference equations described in section 55 c Show that for all of these equilibria except the one that is associated with the minimal p0 that you calculated in part b the gross inflation rate and the gross money creation rate both eventually converge to the same value Compute this value d Show that there is a unique equilibrium with a lower inflation rate than the one that you computed in part b Compute this inflation rate e Increase the level of g to 075 Compare the eventual or asymptotic infla tion rate that you computed in part b and the inflation rate that you computed in part c Are your results consistent with the view that larger permanent deficits cause larger inflation rates f Discuss your results from the standpoint of the Laffer curve Hint A Matlab program dlqrmonm performs the calculations It is available from the web site for the book Exercises 155 subject to at1 ct Rat yt yt1 1 ρ1 ρ2 ρ1yt ρ2yt3 σyǫt1 where ct is consumption b 0 is a bliss level of consumption at is financial assets at the beginning of t R β1 is the gross rate of return on assets held from t to t 1 and ǫt1 is an iid scalar process with ǫt1 N0 1 The household faces known initial conditions a0 y0 y1 y2 y3 a Write a Bellman equation for the households problem b Compute the households value function and optimal decision rule for the following parameter values b 1000 β 95 R β1 ρ1 55 ρ2 3 σy 05 ǫ 000001 c Compute the eigenvalues of A BF d Compute the households value function and optimal decision rule for the following parameter values b 1000 β 95 R β1 ρ1 55 ρ2 3 σy 05 ǫ 0 Compare what you obtain with your answers in part b Chapter 6 Search and Unemployment 61 Introduction This chapter applies dynamic programming to a choice between two actions to accept or reject a takeitorleaveit job offer An unemployed worker faces a probability distribution of wage offers or job characteristics from which a limited number of offers are drawn each period Given his perception of the probability distribution of offers the worker must devise a strategy for deciding when to accept an offer The theory of search is a tool for studying unemployment Search theory puts unemployed workers in a setting where they sometimes choose to reject available offers and to remain unemployed now because they prefer to wait for better offers later We use the theory to study how workers respond to variations in the rate of unemployment compensation the perceived riskiness of wage distributions the probability of being fired the quality of information about jobs and the frequency with which a wage distribution can be sampled This chapter provides an introduction to the techniques used in the search literature and a sampling of search models The chapter studies ideas intro duced in two important papers by McCall 1970 and Jovanovic 1979a These papers differ in the search technologies with which they confront an unemployed worker1 We also study a related model of occupational choice by Neal 1999 We hope to convey some of the excitement that Robert E Lucas Jr 1987 p57 expressed when he wrote this about the McCall search model Question ing a McCall worker is like having a conversation with an outofwork friend Maybe you are setting your sights too high or Why did you quit your old job before you had a new one lined up This is real social science an attempt to model to understand human behavior by visualizing the situations people find 1 Stiglers 1961 important early paper studied a search technology different from both McCalls and Jovanovics In Stiglers model an unemployed worker has to choose in advance a number n of offers to draw from which he takes the highest wage offer Stiglers formulation of the search problem was not sequential 157 A lake model 171 can mimic outcomes in situations where they would facing possible firings by occasionally firing themselves by quitting into unemployment they choose not to do so because that would lower their expected present value of income Since the employed workers in the situation where they face possible firings are worse off than employed workers in the situation without possible firings it fol lows that ˆvw lies strictly below vw over the whole domain because even at wages that are rejected the value function partly reflects a stream of future outcomes whose expectation is less favorable in the situation in which workers face a chance of being fired Since the value function ˆvw with firings lies strictly below the value func tion vw without firings it follows from 638 and 637 that the reservation wage w is strictly lower with firings There is less of a reason to hold out for highpaying jobs when a job is expected to last for a shorter period of time That is unemployed workers optimally invest less in search when the payoffs associated with wage offers have gone down because of the probability of being fired 64 A lake model Consider an economy consisting of a continuum of ex ante identical workers living in the environment described in the previous section These workers move recurrently between unemployment and employment The mean duration of each spell of employment is α1 and the mean duration of unemployment is 1 Fw1 The average unemployment rate Ut across the continuum of workers obeys the difference equation Ut1 α 1 Ut F w Ut where α is the hazard rate of escaping employment and 1Fw is the hazard rate of escaping unemployment Solving this difference equation for a stationary solution ie imposing Ut1 Ut U gives U α α 1 F w U 1 1 F w 1 1 F w 1 α 641 172 Search and Unemployment Equation 641 expresses the stationary unemployment rate in terms of the ratio of the average duration of unemployment to the sum of average durations of unemployment and employment The unemployment rate being an average across workers at each moment thus reflects the average outcomes experienced by workers across time This way of linking economywide averages at a point in time with the timeseries average for a representative worker is our first en counter with a class of models sometimes referred to as Bewley models which we shall study in depth in chapter 18 This model of unemployment is sometimes called a lake model and can be depicted as in Figure 641 with two lakes denoted U and 1 U representing volumes of unemployment and employment and streams of rate α from the 1 U lake to the U lake and of rate 1 Fw from the U lake to the 1 U lake Equation 641 allows us to study the determinants of the unemployment rate in terms of the hazard rate of becoming unemployed α and the hazard rate of escaping unemployment 1 Fw 1U U 1Fw α Figure 641 Lake model with flows of rate α from em ployment state 1 U to unemployment state U and of rate 1 Fw from U to 1 U 174 Search and Unemployment to convergence on the Bellman equation The optimal policy is characterized by three regions in the θ ǫ space For high enough values of ǫ θ the worker stays put For high θ but low ǫ the worker retains his career but searches for a better job For low values of θ ǫ the worker finds a new career and a new job In figures 651 and 652 the decision to retain both job and career occurs in the high θ high ǫ region of the state space the decision to retain career θ but search for a new job ǫ occurs in the high θ and low ǫ region of the state space and the decision to get a new life by drawing both a new θ and a new ǫ occurs in the low θ low ǫ region7 0 1 2 3 4 5 0 1 2 3 4 5 155 160 165 170 175 180 185 190 195 200 career choice θ job choice ε vθε Figure 651 Optimal value function for Neals model with β 95 The value function is flat in the reject θ ǫ region increasing in θ only in the keepcareerbutdrawnewjob re gion and increasing in both θ and ǫ in the stayput region When the careerjob pair θ ǫ is such that the worker chooses to stay put the value function in 651 attains the value θ ǫ1 β Of course this happens when the decision to stay put weakly dominates the other two actions which occurs when θ ǫ 1 β max C θ Q 652 7 The computations were performed by the Matlab program neal2m 180 Search and Unemployment 0 02 04 06 08 1 12 14 16 18 2 0 02 04 06 08 1 12 14 w gwfw Figure 661 Two densities for wages a uniform fw and gw that is a beta distribution with parameters 3 12 0 01 02 03 04 05 06 07 08 09 1 154 156 158 16 162 164 166 168 wπ π Figure 662 The reservation wage as a function of the posterior probability π that the worker thinks that the wage is drawn from the uniform density f of the two densities from which nature draws wage offers Thus workers usually Offer distribution unknown 181 choose not to collect enough observations for them to learn for sure which distri bution governs wage offers In both panels the lower line shows the cumulative distribution function when nature draws from F and the lower panel shows the cdf when nature draws from G13 0 10 20 30 40 50 60 0 02 04 06 08 1 time Probtime 0 01 02 03 04 05 06 07 08 09 1 0 02 04 06 08 1 π Probπ Figure 663 Top panel CDF of duration of unemploy ment bottom panel CDF of π at time worker accepts wage and leaves unemployment In each panel the lower filled line is the CDF when nature permanently draws from the uni form density f while the dotted line is the CDF when nature permanently draws from the beta density g A comparison of the CDFs when nature draws from F and G respectively is revealing When G prevails the cumulative distribution functions in the top panel reveal that workers typically accept jobs earlier than when F prevails This captures what the interrogator of an unemployed McCall worker in the passage of Lucas cited in the introduction might have had in mind when he said Maybe you are setting your sights too high The bottom panel reveals that when nature permanently draws from G employed workers put a higher 13 It is a useful exercise to use recall formula 622 for the mean of a nonnegative random variable and then glance at the CDFs in the bottom panel to approximate the mean πt at time of job acceptance 182 Search and Unemployment probability on their having actually sampled from G than from F while the reverse is true when nature draws permanently from F 67 An equilibrium price distribution The McCall search model confronts a worker with a given distribution of wages In this section we ask why firms might conceivably choose to confront an ex ante homogenous collection of workers with a nontrivial distribution of wages Knowing that the workers have a reservation wage policy why would a firm ever offer a worker more than the reservation wage That question challenges us to think about whether it is possible to conceive of a coherent setting in which it would be optimal for a collection of profit maximizing firms somehow to make decisions that generate a distribution of wages In this section we take up this question but for historical reasons inves tigate it in the context of a sequential search model in which buyers seek the lowest price14 Buyers can draw additional offers from a known distribution at a fixed cost c for each additional batch of n independent draws from a known price distribution Both within and across batches successive draws are in dependent The buyers optimal strategy is to set a reservation price and to continue drawing until the first time a price less than the reservation price has been offered Let p be the reservation price Rothschild 1973 posed the following challenge for a model in which there is a large number of identical buyers each of whom has reservation price p If all sellers know the reservation price p why would any of them offer a price less than p This cogent question points to a force for the price distribution to collapse an outcome that would destroy the motive for search behavior on the part of buyers Thus the challenge is to construct an equilibrium version of a search model in which it is in firms interest to generate the nontrivial price distribution that sustains buyers search activities Burdett and Judd 1983 met this challenge by creating an environment in which ex ante identical buyers ex post receive differing numbers of price offers that are drawn from a common distribution set by firms They construct 14 See Burdett and Mortensen 1998 for a parallel analysis of the analogous issues in a model of job search An equilibrium price distribution 183 an equilibrium in which a continuum of profit maximizing sellers are content to generate this distribution of prices Sellers set their prices to maximize expected profit per customer But sellers dont know the number of other offers that a prospective customer has received Heterogeneity in the number of offers received by buyers together with sellers ignorance of the number and nature of other offers received by a particular customer creates a tradeoff between profit per customer and volume that makes possible a nondegenerate equilibrium price distribution Firms that post higher prices are lowervolume sellers Firms that post lower prices are highervolume sellers There exists an equilibrium distribution of prices in which all types of firms expect to earn the same profit per potential customer 671 A BurdettJudd setup A continuum of buyers purchases a single good from one among a continuum of firms Each firm contacts a fixed measure ν of potential buyers The firms produce a homogeneous good at zero marginal cost Each firm takes the cdf of prices charged by other firms as given and chooses a price The firm wants to maximize its expected profits per consumer A firms expected profit per consumer equals its price times the probability that its price is the minimum among the set of acceptable offers received by the buyer The distribution of prices set by other firms impinges on a firms expected profits because it affects the probability that its offer will be accepted by a buyer 672 Consumer problem with noisy search A consumer wants to purchase a good for a minimum price Firms make offers that buyers can view as being drawn from a distribution of nonnega tive prices with cumulative distribution function GP Probp P with Gp 0 GB 1 Assume that G is continuously differentiable and so has an associated probability density A buyers search activity is divided into batches Within each batch the buyer receives a random number of offers drawn from the same distribution G Burdett and Judd call this structure noisy An equilibrium price distribution 185 673 Firms For simplicity and to focus our attention entirely on the search problem we assume that the good costs firms nothing to produce In setting its price we assume that a firm seeks to maximize expected profit per customer A firm makes an offer to a customer without knowing whether this is the only offer available to the customer or whether the customer having drawn two offers possibly has a lower offer in hand The firm begins by computing the fraction of its customers who will have received one offer and the fraction of its customers who will have received only one offer Let there be a large number ν of total potential buyers per batch consisting of νq persons each of whom receives one offer and ν1 q people each of whom receives two offers The total number of offers is evidently ν1q21q ν2q Evidently the fraction of all offers that is received by customers who have received one offer is νq ν2q q 2q This calculation induces a typical firm to believe that the fraction of its customers who receive one offer is ˆq q 2 q 674 and the fraction who receive two offers is 1 ˆq 21q 2q The firm regards ˆq as its estimate of the probability that a given customer has received only its offer while it thinks that a fraction 1 ˆq of its customers has also received a competing offer from another firm There is a continuum of firms each of which takes as given a price offer distribution of other firms with cdf Gp where Gp 0 Gp 1 We have assume that G is differentiable16 This distribution satisfies the outcome that in equilibrium no firm makes an offer exceeding the buyers reservation price p Let Qp be the probability that a consumer will accept an offer p where p p p Evidently a consumer who receives one offer p p will accept it with probability 1 But only a fraction 1 Gp of consumer who receive two offers will accept an offer p p Why because 1 Gp is the 16 Burdett and Judd 1983 p 959 lemma 1 show that an equilibrium G is differentiable when q 0 1 and p 0 Their argument goes as follows Suppose to the contrary that there is a positive probability attached to a single price p 0 p Consider a firm that contemplates charging p When q 1 the firm knows that there is a positive probability that a prospective consumer has received another offer also of p If the firm lowers its offer infinitesimally it can expect to steal that customer and thereby increase its expected profits Therefore a decision to charge p cant maximize expected profits for a typical firm We have been led to a contradiction by assuming that G has a discontinuity at p An equilibrium price distribution 187 Therefore the expected profit per customer for a firm that sets price p p p is pQ p pˆq 678 which is evidently independent of the firms choice of offer p in the interval p p The firm is indifferent about the price it offers on this interval In particular notice that The right side of equality 678 is the product of the fraction of a firms buyers receiving one offer ˆq times the reservation price p This is the expected profit per customer of a firm that charges the reservation price The left side of equality 678 is the product of the price p times probability Qp that a buyer will accept price p which as we have noted equals the expected profit per customer for a firm that sets price p We assume that firms randomize over choices of p in such a way that Gp given by 677 emerges as the cdf for prices 675 Special cases The BurdettJudd model isolates forces for the price distribution to collapse and countervailing forces that can sustain a nontrivial price distribution 1 Consider the special case in which q 1 and therefore ˆq 1 Here p p The formula 677 shows that the distribution of prices collapses This case exhibits the Rothschild challenge with which we began 2 Next consider the opposite special case in which q 0 and therefore ˆq 0 Here p 0 and the cdf Gp 1p p p Bertrand com petition drives all prices down to the marginal cost of production which we have assumed to be zero This case exhibits another force for the price distribution to collapse again in the spirit of Rothschilds challenge 3 Finally consider the general case in which q 0 1 and therefore ˆq 0 1 When q is strictly in the interior of p p we can sustain a nontrivial distribution of prices Firms are indifferent between being high volume low price sellers and high price low volume sellers The equilibrium price distribution Gp renders a firms expected profits per prospective customer pQp independent of p 188 Search and Unemployment 68 Jovanovics matching model Another interesting effort to confront Rothschilds questions about the source of the equilibrium wage or price distribution comes from matching models in which the main idea is to reinterpret w not as a wage but instead more broadly as a parameter characterizing the entire quality of a match occurring between a pair of agents The variable w is regarded as a summary measure of the productivities or utilities jointly generated by the activities of the match We can consider pairs consisting of a firm and a worker a man and a woman a house and an owner or a person and a hobby The idea is to analyze the way in which matches form and maybe also dissolve by viewing both parties to the match as being drawn from populations that are statistically homogeneous to an outside observer even though the match is idiosyncratic from the perspective of the parties to the match Jovanovic 1979a used a model of this kind supplemented by an hypothesis that both sides of a match behave optimally but only gradually learn about the quality of the match Jovanovic was motivated by a desire to explain three features of labor market data 1 on average wages rise with tenure on the job 2 quits are negatively correlated with tenure that is a quit has a higher probability of occurring earlier in tenure than later and 3 the probability of a subsequent quit is negatively correlated with the current wage rate Jovanovics insight was that each of these empirical regularities could be interpreted as reflecting the operation of a matching process with gradual learning about match quality We consider a simplified version of Jovanovics model of matching Prescott and Townsend 1980 describe a discretetime version of Jovanovics model which has been simplified here A market has two sides that could be var iously interpreted as consisting of firms and workers or men and women or owners and renters or lakes and fishermen Following Jovanovic we shall adopt the firmworker interpretation here An unmatched worker and a firm form a pair and jointly draw a random match parameter θ from a probability distri bution with cumulative distribution function Probθ s Fs Here the match parameter reflects the marginal productivity of the worker in the match In the first period before the worker decides whether to work at this match or to wait and to draw a new match next period from the same distribution F the worker and the firm both observe only y θ u where the random noise u Jovanovics matching model 189 is uncorrelated with θ Thus in the first period the workerfirm pair receives only a noisy observation on θ This situation corresponds to that when both sides of the market form only an errorridden impression of the quality of the match at first On the basis of this noisy observation the firm which is imagined to operate competitively under constant returns to scale offers to pay the worker the conditional expectation of θ given θ u for the first period with the understanding that in subsequent periods it will pay the worker the expected value of θ depending on whatever additional information both sides of the match receive18Given this policy of the firm the worker decides whether to accept the match and to work this period for Eθθ u or to refuse the offer and draw a new match parameter θ and noisy observation on it θ u next period If the worker decides to accept the offer in the first period then in the second period both the firm and the worker are assumed to observe the true value of θ This situation corresponds to that in which both sides learn about each other and about the quality of the match In the second period the firm offers to pay the worker θ then and forever more The worker next decides whether to accept this offer or to quit be unemployed this period and draw a new match parameter and a noisy observation on it next period We can conveniently think of this process as having three stages Stage 1 is the predraw stage in which a previously unemployed worker has yet to draw the one match parameter and the noisy observation on it that he is entitled to draw after being unemployed the previous period We let Q denote the expected present value of wages before drawing of a worker who was unemployed last period and who behaves optimally The second stage of the process occurs after the worker has drawn a match parameter θ has received the noisy observation of θ u on it and has received the firms wage offer of Eθθ u for this period At this stage the worker decides whether to accept this wage for this period and the prospect of receiving θ in all subsequent periods The third stage occurs in the next period when the worker and firm discover the true value of θ and the worker must decide whether to work at θ this period and in all subsequent periods that he remains at this job match 18 Jovanovic assumed firms to be risk neutral and to maximize the expected present value of profits They compete for workers by offering wage contracts In a longrun equilibrium the payments practices of each firm would be well understood and this fact would support the described implicit contract as a competitive equilibrium A longer horizon version of Jovanovics model 197 which is evidently negatively correlated with m0 the firstperiod wage Thus the model explains each observation that Jovanovic sought to interpret In the version of the model that we have studied a worker eventually becomes perma nently matched with probability 1 If we were studying a population of such workers of fixed size all workers would eventually be absorbed into the state of being permanently matched To provide a mechanism for replenishing the stock of unmatched workers one could combine Jovanovics model with the firing model in section 635 By letting matches θ go bad with probability λ each period one could presumably modify Jovanovics model to get the implication that with a fixed population of workers a fraction would remain unmatched each period because of the dissolution of previously acceptable matches 69 A longer horizon version of Jovanovics model Here we consider a T 1 period version of Jovanovics model in which learning about the quality of the match continues for T periods before the quality of the match is revealed by nature Jovanovic assumed that T We use the recursive projection technique the Kalman filter of chapter 2 to handle the firms and workers sequential learning The prediction of the true match quality can then easily be updated with each additional noisy observation A firmworker pair jointly draws a match parameter θ at the start of the match which we call the beginning of period 0 The value θ is revealed to the pair only at the beginning of the T 1th period of the match After θ is drawn but before the match is consummated the firmworker pair observes y0 θ u0 where u0 is random noise At the beginning of each period of the match the workerfirm pair draws another noisy observation yt θ ut on the match parameter θ The worker then decides whether or not to continue the match for the additional period Let yt y0 yt be the firms and workers information set at time t We assume that θ and ut are independently distributed random variables with θ Nµ Σ0 and ut N0 σ2 u For t 0 define mt Eθyt and m1 µ The conditional means mt and variances Eθ mt2 Σt1 can be computed with the Kalman filter via the formulas from chapter 2 mt 1 Kt mt1 Ktyt 691a 200 Search and Unemployment 610 Concluding remarks The situations analyzed in this chapter are ones in which a currently unem ployed worker rationally chooses to refuse an offer to work preferring to remain unemployed today in exchange for better prospects tomorrow The worker is voluntarily unemployed in one sense having chosen to reject the current draw from the distribution of offers In this model the activity of unemployment is an investment incurred to improve the situation faced in the future A theory in which unemployment is voluntary permits an analysis of the forces imping ing on the choice to remain unemployed Thus we can study the response of the workers decision rule to changes in the distribution of offers the rate of unemployment compensation the number of offers per period and so on Chapter 23 studies the optimal design of unemployment compensation That issue is a trivial one in the present chapter with riskneutral agents and no externalities Here the government should avoid any policy that affects the workers decision rules since it would harm efficiency and the firstbest way of pursuing distributional goals is through lumpsum transfers In contrast chapter 23 assumes riskaverse agents and incomplete insurance markets which together with information asymmetries make for an intricate contract design problem in the provision of unemployment insurance Chapter 29 presents various equilibrium models of search and matching We study workers searching for jobs in an island model workers and firms forming matches in a model with a matching function and how a medium of exchange can overcome the problem of double coincidence of wants in a search model of money 202 Search and Unemployment 6A2 Example 5 a Jovanovic model Here is a simplified version of the search model of Jovanovic 1979a A newly unemployed worker draws a job offer from a distribution given by µi Probw1 wi where w1 is the firstperiod wage Let µ be the n 1 vector with ith component µi After an offer is drawn subsequent wages associated with the job evolve according to a Markov chain with timevarying transition matrices Pt i j Prob wt1 wjwt wi for t 1 T We assume that for times t T the transition matrices Pt I so that after T a jobs wage does not change anymore with the passage of time We specify the Pt matrices to capture the idea that the workerfirm pair is learning more about the quality of the match with the passage of time For example we might set Pt 1 qt qt 0 0 0 0 qt 1 2qt qt 0 0 0 0 qt 1 2qt qt 0 0 0 0 0 0 1 2qt qt 0 0 0 0 qt 1 qt where q 0 1 In the following numerical examples we use a slightly more general form of transition matrix in which except at endpoints of the distribu tion Probwt1 wkmwt wk Pt k k m qt Pt k k 1 2qt 6A1 Here m 1 is a parameter that indexes the spread of the distribution At the beginning of each period a previously matched worker is exposed with probability λ 0 1 to the event that the match dissolves We then have a set of Bellman equations vt max w β 1 λ Ptvt1 βλQ βQ c 6A2a for t 1 T and vT 1 max w β 1 λ vT 1 βλQ βQ c 6A2b Exercises 205 5 10 15 20 35 4 45 5 55 6 65 7 Figure 6A1 Reservation wages as a function of tenure for model with three different parameter settings m 6 λ 0 the dots m 10 λ 0 the line with circles and m 10 λ 1 the dashed line 5 10 15 20 62 64 66 68 7 72 74 76 78 8 Figure 6A2 Mean wages as a function of tenure for model with three different parameter settings m 6 λ 0 the dots m 10 λ 0 the line with circles and m 10 λ 1 the dashed line Exercises 217 Exercise 619 Value function iteration and policy improvement algo rithm donated by PierreOlivier Weill The goal of this exercise is to study in the context of a specific problem two methods for solving dynamic programs value function iteration and Howards policy improvement Consider McCalls model of intertemporal job search An unemployed worker draws one offer from a cdf F with F0 0 and FB 1 B If the worker rejects the offer she receives unemployment compensation c and can draw a new wage offer next period If she accepts the offer she works forever at wage w The objective of the worker is to maximize the expected discounted value of her earnings Her discount factor is 0 β 1 a Write the Bellman equation Show that the optimal policy is of the reser vation wage form Write an equation for the reservation wage w b Consider the value function iteration method Show that at each iteration the optimal policy is of the reservation wage form Let wn be the reservation wage at iteration n Derive a recursion for wn Show that wn converges to w at rate β c Consider Howards policy improvement algorithm Show that at each it eration the optimal policy is of the reservation wage form Let wn be the reservation wage at iteration n Derive a recursion for wn Show that the rate of convergence of wn towards w is locally quadratic Specifically use a Taylor expansion to show that for wn close enough to w there is a constant K such that wn1 w Kwn w2 Exercise 620 Different types of unemployed workers are identical except that they sample from different wage distributions Each period an unemployed worker of type α draws a single new offer to work forever at a wage w from a cumulative distri bution function Fα that satisfies Fαw 0 for w 0 Fα0 α FαB 1 where B 0 and Fα is a right continuous function mapping 0 B into 0 1 The cdf of a type α worker is given by Fα w α for 0 w αB wB for αB w B αB 1 α for B αB w B 1 for w B Exercises 221 c Assume that β 95 c 1 w1 w2 wn 1 2 3 4 5 and P 8 2 0 0 0 18 8 02 0 0 25 25 0 25 25 0 0 02 8 18 0 0 0 2 8 Please write a Matlab or R or C program to solve the Bellman equation Show the optimal policy function and the value function d Assume that all parameters are the same as in part c except for β which now equals 99 Please find the optimal policy function and the optimal value function e Please discuss whether why and how your answers to parts c and d differ Exercise 626 Neal models Markov implications This will be yet another exercise that illustrates the theme that finding the state is an art Consider the version of the Neal 1999 career choice model that we analyzed in the text In the text a workers state is the job career pair ǫ θ with which the worker enters the period Knowing the structure of the outcome and the optimal decision rule lets use figure 652 to partition the state space ǫ θ into three sets that define the three new states st 1 2 3 that well use to define the states in a Markov chain We say that st 1 if the worker wants a new life ie he wants to draw a new job career pair next period We say that st 2 if the worker wants a new job ie if he is content with his career θ but wants to draw a new job ǫ next period We say that st 3 if the worker plans to remain in his current job career pair next period Let Pij Probst1 jst i Assume that the initial probability distribution is π0 Probs0 1 1 a Show that P has the following structure P 1 P12 P13 P12 P13 0 1 P23 P23 0 0 1 and please tell how to compute the nontrivial elements of P from the information used to compute figure 652 Part III Competitive Equilibria and Applications Chapter 7 Recursive Competitive Equilibrium I 71 An equilibrium concept This chapter formulates competitive and oligopolistic equilibria in some dynamic settings Up to now we have studied singleagent problems where components of the state vector not under the control of the agent were taken as given In this chapter we describe multipleagent settings in which components of the state vector that one agent takes as exogenous are determined by the deci sions of other agents We study partial equilibrium models of a kind applied in microeconomics1 We describe two closely related equilibrium concepts for such models a rational expectations or recursive competitive equilibrium and a Markov perfect equilibrium The first equilibrium concept jointly restricts a Bellman equation and a transition law that is taken as given in that Bellman equation The second equilibrium concept leads to pairs in the duopoly case or sets in the oligopoly case of Bellman equations and transition equations that are to be solved by simultaneous backward induction Though the equilibrium concepts introduced in this chapter transcend linear quadratic setups we choose to present them in the context of linear quadratic examples because this renders the Bellman equations tractable 1 For example see Rosen and Topel 1988 and Rosen Murphy and Scheinkman 1994 225 228 Recursive Competitive Equilibrium I 728 Solving Bellman equation 725 by backward induction automatically incorporates both equations 727 and 728 The firms optimal policy function is yt1 h yt Yt 729 Then with n identical firms setting Yt nyt makes the actual law of motion for output for the market Yt1 nh Ytn Yt 7210 Thus when firms believe that the law of motion for marketwide output is equation 724 their optimizing behavior makes the actual law of motion equation 7210 For this model we adopt the following definition Definition A recursive competitive equilibrium4 of the model with adjust ment costs is a value function vy Y an optimal policy function hy Y and a law of motion HY such that a Given H vy Y satisfies the firms Bellman equation and hy Y is the optimal policy function b The law of motion H satisfies HY nhYn Y A recursive competitive equilibrium equates the actual and perceived laws of motion 724 and 7210 The firms optimum problem induces a mapping M from a perceived law of motion for output H to an actual law of motion MH The mapping is summarized in equation 7210 The H component of a rational expectations equilibrium is a fixed point of the operator M This is a special case of a recursive competitive equilibrium to be defined more generally in section 73 How might we find an equilibrium The mapping M is not a contraction and there is no guarantee that direct iterations on M will converge5 In fact in many contexts including the present one there exist admissible parameter values for which divergence of iterations on M prevails 4 This is also often called a rational expectations equilibrium 5 A literature that studies whether models populated with agents who learn can converge to rational expectations equilibria features iterations on a modification of the mapping M that can be approximated as γM1γI where I is the identity operator and γ 0 1 is a relaxation parameter See Marcet and Sargent 1989 and Evans and Honkapohja 2001 for 230 Recursive Competitive Equilibrium I Applying the BenvenisteScheinkman formula gives V Y A0 A1Y d Y Y 7215 Substituting this into equation 7214 and rearranging gives βA0 dYt βA1 d 1 β Yt1 dβYt2 0 7216 Return to equation 727 and set yt Yt for all t Remember that we have set n 1 When n 1 we have to adjust pieces of the argument for n Notice that with yt Yt equations 7216 and 727 are identical The Euler equation for the planning problem matches the secondorder difference equation that we derived by first finding the Euler equation of the representative firm and substituting into it the expression Yt nyt that makes the representative firm representative Thus if it is appropriate to apply the same terminal conditions for these two difference equations which it is then we have verified that a solution of the planning problem also is an equilibrium Setting yt Yt in equation 727 amounts to dropping equation 724 and instead solving for the coefficients H0 H1 that make yt Yt true and that jointly solve equations 724 and 727 It follows that for this example we can compute an equilibrium by forming the optimal linear regulator problem corresponding to the Bellman equation 7213 The optimal policy function for this problem is the law of motion Y HY that a firm faces within a rational expectations equilibrium6 6 Lucas and Prescott 1971 used the method of this section The method exploits the connection between equilibrium and Pareto optimality expressed in the fundamental theorems of welfare economics See MasColell Whinston and Green 1995 Recursive competitive equilibrium 231 73 Recursive competitive equilibrium The equilibrium concept of the previous section is widely used Following Prescott and Mehra 1980 it is useful to define the equilibrium concept more generally as a recursive competitive equilibrium Let x be a vector of state variables under the control of a representative agent and let X be the vector of those same variables chosen by the market Let Z be a vector of other state variables chosen by nature that is determined outside the model The representative agents problem is characterized by the Bellman equation v x X Z max u R x X Z u βv x X Z 731 where denotes next periods value and where the maximization is subject to the restrictions x g x X Z u 732 X G X Z 733 Z ζ Z 734 Here g describes the impact of the representative agents controls u on his state x G and ζ describe his beliefs about the evolution of the aggregate state The solution of the representative agents problem is a decision rule u h x X Z 735 To make the representative agent representative we impose X x but only after we have solved the agents decision problem Substituting equation 735 and X xt into equation 732 gives the actual law of motion X GA X Z 736 where GAX Z gX X Z hX X Z We are now ready to propose a definition Definition A recursive competitive equilibrium is a policy function h an actual aggregate law of motion GA and a perceived aggregate law G such that a given G h solves the representative agents optimization problem and b h implies that GA G Equilibrium occupational choice 235 where Ut is a stock of skilled labor and St is a stock of unskilled labor and f2 is a positive semidefinite matrix parameterizing whether skilled and unskilled labor are complements or substitutes in production Stocks of the two types of labor evolve according to the laws of motion Ut1 δUUt nUt St1 δSSt nSt2 752 where flows into the two types of skills are restricted by nUt nSt nt 753 where nt is an exogenous flow of new entrants into the labor market governed by the stochastic process nt1 µn 1 ρ ρnt σnǫt1 754 where ǫt1 is an iid scalar stochastic process with time t 1 component dis tributed as N0 1 Equations 752 753 754 express a timetobuild or schooling technology for converting new entrants nt into increments in stocks of unskilled labor this takes one period of waiting and of skilled labor this takes three periods of waiting Stocks of skilled and unskilled labors depreciate say through death or retirement at the rates 1 δS 1 δU respectively where δS 0 1 and δU 0 1 In addition we assume that there is an output cost of e 2n2 st associated with allocating new workers or students to the skilled worker pool 242 Recursive Competitive Equilibrium I The equilibrium sequences F1t F2t t t0 t0 1 t1 1 can be calcu lated from the pair of coupled Riccati difference equations 775 and 777 In particular we use equations 774 775 776 and 777 to work backward from time t1 1 Notice that given P1t1 and P2t1 equations 774 and 776 are a system of k2 n k1 n linear equations in the k2 n k1 n unknowns in the matrices F1t and F2t Notice how j s control law Fjt is a function of Fis s t i j Thus agent is choice of Fit t t0 t1 1 influences agent j s choice of control laws However in the Markov perfect equilibrium of this game each agent is assumed to ignore the influence that his choice exerts on the other agents choice11 We often want to compute the solutions of such games for infinite horizons in the hope that the decision rules Fit settle down to be time invariant as t1 In practice we usually fix t1 and compute the equilibrium of an infinite horizon game by driving t0 Judd followed that procedure in the following example 771 An example This section describes the Markov perfect equilibrium of an infinite horizon linear quadratic game proposed by Kenneth Judd 1990 The equilibrium is computed by iterating to convergence on the pair of Riccati equations defined by the choice problems of two firms Each firm solves a linear quadratic op timization problem taking as given and known the sequence of linear decision rules used by the other player The firms set prices and quantities of two goods interrelated through their demand curves There is no uncertainty Relevant variables are defined as follows Iit inventories of firm i at beginning of t qit production of firm i during period t pit price charged by firm i during period t Sit sales made by firm i during period t Eit costs of production of firm i during period t 11 In an equilibrium of a Stackelberg or dominant player game the timing of moves is so altered relative to the present game that one of the agents called the leader takes into account the influence that his choices exert on the other agents choices See chapter 19 Exercises 245 where Yt is the market level of output which the firm takes as exogenous and which the firm believes follows the law of motion Yt1 H0 H1Yt 4 with Y0 as a fixed initial condition a Formulate the Bellman equation for the firms problem b Formulate the firms problem as a discounted optimal linear regulator prob lem being careful to describe all of the objects needed What is the state for the firms problem c Use the Matlab program olrpm to solve the firms problem for the following parameter values A0 100 A1 05 β 95 d 10 H0 955 and H1 95 Express the solution of the firms problem in the form yt1 h0 h1yt h2Yt 5 giving values for the hj s d If there were n identical competitive firms all behaving according to equation 5 what would equation 5 imply for the actual law of motion 4 for the market supply Y e Formulate the Euler equation for the firms problem Exercise 72 Rational expectations Now assume that the firm in problem 1 is representative We implement this idea by setting n 1 In equilibrium we will require that yt Yt but we dont want to impose this condition at the stage that the firm is optimizing because we want to retain competitive behavior Define a rational expectations equilibrium to be a pair of numbers H0 H1 such that if the representative firm solves the problem ascribed to it in problem 1 then the firms optimal behavior given by equation 5 implies that yt Yt t 0 a Use the program that you wrote for exercise 71 to determine which if any of the following pairs H0 H1 is a rational expectations equilibrium i 940888 9211 ii 9322 9433 and iii 9508187459215024 95245906270392 b Describe an iterative algorithm that uses the program that you wrote for exercise 71 to compute a rational expectations equilibrium You are not being asked actually to use the algorithm you are suggesting Chapter 8 Equilibrium with Complete Markets 81 Time 0 versus sequential trading This chapter describes competitive equilibria of a pure exchange infinite horizon economy with stochastic endowments These are useful for studying risk shar ing asset pricing and consumption We describe two systems of markets an ArrowDebreu structure with complete markets in dated contingent claims all traded at time 0 and a sequentialtrading structure with complete oneperiod Arrow securities These two entail different assets and timings of trades but have identical consumption allocations Both are referred to as complete markets economies They allow more comprehensive sharing of risks than do the incom plete markets economies to be studied in chapters 17 and 18 or the economies with imperfect enforcement or imperfect information studied in chapters 21 and 22 82 The physical setting preferences and endowments In each period t 0 there is a realization of a stochastic event st S Let the history of events up and until time t be denoted st s0 s1 st The unconditional probability of observing a particular sequence of events st is given by a probability measure πtst For t τ we write the probability of observing st conditional on the realization of sτ as πtstsτ In this chapter we shall assume that trading occurs after observing s0 which we capture by setting π0s0 1 for the initially given value of s0 1 In section 89 we shall follow much of the literatures in macroeconomics and econometrics and assume that πtst is induced by a Markov process We wait to impose that special assumption until section 89 because some important findings do not require making that assumption 1 Most of our formulas carry over to the case where trading occurs before s0 has been realized just postulate a nondegenerate probability distribution π0s0 over the initial state 249 Alternative trading arrangements 251 83 Alternative trading arrangements For a twoevent stochastic process st S 0 1 the trees in Figures 831 and 832 give two portraits of how histories st unfold From the perspective of time 0 given s0 0 Figure 831 portrays all prospective histories possible up to time 3 Figure 832 portrays a particular history that it is known the economy has indeed followed up to time 2 together with the two possible oneperiod continuations into period 3 that can occur after that history 0111 0110 0101 0100 0011 0010 0001 0000 t0 t1 t2 t3 Figure 831 The ArrowDebreu commodity space for a twostate Markov chain At time 0 there are trades in time t 3 goods for each of the eight nodes that signify histories that can possibly be reached starting from the node at time 0 In this chapter we shall study two distinct trading arrangements that cor respond respectively to the two views of the economy in Figures 831 and 832 One is what we shall call the ArrowDebreu structure Here markets meet at time 0 to trade claims to consumption at all times t 0 and that are contingent on all possible histories up to t st In that economy at time 0 Time 0 trading ArrowDebreu securities 257 851 Equilibrium pricing function Suppose that ci i 1 I is an equilibrium allocation Then the marginal condition 852 or 854 can be regarded as determining the price system q0 t st as a function of the equilibrium allocation assigned to consumer i for any i But to exploit this fact in computation we need a way first to compute an equilibrium allocation without simultaneously computing prices As we shall see soon solving the planning problem provides a convenient way to do that Because the units of the price system are arbitrary one of the prices can be normalized at any positive value We shall set q0 0s0 1 putting the price system in units of time 0 goods This choice implies that µi u ici 0s0 852 Optimality of equilibrium allocation A competitive equilibrium allocation is a particular Pareto optimal allocation one that sets the Pareto weights λi µ1 i These weights are unique up to multiplication by a positive scalar Furthermore at a competitive equilibrium allocation the shadow prices θtst for the associated planning problem equal competitive equilibrium prices q0 t st for goods to be delivered at date t at history st That allocations for the planning problem and the competitive equi librium are identical reflects the two fundamental theorems of welfare economics see MasColell Whinston and Green 1995 The first welfare theorem states that a competitive equilibrium allocation is efficient The second welfare theo rem states that there exist a price system and an initial distribution of wealth that can support an efficient allocation as a competitive equilibrium allocation 258 Equilibrium with Complete Markets 853 Interpretation of trading arrangement In the competitive equilibrium with ArrowDebreu timing all trades occur at t 0 in one market Deliveries occur after t 0 but no more trades A vast clearing or credit system operates at t 0 It ensures that condition 851 holds for each consumer i A symptom of the onceandforall and netclearing trading arrangement is that each consumer faces one budget constraint that restricts trades across all dates and histories In section 88 we describe another trading arrangement with more trading dates 854 Equilibrium computation To compute an equilibrium we have somehow to determine ratios of the La grange multipliers µiµ1 i 1 I that appear in equations 856 and 857 The following Negishi algorithm accomplishes this5 1 Fix a positive value for one µi say µ1 throughout the algorithm Guess positive values for the remaining µi s Then solve equations 856 and 857 for a candidate consumption allocation ci i 1 I 2 Use 854 for any consumer i to solve for the price system q0 t st 3 For i 1 I check the budget constraint 851 For those is for which the cost of consumption exceeds the value of their endowment raise µi while for those is for which the reverse inequality holds lower µi 4 Iterate to convergence on steps 13 Multiplying all of the µi s by a positive scalar simply changes the units of the price system That is why we are free to normalize as we have in step 1 In general the equilibrium price system and distribution of wealth are mu tually determined Along with the equilibrium allocation they solve a vast system of simultaneous equations The Negishi algorithm provides one way to solve those equations In applications it can be complicated to implement Therefore in order to simplify things most of the examples and exercises in this chapter specialize preferences in a way that eliminates the dependence of equilibrium prices on the distribution of wealth 5 See Negishi 1960 Sequential trading 267 882 Financial wealth as an endogenous state variable A key step in constructing a sequentialtrading arrangement is to identify a variable to serve as the state in a value function for the consumer at date t and history st We find this state by taking an equilibrium allocation and price system for the ArrowDebreu time 0 trading structure and applying a guessandverify method We begin by asking the following question In the competitive equilibrium where all trading takes place at time 0 what is the implied continuation wealth of consumer i at time t after history st The answer is obtained by summing up the value of the consumers holdings of claims to current and future consumption at time t and history st Since history st has been realized we discard all claims contingent on time t histories st st that were not realized Hence the implied wealth is determined simply by the trades that were undertaken by consumer i at the outset of a time 0 trading equilibrium when the consumer can be thought of as having sold the entire endowment stream on the right side of budget constraint 851 in order to acquire the contingent consumption claims on the left side of budget constraint 851 The differences in a sequentialtrading arrangement are that Arrow one period securities are traded period by period and that consumers retain the ownership to their endowment processes throughout time Hence from the per spective of a sequentialtrading arrangement the wealth of consumer i at a point in time can be decomposed into financial wealth and nonfinancial wealth9 Fi nancial wealth at time t after history st is the consumers beginningofperiod holdings of Arrow securities that are contingent on the current state st being realized while the present value of the consumers current and future endow ment constitutes nonfinancial wealth From Arrows 1964 insight that the two trading arrangements yield identical equilibrium allocations a consumers financial wealth in a sequential trading equilibrium should be equal to its con tinuation wealth in a time 0 trading equilibrium minus the continuation value of its current and future endowment ie its nonfinancial wealth also evalu ated in terms of prices for a time0trading competitive equilibrium Thus the financial wealth of consumer i at time t after history st expressed in terms of 9 In some applications financial wealth is also called nonhuman wealth and nonfinancial wealth is called human wealth 272 Equilibrium with Complete Markets 886 Equivalence of allocations By making an appropriate guess about the form of the pricing kernels it is easy to show that a competitive equilibrium allocation of the complete markets model with time 0 trading is also an allocation for a competitive equilibrium with sequential trading of oneperiod Arrow securities one with a particular initial distribution of wealth Thus take q0 t st as given from the ArrowDebreu equilibrium and suppose that the pricing kernel Qtst1st makes the following recursion true q0 t1st1 Qtst1stq0 t st or Qtst1st qt t1st1 887 where recall that qt t1st1 q0 t1st1 q0 t st Let ci tst be a competitive equilibrium allocation in the ArrowDebreu economy If equation 887 is satisfied that allocation is also a sequential trading competitive equilibrium allocation To show this fact take the con sumers firstorder conditions 854 for the ArrowDebreu economy from two successive periods and divide one by the other to get βu ici t1st1πst1st u ici tst q0 t1st1 q0 t st Qtst1st 888 If the pricing kernel satisfies equation 887 this equation is equivalent with the firstorder condition 886 for the sequentialtrading competitive equilibrium economy It remains for us to choose the initial wealth of the sequentialtrading equilibrium so that the sequentialtrading competitive equilibrium duplicates the ArrowDebreu competitive equilibrium allocation We conjecture that the initial wealth vector a0s0 of the sequentialtrading economy should be chosen to be the zero vector This is a natural conjecture because it means that each consumer must rely on its own endowment stream to finance consumption in the same way that consumers are constrained to finance their historycontingent purchases for the infinite future at time 0 in the ArrowDebreu economy To prove that the conjecture is correct we must show that the zero initial wealth vector enables consumer i to finance ci tst and leaves no room to increase consumption in any period after any history Recursive competitive equilibrium 275 and current information This leads us to make the following specialization of the exogenous forcing processes that facilitates a recursive formulation of the sequentialtrading equilibrium 891 Endowments governed by a Markov process Let πss be a Markov chain with given initial distribution π0s and state space s S That is Probst1 sst s πss and Probs0 s π0s As we saw in chapter 2 the chain induces a sequence of probability measures πtst on histories st via the recursions πtst πstst1πst1st2 πs1s0π0s0 891 In this chapter we have assumed that trading occurs after s0 has been observed which we capture by setting π0s0 1 for the initially given value of s0 Because of the Markov property the conditional probability πtstsτ for t τ depends only on the state sτ at time τ and does not depend on the history before τ πtstsτ πstst1πst1st2 πsτ1sτ 892 Next we assume that consumers endowments in period t are time invariant measurable functions of st yi tst yist for each i All of our previous results continue to hold but the Markov assumption for st imparts further structure to equilibrium prices and quantities 276 Equilibrium with Complete Markets 892 Equilibrium outcomes inherit the Markov property Proposition 2 asserted a particular kind of history independence of the equilib rium allocation that prevails under any stochastic process for the endowments In particular each individuals consumption is a function only of the current re alization of the aggregate endowment and does not depend on the specific history leading to that outcome11 Under our present assumption that yi tst yist for each i it follows immediately that ci tst cist 893 Substituting 892 and 893 into 886 shows that the pricing kernel in the sequentialtrading equilibrium is a function only of the current state Qtst1st β u icist1 u icist πst1st Qst1st 894 After similar substitutions with respect to equation 875 we can also establish history independence of relative prices in the ArrowDebreu economy Proposition 4 If time t endowments are a function of a Markov state st the ArrowDebreu equilibrium price of datet 0 history st consumption goods expressed in terms of date τ 0 τ t history sτ consumption goods is not history dependent qτ t st qj ksk for j k 0 such that t τ k j and sτ sτ1 st sj sj1 sk Using this proposition we can verify that both the natural debt limits 882 and consumers wealth levels 881 exhibit history independence Ai tst Aist 895 Υi tst Υist 896 The finding concerning wealth levels 896 conveys a useful insight into how the sequentialtrading competitive equilibrium attains the firstbest outcome in which no idiosyncratic risk is borne by individual consumers In particular each consumer enters every period with a wealth level that is independent of past realizations of his endowment That is his past trades have fully insured him 11 Of course the equilibrium allocation also depends on the distribution of yi tst pro cesses across agents i as reflected in the relative values of the Lagrange multipliers µi 284 Equilibrium with Complete Markets Represent this equation for our collection of bonds as i 1 I as v1t v2t vIt d1t1 d1t2 d1tτ d2t1 d2t2 d2tτ dIt1 dIt2 dItτ p1st p2st pIst or Vt DtPt If we observe Vt and Dt we can recover the prices Pt by applying an appropriate inverse or generalized inverse to each side of this matrix equation If I T and Dt is of full rank we use Pt D1 t Vt while if I T we use the least squares formula ˆPt D tDt1D tVt 8112 and if T I we use the formula ˆPt D tDtD t1Vt 8113 We use formula 8112 when there are too many securities and formula 8113 when there are too few securities relative to the primitive securities whose prices Pt we want to infer After we have constructed Pt or ˆPt we construct yields from equation 8111 Concluding remarks 287 Therefore any solution of the Pareto problem leaves the continuation value ws independent of the state s Equation 8123a implies that u 21 cs u 1cs P v 8124 Since the right side of 8124 is independent of s so is the left side and therefore c is independent of s And since v is constant over time because ws v for all s it follows that c is constant over time Notice from 8124 that P v serves as a relative Pareto weight on the type 1 person The recursive formulation brings out that because P ws P v the relative Pareto weight remains constant over time and is independent of the realization of st The planner imposes complete risk sharing In chapter 21 we shall encounter recursive formulations again Impedi ments to risk sharing that occur in the form either of enforcement or of informa tion constraints will impel the planner sometimes to make continuation values respond to the current realization of shocks to endowments or preferences 813 Concluding remarks The framework in this chapter serves much of macroeconomics either as foun dation or straw man benchmark is a kinder phrase than straw man It is the foundation of extensive literatures on asset pricing and risk sharing We describe the literature on asset pricing in more detail in chapters 13 and 14 The model also serves as benchmark or point of departure for a variety of models designed to confront observations that seem inconsistent with complete markets In particular for models with exogenously imposed incomplete mar kets see chapters 17 on precautionary saving and 18 on incomplete markets For models with endogenous incomplete markets see chapters 21 and 22 on en forcement and information problems For models of money see chapters 27 and 28 To take monetary theory as an example complete markets models assign no role to money because they contain an efficient multilateral trading mechanism with such extensive netting of claims that no additional asset is required to facil itate bilateral exchanges Any modern model of money introduces frictions that impede complete markets Some monetary models eg the cashinadvance model of Lucas 1981 impose minimal impediments to complete markets in 288 Equilibrium with Complete Markets ways that preserve many of the assetpricing implications of complete markets models while also activating classical monetary doctrines like the quantity the ory of money The shopping time model of chapter 27 is constructed in a similar spirit Other monetary models such as the Townsend turnpike model of chapter 28 or the KiyotakiWright search model of chapter 29 impose more extensive frictions on multilateral exchanges and leave the complete markets model far ther behind Before leaving the complete markets model well put it to work in several of the following chapters 294 Equilibrium with Complete Markets 8B2 Equilibrium prices reflect beliefs Do competitive equilibrium prices accurately reflect available information If accurately means embed correct probability assessments the theory pre sented in this appendix answers no not at first but yes asymptotically The yes asymptotically answer formalizes Milton Friedmans assertion that com petition and survival of the fittest will eventually align the personal beliefs re flected in competitive equilibrium prices of risky securities with the objective probabilities that generate the data 8B3 Mispricing In our example what drives the divergence outcome is that the consumer with the less accurate beliefs pays too much when buying insurance and accepts too little when selling insurance The inexorable working of the law of large numbers eventually transfers more and more wealth to consumers with more accurate beliefs 8B4 Learning While we have presented simple examples in which agents dont learn about probabilities the same basic force continues to drive outcomes when consumers can learn Thus Blume and Easley 2006 presented richer examples with heterogeneous beliefs across agents who update using Bayes rule Blume and Easleys analysis covers cases in which the sole source of heterogeneity is that dif ferent Bayesian agents have different priors They construct examples in which agents with either a looser or a less accurate prior receive equilibrium allocations that approach zero asymptotically Relative entropies again play a key role Incomplete markets 295 8B5 Role of complete markets In the body of this chapter we showed that Pareto optimal consumption alloca tions are competitive equilibrium allocations for two alternative trading struc tures with complete markets one with trading of many securities only at time 0 and another with trading each period t 0 of far fewer oneperiod securities In studying these structures we maintained the homogeneous beliefs of preference specification 821 Equivalence of Pareto optimal allocations to competitive equilibrium allocations also applies to the heterogenous beliefs setting of this appendix The assertions about limiting allocations that we have made in this ap pendix all come from manipulating firstorder condition 8B2 for our Pareto problem These assertions about outcomes in complete markets economies dont carry over to incomplete market economies for example of the type to be analyzed in chapter 18 Indeed there exist examples of incomplete markets economies in which the consumption of the consumer with less accurate beliefs grows over time19 C Incomplete markets Beker and Chattopadhyay 2010 analyze infinite horizon economies with two consumers one good and incomplete markets So long as an equilibrium re mains effectively constrained by market incompleteness Beker and Chattopad hyay prove that either a the consumption of both consumers is arbitrarily close to zero infinitely often or b the consumption of one consumer converges to zero The result prevails whether or not beliefs are heterogeneous Moreover a consumer whose consumption eventually vanishes can be marginally more pa tient or have more accurate beliefs than another consumer whose consumption remains positive These outcomes stand in contrast to those with complete markets as illustrated by an example to be presented in section 8C2 To attain an outcome in which both consumers consumptions remain pos itive Beker and Chattopadhyay show that it is sufficient to assume that indi viduals endowments are uniformly positive and governed by a Markov process Imposition of a uniform bound on the value of a consumers debt prevents a 19 See Blume and Easley 2006 and Cogley Sargent and Tsyrennikov 2014 Incomplete markets 301 It now follows from 8C13 that if both consumers have the same discount factor β1 β2 and if both hold the same true beliefs π1s π2s πs for all s S then the ratio of consumer 1s consumption to consumer 2s consumption diverges to Given that feasibility market clearing imposes an upper bound on consumer 1s consumption c1 tst Ytst we conclude that c2 T sT 0 as T goes to infinity ie consumer 2s consumption vanishes with probability one In contrast in a complete market economy the consumption allocation would be invariant to calender time and just depend on the aggregate endowment realization and a set of timeinvariant Pareto weights Actually our finding in this example does not depend on the beliefs held by consumer 2 since his subjective probabilities π2s s S are absent from equilibrium expression 8C13 For the sake of the argument suppose that consumer 2 holds the true beliefs but that consumer 1 has incorrect beliefs As shown above for π1s s S sufficiently close to πs s S consumer 2s consumption eventually vanishes This differs from the outcome in the complete market economy that we studied in section 8B1 where the consumer with the true beliefs would eventually consume the entire aggregate endowment Using the same line of reasoning another implication of 8C13 is that even if consumer 1 is marginally less patient than consumer 2 β1 β2 and allowing for the possibility that consumer 1 also has marginally incorrect beliefs it still follows that consumer 2s consumption goes to zero as time goes to infinity The consumption of a more patient consumer with more accurate beliefs could not vanish in a complete market economy 8C3 Beneficial market incompleteness Modifying the preceding example we now assume that the preference param eter of consumer 1 is γ 1 ie both consumers have a logarithmic utility function and that discount factors are identical β1 β2 β According to 8C11 ξ1s 1 for all s S and hence the consumption ratio c1 tstc2 tst in 8C13 remains constant over time In this incomplete markets economy it turns out that the consumption allocation does not depend on whether con sumers beliefs are correct or incorrect Furthermore the allocation equals the allocation that would prevail in a complete markets economy with correct beliefs This outcome motivates our section title beneficial market incompleteness we Incomplete markets 303 economy We have already verified equality for t 0 because the allocation c2 0s0 λ2Y0s0 1 βY0s0 is indeed the consumption of consumer 2 in an incomplete market economy as given by 8C5a where consumer 2 con sumes a fraction 1β of his initially accumulated wealth a2 0s0 Y0s0 To confirm that it is the equilibrium allocation for all future periods we conjecture that consumption shares λ1 λ2 constitute an incomplete market equilibrium then compute the implied equilibrium prices ptst from Euler equations Next given those prices we compute consumers choices of consumption and verify that they coincide with the conjectured consumption shares λ1 λ2 We do this with consumer 2s Euler equation 8C6 ptst β Yt1st1λ2Yt1st11 λ2Ytst1 βYtst 8C17 Given these prices we use consumer 2s decision rule in 8C5b to compute his asset choice at time 0 b2 1s0 βY0s0p0s0 1 ie consumer 2 pur chases one unit of the asset with asset payoffs next period equal to µ1s1 Y1s1 It follows that consumer 2s beginningofperiod wealth in period 1 is a2 1s1 Y1s1 so we can apply the same reasoning to period 1 according to decision rules 8C5 consumer 2 consumes a fraction 1 β of a2 1s1 and saves the rest by purchasing assets b2 2s1 βa2 1s1p1s1 1 This continues ad infinitum where in each period consumer 2 lends a fraction β of his accu mulated wealth to consumer 1 in exchange for consumer 1s entire endowment next period In this way we verify our conjecture that the incomplete mar ket economy has an equilibrium allocation with constant consumption shares λ1 λ2 β 1 β Because our argument has not involved any mathematical expectations consumers beliefs can be either correct or incorrect23 What features of the example explain why the equilibrium allocation of the incomplete market economy with or without correct beliefs equals that of the complete market economy with correct beliefs Starting with the case of correct beliefs we know from section 861 that a common utility function of 23 Why dont consumers beliefs matter in the incomplete markets economy Actually our assertion that no expectations were involved in the above reasoning is subject to a qualification That qualification is best seen by replacing consumer 2s Euler equation in the above reasoning with that of consumer 1 as given by 8C4 This switch reinserts expectations into our reasoning but as before for consumer 2 in 8C6 these expectations now also vanish for consumer 1 when the product µt1st1c1 t1st11 is constant across realizations of next periods stochastic event st1 304 Equilibrium with Complete Markets the constant relative riskaversion CRRA form implies that an efficient alloca tion prescribes that each consumer consumes a constant share of the aggregate endowment in all periods and all states So it seems to be important that the ex ogenously specified payoffs of the single asset in the incomplete market economy are perfectly correlated with the aggregate endowment What role is played by the exogenous package of Arrow securities implicit in the single asset in the incomplete market economy How can we be certain that this mix can support the same allocation that would prevail in a complete market economy The answer is that consumer 1 owns the entire aggregate endowment after period 0 and hence the efficient risk sharing characterized here by having each distinct consumer consume a constant fraction of the aggregate endowment can be achieved by trading Arrow securities for each possible state next period in proportion to the realizations of the aggregate endowment in those states The trades implicit in the bundle associated with the single asset in the incomplete market economy accomplish exactly this Exercises Exercise 81 Family economics I There is one consumption good and one input labor A family has two members named 1 and 2 The family is run by person 1 His welfare function is λ1 log c1 λ2log c2 n2 where λ1 and λ2 are positive Pareto weights c1 c2 are consumption of person 1 and person 2 respectively and n2 is labor supplied by person 2 Person 2 has no labor but is endowed with s units of the consumption good where s 0 Feasible allocations satisfy c1 c2 n2 s a Formulate the Pareto problem as a Lagrangian b Solve the Pareto problem for an optimal allocation and Lagrange multiplier Exercises 305 c Interpret the Lagrange multiplier as a shadow price and tell the object of which it is the shadow price d Describe how the family could be reorganized as a competitive economy being careful to identify an initial distribution of property and a price system e Compute a competitive equilibrium of the economy that you identified in part d f Please tell how n2 would respond to different values of s Exercise 82 Family economics II Modify the economy in exercise 81 in the following way only Instead of being endowed with s units of consumption household 1 is endowed with 1 unit of the consumption good and one unit of labor now produces sn2 units of the consumption good where s 0 So a feasible allocation now satisfies c1 c2 sn2 1 Preferences are identical with those described in exercise 825 Please answer counterparts of parts a f for this family Exercise 83 Existence of representative consumer Suppose consumers 1 and 2 have oneperiod utility functions uc1 and wc2 respectively where u and w are both increasing strictly concave twice differ entiable functions of a scalar consumption rate Let c 0 be the total amount the single consumption good available to be allocated between consumers 1 and 2 Where θ 0 1 is a Pareto weight consider the Pareto problem vθc max c1c2 θuc1 1 θwc2 subject to the constraint c1 c2 c Show that the solution of this problem has the form of a concave utility function vθc which depends on the Pareto weight θ Where c1c θ c2c θ is a Pareto optimal allocation show that v θc θuc1c θ 1 θwc2c θ The function vθc is the utility function of a representative consumer A representative consumer always lurks within a complete markets competitive equilibrium even with heterogeneous preferences At a competitive equilibrium 330 Equilibrium with Complete Markets into a 2 2 matrix Q2 whose i j component Q2 ij is the price of one unit of consumption when Markov state st2 sj two periods ahead when the Markov state st today is in state si Please give formulas for all elements of Q2 in terms of the fundamental parameters of the economy β γ λ δ y1 y2 d inverse problem An outsider observes this economy The outsider knows the theoretical structure of the economy but does not know the parameter val ues λ δ β γ y1 y2 But at a particular date at which st s1 the outsider observes Q11 1 Q11 2 Q21 1 Q21 2 Please interpret these observations From these observations alone can the outsider infer λ δ β Please explain your answer Chapter 9 Overlapping Generations This chapter describes the pure exchange overlapping generations model of Paul Samuelson 1958 We begin with an abstract presentation that treats the over lapping generations model as a special case of the chapter 8 general equilibrium model with complete markets and all trades occurring at time 0 A peculiar type of heterogeneity across agents distinguishes the model Each individual cares about consumption only at two adjacent dates and the set of individuals who care about consumption at a particular date includes some who care about consumption one period earlier and others who care about consumption one pe riod later We shall study how this special preference and demographic pattern affects some of the outcomes of the chapter 8 model While it helps to reveal the fundamental structure allowing complete mar kets with time 0 trading in an overlapping generations model strains credulity The formalism envisions that equilibrium price and quantity sequences are set at time 0 before the participants who are to execute the trades have been born For that reason most applied work with the overlapping generations model adopts a sequentialtrading arrangement like the sequential trade in Arrow securities described in chapter 8 The sequentialtrading arrangement has all trades executed by agents living in the here and now Nevertheless equilibrium quantities and intertemporal prices are equivalent between these two trading arrangements Therefore analytical results found in one setting transfer to the other Later in the chapter we use versions of the model with sequential trading to tell how the overlapping generations model provides a framework for thinking about equilibria with government debt andor valued fiat currency intergener ational transfers and fiscal policy 331 332 Overlapping Generations 91 Endowments and preferences Time is discrete starts at t 1 and lasts forever so t 1 2 There is an infinity of agents named i 0 1 We can also regard i as agent is period of birth There is a single good at each date The good is not storable There is no uncertainty Each agent has a strictly concave twice continuously differentiable oneperiod utility function uc which is strictly increasing in consumption c of the one good Agent i consumes a vector ci ci t t1 and has the special utility function U ici uci i uci i1 i 1 911a U 0c0 uc0 1 911b Notice that agent i only wants goods dated i and i 1 The interpretation of equations 911 is that agent i lives during periods i and i 1 and wants to consume only when he is alive Each household has an endowment sequence yi satisfying yi i 0 yi i1 0 yi t 0 t i or i 1 Thus households are endowed with goods only when they are alive 92 Time 0 trading We use the definition of competitive equilibrium from chapter 8 Thus we temporarily suspend disbelief and proceed in the style of Debreu 1959 with time 0 trading Specifically we imagine that there is a clearinghouse at time 0 that posts prices and at those prices aggregates demands and supplies for goods in different periods An equilibrium price vector makes markets for all periods t 2 clear but there may be excess supply in period 1 that is the clearinghouse might end up with goods left over in period 1 Any such excess supply of goods in period 1 can be given to the initial old generation without any effects on the equilibrium price vector since those old agents optimally consume all their wealth in period 1 and do not want to buy goods in future periods The reason for our special treatment of period 1 will become clear as we proceed 334 Overlapping Generations notice that each households firstorder conditions are satisfied and that the allocation is feasible Extensive intergenerational trade occurs at time 0 at the equilibrium price vector q0 t Constraint 923 holds with equality for all t 2 but with strict inequality for t 1 Some of the t 1 consumption good is left unconsumed 2 Equilibrium L a lowinterestrate equilibrium Set q0 1 1 q0 t1 q0 t uǫ u1ǫ α 1 Set ci t yi t for all i t This equilibrium is autarkic with prices being set to eradicate all trade 922 Relation to welfare theorems As we shall explain in more detail later equilibrium H Pareto dominates equi librium L In equilibrium H every generation after the initial old one is better off and no generation is worse off than in equilibrium L The equilibrium H alloca tion is strange because some of the time 1 good is not consumed leaving room to set up a giveaway program to the initial old that makes them better off and costs subsequent generations nothing We shall see how the institution of either perpetual government debt or of fiat money can accomplish this purpose1 Equilibrium L is a competitive equilibrium that evidently fails to satisfy one of the assumptions needed to deliver the first fundamental theorem of welfare economics which identifies conditions under which a competitive equilibrium allocation is Pareto optimal2 The condition of the theorem that is violated by equilibrium L is the assumption that the value of the aggregate endowment at the equilibrium prices is finite3 1 See Karl Shell 1971 for an investigation that characterizes why some competitive equi libria in overlapping generations models fail to be Pareto optimal Shell cites earlier studies that had sought reasons why the welfare theorems seem to fail in the overlapping generations structure 2 See MasColell Whinston and Green 1995 and Debreu 1954 3 Note that if the horizon of the economy were finite then the counterpart of equilibrium H would not exist and the allocation of the counterpart of equilibrium L would be Pareto optimal Time 0 trading 335 923 Nonstationary equilibria Our example economy has more equilibria To construct more equilibria we summarize preferences and consumption decisions in terms of an offer curve We describe a graphical apparatus proposed by David Gale 1973 and used to good advantage by William Brock 1990 Definition The households offer curve is the locus of ci i ci i1 that solves max ci ici i1 Uci subject to ci i αici i1 yi i αiyi i1 Here αi q0 i1 q0 i the reciprocal of the oneperiod gross rate of return from period i to i 1 is treated as a parameter Evidently the offer curve solves the following pair of equations ci i αici i1 yi i αiyi i1 925a uci i1 uci i αi 925b for αi 0 We denote the offer curve by ψci i ci i1 0 The graphical construction of the offer curve is illustrated in Figure 921 We trace it out by varying αi in the households problem and reading tangency points between the households indifference curve and the budget line The resulting locus depends on the endowment vector and lies above the indifference curve through the endowment vector By construction the following property is also true at the intersection between the offer curve and a straight line through the endowment point the straight line is tangent to an indifference curve4 4 Given our assumptions on preferences and endowments the conscientious reader will note that Figure 921 appears distorted because the offer curve really ought to intersect the feasibility line along the 45 degree line with ct t ct t1 ie at the allocation affiliated with equilibrium H above 336 Overlapping Generations Offer curve Feasibility line to the endowment corresponding Indifference curve c t t1 c t1 t yt t y t t1 c t t Figure 921 The offer curve and feasibility line Following Gale 1973 we can use the offer curve and a straight line de picting feasibility in the ci i ci1 i plane to construct a machine for computing equilibrium allocations and prices In particular we can use the following pair of difference equations to solve for an equilibrium allocation For i 1 the equations are5 ψci i ci i1 0 926a ci i ci1 i yi i yi1 i 926b We take c1 1 as an initial condition After the allocation has been computed the equilibrium price system can be computed from q0 i uci i for all i 1 5 By imposing equation 926b with equality we are implicitly possibly including a give away program to the initial old 340 Overlapping Generations In terms of the logarithmic preference example 5 below the difference equa tion 929 becomes modified to αi 1 2d ǫ ǫ1 1 αi1 927 Example 4 Government expenditures Take the preferences and endowments to be as in example 1 again but now alter the feasibility condition to be ci i ci1 i g yi i yi1 i for all i 1 where g 0 is a positive level of government purchases The clearinghouse is now looking for an equilibrium price vector such that this feasibility constraint is satisfied We assume that government purchases do not give utility The offer curve and the feasibility line look as in Figure 924 Notice that the endowment point yi i yi i1 lies outside the relevant feasibility line Formally this graph looks like example 3 but with a negative dividend d Now there are two stationary equilibria with α 1 and a continuum of equilibria converging to the higher α equilibrium the one with the lower slope α1 of the associated budget line Equilibria with α 1 cannot be ruled out by the argument in example 3 because no ones endowment sequence receives infinite value when α 1 Later we shall interpret this example as one in which a government finances a constant deficit either by money creation or by borrowing at a negative real net interest rate We shall discuss this and other examples in a setting with sequential trading Example 5 Log utility Suppose that uc ln c and that the endowment is described by equations 924 Then the offer curve is given by the recursive formulas ci i 51 ǫ αiǫ ci i1 α1 i ci i Let αi be the gross rate of return facing the young at i Feasibility at i and the offer curves then imply 1 2αi1 1 ǫ αi1ǫ 51 ǫ αiǫ 1 928 This implies the difference equation αi ǫ1 ǫ1 1 αi1 929 Time 0 trading 341 c t t1 c t1 t y t t1 yt t without government spendings Offer curve Feasibility line government spendings spendings government with Feasibility line low inflation equilibrium High interest rate Low interest rate equilibrium high inflation c t t Figure 924 Equilibria with debt or moneyfinanced gov ernment deficit finance See Figure 922 An equilibrium αi sequence must satisfy equation 928 and have αi 0 for all i Evidently αi 1 for all i 1 is an equilibrium α sequence So is any αi sequence satisfying equation 928 and α1 1 α1 1 will not work because equation 928 implies that the tail of αi is an unbounded negative sequence The limiting value of αi for any α1 1 is 1ǫ ǫ uǫu1ǫ which is the interest factor associated with the stationary autarkic equilibrium Notice that Figure 922 suggests that the stationary αi 1 equilibrium is not stable while the autarkic equilibrium is 342 Overlapping Generations 93 Sequential trading We now alter the trading arrangement to bring them into line with standard presentations of the overlapping generations model We abandon the time 0 complete markets trading arrangement and replace it with sequential trading in which a durable asset either government debt or unbacked fiat money or claims on a Lucas tree is passed from old to young Some crossgeneration transfers occur with voluntary exchanges while others are engineered by government tax and transfer programs 94 Money In Samuelsons 1958 version of the model trading occurs sequentially through a medium of exchange an inconvertible or fiat currency In Samuelsons model preferences and endowments are as described above with one impor tant additional component of the endowment At date t 1 old agents are endowed in the aggregate with M 0 units of intrinsically worthless currency No one has promised to redeem the currency for goods The currency is not backed by any government promise to redeem it for goods But as Samuelson showed there exists a system of expectations that makes unbacked currency be valued Currency will be valued today if people expect it to be valued to morrow Samuelson thus envisioned a situation in which currency is backed by expectations without promises For each date t 1 young agents purchase mi t units of currency at a price of 1pt units of the time t consumption good Here pt 0 is the time t price level At each t 1 each old agent exchanges his holdings of currency for the time t consumption good The budget constraints of a young agent born in period i 1 are ci i mi i pi yi i 941 ci i1 mi i pi1 yi i1 942 mi i 0 943 Deficit finance 345 In the monetary equilibrium time t real balances equal the per capita saving of the young and the per capita dissaving of the old To be a monetary equilibrium both quantities must be positive for all t 1 A converse of the proposition is true Proposition Let ci be an equilibrium allocation for the fiat money economy Then there is a competitive equilibrium with time 0 trading with the same allocation provided that the endowment of the initial old is augmented with an appropriate transfer from the clearinghouse To verify this proposition we have to construct the required transfer from the clearinghouse to the initial old Evidently it is y1 1 c1 1 We invite the reader to complete the proof 95 Deficit finance For the rest of this chapter we shall assume sequential trading With sequential trading of fiat currency this section reinterprets one of our earlier examples with time 0 trading the example with government spending Consider the following overlapping generations model The population is constant At each date t 1 N identical young agents are endowed with yt t yt t1 w1 w2 where w1 w2 0 A government levies lumpsum taxes of τ1 on each young agent and τ2 on each old agent alive at each t 1 There are N old people at time 1 each of whom is endowed with w2 units of the consumption good and M0 0 units of inconvertible perfectly durable fiat currency The initial old have utility function c0 1 The young have utility function uct t uct t1 For each date t 1 the government augments the currency supply according to Mt Mt1 ptg τ1 τ2 951 where g is a constant stream of government expenditures per capita and 0 pt is the price level If pt we intend that equation 951 be interpreted as g τ1 τ2 952 For each t 1 each young persons behavior is summarized by st fRt τ1 τ2 arg max s0 uw1 τ1 s uw2 τ2 Rts 953 346 Overlapping Generations Definition An equilibrium with valued fiat currency is a pair of positive sequences Mt pt such that a given the price level sequence Mtpt fRt the dependence on τ1 τ2 being understood b Rt ptpt1 and c the government budget constraint 951 is satisfied for all t 1 The condition fRt Mtpt can be written as fRt Mt1pt Mt Mt1pt The left side is the saving of the young The first term on the right side is the dissaving of the old the real value of currency that they exchange for time t consumption The second term on the right is the dissaving of the government its deficit which is the real value of the additional currency that the government prints at t and uses to purchase time t goods from the young To compute an equilibrium define d g τ1 τ2 and write equation 951 as Mt pt Mt1 pt1 pt1 pt d for t 2 and M1 p1 M0 p1 d for t 1 Substitute the equilibrium condition Mtpt fRt into these equations to get fRt fRt1Rt1 d 954a for t 2 and fR1 M0 p1 d 954b Given p1 which determines an initial R1 by means of equation 954b equations 954 form an autonomous difference equation in Rt With ap propriate transformations of variables this system can be solved using Figure 924 Deficit finance 347 951 Steady states and the Laffer curve Lets seek a stationary solution of equations 954 a quest rendered reasonable by the fact that fRt is time invariant because the endowment and the tax patterns as well as the government deficit d are timeinvariant Guess that Rt R for t 1 Then equations 954 become fR1 R d 955a fR M0 p1 d 955b For example suppose that uc lnc Then fR w1τ1 2 w2τ2 2R We have graphed fR1 R against d in Figure 951 Notice that if there is one solution for equation 955a then there are at least two Reciprocal High inflation equilibrium low interest rate Low inflation equilibrium high interest rate government spendings Seigneuriage earnings of the gross inflation rate Figure 951 The Laffer curve in revenues from the inflation tax Here 1R can be interpreted as a tax rate on real balances and fR1 R is a Laffer curve for the inflation tax rate The highreturn lowtax R R is associated with the good Laffer curve stationary equilibrium and the low return hightax R R comes with the bad Laffer curve stationary equilibrium Once R is determined we can determine p1 from equation 955b Figure 951 is isomorphic with Figure 924 The saving rate function fR can be deduced from the offer curve Thus a version of Figure 924 can be used to solve the difference equation 954a graphically If we do so we discover a continuum of nonstationary solutions of equation 954a all but one of which have Rt R as t Thus the bad Laffer curve equilibrium is stable Optimality and the existence of monetary equilibria 351 where τ s t is the time t tax on a person born in period s 97 Optimality and the existence of monetary equilibria Wallace 1980 discusses the connection between nonoptimality of the equilib rium without valued money and existence of monetary equilibria Abstracting from his assumption of a storage technology we study how the arguments ap ply to a pure endowment economy The environment is as follows At any date t the population consists of Nt young agents and Nt1 old agents where Nt nNt1 with n 0 Each young person is endowed with y1 0 goods and an old person receives the endowment y2 0 Preferences of a young agent at time t are given by the utility function uct t ct t1 which is twice differentiable with indifference curves that are convex to the origin The two goods in the utility function are normal goods and θc1 c2 u1c1 c2u2c1 c2 the marginal rate of substitution function approaches infinity as c2c1 ap proaches infinity and approaches zero as c2c1 approaches zero The welfare of the initial old agents at time 1 is strictly increasing in c0 1 and each one of them is endowed with y2 goods and m0 0 0 units of fiat money Thus the beginningofperiod aggregate nominal money balances in the initial period 1 are M0 N0m0 0 For all t 1 Mt the posttransfer time t stock of fiat money obeys Mt zMt1 with z 0 The time t transfer or tax z 1Mt1 is divided equally at time t among the Nt1 members of the current old generation The transfers or taxes are fully anticipated and are viewed as lumpsum they do not depend on consumption and saving behavior The budget constraints of a young agent born in period t are ct t mt t pt y1 971 ct t1 y2 mt t pt1 z 1 Nt Mt pt1 972 mt t 0 973 352 Overlapping Generations where pt 0 is the time t price level In a nonmonetary equilibrium the price level is infinite so the real values of both money holdings and transfers are zero Since all members in a generation are identical the nonmonetary equilibrium is autarky with a marginal rate of substitution equal to θaut u1y1 y2 u2y1 y2 We ask two questions about this economy Under what circumstances does a monetary equilibrium exist And when it exists under what circumstances does it improve matters Let ˆmt denote the equilibrium real money balances of a young agent at time t ˆmt MtNtpt Substitution of equilibrium money holdings into budget constraints 971 and 972 at equality yield ct t y1 ˆmt and ct t1 y2 n ˆmt1 In a monetary equilibrium ˆmt 0 for all t and the marginal rate of substitution θct t ct t1 satisfies θy1 ˆmt y2 n ˆmt1 pt pt1 θaut t 1 974 The equality part of 974 is the firstorder condition for money holdings of an agent born in period t evaluated at the equilibrium allocation Since ct t y1 and ct t1 y2 in a monetary equilibrium the inequality in 974 follows from the assumption that the two goods in the utility function are normal goods Another useful characterization of the equilibrium rate of return on money ptpt1 can be obtained as follows By the rule generating Mt and the equi librium condition Mtpt Nt ˆmt we have for all t pt pt1 Mt1 zMt pt pt1 Nt1 ˆmt1 zNt ˆmt n z ˆmt1 ˆmt 975 We are now ready to address our first question under what circumstances does a monetary equilibrium exist Proposition θautz n is necessary and sufficient for the existence of at least one monetary equilibrium Proof We first establish necessity Suppose to the contrary that there is a monetary equilibrium and θautzn 1 Then by the inequality part of 974 and expression 975 we have for all t ˆmt1 ˆmt zθaut n 1 976 Optimality and the existence of monetary equilibria 353 If zθautn 1 one plus the net growth rate of ˆmt is bounded uniformly above one and hence the sequence ˆmt is unbounded which is inconsistent with an equilibrium because real money balances per capita cannot exceed the en dowment y1 of a young agent If zθautn 1 the strictly increasing sequence ˆmt in 976 might not be unbounded but converge to some constant ˆm According to 974 and 975 the marginal rate of substitution will then converge to nz which by assumption is now equal to θaut the marginal rate of substitution in autarky Thus real balances must be zero in the limit which contradicts the existence of a strictly increasing sequence of positive real bal ances in 976 To show sufficiency we prove the existence of a unique equilibrium with constant per capita real money balances when θautz n Substitute our can didate equilibrium ˆmt ˆmt1 ˆm into 974 and 975 which yields two equilibrium conditions θy1 ˆm y2 n ˆm n z θaut The inequality part is satisfied under the parameter restriction of the proposi tion and we only have to show the existence of ˆm 0 y1 that satisfies the equality part But the existence and uniqueness of such a ˆm is trivial Note that the marginal rate of substitution on the left side of the equality is equal to θaut when ˆm 0 Next our assumptions on preferences imply that the marginal rate of substitution is strictly increasing in ˆm and approaches infinity when ˆm approaches y1 The stationary monetary equilibrium in the proof will be referred to as the ˆm equilibrium In general there are other nonstationary monetary equilibria when the parameter condition of the proposition is satisfied For example in the case of logarithmic preferences and a constant population recall the con tinuum of equilibria indexed by the scalar c 0 in expression 948 But here we choose to focus solely on the stationary ˆm equilibrium and its welfare implications The ˆm equilibrium will be compared to other feasible allocations using the Pareto criterion Evidently an allocation C c0 1 ct t ct t1 t 1 is feasible if Ntct t Nt1ct1 t Nty1 Nt1y2 t 1 or equivalently 354 Overlapping Generations nct t ct1 t ny1 y2 t 1 977 The definition of Pareto optimality is Definition A feasible allocation C is Pareto optimal if there is no other feasible allocation C such that c0 1 c0 1 uct t ct t1 uct t ct t1 t 1 and at least one of these weak inequalities holds with strict inequality We first examine under what circumstances the nonmonetary equilibrium autarky is Pareto optimal Proposition θaut n is necessary and sufficient for the optimality of the nonmonetary equilibrium autarky Proof To establish sufficiency suppose to the contrary that there exists an other feasible allocation C that is Pareto superior to autarky and θaut n Without loss of generality assume that the allocation C satisfies 977 with equality Given an allocation that is Pareto superior to autarky but that does not satisfy 977 one can easily construct another allocation that is Pareto superior to the given allocation and hence to autarky Let period t be the first period when this alternative allocation C differs from the autarkic allocation The requirement that the old generation in this period is not made worse off ct1 t y2 implies that the first perturbation from the autarkic allocation must be ct t y1 with the subsequent implication that ct t1 y2 It follows that the consumption of young agents at time t 1 must also fall below y1 and we define ǫt1 y1 ct1 t1 0 978 Now given ct1 t1 we compute the smallest number ct1 t2 that satisfies uct1 t1 ct1 t2 uy1 y2 Let ct1 t2 be the solution to this problem Since the allocation C is Pareto superior to autarky we have ct1 t2 ct1 t2 Before using this inequality though we want to derive a convenient expression for ct1 t2 356 Overlapping Generations To establish necessity we prove the existence of an alternative feasible al location ˆC that is Pareto superior to autarky when θaut n First pick an ǫ 0 sufficiently small so that θaut fǫ n 9713 where f is defined implicitly by equation 979 Second set ˆct t y1 ǫ ˆc1 and ˆct t1 y2 ǫθaut fǫ ˆc2 t 1 9714 That is we have constructed a consumption bundle ˆc1 ˆc2 that lies on the same indifference curve as y1 y2 and from 9713 and 9714 we have ˆc2 y2 nǫ which ensures that the condition for feasibility 977 is satisfied for t 2 By setting ˆc0 1 y2 nǫ feasibility is also satisfied in period 1 and the initial old generation is then strictly better off under the alternative allocation ˆC With a constant nominal money supply z 1 the two propositions show that a monetary equilibrium exists if and only if the nonmonetary equilibrium is suboptimal In that case the following proposition establishes that the sta tionary ˆm equilibrium is optimal Proposition Given θautz n then z 1 is necessary and sufficient for the optimality of the stationary monetary equilibrium ˆm Proof The class of feasible stationary allocations with ct t ct t1 c1 c2 for all t 1 is given by c1 c2 n y1 y2 n 9715 ie the condition for feasibility in 977 It follows that the ˆm equilibrium satisfies 9715 at equality and we denote the associated consumption alloca tion of an agent born at time t 1 by ˆc1 ˆc2 It is also the case that ˆc1 ˆc2 maximizes an agents utility subject to budget constraints 971 and 972 The consolidation of these two constraints yields c1 z nc2 y1 z ny2 z n z 1 Nt Mt pt1 9716 Optimality and the existence of monetary equilibria 357 where we have used the stationary rate or return in 975 ptpt1 nz After also invoking zMt Mt1 n Nt1Nt and the equilibrium condition Mt1pt1Nt1 ˆm expression 9716 simplifies to c1 z nc2 y1 z ny2 z 1 ˆm 9717 To prove the statement about necessity Figure 971 depicts the two curves 9715 and 9717 when condition z 1 fails to hold ie we assume that z 1 The point that maximizes utility subject to 9715 is denoted c1 c2 Transitivity of preferences and the fact that the slope of budget line 9717 is flatter than that of 9715 imply that ˆc1 ˆc2 lies southeast of c1 c2 By revealed preference then c1 c2 is preferred to ˆc1 ˆc2 and all generations born in period t 1 are better off under the allocation C The initial old generation can also be made better off under this alternative allocation since it is feasible to strictly increase their consumption c0 1 y2 ny1 c1 1 y2 ny1 ˆc1 1 ˆc0 1 Thus we have established that z 1 is necessary for the optimality of the stationary monetary equilibrium ˆm To prove sufficiency note that 974 975 and z 1 imply that θˆc1 ˆc2 n z n We can then construct an argument that is analogous to the sufficiency part of the proof to the preceding proposition As pointed out by Wallace 1980 the proposition implies no connection be tween the path of the price level in an ˆm equilibrium and the optimality of that equilibrium Thus there may be an optimal monetary equilibrium with positive inflation for example if θaut n z 1 and there may be a nonoptimal mon etary equilibrium with a constant price level for example if z n 1 θaut What counts is the nominal quantity of fiat money The proposition suggests that the quantity of money should not be increased In particular if z 1 then an optimal ˆm equilibrium exists whenever the nonmonetary equilibrium is nonoptimal 362 Overlapping Generations The left side is the demand for savings or the demand for currency while the right side is the supply consisting of privately issued IOUs the first term and governmentissued currency The right side is thus an abstract version of what is called M1 which is a sum of privately issued IOUs demand deposits and governmentissued reserves and currency 983 Nonstationary equilibria Mathematically the equilibrium conditions for the model with log preferences and two groups have the same structure as the model analyzed previously in equations 947 and 948 with simple reinterpretations of parameters We leave it to the reader here and in an exercise to show that if there exists a stationary equilibrium with valued fiat currency then there exists a continuum of equilibria with valued fiat currency all but one of which have the real value of government currency approaching zero asymptotically A linear difference equation like 947 supports this conclusion 984 The real bills doctrine In nineteenthcentury Europe and the early days of the Federal Reserve system in the United States central banks conducted open market operations not by purchasing government securities but by purchasing safe riskfree shortterm private IOUs We now analyze this oldfashioned type of open market operation We allow the government to issue additional currency each period It uses the proceeds exclusively to purchase private IOUs make loans to private agents in the amount Lt at time t Such open market operations are subject to the sequence of restrictions Lt Rt1Lt1 Ht Ht1 pt 982 for t 1 and H0 H 0 given L0 0 Here Lt is the amount of the time t consumption good that the government lends to the private sector from period t to period t1 Equation 982 states that the government finances these loans in two ways first by rolling over the proceeds Rt1Lt1 from the repayment of last periods loans and second by injecting new currency in the amount 366 Overlapping Generations 910 Concluding remarks The overlapping generations model is a workhorse in analyses of public finance welfare economics and demographics Diamond 1965 studied some fiscal pol icy issues within a version of the model with a neoclassical production He showed that depending on preference and productivity parameters equilibria of the model can have too much capital and that such capital overaccumula tion can be corrected by having the government issue and perpetually roll over unbacked debt11 Auerbach and Kotlikoff 1987 formulated a longlived over lapping generations model with capital labor production and various kinds of taxes They used the model to study a host of fiscal issues RiosRull 1994a used a calibrated overlapping generations growth model to examine the quanti tative importance of market incompleteness for insuring against aggregate risk See Attanasio 2000 for a review of theories and evidence about consumption within lifecycle models Several authors in a 1980 volume edited by John Kareken and Neil Wallace argued through example that the overlapping generations model is useful for analyzing a variety of issues in monetary economics We refer to that volume McCandless and Wallace 1992 Champ and Freeman 1994 Brock 1990 and Sargent 1987b for a variety of applications of the overlapping generations model to issues in monetary economics Exercises Exercise 91 At each date t 1 an economy consists of overlapping generations of a constant number N of twoperiodlived agents Young agents born in t have preferences over consumption streams of a single good that are ordered by uct t uct t1 where uc c1γ1 γ and where ci t is the consumption of an agent born at i in time t It is understood that γ 0 and that when γ 1 uc ln c Each young agent born at t 1 has identical preferences and endowment pattern w1 w2 where w1 is the endowment when young and w2 is the endowment when old Assume 0 w2 w1 In addition there are some initial old agents at time 1 who are endowed with w2 of the time 1 11 Abel Mankiw Summers and Zeckhauser 1989 propose an empirical test of whether there is capital overaccumulation in the US economy and conclude that there is not Exercises 367 consumption good and who order consumption streams by c0 1 The initial old ie the old at t 1 are also endowed with M units of unbacked fiat currency The stock of currency is constant over time a Find the saving function of a young agent b Define an equilibrium with valued fiat currency c Define a stationary equilibrium with valued fiat currency d Compute a stationary equilibrium with valued fiat currency e Describe how many equilibria with valued fiat currency there are You are not being asked to compute them f Compute the limiting value as t of the rate of return on currency in each of the nonstationary equilibria with valued fiat currency Justify your calculations Exercise 92 Consider an economy with overlapping generations of a constant population of an even number N of twoperiodlived agents New young agents are born at each date t 1 Half of the young agents are endowed with w1 when young and 0 when old The other half are endowed with 0 when young and w2 when old Assume 0 w2 w1 Preferences of all young agents are as in problem 1 with γ 1 Half of the N initial old are endowed with w2 units of the consumption good and half are endowed with nothing Each old person orders consumption streams by c0 1 Each old person at t 1 is endowed with M units of unbacked fiat currency No other generation is endowed with fiat currency The stock of fiat currency is fixed over time a Find the saving function of each of the two types of young person for t 1 b Define an equilibrium without valued fiat currency Compute all such equi libria c Define an equilibrium with valued fiat currency d Compute all the nonstochastic equilibria with valued fiat currency e Argue that there is a unique stationary equilibrium with valued fiat currency f How are the various equilibria with valued fiat currency ranked by the Pareto criterion 368 Overlapping Generations Exercise 93 Take the economy of exercise 91 but make one change Endow the initial old with a tree that yields a constant dividend of d 0 units of the consumption good for each t 1 a Compute all the equilibria with valued fiat currency b Compute all the equilibria without valued fiat currency c If you want you can answer both parts of this question in the context of the following particular numerical example w1 10 w2 5 d 000001 Exercise 94 Take the economy of exercise 91 and make the following two changes First assume that γ 1 Second assume that the number of young agents born at t is Nt nNt 1 where N0 0 is given and n 1 Everything else about the economy remains the same a Compute an equilibrium without valued fiat money b Compute a stationary equilibrium with valued fiat money Exercise 95 Consider an economy consisting of overlapping generations of two periodlived consumers At each date t 1 there are born Nt identical young people each of whom is endowed with w1 0 units of a single consumption good when young and w2 0 units of the consumption good when old Assume that w2 w1 The consumption good is not storable The population of young people is described by Nt nNt 1 where n 0 Young people born at t rank utility streams according to lnct t lnct t1 where ci t is the consumption of the time t good of an agent born in i In addition there are N0 old people at time 1 each of whom is endowed with w2 units of the time 1 consumption good The old at t 1 are also endowed with one unit of unbacked pieces of infinitely durable but intrinsically worthless pieces of paper called fiat money a Define an equilibrium without valued fiat currency Compute such an equi librium b Define an equilibrium with valued fiat currency c Compute all equilibria with valued fiat currency d Find the limiting rates of return on currency as t in each of the equilibria that you found in part c Compare them with the oneperiod interest rate in the equilibrium in part a Exercises 369 e Are the equilibria in part c ranked according to the Pareto criterion Exercise 96 Exchange rate determinacy The world consists of two economies named i 1 2 which except for their governments policies are copies of one another At each date t 1 there is a single consumption good which is storable but only for rich people Each economy consists of overlapping generations of twoperiodlived agents For each t 1 in economy i N poor people and N rich people are born Let ch t s yh t s be the time s consumption endowment of a type h agent born at t Poor agents are endowed with yh t t yh t t 1 α 0 rich agents are endowed with yh t t yh t t 1 β 0 where β α In each country there are 2N initial old who are endowed in the aggregate with Hi0 units of an unbacked currency and with 2Nǫ units of the time 1 consumption good For the rich people storing k units of the time t consumption good produces Rk units of the time t 1 consumption good where R 1 is a fixed gross rate of return on storage Rich people can earn the rate of return R either by storing goods or by lending to either government by means of indexed bonds We assume that poor people are prevented from storing capital or holding indexed government debt by the sort of denomination and intermediation restrictions described by Sargent and Wallace 1982 For each t 1 all young agents order consumption streams according to ln ch t t ln ch t t 1 For t 1 the government of country i finances a stream of purchases to be thrown into the ocean of Git subject to the following budget constraint 1 Git RBit 1 Bit Hit Hit 1 pit Tit where Bi0 0 pit is the price level in country i Tit are lumpsum taxes levied by the government on the rich young people at time t Hit is the stock of is fiat currency at the end of period t Bit is the stock of indexed government interestbearing debt held by the rich of either country The government does not explicitly tax poor people but might tax through an inflation tax Each government levies a lumpsum tax of TitN on each young rich citizen of its own country Poor people in both countries are free to hold whichever currency they prefer Rich people can hold debt of either government and can also store storage and both government debts bear a constant gross rate of return R 370 Overlapping Generations a Define an equilibrium with valued fiat currencies in both countries b In a nonstochastic equilibrium verify the following proposition if an equilib rium exists in which both fiat currencies are valued the exchange rate between the two currencies must be constant over time c Suppose that government policy in each country is characterized by specified exogenous levels Git Gi Tit Ti Bit 0 t 1 The remaining elements of government policy adjust to satisfy the government budget con straints Assume that the exogenous components of policy have been set so that an equilibrium with two valued fiat currencies exists Under this descrip tion of policy show that the equilibrium exchange rate is indeterminate d Suppose that government policy in each country is described as follows Git Gi Tit Ti Hit 1 Hi1 Bit Bi1 t 1 Show that if there exists an equilibrium with two valued fiat currencies the exchange rate is determinate e Suppose that government policy in country 1 is specified in terms of exoge nous levels of s1 H1t H1t 1p1t t 2 and G1t G1 t 1 For country 2 government policy consists of exogenous levels of B2t B21 G2t G2t 1 Show that if there exists an equilibrium with two valued fiat currencies then the exchange rate is determinate Exercise 97 Credit controls Consider the following overlapping generations model At each date t 1 there appear N twoperiodlived young people said to be of generation t who live and consume during periods t and t 1 At time t 1 there exist N old people who are endowed with H0 units of paper dollars which they offer to supply inelastically to the young of generation 1 in exchange for goods Let pt be the price of the one good in the model measured in dollars per time t good For each t 1 N2 members of generation t are endowed with y 0 units of the good at t and 0 units at t 1 whereas the remaining N2 members of generation t are endowed with 0 units of the good at t and y 0 units when they are old All members of all generations have the same utility function uch t t ch t t 1 ln ch t t ln ch t t 1 where ch t s is the consumption of agent h of generation t in period s The old at t 1 simply maximize ch 01 The consumption good is nonstorable The currency supply is constant through time so Ht H0 t 1 Exercises 371 a Define a competitive equilibrium without valued currency for this model Who trades what with whom b In the equilibrium without valued fiat currency compute competitive equi librium values of the gross return on consumption loans the consumption al location of the old at t 1 and that of the borrowers and lenders for t 1 c Define a competitive equilibrium with valued currency Who trades what with whom d Prove that for this economy there does not exist a competitive equilibrium with valued currency e Now suppose that the government imposes the restriction that lh t t1 rt y4 where lh t t1 rt represents claims on t 1period con sumption purchased if positive or sold if negative by household h of gener ation t This is a restriction on the amount of borrowing For an equilibrium without valued currency compute the consumption allocation and the gross rate of return on consumption loans f In the setup of part e show that there exists an equilibrium with valued currency in which the price level obeys the quantity theory equation pt qH0N Find a formula for the undetermined coefficient q Compute the consumption allocation and the equilibrium rate of return on consumption loans g Are lenders better off in economy b or economy f What about borrowers What about the old of period 1 generation 0 Exercise 98 Inside money and real bills Consider the following overlapping generations model of twoperiodlived people At each date t 1 there are born N1 individuals of type 1 who are endowed with y 0 units of the consumption good when they are young and zero units when they are old there are also born N2 individuals of type 2 who are endowed with zero units of the consumption good when they are young and Y 0 units when they are old The consumption good is nonstorable At time t 1 there are N old people all of the same type each endowed with zero units of the consumption good and H0N units of unbacked paper called fiat currency The populations of type 1 and 2 individuals N1 and N2 remain constant for all t 1 The young of each generation are identical in preferences and maximize 372 Overlapping Generations the utility function ln ch t tln ch t t 1 where ch t s is consumption in the sth period of a member h of generation t a Consider the equilibrium without valued currency that is the equilibrium in which there is no trade between generations Let 1 rt be the gross rate of return on consumption loans Find a formula for 1 rt as a function of N1 N2 y and Y b Suppose that N1 N2 y and Y are such that 1rt 1 in the equilibrium without valued currency Then prove that there can exist no quantitytheory style equilibrium where fiat currency is valued and where the price level pt obeys the quantity theory equation pt q H0 where q is a positive constant and pt is measured in units of currency per unit good c Suppose that N1 N2 y and Y are such that in the nonvaluedcurrency equilibrium 1 rt 1 Prove that there exists an equilibrium in which fiat currency is valued and that there obtains the quantity theory equation pt q H0 where q is a constant Construct an argument to show that the equilibrium with valued currency is not Pareto superior to the nonvalued currency equilibrium d Suppose that N1 N2 y and Y are such that in the preceding nonvalued currency economy 1 rt 1 there exists an equilibrium in which fiat currency is valued Let p be the stationary equilibrium price level in that economy Now consider an alternative economy identical with the preceding one in all respects except for the following feature a government each period purchases a constant amount Lg of consumption loans and pays for them by issuing debt on itself called inside money MI in the amount MIt Lgpt The government never retires the inside money using the proceeds of the loans to finance new purchases of consumption loans in subsequent periods The quantity of outside money or currency remains H0 whereas the total high power money is now H0 MIt i Show that in this economy there exists a valuedcurrency equilibrium in which the price level is constant over time at pt p or equivalently with p qH0 where q is defined in part c ii Explain why government purchases of private debt are not inflationary in this economy Exercises 373 iii In many models onceandforall government openmarket operations in private debt normally affect real variables andor price level What ac counts for the difference between those models and the one in this exercise Exercise 99 Social security and the price level Consider an economy economy I that consists of overlapping generations of twoperiodlived people At each date t 1 there is born a constant number N of young people who desire to consume both when they are young at t and when they are old at t 1 Each young person has the utility function ln ctt ln ctt 1 where cst is time t consumption of an agent born at s For all dates t 1 young people are endowed with y 0 units of a single nonstorable consumption good when they are young and zero units when they are old In addition at time t 1 there are N old people endowed in the aggregate with H units of unbacked fiat currency Let pt be the nominal price level at t denominated in dollars per time t good a Define and compute an equilibrium with valued fiat currency for this econ omy Argue that it exists and is unique Now consider a second economy economy II that is identical to economy I except that economy II possesses a social security system In particular at each date t 1 the government taxes τ 0 units of the time t consumption good away from each young person and at the same time gives τ units of the time t consumption good to each old person then alive b Does economy II possess an equilibrium with valued fiat currency De scribe the restrictions on the parameter τ if any that are needed to ensure the existence of such an equilibrium c If an equilibrium with valued fiat currency exists is it unique d Consider the stationary equilibrium with valued fiat currency Is it unique Describe how the value of currency or price level would vary across economies with differences in the size of the social security system as measured by τ Exercise 910 Seignorage Consider an economy consisting of overlapping generations of twoperiodlived agents At each date t 1 there are born N1 lenders who are endowed with α 0 units of the single consumption good when they are young and zero units when they are old At each date t 1 there are also born N2 borrowers who 374 Overlapping Generations are endowed with zero units of the consumption good when they are young and β 0 units when they are old The good is nonstorable and N1 and N2 are constant through time The economy starts at time 1 at which time there are N old people who are in the aggregate endowed with H0 units of unbacked intrinsically worthless pieces of paper called dollars Assume that α β N1 and N2 are such that N2β N1α 1 Assume that everyone has preferences uch t t ch t t 1 ln ch t t ln ch t t 1 where ch t s is consumption of time s good of agent h born at time t a Compute the equilibrium interest rate on consumption loans in the equilib rium without valued currency b Construct a brief argument to establish whether or not the equilibrium without valued currency is Pareto optimal The economy also contains a government that purchases and destroys Gt units of the good in period t t 1 The government finances its purchases entirely by currency creation That is at time t Gt Ht Ht 1 pt where Ht Ht 1 is the additional dollars printed by the government at t and pt is the price level at t The government is assumed to increase Ht according to Ht zHt 1 z 1 where z is a constant for all time t 1 At time t old people who carried Ht 1 dollars between t 1 and t offer these Ht 1 dollars in exchange for time t goods Also at t the government offers Ht Ht 1 dollars for goods so that Ht is the total supply of dollars at time t to be carried over by the young into time t 1 c Assume that 1z N2βN1α Show that under this assumption there exists a continuum of equilibria with valued currency Exercises 375 d Display the unique stationary equilibrium with valued currency in the form of a quantity theory equation Compute the equilibrium rate of return on currency and consumption loans e Argue that if 1z N2βN1α then there exists no valuedcurrency equilib rium Interpret this result Hint Look at the rate of return on consumption loans in the equilibrium without valued currency f Find the value of z that maximizes the governments Gt in a stationary equilibrium Compare this with the largest value of z that is compatible with the existence of a valuedcurrency equilibrium Exercise 911 Unpleasant monetarist arithmetic Consider an economy in which the aggregate demand for government currency for t 1 is given by Mtptd gR1t where R1t is the gross rate of return on currency between t and t 1 Mt is the stock of currency at t and pt is the value of currency in terms of goods at t the reciprocal of the price level The function gR satisfies 1 gR1 R hR 0 for R R 1 where hR 0 for R R R 1 R 0 and where hR 0 for R Rm hR 0 for R Rm hRm D where D is a positive number to be defined shortly The government faces an infinitely elastic demand for its interestbearing bonds at a constantovertime gross rate of return R2 1 The government finances a budget deficit D defined as government purchases minus explicit taxes that is constant over time The governments budget constraint is 2 D ptMt Mt 1 Bt Bt 1R2 t 1 subject to B0 0 M0 0 In equilibrium 3 Mtpt gR1t The government is free to choose paths of Mt and Bt subject to equations 2 and 3 a Prove that for Bt 0 for all t 0 there exist two stationary equilibria for this model Exercises 377 with equality if currency is valued 1 rt 1 rtptp 0 pt The loan marketclearing condition in this economy is f1 rt Htpt a Define an equilibrium b Prove that there exists a unique monetary equilibrium in this economy and compute it Exercise 913 BryantKeynesWallace Consider an economy consisting of overlapping generations of twoperiodlived agents There is a constant population of N young agents born at each date t 1 There is a single consumption good that is not storable Each agent born in t 1 is endowed with w1 units of the consumption good when young and with w2 units when old where 0 w2 w1 Each agent born at t 1 has identical preferences ln ch t t ln ch t t 1 where ch t s is time s consumption of agent h born at time t In addition at time 1 there are alive N old people who are endowed with H0 units of unbacked paper currency and who want to maximize their consumption of the time 1 good A government attempts to finance a constant level of government purchases Gt G 0 for t 1 by printing new base money The governments budget constraint is G Ht Ht 1pt where pt is the price level at t and Ht is the stock of currency carried over from t to t 1 by agents born in t Let g GN be government purchases per young person Assume that purchases Gt yield no utility to private agents a Define a stationary equilibrium with valued fiat currency b Prove that for g sufficiently small there exists a stationary equilibrium with valued fiat currency c Prove that in general if there exists one stationary equilibrium with valued fiat currency with rate of return on currency 1rt 1r1 then there exists 378 Overlapping Generations at least one other stationary equilibrium with valued currency with 1 rt 1 r2 1 r1 d Tell whether the equilibria described in parts b and c are Pareto optimal among allocations among private agents of what is left after the government takes Gt G each period A proof is not required here an informal argument will suffice Now let the government institute a forced saving program of the following form At time 1 the government redeems the outstanding stock of currency H0 exchanging it for government bonds For t 1 the government offers each young consumer the option of saving at least F worth of time t goods in the form of bonds bearing a constant rate of return 1r2 A legal prohibition against private intermediation is instituted that prevents two or more private agents from sharing one of these bonds The governments budget constraint for t 2 is GN Bt Bt 11 r2 where Bt F Here Bt is the saving of a young agent at t At time t 1 the governments budget constraint is GN B1 H0 Np1 where p1 is the price level at which the initial currency stock is redeemed at t 1 The government sets F and r2 Consider stationary equilibria with Bt B for t 1 and r2 and F constant e Prove that if g is small enough for an equilibrium of the type described in part a to exist then a stationary equilibrium with forced saving exists Either a graphical argument or an algebraic argument is sufficient f Given g find the values of F and r2 that maximize the utility of a repre sentative young agent for t 1 g Is the equilibrium allocation associated with the values of F and 1 r2 found in part f optimal among those allocations that give Gt G to the government for all t 1 Here an informal argument will suffice Chapter 10 Ricardian Equivalence 101 Borrowing limits and Ricardian equivalence This chapter studies whether the timing of taxes matters Under some assump tions it does and under others it does not The Ricardian doctrine describes assumptions under which the timing of lump taxes does not matter In this chapter we will study how the timing of taxes interacts with restrictions on the ability of households to borrow We study the issue in two equivalent settings 1 an infinite horizon economy with an infinitely lived representative agent and 2 an infinite horizon economy with a sequence of oneperiodlived agents each of whom cares about its immediate descendant We assume that the interest rate is exogenously given For example the economy might be a small open economy that faces a given interest rate determined in the international capital market Chapters 11 amd 13 will describe general equilibrium analyses of the Ricardian doctrine where the interest rate is determined within the model The key findings of the chapter are that in the infinite horizon model Ricar dian equivalence holds under what we earlier called the natural borrowing limit but not under more stringent ones The natural borrowing limit lets households borrow up to the capitalized value of their endowment sequences These results have limited counterparts in the overlapping generations model since that model is equivalent to an infinite horizon model with a noborrowing constraint1 In the overlapping generations model a noborrowing constraint translates into a requirement that bequests be nonnegative Thus in the overlapping generations model the domain of the Ricardian proposition is restricted at least relative to the infinite horizon model under the natural borrowing limit 1 This is one of the insights in the influential paper of Barro 1974 that reignited modern interest in Ricardian equivalence 379 Linked generations interpretation 387 104 Linked generations interpretation Much of the preceding analysis with borrowing constraints applies to a setting with overlapping generations linked by a bequest motive Assume that there is a sequence of oneperiodlived agents For each t 0 there is a oneperiodlived agent who values consumption and the utility of his direct descendant a young person at time t 1 Preferences of a young person at t are ordered by uct βV bt1 where uc is the same utility function as in the previous section bt1 0 are bequests from the time t person to the time t 1 person and V bt1 is the maximized utility function of a time t1 agent The maximized utility function is defined recursively by V bt max ctbt1uct βV bt1 1041 where the maximization is subject to ct R1bt1 yt τt bt 1042 and bt1 0 The constraint bt1 0 requires that bequests cannot be negative Notice that a person cares about his direct descendant but not vice versa We continue to assume that there is an infinitely lived government whose taxes and purchasing and borrowing strategies are as described in the previous section In consumption outcomes this model is equivalent to the previous model with a noborrowing constraint Bequests here play the role of savings bt1 in the previous model A positive savings condition bt1 0 in the previous version of the model becomes an operational bequest motive in the overlapping generations model It follows that we can obtain a restricted Ricardian equivalence proposition qualified as in Proposition 2 The qualification is that the initial equilibrium must have an operational bequest motive for all t 0 and that the new tax policy must not be so different from the initial one that it renders the bequest motive inoperative 388 Ricardian Equivalence 105 Concluding remarks The arguments in this chapter were cast in a setting with an exogenous interest rate R and a capital market that is outside of the model When we discussed potential failures of Ricardian equivalence due to households facing noborrowing constraints we were also implicitly contemplating changes in the governments outside asset position For example consider an altered tax plan ˆτt t0 that satisfies 1036 and shifts taxes away from the future toward the present A large enough change will definitely ensure that the government is a lender in early periods But since the households are not allowed to become indebted the government must lend abroad and we can show that Ricardian equivalence breaks down The readers might be able to anticipate the nature of the general equilibrium proof of Ricardian equivalence in chapter 13 First private consumption and government expenditures must then be consistent with the aggregate endowment in each period ct gt yt which implies that an altered tax plan cannot affect the consumption allocation as long as government expenditures are kept the same Second interest rates are determined by intertemporal marginal rates of substitution evaluated at the equilibrium consumption allocation as studied in chapter 8 Hence an unchanged consumption allocation implies that interest rates are also unchanged Third at those very interest rates it can be shown that households would like to choose asset positions that exactly offset any changes in the governments asset holdings implied by an altered tax plan For example in the case of the tax change contemplated in the preceding paragraph the households would demand loans exactly equal to the rise in government lending generated by budget surpluses in early periods The households would use those loans to meet the higher taxes and thereby finance an unchanged consumption plan The finding of Ricardian equivalence in the infinitely lived agent model is a useful starting point for identifying alternative assumptions under which the irrelevance result might fail to hold7such as our imposition of borrowing con straints that are tighter than the natural debt limit Another deviation from the benchmark model is finitely lived agents as analyzed by Diamond 1965 and Blanchard 1985 But as suggested by Barro 1974 and shown in this 7 Seater 1993 reviews the theory and empirical evidence on Ricardian equivalence Concluding remarks 389 chapter Ricardian equivalence will continue to hold if agents are altruistic to ward their descendants and there is an operational bequest motive Bernheim and Bagwell 1988 take this argument to its extreme and formulate a model where all agents are interconnected because of linkages across dynastic families They show how those linkages can become extensive enough to render neutral all redistributive policies including ones attained via distortionary taxes But in general replacing lumpsum taxes by distortionary taxes is a surefire way to undo Ricardian equivalence see eg Barsky Mankiw and Zeldes 1986 We will return to the question of the timing of distortionary taxes in chapter 16 Kimball and Mankiw 1989 describe how incomplete markets can make the tim ing of taxes interact with a precautionary savings motive in a way that disarms Ricardian equivalence We take up precautionary savings and incomplete mar kets in chapters 17 and 18 Finally by allowing distorting taxes to be history dependent Bassetto and Kocherlakota 2004 attain a Ricardian equivalence result for a variety of taxes Chapter 11 Fiscal Policies in a Growth Model 111 Introduction This chapter studies effects of technology and fiscal shocks on equilibrium out comes in a nonstochastic growth model We use the model to state some classic doctrines about the effects of various types of taxes and also as a laboratory to exhibit numerical techniques for approximating equilibria and to display the structure of dynamic models in which decision makers have perfect foresight about future government decisions Foresight imparts effects on prices and al locations that precede government actions that cause them Following Hall 1971 we augment a nonstochastic version of the standard growth model with a government that purchases a stream of goods and that finances itself with an array of distorting flatrate taxes We take government behavior as exogenous1 which means that for us a government is simply a list of sequences for government purchases gt t0 and taxes τct τkt τnt τht t0 Here τct τkt τnt are respectively timevarying flatrate rates on consumption earnings from capital and labor earnings and τht is a lumpsum tax a head tax or poll tax Distorting taxes prevent a competitive equilibrium allocation from solving a planning problem Therefore to compute an equilibrium allocation and price system we solve a system of nonlinear difference equations consisting of the firstorder conditions for decision makers and the other equilibrium conditions We first use a method called shooting It produces an accurate approximation Less accurate but in some ways more revealing approximations can be found by following Hall 1971 who solved a linear approximation to the equilibrium conditions We apply the lag operators described in appendix A of chapter 2 to find and represent the solution in a way that is especially helpful in revealing the dynamic effects of perfectly foreseen alterations in taxes and expenditures and 1 In chapter 16 we take up a version of the model in which the government chooses taxes to maximize the utility of a representative consumer 391 Digression sequential version of government budget constraint 395 A graph of rtts against s for s 1 2 S is called the real yield curve at t An insight about the expectations theory of the term structure of interest rates can be gleaned from computing gross oneperiod holding period returns on zero coupon bonds of maturities 1 2 Consider the gross return earned by someone who at time 0 purchases one unit of time t consumption for qt units of the numeraire and then sells it at time 1 The person pays qt q0 units of time 0 consumption goods to earn qt q1 units of time 1 consumption goods The gross rate of return from this trade measured in time 1 consumption goods per unit of time 0 consumption goods is q0 q1 which does not depend on the date t of the good bought at time 0 and then sold at time 1 Evidently at time 0 the oneperiod return is identical for pure discount bonds of all maturities t 1 More generally at time t the oneperiod holding period gross return on zero coupon bonds of all maturities equals qt qt1 A way to characterize the expectations theory of the term structure of interest rates is by the requirement that the price vector qt t0 of zero coupon bonds must be such that oneperiod holding period yields are equated across zero coupon bonds of all maturities Note also how the price system qt t0 contains forecasts of oneperiod holding period yields on zero coupon bonds of all maturities at all dates t 0 In subsequent sections well indicate how the growth model with taxes and government expenditures links the term structure of interest rates to aspects of government fiscal policy 114 Digression sequential version of government budget constraint We have used the time 0 trading abstraction described in chapter 8 Sequential trading of oneperiod riskfree debt can also support the equilibrium allocations that we shall study in this chapter It is especially useful explicitly to describe the sequence of oneperiod government debt that is implicit in the equilibrium tax policies here 400 Fiscal Policies in a Growth Model The household inherits a given k0 that it takes as an initial condition and it is free to choose any sequence ct nt kt1 t0 that satisfies 1151 where all prices and tax rates are taken as given The objective of the household is to maximize lifetime utility 1121 which is increasing in consumption ct t0 and for one of our preference specifications below also increasing in leisure 1 nt t0 All else equal the household would be happier with larger values on the right side of 1151 preferably plus infinity which would enable it to purchase unlimited amounts of consumption goods Because resources are finite we know that the right side of the households budget constraint must be bounded in an equilibrium This fact leads to an important restriction on the price and tax sequences If the right side of the households budget constraint is to be bounded then the terms multiplying kt for t 1 must all equal zero because if any of them were strictly positive negative for some date t the household could make the right side of 1151 an arbitrarily large positive number by choosing an arbitrarily large positive negative value of kt On the one hand if one such term were strictly positive for some date t the household could purchase an arbitrarily large capital stock kt assembled at time t 1 with a presentvalue cost of qt1kt and then sell the rental services and the undepreciated part of that capital stock to be delivered at time t with a presentvalue income of 1 τktηt δ 1qtkt If such a transaction were to yield a strictly positive profit it would offer the consumer a pure arbitrage opportunity and the right side of 1151 would become unbounded On the other hand if there is one term multiplying kt that is strictly negative for some date t the household can make the right side of 1151 arbitrarily large and positive by short selling capital by setting kt 0 The household could turn to purchasers of capital assembled at time t 1 and sell synthetic units of capital to them Such a transaction need not involve any actual physical capital the household could merely undertake trades that would give the other party to the transaction the same costs and incomes as those associated with purchasing capital assembled at time t1 If such short sales of capital yield strictly positive profits it would provide the consumer with a pure arbitrage opportunity and the right side of 1151 would become unbounded Therefore the terms multiplying kt must equal zero for all t 1 so that qt qt1 1 τkt1ηt1 δ 1 1152 402 Fiscal Policies in a Growth Model The user cost of capital takes into account the rate of taxation of capital earn ings the capital gain or loss from t to t 1 and a depreciation cost8 1153 Household firstorder conditions So long as the noarbitrage conditions 1152 prevail households are indifferent about how much capital they hold Recalling that the oneperiod utility function is Uc 1 n let U1 U c and U2 U 1n so that U n U2 Then we have that the households firstorder conditions with respect to ct nt are βtU1t µqt1 τct 1155a βtU2t µqtwt1 τnt if nt 1 1155b where µ is a nonnegative Lagrange multiplier on the households budget con straint 1124 Multiplying the price system by a positive scalar simply rescales the multiplier µ so we are free to choose a numeraire by setting µ to an arbi trary positive number 1154 A theory of the term structure of interest rates Equation 1155a allows us to solve for qt as a function of consumption µqt βtU1t1 τct 1156a or in the special case that Uct 1 nt uct µqt βtuct1 τct 1156b In conjunction with the observations made in subsection 113 these formulas link the term structure of interest rates to the paths of ct τct The government policy gt τct τnt τkt τht t0 affects the term structure of interest rates directly via τct and indirectly via its impact on the path for ct t0 8 This is a discretetime version of a continuoustime formula derived by Hall and Jorgenson 1967 A digression on backsolving 409 many equilibria the class of tax and expenditure processes has to be restricted drastically to narrow the search for an equilibrium12 117 A digression on backsolving The shooting algorithm takes sequences for gt and the various tax rates as given and finds paths of the allocation ct kt1 t0 and the price system that solve the system of difference equations formed by 1163 and 1168 Thus the shooting algorithm views government policy as exogenous and the price system and allocation as endogenous Sims 1989 proposed another way to solve the growth model that exchanges the roles of some exogenous and endogenous variables In particular his backsolving approach takes a path ct t0 as given and then proceeds as follows Step 1 Given k0 and sequences for the various tax rates solve 1163 for a sequence kt1 Step 2 Given the sequences for ct kt1 solve the feasibility condition 1168a for a sequence of government expenditures gt t0 Step 3 Solve formulas 1168b1168e for an equilibrium price system The present model can be used to illustrate other applications of back solving For example we could start with a given process for qt use 1168b to solve for ct and proceed as in steps 1 and 2 above to determine processes for kt1 and gt and then finally compute the remaining prices from the as yet unused equations in 1168 Sims recommended this method because it adopts a flexible or symmetric attitude toward exogenous and endogenous variables DiazGimenez Prescott Fitzgerald and Alvarez 1992 Sargent and Smith 1997 and Sargent and Velde 1999 have all used the method We shall not use it in the remainder of this chapter but it is a useful method to have in our toolkit13 12 See chapter 16 for theories about how to choose taxes in socially optimal ways 13 Constantinides and Duffie 1996 used backsolving to reverse engineer a crosssection of endowment processes that with incomplete markets would prompt households to consume their endowments at a given stochastic process of asset prices Transition experiments with inelastic labor supply 411 119 Transition experiments with inelastic labor supply We continue to study the special case with Uc 1 n uc Figures 1191 through 1195 apply the shooting algorithm to an economy with uc 1 γ1c1γ fk kα with parameter values α 33 δ 2 β 95 and an initial constant level of g of 2 All of the experiments except one to be described in figure 1192 set the critical utility curvature parameter γ 2 We initially set all distorting taxes to zero and consider perturbations of them that we describe in the experiments below Figures 1191 to 1195 show responses to foreseen onceandforall increases in g τc and τk that occur at time T 10 where t 0 is the initial time period Prices induce effects that precede the policy changes that cause them We start all of our experiments from an initial steady state that is appropriate for the prejump settings of all government policy variables In each panel a dashed line displays a value associated with the steady state at the initial constant values of the policy vector A solid line depicts an equilibrium path under the new policy It starts from the value that was associated with an initial steady state that prevailed before the policy change at T 10 was announced Before date t T 10 the response of each variable is entirely due to expectations about future policy changes After date t 10 the response of each variable represents a purely transient response to a new stationary level of the forcing function in the form of the exogenous policy variables That is before t T the forcing function is changing as date T approaches after date T the policy vector has attained its new permanent level so that the only sources of dynamics are transient Discounted future values of fiscal variables impinge on current outcomes where the discount rate in question is endogenous while departures of the capital stock from its terminal steadystate value set in place a force for it to decay toward its steady state rate at a particular rate These two forces discounting of the future and transient decay back toward the terminal steady state are evident in the experiments portrayed in Figures 11911195 In section 11106 we express the decay rate as a function of the key curvature parameter γ in the oneperiod utility function uc 1 γ1c1γ and we note that the endogenous rate at which future fiscal variables are discounted is tightly linked to that decay rate 412 Fiscal Policies in a Growth Model 0 20 40 14 16 18 2 22 k 0 20 40 04 045 05 055 06 065 c 0 20 40 1 102 104 106 108 R 0 20 40 02 022 024 026 η 0 20 40 01 0 01 02 03 04 g Figure 1191 Response to foreseen onceandforall in crease in g at t 10 From left to right top to bottom k c R η g The dashed line is the original steady state 0 20 40 13 135 14 145 15 k 0 20 40 06 062 064 066 068 c 0 20 40 085 09 095 1 105 11 R 0 20 40 025 026 027 028 η 0 20 40 01 0 01 02 03 04 τc Figure 1194 Response to foreseen onceandforall in crease in τc at t 10 From left to right top to bottom k c R η τc Transition experiments with inelastic labor supply 413 0 20 40 14 16 18 2 22 k 0 20 40 04 045 05 055 06 065 c 0 20 40 1 102 104 106 108 R 0 20 40 02 022 024 026 η 0 20 40 01 0 01 02 03 04 g Figure 1192 Response to foreseen onceandforall in crease in g at t 10 From left to right top to bottom k c R η g The dashed lines show the original steady state The solid lines are for γ 2 while the dasheddotted lines are for γ 2 Foreseen jump in gt Figure 1191 shows the effects of a foreseen permanent increase in g at t T 10 that is financed by an increase in lumpsum taxes Although the steadystate value of the capital stock is unaffected this follows from the fact that g disappears from the steady state version of the Euler equation 1162 consumers make the capital stock vary over time If the government consumes more the household must consume less The competitive economy sends a signal to consumers that they must consume less in the form of an increase in the stream of lump sum taxes that the government uses to finance the increase in its expenditures Because consumers care about the present value of lumpsum taxes and are indifferent to their timing an adverse wealth effect on consumption precedes the actual rise in government expenditures Consumers choose immediately to increase their saving in response to the adverse wealth effect that they suffer from the increase in lumpsum taxes that finances the permanently higher level of government expenditures Because the present value of lumpsum taxes jumps immediately consumption also falls immediately in anticipation of the increase in government expenditures This leads to a gradual 414 Fiscal Policies in a Growth Model 0 20 40 04 045 05 055 06 065 c 0 20 40 0 02 04 06 08 1 q 0 20 40 0 002 004 006 rtt1 0 20 40 0 002 004 006 rtts s 0 20 40 01 0 01 02 03 04 g Figure 1193 Response to foreseen onceandforall in crease in g at t 10 From left to right top to bottom c q rtt1 and yield curves rtts for t 0 solid line t 10 dashdotted line and t 60 dashed line term to maturity s is on the x axis for the yield curve time t for the other panels buildup of capital in the dates between 0 and T followed by a gradual fall after T Variation over time in the capital stock helps smooth consumption over time so that the main force at work is the consumptionsmoothing motive featured in Milton Friedmans permanent income theory The variation over time in R reconciles the consumer to a consumption path that is not completely smooth According to 1169 the gradual increase and then the decrease in capital are inversely related to variations in the gross interest rate that reconcile the household to a consumption path that varies over time Figure 1192 compares the responses to a foreseen increase in g at t 10 for two economies our original economy with γ 2 shown in the solid line and an otherwise identical economy with γ 2 shown in the dasheddotted line The utility curvature parameter γ governs the households willingness to substitute consumption across time Lowering γ increases the households will ingness to substitute consumption across time This shows up in the equilibrium Transition experiments with inelastic labor supply 417 declining immediately due to a rise in current consumption and a growing flow of consumption The aftertax gross rate of return on capital starts rising at t 0 and increases until t 9 It falls precipitously at t 10 see formula 1168e because of the foreseen jump in τk Thereafter R rises as required by the transition dynamics that propel kt toward its new lower steady state Consumption is lower in the new steady state because the new lower steady state capital stock produces less output Consumption is smoother when γ 2 than when γ 2 Alterations in R accompany effects of the tax increase at t 10 on consumption at earlier and later dates So far we have explored consequences of foreseen onceandforall changes in government policy Next we describe some experiments in which there is a foreseen onetime change in a policy variable a pulse Foreseen onetime pulse in g10 Figure 1196 shows the effects of a foreseen onetime increase in gt at date t 10 that is financed entirely by alterations in lump sum taxes Consumption drops immediately then falls further over time in anticipation of the onetime surge in g Capital is accumulated before t 10 At t T 10 capital jumps downward because the government consumes it The reduction in capital is accompanied by a jump in R above its steadystate value The gross return R then falls toward its steady rate level and consumption rises at a diminishing rate toward its steadystate value This experiment highlights what again looks like a version of a permanent income theory response to a foreseen decrease in the resources available for the public to spend that is what the increase in g is about with effects that are modified by the general equilibrium adjustments of the gross return R 418 Fiscal Policies in a Growth Model 0 20 40 14 145 15 155 16 k 0 20 40 062 063 064 065 c 0 20 40 104 1045 105 1055 106 1065 R 0 20 40 024 025 026 027 η 0 10 20 30 01 0 01 02 03 04 g Figure 1196 Response to foreseen onetime pulse increase in g at t 10 From left to right top to bottom k c R η g 1110 Linear approximation The present model is simple enough that it is very easy to apply the shooting algorithm But for models with larger state spaces it can be more difficult to apply the shooting algorithm For those models a frequently used procedure is to obtain a linear or log linear approximation around a steady state of the difference equation for capital then to solve it to get an approximation of the dynamics in the vicinity of that steady state The present model is a good lab oratory for illustrating how to construct linear approximations In addition to providing an easy way to approximate a solution the method illuminates impor tant features of the solution by partitioning it into two parts14 1 a feedback part that portrays the transient response of the system to an initial condition k0 that deviates from an asymptotic steady state and 2 a feedforward part that shows the current effects of foreseen tax rates and expenditures15 To obtain a linear approximation perform the following steps16 14 Hall 1971 employed linear approximations to exhibit some of this structure 15 Vector autoregressions embed the consequences of both backwardlooking transient and forwardlooking foresight responses to government policies 16 For an extensive treatment of lag operators and their uses see Sargent 1987a Linear approximation 425 Furthermore as mentioned above because there are no distorting taxes in the initial steady state we know that λ1 1 βλ2 so that according to 11108 the feedforward response to future zs is a discounted sum that decays at rate βλ2 Thus when γ 0 anticipations of future zs have no effect on current k This is the other side of the coin of the immediate adjustment associated with the feedback part As the curvature parameter γ increases λ2 increases more rapidly at first more slowly later As γ increases the household values a smooth consumption path more and more highly Higher values of γ impart to the equilibrium capital sequence both a more sluggish feedback response and a feedforward response that puts relatively more weight on prospective values of the zs in the more distant future 0 1 2 3 4 5 6 7 0 01 02 03 04 05 06 07 08 09 1 λ2 γ Figure 11101 Feedback coefficient λ2 as a function γ evaluated at α 33 β 95 δ 2 g 2 426 Fiscal Policies in a Growth Model 11107 A remark about accuracy Euler equation errors It is important to estimate the accuracy of approximations One simple diag nostic tool is to take a candidate solution for a sequence ct kt1 substitute them into the two Euler equations 11121 and 11122 and call the devia tions between the left sides and the right sides the Euler equation errors26 An accurate method makes these errors small27 1111 Growth It is straightforward to alter the model to allow for exogenous growth We modify the production function to be Yt FKt Atnt 11111 where Yt is aggregate output Nt is total employment At is laboraugmenting technical change and FK AN is the same linearly homogeneous production function as before We assume that At follows the process At1 µt1At 11112 and will usually but not always assume that µt1 µ 1 We exploit the linear homogeneity of 11111 to express the production function as yt fkt 11113 where fk Fk 1 and now kt Kt ntAt yt Yt ntAt We say that kt and yt are measured per unit of effective labor Atnt We also let ct Ct Atnt and gt Gt Atnt where Ct and Gt are total consumption and total government expen ditures respectively We consider the special case in which labor is inelastically supplied Then feasibility can be summarized by the following modified version of 1161 kt1 µ1 t1fkt 1 δkt gt ct 11114 26 For more about this method see Den Haan and Marcet 1994 and Judd 1998 27 Calculating Euler equation errors but for a different purpose goes back a long time In chapter 2 of The General Theory of Interest Prices and Money John Maynard Keynes noted that plugging in data not a candidate simulation into 11122 gives big residuals Keynes therefore assumed that 11122 does not hold workers are off their labor supply curve 428 Fiscal Policies in a Growth Model 0 20 40 11 115 12 125 k 0 20 40 058 0585 059 0595 06 c 0 20 40 1095 11 1105 111 R 0 20 40 029 0295 03 0305 031 η 0 10 20 30 1 101 102 103 104 µ Figure 11111 Response to foreseen onceandforall in crease in rate of growth of productivity µ at t 10 From left to right top to bottom k c R η µ where now k c are measured in units of effective unit of labor Foreseen jump in productivity growth at t 10 Figure 11111 shows effects of a permanent increase from 102 to 1025 in the productivity gross growth rate µt at t 10 This figure and also Figure 11112 now measure c and k in effective units of labor The steadystate Euler equation 11117 guides main features of the outcomes and implies that a permanent increase in µ will lead to a decrease in the steadystate value of capital per unit of effective labor Because capital is more efficient even with less of it consumption per capita can be raised and that is what individuals care about Consumption jumps immediately because people are wealthier The increased productivity of capital spurred by the increase in µ leads to an increase in the gross return R Perfect foresight makes the effects of the increase in the growth of capital precede it Immediate unforeseen jump in productivity growth at t 1 Figure 11112 shows effects of an immediate jump in µ at t 0 It is instructive to compare these with the effects of the foreseen increase in Figure 11111 In Figure 11112 the paths of all variables are entirely dominated by the feedback Growth 429 0 20 40 11 115 12 125 k 0 20 40 058 0585 059 0595 06 0605 c 0 20 40 1095 11 1105 111 R 0 20 40 029 0295 03 0305 031 η 0 10 20 30 1 101 102 103 104 µ Figure 11112 Response to increase in rate of growth of productivity µ at t 0 From left to right top to bottom k c R η µ where now k c are measured in units of effective unit of labor part of the solution while before t 10 those in Figure 11111 have contribu tions from the feedforward part The absence of feedforward effects makes the paths of all variables in Figure 11112 smooth Consumption per effective unit of labor jumps immediately then declines smoothly toward its steady state as the economy moves to a lower level of capital per unit of effective labor The aftertax gross return R once again comoves with the consumption growth rate to verify the Euler equation 11117 Elastic labor supply 431 11121 Steadystate calculations To compute a steady state for this version of the model assume that government expenditures and all flatrate taxes are constant over time Steadystate versions of 11121 11122 are 1 β1 1 τkFkk n δ 11125 U2c 1 n U1c 1 n 1 τn 1 τc Fnk n 11126 and the steady state version of the feasibility condition 1122 is c g δk Fk n 11127 The linear homogeneity of Fk n means that equation 11125 by itself de termines the steadystate capitallabor ratio k n In particular where k k n notice that Fk n nfk and Fkk n f k It is helpful to use these facts to write 11127 as c g n fk δk 11128 Next letting β 1 1ρ 11125 can be expressed as δ ρ 1 τk f k 11129 an equation that determines a steadystate capitallabor ratio k An increase in 1 1τk decreases the capitallabor ratio but the steadystate capitallabor ratio is independent of the steady state values of τc τn However given the steady state value of the capitallabor ratio k flat rate taxes on consumption and labor income influence the steadystate levels of consumption and labor via the steady state equations 11126 and 11127 Formula 11126 reveals how both τc and τn distort the same laborleisure margin If we define ˇτc τnτc 1τc and ˇτk τk 1τk then it follows that 1τn 1τc 1 ˇτc and 1 1τk 1 ˇτk The wedge 1 ˇτc distorts the steadystate laborleisure decision via 11126 and the wedge 1 ˇτk distorts the steadystate capital labor ratio via 11129 Elastic labor supply 433 These asymptotic outcomes immediately drop out of our steady state equa tions The increase in g is accompanied by increases in k and n that leave the steady state capitallabor ratio unaltered as required by equation 11129 Equation 111211 then dictates that steadystate consumption per capita also remain unaltered 0 20 40 08 09 1 11 12 13 k 0 20 40 02 022 024 026 c 0 20 40 05 06 07 08 09 1 n 0 20 40 1 105 11 115 R 0 20 40 065 07 075 08 w 0 20 40 01 0 01 02 03 04 g Figure 11121 Elastic labor supply response to unfore seen increase in g at t 0 From left to right top to bottom k c n R w g The dashed line is the original steady state Unforeseen jump in τn Figure 11122 shows outcomes from an unforeseen increase in the marginal tax rate on labor τn once again accompanied by an adjustment in the present value of lump sum taxes required to balance the governments budget Here the effect is to shrink the economy As required by equation 11129 the steady state capital labor ratio is unaltered But equation 111211 then requires that steady state consumption per capita must fall in response to the increase in τn Both labor supplied n and capital fall in the new steady state Countervailing forces contributing to Prescott 2002 The preceding two experiments isolate forces that Prescott 2002 combines to reach his con clusion that Europes economic activity has been depressed relative to the US 434 Fiscal Policies in a Growth Model because its tax rates have been higher Prescotts numerical calculations acti vate the forces that shrink the economy in our second experiment that increases τn while shutting down the force to grow the economy implied by a larger g In particular Prescott assumes that crosscountry outcomes are generated by second experiment with lump sum transfers being used to rebate the revenues raised from the larger labor tax rate τn that he estimates to prevail in Europe If instead one assumes that higher taxes in Europe are used to pay for larger per capita government purchases then forces to grow the economy identified in our first experiment are unleashed making the adverse consequences for the level of economic activity of larger g τn pairs in Europe become much smaller than Prescott calculated 0 20 40 07 075 08 085 09 k 0 20 40 02 022 024 026 c 0 20 40 04 045 05 055 n 0 20 40 103 1035 104 1045 105 1055 R 0 20 40 076 078 08 082 084 w 0 20 40 01 0 01 02 03 04 τn Figure 11122 Elastic labor supply response to unfore seen increase in τn at t 0 From left to right top to bottom k c n R w τn The dashed line is the original steady state 436 Fiscal Policies in a Growth Model These equations teach us that the foreseen increase in τn sparks a substantial rearrangement in how the household distributes its work over time The effect of the permanent increase in τn at t 10 is to reduce the aftertax wage from t 10 onward though initially the real wage falls by less than the decrease in 1 τn because of the increase in the capital labor ratio induced by the drastic fall in n at t 10 Eventually as the pretax real wage w returns to its initial value the real wage falls by the entire amount of the decrease in 1 τn The decrease in the aftertax wage after t 10 makes it relatively more attractive to work before t 10 As a consequence nt rises above its initial steady state value before t 10 The household uses the extra income to purchase enough capital to keep the capitallabor ratio and consumption equal to their respective initial steady state values for the first nine periods This force increases nt in the periods before t 10 The effect of the build up of capital in the periods before t 0 is to attenuate the decrease in the after tax wage that occurs at t 10 because the equilibrium marginal product of labor has been raised higher than it would have been if capital had remained at its initial steady state value From t 10 onward the capital stock is drawn down and the marginal product of labor falls making the pretax real wage eventually return to its value in the initial steady state Mertens and Ravn 2011 use these effects to offer an interpretation of contractionary contributions that the Reagan tax cuts made to the US recession of the early 1980s 1113 A twocountry model This section describes a two country version of the basic model of this chapter The model has a structure similar to ones used in the international real business cycle literature eg Backus Kehoe and Kydland 1992 and is in the spirit of an analysis of distorting taxes by Mendoza and Tesar 1998 though our presentation differs from theirs We paste two countries together and allow them freely to trade goods claims on future goods but not labor We shall have to be careful in how we specify taxation of earnings by non residents There are now two countries like the one in previous sections Objects for the first country are denoted without asterisks while those for the second country bear asterisks There is international trade in goods capital and debt 438 Fiscal Policies in a Growth Model which together imply that aftertax rental rates on capital are equalized across the two countries 1 τ ktη t δ 1 τktηt δ 11134 No arbitrage conditions for Bf t for t 0 are qt qt1Rtt1 which implies that qt1 qtRt1t 11135 for t 1 Since domestic capital foreign capital and consumption loans bear the same rates of return by virtue of 11134 and 11135 portfolios are inde terminate We are free to set holdings of foreign capital equal to zero in each country if we allow Bf t to be nonzero Adopting this way of resolving portfolio indeterminacy is convenient because it economizes on the number of initial con ditions we have to specify Therefore we set holdings of foreign capital equal to zero in both countries but allow international lending Then given an initial level Bf 1 of debt from the domestic country to the foreign country and where Rt1t qt1 qt international debt dynamics satisfy Bf t Rt1tBf t1 ct kt1 1 δkt gt fkt 11136 and c t k t1 1 δk t g t Rt1tBf t1 fk t Bf t 11137 Firms firstorder conditions in the two countries are ηt f kt wt fkt ktf kt η t f k t w t fk t k t f k t 11138 International trade in goods establishes qt βt uct 1 τct µ uc t 1 τ ct 11139 where µ is a nonnegative number that is a function of the Lagrange multi plier on the budget constraint for a consumer in country and where we have normalized the Lagrange multiplier on the budget constraint of the domestic 440 Fiscal Policies in a Growth Model 11133 Initial equilibrium values Trade in physical capital and time 0 debt takes place before production and trade in other goods occurs at time 0 We shall always initialize international debt at zero Bf 1 0 a condition that we use to express that international trade in capital begins at time 0 Given an initial total worldwide capital stock ˇk0 ˇk 0 initial values of k0 and k 0 satisfy k0 k 0 ˇk0 ˇk 0 111316 1 τk0f k0 δτk0 1 τ k0f k 0 δτ k0 111317 The price of a unit of capital in either country at time 0 is pk0 1 τk0f k0 1 δ δτk0 111318 It follows that Bk0 pk0k0 ˇk0 111319 which says that the domestic country finances imports of physical capital from abroad by borrowing from the foreign country 11134 Shooting algorithm To apply a shooting algorithm we would search for pairs c0 µ that yield a pair k0 k 0 and paths ct c t kt k t Bf t T t0 that solve equations 111316 111317 111318 111319 11136 11139 and 111318 The shooting algorithm aims for k k that satisfy the steadystate equations 111312 111313 A twocountry model 441 10 20 30 08 1 12 14 16 k 10 20 30 045 05 055 06 065 07 c 10 20 30 105 11 115 12 125 R 10 20 30 025 03 035 04 045 η 0 10 20 30 0 02 04 06 x 10 20 30 1 08 06 04 02 0 Bf Figure 11131 Response to unforeseen opening of trade at time 1 From left to right top to bottom k c R η x and Bf The solid line is the domestic country the dashed line is the foreign country and the dashed dotted line is the original steady state 11135 Transition exercises In the onecountry exercises earlier in this chapter announcements of new poli cies always occurred at time 0 In the twocountry exercises to follow we assume that announcements of new paths of tax rates andor expenditures or trade regimes all occur at time 1 We do this to show some dramatic jumps in partic ular variables that occur at time 1 in response to announcements about changes that will occur at time 10 and later Showing variables at times 0 and 1 helps display some of the outcomes on which we shall focus here The production func tion is fk Akα Parameter values are β 95 γ 2 δ 2 α 33 A 1 g is initially 2 in both countries and all distorting taxes are initially 0 We describe outcomes from three exercises that illustrate two economic forces The first force is consumers desire to smooth consumption over time expressed through households consumption Euler equations The second force is that equilibrium outcomes must offer no opportunities for arbitrage expressed through equations that equate rates of returns on bonds and capital 442 Fiscal Policies in a Growth Model In the first two experiments all taxes are lump sum in both countries In the third experiment we activate a tax on capital in the domestic but not the foreign country In all experiments we allow lump sum taxes in both countries to adjust to satisfy government budget constraints in both countries 111351 Opening International Flows In our first example we study the transition dynamics for two countries when in period one newly produced output and stocks of capital but not labor suddenly become internationally mobile The two economies are initially identical in all aspects except for one we start the domestic economy at its autarkic steady state while we start the foreign economy at an initial capital stock below its au tarkic steady state Because there are no distorting taxes on returns to physical capital capital stocks in both economies converge to the same level In this experiment the domestic country is at its steady state capital stock while the poorer foreign country has a capital stock that is 5 less This means that initially before trade is opened at t 1 the marginal product of capital in the foreign country exceeded the marginal product capital in the domestic country that the foreign interest rate R 01 exceeded the domestic rate R01 and that consequently the foreign consumption growth rate exceeded the do mestic consumption growth rate The disparity of interest rates before trade is opened is a force for physical capital to flow from the domestic country to the foreign country once when trade is opened at t 1 Figure 11132 presents the transitional dynamics When countries become open to trade in goods and capital in period one there occurs an immediate reallocation of capital from the capitalrich domestic country to the capitalpoor foreign country This transfer of capital has to take place because if it didnt capital in different countries would yield different returns providing consumers in both countries with arbi trage opportunities Those cannot occur in equilibrium Before international trade had opened rental rates on capital and interest rates differed across country because marginal products of capital differed and consumption growth rates differed When trade opens at time 1 and capital is reallocated across countries to equalize returns the interest rate in the domestic country jumps at time one Because γ 2 this means that consumption c in the domestic country must fall The opposite is true for the foreign economy A twocountry model 443 Notice also that figure 11132 shows an investment spike abroad while there is a large decline in investment in the domestic economy This occurs because capital is reallocated from the domestic country to the foreign one This transfer is feasible because investment in capital is reversible The foreign country finances this import of physical capital by borrowing from the domestic country so Bf increases Foreign debt Bf continues to increase as both economies converge smoothly towards a steadystate with a positive level of B f Ultimately these differences account for differences in steadystate consumption by 2ρBf Opening trade in goods and capital at time 1 benefits consumers in both economies By opening up to capital flows the foreign country achieves conver gence to a steadystate consumption level at an accelerated rate This steady state consumption rate is lower than what it would be had the economy remained closed but this reduction in longrun consumption is more than compensated by the rapid increase in consumption and output in the shortrun In contrast domestic consumption falls in the shortrun as trade allows domestic consumers to accumulate foreign assets that support greater steadystate consumption This experiment shows the importance of studying transitional dynamics for welfare analysis In this example focusing only on steadystate consumption would lead to the false conclusion that opening markets are detrimental for poorer economies 111352 Foreseen Increase in g Figure 11132 presents transition dynamics after an increase in g in the domes tic economy from 2 to 4 that is announced ten periods in advance We start both economies from a steadystate with Bf 0 0 When the new g path is an nounced at time 1 consumption smoothing motives induce domestic households to increase their savings in response to the adverse shock to domestic private wealth that is caused at time 1 by the foreseen increase in domestic government purchases g Domestic households plan to use those savings to dampen the im pact on consumption in periods after g will have increased ten periods ahead Households save partly by accumulating more domestic capital in the shortrun their only source of assets in the closed economy version of this experiment In an open economy they have other ways to save namely by lending abroad The noarbitrage conditions connect adjustments of both types of saving the 444 Fiscal Policies in a Growth Model 10 20 30 40 14 15 16 17 18 19 k 10 20 30 40 045 05 055 06 065 07 c 10 20 30 40 102 103 104 105 106 107 R 10 20 30 40 025 03 035 04 045 x 0 20 40 01 0 01 02 03 04 g 10 20 30 40 1 08 06 04 02 0 Bf Figure 11132 Response to increase in g at time 10 fore seen at time 1 From left to right top to bottom k c R x g Bf The dasheddotted line is the original steady state in the do mestic country The dashed line denotes the foreign country increase in savings by domestic households will reduce the equilibrium return on bonds and capital in the foreign economy to prevent arbitrage opportunities Confronting the revised interest rate path that now begins with lower interest rates foreign households increase their rates of consumption and investment in physical capital These increases in foreign absorbtion are funded by increases in foreign consumers external debt After the announcement of the increase in g the paths for consumption and capital in both countries follow the same patterns because noarbitrage conditions equate the ratios of their marginal util ities of consumption Both countries continue to accumulate capital until the increase in g occurs After that domestic households begin consuming some of their capital Again by noarbitrage conditions when g actually increases both countries reduce their investment rates The domestic economy in turn starts running currentaccount deficits partially to fund the increase in g This means that foreign households begin repaying part of their external debt by reducing their capital stock Although not plotted in figure 11132 there is a sharp re duction in gross investment x in both countries when the increase in g occurs After t 10 all variable converge smoothly towards a new steady state where 446 Fiscal Policies in a Growth Model 111353 Foreseen increase in τk We now explore the impact of an increase in capital taxation in the domestic economy 10 periods after its announcement at t 1 Figure 11133 shows equi librium outcomes When the increase in τk is announced domestic households become aware that the domestic capital stock will eventually decline to increase gross returns to equalize aftertax returns across countries despite a higher do mestic tax rate on returns from capital Domestic households will reduce their capital stock by increasing their rate of consumption The consequent higher equilibrium world interest rates then also induces foreign households to increase consumption Prior to the increase in τk the domestic country runs a current account deficit When τk is eventually increased capital is rapidly reallocated across borders to preclude arbitrage opportunities leading to a lower interest rate on bonds The fall in the return on bonds occurs because the capital re turns tax τk in the domestic country will reduce the aftertax return on capital and because the foreign economy has a higher capital stock Foreign households fund this large purchase of capital with a sharp increase in external debt to be interpreted as a current account deficit After τk has increased the economies smoothly converge to a new steady state that features lower consumption rates in both countries and where the differences in the capital stock equate after tax returns It is useful to note that steadystate consumption in the foreign economy is higher than in the domestic country despite its perpetually having positive liabilities This occurs because foreign output is larger because the capital stock held abroad is also larger This example shows how via the noarbitrage conditions both countries share the impact of the shock and how fluctuations in capital stocks smooth over time the adjustments in consumption in both countries Concluding remarks 447 1114 Concluding remarks In chapter 12 we shall describe a stochastic version of the basic growth model and alternative ways of representing its competitive equilibrium29 Stochastic and nonstochastic versions of the growth model are widely used throughout aggregative economics to study a range of policy questions Brock and Mirman 1972 Kydland and Prescott 1982 and many others have used a stochastic version of the model to approximate features of the business cycle In much of the earlier literature on real business cycle models the phrase features of the business cycle has meant particular moments of some aggregate time series that have been filtered in a particular way to remove trends Lucas 1990 uses a nonstochastic model like the one in this chapter to prepare rough quantitative estimates of the eventual consequences of lowering taxes on capital and raising those on consumption or labor Prescott 2002 uses a version of the model in this chapter with leisure in the utility function together with some illustrative high labor supply elasticities to construct the argument that in the last two decades Europes economic activity has been depressed relative to that of the United States because Europe has taxed labor more highly that the United States Ingram Kocherlakota and Savin 1994 and Hall 1997 use actual data to construct the errors in the Euler equations associated with stochastic versions of the basic growth model and interpret them not as computational errors as in the procedure recommended in section 11107 but as measures of additional shocks that have to be added to the basic model to make it fit the data In the basic stochastic growth model described in chapter 12 the technology shock is the only shock but it cannot by itself account for the discrepancies that emerge in fitting all of the models Euler equations to the data A message of Ingram Kocherlakota and Savin 1994 and Hall 1997 is that more shocks are required to account for the data Wen 1998 and Otrok 2001 build growth models with more shocks and additional sources of dynamics fit them to US time series using likelihood functionbased methods and discuss the additional shocks and sources of data that are required to match the data See Christiano Eichenbaum and Evans 2003 and Christiano Motto and Rostagno 2003 for papers that add a number of additional shocks and measure their importance 29 It will be of particular interest to learn how to achieve a recursive representation of an equilibrium by finding an appropriate formulation of a state vector in terms of which to cast an equilibrium Because there are endogenous state variables in the growth model we shall have to extend the method used in chapter 8 448 Fiscal Policies in a Growth Model Greenwood Hercowitz and Krusell 1997 introduced what seems to be an important additional shock in the form of a technology shock that impinges directly on the relative price of investment goods Jonas Fisher 2006 develops econometric evidence attesting to the importance of this shock in accounting for aggregate fluctuations Davig Leeper and Walker 2012 use stochastic versions of the types of models discussed in this chapter to study issues of intertemporal fiscal balance SchmittGrohe and Uribe 2004b and Kim and Kim 2003 warn that the linear and log linear approximations described in this chapter can be treach erous when they are used to compare the welfare under alternative policies of economies like the ones described in this chapter in which distortions prevent equilibrium allocations from being optimal ones They describe ways of at taining locally more accurate welfare comparisons by constructing higher order approximations to decision rules and welfare functions A Log linear approximations Following Christiano 1990 a widespread practice is to obtain log linear rather than linear approximations Here is how this would be done for the model of this chapter Let log kt kt so that kt exp kt similarly let log gt gt Represent zt as zt expgt τkt τct note that only gt has been replaced by its log here Then proceed as follows to get a log linear approximation 1 Compute the steady state as before Set the government policy zt z a constant level Solve Hexpk expk expk z z 0 for a steady state k Of course this will give the same steady state for the original unlogged variables as we got earlier 2 Take firstorder Taylor series approximation around k z Hktkt k Hkt1kt1 k Hkt2kt2 k Hztzt z Hzt1zt1 z 0 11A1 But please remember here that the first component of zt is now gt 3 Write the resulting system as φ0kt2 φ1kt1 φ2kt A0 A1zt A2zt1 11A2 Exercises 455 a Define a competitive equilibrium with time 0 trading b Suppose that before time 0 the economy had been in a steady state in which g had always been zero and had been expected always to equal zero Find a formula for the initial steady state capital stock in a competitive equilibrium with time zero trading Let this value be k0 c At time 0 everyone suddenly wakes up to discover that from time 0 on government expenditures will be g 0 where g δk0 fk0 which implies that the new level of government expenditures would be feasible in the old steady state Suppose that the government finances the new path of expenditures by a capital levy at time T 0 The government imposes a capital levy by sending the household a bill for a fraction of the value of its capital at the time indicated Find the new steady state value of the capital stock in a competitive equilibrium Describe an algorithm to compute the fraction of the capital stock that the government must tax away at time 0 to finance its budget Find the new steady state value of the capital stock in a competitive equilibrium Describe the time paths of capital consumption and the interest rate from t 0 to t in the new equilibrium and compare them with their counterparts in the initial gt 0 equilibrium d Assume the same new path of government expenditures indicated in part c but now assume that the government imposes the onetime capital levy at time T 10 and that this is foreseen at time 0 Find the new steady state value of the capital stock in a competitive equilibrium that is associated with this tax policy Describe an algorithm to compute the fraction of the capital stock that the government must tax away at time T 10 to finance its budget Describe the time paths of capital consumption and the interest rate in this new equilibrium and compare them with their counterparts in part b and in the initial gt 0 equilibrium e Define a competitive equilibrium with sequential trading of oneperiod Arrow securities Describe how to compute such an equilibrium Describe the time path of the consumers holdings of oneperiod securities in a competitive equilibrium with one period Arrow securities under the government tax policy assumed in part d Describe the time path of government debt Exercises 459 Let qt ηt wt t0 be a price system Exercise 117 Consider an economy in which gt g 0 t 0 and in which initially the government finances all expenditures by lump sum taxes a Find a formula for the steady state capital labor ratio kt for this economy Find formulas for the steady state level of ct and Rt 1 δ f kt1 b Now suppose that starting from k0 k ie the steady state that you computed in part a the government suddenly increases the tax on earnings from capital to a constant level τk 0 The government adjusts lump sum taxes to keep the government budget balances Describe competitive equilibrium time paths for ct kt1 Rt and their relationship to corresponding values in the old steady state that you described in part a c Describe how the shapes of the paths that you found in part b depend on the curvature parameter γ in the utility function uc c1γ 1γ Higher values of γ imply higher curvature and more aversion to consumption path that fluctuate Higher values of γ imply that the consumer values smooth consumption paths even more d Starting from the steady state k that you computed in part a now consider a situation in which the government announces at time 0 that starting in period 10 the tax on earnings from capital τk will rise permanently to τk 0 The government adjusts its lump sum taxes to balance its budget i Find the new steady state values for kt ct Rt ii Describe the shapes of the transition paths from the initial steady states to the new one for kt ct Rt iii Describe how the shapes of the transition paths depend on the curvature parameter γ in the utility function uc Hint When γ is bigger consumers more strongly prefer smoother con sumption paths Recall the forces behind formula 111016 in section 11106 Exercises 463 c Plot r0t for this economy for t 1 2 10 this is what Bloomberg plots d Now assume that at time 0 starting from k0 k for the steady state you computed in part a the government unexpectedly and permanently raises the tax rate on income from capital τkt τk 0 to a positive rate i Plot rt1t for this economy for t 1 2 10 Explain how you got this outcome ii Plot r0t for this economy for t 1 2 10 Explain how you got this outcome Exercise 1111 This problem assumes the same economic environment as the previous exercise ie the growth model with fiscal policy Suppose that you observe the path for consumption per capita in figure 111 Say what you can about the likely behavior over time of kt Rt 1 1 τktf kt δ gt and τkt You are free to make up any story that is consistent with the model ct 0 t Figure 111 Consumption per capita Exercise 1112 Assume the same economic environment as in the previous two problems As sume that someone has observed the time path for ct in figure 112 a Describe a consistent set of assumptions about the fiscal policy that explains this time path for ct In doing so please distinguish carefully between changes in taxes and expenditures that are foreseen versus unforeseen 466 Fiscal Policies in a Growth Model 0 5 10 15 20 25 30 35 40 Time t r0t Figure 113 Yield to maturity r0t at time 0 as a function of term to maturity t finances its purchases by imposing lump sum taxes Describe ie draw graphs showing the time paths of ct kt1 Rt1 t0 and compare them to the outcomes that you obtained in part c What outcomes differ What outcomes if any are identical across the two economies Please explain e Starting from the same initial k0 assumed in part c assume now that gt g φfk0 0 for all t 0 where φ 0 1 δ Assume that the government must now finance these purchases by imposing a timeinvariant tax rate τk on capital each period The government cannot impose lump sum taxes or any other kind of taxes to balance its budget Please describe how to find a competitive equilibrium Exercise 1115 The structure of the economy is identical to that described in the previous exercise Let r0t be the yield to maturity on a t period bond at time 0 t 1 2 At time 0 Bloomberg reports the term structure of interest rates in figure 113 Please say what you can about the evolution of ct kt1 in this economy Feel free to make any assumptions you need about fiscal policy gt τkt τct τht t0 to make your answer coherent Exercises 467 a Time t kt b kt Time t k0 k0 Figure 114 Capital stock as function of time in two economies with different values of γ Exercise 1116 The structure of the economy is identical to that described in exercise 1114 As sume that gt τct τkt t0 are all constant sequences their values dont change over time In this problem we ask you to infer differences across two economies in which all aspects of the economy are identical except the parameter γ in the utility function30 In both economies γ 0 In one economy γ 0 is high and in the other it is low Among other identical features the two economies have identical government policies and identical initial capital stocks a Please look at figure 114 Please tell which outcome for kt1 t0 describes the low γ economy and which describes the high γ economy Please explain your reasoning b Please look at figure 115 Please tell which outcome for f kt t0 describes the low γ economy and which describes the high γ economy Please explain your reasoning c Please plot time paths of consumption for the low γ and the high γ economies 30 It is possible that lump sum taxes differ across the two economies Assume that lump sum taxes are adjusted to balance the government budget 470 Fiscal Policies in a Growth Model the market also known as the invisible hand presents as exogenous to the representative household b Please describe how the Big K part of a Big K little k argument is used to determine all of those objects exogenous to the household Hint This is accomplished by applying the shooting algorithm in section 119 c Please explain thoroughly how the representative household chooses to re spond to the signals presented to it by the market at time 0 d Given the objects that the market presents to the representative household please tell how you would could a shooting algorithm to compute the path of ct kt1 t0 chosen by the household e Please describe how to complete a Big K little k argument using your answers to parts c and d Exercise 1120 The Invisible Hand II Please consider again the Foreseen jump in τn experiment in section 1112 Like the previous problem this one puts you into the shoes of the representative household and asks you to think through the optimum problem that it faces within a competitive equilibrium with distorting taxes at time 0 a Please describe the signals that the market sends to the household after the new τnt t0 policy materializes at time 0 Please list all objects that the market presents as exogenous to the representative household b Keeping in mind that there is a Big K little k argument in the back ground please provide a complete explanation for why the household chooses the paths of ct nt kt t0 displayed in figure 11123 Chapter 12 Recursive Competitive Equilibrium II 121 Endogenous aggregate state variable For pure endowment stochastic economies chapter 8 described two types of com petitive equilibria one in the style of Arrow and Debreu with markets that con vene at time 0 and trade a complete set of historycontingent securities another with markets that meet each period and trade a complete set of oneperiodahead statecontingent securities called Arrow securities Though their price systems and trading protocols differ both types of equilibria support identical equilib rium allocations Chapter 8 described how to transform the ArrowDebreu price system into one for pricing Arrow securities The key step in transforming an equilibrium with time 0 trading into one with sequential trading was to account for how individuals wealth evolve as time passes in a time 0 trading economy In a time 0 trading economy individuals do not make any trades other than those executed in period 0 but the present value of those portfolios change as time passes and as uncertainty gets resolved So in period t after some history st we used the ArrowDebreu prices to compute the value of an individuals purchased claims to current and future goods net of his outstanding liabilities We could then show that these wealth levels and the associated consumption choices could also be attained in a sequentialtrading economy where there are only markets in oneperiod Arrow securities that reopen in each period In chapter 8 we also demonstrated how to obtain a recursive formulation of the equilibrium with sequential trading This required us to assume that individuals endowments were governed by a Markov process Under that as sumption we could identify a state vector in terms of which the Arrow securities could be cast This aggregate state vector then became a component of the state vector for each individuals problem This transformation of price systems is easy in the pure exchange economies of chapter 8 because in equilibrium the relevant state variable wealth is a function solely of the current realization of the exogenous Markov state variable The transformation is more subtle in economies in which part of the aggregate state is endogenous in the sense that it 471 The stochastic growth model 473 In each period the representative household is endowed with one unit of time that can be devoted to leisure ℓtst or labor ntst 1 ℓtst ntst 1222 The only other endowment is a capital stock k0 at the beginning of period 0 The technology is ctst xtst AtstFktst1 ntst 1223a kt1st 1 δktst1 xtst 1223b where F is a twice continuously differentiable constantreturnstoscale pro duction function with inputs capital ktst1 and labor ntst and Atst is a stochastic process of Harrodneutral technology shocks Outputs are the consumption good ctst and the investment good xtst In 1223b the investment good augments a capital stock that is depreciating at the rate δ Negative values of xtst are permissible which means that the capital stock can be reconverted into the consumption good We assume that the production function satisfies standard assumptions of positive but diminishing marginal products Fik n 0 Fiik n 0 for i k n and the Inada conditions lim k0 Fkk n lim n0 Fnk n lim k Fkk n lim n Fnk n 0 Since the production function has constant returns to scale we can define Fk n nfˆk where ˆk k n 1224 Another property of a linearly homogeneous function Fk n is that its first derivatives are homogeneous of degree 0 and thus the first derivatives are func tions only of the ratio ˆk In particular we have Fkk n nf kn k f ˆk 1225a Fnk n nf kn n fˆk f ˆkˆk 1225b 486 Recursive Competitive Equilibrium II Note that the economys endofperiod wealth as embodied in kII t1st in period t after history st is willingly held by the representative household This follows immediately from fact that the households desired beginningofperiod wealth next period is given by at1st1 and is equal to Υt1st1 as given by 12416 Thus the equilibrium prices entice the representative household to enter each future period with a strictly positive net asset level that is equal to the value of the type II firm We have then confirmed the correctness of our earlier conjecture that the arbitrary debt limit of zero is not binding in the households optimization problem 126 Recursive formulation Following the approach taken in chapter 8 we have established that the equi librium allocations are the same in the ArrowDebreu economy with complete markets at time 0 and in a sequentialtrading economy with complete oneperiod Arrow securities This finding holds for an arbitrary technology process Atst defined as a measurable function of the history of events st which in turn are governed by some arbitrary probability measure πtst At this level of general ity all prices Qtst1st wtst rtst and the capital stock kt1st in the sequentialtrading economy depend on the history st That is these objects are timevarying functions of all past events sτt τ0 In order to obtain a recursive formulation and solution to both the social planning problem and the sequentialtrading equilibrium we make the following specialization of the exogenous forcing process for the technology level 490 Recursive Competitive Equilibrium II posing the decision problems of the household and firms to impose equilibrium we set K k after firms and consumers have optimized 1282 Price system To decentralize the economy in terms of oneperiod Arrow securities we need a description of the aggregate state in terms of which oneperiod statecontingent payoffs are defined We proceed by guessing that the appropriate description of the state is the same vector X that constitutes the state for the plan ning problem We temporarily forget about the optimal policy functions for the planning problem and focus on a decentralized economy with sequential trading and oneperiod prices that depend on X We specify price functions rX wX QXX that represent respectively the rental price of capital the wage rate for labor and the price of a claim to one unit of consumption next period when next periods state is X and this periods state is X Forgive us for recycling the notation for r and w from the previous sections on the formulation of historydependent competitive equilibria with commodity space st The prices are all measured in units of this periods consumption good We also take as given an arbitrary candidate for the law of motion for K K GX 1281 Equation 1281 together with 1272b and a given subjective transition den sity ˆπss induce a subjective transition density ˆΠXX for the state X For now G and ˆπss are arbitrary We wait until later to impose other equilib rium conditions including rational expectations in the form of some restrictions on G and ˆπ 494 Recursive Competitive Equilibrium II rX 1 δK wXn c 1292 Next by recalling equilibrium condition 12811 and the fact that K is a predetermined variable when entering next period it follows that the left side of 1292 is equal to K After also substituting equilibrium prices 1289 into the right side of 1292 we obtain K AsFkk n 1 δ K AsFnk nn c AsFK σna X 1 δK σca X 1293 where the second equality invokes Eulers theorem on linearly homogeneous functions and equilibrium conditions K k N n σna X and C c σca X To express the right side of equation 1293 solely as a function of the current aggregate state X K A s we also impose equilibrium condition 1291b K AsF K σnrX 1 δK X 1 δK σcrX 1 δK X 1294 Given the arbitrary perceived law of motion 1281 for K that underlies the households optimum problem the right side of 1294 is the actual law of motion for K that is implied by the households and firms optimal decisions In equilibrium we want G in 1281 not to be arbitrary but to be an outcome We want to find an equilibrium perceived law of motion 1281 By way of imposing rational expectations we require that the perceived and actual laws of motion be identical Equating the right sides of 1294 and the perceived law of motion 1281 gives GX AsF K σnrX 1 δK X 1 δK σcrX 1 δK X 1295 Please remember that the right side of this equation is itself implicitly a func tion of G so that 1295 is to be regarded as instructing us to find a fixed point equation of a mapping from a perceived G and a price system to an ac tual G This functional equation requires that the perceived law of motion for the capital stock GX equals the actual law of motion for the capital stock that is determined jointly by the decisions of the household and the firms in a competitive equilibrium 496 Recursive Competitive Equilibrium II In an equilibrium it will turn out that the households decision rules for con sumption and labor supply will match those chosen by the planner7 ΩCX σcrX 1 δK X 1297a ΩNX σnrX 1 δK X 1297b The key to verifying these guesses is to show that the firstorder conditions for both types of firms and the household are satisfied at these guesses We leave the details to an exercise Here we are exploiting some consequences of the welfare theorems transported this time to a recursive setting with an endogenous aggregate state variable 1210 Concluding remarks The notion of a recursive competitive equilibrium was introduced by Lucas and Prescott 1971 and Mehra and Prescott 1979 The application in this chapter is in the spirit of those papers but differs substantially in details In particular neither of those papers worked with Arrow securities while the focus of this chapter has been to manage an endogenous state vector in terms of which it is appropriate to cast Arrow securities 7 The two functional equations 1297 state restrictions that a recursive competitive equilibrium imposes across the households decision rules σ and the planners decision rules Ω The permanent income model revisited 501 12A4 Interpretation As we saw in section 212 of chapter 2 and also in representation 12A4 12A5 here what is now equilibrium consumption is a random walk Why despite his preference for a smooth consumption path does the representative consumer accept fluctuations in his consumption In the complete markets economy of this appendix the consumer believes that it is possible for him com pletely to smooth consumption over time and across histories by purchasing and selling history contingent claims But at the equilibrium prices facing him the consumer prefers to tolerate fluctuations in consumption over time and across histories Chapter 13 Asset Pricing Theory 131 Introduction Chapter 8 showed how an equilibrium price system for an economy with a com plete markets model could be used to determine the price of any redundant asset That approach allowed us to price any asset whose payoff could be syn thesized as a measurable function of the economys state We could use either the ArrowDebreu time 0 prices or the prices of oneperiod Arrow securities to price redundant assets We shall use this complete markets approach again later in this chapter and in chapter 14 However we begin with another frequently used approach one that does not require the assumption that there are complete markets This ap proach spells out fewer aspects of the economy and assumes fewer markets but nevertheless derives testable intertemporal restrictions on prices and returns of different assets and also across those prices and returns and consumption alloca tions This approach uses only the Euler equations for a maximizing consumer and supplies stringent restrictions without specifying a complete general equi librium model In fact the approach imposes only a subset of the restrictions that would be imposed in a complete markets model As we shall see in chapter 14 even these restrictions have proved difficult to reconcile with the data the equity premium being a widely discussed example Assetpricing ideas have had diverse ramifications in macroeconomics In this chapter we describe some of these ideas including the important Modigliani Miller theorem asserting the irrelevance of firms asset structures We describe a closely related kind of Ricardian equivalence theorem1 1 See Duffie 1996 for a comprehensive treatment of discrete and continuoustime asset pricing theories See Campbell Lo and MacKinlay 1997 for a summary of recent work on empirical implementations 503 Euler equations 505 and next periods wealth is At1 Lt pt1 yt1Nt 1323 The stochastic dividend is the only source of exogenous fundamental uncer tainty with properties to be specified as needed later The agents maximization problem is then a dynamic programming problem with the state at t being At and current and past y3 and the controls being Lt and Nt At interior solu tions the Euler equations associated with controls Lt and Nt are uctR1 t Etβuct1 1324 uctpt Etβyt1 pt1uct1 1325 These Euler equations give a number of insights into asset prices and consump tion Before turning to these we first note that an optimal solution to the agents maximization problem must also satisfy the following transversality conditions4 lim k EtβkuctkR1 tkLtk 0 1326 lim k EtβkuctkptkNtk 0 1327 Heuristically if any of the expressions in equations 1326 and 1327 were strictly positive the agent would be overaccumulating assets so that a higher expected lifetime utility could be achieved by for example increasing consumption today The counterpart to such nonoptimality in a finite horizon model would be that the agent dies with positive asset holdings For reasons like those in a finite horizon model the agent would be happy if the two conditions 1326 and 1327 could be violated on the negative side But the market would stop the agent from financing consumption by accumulating the debts that would be associated with such violations of 1326 and 1327 No other agent would want to make those loans 3 Current and past y s enter as information variables How many past y s appear in the Bellman equation depends on the stochastic process for y 4 For a discussion of transversality conditions see Benveniste and Scheinkman 1982 and Brock 1982 Equilibrium asset pricing 511 135 Equilibrium asset pricing The preceding discussion of the Euler equations 1324 and 1325 leaves open how the economy generates for example the constant gross interest rate assumed in Halls work We now explore equilibrium asset pricing in a simple representative agent endowment economy Lucass assetpricing model10 We imagine an economy consisting of a large number of identical agents with prefer ences as specified in expression 1321 The only durable good in the economy is a set of identical trees one for each person in the economy At the be ginning of period t each tree yields fruit or dividends in the amount yt The fruit is not storable but the tree is perfectly durable Each agent starts life at time zero with one tree The dividend yt is assumed to be governed by a Markov process and the dividend is the sole state variable st of the economy ie st yt The timeinvariant transition probability distribution function is given by Probst1 sst s Fs s All agents maximize expression 1321 subject to the budget constraint 13221323 and transversality conditions 13261327 In an equi librium asset prices clear the markets That is the bond holdings of all agents sum to zero and their total stock positions are equal to the aggregate number of shares As a normalization let there be one share per tree Due to the assumption that all agents are identical with respect to both preferences and endowments we can work with a representative agent11 Lu cass model shares features with a variety of representative agent assetpricing models see Brock 1982 and Altug 1989 for example These use versions of stochastic optimal growth models to generate allocations and price assets Such assetpricing models can be constructed by the following steps 1 Describe the preferences technology and endowments of a dynamic econ omy then solve for the equilibrium intertemporal consumption allocation Sometimes there is a particular planning problem whose solution equals the competitive allocation 2 Set up a competitive market in some particular asset that represents a specific claim on future consumption goods Permit agents to buy and 10 See Lucas 1978 Also see the important early work by Stephen LeRoy 1971 1973 Breeden 1979 was an early work on the consumptionbased capitalassetpricing model 11 In chapter 8 we showed that some heterogeneity is also consistent with the notion of a representative agent 516 Asset Pricing Theory where wi represents the pricedividend ratio Equation 1374 was used by Mehra and Prescott 1985 to compute equilibrium prices 138 Term structure of interest rates We will now explore the term structure of interest rates by pricing bonds with different maturities14 We continue to assume that the time t state of the economy is the current dividend on a Lucas tree yt st which is Markov with transition Fs s The riskfree real gross return between periods t and t j is denoted Rjt measured in units of time t j consumption good per time t consumption good Thus R1t replaces our earlier notation Rt for the one period gross interest rate At the beginning of t the return Rjt is known with certainty and is risk free from the viewpoint of the agents That is at t R1 jt is the price of a perfectly sure claim to one unit of consumption at time t j For simplicity we only consider such zerocoupon bonds and the extra subscript j on gross earnings Ljt now indicates the date of maturity The subscript t still refers to the agents decision to hold the asset between period t and t 1 As an example with one and twoperiod safe bonds the budget constraint and the law of motion for wealth in 1322 and 1323 are augmented as follows ct R1 1t L1t R1 2t L2t ptNt At 1381 At1 L1t R1 1t1L2t pt1 yt1Nt 1382 Even though safe bonds represent sure claims to future consumption these assets are subject to price risk prior to maturity For example twoperiod bonds from period t L2t are traded at the price R1 1t1 in period t 1 as shown in wealth expression 1382 At time t an agent who buys such assets and plans to sell them next period would be uncertain about the proceeds since R1 1t1 is not known at time t The price R1 1t1 follows from a simple arbitrage argument since in period t 1 these assets represent identical sure claims to time t 2 consumption goods as newly issued oneperiod bonds in period t 1 The variable Ljt should therefore be understood as the agents net holdings between 14 Dynamic assetpricing theories for the term structure of interest rates have been devel oped by Cox Ingersoll and Ross 1985a 1985b and by LeRoy 1982 534 Asset Pricing Theory 0 002 004 006 008 01 0 1 2 3 4 5 092 094 096 098 1 102 104 106 108 Rel risk aversion Stand dev of growth Riskfree interest rate Figure 13101 The riskfree interest rate R1 as a function of the coefficient of relative risk aversion γ and the standard deviation of dividend growth There are two states of divi dend growth that are equally likely to occur with a mean of 1 percent Ey 1 01 and the subjective discount factor is β 98 If R1 Ey so that the expected value of future debt discounted at the safe interest rate does not converge to zero in equation 131019 it follows that the expected sum of all future government surpluses discounted at the safe interest rate in equation 131015 falls short of the initial debt In fact our example is then associated with negative expected surpluses at all future horizons Et τtj gtj Et btj1 btjR1tj Et R1 ytj bytj1 R1 E y b E yj1 yt 0 if R1 E y 0 if R1 E y 0 if R1 E y 131020 where the first equality invokes budget constraint 13106 Thus for R1 Ey the sum of covariance terms in equation 131015 must be positive The described debt policy also clearly has this implication where for example a low realization of ytj implies a relatively high marginal utility of consumption and Gaussian assetpricing model 541 houses bubbles end when the supply of the asset has grown enough to outstrip optimistic investors resources for purchasing the asset 3 If optimistic investors finance purchases by borrowing tightening leverage constraints can extinguish a bubble Scheinkman extracts insights about effects of financial regulations on bub bles He emphasizes how limiting short sales and limiting leverage have opposite effects Please notice key differences in the assumptions of the HarrisonKreps model presented in this appendix and the Blume and Easley model of appendix B of chapter 8 The chapter 8 model assumes complete markets and risk averse consumers it focuses on the dynamics of continuation wealth in a competitive equilibrium There is zero volume in the sense that no trades occur after date 0 By way of contrast the HarrisonKreps model of this appendix assumes in complete markets riskneutral consumers and restrictions on short sales By assuming that both types of agent always have deep enough pockets to pur chase all of the asset the model takes wealth dynamics off the table The HarrisonKreps model generates high trading volume when the state changes either from 1 to 2 or from 2 to 1 B Gaussian assetpricing model The theory of chapter 8 can readily be adapted to a setting in which the state of the economy evolves according to a continuousstate Markov process We use such a version in chapter 14 Here we give a taste of how such an adaptation can be made by describing an economy in which the state follows a linear stochastic difference equation driven by a Gaussian disturbance If we supplement this with the specification that preferences are quadratic we get a setting in which asset prices can be calculated swiftly Suppose that the state evolves according to the stochastic difference equa tion st1 Ast Cwt1 13B1 where A is a matrix whose eigenvalues are bounded from above in modulus by 1β and wt1 is a Gaussian martingale difference sequence adapted to the 552 Asset Pricing Empirics γ y 10 100 1000 5000 2 02 2 20 500 5 05 5 50 1217 10 1 1 100 2212 Table 1421 Risk premium Cy C for various values of y and γ when C 50 000 143 The equity premium puzzle Table 1431 depicts empirical first and second moments of yields on relatively riskless bonds and risky equity in the US data over the 90year period 1889 1978 The average real yield on the Standard Poors 500 index was 7 percent while the average yield on shortterm debt was only 1 percent The equity premium puzzle is that with aggregate consumption data it takes an extraor dinarily large value of the coefficient of relative risk aversion to generate such a large gap between the returns on equities and riskfree securities2 Mean VarianceCovariance 1 rs t1 1 rb t1 Ct1Ct 1 rs t1 1070 00274 000104 000219 1 rb t1 1010 000308 0000193 Ct1Ct 1018 000127 Table 1431 Summary statistics for US annual data 18891978 The quantity 1rs t1 is the real return to stocks 1 rb t1 is the real return to relatively riskless bonds and Ct1Ct is the growth rate of per capita real consumption of nondurables and services Source Kocherlakota 1996a Ta ble 1 who uses the same data as Mehra and Prescott 1985 2 For insightful reviews and lists of possible resolutions of the equity premium puzzle see Aiyagari 1993 Kocherlakota 1996a and Cochrane 1997 HansenJagannathan bounds 557 145 HansenJagannathan bounds The section 143 HansenSingleton 1983 exposition of the equity premium puz zle based on the log normal specification of returns was tied to particular para metric specifications of preferences and the distribution of asset returns Hansen and Jagannathan 1991 described a less structured way of stating an equity premium puzzle Their work can be regarded as extending Robert Shillers and Stephen LeRoys earlier work on variance bounds8 We present one of Hansen and Jagannathans bounds Until now we have worked with theories that price assets by using a par ticular stochastic discount factor mt1 β uCt1 uCt The theories assert that the price at t of an asset with oneperiod random payoff pt1 is Etmt1pt1 Hansen and Jagannathan were interested in settings in which the stochastic discount factor can assume other forms 1451 Law of one price implies that EmR 1 This section briefly indicates how a very weak theoretical restriction on prices and returns implies that there exists a stochastic discount factor m that satisfies EmRj 1 for the return Rj on any asset j In fact when markets are incomplete there exist many different random variables m that satisfy EmRj 1 We have to say very little about consumers preferences to get this result a law of one price being enough Following Hansen and Jagannathan let xj be a random payoff on a security Let there be J primitive securities so j 1 J Let x be a J 1 vector of random payoffs on the primitive securities Assume that the J J matrix Exx exists and that so does its inverse Exx1 Also assume that a J 1 vector q of prices of the primitive securities is observed where the j th component of q is the price of the j th component of the payoff vector x Consider forming portfolios ie linear combinations of the primitive securities How do prices of portfolios relate to the prices of the primitive securities from which they have been formed Let c IRJ be a vector of portfolio weights The random payoff on a portfolio with weights c is c x Define the space of payoffs attainable from 8 See Hansens 1982a early call for such a generalization HansenJagannathan bounds 559 1452 Inner product representation of price functional Hansen and Jagannathan used a convenient representation of a linear functional If y is a scalar random variable Eyx is the vector whose j th component is Eyxj The crossmoments Eyx are called the inner product of x and y According to the Riesz representation theorem a linear functional φ can be represented as the inner product of the random payoff x with some scalar random variable y that we call a stochastic discount factor9 Thus a stochastic discount factor is a scalar random variable y that verifies φ p E yp p P 1452 Equality 1452 implies that the vector q of prices of the primitive securities satisfies q E yx 1453 The law of one price implies that a pricing functional is linear and that there fore there exists a stochastic discount factor When markets are not complete there exist many stochastic discount factors Hansen and Jagannathan sought to characterize a set of admissible stochastic discount factors meaning scalar random variables y that satisfy 1452 Note cov y p E yp E y E p which implies that the price functional can be represented as φ p E y E p cov y p This expresses the price of a portfolio as the expected value of the stochas tic discount factor times the expected payoff plus the covariance between the stochastic discount factor and the payoff Notice that φ1 Ey so that the expected value of the stochastic discount factor is simply the price of a sure scalar payoff of unity The linearity of the pricing functional leaves open the possibility that prices of some portfolios are negative That would open arbitrage opportunities David Kreps 1979 showed that the principle that the price system should offer no arbitrage opportunities requires that the stochastic discount factor be strictly 9 See appendix A for a statement and proof of the Riesz representation theorem HansenJagannathan bounds 561 where e is orthogonal to x and b cov x x1 cov x y a Ey Exb Here covx x Exx ExEx We have data that allow us to estimate the secondmoment matrix of x but no data on y and therefore no data on covx y But we do have data on q the vector of security prices So Hansen and Jagannathan proceeded to use the data on q x to infer something but not everything about covx y Notice that q Eyx implies covx y q EyEx Therefore b cov x x1 q E y E x 1456 Given a guess about Ey asset payoffs x and prices q can be used to estimate b That the residuals in the projection equation 1455 are orthogonal to x induces the variance decomposition var y var xb var e Therefore var xb5 σ y 1457 where σy denotes the standard deviation of the random variable y The left side of 1457 is a lower bound on the standard deviation of all stochas tic discount factors with assumed mean Ey used to compute b in equation 145612 For various specifications Hansen and Jagannathan used expres sions 1456 and 1457 to compute a lower bound on σy as a function of Ey thereby tracing out a frontier of admissible stochastic discount factors in terms of their means and standard deviations We focus on the case in which no riskfree asset is included among the basis securities underlying x For this case Hansen and Jagannathan calculate a lower bound on σy as a function of an unknown value of Ey They do this for data on gross returns on a set of assets For a set of returns q 1 so that equation 1456 becomes b cov x x1 1 E y E x 1458 12 The stochastic discount factors are not necessarily positive Hansen and Jagannathan 1991 derive another bound that imposes positivity 564 Asset Pricing Empirics 08 085 09 095 1 0 005 01 015 02 025 03 Em σm Time separable CRRA preferences Figure 1461 Solid line HansenJagannathan volatility bounds for quarterly returns on the valueweighted NYSE and Treasury Bill 19482005 Crosses Mean and standard deviation for intertemporal marginal rate of substitution for CRRA time separable preferences The coefficient of relative risk aversion γ takes values 1 5 10 15 20 25 30 35 40 45 50 and the discount factor β 0995 Notice that Emt1 exprt so rt is the oneperiod net rate of return on a riskfree claim often appropriately called the short rate Equation 1469 shows how discounting in preferences ρ consumption growth µ taste for smooth consumption γ and a precautionary savings motive 1 2σ2 cγ2 all affect the short rate In a literature on exponential quadratic stochastic discount fac tors to be discussed in section 1411 the loading γσc of the log of the stochastic discount factor on the innovation εt1 is called the price of consumption growth risk That loading equals the market price of risk computed above Figure 1461 plots HansenJagannathan bounds They form the parabola in the upper right corner and were constructed using quarterly data on two returns the real return on a valueweighted NYSE stock return and the real return on US Treasury bills over the period 19482005 in conjunction with inequality 1458 The figure also reports the locus of Em and σm implied by equations 1466 and 1467 traced out by different values of γ The figure shows that while high values of γ deliver high σm high values of γ also push 566 Asset Pricing Empirics Table 1461 Estimates from quarterly US data 19482 20054 Parameter Estimate µ 0004952 σc 0005050 In conclusion the fact that the same parameter γ expresses two attitudes atemporal risk aversion and intertemporal substitution aversion leads to Weils riskfree rate puzzle as captured by our figure 1461 In the next section we describe how Tallarini 2000 made progress by assigning to γ only the single job of describing risk aversion while using a new parameter η to describe attitudes toward intertemporal substitution By proceeding this way Tallarini was able to find values of the risk aversion parameter γ that push the Em σm pair toward the Hansen and Jagannathan bounds 147 Nonexpected utility To separate risk aversion from intertemporal substitution Tallarini 2000 as sumed preferences that can be described by a recursive nonexpected utility value function iteration a la Kreps and Porteus 1978 Epstein and Zin 1989 and Weil 1990 namely16 Vt W Ct ξ Vt1 1471 Here W is an aggregator function that maps todays consumption C and a function ξ of tomorrows random continuation value Vt1 into a value Vt today ξ is a certainty equivalent function that maps a random variable Vt1 that is measurable with respect to next periods information into a random variable that is measurable with respect to this periods information ξ Vt1 f 1 Etf Vt1 random walk model of log consumption and CRRA timeseparable preferences thus explaining both the equity premium and the riskfree rate Doing so requires a very high γ and β 1 16 Obstfeld 1994 and Dolmas 1998 used recursive preferences to study costs of consump tion fluctuations Nonexpected utility 569 1471 Another representation of the utility recursion When log consumption follows the random walk with drift 1463 or more broadly is a member of a class of models that makes the conditional distribution of ct1 be Gaussian another way to express recursion 1476 is Ut ct βEtUt1 β 2θvart Ut1 14710 where vartUt1 denotes the conditional variance of continuation utility Ut1 Using 1475 to eliminate θ in favor of γ we can also express 14710 as Ut ct βEtUt1 β 1 γ 1 β 2 vart Ut1 14711 When θ or γ 1 representation 14710 generalizes the ordinary time separable expected utility recursion by making the consumer care not only about the conditional expectation of continuation utility but also its conditional variance According to 14710 when θ the consumer dislikes conditional variance in continuation utility18 This means that the consumer cares about both the timing of the resolution of uncertainty and the persistence of risk When θ he cares about neither Figures 1471 and 1472 show payoffs displayed above the nodes and transition probabilities the fractions above the lines connecting nodes for four plans When 0 θ the consumer prefers early resolution of risk he prefers plan A to plan B in figure 1471 while he is indifferent to the timing of risk when θ 19 When 0 θ the consumer dislikes persistence of risk he prefers plan C to plan D in figure 1472 while when θ he is indifferent to the persistence of risk20 18 Equation 14710 is a discrete time version of the stochastic differential utility model of Duffie and Epstein 1992 19 See Kreps and Porteus 1978 20 See Duffie and Epstein 1992 572 Asset Pricing Empirics 08 085 09 095 1 0 005 01 015 02 025 03 Em σm Figure 1473 Solid line HansenJagannathan volatility bounds for quarterly returns on the valueweighted NYSE and Treasury bill 19482005 Circles Mean and standard devi ation for intertemporal marginal rate of substitution gener ated by EpsteinZin preferences with random walk consump tion Crosses Mean and standard deviation for intertemporal marginal rate of substitution for CRRA time separable pref erences The coefficient of relative risk aversion γ takes on the values 1 5 10 15 20 25 30 35 40 45 50 and the discount factor β 0995 Please compare these two to the corresponding formulas 1466 1467 for the timeseparable CRRA specification The salient difference is that γ no longer appears in the key component expµ of Em in 14719 while it does appear in the corresponding term in formula 1466 coming from time separable CRRA preferences Tallarini made γ disappear there by locking the inverse intertemporal rate of substitution parameter η at unity while still al lowing what is now a pure risk aversion parameter γ to vary This arrests the force causing Em in 1466 to fall as γ rises and allows Tallarini to avoid the riskfree rate puzzle and to approach the HansenJagannathan bounds as the risk aversion parameter γ increases To connect to the exponential quadratic stochastic discount factor models of section 1411 we can represent Tallarinis stochastic discount factor 14717 Reinterpretation of the utility recursion 575 148 Reinterpretation of the utility recursion Is Tallarinis explanation convincing Not to Robert E Lucas Jr To succeed in approaching the HansenJagannathan bounds required that Tallarini set the riskaversion parameter γ to such a high value namely 50 that it provoked Robert E Lucas Jr to disregard Tallarinis evidence for high risk aversion No one has found risk aversion parameters of 50 or 100 in the diversification of individual portfolios in the level of insurance deductibles in the wage premiums associated with occupations with high earnings risk or in the revenues raised by stateoperated lotteries It would be good to have the equity premium resolved but I think we need to look beyond high estimates of risk aversion to do it Macroeconomic Priorities 2003 To measure the costs of aggregate fluctuations along lines to be described in section 149 Lucas 1987 2003 preferred to use a value of γ of 1 or 2 rather than the γ of 50 that Tallarini required to reconcile his model of preferences with both the consumption data and the asset returns data as summarized by the HansenJagannathan bounds24 1481 Risk aversion versus model misspecification aversion To respond to Lucass reluctance to use Tallarinis findings as a source of evi dence about a representative consumers distaste for consumption fluctuations we now reinterpret γ as a parameter that expresses not risk aversion but in stead distress about model specification doubts Fearing risk means disliking randomness described by a known probability distribution Fearing uncertainty also called model misspecification means disliking not knowing a proba bility distribution We will reinterpret the forwardlooking term25 gεt1 exp1β1γUt1 Etexp1β1γUt1 that multiplies the ordinary logarithmic stochastic dis count factor β Ct Ct1 in 14713 as an adjustment of the stochastic discount factor that reflects a consumers fears about model misspecification While a 24 For another perspective on the evidence see Barseghyan Molinari ODonoghue and Teitelbaum 2013 Their findings challenge the notion that purchasers of insurance know probability distributions 25 The presence of the continuation value Ut1 is our reason for saying forwardlooking 580 Asset Pricing Empirics Recall formula 1475 that for risksensitive preferences defines θ in terms of the elementary parameters β and γ θ 1 1 β 1 γ For Tallarini γ 1 θ1 1β is the fundamental parameter For him it de scribes the consumers attitude toward atemporal risky choices under a known probability distribution But under the probability ambiguity or robustness in terpretation θ is an elementary parameter in its own right one that measures the consumers doubts about the probability model that describes consumption growth risk The evidence cited in the above quote from Lucas 2003 and the introspective reasoning of Cochrane 1997 and Pratt 1964 that we described above on page 551 explain why many economists think that only small positive values of γ are plausible when it is interpreted as a riskaversion parameter Pratts experiment confronts a decision maker with choices between gambles with known probability distributions How should we think about plausible values of γ or rather θ when it is instead interpreted as encoding responses to gambles that involve unknown probability distributions Hansen Sargent and Wang 2002 and Anderson Hansen and Sargent 2003 answer this question by recognizing the role of entropy in statistical tests for discriminating one probability distribution from another based on a sample of size T drawn from one or the other of the two distributions They use the probability of making an error in discriminating between the two models as a way of disciplining the calibration of θ That leads them to argue that it is not appropriate to regard θ as a parameter that remains fixed across alternative hypothetical stochastic processes for consumption We take up this issue again in section 1487 582 Asset Pricing Empirics they regard as moderate and plausible amounts of model uncertainty goes a long way toward pushing what Tallarini would measure as the market price of uncertainty but which they instead interpret as the market price of model uncertainty toward the HansenJagannathan bounds 1487 Measuring model uncertainty Anderson Hansen and Sargent 2003 took the following approach to measuring plausible amounts of model uncertainty The decision makers baseline approx imating model is the random walk with drift 1463 However the decision maker doubts this model and surrounds it with a cloud of models characterized by likelihood ratios gε To get a robust valuation he constructs a worst case model namely the model associated with the minimizing likelihood ratio gε in the appropriate version of 1485 When his approximating model is 1463 this worstcase model for log consumption growth is ct1 ct µ σcw σcεt1 14811 where εt1 is again distributed according to a Gaussian density with mean zero and unit variance Equation 14811 says that the mean of consumption growth is not µ but µ σcw where w is again given by 14715 or 14716 Evidently the approximating model is the γ 1 version of 14811 The ambiguity averse consumer has a stochastic discount factor with respect to the approximating model 1463 that looks as if he believes 14811 instead of 1463 It is as if he evaluates utility according to the ordinary utility recursion Ut ctβ EtUt1 where Et is the mathematical expectation taken with respect to the probability distribution generated by 14811 When it is interpreted as a measure of model uncertainty rather than risk aversion Anderson Hansen and Sargent recommend calibrating γ or θ by using an object called a detection error probability In the present context this object answers the following question Given a sample of size T drawn from either 1463 call it model A or 14811 call it model B what is the probability that a likelihood ratio test would incorrectly testify either that model A generated the data when in fact model B generated the data or that model B generated the data when in fact model A did31 It is easy to compute 31 Anderson Hansen and Sargent 2003 describe the close links between entropy and such detection error probabilities Reinterpretation of the utility recursion 583 detection error probabilities by simulating likelihood ratios for samples of size T and counting the frequency of such model discrimination mistakes32 Evidently when γ 1 which means θ w 0 so models A and B are identical and therefore statistically indistinguishable In this case the detection error probability is 5 signifying that via rounding error the computer essentially flips a coin in deciding which model generated the data A detection error probability of 5 thus means that it is impossible to distinguish the models from sample data But as we increase γ above 1 ie drive the penalty parameter θ below the detection error probability falls The idea here is to guide our choice of γ or θ as follows Set a detection error probability that reflects an amount of model specification uncertainty about which it seems plausible for the decision maker to be concerned then in the context of the particular approximating model at hand which for us is 1463 find the γ associated with that detection error probability 08 085 09 095 1 0 005 01 015 02 025 03 Em σm Figure 1481 Reciprocal of risk free rate market price of risk pairs for the random walk model for values of detection error probabilities of 50 45 40 35 30 25 20 15 10 5 and 1 percent A plausible value for the detection error probability is a matter of judge ment If the detection error probability is 5 it means that the two models are 32 See Barillas Hansen and Sargent 2009 584 Asset Pricing Empirics statistically identical and cant be distinguished A detection error probability of 25 means that there is a one in four chance of making the wrong decision about which model is generating the data From our own experiences fitting models to data a person whose specification doubts include perturbed models with a detection error of 25 or 1 or even 05 could be said to have a plausible amount of model uncertainty Figure 1481 redraws Tallarinis figure in terms of detection error proba bilities for a sample size equal to the number of quarterly observations between 1948 and 2005 used to compute the Hansen and Jagannathan bounds The fig ure again plots Em σm pairs given by formulas 14719 14720 for γ s chosen to deliver the indicated detection error probabilities The figure shows that moderate detection error probabilities of 10 or 15 percent take us more than half way to the Hansen and Jagannathan bounds while 1 percent gets us there The sense of these calculations is that moderate amounts of aversion to model uncertainty can substitute for huge amounts of risk aversion from the point of view of pushing the Em σm toward the HansenJagannathan bounds In the next section we revisit the quote from Lucas in light of this finding 149 Costs of aggregate fluctuations We now take up the important substantive issue that prompted Lucas to dis miss Tallarinis evidence about γ for the particular purpose then at hand for Lucas 1987 2003 Lucas wanted to measure the gains to eliminating further unpredictable fluctuations in aggregate US per capita consumption beyond the reductions that had already been achieved by post World War II aggregate sta bilization policies His method was to find an upper bound on possible gains by computing the reduction in initial consumption that a representative con sumer with timeseparable preferences would be willing to accept in exchange for eliminating all unpredictable fluctuations that post WWII consumption has exhibited In this section we describe Tallarinis version of Lucass calculation and spotlight how γ affects conclusions33 For the random walk with drift model of log consumption described by equation 1463 the level of consumption Ct expct obeys Ct1 expµ 33 For another perspective on the topic of this section see Alvarez and Jermann 2004 586 Asset Pricing Empirics 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 detection error probability Proportion of consumption Figure 1491 Proportions c0 cd 0 of initial consumption that a representative consumer with modeluncertainty averse multiplier preferences would surrender not to confront risk dotted line and model uncertainty solid line for random walk model of log consumption growth plotted as a function of detection error probability In section 1486 we argued that most of what Tallarini interpreted as the market price of risk should instead be interpreted as a market price of model uncertainty The section 1486 argument is one possible way of fulfilling Lu cass hope that It would be good to have the equity premium resolved but I think we need to look beyond high estimates of risk aversion to do it And it is compatible with Lucass judgement that Tallarinis values of γ s calibrated to get into the HansenJagannathan bounds are not suitable for mental exper iments about risks with known probabilities of the kind that Lucas performed Those high estimates of γ are relevant to other mental experiments about elim inating the consumers concern about model uncertainty but not about Lucass experiment Figure 1491 shows Barillas Hansen and Sargents 2009 measures of the costs of removing random fluctuations in aggregate consumption per capita the dotted line as well as costs of removing model uncertainty the solid line The figure reports these costs as a function of the detection error probability described in subsection 1487 The costs of consumption risk drawn from a Reverse engineered consumption heterogeneity 587 known distribution are small as Lucas asserted but for moderate values of detection error probabilities the costs of model uncertainty are substantial35 1410 Reverse engineered consumption heterogeneity In earlier sections we explored how risksensitive preferences or a fear of model misspecification would increase the volatility of the stochastic discount factor by multiplying the ordinary stochastic discount factor mt1 β expct1 ct with a random variable gεt1 that can be interpreted as a likelihood ratio In this section we describe how Constantinides and Duffie 1996 constructed such a volatilityincreasing multiplicative adjustment in another way namely by introducing incomplete markets and stochastic volatility in the crosssectional distribution of consumption growth Let Rjt1 j 1 J be a list of returns on assets and let mt1 0 be a stochastic discount factor for which Etmt1Rjt1 1 14101 for j 1 J As discussed in section 145 we know that such a discount factor exists under the weak assumptions that returns obey the law of one price and a noarbitrage outcome Constantinides and Duffie 199636 assumed that 14101 holds for some mt1 They then reverse engineered consumption processes Ci t and personal stochastic discount factors mi t1 for a collection of heterogeneous consumers i I with the properties that 1 For each i the personal stochastic discount factor mi t1 satisfies Etmi t1Rjt1 1 j 1 J 35 See De Santis 2007 for a modification of the baseline around which the costs of aggre gate fluctuations are measured De Santis adopts a specification according to which a typical consumers consumption process consists of an aggregate component and an uninsurable id iosyncratic component modeled in the same fashion Constantinides and Duffie 1996 do in the model described in the next section De Santis describes the welfare consequences of eliminating aggregate fluctuations while leaving idiosyncratic fluctuations unaltered at their calibrated value For a coefficient of relative risk aversion of 3 De Santis finds that the ben efits of removing aggregate fluctuations are much larger when idiosyncratic fluctuations are not removed first If one were to repeat De Santiss exercise for a coefficient of risk aversion of 1 the effect that he finds would disappear 36 Also see Mankiw 1986 and Attanasio and Weber 1993 for analyses that anticipate elements of the setup of this section 592 Asset Pricing Empirics 1464 or 14681469 and b can be calibrated to fit observed as set prices without provoking skeptical comments about implausible parameter values and magnitudes of risk prices a test critics like Lucas say 1464 or Tallarinis 1472114722 fails to pass The alternative approach imposes the law of one price via Emt1Rt1 1 and often also the noarbitrage principle via mt1 0 but abandons a link between the stochastic discount factor and a consumption growth process Instead it posits a stochastic process for the stochastic discount factor that is not tightly linked to a theory about consumers preferences and then uses overidentifying restrictions from Emt1Rjt1 1 for a set of N returns Rjt1 i 1 N to let the data reveal risks and their prices The model has two components The first is a vector autoregression that describes underlying risks εt1 and the evolution of the yield rt on a one period risk free claim43 zt1 µ φzt Cεt1 14111 rt δ0 δ 1zt 14112 where φ is a stable m m matrix C is an m m matrix εt1 N0 I is an iid m 1 random vector and zt is an m 1 state vector The second component is a vector of risk prices λt and an associated stochastic discount factor mt1 defined via λt λ0 λzzt 14113 log mt1 rt 1 2λ tλt λ tεt1 14114 Here λ0 is m1 and λz is mm The entries of λt that multiply corresponding entries of the risks εt1 are called risk prices for reasons that we explain in the next subsection The stochastic discount factor mt1 is exponential quadratic in the state zt as a result of the risk prices λt being affine in zt Evidently Et mt1 exp δ0 δ 1zt exp rt 14115 stdt mt1 Et mt1 exp λ tλt 1 1 2 λt 14116 Equation 14115 confirms that rt is the yield on a riskfree oneperiod bond That is why it is often called the short rate in the literature on exponential 43 Note that we are recycling notation by redefining εt1 here 596 Asset Pricing Empirics how the formula Etmt1Rjt1 1 adjusts expected returns for exposures to the vector of risks εt1 14121 Asset pricing in a nutshell Let EP denote an expectation under the physical measure that nature uses to generate the data Our key asset pricing equation is EP t mt1Rjt1 1 for all returns Rjt1 Using 14121 it is convenient to express the exponential quadratic stochastic discount factor 14114 as mt1 ξQ t1 ξQ t exp rt where remember that rt is the riskfree net short rate Then the condition EP t mt1Rjt1 1 is equivalent with EP t exprt ξQ t1 ξQ t Rjt1 1 or EQ t Rjt1 exp rt where EQ t is the conditional expectation under the risk neutral measure 14122 Under the risk neutral measure expected returns on all assets equal the riskfree return 1413 Distorted beliefs Piazzesi Salomao and Schneider 2015 assemble survey evidence that suggests that economic experts forecasts are systematically biased Let ztT t1 be a record of observations on the state z and let ˇzt1T t1 be a record of one period ahead expert forecasts Let ˇµ ˇφ be regression coefficients in the least squares regression ˇzt1 ˇµ ˇφzt et1 14131 where the residual et1 has mean zero is orthogonal to zt and we assume that Eet1e t1 CC By comparing estimates of the regression coefficients µ φ in equation 14111 that nature uses to generate the data with estimates of ˇµ ˇφ in 14131 that describe the subjective beliefs of the experts Piazzesi 600 Asset Pricing Empirics possible only if ax1 bx2 0 contradicting the premise that x1 and x2 are linearly independent Thus there is a unique linearly independent vector x1 that is a basis for N We propose the following scaled version of the basis vector x1 as our can didate for the vector y in representation 14A1 y φ x1 x1 x1 x1 14A2 Evidently y N By computing a population linear least squares regression of x X on y X we can represent x as the sum of the linear least squares projection of x1 on y and an orthogonal residual x ay x ay 14A3 where a is the scalar regression coefficient a x y y y 14A4 Both ay N and x ay N are unique in representation 14A4 In 14A3 ay N and x ay N because the least squares residual x ay is orthogonal to the regressor y Therefore applying φ to both sides of 14A3 gives φ x aφ y by the linearity of φ and the fact that φx ay 0 because x ay N Direct computations show that a xyx1x1 φx12 and from definition 14A2 that φy xy x1x1 Therefore φx aφy x y Remark Suppose that x M where M is a closed linear subspace of X Then a corollary of Theorem 14A1 asserts that there exist multiple random variables y X for which φ x E yx The random variable y can be constructed as y y ε where y is constructed as in Theorem 14A1 except that now it is required to be in the linear sub space M and ε is any random vector in the orthogonal complement of M ie A log normal bond pricing model 605 14C1 Slope of yield curve From 14C7 it follows immediately that the unconditional mean of the term structure is Ey t δ σ2 1 2 δ σ2 n 2n so that the term structure on average rises with horizon only if σ2 j j falls as j increases By interpreting our formulas for the σ2 j s it is possible to show that a term structure that on average rises with maturity implies that the log of the stochastic discount factor is negatively serially correlated Thus it can be verified from 14C3 that the term σ2 j in 14C4 and 14C5 satisfies σ2 j vart log mt1 log mtj where vart denotes a variance conditioned on time t information zt Notice for example that vart log mt1 log mt2 vart log mt1 vart log mt2 2covt log mt1 log mt2 14C8 where covt is a conditional covariance It can then be established that σ2 1 σ2 2 2 can occur only if covtlog mt1 log mt2 0 Thus a yield curve that is upward sloping on average reveals that the log of the stochastic discount factor is negatively serially correlated See the spectrum of the log stochastic discount factor in Figure 14C5 14C2 Backus and Zins stochastic discount factor For a specification of Az Cz δ for which the eigenvalues of Az are all less than unity we can use the formulas presented above to compute moments of the stationary distribution EYt as well as the autocovariance function CovY τ and the impulse response function given in 2414 or 2415 For the term structure of nominal US interest rates over much of the postWorld War II period Backus and Zin 1994 provide us with an empirically plausible speci fication of Az Cz ez In particular they specify that log mt1 is a stationary autoregressive moving average process φ L log mt1 φ 1 δ θ L σwt1 606 Asset Pricing Empirics where wt1 is a scalar Gaussian white noise with Ew2 t1 1 and φ L 1 φ1L φ2L2 14C9a θ L 1 θ1L θ2L2 θ3L3 14C9b Backus and Zin specified parameter values that imply that all of the zeros of both φL and θL exceed unity in modulus51 a condition that ensures that the eigenvalues of Ao are all less than unity in modulus Backus and Zins specification can be captured by setting zt log mt log mt1 wt wt1 wt2 and Az φ1 φ2 θ1σ θ2σ θ3σ 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 and Cz σ 0 1 0 0 where σ 0 is the standard deviation of the innovation to log mt1 and ez 1 0 0 0 0 14C3 Reverse engineering a stochastic discount factor Backus and Zin use time series data on yt together with the restrictions im plied by the log normal bond pricing model to deduce implications about the stochastic discount factor mt1 They call this procedure reverse engineering the yield curve but what they really do is use time series observations on the yield curve to reverse engineer a stochastic discount factor They used the gen eralized method of moments to estimate some people say calibrate the fol lowing values for monthly United States nominal interest rates on pure discount bonds δ 528 σ 1023 θL 1 1031448L 073011L2 000322L3 φL 11031253L073191L2 Why do Backus and Zin carry along so many digits To explain why first notice that with these particular values θL φL 1 so that the log of the stochastic discount factor is well approximated by an iid process log mt1 δ σwt1 51 A complex variable z0 is said to be a zero of φz if φz0 0 A log normal bond pricing model 607 This means that fluctuations in the log stochastic discount factor are difficult to predict Backus and Zin argue convincingly that to match observed features that are summarized by estimated first and second moments of the nominal term structure yt process and for yields on other risky assets for the United States after World War II it is important that θL φL have two properties a first θL φL so that the stochastic discount factor is a volatile vari able whose fluctuations are difficult to predict variable and b nevertheless that θL φL so that the stochastic discount factor has subtle predictable components Feature a is needed to match observed prices of risky securities as we shall discuss in chapter 14 In particular observations on returns on risky securities can be used to calculate a socalled market price of risk that in the ory should equal σtmt1 Etmt1 where σt denotes a conditional standard deviation and Et a conditional mean conditioned on time t information Empirical es timates of the stochastic discount factor from the yield curve and other asset returns suggest a value of the market price of risk that is relatively large in a sense that we explore in depth in chapter 14 A high volatility of mt1 deliv ers a high market price of risk Backus and Zin use feature b to match the shape of the yield curve over time Backus and Zins estimates of φL θL imply term structure outcomes that display both features a and b For their values of θL φL σ Figures 14C114C5 show various aspects of the theoretical yield curve Figure 14C1 shows the theoretical value of the mean term structure of interest rates which we have calculated by applying our chap ter 2 formula for µY GµX to 14C7 The theoretical value of the yield curve is on average upward sloping as is true also in the data For yields of durations j 1 3 6 12 24 36 48 60 120 360 where duration is measured in months Figure 14C2 shows the impulse response of yjt to a shock wt1 in the log of the stochastic discount factor We use formula 2415 to compute this impulse response function In Figure 14C2 bigger impulse response functions are associated with shorter horizons The shape of the impulse response func tion for the short rate differs from the others it is the only one with a humped shape Figures 14C3 and 14C4 show the impulse response function of the log of the stochastic discount factor Figure 14C3 confirms that log mt1 is approximately iid the impulse response occurs mostly at zero lag but Fig ure 14C4 shows the impulse response coefficients for lags of 1 and greater and confirms that the stochastic discount factor is not quite iid Since the initial response is a large negative number these small positive responses for positive 608 Asset Pricing Empirics lags impart negative serial correlation to the log stochastic discount factor As noted above and as stressed by Backus and Zin 1992 negative serial correla tion of the stochastic discount factor is needed to account for a yield curve that is upward sloping on average 0 50 100 150 200 250 300 350 400 0054 0056 0058 006 0062 0064 0066 0068 007 0072 0074 Figure 14C1 Mean term structure of interest rates with BackusZin stochastic discount factor months on horizontal axis 0 5 10 15 20 25 30 5 45 4 35 3 25 2 15 1 05 0 x 10 3 Figure 14C2 Impulse response of yields ynt to innovation in stochastic discount factor Bigger responses are for shorter maturity yields A log normal bond pricing model 609 0 5 10 15 20 25 30 35 40 45 50 14 12 10 8 6 4 2 0 2 log sdf Figure 14C3 Impulse response of log of stochastic dis count factor 1 2 3 4 5 6 7 8 9 10 0 05 1 15 2 25 3 35 4 45 5 x 10 3 log sdf after 0 Figure 14C4 Impulse response of log stochastic discount factor from lag 1 on Figure 14C5 applies the Matlab program bigshow3 to Backus and Zins specified values of σ δ θL φL The panel on the upper left is the im pulse response again The panel on the lower left shows the covariogram which as expected is very close to that for an iid process The spectrum of the log stochastic discount factor is not completely flat and so reveals that the log stochastic discount factor is serially correlated Remember that the spectrum for a serially uncorrelated process a white noise is perfectly flat That the Exercises 623 where UC C1γ 1γ γ 0 and β 0 1 Consumer i has a stochastic endowment of the consumption good described by log Y i t1 log Y i t µ σǫi t1 where ǫi t1 N0 1 for i 1 2 and Y i 0 is given for i 1 2 The consumers trade a single asset a riskfree bond whose gross rate of return between t and t 1 is Rt exprt Each consumer faces a sequence of budget constraints 2 Ci t R1 t bi t1 Y i t bi t t 0 where bi 0 0 for i 1 2 and where bi t is person is holdings of free bonds at the beginning of time t The gross interest rate Rt is known at time t Each consumer faces Rt as a price taker and chooses a stochastic process for Ci t bi t1 to maximize 1 subject to 2 and the initial conditions Y i 0 bi 0 a For consumer i please compute a personal stochastic discount factor evaluated at a notrade Ci t Y i t allocation For i j does person is personal stochastic discount factor equal person j s Please explain why or why not b Where log β ρ verify that Ci t Y i t bi t1 0 rt ρ γµ 5σ2γ2 are competitive equilibrium objects for the incomplete markets economy with financial trades only in a risk free bond c In what sense does this economy display multiple stochastic discount factors Is there anything that is unique about the stochastic discount factors Exercise 1417 Incomplete Markets II Consider a version of the economy described in exercise 1416 but in which the endowment processes for the two consumers are now described by log Y i t1 log Y i t µ xt σǫi t1 xt1 λxt σxut1 where λ 1 ǫt1 N0 1 and ut1 N0 1 Everything else about the economy is the same a Show that equilibrium objects are now Ci t Y i t bi t1 0 and rt ρ γ µ xt 5σ2γ2 626 Asset Pricing Empirics 1 5 100 1000 10000 Argue informally that as t ξt converges in dis tribution to 0 despite the fact that Eξt 1 for all t 1 How can this be Hint Compute the skewness of the distribution of ξt or plot the cdf of ξt for various values of the nonnegative integer t Exercise 1421 Distorted beliefs A representative agent with distorted beliefs prices assets The physical measure that governs the log of consumption growth is conistent with the law of motion 1 ct1 ct µ σcεt1 t 0 where εt1 N0 1 However the representative agent believes that the disturbance εt1 in equation 1 actually has distribution Nw 1 where w is a constant scalar Call the probability measure under this distorted belief the subjective measure and denote the mathematical expectations taken under this measure ES The representative agent uses a stochastic discount factor described by m t1 exp ρ exp ct1 ct so he acts as if he has time separable preferences with CRRA preferences and coefficient of risk aversion equal to 1 Assets are priced so that the return Rjt1 on any asset j 1 J obeys 3 ES t m t1Rjt1 1 where ES t is a conditional expectation with respect to subjective biased beliefs rather than the physical measure a Please explain the role of the parameters ρ and w in helping specification 2 attain the HansenJagannathan bounds b A macroeconomist from Minnesota always imposes rational expectations and so instead of fitting equation 2 to the data taking into account the gap between the subjective and physical measure fits the model EP t mt1Rjt1 1 where EP t denotes a conditional expectation under the physical measure and mt1 is the stochastic discount factor fit by this rational expectations econo metrician Please reverse engineer a stochastic discount factor mt1 that this 630 Asset Pricing Empirics Here α0j and η0j are m 1 vectors and αzj and ηzj are m m matrices a Please verify that EtRjt1 expνtj b Please find a formula for νtj as a function of zt that verifies Emt1Rjt1 1 c Please explain why νtj depends on αtj the way it does d Please explain why νtj depends on ηtj the way it does Chapter 15 Economic Growth 151 Introduction This chapter describes basic nonstochastic models of sustained economic growth We begin by describing a benchmark exogenous growth model in which sustained growth is driven by exogenous growth in labor productivity Then we turn our attention to several endogenous growth models in which sustained growth of labor productivity is somehow chosen by the households in the economy We describe several models that differ in whether the equilibrium market economy matches what a benevolent planner would choose Where the market outcome doesnt match the planners outcome there can be room for welfareimproving government interventions The objective of the chapter is to shed light on the mechanisms at work in different models We facilitate comparison by using the same production function and simply changing the meaning of one argument In the spirit of Arrows 1962 model of learning by doing Romer 1986 presents an endogenous growth model in which the accumulation of capital or knowledge is associated with a positive externality on the available technology The aggregate of all agents holdings of capital is positively related to the level of technology which in turn interacts with individual agents savings decisions and thereby determines the economys growth rate Thus the households in this economy are choosing how fast the economy is growing but they do so in an unintentional way The competitive equilibrium growth rate is less than the socially optimal one Another approach assumes that all production factors are reproducible Following Uzawa 1965 Lucas 1988 formulates a model with accumulation of both physical and human capital The joint accumulation of all inputs ensures that growth will not come to a halt even though each individual factor in the finalgood production function is subject to diminishing returns In the absence of externalities the growth rate in the competitive equilibrium coincides with the social optimum 631 632 Economic Growth Romer 1987 constructs a model in which agents can choose to engage in research that produces technological improvements Each invention represents a technology for producing a new type of intermediate input that can be used in the production of final goods without affecting the marginal product of existing intermediate inputs The introduction of new inputs enables the economy to ex perience sustained growth even though each intermediate input taken separately is subject to diminishing returns In a decentralized equilibrium private agents will expend resources on research only if they are granted property rights over their inventions Under the assumption of infinitely lived patents Romer solves for a monopolistically competitive equilibrium that exhibits the classic tension between static and dynamic efficiency Patents and the associated market power are necessary for there to be research and new inventions in a decentralized equilibrium while the efficient production of existing intermediate inputs would require marginalcost pricing that is the abolition of granted patents The monopolistically competitive equilibrium is characterized by smaller supplies of intermediate inputs and a lower growth rate than is socially optimal Finally we revisit the question of when nonreproducible factors may not pose an obstacle to growth Rebelo 1991 shows that even if there are nonrepro ducible factors in fixed supply in a neoclassical growth model sustained growth is possible if there is a core of capital goods that is produced without direct or indirect use of the nonreproducible factors Because of the everincreasing relative scarcity of a nonreproducible factor Rebelo finds that its price increases over time relative to a reproducible factor Romer 1990 assumes that research requires the input of labor and not only goods as in his earlier model 1987 Now if labor is in fixed supply and workers innate productivity is constant it follows immediately that growth must asymptotically come to an halt To make sustained growth feasible we can take a cue from our earlier discussion One modeling strategy would be to introduce an externality that enhances re searchers productivity and an alternative approach would be to assume that researchers can accumulate human capital Romer adopts the first type of as sumption and we find it instructive to focus on its role in overcoming a barrier to growth that nonreproducible labor would otherwise pose 642 Economic Growth which directly implies that human capital must also grow at the rate 1µ along a balanced growth path Moreover by equation 1555b the growth rate is 1 µ 1 A 1 φ 1558 so it remains to determine the steadystate value of φ The equilibrium value of φ has to be such that a unit of human capital receives the same factor payment in both sectors that is the marginal products of human capital must be the same ptA 1 α Kα t φXtα where pt is the relative price of human capital in terms of the composite con sumptioncapital good Since the ratio KtXt is constant along a balanced growth path it follows that the price pt must also be constant over time Fi nally the remaining equilibrium condition is that the rates of return on human and physical capital be equal pt 1 A pt1 αKα1 t φXt1α 1 δ and after invoking a constant steadystate price of human capital and equilib rium condition 1556 we obtain 1 µ β 1 A1σ 1559 Thus the growth rate is positive as long as β1A 1 but feasibility requires also that solution 1559 fall below 1A which is the maximum growth rate of human capital in equation 1555b This parameter restriction β1A1σ 1 A also ensures that the growth rate in equation 1559 yields finite lifetime utility As in the onesector model there is no discrepancy between private and social rates of return so the competitive equilibrium is Pareto optimal Lucas 1988 does allow for an externality in the spirit of our earlier section where the economywide average of human capital per worker enters the production function in the goods sector but as he notes the externality is not needed to generate endogenous growth Lucas provides an alternative interpretation of the technologies in equations 1555 Each worker is assumed to be endowed with one unit of time The time 646 Economic Growth 0 002 004 006 008 01 0 0005 001 0015 002 0025 003 0035 004 Net interest rate R Figure 1561 Interest rates in a version of Romers 1987 model of research and monopolistic competition The dot ted line is the linear relationship κR while the solid and dashed curves depict ΩmR and ΩsR respectively The intersection between κR and ΩmR ΩsR determines the interest rate along a balanced growth path for the laissezfaire economy planner allocation as long as R β1 1 The parameterization is α 09 κ 03 and L 1 Note that the solution to equation 15610 exhibits positive scale effects where a larger labor force L implies a higher interest rate and therefore a higher growth rate in equation 15611 The reason is that a larger economy enables input producers to profit from a larger sales volume in equation 1567b which spurs more inventions until the discounted stream of profits of an input is driven down to the invention cost κ by means of the higher equilibrium interest rate In other words it is less costly for a larger economy to expand its range of inputs because the cost of an additional input is smaller in per capita terms 648 Economic Growth We can also show that the laissezfaire supply of an input falls short of the socially optimal one Zm Zs α 1 Rs 1 Rm 1 15616 To establish condition 15616 divide equation 1567b by equation 15612 Thus the laissezfaire equilibrium is characterized by a smaller supply of each intermediate input and a lower growth rate than would be socially optimal These inefficiencies reflect the fact that suppliers of intermediate inputs do not internalize the full contribution of their inventions and so their monopolistic pricing results in less than socially efficient quantities of inputs 157 Growth in spite of nonreproducible factors 1571 Core of capital goods produced without nonreproducible inputs It is not necessary that all factors be producible in order to experience sustained growth through factor accumulation in the neoclassical framework Instead Rebelo 1991 shows that the critical requirement for perpetual growth is the existence of a core of capital goods that is produced with constant returns technologies and without the direct or indirect use of nonreproducible factors Here we will study the simplest version of his model with a single capital good that is produced without any input of the economys constant labor endowment Jones and Manuelli 1990 provide a general discussion of convex models of economic growth and highlight the crucial feature that the rate of return to accumulated capital must remain bounded above the inverse of the subjective discount factor in spite of any nonreproducible factors in production Rebelo 1991 analyzes the competitive equilibrium for the following tech nology Ct L1α φtKtα 1571a It A 1 φt Kt 1571b Kt1 1 δ Kt It 1571c 650 Economic Growth Thus the growth rate of capital and therefore the growth rate of consumption are positive as long as β 1 δ A 1 1578a Moreover the maintained assumption of this chapter that parameters are such that derived growth rates yield finite lifetime utility βct1ct1σ 1 imposes here the parameter restriction ββ1 δ Aα1σ1α1σ 1 which can be simplified to read β 1 δ Aα1σ 1 1578b Given that conditions 1578 are satisfied there is a unique equilibrium value of φ because the left side of equation 1577 is monotonically decreasing in φ 0 1 and it is strictly greater smaller than the right side for φ 0 φ 1 The outcome is socially efficient because private and social rates of return are the same as in the previous models with all factors reproducible 1572 Research labor enjoying an externality Romers 1987 model includes labor as a fixed nonreproducible factor but similar to the last section an important assumption is that this nonreproducible factor is not used in the production of inventions that expand the input variety which constitutes a kind of reproducible capital in that model In his sequel Romer 1990 assumes that the input variety At is expanded through the effort of researchers rather than the resource cost κ in terms of final goods Suppose that we specify this new invention technology as At1 At η 1 φt L where 1 φt is the fraction of the labor force employed in the research sector and φt is working in the finalgoods sector After dividing by At it becomes clear that this formulation cannot support sustained growth since new inven tions bounded from above by ηL must become a smaller fraction of any growing range At Romer solves this problem by assuming that researchers productivity grows with the range of inputs ie an externality as discussed previously At1 At ηAt 1 φt L 652 Economic Growth to determine which consumption growth rate given by equation 15712 is supported by Euler equation 15611 1 ηL Rm α β 1 Rm1σ 15713 The left side of equation 15713 is monotonically decreasing in Rm and the right side is increasing It is also trivially true that the left side is strictly greater than the right side for Rm 0 Thus a unique solution exists as long as the technology is sufficiently productive in the sense that β1 αηL 1 This parameter restriction ensures that the left side of equation 15713 is strictly less than the right side at the interest rate Rm αηL corresponding to a situation with zero growth since no labor is allocated to the research sector φ 1 Equation 15713 shows that this alternative model of research shares the scale implications described earlier that is a larger economy in terms of L has a higher equilibrium interest rate and therefore a higher growth rate It can also be shown that the laissezfaire outcome continues to produce a smaller quantity of each input and to yield a lower growth rate than what is socially optimal An additional source of underinvestment is now that agents who invent new inputs do not take into account that their inventions will increase the productivity of all future researchers 158 Concluding remarks This chapter has focused on the mechanics of endogenous growth models with only limited motivation for assumptions For example we have examined how externalities might enter models to overcome the onset of diminishing returns from nonreproducible factors without referring too much to the authors inter pretation of those externalities The formalism of models is of course silent on why the assumptions are made but the conceptual ideas behind the models contain valuable insights In the last setup Paul Romer argues that input de signs represent excludable factors in the monopolists production of inputs but the input variety A is also an aggregate stock of knowledge that enters as a nonexcludable factor in the production of new inventions That is the patent holder of an input type has the sole right to produce and sell that particular in put but she cannot stop inventors from studying the input design and learning Concluding remarks 653 knowledge that helps to invent new inputs This multiple use of an input design hints at the nonrival nature of ideas and technology ie a nonrival object has the property that its use by one person in no way limits its use by another Romer 1990 p S75 emphasizes this fundamental nature of technology and its implication If a nonrival good has productive value then output cannot be a constantreturnstoscale function of all its inputs taken together The standard replication argument used to justify homogeneity of degree one does not apply because it is not necessary to replicate nonrival inputs Thus an endogenous growth model that is driven by technological change must be one where the advancement enters the economy as an externality or the assumption of perfect competition must be abandoned Besides technological change an alternative approach in the endogenous growth literature is to assume that all production factors are reproducible or that a core of capital goods is produced without direct or indirect use of nonreproducible factors Much effort in the endogenous growth literature has been expended to spec ify an appropriate technology Even though growth is an endogenous outcome in these models its manifestation ultimately hinges on technology assumptions In the case of the last setup as pointed out by Romer 1990 p S84 Linearity in A is what makes unbounded growth possible and in this sense unbounded growth is more like an assumption than a result of the model It follows that various implications of the analyses stand and fall with the assumptions on technology For example the preceding model of research and monopolistic competition implies that the laissezfaire economy grows at a slower rate than the social optimum but Benassy 1998 shows how this result can be over turned if the production function for final goods on the right side of equation 1561 is multiplied by the input range raised to some power ν Aν t The laissezfaire growth rate can exceed the socially optimal rate because of how the new production function rearranges input producers market power measured by the parameter α and the economys returns to specialization measured by the parameter ν Segerstrom Anant and Dinopoulos 1990 Grossman and Helpman 1991 and Aghion and Howitt 1992 provide early attempts to explore endogenous growth arising from technologies that allow for product improvements and there fore product obsolescence These models open the possibility that the laissez faire growth rate is excessive because of a businessstealing effect where agents 658 Economic Growth is increasing differentiable concave and homogeneous of degree 1 Firms max imize the present discounted value of profits Assume that initial ownership of firms is uniformly distributed across households a Define a competitive equilibrium b Discuss i and ii and justify your answer Be as formal as you can i Economist A argues that the steady state of this economy is unique and independent of the ui functions while B says that without knowledge of the ui functions it is impossible to calculate the steadystate interest rate ii Economist A says that if k0 is the steadystate aggregate stock of capital then the pattern of consumption inequality will mirror exactly the pattern of initial capital inequality ie ki0 even though capital markets are perfect Economist B argues that for all k0 in the long run per capita consumption will be the same for all households c Assume that the economy is at the steady state Describe the effects of the following three policies i At time zero capital is redistributed across households ie some people must surrender capital and others get their capital ii Half of the households are required to pay a lumpsum tax The proceeds of the tax are used to finance a transfer program to the other half of the population iii Twothirds of the households are required to pay a lumpsum tax The proceeds of the tax are used to finance the purchase of a public good say g which does not enter in either preferences or technology Exercise 156 Taxes and growth donated by Rodolfo Manuelli Consider a simple twoplanner economy The first planner picks tax rates τt and makes transfers to the representative agent vt The second planner takes the tax rates and the transfers as given That is even though we know the connection between tax rates and transfers the second planner does not he or she takes the sequence of tax rates and transfers as given and beyond his or her control when solving for the optimal allocation Thus the problem faced by the Chapter 16 Optimal Taxation with Commitment 161 Introduction This chapter formulates a dynamic optimal taxation problem called a Ramsey problem whose solution is called a Ramsey plan The governments goal is to maximize households welfare subject to raising prescribed revenues through distortionary taxation When designing an optimal policy the government takes into account the competitive equilibrium reactions by consumers and firms to the tax system We first study a nonstochastic economy then a stochastic economy The model is a competitive equilibrium version of the basic neoclassical growth model with a government that finances an exogenous stream of govern ment purchases In the simplest version the production factors are raw labor and physical capital on which the government levies distorting flatrate taxes The problem is to determine optimal sequences for the two tax rates In a non stochastic economy Chamley 1986 and Judd 1985b show in related settings that if an equilibrium has an asymptotic steady state then the optimal policy is eventually to set the tax rate on capital to zero1 This remarkable result asserts that capital income taxation serves neither efficiency nor redistributive pur poses in the long run The conclusion follows immediately from timeadditively separable utility a constantreturnstoscale production technology competitive markets and a complete set of flatrate taxes However if the tax system is in complete the limiting value of the optimal capital tax can differ from zero To illustrate this possibility we follow Correia 1996 and study a case with an additional fixed production factor that cannot be taxed by the government 1 Straub and Werning 2015 offer corrections to Chamleys 1986 and Judds 1985b results about an asymptotically zero tax rate on capital for specifications in which preferences are nonadditive intertemporally the government budget must be balanced each period and an infinite sequence of restrictions is imposed on the sequence of tax rates on capital Our treatment here steers clear of these situations by assuming timeadditively separable utility a government that can freely access debt markets subject to the usual noPonzi constraints and a restriction on the capital tax rate only in the initial period 661 662 Optimal Taxation with Commitment In a stochastic version of the model with complete markets we find in determinacy of statecontingent debt and capital taxes Infinitely many plans implement the same competitive equilibrium allocation For example two such plans are 1 that the government issues riskfree bonds and lets the capital tax rate depend on the current state or 2 that the government fixes the capital tax rate one period ahead and lets debt be state contingent While the stateby state capital tax rates cannot be pinned down an optimal plan does determine the current market value of next periods tax payments across states of nature Dividing by the current market value of capital income gives a measure that we call the ex ante capital tax rate If there exists a stationary Ramsey alloca tion Zhu 1992 shows that for some special utility functions the Ramsey plan prescribes a zero ex ante capital tax rate that can be implemented by setting a zero tax on capital income But except for those preferences Zhu concludes that the ex ante capital tax rate should vary around zero in the sense that there is a positive measure of states with positive tax rates and a positive measure of states with negative tax rates Chari Christiano and Kehoe 1994 perform numerical simulations and conclude that an optimal ex ante capital tax rate is approximately zero To gain further insights we turn to Lucas and Stokey 1983 who analyze a completemarkets model without physical capital Examples of deterministic and stochastic government expenditure streams bring out the important role of government debt in smoothing tax distortions over both time and states State contingent government debt is used as an insurance policy that allows the government to smooth taxes across states In this complete markets model the current value of the governments debt reflects the current and likely future path of government expenditures rather than anything about its past This feature of an optimal debt policy is especially apparent when government expenditures follow a Markov process because then the beginningofperiod statecontingent government debt is a function of the current state only and hence there are no lingering effects of past government expenditures Aiyagari Marcet Sargent and Seppala 2002 alter that outcome by assuming that the government can issue only riskfree debt Not having access to statecontingent debt constrains the governments ability to smooth taxes over states and allows past values of government expenditures to have persistent effects on both future tax rates and debt levels Reasoning by analogy from the savings problem of chapter 17 to an optimal taxation problem Barro 1979 asserted that tax revenues would be a Introduction 663 martingale that is cointegrated with government debt Barro thus predicted persistent effects of government expenditures that are absent from the Ramsey plan in Lucas and Stokeys model Aiyagari et als suspension of complete markets goes a long way toward rationalizing outcomes Barro had described In a monetary economy in which a government can trade only nominal debt bearing a riskfree nominal interest rate Chari Christiano and Kehoe 1996 construct an optimal monetary policy that implements the same Ramsey allocation that would prevail if the government were also able to issue state contingent real debt An optimal monetary policy accomplishes that by engi neering statecontingent inflation that transforms nonstatecontingent nominal debt into statecontingent real debt Systematic variations in the nominal price level rearrange real government obligations across states in the following ways In bad times that are associated with high government expenditures it is opti mal to raise the price level so that real returns on nominal debt are relatively small while for symmetric reasons it is optimal to lower the price level in good times that are associated with low government expenditures We also use the Chari Christiano and Kehoe 1996 framework to compare views about the fis cal theory of the price level Because it spells out all of the details the model of Chari el al delivers a coherent general equilibrium analysis of the determinants of the nominal price level and real government indebtedness at each node of an ArrowDebreu event tree We return to the fiscal theory of the price level in chapter 27 Jones Manuelli and Rossi 1997 augment a nonmonetary nonstochastic growth model by allowing human capital accumulation They make the partic ular assumption that the technology for human capital accumulation is linearly homogeneous in a stock of human capital and a flow of inputs coming from current output Under this special constant returns assumption they show that a zero limiting tax applies also to labor income that is the return to human capital should not be taxed in the limit Instead the government should re sort to a consumption tax But for a particular class of preferences even this consumption tax and therefore all taxes should be zero in the limit when it is optimal during a transition period for the government to amass enough claims on the private economy that interest earnings suffice to finance government expenditures While these successive results on optimal taxation require ever more stringent assumptions the basic prescription for a zero capital tax in a nonstochastic steady state is an implication of timeadditively separable utility 668 Optimal Taxation with Commitment 163 The Ramsey problem We shall use symbols without subscripts to denote the onesided infinite sequence for the corresponding variable eg c ct t0 Definition A feasible allocation is a sequence k c ℓ g that satisfies equation 1623 Definition A price system is a 3tuple of nonnegative bounded sequences w r R Definition A government policy is a 4tuple of sequences g τ k τ n b Definition A competitive equilibrium is a feasible allocation a price system and a government policy such that a given the price system and the govern ment policy the allocation solves both the firms problem and the households problem and b given the allocation and the price system the government policy satisfies the sequence of government budget constraints 1625 There are many competitive equilibria indexed by different government policies This multiplicity motivates the Ramsey problem Definition Given k0 and b0 the Ramsey problem is to choose a competitive equilibrium that maximizes expression 1621 To make the Ramsey problem interesting we always impose a restriction on τ k 0 for example by taking it as given at a small number say 0 This approach rules out taxing the initial capital stock via a socalled capital levy that would constitute a lumpsum tax since k0 is in fixed supply2 2 According to our assumption on the technology in equation 1623 capital is reversible and can be transformed back into the consumption good Thus the capital stock is a fixed factor for only one period at a time so τk 0 is the only tax that we need to restrict to ensure an interesting Ramsey problem 670 Optimal Taxation with Commitment The equation has a straightforward interpretation A marginal increment of capital investment in period t increases the quantity of available goods at time t 1 by the amount Fkt 1 1 δ which has a social marginal value θt1 In addition there is an increase in tax revenues equal to Fkt1rt1 which enables the government to reduce its debt or other taxes by the same amount The reduction of the excess burden equals Ψt1Fkt 1 rt1 The sum of these two effects in period t 1 is discounted by the discount factor β and set equal to the social marginal value of the initial investment good in period t which is given by θt Suppose that government expenditures stay constant after some period T and assume that the solution to the Ramsey problem converges to a steady state that is all endogenous variables remain constant Using equation 16218a the steadystate version of equation 1642 is θ β Ψ r r θ r 1 δ 1643 Now with a constant consumption stream the steadystate version of the house holds optimality condition for the choice of capital in equation 16211b is 1 β r 1 δ 1644 A substitution of equation 1644 into equation 1643 yields θ Ψ r r 0 1645 Since the marginal social value of goods θ is strictly positive and the marginal social value of reducing government debt or taxes Ψ is nonnegative it follows that r must be equal to r so that τ k 0 This analysis establishes the following celebrated result versions of which were attained by Chamley 1986 and Judd 1985b Proposition 1 If there exists a steadystate Ramsey allocation the associated limiting tax rate on capital is zero It is important to keep in mind that the zero tax on capital result pertains only to the limiting steady state Our analysis is silent about how much capital is taxed in the transition period Nonzero capital tax due to incomplete taxation 677 In contrast to equation 1657 kt enters now as an argument in V because of the presence of the marginal product of the factor Z but we have chosen to suppress the quantity Z itself since it is in fixed supply Except for these changes of the functions F and V the Lagrangian of the Ramsey problem is the same as equation 1658 The firstorder condition with respect to kt1 is θt βVk t 1 βθt1 Fk t 1 1 δ 1673 Assuming the existence of a steady state the stationary version of equation 1673 becomes 1 β Fk 1 δ β Vk θ 1674 Condition 1674 and the noarbitrage condition for capital 16512 imply an optimal value for τ k τ k Vk θFk ΦucZ θFk Fzk As discussed earlier in a secondbest solution with distortionary taxation Φ 0 so the limiting tax rate on capital is zero only if Fzk 0 Moreover the sign of τ k depends on the direction of the effect of capital on the marginal product of the untaxed factor Z If k and Z are complements the limiting capital tax is positive and it is negative in the case where the two factors are substitutes Other examples of a nonzero limiting capital tax are presented by Stiglitz 1987 and Jones Manuelli and Rossi 1997 who assume that two types of labor must be taxed at the same tax rate Once again the incompleteness of the tax system makes the optimal capital tax depend on how capital affects the marginal products of the other factors 686 Optimal Taxation with Commitment Proposition 2 If there exists a stationary Ramsey allocation the ex ante capital tax rate is such that a either P τ k t 0 1 or P τ k t 0 0 and P τ k t 0 0 b P τ k t 0 1 if and only if P Vcct nt Φucct ℓt Λ 1 for some constant Λ A sketch of the proof is provided in the next subsection Let us just add here that the two possibilities with respect to the ex ante capital tax rate are not vacuous One class of utilities that imply P τ k t 0 1 is u ct ℓt c1σ t 1 σ v ℓt for which the ratio Vcct nt Φucct ℓt is equal to 1 Φ1 σ which plays the role of the constant Λ required by Proposition 2 Chari Christiano and Kehoe 1994 solve numerically for Ramsey plans when the preferences do not satisfy this condition In their simulations the ex ante tax on capital income remains approximately equal to zero To revisit the result on the optimality of a zero capital tax in a nonstochastic economy it is trivially true that the ratio Vcct nt Φucct ℓt is constant in a nonstochastic steady state In a stationary equilibrium of a stochastic economy Proposition 2 extends this result for some utility functions the Ramsey plan prescribes a zero ex ante capital tax rate that can be implemented by setting a zero tax on capital income But except for such special classes of preferences Proposition 2 states that the ex ante capital tax rate should fluctuate around zero in the sense that P τ k t 0 0 and P τ k t 0 0 Examples of labor tax smoothing 697 1613 Examples of labor tax smoothing Following Lucas and Stokey 1983 we now exhibit examples of government expenditure streams and how they affect optimal tax policies We assume that b0 0 16131 Example 1 gt g for all t 0 Given constant government purchases gt g the firstorder condition 16124 is the same for all t st and we conclude that the optimal allocation is constant over time ct nt ˆc ˆn for t 0 It then follows from condition 1685a or 161210a that the tax rate that implements the optimal allocation is also constant over time τ n t ˆτ n for t 0 The government budget is balanced each period Government debt issues in this economy serve to smooth distortions over time Because government expenditures are already smooth in this economy they are optimally financed from contemporaneous taxes Nothing is gained by using debt to change the timing of tax collections 16132 Example 2 gt 0 for t T and nonstochastic gT 0 Setting g 0 in expression 16124 the optimal allocation ct nt ˆc ˆn is the same for all t T and consequently from condition 1685a the tax rate is also constant over these periods τ n t ˆτ n for t T Using equations 16127 we can deduce tax revenues Recall that ct nt 0 for t T and that b0 0 Thus the last terms in equations 16127 drop out Since Φ 0 the second quadratic term is negative so the first term must be positive Since 1 Φ 0 this fact implies 0 ˆc uℓ uc ˆn ˆc 1 ˆτ n ˆn ˆτ nˆn where the first equality invokes condition 1685a We conclude that tax rev enue is positive for t T For period T the last term in equation 16127 θT gT is positive Therefore the sign of the first term is indeterminate labor may be either taxed or subsidized in period T 698 Optimal Taxation with Commitment This example is a stark illustration of tax smoothing in which the Ramsey planner uses government debt to redistribute tax distortions over time With the same tax revenues in all periods before and after time T the optimal debt policy is as follows in each period t 0 1 T 1 the government runs a surplus using it to accumulate bonds issued by the representative household So bt bt1 0 for t 1 T In period T the expenditure gT is met by selling all of these bonds possibly levying a tax on current labor income and issuing new bonds that are thereafter rolled over forever Interest payments on that constant outstanding government debt are equal to the constant tax revenue for t T ˆτ nˆn Thus the tax distortion is the same in all periods surrounding period T regardless of their proximity to date T 16133 Example 3 gt 0 for t T and gT is stochastic We assume that gT g 0 with probability α 0 1 and gT 0 with probability 1 α As in the previous example there is an optimal constant allocation ct nt ˆc ˆn for all periods t T although the optimum values of ˆc and ˆn will not in general be the same as in example 2 In addition equation 16124 implies that cT nT ˆc ˆn if gT 0 The argument in example 2 shows that tax revenue is positive Debt issues are as follows At t 0 1 T 2 the government runs a surplus and uses it to ac cumulate riskfree oneperiod bonds issued by the private sector A significant difference from example 2 occurs in period T 1 In the present case the gov ernment now sells all of the bonds that it has accumulated and uses the proceeds plus current labor tax revenue to buy oneperiod Arrow securities that pay off at T only if gT g In addition the government buys more of these contingent claims in period T 1 It finances these additional purchases of Arrow securi ties by simultaneously issuing oneperiod riskfree claims As in example 2 at t T the government just rolls over its riskfree debt and pays out net pay ments equal to ˆτ nˆn only here the riskfree debt is issued one period earlier At time t T there are two cases to consider depending on the realization of gov ernment expenditures at date T a random variable If gT 0 the government clearly satisfies its intertemporal budget constraint If gT g the construction of our Ramsey equilibrium ensures that the payoff on the governments holdings of contingent claims against the private sector equal g plus interest payments of Examples of labor tax smoothing 699 ˆτ nˆn on government debt net of any current labor taxsubsidy in period T In periods T 1 T 2 the situation is as in example 2 regardless of whether gT 0 or gT g This is another example of tax smoothing over time in which the tax dis tortion is the same in all periods around time T It also demonstrates the risksharing aspects of fiscal policy under uncertainty In effect the government in period T 1 buys insurance from the private sector against the event that gT g 16134 Time 0 is special with b0 0 To illustrate how period 0 is special we revisit example 1 with b0 0 We assume preferences uc ℓ logc κℓ with a value of κ 0 that is large enough to guarantee an interior solution to labor n 1 ℓ 0 111 Since government purchases are deterministic and constant over time in example 1 a Ramsey plan brings a timeinvariant allocation c n for t 1 determined by condition 16124 and possibly different values c0 n0 in period 0 determined by condition 16125 In section 16131 the initial condition was b0 0 and hence the two conditions 16124 and 16125 were the same and the Ramsey plan prescribed a timeinvariant allocation ˆc c0 c and ˆn n0 n Given b0 0 and the assumed preference specification expressions 16124 and 16125 become 1 Φ 1 c Φc 1 c2 1 Φ κ 1 c 1 Φ κ 16131a 1 c0 Φ 1 c2 0 b0 1 Φ κ 16131b Since the multiplier Φ is strictly positive in a Ramsey plan with distortionary taxation it follows from expressions 16131 that c0 c c0 c when b0 0 b0 0 Thus according to firstorder condition 161210a τ n 0 τn τ n 0 τn when b0 0 b0 0 where τ n is the labor tax rate for t 1 and τ n 0 is the tax rate in period 0 or a subsidy rate if τ n 0 0 11 A sufficiently large parameter value κ ensures that n 1 and because the utility function satisfies an Inada condition uc as c 0 consumption will be strictly positive in an equilibrium and hence n 0 700 Optimal Taxation with Commitment Consider first the case with initial government debt b0 0 why does the Ramsey plan depart from the policy of perfect tax smoothing in section 16131 If the government so wanted it could roll over a constant debt level forever and in each period finance a constant interest payment in addition to the government expenditure g which would allow for a timeinvariant tax rate ˆτ n to be levied in all periods just as in section 16131 But the Ramsey planner does not choose that feasible policy Instead the Ramsey planner trades off the welfare cost of a timevarying labor tax wedge τ n 0 τn against a welfare gain from manipulating an asset price Using expression 161210b the price of an asset paying one good next period under the Ramsey plan in this nonstochastic economy is p0 β uc c n uc c0 n0 β c0 c β p where p0 is the price in period 0 and p is the price in all future periods t 1 when the allocation c n stays constant Hence the lower labor tax rate τ n 0 induces higher consumption c0 in period 0 and therefore increases the asset price p0 ie lowers the interest rate which reduces the burden of an initially indebted government And the higher the initial debt b0 0 is the more the Ramsey planner would choose to raise c0 relative to c according to expressions 1613112 As one would expect reductions in the interest rate become more valuable when applied to an implied larger stock of government debt carried over between periods 0 and 1 while the welfare loss of too low of a labor tax wedge τ n 0 at time 0 applies to a single period By so manipulating the asset price p0 the Ramsey planner de facto reduces government indebtedness at the beginning of period 1 and hence lowers future debt burden For symmetric reasons given initial government assets b0 0 the Ramsey planner reduces consumption in period 0 c0 c so as to lower the asset price in period 0 p0 p ie increase the interest rate in order to improve the return on the implied stock of government assets carried over between periods 0 and 1 The welfare loss of too high of a labor tax wedge in period 0 τ n 0 τn is outweighed by the increase in the value of government assets at the beginning of period 1 which implies a lower distortionary tax rate τ n relative to what it would have had to be without the manipulation of the asset price p0 12 A higher b0 is associated with a higher multiplier Φ so the product Φb0 in expression 16131b is unequivocally increasing in b0 Lessons for optimal debt policy 701 Now since there are such welfare gains from manipulating the price p0 why do firstorder conditions 16123 of the Ramsey plan not involve yet more similar manipulations of future asset prices The simple answer is that the Ramsey problem solves for an optimal plan under commitment That is the Ramsey planner is not allowed to reoptimize in the future to debase government debt or enhance its assets because that would amount to reneging on rates of return in the original time0 Ramsey plan This reasoning is elucidated in chapter 20 when the Ramsey problem is formulated recursively Then there is a Ramsey planner at time 0 with state variable b0 and in each future period t and history st there is a continuation Ramsey planner with state variable xtst ucstbtstst1 who is constrained to satisfy an implementability condition in the form of expression 161211 An implication being that the continuation Ramsey planners are prevented from manipulating asset prices in the way that is available to the Ramsey planner in period 0 as exemplified in this subsection 1614 Lessons for optimal debt policy Lucas and Stokey 1983 draw three lessons from their analysis of the model in our previous section The first is built into the model at the outset budget balance in a presentvalue sense must be respected In a stationary economy deficits in some states and dates must necessarily be offset by surpluses at other dates and states Thus in examples with erratic government expenditures good times are associated with budget surpluses Second in the face of erratic gov ernment spending the role of government debt is to smooth tax distortions over time and the government should not seek to balance its budget on a continual basis Third the contingentclaim character of government debt is important for an optimal policy13 13 Aiyagari Marcet Sargent and Seppala 2002 offer a qualification to the importance of statecontingent government debt in the model of Lucas and Stokey 1983 In numerical simulations they explore Ramsey outcomes under the assumption that contingent claims cannot be traded We present their setup in section 1615 They find that the incomplete markets Ramsey allocation is very close to the complete markets Ramsey allocation This proximity comes from the Ramsey policys use of selfinsurance through riskfree borrowing and lending with households Compare this outcome to our chapter 18 on heterogeneous agents and how selfinsurance can soften the effects of market incompleteness 704 Optimal Taxation with Commitment capital stock and is the only information needed to form conditional expecta tions of future states Putting together the lessons of this section with earlier ones reliance on statecontingent debt andor statecontingent capital taxes en ables the government to avoid any lingering effects on indebtedness from past shocks to government expenditures and past productivity shocks that affected labor tax revenues This striking lack of history dependence contradicts the extensive history dependence of the stock of government debt that Robert Barro 1979 identified as one of the salient characteristics of his model of optimal fiscal policy Ac cording to Barro government debt should be cointegrated with tax revenues which in turn should follow a random walk with innovations that are perfectly correlated with innovations in the government expenditure process Important aspects of such behavior of government debt seem to be observed For example Sargent and Velde 1995 display long series of government debt for eighteenth century Britain that more closely resembles the outcome from Barros model than from Lucas and Stokeys Partly inspired by those observations Aiyagari Marcet Sargent and Seppala 2002 returned to the environment of Lucas and Stokeys model and altered the market structure in a way that brought outcomes closer to Barros We create their model by closing almost all of the markets that Lucas and Stokey had allowed15 15 Werning 2007 extends the LucasStokey model in another interesting direction He assumes that there are complete markets in consumption that agents are heterogeneous in the efficiencies of their labor supplies and that taxes can be nonlinear functions of labor earnings For example with affine rather than linear taxes he explores how distorting taxes on labor are imposed to redistribute income as well as to raise revenues for financing expenditures Without heterogeneity of labor efficiencies no distorting taxes on labor are imposed 710 Optimal Taxation with Commitment 16151 Future values of gt become deterministic Aiyagari et al 2002 prove that if gtst has absorbing states in the sense that gt gt1 almost surely for t large enough then Ψtst converges when gtst enters an absorbing state The optimal tail allocation for this economy without statecontingent government debt coincides with the allocation of an economy with statecontingent debt that would have occurred under the same shocks but for different initial debt That is the limiting random variable Ψ would then play the role of the single multiplier in an economy with statecontingent debt because as noted above the firstorder condition 161511a would then be the same as expression 16124 where Φ Ψ The value of Ψ depends on the realization of the government expenditure path If the absorbing state is reached after many bad shocks high values of gtst the government would have accumulated high debt and convergence would occur to a contingentdebt economy with high initial debt and therefore a high value of the multiplier Φ This particular result about convergence can be stated in more general terms ie Ψtst can be shown to converge if the future path of government expenditures eventually becomes deterministic for example if government ex penditures eventually become constant Once uncertainty about future gov ernment expenditures ceases the government can thereafter attain the Ramsey allocation with oneperiod riskfree bonds as described at the beginning of this chapter In the present setup this becomes apparent from examining firstorder condition 161511b when there is no uncertainty next periods nonstochastic marginal utility of consumption must be multiplied by a nonstochastic mul tiplier γt1 0 in order for that firstorder condition to be satisfied under certainty The zero value of all future multipliers γt implies convergence of Ψtst Ψ and we return to our earlier logic where expression 16124 with Φt Ψ characterizes the optimal tail allocation for an economy without statecontingent government debt when there is no uncertainty 712 Optimal Taxation with Commitment When setting Ψ γ 0 in firstorder condition 161511a it follows that a Ramsey tax policy must eventually lead to a firstbest allocation with ucst uℓst ie τ n 0 This implies that government assets converge to a level always sufficient to support government expenditures from interest earn ings alone Unspent interest earnings on governmentowned assets are returned to the households as positive lumpsum transfers Such transfers occur when government expenditures fall below their maximum possible level A proof that Ψtst converges to zero and that government assets eventu ally become large enough to finance all future government expenditures can be constructed along lines used in our chapter 17 analysis of selfinsurance with incomplete markets Like the analysis there we can appeal to a martingale convergence theorem and use an argument based on contradictions to rule out convergence to any number other than zero To establish a contradiction in the present setting suppose that Ψtst does not converge to zero but rather to a strictly positive limit Ψ 0 According to our argument above the Ramsey tail allocation for this economy without statecontingent government debt will then coincide with the allocation of an economy that has statecontingent debt and a particular initial debt level It follows that these two economies should have identical labor tax rates supporting that optimal tail allocation But Aiya gari et al 2002 show that a government that follows such a tax policy and has access only to riskfree bonds to absorb stochastic surpluses and deficits will with positive probability either see its debt grow without bound or watch its assets grow without bound two outcomes that are inconsistent with an op timal allocation A heuristic explanation is as follows The government in an economy with statecontingent debt uses these debt instruments as an insur ance policy to smooth taxes across realizations of the state The governments lack of access to such insurance when only riskfree bonds are available means that implementing those very same tax rates unresponsive as they are to re alizations of the state would expose the government to a positive probability of seeing either its debt level or its asset level drift off to infinity But that contradicts a supposition that such a tax policy would be optimal in an econ omy without statecontingent debt First it is impossible for government debt to grow without bound because households would not be willing to lend to a government that violates its natural borrowing limit Second it is not optimal for the government to accumulate assets without bound because welfare could then be increased by cutting tax rates in some periods and thereby reducing the 714 Optimal Taxation with Commitment As in our analysis of this example when there are complete statecontingent debt markets we assume that gT g 0 with probability α and gT 0 with probability 1α We also retain our assumption that the government starts with no assets or debt b0s1 0 so that the multiplier on constraint 16159a is strictly positive γ0s0 Ψ0s0 0 Since no additional information about future government expenditures is revealed in periods t T it follows that the multiplier Ψtst Ψ0s0 Ψ0 0 for t T Given the multiplier Ψ0 the optimal consumption level for t T denoted c0 satisfies the following version of firstorder condition 161513 1 Ψ0 1 Hℓ1 c0 Ψ0 Hℓℓ1 c0 c0 161514 In period T there are two possible values of gT and hence the stochastic multiplier γT sT can take two possible values one negative value and one positive value according to firstorder condition 161511b γT sT is negative if gT 0 because that represents good news that should cause the multiplier ΨT sT to fall In fact the multiplier ΨT sT falls all the way to zero if gT 0 because the government would then never again have to resort to distortionary taxation And any tax revenues raised in earlier periods and carried over as governmentowned assets would then also be handed back to the households as a lumpsum transfer If on the other hand gT g 0 then γT sT γT is strictly positive and the optimal consumption level for t T denoted c would satisfy the following version of firstorder condition 161513 1 Ψ0 γT 1 Hℓ1 c Ψ0 γT Hℓℓ1 c c 161515 In response to γT 0 the multiplicative factors within square brackets have increased on both sides of equation 161515 but proportionately more on the right side Because both equations 161514 and 161515 must hold with equality at the optimal allocation it follows that the change from c0 to c has to be such that 1 Hℓ1 c increases proportionately more than Hℓℓ1 c c Since the former expression is decreasing in c and the latter expression is increasing in c we can conclude that c c0 and hence that the implied labor tax rate is raised for all periods t T if government expenditures turn out to be strictly positive in period T It is evident from this example that a government with access to riskfree bonds only cannot smooth tax rates across different realizations of the state Taxation without statecontingent debt 715 Recall that the optimal tax policy with statecontingent debt prescribed a con stant tax rate for all t T regardless of the realization of gT Note also that as promised earlier the multiplier Ψtst in the economy without statecontingent debt does converge when the future path of government expenditures becomes deterministic in period T In our example Ψtst converges either to zero or to Ψ0 γT 0 depending on the realization of government expenditures Starting from period T 1 the tail of the Ramsey allocation coincides with the allocation of an economy with statecontingent debt that would have occurred under the same shocks but for different initial debt either a strictly negative debt level associated with Φ 0 if gT 0 or a strictly positive debt level that would correspond to Φ Ψ0 γT if gT g 0 It is instructive to consider two realizations of such a statecontingentdebt economy in which timeinvariant multipliers satisfy Φ 0 and Φ Ψ0 γT respectively These are two economies whose tail allocations after period T coincide with those of our economy without statecontingent debt under the realizations gT 0 and gT g 0 respectively A statecontingentdebt economy with multiplier Φ 0 means that its government has never had to rely on any distortionary taxation and hence the government must initially have owned enough claims against the private sector to finance the stochastic government expenditure in period T Thus by using the equilibrium prices of statecontingent claims at a firstbest allocation ie when there is never any distortionary taxation we can compute the strictly negative initial debt level that would suffice to finance a government expenditure of g 0 with probability α in period T If the governments initial claims against the private sector exceed that critical number excess government assets would be handed back lumpsum to the representative household Now consider the statecontingentdebt economy with multiplier Φ Ψ0 γT Its initial government debt level must be strictly positive for the following reasons Recall that the economy without statecontingent debt starts with a zero initial debt level and as we have shown its Ramsey plan involves an increase in the tax rate after the unfortunate realization of gT g 0 In contrast the government in the statecontingentdebt economy levies the same tax rate for all t T and for it to have chosen in all those periods the proposed eventual high tax rate of the economy without statecontingent debt the initial government debt level of the statecontingentdebt economy must have been larger than zero 716 Optimal Taxation with Commitment We end with some reflections on welfare and commitment The representa tive household is ex ante better off when the government can use statecontingent claims because the Pareto problem for that economy has fewer constraints than it does in an economy in which the government is constrained to use only risk free bonds It is also true that the representative household of the riskfree bonds only economy is ex post better off under the tail allocation after period T if the state gT 0 is realized after which there would never be any distor tionary taxation in the economy without statecontingent debt A tail allocation without distortionary taxation would tempt a government facing that tail gov ernment expenditure sequence but finding itself indebted after period T Since the government will never again need access to financial markets to finance any government expenditures if it were offered a choice to do so the government would want to renege on any existing government debt in order to make the representative household better off Yes the representative household would lose from not receiving payments on its holdings of government debt but that loss would be outweighed by the gain from never again having to pay any distor tionary taxes This mental experiment is simply not allowed here because the Ramsey problem has been structured from the beginning so that a government can never renege on its liabilities and the households behavior relies on that If households had anticipated that a government would ever renege on its liabili ties it would not have bought the government debt in the first place This circle of concerns will be the topic of chapters 24 and 25 when we study governments that lack the ability to commit and must therefore rely on credible government policies A government policy is credible if the government has incentives to adhere to it in all periods and under all circumstances Nominal debt as statecontingent real debt 717 1616 Nominal debt as statecontingent real debt We now turn to a monetary economy of Lucas and Stokey 1983 that with special preferences Chari Christiano and Kehoe 1996 used to study the opti mality of the Friedman rule and whether equilibrium price level adjustments can transform nominal nonstatecontingent debt into statecontingent real debt In particular Chari Christiano and Kehoe restricted preferences to satisfy As sumption 1 below in order to show optimality of the Friedman rule They stated no additional conditions to reach their conclusion that under an appro priate policy nonstatecontingent nominal government debt can be transformed into statecontingent real debt However we find that their statements about the equivalence of allocations under these two debt structures require stronger assumptions because of a potential signswitching problem with optimal debt across state realizations at a point in time To obtain Chari Christiano and Kehoes conclusion we add our Assumption 2 below Our strategy is to follow Chari Christiano and Kehoe First in subsection 16162 we find a Ramsey plan and an associated Ramsey allocation for a non monetary economy with statecontingent government debt Then in subsection 16163 we state conditions on fundamentals that allow that same allocation to be supported by a Ramsey plan for a monetary economy having only nominal nonstatecontingent debt A key outcome here is that the Ramsey equilibrium in the monetary econ omy makes the price level fluctuate in response to government expenditure shocks The price level adjusts to deliver historycontingent returns to holders of government debt required to support the Ramsey allocation This structure includes many if not most components of a fiscal theory of the price level20 That it does so in a coherent way can help us describe a set of contentious issues associated with some expositions of fiscal theories of the price level We take up this theme in section 1617 and again in chapter 27 20 We admit that some writers could legitimately beg to differ here because we have chosen to express the fiscal theory of the price level within the straightjacket of a rational expectations competitive equilibrium in an ArrowDebreu complete markets economy Some would argue that the fiscal theory of the price level can dispense with auxiliary assumptions like rational expectations and complete markets Relation to fiscal theories of the price level 727 16164 Sticky prices SchmittGrohe and Uribe 2004a and Siu 2004 analyze optimal monetary and fiscal policies in economies in which the government can issue only nominal risk free debt Unanticipated inflation makes riskfree nominal debt state contingent in real terms and provides a motive for the government to make inflation vary SchmittGrohe and Uribe and Siu both focus on how price stickiness would affect the governments use of fluctuations in inflation as an indirect way of introducing statecontingent debt They find that even a very small amount of price stickiness causes the volatility of the optimal inflation rate to become very small Thus the government abstains from using the indirect inflation channel for synthesizing statecontingent debt The authors relate their finding to the aspect of Aiyagaris et als 2002 calculations for an economy with no state contingent debt mentioned in footnote 11 of this chapter that the Ramsey allocation in their economy without statecontingent debt closely approximates that for the economy with complete markets 1617 Relation to fiscal theories of the price level In chapter 27 we take up monetaryfiscal theories of inflation including one that has been christened a fiscal theory of the price level The model of the previous section includes components that combine to give rise to all of the forces active in that theory As emphasized by Niepelt 2004 accounts of that theory are too often at best incomplete because they leave implicit aspects of an underlying general equilibrium model For that reason we find it enlightening to interpret statements about the fiscal theory of the price level within the context of a coherent general equilibrium model like that of Chari Christiano and Kehoe 1996 Relation to fiscal theories of the price level 729 16172 Disappearance of quantity theory Actually a version of the quantity theory coexists along with equation 16171 in the Chari Christiano and Kehoe model To see this note that the govern ment can set the initial price level P0s0 to any positive number by executing an appropriate time 0 open market operation24 Thus we can regard the time 0 government budget constraint 16169 as a constraint on a time 0 open mar ket operation The government can set the nominal money supply M1s0 0 to an arbitrary positive number subject to the constraint B1s0 M1s0 that holds under the Friedman rule The government issues money to purchase nominally denominated bonds subject to B1s0 M1s0 as given by gov ernment budget constraint 16169 under the Friedman rule The household is willing to issue these bonds in exchange for money Presuming that the cash inadvance constraint is satisfied with equality the price level and money supply then conform to P0s0c10s0 M1s0 which is a version of the quantity the ory equation for the price level at time 025 This equation provides the basis for a sharp statement of the quantity theory with fiscal policy being held constant a time 0 open market operation that increases M1s0 leads to a proportionate increase in the price level at all histories and dates while leaving the equilibrium allocation and real rates of return unaltered 24 The argument of this subsection treats time 0 in a peculiar way because no endogenous variables inherited from the past impede independently manipulating time 0 and all subse quent nominal quantities This special treatment of time 0 also characterizes many other presentations of the quantity theory of money as a pure units change experiment that multi plies nominal quantities at all dates and all histories by the same positive scalar Commenting on a paper by Robert Townsend at the Minneapolis Federal Reserve Bank in 1985 Ramon Marimon asked when is time 0 thereby anticipating doubts expressed by Niepelt 2004 25 Under the Friedman rule with a zero nominal interest rate the household would be indifferent between holding excess balances of money above and beyond cashinadvance con straint 16164 or holding of nominal government bonds Likewise the government would be indifferent about whether to issue nominal indebtedness in the form of nominal bonds or money because both liabilities carry the same cost to the government either in the form of interest payments on bonds or openmarket repurchases of money to deliver a deflation that amounts to the same real return on money as on bonds However while the composition of nominal government liabilities is indeterminate under the Friedman rule the total amount of such liabilities and hence the price level is determinate Zero tax on human capital 735 However the term in braces is zero by firstorder condition 16186 so the sum on the right side of equation 16184 simplifies to the very first term in this expression Following our standard scheme of constructing the Ramsey plan a few more manipulations of the households firstorder conditions are needed to solve for prices and taxes in terms of the allocation We first assume that τ c 0 τ k 0 τ n 0 τ m 0 0 If the numeraire is q0 0 1 then condition 16185a implies q0 t βt uc t uc 0 1 1 τ c t 16188a From equations 16185b and 16188a and wt Fet we obtain 1 τ c t uℓ t uc t 1 τ n t Fe t Mn t 16188b and by equations 16185c 16185e and 16188a 1 τ c t uℓ t uc t Hn t Hx t 16188c and equation 16185d with wt Fet yields 1 τ m t 1 τ n t Fe t Mx t 16188d For a given allocation expressions 16188 allow us to recover prices and taxes in a recursive fashion 16188c defines τ c t and 16188a can be used to compute q0 t 16188b sets τ n t and 16188d pins down τ m t Only one task remains to complete our strategy of determining prices and taxes that achieve any allocation The additional condition 16186 charac terizes the households intertemporal choice of human capital which imposes still another constraint on the price q0 t and the tax τ n t Our determination of τ n t in equation 16188b can be thought of as manipulating the margin that the household faces in its static choice of supplying effective labor et but the tax rate also affects the households dynamic choice of human capital ht Thus in the Ramsey problem we will have to impose the extra constraint that the allocation is consistent with the same τ n t entering both equations 16188b and 16186 To find an expression for this extra constraint solve for 1 τ n t from equation 16188b and a lagged version of equation 16186 which are 738 Optimal Taxation with Commitment It follows immediately from equations 161816 and 161817 that τ n 0 Given τ n 0 conditions 16188d and 161813d imply τ m 0 We conclude that in the present model neither labor nor capital should be taxed in the limit 1619 Should all taxes be zero The optimal steadystate tax policy of the model in the previous section is to set τ k τ n τ m 0 However in general this implies τ c 0 To see this point use equation 16188b and τ n 0 to get 1 τ c uc uℓ FeMn 16191 From equations 161813a and 161813b FeMn Vnm Vc uℓ Φucℓc uc Φ uc uccc 16192 Hence 1 τ c ucuℓ Φucucℓc ucuℓ Φ ucuℓ uccuℓc 16193 As discussed earlier a firstbest solution without distortionary taxation has Φ 0 so τ c should trivially be set equal to zero In a secondbest solution Φ 0 and we get τ c 0 if and only if ucucℓc ucuℓ uccuℓc 16194 which is in general not satisfied However Jones Manuelli and Rossi 1997 point out one interesting class of utility functions that is consistent with equation 16194 u c ℓ c1σ 1 σ v ℓ if σ 0 σ 1 ln c v ℓ if σ 1 If a steady state exists the optimal solution for these preferences is eventually to set all taxes equal to zero It follows that the optimal plan involves collecting tax revenues in excess of expenditures in the initial periods When the government has amassed claims against the private sector so large that the interest earnings Concluding remarks 739 suffice to finance g all taxes are set equal to zero Since the steadystate interest rate is R β1 we can use the governments budget constraint 1625 to find the corresponding value of government indebtedness b β β 1g 0 1620 Concluding remarks Perhaps the most startling finding of this chapter is that the optimal steady state tax on physical capital in a nonstochastic economy is equal to zero The conclusion follows immediately from timeadditively separable utility a stan dard constantreturnstoscale production technology competitive markets and a complete set of flatrate taxes It is instructive to consider Jones Manuelli and Rossis 1997 extension of the notax result to labor income or more pre cisely human capital They ask rhetorically Is physical capital special We are inclined to answer yes to this question for the following reason The zero tax on human capital is derived in a model where the production of both hu man capital and efficiency units of labor show constant returns to scale in the stock of human capital and the use of final goods but not raw labor which otherwise enters as an input in the production functions These assumptions explain why the stream of future labor income in the households presentvalue budget constraint in equation 16184 is reduced to the first term in equation 16187 which is the value of the households human capital at time 0 Thus the functional forms have made raw labor disappear as an object for taxation in future periods Or in the words of Jones Manuelli and Rossi 1997 pp 103 and 99 Our zero tax results are driven by zero profit conditions Zero profits follow from the assumption of linearity in the accumulation technolo gies Since the activity capital income and the activity labor income display constant returns to scale in reproducible factors their profits cannot enter the budget constraint in equilibrium But for alternative production functions that make the endowment of raw labor reappear the optimal labor tax would not be zero It is for this reason that we think physical capital is special because the zerotax result arises with the minimal assumptions of the standard neoclassical 740 Optimal Taxation with Commitment growth model while the zerotax result on labor income requires that raw labor vanishes from the agents presentvalue budget constraint27 Our optimal steadystate tax analysis is silent about how long it takes to reach the zero tax on capital income and how taxes and redistributive transfers are set during the transition to a steady state These issues have been studied numerically by Chari Christiano and Kehoe 1994 though their paper in volves no redistributional concerns because they assume a representative agent Domeij and Heathcote 2000 construct a model with heterogeneous agents and incomplete insurance markets to study the welfare implications of eliminating capital income taxation Using earnings and wealth data from the United States they calibrate a stochastic process for labor earnings that implies a wealth dis tribution of asset holdings resembling the empirical one Setting initial tax rates equal to estimates of present taxes in the United States they study the effects of an unexpected policy reform that sets the capital tax permanently equal to zero and raises the labor tax to maintain longrun budget balance They find that a majority of households prefers the status quo to the tax reform because of the distributional implications This example illustrates the importance of a welldesigned tax and transfer policy in the transition to a new steady state In addition as shown by Aiyagari 1995 the optimal capital tax in a heterogeneousagent model with incomplete insurance markets is actually positive even in the long run A positive capital tax is used to counter the tendency of such an economy to overaccumulate capital because of too much precautionary saving We say more about these heterogeneousagent models in chapter 18 Golosov Kocherlakota and Tsyvinski 2003 pursue another way of dis rupting the connection between stationary values of the two key Euler equations that underlie Chamley and Judds zerotaxoncapital outcome They put the Ramsey planner in a private information environment in which it cannot observe the hidden skill levels of different households That impels the planner to design the tax system as an optimal dynamic incentive mechanism that trades off cur rent and continuation values in an optimal way We discuss such mechanisms for 27 One special case of Jones Manuelli and Rossis 1997 framework with its zerotax result for labor is Lucass 1988 endogenous growth model studied in chapter 15 Recall our alternative interpretation of that model as one without any nonreproducible raw labor but just two reproducible factors physical and human capital No wonder that raw labor in Lucass model does not affect the optimal labor tax since the model can equally well be thought of as an economy without raw labor Exercises 741 coping with private information in chapter 21 Because the information problem alters the planners Euler equation for the households consumption Chamley and Judds result does not hold for this environment Throughout this chapter we have assumed that a government can commit to future tax rates at time 0 As noted earlier taxing the capital stock at time 0 amounts to lumpsum taxation and therefore dispenses with distortionary taxation It follows that a government without a commitment technology would be tempted in future periods to renege on its promises and levy a confiscatory tax on capital An interesting question arises can the incentive to maintain a good reputation replace a commitment technology That is can a promised policy be sustained in an equilibrium because the government wants to preserve its reputation Reputation involves history dependence and incentives and will be studied in chapter 27 Exercises Exercise 161 A small open economy Razin and Sadka 1995 Consider the nonstochastic model with capital and labor in this chapter but assume that the economy is a small open economy that cannot affect the in ternational rental rate on capital r t Domestic firms can rent any amount of capital at this price and the households and the government can choose to go short or long in the international capital market at this rental price There is no labor mobility across countries We retain the assumption that the government levies a tax τ n t on each households labor income but households no longer have to pay taxes on their capital income Instead the government levies a tax ˆτ k t on domestic firms rental payments to capital regardless of the capitals origin domestic or foreign Thus a domestic firm faces a total cost of 1 ˆτ k t r t on a unit of capital rented in period t a Solve for the optimal capital tax ˆτ k t b Compare the optimal tax policy of this small open economy to that of the closed economy of this chapter Exercises 743 Exercise 164 Two labor inputs Jones Manuelli and Rossi 1997 Consider the nonstochastic model with capital and labor in this chapter but assume that there are two labor inputs n1t and n2t entering the production function Fkt n1t n2t The households period utility function is still given by uct ℓt where leisure is now equal to ℓt 1 n1t n2t Let τ n it be the flatrate tax at time t on wage earnings from labor nit for i 1 2 and τ k t denotes the tax on earnings from capital a Solve for the Ramsey plan What is the relationship between the optimal tax rates τ n 1t and τ n 2t for t 1 Explain why your answer is different for period t 0 As an example assume that k and n1 are complements while k and n2 are substitutes We now assume that the period utility function is given by uct ℓ1t ℓ2t where ℓ1t 1 n1t and ℓ2t 1 n2t Further the government is now constrained to set the same tax rate on both types of labor ie τ n 1t τ n 2t for all t 0 b Solve for the Ramsey plan Hint Using the households firstorder condi tions we see that the restriction τ n 1t τ n 2t can be incorporated into the Ramsey problem by adding the constraint uℓ1tFn2t uℓ2tFn1t c Suppose that the solution to the Ramsey problem converges to a steady state where the constraint that the two labor taxes should be equal is binding Show that the limiting capital tax is not zero unless Fn1Fn2k Fn2Fn1k Exercise 165 Another specific utility function Consider the following optimal taxation problem There is no uncertainty There is one good that is produced by labor xt of the representative household and that can be divided among private consumption ct and government consumption gt subject to ct gt 1 xt 0 The good is produced by zeroprofit competitive firms that pay the worker a pretax wage of 1 per unit of 1 xt ie the wage is tied down by the linear Exercises 747 rt is the rental rate on capital We assume that the tax rates in period 0 cannot be chosen by the government but must be set equal to zero τ n 0 τ a 0 0 The government can trade oneperiod bonds We assume that there is no outstanding government debt at time 0 a Formulate the Ramsey problem and characterize the optimal government policy using the primal approach to taxation b Show that if there exists a steady state Ramsey allocation the limiting tax rate τ a is zero Consider another economy with identical preferences endowment technology and government expenditures but where labor taxation is forbidden Instead of a labor tax this economy must use a consumption tax τ c t We use a tilde to distinguish outcomes in this economy as compared to the previous economy Hence this economys tax revenues in period t are equal to τ c t ctτ a t rt1δkt We assume that the tax rates in period 0 cannot be chosen by the government but must be set equal to zero τ c 0 τ a 0 0 And as before the government can trade in oneperiod bonds and there is no outstanding government debt at time 0 c Formulate the Ramsey problem and characterize the optimal government policy using the primal approach to taxation Let the allocation and tax rates that solve the Ramsey problem in question a be given by Ω ct ℓt nt kt1 τ n t τ a t t0 And let the allocation and tax rates that solve the Ramsey problem in question c be given by Ω ct ℓt nt kt1 τ c t τ a t t0 d Make a careful argument for how the allocation ct ℓt nt kt1 t0 compares to the allocation ct ℓt nt kt1 t0 e Find expressions for the tax rates τ c t τ a t t1 solely in terms of τ n t τ a t t1 f Write down the governments present value budget constraint in the first economy which holds with equality for the allocation and tax rates as given by Ω Can you manipulate this expression so that you arrive at the govern ments present value budget constraint in the second economy by only using your characterization of Ω in terms of Ω in questions d and e 752 Optimal Taxation with Commitment Describe outcomes Arrow securities prices and the allocation in a competitive equilibrium with sequential trading of Arrow securities f Now suppose that b0 0 Consider a government policy that always runs a balanced budget budget ie that sets τ c t stctst at a level that guarantees that total government debt owed at time t 1 equals b0 for all t st Here the government always runs what the IMF calls a balanced budget netofinterest In doing this it always rolls over its oneperiod debt bt b0 Please find a formula for the rate of return paid on government debt and describe precisely the quantities of historycontingent bonds issued by the government at each t st Please describe how the tax rate τ c t st depends on b0 g Continuing to assume that b0 0 now consider another government fiscal policy Here the government sets a constant tax rate τ c τ c t st for all t st The tax rate is set to satisfy the time 0 ArrowDebreu budget constraint 2 Please tell how to compute τ c Please describe the Arrow securities that the government issues or purchases at each state Please carefully take into account how you have defined states in part a h Using the labeling of states described in part a please describe the payouts on the government securities for the following two histories for this economy 1 2 3 5 5 5 5 and 1 2 4 5 5 5 5 i Please formulate and solve a Ramsey problem for this economy assuming that τ c t st is the only tax that the government can impose Please state the Ramsey problem carefully and describe in detail an algorithm for computing all of the objects that comprise a Ramsey plan j Please compare the Ramsey plan that you computed in part i with the arbitrary policies studied in parts f and g k Define a continuation of a Ramsey plan For this economy is a continuation of a Ramsey plan a Ramsey plan Please explain Exercise 1612 Yet another LucasStokey economy Consider the following economy without capital There is an exogenous state st governed by an S state Markov chain with initial distribution over states π0 754 Optimal Taxation with Commitment a Finding the state is an art Please define a state space and correspond ing Markov chain for this economy Please completely specify the state space S the initial distribution π0 and the transition matrix P Hint Try defining the state as a t g pair Try getting by with these 5 states 0 gL 1 gL 2 gL 2 gH t 3 gL say and call them states 1 2 3 4 5 respectively Then take the data supplied and create π0 and P b For the Markov chain that you created in part a please compute un conditional probabilities over histories at dates t 1 ie please compute Πtst t 1 c Please define an ArrowDebreu style competitive equilbrium with distorting taxes for this environment with all trades at time 0 d Please define an Arrowsecurities style competitive equilibrium with distort ing taxes with trades each period of oneperiod ahead Arrow securities e Temporarily suppose that b0 0 Consider a government policy that always runs a balanced budget budget ie that sets gtst τ n t stwtstntst for all t st Describe outcomes Arrow securities prices and the allocation in a competitive equilibrium with sequential trading of Arrow securities f Now suppose that b0 0 Consider a government policy that always runs a balanced budget budget ie that sets τ n t stwtstntst at a level that guar antees that total government debt owed at time t 1 equals b0 for all t st Here the government always runs what the IMF calls a balanced budget netof interest In doing this it always rolls over its oneperiod debt bt b0 Please find a formula for the rate of return paid on government debt and describe pre cisely the quantities of historycontingent bonds issued by the government at each t st Please describe how the tax rate τ n t st depends on b0 g Continuing to assume that b0 0 now consider another government fiscal policy Here the government sets a constant tax rate τ n τ n t st for all t st The tax rate is set to satisfy the time 0 ArrowDebreu budget constraint 2 Please tell how to compute τ n Please describe the Arrow securities that the government issues or purchases at each state Please carefully take into account how you have defined states in part a h Please describe the payouts on the government securities for all possible histories for this economy Exercises 755 i Please formulate and solve a Ramsey problem for this economy assuming that τ n t st is the only tax that the government can impose Please state the Ramsey problem carefully and describe in detail an algorithm for computing all of the objects that comprise a Ramsey plan j Please compare the Ramsey plan that you computed in part i with the arbitrary policies studied in parts f and g k Define a continuation of a Ramsey plan For this economy is a continuation of a Ramsey plan a Ramsey plan Please explain Exercise 1613 Positive initial debt Please describe outcomes in modified versions of examples 1 2 and 3 of section 1613 in which b0 0 rather than b0 0 Part IV Savings Problems and Bewley Models Chapter 17 SelfInsurance 171 Introduction This chapter describes a version of what is sometimes called a savings problem eg Chamberlain and Wilson 2000 A consumer wants to maximize the expected discounted sum of a concave function of oneperiod consumption rates as in chapter 8 However the consumer is cut off from all insurance markets and almost all asset markets The consumer can purchase only nonnegative amounts of a single riskfree asset The absence of insurance opportunities induces the consumer to use variations over time in his asset holdings to acquire selfinsurance This model is interesting to us partly as a benchmark to compare with the complete markets model of chapter 8 and some of the recursive contracts models of chapters 21 and 22 where information and enforcement problems restrict allocations relative to chapter 8 but nevertheless permit more insurance than is allowed in this chapter A version of the singleagent model of this chapter will also be an important component of the incomplete markets models of chapter 18 Finally the chapter provides our first encounter with the powerful supermartingale convergence theorem To highlight the effects of uncertainty and borrowing constraints we shall study versions of the savings problem under alternative assumptions about the stringency of the borrowing constraint and about whether the households en dowment stream is known or uncertain 759 Nonstochastic endowment 761 Selfinsurance occurs when the agent uses savings to insure himself against income fluctuations On the one hand in response to low income realizations an agent can draw down his savings and avoid temporary large drops in con sumption On the other hand the agent can partly save high income realizations in anticipation of poor outcomes in the future We are interested in the long run properties of an optimal selfinsurance scheme Will the agents future consumption settle down around some level c1 Or will the agent eventually be come impoverished2 Following the analysis of Chamberlain and Wilson 2000 and Sotomayor 1984 we will show that neither of these outcomes occurs consumption diverges to infinity We begin by studying the savings problem under the assumption that the endowment is a nonrandom sequence that does not grow perpetually In this case consumption does converge 173 Nonstochastic endowment Without uncertainty the question of insurance is moot However it is instruc tive to study the optimal consumption decisions of an agent with an uneven income stream who faces a borrowing constraint We break our analysis of the nonstochastic case into two parts depending on the stringency of the borrow ing constraint We begin with the least stringent possible borrowing constraint namely the natural borrowing constraint on oneperiod Arrow securities which are risk free in the current context After that well arbitrarily tighten the bor rowing constraint to arrive at the noborrowing condition at1 yt1 imposed in the statement of the problem in the previous section With the natural bor rowing constraint the outcome is that the agent completely smooths consump tion having a constant consumption rate over time With the more stringent noborrowing constraint in general the outcome will be different Here con sumption will be a monotonic increasing sequence with jumps in consumption at times when the noborrowing constraint binds 1 As will occur in the model of social insurance without commitment to be analyzed in chapter 21 2 As in the case of social insurance with asymmetric information to be analyzed in chapter 21 772 SelfInsurance annuity return on the endowment process is sufficiently stochastic Instead the optimal consumption path will converge to infinity This stark difference between the case of certainty and uncertainty is quite remarkable10 177 Intuition Imagine that you perturb any constant endowment stream by adding the slight est iid component Our two propositions then say that the optimal consump tion path changes from being a constant to becoming a stochastic process that goes to infinity Beyond appealing to martingale convergence theorems Cham berlain and Wilson 2000 p 381 comment on the difficulty of developing eco nomic intuition for this startling finding Unfortunately the line of argument used in the proof does not provide a very convincing economic explanation Clearly the strict concavity of the utility function must play a role The result does not hold if for instance u is a linear function over a sufficiently large domain and xt is bounded But to simply attribute the result to risk aversion on the grounds that uncertain future returns will cause riskaverse con sumers to save more given any initial asset level is not a completely satisfactory explanation either In fact it is a bit misleading First that argument only explains why expected accumulated assets would tend to be larger in the limit It does not really explain why consump tion should grow without bound Second over any finite time horizon the argument is not even necessarily correct Given a finite horizon Chamberlain and Wilson proceed to discuss how mean preserving spreads of future income leave current consumption unaffected when the agents utility function is quadratic over a sufficiently large domain We believe that the economic intuition is to be found in the strict concavity of the utility function and the assumption that the marginal utility of consump tion must remain positive for any arbitrarily high consumption level This rules out quadratic utility for example To advance this explanation we first focus 10 In exercise 173 you will be asked to prove that the divergence of consumption to also occurs under a stochastic counterpart to the natural borrowing limits These are less stringent than the noborrowing condition Concluding remarks 777 random labor productivity process and extinguishes subsequent randomness in his income process Whether case 1 or 2 prevails depends on the shape of the consumers utility function uc h Zhu 2009 considers the following two assumptions that tilt things toward case 2 Assumption A4 asserts that u12u1 u11u2 0 and u12u2 u22u1 0 This assumption makes c and h both be normal goods and also implies that u2 u1 is increasing in c and decreasing in h It also implies that cY e and hY e are both increasing in Y Zhu also makes the stronger assumption A4 that u12 0 which makes c and h be complements and implies A4 Zhu 2009 establishes the following Proposition Under assumptions A1A4 a cA Y and hA Y are both continuous and increasing in A b hA e 1 e when A is sufficiently large Zhu shows how this proposition is the heart of an argument that generates suf ficient conditions for case 2 to prevail In this way he constructs circumstances that disarm the divergence outcomes of Chamberlain and Wilson11 179 Concluding remarks This chapter has maintained the assumption that β1 r 1 which is a very important ingredient in delivering the divergence toward infinity of the agents asset and consumption level Chamberlain and Wilson 1984 study a much more general version of the model where they relax this condition To build some incomplete markets models chapter 18 will put together continua of agents facing generalizations of the savings problems The models of that chapter will determine the interest rate 1 r as an equilibrium object In these models to define a stationary equilibrium we want the sequence of distributions of each agents asset holdings to converge to a welldefined invariant distribution with finite first and second moments For there to exist a stationary 11 Zhu also provides examples of preferences that push things toward case 1 An example is preferences of a type used by Greenwood Hercowitz and Huffman 1988 uc h Uc G1 h where U 0 U 0 G 0 G 0 with U bounded above Here the marginal rate of substitution between c and h depends only on h and labor supplied is independent of the intertemporal consumptionsavings choice 778 SelfInsurance equilibrium without aggregate uncertainty the findings of the present chapter would lead us to anticipate that the equilibrium interest rate in those models must fall short of β1 In a production economy with physical capital that result implies that the marginal product of capital will be less than the one that would prevail in a complete markets world when the stationary interest rate would be given by β1 In other words an incomplete markets economy is characterized by an overaccumulation of capital that drives the interest rate below β1 which serves to choke off the desire to accumulate an infinite amount of assets that agents would have had if the interest rate had been equal to β1 Chapters 21 and 22 will consider several models in which the condition β1 r 1 is maintained There the assumption will be that a social planner has access to riskfree loans outside the economy and seeks to maximize agents welfare subject to enforcement andor information problems The environment is once again assumed to be stationary without aggregate uncertainty so in the absence of enforcement and information problems the social planner would just redistribute the economys resources in each period without any intertemporal trade with the outside world But when agents are free to leave the economy with their endowment streams and forever live in autarky optimality prescribes that the planner amass sufficient outside claims so that each agent is granted a constant consumption stream in the limit at a level that weakly dominates autarky for all realizations of an agents endowment In the case of asymmet ric information where the planner can induce agents to tell the truth only by manipulating promises of future utilities we obtain a conclusion that is diamet rically opposite to the selfinsurance outcome of the present chapter Instead of consumption approaching infinity in the limit the optimal solution has all agents consumption approaching its lower bound 782 SelfInsurance where σ 0 and ǫt1 is an iid process Gaussian process with mean 0 and variance 1 The household of type 2 has endowment 3 y2t1 y2t σǫt1 where ǫt1 is the same random process as in 2 At time t yit is realized before consumption at t is chosen Assume that at time 0 y10 y20 and that y10 is substantially less than the bliss point u1u2 To make the computation easier please assume that there is no disposal of resources Part I In this part please assume that there are complete markets in history and datecontingent claims a Define a competitive equilibrium being careful to specify all of the objects of which a competitive equilibrium is composed b Define a Pareto problem for a fictitious planner who attaches equal weight to the two households Find the consumption allocation that solves the Pareto or planning problem c Compute a competitive equilibrium Part II Now assume that markets are incomplete There is only one traded asset a oneperiod riskfree bond that both households can either purchase or issue The gross rate of return on the asset between date t and date t 1 is Rt Household is budget constraint at time t is 4 cit R1 t bit1 yit bit where bit is the value in terms of time t consumption goods of households i holdings of oneperiod riskfree bonds We require that a consumerss holdings of bonds are subject to the restriction 5 lim t βtucitEbit1 0 Assume that b10 b20 0 An incomplete markets competitive equilibrium is a gross interest rate sequence Rt sequences of bond holdings bit for i 1 2 and feasible allocations cit i 1 2 such that given Rt household i 1 2 is maximizing 1 subject to the sequence of budget constraints 4 and the given initial levels of b10 b20 Chapter 18 Incomplete Markets Models 181 Introduction In the complete markets model of chapter 8 the optimal consumption alloca tion is not history dependent the allocation depends on the current value of the Markov state variable only This outcome reflects the comprehensive opportu nities to insure risks that markets provide This chapter and chapters 21 and 22 describe settings with more impediments to exchanging risks These reduced opportunities make allocations history dependent In this chapter the history dependence is encoded in the dependence of a households consumption on the households current asset holdings In chapters 21 and 22 history dependence is encoded in the dependence of the consumption allocation on a continuation value promised by a planner or principal The present chapter describes a particular type of incomplete markets model The models have a large number of ex ante identical but ex post het erogeneous agents who trade a single security For most of this chapter we study models with no aggregate uncertainty and no variation of an aggregate state variable over time so macroeconomic time series variation is absent But there is much uncertainty at the individual level Households only option is to selfinsure by managing a stock of a single asset to buffer their consumption against adverse shocks We study several models that differ mainly with respect to the particular asset that is the vehicle for selfinsurance for example fiat currency or capital The tools for constructing these models are discretestate discounted dy namic programming used to formulate and solve problems of the individuals and Markov chains used to compute a stationary wealth distribution The models produce a stationary wealth distribution that is determined simultane ously with various aggregates that are defined as means across corresponding individuallevel variables 785 786 Incomplete Markets Models We begin by recalling our discretestate formulation of a singleagent infinite horizon savings problem We then describe several economies in which house holds face some version of this infinite horizon savings problem and where some of the prices taken parametrically in each households problem are determined by the average behavior of all households This class of models was invented by Bewley 1977 1980 1983 1986 partly to study a set of classic issues in monetary theory The second half of this chapter joins that enterprise by using the model to represent inside and outside money a free banking regime a subtle limit to the scope of Friedmans optimal quantity of money a model of international exchange rate indeterminacy and some related issues The chapter closes by describing work of Krusell and Smith 1998 that extended the domain of such models to include a timevarying stochastic aggregate state variable As we shall see this innovation makes the state of the households problem include the time t crosssection distribution of wealth an immense object Researchers have used calibrated versions of Bewley models to give quanti tative answers to questions including the welfare costs of inflation Imrohoroglu 1992 the risksharing benefits of unfunded social security systems Imrohoroglu Imrohoroglu and Joines 1995 the benefits of insuring unemployed people Hansen and Imrohoroglu 1992 and the welfare costs of taxing capital Aiya gari 1995 Also see Heathcote Storesletten and Violante 2008 and Krueger Perri Pistaferri and Violante 2010 See Kaplan and Violante 2010 for a quantitative study of how much insurance consumers seem to attain beyond the selfinsurance allowed in Bewley models Heathcote Storesletten and Vi olante 2012 combine ideas of Bewley with those of Constantinides and Duffie 1996 to build a model of partial insurance Heathcote Perri and Violante 2010 present an enlightening account of recent movements in the distributions of wages earnings and consumption across people and across time in the US A savings problem 789 element of x be the pair ai sh where j i 1m h Denote x a1 s1 a1 s2 a1 sm a2 s1 a2 sm an s1 an sm The optimal policy function a ga s and the Markov chain P on s induce a Markov chain for x via the formula Probat1 a st1 sat a st s Probat1 aat a st s Probst1 sst s Ia a sPs s where Ia a s 1 is defined as above This formula defines an N N matrix P where N n m This is the Markov chain on the households state vector x3 Suppose that the Markov chain associated with P is asymptotically sta tionary and has a unique invariant distribution π Typically all states in the Markov chain will be recurrent and the individual will occasionally revisit each state For long samples the distribution π tells the fraction of time that the household spends in each state We can unstack the state vector x and use π to deduce the stationary probability measure λai sh over ai sh pairs where λai sh Probat ai st sh πj and where πj is the j th component of the vector π and j i1mh 1822 Reinterpretation of the distribution λ The solution of the households optimum savings problem induces a stationary distribution λa s that tells the fraction of time that an infinitely lived agent spends in state a s We want to reinterpret λa s Thus let a s index the state of a particular household at a particular time period t and assume that there is a crosssection of households distributed over states a s We start the economy at time t 0 with a crosssection λa s of households that we want to repeat over time The models in this chapter arrange the initial distribution and other things so that the crosssection distribution of agents over individual state variables a s remains constant over time even though the state of the individual household is a stochastic process 3 Matlab programs to be described later in this chapter create the Markov chain for the joint a s state 798 Incomplete Markets Models 1851 A candidate for a single state variable For the special case in which s is iid Aiyagari showed how to cast the model in terms of a single state variable to appear in the households value function To synthesize a single state variable note that the disposable resources available to be allocated at t are zt wst 1 rat φ Thus zt is the sum of the current endowment current savings at the beginning of the period and the maximal borrowing capacity φ This can be rewritten as zt wst 1 rˆat rφ where ˆat at φ In terms of the single state variable zt the households budget set can be represented recursively as ct ˆat1 zt 1855a zt1 wst1 1 rˆat1 rφ 1855b where we must have ˆat1 0 The Bellman equation is vzt st max ˆat10 uzt ˆat1 βEvzt1 st1 1856 Here st appears in the state vector purely as an information variable for predict ing the employment component st1 of next periods disposable resources zt1 conditional on the choice of ˆat1 made this period Therefore it disappears from both the value function and the decision rule in the iid case More generally with a serially correlated state associated with the solution of the Bellman equation is a policy function ˆat1 Azt st 1857 Borrowing limits natural and ad hoc 799 1852 Supermartingale convergence again Lets revisit a main issue from chapter 17 but now consider the possible case β1 r 1 From equation 1855a optimal consumption satisfies ct zt Azt st The optimal policy obeys the Euler inequality uct β1 rEtuct1 if ˆat1 0 1858 We can use equation 1858 to deduce significant aspects of the limiting be havior of mean assets as a function of r Following Chamberlain and Wilson 2000 and others to deduce the effect of r on the mean of assets we analyze the limiting behavior of consumption implied by the Euler inequality 1858 Define Mt βt1 rtuct 0 Then Mt1 Mt βt1 rtβ1 ruct1 uct Equation 1858 can be written EtMt1 Mt 0 1859 which asserts that Mt is a supermartingale Because Mt is nonnegative the supermartingale convergence theorem applies It asserts that Mt converges almost surely to a nonnegative random variable M Mt as M It is interesting to consider three cases 1 β1 r 1 2 β1 r 1 and 3 β1 r 1 In case 1 the fact that Mt converges implies that uct converges to zero almost surely Because uct 0 and uct 0 this fact then implies that ct and that the consumers asset holdings diverge to Chamberlain and Wilson 2000 show that such results also characterize the borderline case 3 see chapter 17 In case 2 convergence of Mt leaves open the possibility that uc does not converge almost surely To take a simple example of nonconvergence in case 2 consider the case of a nonstochastic endowment Under the natural borrowing constraint the consumer chooses to drive uc as time passes and so asymptotically chooses to impoverish himself The marginal utility uc diverges It is easier to analyze the borderline case β1 r 1 in the special case that the employment process is independently and identically distributed meaning that the stochastic matrix P has identical rows7 In this case st provides no information about zt1 and so st can be dropped as an argument 7 See chapter 17 for a closely related proof Average assets as a function of r 801 r ρ r1 E a r K1 K FK δ r2 0 w s r 1 b K0 Figure 1861 Demand for capital and determination of interest rate The Ear curve is constructed for a fixed wage that equals the marginal product of labor at level of capital K1 In the nonstochastic version of the model with capital the equilibrium interest rate and capital stock are ρ K0 while in the stochastic version they are r K1 For a version of the model without capital in which w is fixed at this same fixed wage the equilibrium interest rate in Huggetts pure credit economy occurs at the intersection of the Ear curve with the raxis Figure 1861 assumes that the wage w is fixed in drawing the Ear curve Later we will discuss how to draw a similar curve making w adjust as the function of r that is induced by the marginal productivity conditions for positive values of K For now we just assume that w is fixed at the value equal to the marginal product of labor when K K1 the equilibrium level of capital in the model The equilibrium interest rate is determined at the intersection of the Ear curve with the marginal productivity of capital curve Notice that the equilibrium interest rate r is lower than ρ its value in the nonstochastic 802 Incomplete Markets Models version of the model and that the equilibrium value of capital K1 exceeds the equilibrium value K0 determined by the marginal productivity of capital at r ρ in the nonstochastic version of the model For a pure credit version of the model like Huggetts but the same Ear curve the equilibrium interest rate is determined by the intersection of the Ear curve with the raxis E ar φ φ φ r E a r0 E ar 0 Figure 1862 The effect of a shift in φ on the Ear curve Both Ear curves are drawn assuming that the wage is fixed For the purpose of comparing some of the models that follow it is useful to note the following aspect of the dependence of Ea0 on φ Proposition 1 When r 0 the optimal rule ˆat1 Azt st is independent of φ This implies that for φ 0 Ea0 φ Ea0 0 φ Proof It is sufficient to note that when r 0 φ disappears from the right side of equation 1855b the consumers budget constraint Therefore the optimal rule ˆat1 Azt st does not depend on φ when r 0 More explicitly when r 0 add φ to both sides of the households budget constraint to get at1 φ ct at φ wst If the households problem with φ 0 is solved by the decision rule at1 gat zt then the households problem with φ 0 is solved with the same decision rule evaluated at at1 φ gat φ zt Average assets as a function of r 803 Thus it follows that at r 0 an increase in φ displaces the Ear curve to the left by the same amount See Figure 1862 We shall use this result to analyze several models In the following sections we use a version of Figure 1861 to compute equilibria of various models For models without capital the figure is drawn assuming that the wage is fixed Typically the Ear curve will have the same shape as Figure 1861 In Huggetts model the equilibrium interest rate is determined by the intersection of the Ear curve with the raxis reflecting that the asset pure consumption loans is available in zero net supply In some models with money the availability of fiat currency as a perfect substitute for consumption loans creates a positive net supply 4 2 0 2 4 6 8 10 12 001 0 001 002 003 004 005 interest rate w 1 b 3 b 6 Figure 1863 Two Ear curves one with b 6 the other with b 3 with w fixed at w 1 Notice that at r 0 the difference between the two curves is 3 the difference in the bs Computed examples 805 0 1 2 3 4 5 6 7 8 9 10 001 0 001 002 003 004 005 interest rate Figure 1871 Two Ear curves when b 0 and the endowment shock s is iid but with different variances the curve with circles belongs to the economy with the higher variance 3 2 1 0 1 2 3 4 001 002 003 004 005 006 007 008 009 01 b3 r0 Figure 1872 The invariant distribution of capital when b 3 806 Incomplete Markets Models 188 Several Bewley models We consider several models in which a continuum of households faces the same savings problem Their behavior generates the asset demand function Ear φ The models share the same family of Ear φ curves as functions of φ but differ in their settings of φ and in their interpretations of the supply of the asset The models are 1 Aiyagaris 1994 1995 model in which the riskfree asset is either physical capital or private IOUs with physical capital being the net supply of the asset 2 Huggetts model 1993 where the asset is private IOUs available in zero net supply 3 Bewleys model of fiat currency 4 modifications of Bewleys model to permit an inflation tax and 5 modifications of Bewleys model to pay interest on currency either explicitly or implicitly through deflation 1881 Optimal stationary allocation Because there is no aggregate risk and the aggregate endowment is constant a stationary optimal allocation would have consumption constant over time for each household Each households consumption plan would have constant consumption over time The implicit riskfree interest rate associated with such an allocation would be r ρ where recall that β 1 ρ1 In the version of the model with capital the stationary aggregate capital stock solves FKK N δ ρ 1881 Equation 1881 restricts the stationary optimal capital stock in the non stochastic optimal growth model of Cass 1965 and Koopmans 1965 The stationary level of capital is K0 in Figure 1861 depicted as the ordinate of the intersection of the marginal productivity net of depreciation curve with a horizontal line r ρ As we saw before the horizontal line at r ρ acts as a longrun demand curve for savings for a nonstochastic version of the sav ings problem The stationary optimal allocation matches the one produced by a nonstochastic growth model We shall use the riskfree interest rate r ρ as a benchmark against which to compare some alternative incomplete market allocations Aiyagaris 1994 model replaces the horizontal line r ρ with an upwardsloping curve Ear causing the stationary equilibrium interest rate to A model with capital and private IOUs 807 fall and the capital stock to rise relative to the savings model with a riskfree endowment sequence 189 A model with capital and private IOUs Figure 1861 can be used to depict the equilibrium of Aiyagaris model described above The single asset is capital There is an aggregate production function Y FK N and w FNK N r δ FKK N We can invert the marginal condition for capital to deduce a downwardsloping curve K Kr This is drawn as the curve labeled FK δ in Figure 1861 We can use the marginal productivity conditions to deduce a factor price frontier w ψr For fixed r we use w ψr as the wage in the savings problem and then deduce Ear We want the equilibrium r to satisfy Ear Kr 1891 The equilibrium interest rate occurs at the intersection of Ear with the FK δ curve See Figure 186110 It follows from the shape of the curves that the equilibrium capital stock K1 exceeds K0 the capital stock required at the given level of total labor to make the interest rate equal ρ There is capital overaccumulation in the stochastic version of the model 10 Recall that Figure 1861 was drawn for a fixed wage w fixed at the value equal to the marginal product of labor when K K1 Thus the new version of Figure 1861 that incorporates w ψr has a new curve Ear that intersects the FK δ curve at the same point r1 K1 as the old curve Ear with the fixed wage Further the new Ear curve would not be defined for negative values of K 808 Incomplete Markets Models 1810 Private IOUs only It is easy to compute the equilibrium of Mark Huggetts 1993 model with Figure 1861 Recall that in Huggetts model the one asset consists of riskfree loans issued by other households There are no outside assets This fits the basic model with at being the quantity of loans owed to the individual at the beginning of t The equilibrium condition is Ear φ 0 18101 which is depicted as the intersection of the Ear curve in Figure 1861 with the raxis There is a family of such curves one for each value of the ad hoc debt limit Relaxing the ad hoc debt limit by driving b sends the equilibrium interest rate upward toward the intersection of the furthest to the left Ear curve the one that is associated with the natural debt limit with the raxis 18101 Limitation of what credit can achieve The equilibrium condition 18101 and limrրρ Ear imply that the equilibrium value of r is less than ρ for all values of the debt limit respecting the natural debt limit This outcome supports the following conclusion Proposition 2 Suboptimality of equilibrium with credit The equilibrium interest rate associated with the natural debt limit is the highest one that Huggetts model can support This interest rate falls short of ρ the interest rate that would prevail in a complete markets world11 11 Huggett used the model to study how tightening the ad hoc debt limit parameter b would reduce the riskfree rate far enough below ρ to explain the riskfree rate puzzle Private IOUs only 809 18102 Proximity of r to ρ Notice how in Figure 1863 the equilibrium interest rate r gets closer to ρ as the borrowing constraint is relaxed How close it can get under the natural borrowing limit depends on several key parameters of the model 1 the discount factor β 2 the curvature of u 3 the persistence of the endowment process and 4 the volatility of the innovations to the endowment process When he selected a plausible β and u and then calibrated the persistence and volatility of the endowment process to US panel data on workers earnings Huggett 1993 found that under the natural borrowing limit r is quite close to ρ and that the household can achieve substantial selfinsurance12 We shall encounter an echo of this finding when we review Krusell and Smiths 1998 finding that under their calibration of idiosyncratic risk a real business cycle model with complete markets does a good job of approximating the prices and the aggregate allocation of a model with identical preferences and technology but in which only a single asset physical capital can be traded 18103 Inside money or free banking interpretation Huggetts can be viewed as a model of pure inside money or of circulating private IOUs Every person is a banker in this setting being entitled to issue notes or evidences of indebtedness subject to the debt limit 1853 A household has issued more IOU notes of its own than it holds of those issued by others whenever at1 0 There are several ways to think about the clearing of notes imposed by equation 18101 Here is one In period t trading occurs in subperiods as follows First households realize their st Second some households choose to set at1 at 0 by issuing new IOUs in the amount at1 at Other households with at 0 may decide to set at1 0 meaning that they want to redeem their outstanding notes and possibly acquire notes issued by others Third households go to the market and exchange goods for notes Fourth notes are cleared or netted out in a centralized clearinghouse positive holdings of 12 This result depends sensitively on how one specifies the left tail of the endowment distri bution Notice that if the minimum endowment s1 is set to zero then the natural borrowing limit is zero However Huggetts calibration permits positive borrowing under the natural borrowing limit 810 Incomplete Markets Models notes issued by others are used to retire possibly negative initial holdings of ones own notes If a person holds positive amounts of notes issued by others some of these are used to retire any of his own notes outstanding This clearing operation leaves each person with a particular at1 to carry into the next period with no owner of IOUs also being in the position of having some notes outstanding There are other ways to interpret the trading arrangement in terms of circulating notes that implement multilateral longterm lending among corre sponding banks notes issued by individual A and owned by B are honored or redeemed by individual C by being exchanged for goods13 In a different setting Kocherlakota 1996b and Kocherlakota and Wallace 1998 describe such trading mechanisms Under the natural borrowing limit we might think of this pure consump tion loans or inside money model as a model of free banking In the model households ability to issue IOUs is restrained only by the requirement that all loans be riskfree and of one period in duration Later well use the equilibrium allocation of this free banking model as a benchmark against which to judge the celebrated Friedman rule in a model with outside money and a severe borrowing limit We now tighten the borrowing limit enough to make room for some outside money 18104 Bewleys basic model of fiat money This version of the model is set up to generate a demand for fiat money an inconvertible currency supplied in a fixed nominal amount by an entity outside the model called the government Individuals can hold currency but not issue it To map the individuals problem into problem 1831 we let mt1p at1 b φ 0 where mt1 is the individuals holding of currency from t to t 1 and p is a constant price level With a constant price level r 0 With b φ 0 ˆat at Currency is the only asset that can be held The fixed supply of currency is M The condition for a stationary equilibrium is Ea0 M p 18102 13 It is possible to tell versions of this story in which notes issued by one individual or group of individuals are extinguished by another 812 Incomplete Markets Models We shall seek a stationary equilibrium with pt1 pt 1 r for t 1 and Mt1 pt a for t 0 These guesses make the previous equation become a G r 18112 For G 0 this is a rectangular hyperbola in the southeast quadrant A sta tionary equilibrium value of r is determined at an intersection of this curve with Ear see Figure 18111 Evidently when G 0 an equilibrium net interest rate r 0 r can be regarded as an inflation tax Notice that if there is one equilibrium net interest rate there is typically more than one This is a con sequence of the Laffer curve present in this model14 Typically if a stationary equilibrium exists there are at least two stationary inflation rates that finance the government budget This conclusion follows from the fact that both curves in Figure 18111 have positive slopes E a r r E a r 1 2 1 r r G r a Figure 18111 Two stationary equilibrium rates of return on currency that finance the constant government deficit G After r is determined the initial price level can be determined by the time 0 version of the government budget constraint 18111 namely a M0p0 G 14 A Laffer curve exists when government revenues from a tax are not a monotonic function of a tax rate 814 Incomplete Markets Models Stationary versions of the two countries budget constraints are a1 a11 r G1 18121 a2 a21 r G2 18122 Sum these to get a1 a2 G1 G2 r Setting this curve against Ea1r Ea2r determines a stationary equilibrium rate of return r To determine the initial price level and exchange rate we use the time 0 budget constraints of the two governments The time 0 budget constraint for country i is Mi1 pi0 Mi0 pi0 Gi or ai Mi0 pi0 Gi 18123 Add these and use p10 ep20 to get a1 a2 G1 G2 M10 eM20 p10 This is one equation in two variables e p10 If there is a solution for some e 0 then there is a solution for any other e 0 In this sense the equilibrium exchange rate is indeterminate Equation 18123 is a quantity theory of money stated in terms of the initial world money supply M10 eM20 Interest on currency 815 1813 Interest on currency Bewley 1980 1983 studied whether Friedmans recommendation to pay in terest on currency could improve outcomes in a stationary equilibrium and possibly even support an optimal allocation He found that when β 1 Fried mans rule could improve things but could not implement an optimal allocation for reasons we now describe As in the earlier fiat money model there is one asset fiat currency issued by a government Households cannot borrow b 0 The consumers budget constraint is mt1 ptct 1 rmt ptwst τpt where mt1 0 is currency carried over from t to t 1 pt is the price level at t r is nominal interest on currency paid by the government and τ is a real lumpsum tax This tax is used to finance the interest payments on currency The governments budget constraint at t is Mt1 Mt rMt τpt where Mt is the nominal stock of currency per person at the beginning of t There are two versions of this model one where the government pays ex plicit interest while keeping the nominal stock of currency fixed another where the government pays no explicit interest but varies the stock of currency to pay interest through deflation For each setting we can show that paying interest on currency where cur rency holdings continue to obey mt 0 can be viewed as a device for weaken ing the impact of this nonnegativity constraint We establish this point for each setting by showing that the households problem is isomorphic with Aiyagaris problem as expressed in 1831 1853 and 1854 816 Incomplete Markets Models 18131 Explicit interest In the first setting the government leaves the money supply fixed setting Mt1 Mt t and undertakes to support a constant price level These settings make the government budget constraint imply τ rMp Substituting this into the households budget constraint and rearranging gives mt1 p ct mt p 1 r wst r M p where the choice of currency is subject to mt1 0 With appropriate trans formations of variables this matches Aiyagaris setup of expressions 1831 1853 and 1854 In particular take r r φ M p mt1 p ˆat1 0 With these choices the solution of the savings problem of a household living in an economy with aggregate real balances of M p and with nominal interest r on currency can be read from the solution of the savings problem with the real interest rate r and a borrowing constraint parameter φ M p Let the solution of this problem be given by the policy function at1 ga s r φ Because we have set mt1 p ˆat1 at1 M p the condition that the supply of real balances equals the demand E mt1 p M p is equivalent with Eˆar φ Note that because at ˆat φ the equilibrium can also be expressed as Ear 0 where as usual Ear is the average of a computed with respect to the invariant distribution λa s The preceding argument shows that an equilibrium of the money economy with mt1 0 equilibrium real balances M p and explicit interest on currency r therefore is isomorphic to a pure credit economy with borrowing constraint φ M p We formalize this conclusion in the following proposition Proposition 3 A stationary equilibrium with interest on currency financed by lumpsum taxation has the same allocation and interest rate as an equilibrium of Huggetts free banking model for debt limit φ equaling the equilibrium real balances from the monetary economy To compute an equilibrium with interest on currency we use a back solving method15 Thus even though the spirit of the model is that the govern ment names r r and commits itself to set the lumpsum tax needed to finance 15 See Sims 1989 and DiazGimenez Prescott Fitgerald and Alvarez 1992 for explana tions and applications of backsolving Interest on currency 817 interest payments on whatever M p emerges we can compute the equilibrium by naming M p first then finding an r that makes things work In particular we use the following steps 1 Set φ to satisfy 0 φ ws1 r We will elaborate on the upper bound in the next section Compute real balances and therefore p by solving M p φ 2 Find r from Eˆar M p or Ear 0 3 Compute the equilibrium tax rate from the government budget constraint τ r M p This construction finds a constant tax that satisfies the government budget constraint and that supports a level of real balances in the interval 0 M p ws1 r Evidently the largest level of real balances that can be supported in equilibrium is the one associated with the natural debt limit The levels of interest rates that are associated with monetary equilibria are in the range 0 r rF B where EarF B 0 and rF B is the equilibrium interest rate in the pure credit economy ie Huggetts model under the natural debt limit 18132 The upper bound on M p To interpret the upper bound on attainable M p note that the governments bud get constraint and the budget constraint of a household with zero real balances imply that τ r M p ws for all realizations of s Assume that the stationary distribution of real balances has a positive fraction of agents with real balances arbitrarily close to zero Let the distribution of employment shocks s be such that a positive fraction of these lowwealth consumers receive income ws1 at any time Then for it to be feasible for the lowest wealth consumers to pay their lumpsum taxes we must have τ rM p ws1 or M p ws1 r In Figure 1861 the equilibrium real interest rate r can be read from the intersection of the Ear curve and the raxis Think of a graph with two Ear curves one with the natural debt limit φ s1w r the other one with an ad hoc debt limit φ minb s1w r shifted to the right The highest interest rate that can be supported by an interest on currency policy is evidently determined by the point where the Ear curve for the natural debt limit passes through the raxis This is higher than the equilibrium interest rate associated with any of the ad hoc debt limits but must be below ρ Note that ρ is the interest 818 Incomplete Markets Models rate associated with the optimal quantity of money Thus we have Aiyagaris 1994 graphical version of Bewleys 1983 result that the optimal quantity of money Friedmans rule cannot be implemented in this setting We summarize this discussion with a proposition about free banking and Friedmans rule Proposition 4 The highest interest rate that can be supported by paying interest on currency equals that associated with the pure credit ie the pure inside money model with the natural debt limit If ρ 0 Friedmans ruleto pay real interest on currency at the rate ρ cannot be implemented in this model The most that can be achieved by paying interest on currency is to eradicate the restriction that prevents households from issuing currency in competition with the government and to implement the free banking outcome 18133 A very special case Levine and Zame 2002 have studied a special limiting case of the preceding model in which the free banking equilibrium which we have seen is equivalent to the best stationary equilibrium with interest on currency is optimal They attain this special case as the limit of a sequence of economies with ρ 0 Heuristically under the natural debt limits the Ear curves converge to a horizontal line at r 0 At the limit ρ 0 the argument leading to Proposition 4 allows for the optimal r ρ equilibrium Interest on currency 819 18134 Implicit interest through deflation There is another arrangement equivalent to paying explicit interest on currency Here the government aspires to pay interest through deflation but abstains from paying explicit interest This purpose is accomplished by setting r 0 and τpt gMt where it is intended that the outcome will be 1 r1 1 g with g 0 The government budget constraint becomes Mt1 Mt1 g This can be written Mt1 pt Mt pt1 pt1 pt 1 g We seek a steady state with constant real balances and inverse of the gross inflation rate pt1 pt 1 r Such a steady state implies that the preceding equation gives 1 r 1 g1 as desired The implied lumpsum tax rate is τ Mt pt1 1 rg Using 1 r 1 g1 this can be expressed τ Mt pt1 r The households budget constraint with taxes set in this way becomes ct mt1 pt mt pt1 1 r wst Mt pt1 r 18131 This matches Aiyagaris setup with Mt pt1 φ With these matches the steadystate equilibrium is determined just as though explicit interest were paid on currency The intersection of the Ear curve with the raxis determines the real interest rate Given the parameter b setting the debt limit the interest rate equals that for the economy with explicit interest on currency 820 Incomplete Markets Models 1814 Precautionary savings As we have seen in the production economy with idiosyncratic labor income shocks the steadystate capital stock is larger when agents have no access to insurance markets as compared to the capital stock in a complete markets econ omy The excessive accumulation of capital can be thought of as the economys aggregate amount of precautionary savingsa point emphasized by Huggett and Ospina 2000 The precautionary demand for savings is usually described as the extra savings caused by future income being random rather than determinate16 In a partial equilibrium savings problem it has been known since Leland 1968 and Sandmo 1970 that precautionary savings in response to risk are associated with convexity of the marginal utility function or a positive third derivative of the utility function In a twoperiod model the intuition can be obtained from the Euler equation assuming an interior solution with respect to consumption u1 ra0 w0 a1 β1 rE0u1 ra1 w1 where 1r is the gross interest rate wt is labor income endowment in period t 0 1 a0 is an initial asset level and a1 is the optimal amount of savings be tween periods 0 and 1 Now compare the optimal choice of a1 in two economies where next periods labor income w1 is either determinate and equal to w1 or random with a mean value of w1 Let an 1 and as 1 denote the optimal choice of savings in the nonstochastic and stochastic economy respectively that satisfy the Euler equations u1 ra0 w0 an 1 β1 ru1 ran 1 w1 u1 ra0 w0 as 1 β1 rE0u1 ras 1 w1 β1 ru1 ras 1 w1 16 Neng Wang 2003 describes an analytically tractable Bewley model with exponential utility He is able to decompose the savings of an infinitely lived agent into three pieces 1 a part reflecting a rainy day motive that would also be present with quadratic preferences 2 a part coming from a precautionary motive and 3 a dissaving component due to impatience that reflects the relative sizes of the interest rate and the consumers discount rate Wang computes the equilibrium of a Bewley model by hand and shows that at the equilibrium interest rate the second and third components cancel effectively leaving the consumer to behave as a permanentincome consumer having a martingale consumption policy 822 Incomplete Markets Models strict concavity include two wellknown cases CARA utility if all of the risk is to labor income no rateofreturn risk and CRRA utility if all of the risk is rateofreturn risk no laborincome risk In the course of the proof Carroll and Kimball generalize the result of Sibley 1975 that a positive third derivative of the utility function is inherited by the value function For there to be precautionary savings the third derivative of the value function with respect to assets must be positive that is the marginal utility of assets must be a convex function of assets The case of the quadratic oneperiod utility is an example where there is no precautionary saving Off corners the value function is quadratic and the third derivative of the value function is zero18 Where precautionary saving occurs and where the marginal utility of con sumption is always positive the consumption function becomes approximately linear for large asset levels19 This feature of the consumption function plays a decisive role in governing the behavior of a model of Krusell and Smith 1998 to which we now turn 1815 Models with fluctuating aggregate variables That the aggregate equilibrium state variables are constant helps makes the preceding models tractable This section describes a way to extend such models to situations with timevarying stochastic aggregate state variables20 Krusell and Smith 1998 modified Aiyagaris 1994 model by adding an aggregate state variable z a technology shock that follows a Markov process Each household continues to receive an idiosyncratic laborendowment shock s 18 In linearquadratic models decision rules for consumption and asset accumulation are independent of the variances of innovations to exogenous income processes 19 Roughly speaking this follows from applying the BenvenisteScheinkman formula and noting that where v is the value function v is increasing in savings and v is bounded 20 See Duffie Geanakoplos MasColell and McLennan 1994 for a general formulation and equilibrium existence theorem for such models These authors cast doubt on whether in general the current distribution of wealth is enough to serve as a complete description of the history of the aggregate state They show that in addition to the distribution of wealth it can be necessary to add a sunspot to the state See Miao 2003 for a later treatment and for an interpretation of the additional state variable in terms of a distribution of continuation values See Marcet and Singleton 1999 for a computational strategy for incomplete markets models with a finite number of heterogeneous agents Models with fluctuating aggregate variables 825 Krusell and Smith make the plausible guess that λtk s is enough to com plete the description of the state22 23 The Bellman equation and the pricing functions induce the household to want to forecast the average capital stock K in order to forecast future prices That desire makes the household want to forecast the crosssection distribution of holdings of capital To do so it consults the law of motion 18157d Definition A recursive competitive equilibrium is a pair of price functions r w a value function a decision rule k fk s λ z and a law of motion H for λk s such that a given the price functions and H the value function solves the Bellman equation 18156 and the optimal decision rule is f b the decision rule f and the Markov processes for s and z imply that todays distribution λk s is mapped into tomorrows λk s by H The curse of dimensionality makes an equilibrium difficult to compute Krusell and Smith propose a way to approximate an equilibrium using simu lations24 First they characterize the distribution λk s by a finite set of moments of capital m m1 mI They assume a parametric functional form for H mapping todays m into next periods value m They assume a form that can be conveniently estimated using least squares They assume initial values for the parameters of H Given H they use numerical dynamic programming to solve the Bellman equation vk s m z max ck uc βEvk s m zs z m subject to the assumed law of motion H for m They take the solution of this problem and draw a single long realization from the Markov process for zt 22 However in general settings this guess remains to be verified Duffie Geanakoplos Mas Colell and McLennan 1994 give an example of an incomplete markets economy in which it is necessary to keep track of a longer history of the distribution of wealth 23 Loosely speaking that the individual moves through the distribution of wealth as time passes indicates that his implicit Pareto weight is fluctuating 24 These simulations can be justified formally using lessons learned from the literature on convergence of least squares learning to rational expectations in selfreferential environments See footnote 5 of chapter 7 the paper by Marcet and Sargent 1989 and the book with extensions and many applications by Evans and Honkapohja 2001 826 Incomplete Markets Models say of length T For that particular realization of z they then simulate paths of kt st of length T for a large number M of households They assemble these M simulations into a history of T empirical crosssection distributions λtk s They use the cross section at t to compute the crosssection moments mt thereby assembling a time series of length T of the crosssection moments mt They use this sample and nonlinear least squares to estimate the transition function H mapping mt into mt 1 They return to the beginning of the procedure use this new guess at H and continue iterating to convergence of the function H Krusell and Smith compare the aggregate time series Kt Nt rt wt from this model with a corresponding representative agent or complete markets model They find that the statistics for the aggregate quantities and prices for the two types of models are very close Krusell and Smith interpret this result in terms of an approximate aggregation theorem that follows from two properties of their parameterized model First consumption as a function of wealth is concave but close to linear for moderate to high wealth levels Second most of the saving is done by the highwealth people These two properties mean that fluctuations in the distribution of wealth have only a small effect on the aggregate amount saved and invested Thus distribution effects are small Also for these high wealth people selfinsurance works quite well so aggregate consumption is not much lower than it would be for the complete markets economy Krusell and Smith compare the distributions of wealth from their model to the US data Relative to the data the model with a constant discount factor generates too few very poor people and too many rich people Krusell and Smith modify the model by making the discount factor an exogenous stochastic process The discount factor switches occasionally between two values Krusell and Smith find that a modest difference between two discount factors can bring the models wealth distribution much closer to the data Patient people become wealthier impatient people eventually become poorer Exercises 827 1816 Concluding remarks The models in this chapter pursue some of the adjustments that households make when their preferences and endowments give a motive to insure but mar kets offer limited opportunities to do so We have studied settings where house holds saving occurs through a single riskfree asset Households use the asset to selfinsure by making intertemporal adjustments of the asset holdings to smooth their consumption Their consumption rates at a given date become a function of their asset holdings which in turn depend on the histories of their endowments In pure exchange versions of the model the equilibrium allocation becomes individual history specific in contrast to the historyindependence of the corresponding complete markets model The models of this chapter arbitrarily shut down or allow markets without explanation The market structure is imposed its consequences then analyzed In chapters 21 and 22 we study a class of models for similar environments that like the models of this chapter make consumption allocations history dependent But the spirit of the models in chapters 21 and 22 differs from those in this chapter in requiring that the trading structure be more firmly motivated by the environment In particular the models in chapters 21 and 22 posit a particular reason that complete markets do not exist coming from enforcement or information problems and then study how risk sharing among people can best be arranged Exercises Exercise 181 Random discount factor BewleyKrusellSmith A household has preferences over consumption of a single good ordered by a value function defined recursively by vβt at st uct βtEtvβt1 at1 st1 where βt 0 1 is the time t value of a discount factor and at is time t holding of a single asset Here v is the discounted utility for a consumer with asset holding at discount factor βt and employment state st The discount factor evolves according to a threestate Markov chain with transition probabilities Pij Probβt1 βjβt βi The discount factor and employment state at t are both known The household faces the sequence of budget constraints at1 ct 1 rat wst 832 Incomplete Markets Models b In the fashion of Bewley define a stationary stochastic equilibrium being careful to define all of the objects composing an equilibrium c Adjust the Bellman equations to accommodate the following modification Assume that every period that a worker finds himself in a bad job there is a probability δupgrade that the following period the bad job is upgraded to a good job conditional on not having been fired d Acemoglu and Shimer calibrate their model to US high school graduates then perform a local analysis of the consequences of increasing the unemploy ment compensation rate b For their calibration they find that there are sub stantial benefits to raising the unemployment compensation rate and that this conclusion prevails despite the presence of a moral hazard problem associated with providing unemployment insurance benefits in their model The reason is that too many workers choose to search for bad rather than good jobs They calibrate β so that workers are sufficiently impatient that most workers with low assets search for bad jobs If workers were more fully insured more workers would search for better jobs That would put a larger fraction of workers in good jobs and raise average productivity In equilibrium unemployed workers with high asset levels do search for good jobs because their assets provide them with the selfinsurance needed to support their investment in search for good jobs Do you think that the modification suggested in part c would affect the outcomes of increasing unemployment compensation b Exercise 186 Gluing stationary equilibria At time 1 there is a continuum of ex ante identical consumer named i 0 1 Just before time 0 net assets ai 0 drawn from a cumulative distribution function F are distributed to agents Net assets may be positive or negative Agent is net assets at the beginning of time 0 are then a0 ai 0 To conserve notation well usually supress the i A typical consumers labor income at time t is wst where where w is a fixed positive number and st evolves according to an mstate Markov chain with transition matrix P Think of initiating the process from the invariant distribution of P over si s If the realization of the process at t is si then at time t the household receives labor income wsi Let at be the households net assets at the beginning of period t For given initial values a0 s0 and a given net riskfree interest rate r a 834 Incomplete Markets Models 0 and T1i is the real value of transfers awarded to consumer i A typical households budget constraint becomes 3 c0 a1 ws0 1 rHa0 T1i at time 0 and remains inequality 2 for t 1 e First please define a stationary equilbrium of an incomplete markets model with valued unbacked fiat currency in the style of Bewley Let G be the cumu lative distribution function of net assets a in this equilbrium f Extra credit Recalling that F is the cross section CDF of net assets in the equliibrium of the Huggett model of part c and that G is the cross section CDF of net assets in the equilibrium of the Bewley model of part e please describe a scheme for awarding the transfers of fiat money before the beginning of time 0 that i preserves the ranks of all assets in the wealth distribution and ii moves the initial asset distribution from F to G where a consumers initial net assets in the Bewley model are aBi 1 r0aHi T1i where aHi were his initial assets in the Huggett model Hint Recall that individuals i are distributed according to a uniform distribu tion on 0 1 Measured in units of time 0 consumption goods let T1i be the transfer to agent i Guess that the transfer is T1i G1i F 1i Let aGi aF i T1i be agent is initial assets after the transfer Verify that the CDF of aGi is G as desired25 g Extra credit Given the above transfer scheme for moving immediately from a stationary equilibrium of a Huggett model to a stationary equilibrium of a Bewley model with valued fiat money taking as given their rankings in the initial wealth distribution do all agents prefer one stationary equilibrium to the other If so which equilibrium do they prefer If not all agents prefer one to the other please describe computations that would allow you to sort agents into those who prefer to stay in the Huggett equilibrium and those who prefer to move to the Bewley equilibrium Exercise 187 Real bills Consider the Bewley model with a constant stock M0 of valued fiat money de scribed in exercise 186 Consider a monetary authority that issues additional 25 In effect the guess recommends applying what is known as an inverse probability inte gral transform or Smirnov transform Exercises 835 currency and that uses it to purchase oneperiod riskfree IOUs issued by con sumers The monetary authoritys budget constraint is 1 At1 At Mt1 Mt p t 0 subject to At1 0 for t 0 and A0 0 where At1 is the stock of one period IOUs that the monetary authority purchases at time t Here we guess that r 0 and that sp is the constant equilibrium price level in the Bewley economy with valued fiat currency from exercise 186 The equilibrium condition in the market for riskfree securities is now 2 At1 Ea0 Mt1 p a Solve the difference equation 1 backwards to verify that 3 At1 Mt1 M0 p b Verify that if a constant price level p satisfies Ea0 M0 p as in the original Bewley economy then equilibrium condition 2 is satisfied at the same price level p for any nonnegative sequence At1 t0 and associated money supply sequence Mt1 t0 that satisfy constraint 1 c Argue that an increase in M0 engineered as described in exercise 186 leads to an increase in the equilibrium p but that increases in Mt for t 1 leave the price level unaffected d Remark Sometimes M0 is called outside money and Mt1 M0 is called inside money It is called inside money because it is backed 100 by safe private IOUs The real bills doctrine asserts that increases in inside money are not inflationary Chapter 19 Dynamic Stackelberg Problems 191 History dependence Except for chapter 16 previous chapters described decision problems that are recursive in what we can call natural state variables meaning state vari ables that describe stocks of capital wealth and information that helps forecast future values of prices and quantities that impinge on utilities or profits In problems that are recursive in the natural state variables optimal decision rules are functions of the natural state variables Kydland and Prescott 1977 and Calvo 1978 gave macroeconomic exam ples of decision problems that are not recursive in natural state variables At time 0 a government chooses actions for all t 0 knowing that it confronts a competitive market composed of many small private agents whose decisions are influenced by their forecasts of the governments future actions In particular what private agents choose to do at date t depends partly on what they expect the government to do at dates t j t 0 In a rational expectations equilib rium in a nonstochastic setting the governments actions at time t 1 equal private agents earlier forecasts of those actions Knowing that the government uses its time t 1 actions to influence earlier actions by private agents The rational expectations equilibrium concept requires that the government confirm private sector forecasts That prevents the governments decision problem from being recursive in natural state variables and makes the governments decision rule at t depend on the history of the natural state variables from time 0 to time t It took time for economists to learn how to formulate policy problems of this type recursively Prescott 1977 asserted that recursive optimal control theory ie dynamic programming did not apply to problems with this struc ture This chapter and chapters 20 21 and 24 describe how Prescotts initial pessimism about the inapplicability of optimal control theory was overturned1 1 The important contribution by Kydland and Prescott 1980 dissipated Prescotts initial pessimism 839 840 Dynamic Stackelberg Problems An important finding is that if the natural state variables are augmented with appropriate forward looking state variables this class of problems can be made recursive This affords computational advantages and yields substantial insights This chapter displays these within the tractable framework of linearquadratic problems 192 The Stackelberg problem To exhibit the essential structure of the decision problems that concerned Kyd land and Prescott 1977 and Calvo 1979 this chapter uses the optimal linear regulator problem of chapter 5 to solve a linearquadratic version of what is known as a dynamic Stackelberg problem2 In some examples the Stackel berg leader is a government or Pareto planner and the Stackelberg follower as a representative agent or private sector In section 195 well give industrial or ganization application with another interpretation of these two types of agent Let zt be an nz 1 vector of natural state variables xt an nx 1 vector of endogenous forwardlooking variables and ut a vector of variables chosen by the Stackelberg leader Included in xt are prices and quantities that adjust instantaneously to clear markets at time t The zt vector is inherited from the past The vector xt is determined purely by future values of z and u Nevertheless at t 1 xt is inherited from the past because values of z and u for all t 0 are set by a Stackelberg plan devised at time 0 Remark For t 1 xt will turn out to be both a forwardlooking and a backward looking variable It is forward looking because it depends on forecasts of future actions of the Stackelberg leader It is backward looking because it is a promise about time t outcomes that was chosen earlier by the Stackelberg leader 2 In some settings it is called a Ramsey problem See chapters 16 and 20 Recursive formulation 843 to add a shock process Cǫt1 to the right side of 1924 where ǫt1 is an iid random vector with mean zero and identity covariance matrix 193 Timing protocol For any vector at define at at at1 Definition Given z0 the Stackelberg problem is to choose u0 x0z1 that maximize criterion 1922 subject to 1924 for t 0 A Stackelberg plan u0 x0z1 solves the Stackelberg problem starting from a given z0 The timing protocol underlying a Stackelberg plan is i Nature chooses z0 ii The Stackelberg leader chooses u0 iii The Stackelberg follower chooses x0 and an outcome path z1 emerges The Stackelberg leader understands how the objects in item iii depend on its choice of u0 This description of a Stackelberg plan views it from a static point of view in terms of sequences chosen at time 0 In the following section we describe a way to decentralize decisions of the Stackelberg leader over time 194 Recursive formulation To assemble a Stackelberg problem recursively we formulate two Bellman equa tions in two sets of state variables 846 Dynamic Stackelberg Problems 1944 Timing protocol Equations 1946 and 1949 form a recursive representation of a Stackelberg plan that features the following timing protocol 1 At times t 0 a continuation Stackelberg leader takes zt xt as given and chooses ut zt1 xt1 2 At time 0 a Stackelberg leader takes z0 as given and chooses x0 u0 z1 x1 Notice how we have distinguished between a continuation Stackelberg leader who chooses at time t 0 and a Stackelberg leader who chooses at time t 0 In this timing protocol the entirely forward looking vector xt that obeys 1926 is part of the state vector confronting a continuation Stackelberg leader at times t 0 but not part of the state confronting a Stackelberg leader at time t 0 It is presented to the continuation leader at time t 0 as a promise to be kept The time t continuation leader delivers xt by choosing ut zt1 xt1 8 1945 Time inconsistency The two subproblems in section 194 express the time inconsistency of the op timal decisions of the Stackelberg leader In the recursive representation of the Stackelberg program different state variables confront a Stackelberg leader at t 0 on the one hand and continuation Stackelberg leaders at dates t 0 on the other At t 0 the leader faces z0 as a state vector and chooses the forward looking vector x0 as well as the forward looking vector x1 that will confront the continuation leader at time 1 At dates t 0 a continuation leader confronts the state vector xt as values promised at time t 1 that must be confirmed at t9 Define a1 as the continuation of the sequence a0 A Stackelberg plan is a u0 x0z1 that solves the Stackelberg problem starting from a given z0 Time inconsistency A concise way to say that a Stackelberg plan is time inconsistent is to note that a continuation of a Stackelberg plan is not a Stack elberg plan10 8 See exercise 192 for a timing protocol that builds in time consistency 9 Another manifestation of timeinconsistency is that µxt is zero at t 0 and different from zero at t 1 10 Why Because x1 does not solve subproblem 2 at z1 Large firm facing a competitive fringe 847 195 Large firm facing a competitive fringe As an example this section studies the equilibrium of an industry with a large firm that acts as a Stackelberg leader with respect to a competitive fringe11The industry produces a single nonstorable homogeneous good One large firm pro duces Qt and a representative firm in a competitive fringe produces qt The representative firm in the competitive fringe acts as a price taker and chooses sequentially The large firm commits to a policy at time 0 taking into account its ability to manipulate the price sequence both directly through the effects of its quantity choices on prices and indirectly through the responses of the competitive fringe to its forecasts of prices12 The costs of production are Ct eQt 5gQ2 t 5cQt1 Qt2 for the large firm and σt dqt 5hq2 t 5cqt1 qt2 for the competitive firm where d 0 e 0 c 0 g 0 h 0 are cost parameters There is a linear inverse demand curve pt A0 A1Qt qt vt 1951 where A0 A1 are both positive and vt is a disturbance to demand governed by vt1 ρvt Cǫˇǫt1 1952 and where ρ 1 and ˇǫt1 is an iid sequence of random variables with mean zero and variance 1 In 1951 qt is equilibrium output of the representative competitive firm In equilibrium qt qt but we must distinguish between qt and qt in posing the optimum problem of a competitive firm 11 Sometimes the large firm is called the monopolist even though there are actually many firms in the industry 12 Hansen and Sargent 2012 use this model as a laboratory to illustrate an equilibrium concept featuring robustness in which at least one of the agents has doubts about the stochastic specification of the demand shock process Large firm facing a competitive fringe 849 for the large firm to influence qt1 by its choice of future Qtj1 s It is this feature that makes the large firms problem fail to be recursive in the natural state variables q Q In effect the large firm arrives at time t j not in the position of being able to take past values of qt as given because these have already been influenced by the large firms choice of Qtj Instead the large firm arrives at period t 0 facing the constraint that it must confirm the expectations about its time t decision upon which the competitive fringe based its decisions at dates before t 1952 The large firms problem The large firm views the competitive firms sequence of Euler equations as con straints on its own opportunities They are implementability constraints on the large firms choices Including the implementability constraints 1955 we can represent the constraints in terms of the transition law facing the large firm 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 A0 d 1 A1 A1 h c 1 vt1 Qt1 qt1 it1 1 0 0 0 0 0 ρ 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 c β 1 vt Qt qt it 0 0 1 0 0 ut 1956 where ut Qt1 Qt is the control of the large firm The last row portrays the implementability constraints 1955 Represent 1956 as yt1 Ayt But 1957 Although we have included the competitive fringes choice variable it as a component of the state yt in the large firms transition law 1957 it is actually a jump variable Nevertheless the analysis in earlier sections of this chapter implies that the solution of the large firms problem is encoded in the Riccati equation associated with 1957 as the transition law Lets decode it Exercises 855 f Is a continuation of a Markov perfect equilibrium a Markov perfect equilib rium Exercise 193 Duopoly An industry with two firms produces a single nonstorable homogeneous good Firm i 1 2 produces Qit Costs of production for firm i are Cit eQit 5gQ2 it5cQit1Qit2 where e 0 g 0 c 0 are cost parameters There is a linear inverse demand curve pt A0 A1Q1t Q2t vt 191 where A0 A1 are both positive and vt is a disturbance to demand governed by vt1 ρvt and where ρ 1 Assume that firm 1 is a Stackelberg leader and that firm 2 is a Stackelberg follower a Please formulate the decision problem of firm 2 and derive Euler equations that relate its current decisions to current and future decisions of firm 1 b Please formulate the decision problem of firm 1 as Stackelberg leader Please tell how to solve it c Describe calculations that answer the following question Starting from an initial state Q10 Q20 and situation in which firm 1 acts as Stackelberg leader and firm 2 acts as follower how much would firm 2 be willing to pay to buy firm 1 and thereby acquire the ability to ac a monopolist The LucasStokey economy 863 Except in these special cases the allocation and the labor tax rate as functions of st differ between dates t 0 and subsequent dates t 1 The presence of the extra term Φ0uccs0 ucℓs0 b0 in the firstorder condition at time 0 expresses the incentive for the Ramsey planner to manipulate ArrowDebreu prices in order to affect ucs0b0 x0 Thankfully the first order conditions here agree with firstorder conditions 16123 derived when we formulated a Ramsey plan in the space of sequences in section 1612 of chapter 16 2027 State variable degeneracy Equations 20217 and 20218 imply that Φ0 Φ1 and that Vx xt st Φ0 20220 for all t 1 When V is concave in x equation 20220 implies state variable degeneracy along a Ramsey plan in the sense that for t 1 xt will be a timeinvariant function of st Given Φ0 this function mapping st into xt can be expressed as a vector x that solves equation 161214 for n and c as functions of g that are associated with Φ Φ0 2028 Symptom and source of time inconsistency While the marginal utility adjusted level of government debt xtst is a key state variable for the continuation Ramsey planners at t 1 it is not a state variable at time 0 The time 0 Ramsey planner faces b0 not x0 ucs0 b0 as a state variable The discrepancy in state variables faced by the time 0 Ramsey planner and the time t 1 continuation Ramsey planners captures the differing obligations and incentives faced by the time 0 Ramsey planner and the time t 1 continuation Ramsey planners While the time 0 Ramsey planner is obligated to honor government debt b0 measured in time 0 consumption goods its choice of a policy can alter the marginal utility of time 0 consumption goods Thus the time 0 Ramsey planner can manipulate the value of government debt 864 Two Ramsey Problems Revisited as measured by ucs0 b0 In contrast time t 1 continuation Ramsey planners are obligated not to alter values of debt as measured by ucst btstst1 that they inherit from an earlier Ramsey planner or continuation Ramsey planner When government expenditures gt are a time invariant function of a Markov state st a Ramsey plan and associated Ramsey allocation feature marginal utilities of consumption ucst that given Φ for t 1 depend only on st but that for t 0 depend on b0 as well This means that ucst will be a time invariant function of st for t 1 but except when b0 0 a different function for t 0 This in turn means that prices of oneperiod Arrow securities pt1st1st pst1st will be the same time invariant functions of st1 st for t 1 but a different function p0s1s0 for t 0 except when b0 0 The differences between these time 0 and time t 1 objects reflect the workings of the Ramsey planners incentive to manipulate Arrow security prices and through them the burden of initial government debt b0 For an illustration see section 16134 of chapter 16 203 Recursive formulation of AMSS model We now describe a recursive version of the Aiyagari Marcet Sargent and Seppala 2002 economy that we studied in section 1615 The AMSS econ omy is identical with the LucasStokey 1983 economy except that instead of trading historycontingent securities or Arrow securities the government and household are allowed to trade only a oneperiod riskfree bond As we saw in section 1615 from the point of view of the Ramsey planner the restriction to oneperiod riskfree securities leaves intact the single implementability con straint on allocations in the LucasStokey economy while adding measurability constraints on functions of tails of allocations at each time and history functions that represent the present values of government surpluses In this section we explore how these measurability constraints alter the Bellman equations for a time 0 Ramsey planner and for time t 1 history st continuation Ramsey planners 868 Two Ramsey Problems Revisited preferences uc ℓ c Hℓ In this case Vxx s 0 is a nonpositive martingale By the martingale convergence theorem Vxx s converges almost surely5 When the Markov chain Πss and the function gs are such that gt is perpetually random Vxx s almost surely converges to zero For quasilinear preferences the firstorder condition with respect to ns becomes 1 µ s 1 uℓ s µ s n s uℓℓ s 0 Since µs βVxxs x converges to zero in the limit uℓs 1 ucs so that the tax rate on labor converges to zero In the limit the government accumulates sufficient assets to finance all expenditures from earnings on those assets returning any excess revenues to the household as nonnegative lump sum transfers Remark Along a Ramsey plan the state variable xt xtst becomes a function of the history st and also the initial government debt b0 Remark In our recursive formulation of the LucasStokey model in section 202 we found that the counterpart to Vxx s is time invariant and equal to the Lagrange multiplier on the single time 0 implementability condition present in the original version of that model cast in terms of choice of sequences We saw that the time invariance of Vxx s in the LucasStokey model is the source of the state variable degeneracy ie xt is an exact function of st a key feature of the LucasStokey model That Vxx s varies over time according to a twisted martingale means that there is no statevariable degeneracy in the AMSS model Both x and s are needed to describe the state This property of the AMSS model is what transmits a twisted martingalelike component to consumption employment and the tax rate 5 For a discussion of the martingale convergence theorem see the appendix to chapter 17 Concluding remarks 869 204 Concluding remarks The next several chapters construct Bellman equations for diverse applications in which implementability conditions inherited from various frictions require us to choose state variables artfully Chapter 21 Incentives and Insurance 211 Insurance with recursive contracts This chapter studies a planner who designs an efficient contract to supply in surance in the presence of incentive constraints We pursue two themes one substantive the other technical The substantive theme is a tension between offering insurance and providing incentives A planner offers stick and carrot incentives that adjust an agents future consumption in ways that provide in centives to adhere to an arrangement at the cost of providing less than ideal insurance Balancing incentives against insurance shapes the evolution of dis tributions of wealth and consumption The technical theme is how memory can be encoded recursively and how incentive problems can be managed with contracts that remember and promise Contracts issue rewards that depend on the history either of publicly observ able outcomes or of an agents announcements about his privately observed outcomes Histories are largedimensional objects But Spear and Srivastava 1987 Thomas and Worrall 1988 Abreu Pearce and Stacchetti 1990 and Phelan and Townsend 1991 discovered that the dimension can be contained by using an accounting system cast solely in terms of a promised value a onedimensional object that summarizes enough aspects of an agents history Working with promised values permits us to formulate contract design problems recursively Three basic models are set within a single physical environment but assume different structures of information enforcement and storage possibilities The first adapts a model of Thomas and Worrall 1988 and Kocherlakota 1996b that has all information being public and focuses on commitment or enforcement problems The second is a model of Thomas and Worrall 1990 that has an incentive problem coming from private information but that assumes away commitment and enforcement problems Common to both of these models is that the insurance contract is assumed to be the only vehicle for households to transfer wealth across states of the world and over time The third model 871 Basic environment 873 this allocation unattainable For each specification of incentive constraints we solve a planning problem Following a tradition started by Green 1987 we assume that a moneylender or planner is the only person in the village who has access to a riskfree loan market outside the village The moneylender can borrow or lend at a constant oneperiod riskfree gross interest rate R β1 Households cannot borrow or lend with each other and can trade only with the moneylender The moneylender is committed to honor his promises We will study three alternative types of incentive constraints a Both the money lender and the household observe the households history of endowments at each time t Although the moneylender can commit to honor a contract households cannot commit and at any time are free to walk away from an arrangement with the moneylender and live in perpetual autarky thereafter They must be induced not to do so by the structure of the contract This is a model of onesided commitment in which the contract must be selfenforcing That is it must be structured to induce the household to prefer to conform to it b Households can make commitments and enter into enduring and binding contracts with the moneylender but they have private information about their own incomes The moneylender can see neither their income nor their consumption Instead exchanges between the moneylender and a household must be based on the households own reports about income realizations An incentivecompatible contract induces a household to report its income truthfully c The environment is the same as b except that now households have access to a storage technology that cannot be observed by the moneylender House holds can store nonnegative amounts of goods at a riskfree gross return of R equal to the interest rate that the moneylender faces in the outside credit market Since the moneylender can both borrow and lend at the interest rate R outside of the village the private storage technology does not change the economys aggregate resource constraint but it does affect the set of incentivecompatible contracts between the moneylender and the households When we compute efficient allocations for each of these three environments we find that the dynamics of the implied consumption allocations differ dramati cally As an indication of the different outcomes that emerge Figures 2121 and 874 Incentives and Insurance 0 5 10 15 20 25 30 35 40 45 50 Time 658 66 662 664 666 668 67 Consumption 0 50 100 150 200 250 300 350 400 450 500 Time 1 2 3 4 5 6 7 8 Consumption Figure 2121 Left panel typical consumption path in en vironment a Right panel typical consumption path in envi ronment b 0 50 100 150 200 250 300 350 400 450 500 Time 5 6 7 8 9 10 11 12 13 Consumption Figure 2122 Typical consumption path in environment c 2122 depict consumption streams that are associated with the same realization of a random endowment stream for households living in environments a b and c respectively1 For all three of these economies we set uc γ1 expγc with γ 7 β 8 y1 y10 6 7 10 and Πs 1λ 1λ10 λs1 with 1 The dotted lines in these figures indicate the consumption allocation under a hypothetical complete markets arrangement that would give each of a continuum of ex ante identical villagers consumption always equal to mean income We thank Sebastian Graves for writing Python code that computes optimal value functions and the policy functions that attain them for these three environments Onesided no commitment 875 λ 4 In all three environments before date 0 the households have entered into efficient contracts with the moneylender We have initiated values for a villager that allow the money lender just to break even of consumption out comes evidently differ substantially across the three environments increasing monotonically and then flattening out in environment a stochastically heading south in environment b and stochastically heading north in environment c These sample path properties reflect how the contract copes with the three different frictions that we have put into the environment relative to the friction less chapter 7 setting This chapter explains why sample paths of consumption differ so much across these three settings 213 Onesided no commitment Our first incentive problem is a lack of commitment A moneylender is com mitted to honor his promises but villagers are free to walk away from their contract with the moneylender at any time The moneylender designs a con tract that the villager wants to honor at every moment and contingency Such a contract is said to be selfenforcing In chapter 22 we shall study another economy in which there is no moneylender only another villager and when no one is able to keep prior commitments Such a contract design problem with participation constraints on both sides of an exchange represents a problem with twosided lack of commitment in contrast to the problem with onesided lack of commitment treated here2 2 For an earlier twoperiod model of a onesided commitment problem see Holmstrom 1983 880 Incentives and Insurance How ws varies with v depends on which of two mutually exclusive and exhaus tive sets of states s v falls into after the realization of ys those in which the participation constraint 2136 binds ie states in which λs 0 or those in which it does not ie states in which λs 0 c g y uc Pw 1 uc w uy v β τ β aut uc w u yv v β β aut τ 1 τ w y l s c g v uc w uy v β β aut s 1 s c w 2 w v τ τ Figure 2131 Determination of consumption and promised utility c w Higher realizations of ys are associated with higher indifference curves uc βw uys βvaut For a given v there is a threshold level yv above which the participation constraint is binding and below which the mon eylender awards a constant level of consumption as a func tion of v and maintains the same promised value w v The cutoff level yv is determined by the indifference curve going through the intersection of a horizontal line at level v with the expansion path ucP w 1 States where λs 0 When λs 0 the participation constraint 2136 holds with equality When λs 0 21313 implies that P ws P v which in turn implies by the concavity of P that ws v Further the participation constraint at equality implies that cs ys because ws v vaut Together these results say that Onesided no commitment 881 when the participation constraint 2136 binds the moneylender induces the household to consume less than its endowment today by raising its continuation value When λs 0 cs and ws solve the two equations ucs βws uys βvaut 21314 ucs P ws1 21315 The participation constraint holds with equality Notice that these equations are independent of v This property is a key to understanding the form of the optimal contract It imparts to the contract what Kocherlakota 1996b calls amnesia when incomes yt are realized that cause the participation constraint to bind the contract disposes of all history dependence and makes both con sumption and the continuation value depend only on the current income state yt We portray amnesia by denoting the solutions of equations 21314 and 21315 by cs g1ys 21316a ws ℓ1ys 21316b Later well exploit the amnesia property to produce a computational algorithm States where λs 0 When the participation constraint does not bind λs 0 and firstorder condi tion 21311 imply that P v P ws which implies that ws v There fore from 21312 we can write ucs P v1 so that consumption in state s depends on promised utility v but not on the endowment in state s Thus when the participation constraint does not bind the moneylender awards cs g2v 21317a ws v 21317b where g2v solves ug2v P v1 882 Incentives and Insurance g v 2 y v c y Figure 2132 The shape of consumption as a function of realized endowment when the promised initial value is v The optimal contract Combining the branches of the policy functions for the cases where the partici pation constraint does and does not bind we obtain c maxg1y g2v 21318 w maxℓ1y v 21319 The optimal policy is displayed graphically in Figures 2131 and 2132 To interpret the graphs it is useful to study equations 2136 and 21312 for the case in which ws v By setting ws v we can solve these equations for a cutoff value call it yv such that the participation constraint binds only when ys yv To find yv we first solve equation 21312 for the value cs associated with v for those states in which the participation constraint is not binding ug2v P v1 and then substitute this value into 2136 at equality to solve for yv uyv ug2v βv vaut 21320 By the concavity of P the cutoff value yv is increasing in v Onesided no commitment 883 Associated with a given level of vt vaut v there are two numbers g2vt yvt such that if yt yvt the moneylender offers the household ct g2vt and leaves the promised utility unaltered vt1 vt The moneylender is thus insuring the villager against the states ys yvt at time t If yt yvt the participation constraint binds prompting the moneylender to induce the household to surrender some of its currentperiod endowment in exchange for a raised promised utility vt1 vt Promised values never decrease They stay constant for lowy states ys yvt and increase in highendowment states that threaten to violate the participation constraint Consumption stays constant during periods when the participation constraint fails to bind and increases during periods when it threatens to bind Whenever the participation binds the household makes a net transfer to the money lender in return for a higher promised continuation utility A household that has ever realized the highest endowment yS is permanently awarded the highest consumption level with an associated promised value v that satisfies ug2v βv uyS βvaut 2133 Recursive computation of contract As we will now show a money lender that takes on a villager whose only alter native is to live in autarky will design a profit maximizing contract that delivers an initial promised value v0 equal to vaut Later we will examine how the optimal contract would be modified if the initial promised value v0 were to be greater than vaut We can compute the optimal contract recursively by using the fact that the villager will ultimately receive a constant welfare level equal to uyS βvaut after ever having experienced the maximum endowment yS We can characterize the optimal policy in terms of numbers cs wsS s1 g1ys ℓ1ysS s1 where g1ys and ℓ1s are given by 21316 These numbers can be computed recursively by working backward as follows Start with s S and compute cS wS from the nonlinear equations ucS βwS uyS βvaut 21321a wS ucS 1 β 21321b 892 Incentives and Insurance A convenient formula links Pv0 to the tail behavior of Bt in particular to the behavior of Bt after the consumption distribution has converged to cS Here we are once again appealing to a law of large numbers so that the expected profits Pv0 becomes a nonstochastic present value of profits associated with making a promise v0 to a large number of households Since the moneylender lets all surpluses and deficits accumulate in the bank account it follows that Pv0 is equal to the present value of the sum of any future balances Bt and the continuation value of the remaining profit stream After all households promised values have converged to wS the continuation value of the remaining profit stream is evidently equal to βPwS Thus for t such that the distribution of c has converged to cs we deduce that Pv0 Bt βPwS 1 rt 21342 Since the term βPwS1 rt in expression 21342 will vanish in the limit the expression implies that the bank balances Bt will eventually change at the gross rate of interest If the initial v0 is set so that Pv0 0 Pv0 0 then the balances will eventually go to plus infinity minus infinity at an expo nential rate The asymptotic balances would be constant only if the initial v0 is set so that Pv0 0 This has the following implications First recall from our calculations above that there can exist an initial promised value v0 vaut wS such that Pv0 0 only if it is true that PwS 0 which by 21328a implies that Ey cS After imposing Pv0 0 and using the expression for PwS in 21328a equation 21342 becomes Bt β EycS 1β or Bt cS Ey r 0 where we have used the definition β1 1r Thus if the initial promised value v0 is such that Pv0 0 then the balances will converge when all households promised values converge to wS The interest earnings on those stationary balances will equal the oneperiod deficit associated with delivering cS to every household while collecting endowments per capita equal to Ey cS After enough time has passed all of the villagers will be perfectly insured because according to 21338 limt Probct cS 1 How much time it takes to converge depends on the distribution Π Eventually everyone will have received the highest endowment realization sometime in the past after A Lagrangian method 893 which his continuation value remains fixed Thus this is a model of temporary imperfect insurance as indicated by the eventual fanning in of the distribution of continuation values 2137 An example Figures 2133 and 2134 summarize aspects of the optimal contract for a version of our economy in which each household has an iid endowment process that is distributed as Probyt ys 1 λ 1 λS λs1 where λ 0 1 and ys s 5 is the sth possible endowment value s 1 S The typical households oneperiod utility function is uc 1 γ1c1γ where γ is the households coefficient of relative risk aversion We have assumed the parameter values β S γ λ 5 20 2 95 The initial promised value v0 is set so that Pv0 0 The moneylenders bank balance in Figure 2133 panel d starts at zero The moneylender makes money at first which he deposits in the bank But as time passes the moneylenders bank balance converges to the point that he is earning just enough interest on his balance to finance the extra payments he must make to pay cS to each household each period These interest earnings make up for the deficiency of his per capita period income Ey which is less than his per period per capita expenditures cS 214 A Lagrangian method Marcet and Marimon 1992 1999 have proposed an approach that applies to most of the contract design problems of this chapter They form a La grangian and use the Lagrange multipliers on incentive constraints to keep track of promises Their approach extends the work of Kydland and Prescott 1980 and is related to Hansen Epple and Roberds 1985 formulation for linear quadratic environments4 We can illustrate the method in the context of the preceding model 4 Marcet and Marimons method is a variant of the method used to compute Stackelberg or Ramsey plans in chapter 19 See chapter 19 for a more extensive review of the history of Insurance with asymmetric information 897 is a nondecreasing random sequence that ct stays constant when the participa tion constraint is not binding and that it rises when the participation constraint binds The numerical computation of a solution to equation 2145 is compli cated by the fact that slackness conditions 2146b and 2146c involve condi tional expectations of future endogenous variables ctj Marcet and Marimon 1992 handle this complication by resorting to the parameterized expectation approach that is they replace the conditional expectation by a parameterized function of the state variables5 Marcet and Marimon 1992 1999 describe a variety of other examples using the Lagrangian method See Kehoe and Perri 2002 for an application to an international trade model 215 Insurance with asymmetric information The moneylendervillager environment of section 213 poses a commitment prob lem because agents are free to choose autarky each period but there is no infor mation problem We now study a contract design problem where the incentive problem comes not from a commitment problem but instead from asymmetric information As before the moneylender or planner can borrow or lend outside the village at the constant riskfree gross interest rate of β1 and each house holds income yt is independently and identically distributed across time and across households However now we assume that the planner and household can enter into an enduring and binding contract At the beginning of time let vo be the expected lifetime utility that the planner promises to deliver to a household The initial promise vo could presumably not be less than vaut since a household would not accept a contract that gives a lower utility than he could attain at time 0 by choosing autarky We defer discussing how vo is determined until the end of the section The other new assumption here is that households have private information about their own income and that the planner can see neither their income nor their consumption It follows that any transfers between the planner and a household must be based on the households 5 For details on the implementation of the parameterized expectations approach in a simple growth model see den Haan and Marcet 1990 The parameterized expectations method was applied by Krusell and Smith 1998 to compute an approximate equilibrium of an incomplete markets model with a fluctuating aggregate state variable See chapter 18 900 Incentives and Insurance v Pv Pv 0 vmax Figure 2151 Value function Pv and the two dashed curves depict the bounds on the value function The vertical solid line indicates vmax sup uc1 β 2151 Efficiency implies bs1 bs ws1 ws An incentivecompatible contract must satisfy bs1 bs insurance and ws1 ws partial insurance This can be established by adding the downward con straint Css1 0 and the upward constraint Cs1s 0 to get uys bs uys1 bs uys bs1 uys1 bs1 where the concavity of uc implies bs bs1 It then follows directly from Css1 0 that ws ws1 Thus for any v a household reporting a lower income receives a higher transfer from the planner in exchange for a lower future utility Insurance with asymmetric information 901 2152 Local upward and downward constraints are enough Constraint set 2153 can be simplified We can show that if the local down ward constraints Css1 0 and upward constraints Css1 0 hold for each s S then the global constraints Csk 0 hold for each s k S The argu ment goes as follows Suppose we know that the downward constraint Csk 0 holds for some s k uys bs βws uys bk βwk 2157 From above we know that bs bk so the concavity of uc implies uys1 bs uys bs uys1 bk uys bk 2158 By adding expressions 2157 and 2158 and using the local downward con straint Cs1s 0 we arrive at uys1 bs1 βws1 uys1 bk βwk that is we have shown that the downward constraint Cs1k 0 holds In this recursive fashion we can verify that all global downward constraints are satisfied when the local downward constraints hold A symmetric reasoning applies to the upward constraints Starting from any upward constraint Cks 0 with k s we can show that the local upward constraint Ck1k 0 implies that the upward constraint Ck1s 0 must also hold and so forth 2153 Concavity of P Thus far we have not appealed to the concavity of the value function but henceforth we shall have to Thomas and Worrall showed that under condition A P is concave Proposition The value function Pv is concave We recommend just skimming the following proof on first reading Proof Let T P be the operator associated with the right side of equation 2151 We could compute the optimum value function by iterating to con vergence on T We want to show that T maps strictly concave P to strictly concave function T P Thomas and Worrall use the following argument Insurance with asymmetric information 905 2157 Comparison to model with commitment problem In the model with a commitment problem studied in section 213 the efficient al location had to satisfy equation 21312 ie uys bs P ws1 As we explained then this condition sets the households marginal rate of substitution equal to the planners marginal rate of transformation with respect to transfers in the current period and continuation values in the next period This condition fails to hold in the present framework with incentivecompatibility constraints associated with telling the truth The efficient tradeoff between current con sumption and a continuation value for a household with income realization ys can not be determined without taking into account the incentives that other households have to report ys untruthfully in order to obtain the corresponding bundle of current and future transfers from the planner It is instructive to note that equation 21312 would continue to hold in the present framework if the incentivecompatibility constraints for truth telling were not binding That is set the multipliers µs s 2 S equal to zero and substitute firstorder condition 21512 into 21511 to obtain uys bs P ws1 2158 Spreading continuation values An efficient contract requires that the promised future utility falls rises when the household reports the lowest highest income realization that is that w1 v wS To show that wS v suppose to the contrary that wS v That this assumption leads to a contradiction is established by the following line of argument Since wS ws for all s S and Pv is strictly concave equation 21514 implies that ws v for all s S Substitution of equation 21513 into equation 21512 then yields a zero on the left side of equation 21512 Moreover the right side of equation 21512 is equal to µ2 when s 1 and µS when s S so we can successively unravel from the constraint set 21512 that µs 0 for all s S Turning to equation 21511 it follows that the marginal utility of consumption is equalized across income real izations uys bs λ1 for all s S Such consumption smoothing requires bs1 bs but from incentive compatibility ws1 ws implies bs1 bs a contradiction We conclude that an efficient contract must have wS v A symmetric argument establishes w1 v 908 Incentives and Insurance 21510 Extension to general equilibrium Atkeson and Lucas 1992 provide examples of closed economies where the con strained efficient allocation also has each households expected utility converg ing to the minimum level with probability 1 Here the planner chooses the incentivecompatible allocation for all agents subject to a constraint that the total consumption handed out in each period to the population of households cannot exceed some constant endowment level Households are assumed to ex perience unobserved idiosyncratic taste shocks ǫ that are iid over time and households The taste shock enters multiplicatively into preferences that take either the logarithmic form uc ǫ ǫ logc the constant relative risk aversion CRRA form uc ǫ ǫcγγ γ 1 γ 0 or the constant absolute risk aversion CARA form uc ǫ ǫ expγc γ 0 The assumption that the utility function belongs to one of these families greatly simplifies the ana lytics of the evolution of the wealth distribution Atkeson and Lucas show that an equilibrium of this model yields an efficient allocation that assigns an ever increasing fraction of resources to an everdiminishing fraction of the economys population 21511 Comparison with selfinsurance We have just seen how in the Thomas and Worrall model the planner re sponds to the incentive problem created by the consumers private information by putting a downward tilt into temporal consumption profiles It is useful to recall how in the savings problem of chapters 17 and 18 the martingale con vergence theorem was used to show that the consumption profile acquired an upward tilt coming from the motive of the consumer to selfinsure Insurance with unobservable storage 909 216 Insurance with unobservable storage In the spirit of an analysis of Franklin Allen 1985 we now augment the model of the previous section by assuming that households have access to a technology that enables them to store nonnegative amounts of goods at a riskfree gross return of R 0 The planner cannot observe private storage The planner can borrow and lend outside the village at a riskfree gross interest rate that also equals R so that private and public storage yield identical rates of return The planner retains an advantage over households of being the only one able to borrow outside of the village The outcome of our analysis will be to show that allowing households to store amounts that are not observable to the planner so impedes the planners ability to manipulate the households continuation valuations that no social insurance can be supplied Instead the planner helps households overcome the nonnegativity constraint on households storage by in effect allowing them to engage also in private borrowing at the riskfree rate R subject to natural borrowing limits Thus outcomes share many features of the allocations studied in chapters 17 and 18 Our analysis partly follows Cole and Kocherlakota 2001 who assume that a households utility function u is strictly concave and twice continuously differentiable over 0 with limc0 uc The domain of u is the entire real line with uc for c 09 They also assume that u satisfies condition A above This preference specification allows Cole and Kocherlakota to characterize an efficient allocation in a finite horizon model Their extension to an infinite horizon involves a few other assumptions including upper and lower bounds on the utility function We retain our earlier assumption that the planner has access to a riskfree loan market outside of the village Cole and Kocherlakota 2001 postulate a closed economy where the planner is constrained to choose nonnegative amounts of storage Hence our concept of feasibility differs from theirs 9 Allowing for negative consumption while setting utility equal to is a convenient device for avoiding having to deal with transfers that exceed the households resources 914 Incentives and Insurance Given the continuous strictly concave objective function and the compact con vex constraint set in problem P2 the solution c is unique and the firstorder conditions are both necessary and sufficient In the efficient allocation the planner chooses transfers that in effect re lax the nonnegativity constraint on a households storage is not binding ie consumption smoothing condition 2168 is satisfied However the optimal transfer scheme offers no insurance across households because the present value of transfers is zero for any history hT ie the netpresent value condition 2169 is satisfied 2164 The twoperiod case In a finite horizon model an immediate implication of the incentive constraints is that transfers in the final period T must be independent of households reported values of yT In the case of two periods we can therefore encode permissible transfer schemes as b1ys bs s S b2ys yj es s j S where bs and es denote the transfer in the first and second period respectively when the household reports income ys in the first period and income yj in the second period Following Cole and Kocherlakota 2001 we will first characterize the so lution to the modified planners problem P3 stated below It has the same objective function as P1 but a larger constraint set In particular we enlarge the constraint set by considering a smaller set of reporting strategies for the households Ω2 R A household strategy ˆy ˆk is an element of Ω2 R if ˆy1ys ys1 ys for s 2 3 S ˆy1y1 y1 That is a household can either tell the truth or lie downward by one notch in the grid of possible income realizations There is no restriction on possible storage strategies Insurance with unobservable storage 919 The proof of Proposition 2 for T 2 is completed by noting that by construction if some allocation c 0 b K solves P3 and c 0 b K is incentive compatible with respect to Ω2 then c 0 b K solves P1 Also since equations 2168 and 2169 fully characterize the consumption allo cation c we have uniqueness with respect to c but there exists a multitude of storage and transfer schemes that the planner can use to implement c in problem P1 2165 Role of the planner Proposition 2 states that any allocation c k b K that solves the planners problem P1 has the same consumption outcome c c as the solution to P2 ie the market outcome when each household can lend or borrow at the riskfree interest rate R This result has both positive and negative messages about the role of the planner Because households have access only to a stor age technology the planner implements the efficient allocation by designing an elaborate transfer scheme that effectively undoes each households nonnegativity constraint on storage while respecting solvency requirements In this sense the planner has an important role to play However the optimal transfer scheme of fers no insurance across households and implements only a selfinsurance scheme tantamount to a borrowingandlending outcome for each household Thus the planners accomplishments as an insurance provider are very limited If we had assumed that households themselves have direct access to the credit market outside of the village it would follow immediately that the plan ner would be irrelevant since the households could then implement the efficient allocation themselves Allen 1985 first made this observation Given any transfer scheme he showed that all households would choose to report the in come that yields the highest present value of transfers regardless of what the actual income is In our setting where the planner has no resources of his own we get the zero net present value condition for the stream of transfers to any individual household Exercises 927 for Cv i i 1 I Briefly discuss the form of the law of motion for v associated with the minimum cost insurance scheme Exercise 212 Wealth dynamics in moneylender model Consider the model in the text of the village with a moneylender The village consists of a large number eg a continuum of households each of which has an iid endowment process that is distributed as Probyt ys 1 λ 1 λS λs1 where λ 0 1 and ys s 5 is the sth possible endowment value s 1 S Let β 0 1 be the discount factor and β1 the gross rate of return at which the moneylender can borrow or lend The typical households one period utility function is uc 1 γ1c1γ where γ is the households coefficient of relative risk aversion Assume the parameter values β S γ λ 5 20 2 95 Hint The formulas given in the section 2133 will be helpful in answering the following questions a Using Matlab compute the optimal contract that the moneylender offers a villager assuming that the contract leaves the villager indifferent between refusing and accepting the contract b Compute the expected profits that the moneylender earns by offering this contract for an initial discounted utility that equals the one that the household would receive in autarky c Let the crosssection distribution of consumption at time t 0 be given by the cdf Probct C FtC Compute Ft Plot it for t 0 t 5 t 10 t 500 d Compute the moneylenders savings for t 0 and plot it for t 0 100 e Now adapt your program to find the initial level of promised utility v vaut that would set Pv 0 Exercise 213 Thomas and Worrall 1988 There is a competitive spot market for labor always available to each of a con tinuum of workers Each worker is endowed with one unit of labor each period 932 Incentives and Insurance c Can you say anything about a typical pattern of government tax collections Tt and distortions WTt over time for a country in an optimal sustainable contract with the IMF What about the average pattern of government surpluses Tt gt across a panel of countries with identical gt processes and W functions Would there be a cohort effect in such a panel ie would the calendar date when the country signed up with the IMF matter d If the optimal sustainable contract gives the country value vaut can the IMF expect to earn anything from the contract Chapter 22 Equilibrium without Commitment 221 Twosided lack of commitment In section 213 of the previous chapter we studied insurance without commit ment That was a small open economy analysis since the moneylender could borrow or lend resources outside of the village at a given interest rate Recall also the asymmetry in the environment where villagers could not make any com mitments while the moneylender was assumed to be able to commit We will now study a closed system without access to an outside credit market Any households consumption in excess of its own endowment must then come from the endowments of other households in the economy We will also adopt the symmetric assumption that no one is able to make commitments That is any contract prescribing an exchange of goods today in anticipation of future ex changes of goods represents a sustainable allocation only if current and future exchanges satisfy participation constraints for all households involved in the con tractual arrangement Households are free to walk away from the arrangement at any point in time and thereafter to live in autarky Such a contract design problem with participation constraints on both sides of an exchange represents a problem with twosided lack of commitment as compared to the problem with onesided lack of commitment in section 213 This chapter draws on the work of Thomas and Worrall 1988 1994 and Kocherlakota 1996b At the end of the chapter we also discuss market arrange ments for decentralizing the constrained Pareto optimal allocation as studied by Kehoe and Levine 1993 and Alvarez and Jermann 2000 933 934 Equilibrium without Commitment 222 A closed system Thomas and Worralls 1988 model of selfenforcing wage contracts is an an tecedent to our villagermoneylender environment The counterpart to our mon eylender in their model is a riskneutral firm that forms a longterm relationship with a riskaverse worker In their model there is also a competitive spot mar ket for labor where a worker is paid yt at time t The worker is always free to walk away from the firm and work in that spot market But if he does he can never again enter into a longterm relationship with another firm The firm seeks to maximize the discounted stream of expected future profits by designing a longterm wage contract that is selfenforcing in the sense that it never gives the worker an incentive to quit In a contract that stipulates a wage ct at time t the firm earns time t profits of yt ct as compared to hiring a worker in the spot market for labor If Thomas and Worrall had assumed a commitment problem only on the part of the worker their model would be formally identical to our villagermoneylender environment However Thomas and Worrall also assume that the firm itself can renege on a wage contract and buy labor at the random spot market wage Hence they require that in a selfenforcing wage contract neither party ever wants to renege Kocherlakota 1996b studies a model similar to Thomas and Worralls1 Kocherlakotas counterpart to Thomas and Worralls firm is a riskaverse second household In Kocherlakotas model two households receive stochastic endow ments The contract design problem is to find an insurancetransfer arrange ment that reduces consumption risk while respecting participation constraints both households must be induced each period not to walk away from the ar rangement Kocherlakota uses his model in an interesting way to help interpret empirically estimated conditional consumptionincome covariances that seem to violate the hypothesis of complete risk sharing Kocherlakota investigates the extent to which those failures reflect impediments to enforcement represented by his participation constraints To create a stationary stochastic environment Kocherlakota assumes two sided lack of commitment In our model of villagers facing a moneylender in section 213 imperfect risk sharing is temporary and so would not prevail in a 1 The working paper of Thomas and Worrall 1994 also analyzed a multiple agent closed model like Kocherlakotas Thomas and Worralls 1994 analysis evolved into an article by Ligon Thomas and Worrall 2002 that we discuss in section 2213 Recursive formulation 937 where expression 2231b is the promisekeeping constraint expression 2231c is the participation constraint for the type 1 agent and expression 2231d is the participation constraint for the type 2 agent The set of feasible c is given by expression 2231e Thomas and Worrall prove the existence of a compact interval that contains all permissible continuation values χj χj 0 xj for j 1 2 S 2231f Thomas and Worrall also show that the Paretofrontier Qj is decreasing strictly concave and continuously differentiable on 0 xj The bounds on χj are motivated as follows The contract cannot award the type 1 agent a value of χj less than zero because that would correspond to an expected future lifetime utility below the agents autarky level There exists an upper bound xj above which the planner would never find it optimal to award the type 1 agent a continuation value conditional on next periods endowment realization being yj It would simply be impossible to deliver a higher continuation value because of the participation constraints In particular the upper bound xj is such that Qjxj 0 2232 Here a type 2 agent receives an expected lifetime utility equal to his autarky level if the next periods endowment realization is yj and a type 1 agent is promised the upper bound xj Our two and threestate examples in sections 2210 and 2211 illustrate what determines xj Attach Lagrange multipliers µ βΠjλj and βΠjθj to expressions 2231b 2231c and 2231d then get the following firstorder conditions for c and χj 5 c u1 c µuc 0 2233a χj βΠjQ jχj µβΠj βΠjλj βΠjθjQ jχj 0 2233b By the envelope theorem Q sx µ 2234 5 Here we are proceeding under the conjecture that the nonnegativity constraints on con sumption in 2231e c 0 and 1 c 0 are not binding This conjecture is confirmed below when it is shown that optimal consumption levels satisfy c y1 yS 938 Equilibrium without Commitment After substituting 2234 into 2233a and 2233b respectively the opti mal choices of c and χj satisfy Q sx u1 c uc 2235a Q sx 1 θjQ jχj λj 2235b 224 Equilibrium consumption 2241 Consumption dynamics From equation 2235a the consumption c of a type 1 agent is an increasing function of the promised value x The properties of the Pareto frontier Qsx imply that c is a differentiable function of x on 0 xs Since x 0 xs c is contained in the nonempty compact interval cs cs where Q s0 u1 cs ucs and Q sxs u1 cs ucs Thus if c cs x 0 so that a type 1 agent gets no gain from the contract from then on If c cs Qsx Qsxs 0 so that a type 2 agent gets no gain Equation 2235a can be expressed as c gQ sx 2241 where g is a continuously and strictly decreasing function By substituting the inverse of that function into equation 2235b we obtain the expression g1c 1 θj g1cj λj 2242 where c is again the current consumption of a type 1 agent and cj is his next periods consumption when next periods endowment realization is yj The optimal consumption dynamics implied by an efficient contract are evidently governed by whether or not agents participation constraints are binding For any given endowment realization yj next period only one of the participation Equilibrium consumption 939 constraints in 2231c and 2231d can bind Hence there are three regions of interest for any given realization yj 1 Neither participation constraint binds When λj θj 0 the consumption dynamics in 2242 satisfy g1c g1cj c cj where c cj follows from the fact that g1 is a strictly decreasing function Hence consumption is independent of the endowment and the agents are offered full insurance against endowment realizations so long as there are no binding participation constraints The constant consumption allocation is determined by the temporary relative Pareto weight µ in equation 2233a 2 The participation constraint of a type 1 person binds λj 0 but θj 0 Thus condition 2242 becomes g1c g1cj λj g1c g1cj c cj The planner raises the consumption of the type 1 agent in order to satisfy his participation constraint The strictly positive Lagrange multiplier λj 0 im plies that 2231c holds with equality χj 0 That is the planner raises the welfare of a type 1 agent just enough to make her indifferent between choosing autarky and staying with the optimal insurance contract In effect the planner minimizes the change in last periods relative welfare distribution that is needed to induce the type 1 agent not to abandon the contract The welfare of the type 1 agent is raised both through the mentioned higher consumption cj c and through the expected higher future consumption Recall our earlier finding that implies that the new higher consumption level will remain unchanged so long as there are no binding participation constraints It follows that the contract for agent 1 displays amnesia when agent 1s participation constraint is binding be cause the previously promised value x becomes irrelevant for the consumption allocated to agent 1 from now on 3 The participation constraint of a type 2 person binds θj 0 but λj 0 Thus condition 2242 becomes g1c 1 θj g1cj g1c g1cj c cj 946 Equilibrium without Commitment For high enough values of β sufficient endowment risk and enough cur vature of u there will exist a set of firstbest sustainable allocations ie cmin cmax If the ex ante division of the gains is then given by an implicit initial consumption level c cmax cmin it follows by the updating rules in 2243 that consumption remains unchanged forever and therefore the asymp totic consumption distribution is degenerate But what happens if the ex ante division of gains is associated with an im plicit initial consumption level outside of this range or if there does not exist any firstbest sustainable allocation cmin cmax To understand the convergence of consumption to an asymptotic distribution in general we make the following observations According to the updating rules in 2243 any increase in the consumption of a type 1 person between two consecutive periods has consump tion attaining the lower bound of some consumption interval It follows that in periods of increasing consumption the consumption level is bounded above by cmax cS and hence increases can occur only if the initial consumption level is less than cmax Similarly any decrease in consumption between two consecutive periods has consumption attain the upper bound of some consump tion interval It follows that in periods of decreasing consumption consumption is bounded below by cmin c1 and hence decreases can only occur if initial consumption is higher than cmin Given a current consumption level c we can then summarize the permissible range for nextperiod consumption c as follows if c cmax then c minc cmin cmax 2262a if c cmin then c cmin maxc cmax 2262b Consumption distribution 947 2262 Temporary imperfect risk sharing We now return to the case that there exist firstbest sustainable allocations cmin cmax but we let the ex ante division of gains be given by an implicit initial consumption level c cmax cmin The permissible range for next period consumption as given in 2262 and the support of the asymptotic consumption becomes if c cmax then c c cmax and lim t ct cmax cS 2263a if c cmin then c cmin c and lim t ct cmin c1 2263b We have monotone convergence in 2263a for two reasons First consumption is bounded from above by cmax Second consumption cannot decrease when c cmin and by assumption cmin cmax so consumption cannot decrease when c cmax It follows immediately that cmax is an absorbing point that is attained as soon as the endowment yS is realized with its consumption level cS cmax Similarly the explanation for monotone convergence in 2263b goes as follows First consumption is bounded from below by cmin Second consumption cannot increase when c cmax and by assumption cmin cmax so consumption cannot increase when c cmin It follows immediately that cmin is an absorbing point that is attained as soon as the endowment y1 is realized with its consumption level c1 cmin These convergence results assert that imperfect risk sharing is at most tem porary if the set of firstbest sustainable allocations is nonempty Notice that when an economy begins with an implicit initial consumption outside of the interval of sustainable constant consumption levels the subsequent monotone convergence to the closest endpoint of that interval is reminiscent of our earlier analysis of the moneylender and the villagers with onesided lack of commitment in section 213 In the current setting the agent who is relatively disadvantaged under the initial welfare assignment will see her consumption weakly increase over time until she has experienced the endowment realization that is most fa vorable to her From there on the consumption level remains constant forever and the participation constraints will never bind again Continuation values a la Kocherlakota 957 for all i S and at least one of them holds with strict inequality The proof proceeds by considering four possible cases for each i S Case a ci cS and cS1i cS According to 2291c cS1i 1 ci so inequality 2295 can then be written as ci 1 cS1i ci which is true since ci ci for all i S as established in section 2244 Case b ci cS and cS1i cS According to 2291c ci 1 cS1i so inequality 2295 can then be written as ci 1 cS1i 1 cS which is true since cS1i cS for all i 1 as established in section 2242 Case c ci cS and cS1i cS According to 2291c cS1i 1 ci so inequality 2295 can then be written as cS 1 cS1i ci which is true since cS ci for all i S as established in section 2242 Case d ci cS and cS1i cS The inequality 2295 can then be written as cS 1 cS which is true since cS 05 as established in 2292 We can conclude that the inequality 2295 holds with strict inequality with only two exceptions 1 when i 1 and case b applies and 2 when i S and case c applies It follows that the difference in 2294 is definitely strictly positive if there are more than two states and hence the asymptotic distribution of continuation values is nondegenerate But what about when there are only two states S 2 Since c1 cS by 2292 and cS cS it follows that case b applies when i 1 and case c applies when i 2 S Therefore the difference in 2294 is zero and thus the continuation value of an agent 958 Equilibrium without Commitment experiencing the highest endowment is equal to that of the other agent who is then experiencing the lowest endowment Since there are no other continuation values in an economy with only two possible endowment realizations it follows that the asymptotic distribution of continuation values is degenerate when there are only two states S 2 A twostate example in section 2210 illustrates our findings The intuition for the degenerate asymptotic distribution of continuation values is straightfor ward On the one hand the planner would like to vary continuation values and thereby avoid large changes in current consumption that would otherwise be needed to satisfy binding participation constraints But on the other hand different continuation values presuppose that there exist intermediate states in which a higher continuation value can be awarded In our twostate example the participation constraint of either one or the other type of agent always binds and the asymptotic distribution is degenerate with only one continuation value 2292 Continuation values do not always respond to binding participation constraints Evidently continuation values will eventually not respond to binding partici pation constraints in a twostate economy since we have just shown that the asymptotic distribution is degenerate with only one continuation value But the outcome that continuation values might not respond to binding participation constraints occurs even with more states when endowments are iid In fact it is present whenever the consumption intervals of two adjacent endowment realizations yk and yk1 do not overlap ie when ck ck1 Here is how the argument goes Since ck ck1 it follows from 2264 that both ck and ck1 belong to the ergodic set of consumption Moreover 2244 implies that Sock Sock1 where So is defined in 2282a Using expression 2283 we can compute a common continuation value vck vck1 ˆv where ˆv is given by 2289 when that expression is evaluated for any c ck ck1 Given this identical continuation value it follows that there are situations where households continuation values will not respond to binding participation con straints A twostate example amnesia overwhelms memory 959 As an example let the current consumption and continuation value of the type 1 household be ck and vck ˆv and suppose that the household next period realizes the endowment yk1 It follows that the participation constraint of the type 1 household is binding and that the optimal solution in 2281 is to award the household a consumption level ck1 and continuation value vck1 That is the household is induced not to defect into autarky by increasing its con sumption ck1 ck but its continuation value is kept unchanged vck1 ˆv Suppose next that the type 1 household experiences yk in the following period This time it will be the participation constraint of the type 2 households that binds and the optimal solution in 2281 prescribes that the type 1 house hold is awarded consumption ck and continuation value vck ˆv Hence only consumption levels but not continuation values are adjusted in these two realizations with alternating binding participation constraints We use a threestate example in section 2211 to elaborate on the point that even though an incoming continuation value lies in the interior of the range of permissible continuation values in 2271f a binding participation constraint still might not trigger a change in the outgoing continuation value because there may not exist any efficient way to deliver a changed continuation value Con tinuation values that do not respond to binding participation constraints are a manifestation of the possibility that the Pareto frontier P need not be differentiable everywhere on the interval vaut vmax as shown in section 2282 2210 A twostate example amnesia overwhelms memory In this example and the threestate example of the following section we use the term continuation value to denote the state variable of Kocherlakota 1996b as described in the preceding section9 That is at the end of a period the continuation value v is the promised expected utility to the type 1 agent that will be delivered at the start of the next period Assume that there are only two possible endowment realizations S 2 with y1 y2 1 y y where y 5 1 Each endowment realization is equally likely to occur Π1 Π2 05 05 Hence the two types of agents 9 See Krueger and Perri 2003b for another analysis of a twostate example 960 Equilibrium without Commitment face the same ex ante welfare level in autarky vaut 5 1 β uy u1 y We will focus on parameterizations for which there exist no firstbest sustainable allocations ie cmin cmax which here amounts to c1 c2 An efficient al location will then asymptotically enter the ergodic consumption set in 2264 that here is given by two points c1 c2 Because of the symmetry in prefer ences and endowments it must be true that c2 1 c1 c where we let c denote the consumption allocated to an agent whose participation constraint is binding and 1 c be the consumption allocated to the other agent Before determining the optimal values 1c c we will first verify that any such stationary allocation delivers the same continuation value to both types of agent Let v be the continuation value for the consumer who last received a high endowment and let v be the continuation value for the consumer who last received a low endowment The promisekeeping constraint for v is v 5uc βv 5u1 c βv and the promisekeeping constraint for v is v 5uc βv 5u1 c βv Notice that the promisekeeping constraints make v and v identical There fore there is a unique stationary continuation value v v v that is independent of the current period endowment as established in section 2291 for S 2 Setting v v v in one of the two equations above and solving gives the stationary continuation value v 5 1 β uc u1 c 22101 To determine the optimal c in this twostate example we use the following two facts First c is the lower bound of the consumption interval c2 c2 c is the consumption level that should be awarded to the type 1 agent when she experiences the highest endowment y2 y and we want to maximize the welfare of the type 2 agent subject to the type 1 agents participation constraint Sec ond c belongs also to the ergodic set c1 c2 that characterizes the stationary A twostate example amnesia overwhelms memory 961 05 052 054 056 058 06 7185 718 7175 717 7165 716 7155 715 7145 714 c u1c β v Figure 22101 Welfare of the agent with low endowment as a function of c efficient allocation and we know that the associated efficient continuation values are then the same for all agents and given by v in 22101 The maximization problem above can therefore be written as max c u1 c βv 22102a subject to uc βv uy βvaut 0 22102b where v is given by 22101 We graphically illustrate how c is chosen in order to maximize 22102a subject to 22102b in Figures 22101 and 22102 for utility function 1 γ1c1γ and parameter values β γ y 85 11 6 It can be verified numerically that c 536 Figure 22101 shows 22102a as a decreasing function of c in the interval 5 6 Figure 22102 plots the left side of 22102b as a function of c Values of c for which the expression is negative are not sustainable ie values less than 536 Values of c for which the expression is nonnegative are sustainable Since the welfare of the agent with a low endowment realization in 22102a is decreasing as a function of c in the interval 5 6 the best sustainable value of c is the lowest value for which the expression in 22102b is nonnegative This value for c gives the most risk sharing that is compatible with the participation constraints 962 Equilibrium without Commitment 05 052 054 056 058 06 006 005 004 003 002 001 0 001 002 c uc β v uy β vautarky Figure 22102 The participation constraint is satisfied for values of c for which the difference uc βv uy βvaut plotted here is positive 22101 Pareto frontier It is instructive to find the entire set of sustainable values V In addition to the value v above associated with a stationary sustainable allocation other values can be sustained for example by promising a value ˆv v to a type 1 agent who has yet to receive a low endowment realization Thus let ˆv be a promised value to such a consumer and let c be the consumption assigned to that consumer in the event that his endowment is high Then promise keeping for the two types of agents requires ˆv 5uc βˆv 5u1 c βv 22103a Pˆv 5u1 c βPˆv 5uc βv 22103b If the type 1 consumer receives the high endowment sustainability of the allo cation requires uc βˆv uy βvaut 22104a u1 c βPˆv u1 y βvaut 22104b A twostate example amnesia overwhelms memory 963 If the type 2 consumer receives the high endowment awarding him c v au tomatically satisfies the sustainability requirements because these are already built into the construction of the stationary sustainable value v Lets solve for the highest sustainable initial value of ˆv namely vmax To do so we must solve the three equations formed by the promisekeeping constraints 22103a and 22103b and the participation constraint 22104b of a type 2 agent when it receives 1 y at equality u1 c βPˆv u1 y βvaut 22105 Equation 22103b and 22105 are two equations in c Pvmax After solving them we can solve 22103a for vmax Substituting 22105 into 22103b gives Pvmax 5u1 y βvaut 5uc βv 22106 But from the participation constraint of a high endowment household in a sta tionary allocation recall that uc βv uy βvaut Substituting this into 22106 and rearranging gives Pvmax vaut and therefore by 22105 c y10 Solving 22103a for vmax we find vmax 1 2 β uy u1 c βv 22107 Now let us study what happens when we set v v vmax and drive v toward v from above Totally differentiating 22103a and 22103b we find dPˆv dˆv u1 c uc Evidently lim vv dPv dv u1 c uc 1 10 According to our general characterization of the ex ante division of the gains of an efficient contract in section 225 it can be viewed as determined by an implicit initial consumption level c y1 yS Notice that the present calculations have correctly computed the upper bound of that interval for our twostate example yS y2 y 964 Equilibrium without Commitment By symmetry lim vv dPv dv uc u1 c 1 Thus there is a kink in the value function Pv at v v At v the value function is not differentiable as established in section 2282 when two adjacent consumption intervals are disjoint At v P v exists only in the sense of a subgradient in the interval u1 cuc ucu1 c Figure 22103 depicts the kink in Pv v v v Pv vmax vmax Pvaut vaut vaut Figure 22103 The kink in Pv at the stationary value of v for the twostate symmetric example A threestate example 965 22102 Interpretation Recall our characterization of the optimal consumption dynamics in 2243 Consumption remains unchanged between periods when neither participation constraint binds and hence the efficient contract displays memory or history dependence When either of the participation constraints binds history depen dence is limited to selecting either the lower or the upper bound of a consumption range cj cj where the range and its bounds are functions of the current en dowment realization yj After someones participation constraint has once been binding history becomes irrelevant because past consumption has no additional impact on the level of current consumption In our twostate example there are only two consumption ranges c1 c1 and c2 c2 And as a consequence the asymptotic consumption distribution has only two points c1 and c2 or in our notation 1 c and c It follows that history becomes irrelevant because consumption is then determined by the endowment realization Thus it can be said that amnesia overwhelms memory in this example and the asymptotic distribution of continuation values becomes degenerate with a single point v11 2211 A threestate example As the twostate example stresses any variation of continuation values in an efficient allocation requires that the environment be such that when a house holds participation constraint is binding the planner has room to increase both the current consumption and the continuation value of that household In the stationary allocation in the twostate example there is no room to adjust the continuation value because of the restrictions that promise keeping imposes We now analyze the stationary allocation of a threestate S 3 example in which the environment still limits the planners ability to manipulate continua tion values but nevertheless sometimes allows adjustments in the continuation value 11 If we adopt the recursive formulation of Thomas and Worrall in 2231 amnesia mani fests itself as a timeinvariant state vector x1 x2 where x1 u1cu1yβv vaut and x2 uc uy βv vaut 968 Equilibrium without Commitment 05 055 06 065 07 05 052 054 056 058 06 062 064 066 068 07 bar C hat C 05 055 06 065 07 05 052 054 056 058 06 062 064 066 068 07 bar C hat C Figure 22111 Left panel Pairs of c ˆc that satisfy uc βw uy βvaut Right panel Pairs of c ˆc that satisfy u1 ˆc βw u05 βvaut endowment y3 y and we want to maximize the welfare of the type 2 agent subject to the type 1 agents participation constraint Second c belongs also to the ergodic set in 22111 that characterizes the stationary efficient allocation and we know that the associated efficient continuation values are w for the agents with high endowment and w for the other agents By invoking functions 22115 that express these continuation values in terms of c ˆc and by using participation constraint 22112 that determines permissible values of ˆc the optimization problem above becomes max c ˆc u1 c βw 22116a subject to uc βw uy βvaut 0 22116b u1 ˆc βw u05 βvaut 0 22116c where w and w are given by 22115 To illustrate graphically how an efficient stationary allocation c ˆc can be computed from optimization problem 22116 we assume a utility function c1γ1 γ and parameter values β γ Π y 07 11 06 07 It should now be evident that we can restrict attention to consumption levels c 05 y and ˆc 05 c Figure shows the sets c ˆc 05 y 05 c that satisfy par ticipation constraint 22116b and 22116c respectively The intersection of these sets is depicted in Figure 22112 where the circle indicates the efficient stationary allocation that maximizes 22116a A threestate example 969 05 055 06 065 07 05 052 054 056 058 06 062 064 066 068 07 bar C hat C Figure 22112 Pairs of c ˆc that satisfy uc βw uy βvaut and u1 ˆc βw u05 βvaut The efficient stationary allocation within this set is marked with a circle 22111 Perturbation of parameter values We also compute efficient stationary allocations for different values of Π 0 1 while retaining all other parameter values As a function of Π the two panels of Figure 22113 depict consumption levels and continuation values respectively For low values of Π we see that there cannot be any risk sharing among the agents so that autarky is the only sustainable allocation The explanation for this is as follows Given a low value of Π an agent who has realized the high endowment y is heavily discounting the insurance value of any transfer in a future state when her endowment might drop to 1 y because such a state occurs only with a small probability equal to Π2 Hence in order for that agent to surrender some of her endowment in the current period she must be promised a significant combined payoff in that unlikely event of a low endowment in the future and a positive transfer in the most common state 2 But such promises are difficult to make compatible with participation constraints because all agents will be discounting the value of any insurance arrangement as soon as the common state 2 is realized since then there is once again only a small probability of experiencing anything else 970 Equilibrium without Commitment When the probability of experiencing extreme values of the endowment realization is set sufficiently high there exist efficient allocations that deliver risk sharing When Π exceeds 04 in the left panel of Figure 22113 the lucky agent is persuaded to surrender some of her endowment and her consumption becomes c y The lucky agent is compensated for her sacrifice not only through the insurance value of being entitled to an equivalent transfer in the future when she herself might realize the low endowment 1y but also through a higher consumption level in state 2 ˆc 0512 In fact if the consumption smoothing motive could operate unhindered in this situation the lucky agents consumption would indeed by equalized across states But what hinders such an outcome is the participation constraint of the unlucky agent when entering state 2 It must be incentive compatible for that earlier unlucky agent to give up parts of her endowment in state 2 when both agents now have the same endowment and the value of the insurance arrangement lies in the future Notice that this participation constraint of the earlier unlucky agent is no longer binding in our example when Π is greater than 094 because the efficient allocation prescribes ˆc c In terms of Thomas and Worralls characterization of the optimal consumption dynamics the parameterization is then such that c2 c3 and the ergodic set in 22111 is given by c1 c3 or in our notation by 1 c c The fact that the efficient allocation raises the consumption of the lucky agent in future realizations of state 2 is reflected in the spread of continuation values in the right panel of Figure 22113 The spread vanishes only in the limit when Π 1 because then the threestate example turns into our twostate example of the preceding section where there is only a single continuation value But while the planner is able to vary continuation values in the threestate ex ample there remains an important limitation to when those continuation values can be varied Consider a parameterization with Π 04 094 for which we know that ˆc c in Figure 22113 The agent who last experienced the highest endowment y is consuming ˆc in state 2 in the efficient stationary allocation and is awarded continuation value w Suppose now that agent once again real izes the highest endowment y and his participation constraint becomes binding 12 Recall that we established in section 2244 that all consumption intervals are nonde generate if there is risk sharing We can use this fact to prove that as soon as the parame ter value for Π exceeds the critical value where risk sharing becomes viable it follows that ˆc c2 y2 05 A threestate example 971 0 02 04 06 08 1 05 052 054 056 058 06 062 064 066 068 07 Π Consumption 0 02 04 06 08 1 3588 3586 3584 3582 358 3578 3576 3574 3572 Π Continuation value Figure 22113 Left panel Consumption levels as a func tion of Π The solid line depicts c ie consumption in states 1 and 3 of a person who realizes the highest endowment y The dashed line depicts ˆc ie consumption in state 2 of the type of person that was the last one to have received y Right panel Continuation values as a function of Π The solid line depicts w ie continuation value of the type of person that was the last one to have received y The dashed line is the continuation value of the other type of person ie w To prevent him from defecting to autarky the planner responds by raising his consumption to c ˆc but keeps his continuation value unchanged at w In other words the optimal consumption dynamics in the efficient stationary allocation leaves no room for increasing the continuation value further The unchanging continuation value is a reflection of the nondifferentiability of the Pareto frontier at v w 974 Equilibrium without Commitment 2212 Empirical motivation Kocherlakota was interested in the case of perpetual imperfect risk sharing be cause he wanted to use his model to think about the empirical findings from panel studies by Mace 1991 Cochrane 1991 and Townsend 1994 Those studies found that after conditioning on aggregate income individual consump tion and earnings are positively correlated belying the risksharing implications of the complete markets models with recursive utility of the type we studied in chapter 8 So long as no firstbest allocation is sustainable the action of the occasionally binding participation constraints lets the model with twosided lack of commitment reproduce that positive conditional covariation In recent work Albarran and Attanasio 2003 and Kehoe and Perri 2003a 2003b pursue more implications of models like Kocherlakotas 2213 Generalization Our formal analysis has followed the approach taken by Thomas and Worrall 1988 We have converted the riskneutral firm into a riskaverse household as suggested by Kocherlakota 1996b Another difference is that our analysis is cast in a general equilibrium setting while Thomas and Worrall formulate a partial equilibrium model where the firm implicitly has access to an outside credit market with a given gross interest rate of β1 when maximizing the ex pected present value of profits However this difference is not material since an efficient contract is such that wages never exceed output13 Hence Thomas and Worralls 1988 analysis can equally well be thought of as a general equilibrium analysis Ligon Thomas and Worrall 2002 further generalize the environment by assuming that the endowment follows a Markov process This allows for the pos sibility of both aggregate and idiosyncratic risk and serial correlation The effi cient contract is characterized by an updating rule for the ratio of the marginal utilities of the two households that resembles our updating rule for consump tion in 2243 Each state of nature is associated with a particular interval 13 The outcome that efficient wages do not exceed output in Thomas and Worralls 1988 analysis is related to our ability to solve optimization problem 2231 without imposing nonnegativity constraints on consumption See footnote 5 976 Equilibrium without Commitment agents take prices as given and budget constraints are the only restrictions on agents consumption sets Instead participation constraints 2221a and 2221b are now modelled as direct restrictions on agents consumption pos sibility sets Partly because of this controversial feature of the KehoeLevine decentralization Alvarez and Jermann use another decentralization one that imposes portfoliosolvency constraints and is cast in terms of sequential trad ing of Arrow securities The endogenously determined solvency constraints are agent and state specific and ensure that the participation constraints are satis fied We turn to the AlvarezJermann decentralization in the next section One can argue that the alternative decentralization simply converts one set of participation constraints into another For both specifications we have a substantial departure from a decentralized equilibrium under full commitment When we remove the assumption of commitment we assign a very demanding task to the invisible hand who now must not find marketclearing prices but must also check participationsolvency constraints for all agents and all states of the world 2215 Endogenous borrowing constraints Alvarez and Jermann 2000 alter Kehoe and Levines decentralization to attain a model with sequentially complete markets in which households face what can be interpreted as endogenous borrowing constraints Essentially they accom plish this by showing how the standard quantity constraints on Arrow securities see chapter 8 can be appropriately tightened to implement the optimal al location as constrained by the participation constraints Their idea is to find borrowing constraints tight enough to make the highest endowment agents ad here to the allocation while letting prices alone prompt lower endowment agents to go along with it For expositional simplicity we let yiy denote the endowment of a house hold of type i when a representative household of type 1 receives y Recall the earlier assumption that y1y y2y y 1 y The state of the econ omy is the current endowment realization y and the beginningofperiod asset holdings A A1 A2 where Ai is the asset holding of a household of type i and A1 A2 0 Because asset holdings add to zero it is sufficient to use A1 to characterize the wealth distribution Define the state of the economy as 978 Equilibrium without Commitment where cia X is the consumption decision rule of a household of type i with beginningofperiod assets a14 People with the highest valuation of an asset buy it Buyers of statecontingent securities are unconstrained so they equate their marginal rate of substitution to the price of the asset At equilibrium prices sellers of statecontingent securities will occasionally like to issue more but are constrained from doing so by statebystate restrictions on the amounts that they can sell Thus the intertemporal marginal rate of substitution of an agent whose participation constraint or borrowing constraint is not binding determines the pricing kernel A binding participation constraint translates into a binding borrowing constraint in the previous period A participation constraint for some state at t restricts the amount of statecontingent debts that can be issued for that state at t 1 In effect constrained and unconstrained agents have their own personal interest rates at which they are just indifferent between borrowing or lending a infinitesimally more A constrained agent wants to consume more today at equilibrium prices ie at the shadow prices 22153 evaluated at the solution of the planning problem and thus has a high personal interest rate He would like to sell more of the statecontingent security than he is allowed to at the equilibrium statedate prices An agent would like to sell statecontingent claims on consumption tomorrow in those states in which he will be well endowed tomorrow But those high endowment states are also the ones in which he will have an incentive to default He must be restrained from doing so by limiting the volume of debt that he is able to carry into those high endowment states This limits his ability to smooth consumption across high and low endowment states Thus his consumption and continuation value increases when he enters one of those high endowment states precisely because he has been prevented from selling enough claims to smooth his consumption over time and across states From a general equilibrium perspective when sellers of a statecontingent security are constrained with respect to the quantities that they can issue it follows that the price is bid up when unconstrained buyers are competing for a smaller volume of that security This tendency of lowering the yield on individual Arrow securities explains Alvarez and Jermanns result that interest rates are 14 For the twostate example with β 85 γ 11 y 6 described in Figure 22101 we computed that c 536 which implies that the riskfree interest rate is 10146 Note that with complete markets the riskfree claim would be β1 11765 Concluding remarks 979 lower when compared to a corresponding complete markets economy a property shared with the Bewley economies studied in chapter 1815 Alvarez and Jermann study how the statecontingent prices 22153 be have as they vary the discount factor and the stochastic process for y They use the additional fluctuation in the stochastic discount factor injected by the par ticipation constraints to explain some asset pricing puzzles See Zhang 1997 and Lustig 2000 2003 for further work along these lines 2216 Concluding remarks The model in this chapter assumes that the economy reverts to an autarkic allocation in the event that a household chooses to deviate from the allocation assigned in the contract Of course assigning autarky continuation values to everyone puts us inside the Pareto frontier and so is inefficient In terms of sustaining an allocation the important feature of the autarky allocation is just the continuation value that it assigns to an agent who is tempted to default ie an agent whose participation constraint binds Kletzer and Wright 2000 recognize that it can be possible to promise an agent who is tempted to default an autarky continuation value while giving those agents whose participation constraints arent binding enough to stay on the Pareto frontier Continuation values that lie on the Pareto frontier are said to be renegotiation proof Further research about how to model the consequences of default in these settings is likely to be fruitful By permitting coalitions of consumers to break away and thereafter share risks among themselves Genicot and Ray 2003 refine a notion of sustainability in a multiconsumer economy 15 In exercise 224 we ask the reader to compute the allocation and interest rate in such an economy 982 Equilibrium without Commitment Assume that β 8 b 5 γ 2 and ǫ 5 a Compute autarky levels of discounted utility v for the two types of house holds Call them vauth and vautℓ b Compute the competitive equilibrium allocation and prices Here assume that there are no enforcement problems c Compute the discounted utility to each household for the competitive equi librium allocation Denote them vCE i for i 1 2 d Verify that the competitive equilibrium allocation is not selfenforcing in the sense that at each t 0 some households would prefer autarky to the competitive equilibrium allocation e Now assume that there are enforcement problems because at the beginning of each period each household can renege on contracts and other social arrange ments with the consequence that it receives the autarkic allocation from that period on Let vi be the discounted utility at time 0 of consumer i Formulate the consumption smoothing problem of a planner who wants to maximize v1 subject to v2 v2 and constraints that make the allocation selfenforcing f Find an efficient selfenforcing allocation of the periodic form c1t ˇc 2 ˇc ˇc and c2t 2ˇc ˇc 2ˇc where continuation utilities of the two agents oscillate between two values vh and vℓ Compute ˇc Compute discounted utilities vh for the agent who receives 1 ǫ in the period and vℓ for the agent who receives 1 ǫ in the period Plot consumption paths for the two agents for i autarky ii complete markets without enforcement problems and iii complete markets with the enforcement constraint Plot continuation utilities for the two agents for the same three allocations Comment on them g Compute oneperiod gross interest rates in the complete markets economies with and without enforcement constraints Plot them over time In which economy is the interest rate higher Explain h Keep all parameters the same but gradually increase the discount factor As you raise β toward 1 compute interest rates as in part g At what value of β do interest rates in the two economies become equal At that value of β is either participation constraint ever binding 984 Equilibrium without Commitment That is please write a programming problem that can be used to compute an optimal sustainable allocation e Under what circumstance will the allocation that you found in part I solve the enforcementconstrained Pareto problem in part d Ie state conditions on u β y that are sufficient to make the enforcement constraints never bind Some useful background For the remainder of this problem please assume that u β y are such that the allocation computed in part I is not sustainable Recall that the amnesia property implies that the consumption allocated to an agent whose participation constraint is binding is independent of the ex ante promised value with which he enters the period With the present iid two state symmetric endowment pattern ex ante each period each of our two agents has an equal chance that it is his participation constraint that is binding In a symmetric sustainable allocation let each agent enter the period with the same ex ante promised value v and let c be the consumption allocated to the high endowment agent whose participation constraint is binding and let 1 c be the consumption allocated to the low endowment agent whose participation constraint is not binding By the above argument c is independent of the promised value v that an agent enters the period with which means that the current allocation to both types of agent does not depend on the promised value with which they entered the period And in a symmetric stationary sustainable allocation both consumers enter each period with the same promised value v f Please give a formula for the promised value v within a symmetric stationary sustainable allocation g Use a graphical argument to show how to determine the v c that are asso ciated with an optimal stationary symmetric allocation h In the optimal stationary sustainable allocation that you computed in part g why doesnt the planner adjust the continuation value of the consumer whose participation constraint is binding i Alvarez and Jermann showed that provided that the usual constraints on issuing Arrow securities are tightened enough the optimal sustainable allocation can be decentralized by trading in a complete set of Arrow securities with price qyy max i12 β uci t1y uci ty 5 Chapter 23 Optimal Unemployment Insurance 231 Historydependent unemployment insurance This chapter applies the recursive contract machinery studied in chapters 21 22 and 24 in contexts that are simple enough that we can go a long way toward computing optimal contracts by hand The contracts encode history dependence by mapping an initial promised value and a random time t observation into a time t consumption allocation and a continuation value to bring into next period We use recursive contracts to study good ways of providing consumption insurance when incentive problems come from the insurance authoritys inability to observe the effort that an unemployed person exerts searching for a job We begin by studying a setup of Shavell and Weiss 1979 and Hopenhayn and Nicolini 1997 that focuses on a single isolated spell of unemployment followed by permanent employment Later we take up settings of Wang and Williamson 1996 and Zhao 2001 with alternating spells of employment and unemployment in which the planner has limited information about a workers effort while he is on the job in addition to not observing his search effort while he is unemployed Here history dependence manifests itself in an optimal contract with intertemporal tieins across these spells Zhao uses her model to rationalize unemployment compensation that replaces a fraction of a workers earnings on his or her previous job 987 A onespell model 991 repeatedly over time V u V makes the continuation value remain constant during the entire spell of unemployment Equation 2327a determines c and equation 2327b determines a both as functions of the promised V That V u V then implies that c and a are held constant during the unemploy ment spell Thus the unemployed workers consumption c and search effort a are both fully smoothed during the unemployment spell But the workers consumption is not smoothed across states of employment and unemployment unless V V e 2323 The incentive problem The preceding efficient insurance scheme requires that the insurance agency con trol both c and a It will not do for the insurance agency simply to announce c and then allow the worker to choose a Here is why The agency delivers a value V u higher than the autarky value Vaut by doing two things It increases the unemployed workers consumption c and decreases his search effort a But the prescribed search effort is higher than what the worker would choose if he were to be guaranteed consumption level c while he remains unemployed This follows from equations 2327a and 2327b and the fact that the insurance scheme is costly CV u 0 which imply βpa1 V e V u But look at the workers firstorder condition 2324 under autarky It implies that if search effort a 0 then βpa1 V e V u which is inconsistent with the preceding inequality βpa1 V e V u that prevails when a 0 under the social insurance arrangement If he were free to choose a the worker would therefore want to fulfill 2324 either at equality so long as a 0 or by setting a 0 otherwise Starting from the a associated with the social insur ance scheme he would establish the desired equality in 2324 by lowering a thereby decreasing the term βpa1 which also lowers V e V u when the value of being unemployed V u increases If an equality can be established be fore a reaches zero this would be the workers preferred search effort otherwise the worker would find it optimal to accept the insurance payment set a 0 and never work again Thus since the worker does not take the cost of the in surance scheme into account he would choose a search effort below the socially optimal one The efficient contract exploits the agencys ability to control both the unemployed workers consumption and his search effort A onespell model 993 η pa pa V e V u 2328b CV u θ η pa 1 pa 2328c where the second equality in equation 2328b follows from strict equality of the incentive constraint 2324 when a 0 As long as the insurance scheme is associated with costs so that CV u 0 firstorder condition 2328b implies that the multiplier η is strictly positive The firstorder condition 2328c and the envelope condition CV θ together allow us to conclude that CV u CV Convexity of C then implies that V u V After we have also used equation 2328a it follows that in order to provide the proper incentives the consumption of the unemployed worker must decrease as the duration of the unemployment spell lengthens It also follows from 2324 at equality that search effort a rises as V u falls ie it rises with the duration of unemployment The duration dependence of benefits is designed to provide incentives to search To see this from 2328c notice how the conclusion that consumption falls with the duration of unemployment depends on the assumption that more search effort raises the prospect of finding a job ie that pa 0 If pa 0 then 2328c and the strict convexity of C imply that V u V Thus when pa 0 there is no reason for the planner to make consumption fall with the duration of unemployment 2325 Computed example For parameters chosen by Hopenhayn and Nicolini Figure 2321 displays the replacement ratio cw as a function of the duration of the unemployment spell4 This schedule was computed by finding the optimal policy functions V u t1 fV u t ct gV u t and iterating on them starting from some initial V u 0 Vaut where Vaut is the autarky level for an unemployed worker Notice how the replacement ratio 4 This figure was computed using the Matlab programs hugom hugo1am hugofoc1m valhugom These are available in the subdirectory hugo which contains a readme file These programs were composed by various members of Economics 233 at Stanford in 1998 especially Eva Nagypal Laura Veldkamp and Chao Wei 994 Optimal Unemployment Insurance 5 10 15 20 25 30 35 40 45 50 0 02 04 06 08 cw 0 5 10 15 20 25 30 35 40 45 50 0 50 100 150 200 250 a duration Figure 2321 Top panel replacement ratio cw as a func tion of duration of unemployment in the ShavellWeiss model Bottom panel effort a as a function of duration declines with duration Figure 2321 sets V u 0 at 16942 a number that has to be interpreted in the context of Hopenhayn and Nicolinis parameter settings We computed these numbers using the parametric version studied by Hopen hayn and Nicolini5 Hopenhayn and Nicolini chose parameterizations and pa rameters as follows They interpreted one period as one week which led them to set β 999 They took uc c1σ 1σ and set σ 5 They set the wage w 100 and specified the hazard function to be pa 1 expra with r chosen to give a hazard rate pa 1 where a is the optimal search effort under autarky To compute the numbers in Figure 2321 we used these same settings 5 In section 473 we described a computational strategy of iterating to convergence on the Bellman equation 2325 subject to expressions 2326 at equality and 2324 996 Optimal Unemployment Insurance 2327 Interpretations The substantial downward slope in the replacement ratio in Figure 2321 comes entirely from the incentive constraints facing the planner We saw earlier that without private information the planner would smooth consumption over the unemployment spell by keeping the replacement ratio constant In the situation depicted in Figure 2321 the planner cant observe the workers search effort and therefore makes the replacement ratio fall and search effort rise as the duration of unemployment increases especially early in an unemployment spell There is a carrotandstick aspect to the replacement rate and search effort schedules the carrot occurs in the forms of high compensation and low search effort early in an unemployment spell The stick occurs in the low compensation and high effort later in the spell We shall see this carrotandstick feature in some of the credible government policies analyzed in chapters 24 25 and 26 The planner offers declining benefits and asks for increased search effort as the duration of an unemployment spell rises in order to provide unemployed workers with proper incentives not to punish an unlucky worker who has been unemployed for a long time The planner believes that a worker who has been unemployed a long time is unlucky not that he has done anything wrong ie not lived up to the contract Indeed the contract is designed to induce the unemployed workers to search in the way the planner expects The falling con sumption and rising search effort of the unlucky ones with long unemployment spells are simply the prices that have to be paid for the common good of pro viding proper incentives 2328 Extension an onthejob tax Hopenhayn and Nicolini allow the planner to tax the worker after he becomes employed and they let the tax depend on the duration of unemployment Giving the planner this additional instrument substantially decreases the rate at which the replacement ratio falls during a spell of unemployment Instead the planner makes use of a more powerful tool a permanent bonus or tax after the worker becomes employed Because it endures this tax or bonus is especially potent when the discount factor is high In exercise 232 we ask the reader to set up the functional equation for Hopenhayn and Nicolinis model A multiplespell model with lifetime contracts 997 2329 Extension intermittent unemployment spells In Hopenhayn and Nicolinis model employment is an absorbing state and there are no incentive problems after a job is found There are not multiple spells of unemployment Wang and Williamson 1996 built a model in which there can be multiple unemployment spells and in which there is also an incentive problem on the job As in Hopenhayn and Nicolinis model search effort affects the probability of finding a job In addition while on a job effort affects the probability that the job ends and that the worker becomes unemployed again Each job pays the same wage In Wang and Williamsons setup the promised value keeps track of the duration and number of spells of employment as well as of the number and duration of spells of unemployment One contract transcends employment and unemployment 233 A multiplespell model with lifetime contracts Rui Zhao 2001 modifies and extends features of Wang and Williamsons model In her model effort on the job affects output as well as the probability that the job will end In Zhaos model jobs randomly end recurrently returning a worker to the state of unemployment The probability that a job ends depends directly or indirectly on the effort that workers expend on the job A planner observes the workers output and employment status but never his effort and wants to insure the worker Using recursive methods Zhao designs a history dependent assignment of unemployment benefits if unemployed and wages if employed that balance a planners desire to insure the worker with the need to provide incentives to supply effort in work and search The planner uses history dependence to tie compensation while unemployed or employed to earlier outcomes that partially inform the planner about the workers efforts while employed or unemployed These intertemporal tieins give rise to what Zhao interprets broadly as a replacement rate feature that we seem to observe in unemployment compensation systems Exercises 1005 234 Concluding remarks The models that we have studied in this chapter isolate the worker from capital markets so that the worker cannot transfer consumption across time or states except by adhering to the contract offered by the planner If the worker in the models of this chapter were allowed to save or issue a riskfree asset bearing a gross oneperiod rate of return approaching β1 it would interfere substantially with the planners ability to provide incentives by manipulating the workers continuation value in response to observed current outcomes In particular forces identical to those analyzed in the Cole and Kocherlakota setup that we analyzed at length in chapter 21 would circumscribe the planners ability to supply insurance In the context of unemployment insurance models like that of this chapter this point has been studied in detail in papers by Ivan Werning 2002 and Kocherlakota 2004 Pavoni and Violante 2007 substantially extended models like those in this chapter to perform positive and normative analysis of a sequence of govern ment programs that try efficiently to provide insurance training and proper incentives for unemployed and undertrained workers to reenter employment Exercises Exercise 231 Optimal unemployment compensation a Write a program to compute the autarky solution and use it to reproduce Hopenhayn and Nicolinis calibration of r as described in text b Use your calibration from part a Write a program to compute the optimum value function CV for the insurance design problem with incomplete infor mation Use the program to form versions of Hopenhayn and Nicolinis table 1 column 4 for three different initial values of V chosen by you to belong to the set Vaut V e Exercise 232 Taxation after employment Show how the functional equation 2325 2326 would be modified if the planner were permitted to tax workers after they became employed 1008 Optimal Unemployment Insurance the worker finds a job he receives a fixed wage w forever sets a 0 and has continuation utility Ve uw 1β The consumption good is not storable and workers can neither borrow nor lend The unemployment agency can borrow and lend at a constant oneperiod riskfree gross interest rate of R β1 The unemployment agency cannot observe the workers effort level Subproblem A a Let V be the value of 1 that the unemployment agency has promised an unemployed worker at the start of a period before he has made his search deci sion Let CV be the minimum cost to the unemployment insurance agency of delivering promised value V Assume that the unemployment insurance agency wants the unemployed worker to set at a for as long as he is unemployed ie it wants to promote high search effort Formulate a Bellman equation for CV being careful to specify any promisekeeping and incentive constraints Assume that there are no participation constraints the unemployed worker must participate in the program b Show that if the incentive constraint binds then the unemployment agency offers the worker benefits that decline as the duration of unemployment grows c Now alter assumption 2 so that πa π0 Do benefits still decline with increases in the duration of unemployment Explain Subproblem B d Now assume that the unemployment insurance agency can tax the worker after he has found a job so that his continuation utility upon entering a state of employment is uwτ 1β where τ is a tax that is permitted to depend on the duration of the unemployment spell Defining V as above formulate the Bellman equation for CV e Show how the tax τ responds to the duration of unemployment Exercise 236 Partially observed search effort Consider the following modification of a model of Hopenhayn and Nicolini An insurance agency wants to insure an infinitely lived unemployed worker against the risk that he will not find a job With probability pa an unemployed worker who searches with effort a this period will find a job that earns wage w in consumption units per period That job will start next period last forever Exercises 1009 and the worker will never quit it With probability 1 pa he will find himself unemployed again at the beginning of next period We assume that pa is an increasing and strictly concave and twice differentiable function of a with pa 0 1 for a 0 and p0 0 The insurance agency is the workers only source of consumption there is no storage or saving available to the worker The worker values consumption according to a twice continuously differentiable and strictly concave utility function uc where u0 is finite While unemployed the workers utility is uca when he is employed it is uw no effort a need be applied when he is working With exogenous probability d 0 1 the insurance agency observes the search effort of a worker who searched last period but did not find a job With probability 1 d the insurance agency does not observe the lastperiod search intensity of an unemployed worker who was not successful in finding a job period Let V be the expected discounted utility of an unemployed worker who is searching for work this period Let CV be the minimum cost to the unem ployment insurance agency of delivering V to the unemployed worker a Formulate a Bellman equation for CV b Get as far as you can in analyzing how the unemployment compensation contract offered to the worker depends on the duration of unemployment and the history of observed search efforts that are detected by the UI agency Hint you might want to allow the continuation value when unemployed to depend on last periods search effort when it is observed Chapter 24 Credible Government Policies I 241 Introduction Kydland and Prescott 1977 opened the modern discussion of time consistency in macroeconomics with some examples that show how outcomes differ in other wise identical economies when the assumptions about the timing of government policy choices are altered1 In particular they compared a timing protocol in which a government chooses its possibly historycontingent policies once and for all at the beginning of time with one in which the government chooses sequen tially Because outcomes are worse when the government chooses sequentially Kydland and Prescotts examples illustrate the superiority of the onceandfor all choice timing protocol for the government Subsequent work on time consistency focused on how a reputation can im prove outcomes when a government chooses sequentially2 The issue is whether constraints confronting the government and private sector expectations can be arranged so that a government adheres to an expected pattern of behavior be cause it would worsen its reputation if it did not A folk theorem from game theory states that if there is no discounting of future payoffs then many firstperiod payoffs can be sustained as equilibria of a repeated version of a game A main purpose of this chapter is to study how discounting of future payoffs affects the set of outcomes that are attainable with a reputational mechanism Modern formulations of reputational models of government policy use and extend ideas from dynamic programming Each period a government faces choices whose consequences include a firstperiod return and a reputation to pass on to next period Under rational expectations any reputation that the 1 Consider two extensiveform versions of the battle of the sexes game described by Kreps 1990 one in which the man chooses first the other in which the woman chooses first Backward induction recovers different outcomes in these two different games Though they share the same choice sets and payoffs these are different games 2 Barro and Gordon 1983a 1983b are early contributors to this literature See Kenneth Rogoff 1989 for a survey 1011 1012 Credible Government Policies I government carries into next period must be one that it will want to confirm We shall study the set of possible values that the government can attain with reputations that it could conceivably want to confirm This chapter and chapter 25 apply an apparatus of Abreu Pearce and Stac chetti 1986 1990 APS to reputational equilibria in a class of macroeconomic models APS use ideas from dynamic programming3 Their work exploits the insight that it is more convenient to work with the set of continuation values associated with equilibrium strategies than it is to work directly with the set of equilibrium strategies We use an economic model like those of Chari Kehoe and Prescott 1989 and Stokey 1989 1991 to exhibit what Chari and Ke hoe 1990 call sustainable government policies and what Stokey calls credible public policies The literature on sustainable or credible government policies in macroeconomics adapts ideas from the literature on repeated games so that they can be applied in contexts in which a single agent a government behaves strategically and in which all other agents behavior can be summarized in terms of a system of expectations about government actions together with competitive equilibrium outcomes that respond to the governments actions4 2411 Diverse sources of history dependence The theory of credible government policy uses particular kinds of history depen dence to render credible a sequence of actions chosen by a sequence of policy makers Here credible means an action that the public rationally expects the government to take because it thinks it is in the governments interest to do so Hence a credible action is one that the government wants to implement By way of contrast in chapter 19 we encountered a distinct source of history dependence in the policy of a Ramsey planner or Stackelberg leader There history dependence came from the requirement that it is necessary to account 3 This chapter closely follows Stacchetti 1991 who applies Abreu Pearce and Stacchetti 1986 1990 to a more general class of models than that treated here Stacchetti also stud ies a class of setups in which the private sector observes only a noiseridden signal of the governments actions 4 For descriptions of theories of credible government policy see Chari and Kehoe 1990 Stokey 1989 1991 Rogoff 1989 and Chari Kehoe and Prescott 1989 For applications of the framework of Abreu Pearce and Stacchetti see Chang 1998 and Phelan and Stacchetti 1999 Oneperiod economy 1013 for constraints that dynamic aspects of private sector behavior put on the time t action of a Ramsey planner or Stackelberg leader who at time 0 makes once andforall choices of intertemporal sequences History dependence came from the requirement that the Ramsey planners time t action must confirm private sector expectations that the Ramsey planner had chosen at time 0 partly to influence private sector outcomes in periods 0 t 1 In settings in which private agents face genuinely dynamic decision problems having their own endogenous state variables like various forms of physical and human capital both sources of history dependence influence a credible policy It can be subtle to disentangle the economic forces contributing to history de pendence in government policies in such settings However for special examples that deprive private agents decision problems of any natural state variables we can isolate the source of history dependence coming from the requirement that a government policy must be credible We consider only such examples in this chapter for the avowed purpose of isolating the source of history de pendence coming from credibility considerations and distinguishing it from the chapter 19 source that instead comes from the need to respect substantial dy namics coming from equilibrium private sector behavior Having isolated one source of history dependence in chapter 19 and another in the present chapter we proceed in chapter 25 to activate both sources of history dependence and then to seek a recursive representation for a credible government policy in that more comprehensive setting 242 Oneperiod economy There is a continuum of households each of which chooses an action ξ X A government chooses an action y Y The sets X and Y are compact The average level of ξ across households is denoted x X The utility of a particular household is uξ x y when it chooses ξ when the average households choice is x and when the government chooses y The payoff function uξ x y is strictly concave and continuously differentiable in ξ and y5 5 The discretechoice examples given later violate some of these assumptions in non essential ways 1014 Credible Government Policies I 2421 Competitive equilibrium For given levels of y and x the representative household faces the problem maxξX uξ x y Let the maximizer be a function ξ fx y When a household believes that the governments choice is y and that the average level of other households choices is x it acts to set ξ fx y Because all house holds are alike this fact implies that the actual level of x is fx y For the representative households expectations about the average to be consistent with the average outcome we require that ξ x or x fx y This makes the representative agent representative We use the following6 Definition 1 A competitive equilibrium or a rational expectations equilibrium is an x X that satisfies x fx y A competitive equilibrium satisfies ux x y maxξX uξ x y For each y Y let x hy denote the corresponding competitive equilib rium We adopt Definition 2 The set of competitive equilibria is C x y ux x y maxξX uξ x y or equivalently C x y x hy 2422 Ramsey problem The following timing of actions underlies a Ramsey plan First the government selects a y Y Then knowing the governments choice of y the aggregate of households responds with a competitive equilibrium The government evaluates policies y Y with the payoff function ux x y that is the government is benevolent In choosing y the government has to forecast how the economy will respond We assume that the government correctly forecasts that the economy will re spond to y with a competitive equilibrium x hy We use these definitions Definition 3 The Ramsey problem is maxyY uhy hy y or equivalently maxxyC ux x y Definition 4 The policy that attains the maximum for the Ramsey problem is denoted yR Let xR hyR Then yR xR is called the Ramsey outcome or Ramsey plan 6 See the definition of a rational expectations equilibrium in chapter 7 Oneperiod economy 1015 Two remarks about the Ramsey problem are in order First the Ramsey outcome is typically inferior to the dictatorial outcome that solves the unre stricted problem maxxX yY ux x y because the restriction x y C is in general binding Second the timing of actions is important The Ramsey prob lem assumes that the government chooses first and must stick with its choice regardless of how private agents subsequently choose x X If the government were granted the opportunity to reconsider its plan after households had chosen x xR the government would in general want to de viate from yR because often there exists an α yR for which uxR xR α uxR xR yR The time consistency problem is the incentive the government would have to deviate from the Ramsey plan if it were allowed to react after households had set x xR In this oneperiod setting to support the Ramsey plan requires a timing protocol that forces the government to choose first 2423 Nash equilibrium Consider an alternative timing protocol that confronts households with a fore casting problem because the government chooses after or simultaneously with the households Assume that households forecast that given x the government will set y to solve maxyY ux x y We use Definition 5 A Nash equilibrium xN yN satisfies 1 xN yN C 2 Given xN uxN xN yN maxηY uxN xN η Condition 1 asserts that xN hyN or that the economy responds to yN with a competitive equilibrium Thus condition 1 says that given xN yN each individual household wants to set ξ xN that is the representative household has no incentive to deviate from xN Condition 2 asserts that given xN the government chooses a policy yN from which it has no incentive to deviate7 7 Much of the language of this chapter is borrowed from game theory but the object under study is not a game because we do not specify all of the objects that formally define a game In particular we do not specify the payoffs to all agents for all feasible choices We only specify the payoffs uξ x y where each private agent chooses the same value of ξ 1016 Credible Government Policies I We can use the solution of the problem in condition 2 to define the govern ments best response function y Hx The definition of a Nash equilibrium can be phrased as a pair x y C such that y Hx There are two timings of choices for which a Nash equilibrium is a natural equilibrium concept One is where households choose first forecasting that the government will respond to the aggregate outcome x by setting y Hx An other is where the government and households choose simultaneously in which case a Nash equilibrium xN yN depicts a situation in which everyone has ra tional expectations given that each household expects the aggregate variables to be xN yN each household responds in a way to make x xN and given that the government expects that x xN it responds by setting y yN We let the values attained by the government under the Nash and Ramsey outcomes respectively be denoted vN uxN xN yN and vR uxR xR yR Because of the additional constraint embedded in the Nash equilibrium out comes are ordered according to vN max xyC yHx ux x y max xyC ux x y vR 243 Nash and Ramsey outcomes To illustrate these concepts we consider two examples taxation within a fully specified economy and a blackbox model with discrete choice sets 2431 Taxation example Each of a continuum of households has preferences over leisure ℓ private con sumption c and per capita government expenditures g The oneperiod utility function is Uℓ c g ℓ logα c logα g α 0 12 Each household is endowed with one unit of time that can be devoted to leisure or labor The production technology is linear in labor and the economys resource constraint is c g 1 ℓ 1018 Credible Government Policies I 0 02 04 06 08 1 16 14 12 1 08 06 04 02 Tax rate Welfare Ramsey Nash Nash Deviation from Ramsey Unconstrained optimum Figure 2431 Welfare outcomes in the taxation example The solid curve depicts the welfare associated with the set of competi tive equilibria W cτ The set of Nash equilibria is the horizontal portion of the solid curve and the equilibrium at τ 12 The Ramsey outcome is marked with an asterisk The time inconsis tency problem is indicated with the triangle showing the outcome if the government were able to reset τ after households had chosen the Ramsey labor supply The dashed line describes the welfare level at the unconstrained optimum W d The graph sets α 03 The objects of the general setup in the preceding section can be mapped into the present taxation example as follows ξ ℓ x ℓ X 0 1 y τ Y 0 1 uξ x y ξlogα1y1ξlogαy1x fx y ℓy hy ℓy and Hx 12 if x 1 and Hx 0 1 if x 1 Nash and Ramsey outcomes 1019 2432 Blackbox example with discrete choice sets Consider a black box example with X xL xH and Y yL yH in which ux x y assume the values given in Table 2431 Assume that values of uξ x y for ξ x are such that the values with asterisks for ξ x are competitive equilibria In particular we might assume that uξ xi yj 0 when ξ xi and i j uξ xi yj 20 when ξ xi and i j These payoffs imply that uxL xL yL uxH xL yL ie 3 0 and uxH xH yH uxL xH yH ie 10 0 Therefore xL xL yL and xH xH yH are competitive equilibria Also uxH xH yL uxL xH yL ie 12 20 so the dictatorial outcome cannot be supported as a competitive equilibrium xL xH yL 3 12 yH 1 10 Table 2431 Oneperiod payoffs uxi xi yj denotes x y C the Ramsey outcome is xH yH and the Nash equilibrium outcome is xL yL Figure 2432 depicts a timing of choices that supports the Ramsey outcome for this example The government chooses first then walks away The Ramsey outcome xH yH is the competitive equilibrium yielding the highest value of ux x y Figure 2433 diagrams a timing of choices that supports the Nash equilib rium Recall that by definition every Nash equilibrium outcome has to be a competitive equilibrium outcome We denote competitive equilibrium pairs x y with asterisks The government sector chooses after knowing that the pri vate sector has set x and chooses y to maximize ux x y With this timing if the private sector chooses x xH the government has an incentive to set y yL a setting of y that does not support xH as a Nash equilibrium The 1020 Credible Government Policies I x h y L L x h y H H yH yL G P P 10 3 Figure 2432 Timing of choices that supports Ramsey outcome Here P and G denote nodes at which the public and the gov ernment respectively choose The government has a commitment technology that binds it to choose first The government chooses the y Y that maximizes uhy hy y where x hy is the function mapping government actions into equilibrium values of x unique Nash equilibrium is xL yL which gives a lower utility ux x y than does the competitive equilibrium xH yH 244 Reputational mechanisms general idea In a finitely repeated economy the government will certainly behave opportunis tically the last period implying that nothing better than a Nash outcome can be supported the last period In a finite horizon economy with a unique Nash equi librium we wont be able to sustain anything better than a Nash equilibrium outcome in any earlier period8 8 If there are multiple Nash equilibria it is sometimes possible to sustain a betterthanNash equilibrium outcome for a while in a finite horizon economy See exercise 241 which uses an idea of Benoit and Krishna 1985 Reputational mechanisms general idea 1021 yH yH y L yL L x H x 10 12 1 3 P G G Figure 2433 Timing of actions in a Nash equilibrium in which the private sector acts first Here G denotes a node at which the government chooses and P denotes a node at which the public chooses The private sector sets x X before knowing the govern ments setting of y Y Competitive equilibrium pairs x y are denoted with an asterisk The unique Nash equilibrium is xL yL We want to study situations in which a government might sustain a Ram sey outcome Therefore we shall study economies repeated an infinite number of times Here a system of historydependent expectations interpretable as a government reputation might be arranged to sustain something better than rep etition of a Nash outcome We strive to set things up so that the government so dearly wants to confirm a good reputation that it will not submit to the temptation to behave opportunistically A reputation is said to be sustainable if it is always in the governments interests to confirm it A state variable that encodes a reputation is both backward looking and forward looking It is backward looking because it remembers salient features of past behavior It is forwardlooking behavior because it measures something about what private agents expect the government to do in the future We are about to study the ingenious machinery of Abreu Pearce and Stacchetti that 1022 Credible Government Policies I astutely exploits these aspects of a reputational variable by recognizing that the ideal reputational state variable is a promised value 2441 Dynamic programming squared A sustainable reputation for the government is one that a the public having rational expectations wants to believe and b the government wants to con firm Rather than finding all possible sustainable reputations Abreu Pearce and Stacchetti henceforth APS 1986 1990 used dynamic programming to characterize all values for the government that are attainable with sustainable reputations This section briefly describes their main ideas while later sections fill in many details First we need some language A strategy profile is a pair of plans one each for the private sector and the government The time t components of the pair of plans maps the observed history of the economy into currentperiod outcomes x y A subgame perfect equilibrium SPE strategy profile has a current period outcome being a competitive equilibrium xt yt whose yt component the government would want to confirm at each t 1 and for every possible history of the economy To characterize SPEs or at least a very interesting subset of them the method of APS is to formulate a Bellman equation that describes the value to the government of a strategy profile and that portrays the idea that the government wants to confirm the private sectors beliefs about y For each t 1 the governments strategy describes its firstperiod action y Y which because the public had expected it determines an associated firstperiod competitive equilibrium x y C Furthermore the strategy implies two continuation values for the government at the beginning of next period a continuation value v1 if it carries out the firstperiod choice y and another continuation value v2 if for any reason the government deviates from the expected firstperiod choice y Associated with the governments strategy is a current value v that obeys the Bellman equation v 1 δux x y δv1 2441a where δ 0 1 is a discount factor x y C v1 is a continuation value awarded for confirming the private sectors expectation that the government 1024 Credible Government Policies I 0 02 04 06 08 1 08 078 076 074 072 07 068 066 064 062 06 Tax rate Value Figure 2441 Mapping of continuation values v1 v2 into val ues v in the infinitely repeated version of the taxation exam ple The solid curve depicts v 1 δuℓτ ℓτ τ δv1 The dashed curve is the right side of the incentive constraint v 1 δuℓτ ℓτ Hℓτ δv2 where H is the govern ments best response function The part of the solid curve that is above the dashed curve shows competitive equilibrium values that are sustainable for continuation values v1 v2 The parameteri zation is α 03 and δ 08 and the continuation values are set as v1 v2 06 063 of the chapter describes details of APSs formulation as applied in our setting We shall see why APS want to get their hands on the entire set of equilibrium values Reputational mechanisms general idea 1025 2442 Etymology of dynamic programming squared Why do we call it dynamic programming squared There are two reasons 1 The construction works by mapping two continuation values into one in contrast to ordinary dynamic programming which maps one continuation value tomorrow into one value function today 2 A continuation value plays a double role one as a promised value that sum marizes expectations of the rewards associated with future outcomes an other as a state variable that summarizes the history of past outcomes In the present setting a subgame perfect equilibrium strategy profile can be represented recursively in terms of an initial value v1 IR and the following 3tuple of functions xt zhvt yt zgvt vt1 Vvt xt ηt ηt Y the first two of which map a promised value into a private sector decision and a government action while the third maps a promised value and an action pair into a promised value to carry into tomorrow By iterating these functions we can deduce that the triple of functions zh zg V induces a strategy profile that maps histories of outcomes into sequences of outcomes The capacity to represent a subgame perfect equilibrium recursively affords immense simplifications in terms of the number of functions we must carry 1028 Credible Government Policies I It might be helpful to write out a few terms for s 0 1 σx1y11 σ2x1 y1 ν1 η1 σx1y12ν1 η1 σ3x1 ν1 y1 η1 ν2 η2 σx1y13ν1 ν2 η1 η2 σ4x1 ν1 ν2 y1 η1 η2 ν3 η3 More generally define the continuation strategy σxtyt1 σt1xt yt σxtyts1νs ηs σts1 x1 xt ν1 νs y1 yt η1 ηs for all s 1 and all νs ηs Xs Y s Here σxtyts1 νs ηs is the induced strategy pair to apply in the s 1th period of the continuation economy We attain this strategy by shifting the original strategy forward t periods and evaluating it at history x1 xt ν1 νs y1 yt η1 ηs for the original economy In terms of the continuation strategy σx1y1 from equation 2452 we know that Vgσ can be represented as Vgσ 1 δrx1 y1 δVgσx1y1 2453 Representation 2453 decomposes the value to the government of strategy profile σ into a oneperiod return and the continuation value Vgσx1y1 as sociated with the continuation strategy σx1y1 Any sequence x y in equation 2452 or any strategy profile σ in equation 2453 can be assigned a value We want a notion of an equilibrium strategy profile Subgame perfect equilibrium SPE 1029 246 Subgame perfect equilibrium SPE Definition 6 A strategy profile σ σh σg is a subgame perfect equilibrium SPE of the infinitely repeated economy if for each t 1 and each history xt1 yt1 Xt1 Y t1 a The private sector outcome xt σh t xt1 yt1 is consistent with compet itive equilibrium when yt σg t xt1 yt1 b For each possible government action η Y 1 δrxt yt δVgσxtyt 1 δ rxt η δVgσxtyt1η Requirement a says two things It attributes a theory of forecasting govern ment behavior to members of the public in particular that they use the time t component σg t of the governments strategy and information available at the end of period t 1 to forecast the governments decision at t Condition a also asserts that a competitive equilibrium appropriate to the publics forecast value for yt is the outcome at time t Requirement b says that at each point in time and following each history the government has no incentive to deviate from the firstperiod action called for by its strategy σg that is the government always wants to choose as the public expects Notice how in condition b the government contemplates setting its time t choice ηt at something other than the value forecast by the public but confronts consequences that deter it from choosing an ηt that fails to confirm the publics expectations of it In section 2415 well discuss the following question who chooses σg the government or the public This question arises naturally because σg is both the governments sequence of policy functions and the private sectors rule for forecasting government behavior Condition b of definition 6 says that the gov ernment chooses to confirm the publics forecasts Definition 6 implies that for each t 2 and each xt1 yt1 Xt1Y t1 the continuation strategy σxt1yt1 is itself an SPE We state this formally for t 2 Proposition 1 Assume that σ is an SPE Then for all ν η X Y σνη is an SPE Proof Write out requirements a and b that Definition 6 asserts that the continuation strategy σνη must satisfy to qualify as an SPE In particular 1030 Credible Government Policies I for all s 1 and for all xs1 ys1 Xs1 Y s1 we require xs ys C 2461 where xs σhνηxs1 ys1 ys σgνηxs1 ys1 We also require that for all η Y 1 δrxs ys δVgσηxsνys 1 δrxs η δVgσνxsηys1η 2462 Notice that requirements a and b of Definition 6 for t 2 3 imply expres sions 2461 and 2462 for s 1 2 The statement that σνη is an SPE for all ν η X Y ensures that σ is almost an SPE If we know that σνη is an SPE for all ν η X Y we must only add two requirements to ensure that σ is an SPE first that the t 1 outcome pair x1 y1 is a competitive equilibrium and second that the governments choice of y1 satisfies the time 1 version of the incentive constraint b in Definition 6 This reasoning leads to the following lemma that is at the heart of the APS analysis Lemma Consider a strategy profile σ and let the associated firstperiod out come be given by x σh 1 y σg 1 The profile σ is an SPE if and only if 1 for each ν η X Y σνη is an SPE 2 x y is a competitive equilibrium 3 η Y 1 δ rx y δ Vgσxy 1 δ rx η δVgσxη Proof First prove the if part Property a of the lemma and properties 2461 and 2462 of Proposition 1 show that requirements a and b of Defi nition 6 are satisfied for t 2 Properties 2 and 3 of the lemma imply that requirements a and b of Definition 6 hold for t 1 Second prove the only if part Part 1 of the lemma follows from Propo sition 1 Parts 2 and 3 of the lemma follow from requirements a and b of Definition 6 for t 1 The lemma is very important because it characterizes SPEs in terms of a firstperiod competitive equilibrium outcome pair x y and a pair of continu ation values a value Vgσxy to be awarded to the government next period Examples of SPE 1031 if it adheres to the y component of the firstperiod pair x y and a value Vgσxη η y to be awarded to the government if it deviates from the ex pected y component Each of these values has to be selected from a set of values Vgσ that are associated with some SPE σ 247 Examples of SPE 2471 Infinite repetition of oneperiod Nash equilibrium It is easy to verify that the following strategy profile σN σh σg forms an SPE σh 1 xN σg 1 yN and for t 2 σh t xN t xt1 yt1 σg t yN t xt1 yt1 These strategies instruct the households and the government to choose the static Nash equilibrium outcomes for all periods for all histories Evidently for these strategies VgσN vN rxN yN Furthermore for these strategies the continuation value Vgσxtyt1η vN for all outcomes η Y These strategies satisfy requirement a of Definition 6 because xN yN is a compet itive equilibrium The strategies satisfy requirement b because rxN yN maxyY rxN y and because the continuation value Vgσ vN is indepen dent of the action chosen by the government in the first period In this SPE σN t σh t σg t xN yN for all t and for all xt1 yt1 and value VgσN and continuation values VgσNxtyt for each history xt yt equal vN It is useful to think about this SPE in terms of the lemma To verify that σN is a SPE we work with the firstperiod outcome pair xN yN and the pair of values VgσxNyN vN Vgσxη vN where vN rxN yN With these settings we can verify that xN yN and vN satisfy requirements 1 2 and 3 of the lemma Examples of SPE 1033 This construction uses the following objects11 1 A proposed firstperiod competitive equilibrium x y C 2 An SPE σ2 with value Vgσ2 that is used as the continuation strategy in the event that the firstperiod outcome does not equal x y so that σxy σ2 if x y x y In the example σ2 σN and Vgσ2 vN 3 An SPE σ1 with value Vgσ1 used to define the continuation value to be assigned after firstperiod outcome x y and an associated continuation strategy σxy σ1 In the example σ1 σ which is defined recursively and selfreferentially via equation 2471 4 A candidate for a new SPE σ and a corresponding value Vgσ In the example Vgσ rx y Note how we have used the lemma in verifying that σ is an SPE We start with the SPE σN with associated value vN We guess a firstperiod outcome pair x y and a value v for a new SPE where v rx y Then we verify requirements 2 and 3 of the lemma with v vN as continuation values and x y as firstperiod outcomes 2473 When reversion to Nash is not bad enough For discount factors δ sufficiently close to one it is typically possible to support repetition of the Ramsey outcome xR yR with a section 2472 trigger strategy of form 2471 This finding conforms with a version of the folk theorem about repeated games However there exist discount factors δ so small that the continuation value associated with infinite repetition of the oneperiod Nash outcome is not low enough to support repetition of Ramsey Anticipating that it will revert to repetition of Nash after a deviation then can at best support a lower value for the government that than that associated with repetition of Ramsey outcome although perhaps its is better than repeating the Nash outcome In this circumstance is there a better SPE than can be supported by an ticipating version to repetition of the oneperiod Nash outcome To support something better evidently requires finding an SPE that has a value worse than that associated with repetition of the oneperiod Nash outcome Following APS we shall soon see that the best and worst equilibrium values are linked 11 In the example objects 3 and 4 are equated 1034 Credible Government Policies I 248 Values of all SPEs The role played by the lemma in analyzing our two examples hints at the central role that it plays in methods that APS developed for describing and computing values for all the subgame perfect equilibria APS build on the way that the lemma characterizes SPE values in terms of a firstperiod competitive equilib rium outcome along with a pair of continuation values each element of which is itself a value associated with some SPE The lemma directs APSs attention away from a set of strategy profiles σ and toward a set of values Vgσ as sociated with those profiles They define the set V of values associated with subgame perfect equilibria V Vg σ σ is an SPE Evidently V IR From the lemma for a given competitive equilibrium x y C there exists an SPE σ for which x σh 1 y σg 1 if and only if there exist two values v1 v2 V V such that 1 δ rx y δv1 1 δ rx η δv2 η Y 2481 Let σ1 and σ2 be subgame perfect equilibria for which v1 Vgσ1 v2 Vgσ2 The SPE σ that supports x y σh 1 σg 1 is completed by specifying the continuation strategies σxy σ1 and σνη σ2 if ν η x y This construction uses two continuation values v1 v2 V V to create an SPE σ with value v V given by v 1 δ rx y δv1 Thus the construction maps pairs of continuation values v1 v2 into a strategy profile σ with firstperiod competitive equilibrium outcome x y and a value v Vgσ APS characterize subgame perfect equilibria by studying a mapping from pairs of continuation values v1 v2 V V into values v V They use the following definitions Definition 7 Let W IR A 4tuple x y w1 w2 is said to be admissible with respect to W if x y C w1 w2 W W and 1 δ rx y δw1 1 δ rx η δw2 η Y 2482 1036 Credible Government Policies I Just as the right side of 2483 takes a candidate continuation value Q for tomorrow and maps it into a value T Q for today APS define a mapping BW that by considering only admissible 4tuples maps a set of values W tomorrow into a new set BW of values today Thus APS use admissible 4tuples to map candidate continuation values tomorrow into new candidate values today In the next section well iterate to convergence on BW but as well see it wont work to start from just any initial set W We have to start from a big enough set 249 APS machinery Definition 8 For each set W IR let BW be the set of possible values w 1 δ rx y δw1 associated with admissible tuples x y w1 w2 Think of W as a set of potential continuation values and BW as the set of values that they support From the definition of admissibility it immediately follows that the operator B is monotone Property monotonicity of B If W W R then BW BW Proof It can be verified directly from the definition of admissible 4tuples that if w BW then w BW simply use the w1 w2 pair that supports w BW to support w BW It can also be verified that B maps compact sets W into compact sets BW The selfsupporting character of subgame perfect equilibria is referred to in the following definition Definition 9 The set W is said to be selfgenerating if W BW Thus a set of continuation values W is said to be selfgenerating if it is contained in the set of values BW that are generated by pairs of continuation values selected from W This description makes us suspect that if a set of values is selfgenerating it must be a set of SPE values Indeed notice that by virtue of the lemma the set V of SPE values Vgσ is selfgenerating Thus we can APS machinery 1037 write V BV APS show that V is the largest selfgenerating set The key to showing this point is the following theorem12 Theorem 1 A selfgenerating set is a subset of V If W IR is bounded and selfgenerating then BW V The proof is based on forward induction and proceeds by taking a point w BW and constructing an SPE with value w Proof Assume W BW Choose an element w BW and transform it as follows into a subgame perfect equilibrium Step 1 Because w BW we know that there exist outcomes x y and values w1 and w2 that satisfy w 1 δ rx y δw1 1 δ rx η δw2 η Y x y C w1 w2 W W Set σ1 x y Step 2 Since w1 W BW there exist outcomes x y and values w1 w2 W that satisfy w1 1 δ rx y δ w1 1 δ rx η δ w2 η Y x y C Set the firstperiod outcome in period 2 the outcome to occur given that y was chosen in period 1 equal to x y that is set σxy1 x y Continuing in this way for each w BW we can create a sequence of continuation values w1 w1 w1 and a corresponding sequence of firstperiod outcomes x y x y x y At each stage in this construction policies are unimprovable which means that given the continuation values oneperiod deviations from the prescribed policies are not optimal It follows that the strategy profile is optimal By construction Vgσ w 12 The unbounded set IR the extended real line is selfgenerating but not meaningful It is selfgenerating because any value v IR can be supported if there are no limits on the continuation values It is not meaningful because most points in IR are values that cannot be attained with any strategy profile 1038 Credible Government Policies I Collecting results we know that 1 V BV by the lemma 2 If W BW then BW V by theorem 1 3 B is monotone and maps compact sets into compact sets Facts 1 and 2 imply that V BV so that the set of equilibrium values is a fixed point of B in particular the largest bounded fixed point Monotonicity of B and the fact that it maps compact sets into compact sets provides an algorithm for computing the set V namely to start with a set W0 for which V BW0 W0 and to iterate on B In more detail we use the following steps 1 Start with a set W0 w0 w0 that we know is bigger than V and for which BW0 W0 It will always work to set w0 maxxyC rx y w0 minxyC rx y 2 Compute the boundaries of the set BW0 w1 w1 The value w1 solves the problem w1 max xyC 1 δ rx y δw0 subject to 1 δ rx y δw0 1 δ rx η δw0 for all η Y The value w1 solves the problem w1 min xyC w1w2w0w02 1 δ rx y δw1 subject to 1 δ rx y δw1 1 δ rx η δw2 η Y With w0 w0 chosen as before it will be true that BW0 W0 3 Having constructed W1 BW0 W0 continue to iterate producing a decreasing sequence of compact sets Wj1 BWj Wj Iterate until the sets converge In section 2413 we will present a direct way to compute the best and worst SPE values one that evades having to iterate on the B operator 1040 Credible Government Policies I The preceding argument thus establishes Proposition 2 A subgame perfect equilibrium σ associated with v minv v V is selfenforcing 24101 The quest for something worse than repetition of Nash outcome Notice that the first subgame perfect equilibrium that we computed whose outcome was infinite repetition of the oneperiod Nash equilibrium is a self enforcing equilibrium However in general the infinite repetition of the one period Nash equilibrium is not the worst subgame perfect equilibrium This fact opens the possibility that even when reversion to Nash after a deviation is not able to support repetition of Ramsey as an SPE we might still support repetition of the Ramsey outcome by reverting to a SPE with a value worse than that associated with repetition of the Nash outcome whenever the government deviates from an expected oneperiod choice 2411 Recursive strategies This section emphasizes similarities between credible government policies and the recursive contracts appearing in chapter 21 We will study situations where the households and the governments strategies have recursive representations This approach substantially restricts the space of strategies because most history dependent strategies cannot be represented recursively Nevertheless this class of strategies excludes no equilibrium payoffs v V We use the following defi nitions Definition 11 Households and the government follow recursive strategies if there is a 3tuple of functions φ zh zg V and an initial condition v1 with the following structure v1 IR is given xt zhvt yt zgvt vt1 Vvt xt yt 24111 Examples of SPE with recursive strategies 1043 members of the same class of objects namely equilibrium values v occur on each side of expression 24114 Circularity comes with recursivity One implication of the work of APS 1986 1990 is that recursive equilib ria of form 24111 can attain all subgame perfect equilibrium values As we have seen APSs innovation was to shift the focus away from the set of equilib rium strategies and toward the set of values V attainable with subgame perfect equilibrium strategies 2412 Examples of SPE with recursive strategies Our two earlier examples of subgame perfect equilibria were already of a recur sive nature But to highlight this property we recast those SPE in the present notation for recursive strategies Equilibria are constructed by using a guess andverify technique First guess v1 zh zg V in equations 24111 then verify parts 1 2 and 3 of Definition 12 The examples parallel the historical development of the theory 1 The first example is infinite repetition of a oneperiod Nash outcome which was Kydland and Prescotts 1977 timeconsistent equilibrium 2 Barro and Gordon 1983a 1983b and Stokey 1989 used the value from infinite repetition of the Nash outcome as a continuation value to deter deviation from the Ramsey outcome For sufficiently high discount factors the continuation value associated with repetition of the Nash outcome can deter the government from deviating from infinite repetition of the Ramsey outcome This is not possible for low discount factors 3 Abreu 1988 and Stokey 1991 showed that Abreus stickand carrot strategy induces more severe consequences than repetition of the Nash outcome 1050 Credible Government Policies I 2 Solve v 1 δrhy y δv2 for continuation value v1 3 For j 1 2 continue solving vj 1 δrhy y δvj1 for the continuation values vj1 as long as vj1 v If vj1 threatens to violate this constraint at step j j then go to step 4 4 Use v as the continuation value and solve vj 1 δrhy y δv for the prescription y to be followed if promised value vj is encountered 5 Set vjs v for s 1 24142 Attaining the worst method 2 To construct another equilibrium supporting the worst SPE value follow steps 1 and 2 and follow step 3 also except that we continue solving vj 1 δrhy y δvj1 for the continuation values vj1 only so long as vj1 vN As soon as vj1 v vN we use v as both the promised value and the continuation value thereafter In terms of our recursive strategy no tation whenever v rhy y is the promised value zhv hy zgv y and vv zhv zgv v 24143 Attaining the worst method 3 Here is another subgame perfect equilibrium that supports v Proceed as in step 1 to find the initial continuation value v1 Now set all subsequent values and continuation values to v1 with associated firstperiod outcome y that solves v1 rhy y It can be checked that the incentive constraint is satisfied with v the continuation value in the event of a deviation Examples alternative ways to achieve the worst 1051 24144 Numerical example We now illustrate the concepts and arguments using the infinitely repeated version of the taxation example To make the problem of finding v nontrivial we impose an upper bound on admissible tax rates given by τ 1 α ǫ where ǫ 0 05 α Given τ Y 0 τ the model exhibits a unique Nash equilibrium with τ 05 For a sufficiently small ǫ the worst oneperiod competitive equilibrium is ℓτ τ Set α δ τ 03 08 06 Compute τ R τ N 03013 05000 vR vN v vabreu 06801 07863 09613 07370 In this numerical example Abreus stickandcarrot strategy fails to attain a value lower than the repeated Nash outcome The reason is that the upper bound on tax rates makes the least favorable oneperiod return the stick not so bad 0 5 10 15 20 25 30 1 095 09 085 08 075 07 065 06 Time Value Figure 24141 Continuation values on coordinate axis of two SPE that attain v Figure 24141 describes two SPEs that attain the worst SPE value v with the depicted sequences of time t promised value tax rate pairs The circles represent the worst SPE attained with method 1 and the xmarks correspond 1052 Credible Government Policies I to method 2 By construction the continuation values of method 2 are less than or equal to the continuation values of method 1 Since both SPEs attain the same promised value v it follows that method 2 must be associated with higher oneperiod returns in some periods Figure 24142 indicates that method 2 delivers those higher oneperiod returns around period 20 when the prescribed tax rates are closer to the Ramsey outcome τ R 03013 When varying the discount factor we find that the cutoff value of δ below which reversion to Nash fails to support Ramsey forever is 02194 0 5 10 15 20 25 30 025 03 035 04 045 05 055 06 065 Time Tax rate Figure 24142 Tax rates associated with the continuation values of Figure 24141 Interpretations 1053 2415 Interpretations The notion of credibility or sustainability emerges from a ruthless and complete application of two principles rational expectations and selfinterest At each moment and for each possible history individuals and the government act in their own best interests while expecting everyone else always to act in their best interests A credible government policy is one that it is in the interest of the government to implement on every occasion The structures that we have studied have multiple equilibria that are indexed by different systems of rational expectations Multiple equilibria are essential because what sustains a good equilibrium is a system of expectations that raises the prospect of reverting to a bad equilibrium if the government chooses to deviate from the good equilibrium For reversion to the bad equilibrium to be credible meaning that it is something that the private agents can expect because the government will want to act accordingly the bad equilibrium must itself be an equilibrium It must always be in the selfinterest of all agents to behave as they are expected to Supporting a Ramsey outcome hinges on finding an equilibrium with outcomes bad enough to deter the government from surrendering to a temporary temptation to deviate17 Is the multiplicity of equilibria a strength or a weakness of such theories Here descriptions of preferences and technologies supplemented by the restric tion of rational expectations dont pin down outcomes There is an indepen dent role for expectations not based solely on fundamentals The theory is silent about which equilibrium will prevail the theory contains no sense in which the government chooses among equilibria Depending on the purpose the multiplicity of equilibria can be regarded either as a strength or as a weakness of these theories In inferior equilibria the government is caught in an expectations trap18 an aspect of the theory that highlights how the government can be regarded as simply resigning itself to affirm the publics expectations about it Within the theory the governments 17 This statement means that an equilibrium is supported by beliefs about behavior at prospective histories of the economy that might never be attained or observed Part of the literature on learning in games and dynamic economies studies situations in which it is not reasonable to expect adaptive agents to learn so much See Fudenberg and Kreps 1993 Kreps 1990 and Fudenberg and Levine 1998 See Sargent 1999 2008 for macroeconomic counterparts 18 See Chari Christiano and Eichenbaum 1998 1054 Credible Government Policies I strategy plays a dual role as it does in any rational expectations model one summarizing the governments choices the other describing the publics rule for forecasting the governments behavior In inferior equilibria the government wishes that it could use a different strategy but nevertheless affirms the publics expectation that it will adhere to an inferior rule 2416 Extensions In chapter 25 we shall describe how Chang 1998 and Phelan and Stacchetti 2001 extended the machinery of this chapter to settings in which private agents problems have natural state variables like stocks of real balances or physical capital so that their best responses to government policies satisfy Eu ler equations or costate equations This will activate an additional source of history dependence The approach of chapter 25 merges aspects of the method described in chapter 19 and 20 with those of this chapter Exercises Exercise 241 Consider the following oneperiod economy Let ξ x y be the choice variables available to a representative agent the market as a whole and a benevolent government respectively In a rational expectations equilibrium or competitive equilibrium ξ x hy where h is the equilibrium response correspondence that gives competitive equilibrium values of x as a function of y that is hy y is a competitive equilibrium Let C be the set of competitive equilibria Let X xM xH Y yM yH For the oneperiod economy when ξi xi the payoffs to the government and household are given by the values of uxi xi yj entered in the following table Exercises 1055 Oneperiod payoffs uxi xi yj xM xH yM 10 20 yH 4 15 Denotes x y C The values of uξk xi yj not reported in the table are such that the competitive equilibria are the outcome pairs denoted by an asterisk a Find the Nash equilibrium in pure strategies and Ramsey outcome for the oneperiod economy b Suppose that this economy is repeated twice Is it possible to support the Ramsey outcome in the first period by reverting to the Nash outcome in the second period in case of a deviation c Suppose that this economy is repeated three times Is it possible to support the Ramsey outcome in the first period In the second period Consider the following expanded version of the preceding economy Y yL yM yH X xL xM xH When ξi xi the payoffs are given by uxi xi yj entered here Oneperiod payoffs uxi xi yj xL xM xH yL 3 7 9 yM 1 10 20 yH 0 4 15 Denotes x y C d What are Nash equilibria in this oneperiod economy e Suppose that this economy is repeated twice Find a subgame perfect equi librium that supports the Ramsey outcome in the first period For what values of δ will this equilibrium work f Suppose that this economy is repeated three times Find an SPE that sup ports the Ramsey outcome in the first two periods assume δ 08 Is it unique Exercise 242 Consider a version of the setting studied by Stokey 1989 Let ξ x y be the choice variables available to a representative agent the market as 1056 Credible Government Policies I a whole and a benevolent government respectively In a rational expectations or competitive equilibrium ξ x hy where h is the equilibrium response correspondence that gives competitive equilibrium values of x as a function of y that is hy y is a competitive equilibrium Let C be the set of competitive equilibria Consider the following special case Let X xL xH and Y yL yH For the oneperiod economy when ξi xi the payoffs to the government are given by the values of uxi xi yj entered in the following table Oneperiod payoffs uxi xi yj xL xH yL 0 20 yH 1 10 Denotes x y C The values of uξk xi yj not reported in the table are such that the competitive equilibria are the outcome pairs denoted by an asterisk a Define a Ramsey plan and a Ramsey outcome for the oneperiod economy Find the Ramsey outcome b Define a Nash equilibrium in pure strategies for the oneperiod economy c Show that there exists no Nash equilibrium in pure strategies for the one period economy d Consider the infinitely repeated version of this economy starting with t 1 and continuing forever Define a subgame perfect equilibrium e Find the value to the government associated with the worst subgame perfect equilibrium f Assume that the discount factor is δ 8913 110120 105 Determine whether infinite repetition of the Ramsey outcome is sustainable as an SPE If it is display the associated subgame perfect equilibrium g Find the value to the government associated with the best SPE h Find the outcome path associated with the worst SPE i Find the oneperiod continuation value v1 and the outcome path associated with the oneperiod continuation strategy σ1 that induces adherence to the worst subgame perfect equilibrium Exercises 1057 j Find the oneperiod continuation value v2 and the outcome path associated with the oneperiod continuation strategy σ2 that induces adherence to the firstperiod outcome of the σ1 that you found in part i k Proceeding recursively define vj and σj respectively as the oneperiod continuation value and the continuation strategy that induces adherence to the firstperiod outcome of σj1 where v1 σ1 were defined in part i Find vj for j 1 2 and find the associated outcome paths l Find the lowest value for the discount factor for which repetition of the Ramsey outcome is an SPE Exercise 243 Finding worst and best SPEs Consider the following model of Kydland and Prescott 1977 A government chooses the inflation rate y from a closed interval 0 10 There is a family of Phillips curves indexed by the publics expectation of inflation x 1 U U θy x where U is the unemployment rate y is the inflation rate set by the government and U 0 is the natural rate of unemployment and θ 0 is the slope of the Phillips curve and where x is the average of private agents setting of a forecast of y called ξ Private agents only decision in this model is to forecast inflation They choose their forecast ξ to maximize 2 5y ξ2 Thus if they know y private agents set ξ y All agents choose the same ξ so that x ξ in a rational expectations equilibrium The government has oneperiod return function 3 rx y 5U 2 y2 5U θy x2 y2 Define a competitive equilibrium as a 3tuple U x y such that given y private agents solve their forecasting problem and 1 is satisfied Chapter 25 Credible Government Policies II 251 Historydependent government policies Chapter 24 began with a static setting in which a pair of private sector gov ernment actions x y belongs to a set C IR2 of competitive equilibria We formed a dynamic economy by infinitely repeating the static economy for t 0 1 so that a competitive equilibrium for the repeated economy was simply a sequence xt yt t0 with xt yt C of competitive equilibria for the static economy We studied dynamics that come from a benevolent continuation governments incentives to confirm private forecasts of its time t actions yt on the basis of histories of outcomes observed through time t 1 In more general settings a competitive equilibrium of an infinite horizon economy is itself a sequence having dynamics coming from private agents de cision making In this chapter we describe how Chang 1998 and Phelan and Stacchetti 2001 studied credible public policies in such economies first by characterizing a competitive equilibrium recursively as we did in chapters 19 and 20 when we posed Stackelberg problems and Ramsey problems and sec ond by adapting arguments of Abreu Pearce and Stachetti APS that we learned in chapter 24 In the model of this chapter there are two sources of history dependence each encoded with its own forward looking state variable One state vari able indexes a continuation competitive equilibrium The other state variable is a discounted present value that an earlier government decision maker had promised that subsequent government decision makers would deliver These state variables bring distinct sources of history dependence In this chapter we describe how recursive methods can be used to analyze both A key message is that to represent credible government plans recursively it is necessary to expand the dimension of the state beyond those used in either chapters 19 and 20 or in chapter 24 Roberto Changs 1998 model is our laboratory 1059 The setting 1061 Chang assumes that u IR IR is twice continuously differentiable strictly concave and strictly increasing that v IR IR is twice continu ously differentiable and strictly concave that ucc0 limm0 vm and that there is a finite level m mf such that vmf 0 The household takes real balances mt qtMt out of period t Inequality 2522 is the households time t budget constraint It tells how real balances qtMt carried out of period t depend on income consumption taxes and real balances qtMt1 carried into the period Equation 2523 imposes an exoge nous upper bound m on the choice of real balances where m mf 2522 Government At each t 0 the government chooses a sequence of inverse money growth rates with time t component ht Mt1 Mt Π π π where 0 π 1 1 β π The government faces a sequence of budget constraints with time t component xt qtMt Mt1 which by using the definitions of mt and ht can also be expressed as xt mt1 ht 2524 The restrictions mt 0 m and ht Π evidently imply that xt X π 1 m π 1 m We define the set E 0 m Π X and require that m h x E To represent the idea that taxes are distorting Chang assumes that per capita output satisfies yt fxt 2525 where f IR IR satisfies fx 0 is twice continuously differentiable f x 0 and fx fx for all x IR so that subsidies and taxes are equally distorting This approach summarizes the consequences of distorting taxes via the function fx and abstains from modeling the distortions more deeply A key part of the specification is that tax distortions are increasing in the absolute value of tax revenues The government is benevolent and chooses a competitive equilibrium that maximizes 2521 Withinperiod timing of decisions is as follows first the 1064 Credible Government Policies II subject to equation 2531 and the budget constraints 2524 and Euler in equalities 2532 for t 0 As in chapters 19 and 20 we break this problem into two subproblems Subproblem 1 confronts a sequence of continuation Ram sey planners one for each t 1 Subproblem 1 takes as given a state variable θ that for t 2 was chosen by the preceding continuation Ramsey planner and for t 1 by the Ramsey planner2 Subproblem 2 confronts the Ramsey planner whose job is to choose m0 x0 h0 as well as a θ Ω to turn over to a time 1 continuation Ramsey planner 2531 Subproblem 1 Continuation Ramsey problem This problem confronts a continuation Ramsey planner at each t 1 The problem takes θ as given Let Jθ be the optimal value function for a con tinuation Ramsey planner facing θ as a state variable that he is obligated to deliver by choosing m x that satisfy equation 2534b The value function Jθ satisfies the Bellman equation Jθ max xmhθ ufx vm βJθ 2533 where maximization is subject to θ ufxx vmm βθ if m m 2534a θ ufxm x 2534b x m1 h 2534c m h x E 2534d θ Ω 2534d The right side of Bellman equation 2533 is attained by policy functions x xθ m mθ h hθ θ gθ 2535 2 We can also think of the entire sequence θt t0 as having been chosen a Ramsey planner at time t 0 Recursive approach to Ramsey problem 1065 2532 Subproblem 2 Ramsey problem A Ramsey planner faces different opportunities and constraints than do con tinuation Ramsey planners The Ramsey planner does not inherit a θ0 that it must deliver with a suitable choice of m0 h0 x0 but instead chooses one Let H be the value of the Ramsey problem It satisfies H max hmxθ ufx vm βJθ 2536 where maximization is subject to mufx vm βθ if m m 2537a x m1 h 2537b m h x E 2537c θ Ω 2537d The maximized value H is attained by a triple h0 m0 x0 of time 0 choices and a continuation θ Ω to pass on to a time 1 continuation Ramsey planner To find remaining settings of the Ramsey plan we iterate on 2535 starting from the θ chosen by the Ramsey planner to deduce a continuation Ramsey tax plan xt t1 and associated continuation inverse money growth real balance sequence ht mt t1 Time inconsistency manifests itself in x0 h0 m0 x1 h1 m1 so that a continuation of a Ramsey plan is not a Ramsey plan as encountered in other contexts in chapters 19 and 20 An equivalent way to express subproblem 2 is H max θmxhJθ 2538 where maximization is subject to θ ufxm x 2539a and restrictions 2537b and 2537c3 3 Please note that in equation 2536 θ is next periods value of θ to be handed over to a continuation Ramsey planner while in equation 2538 θ is a time 0 value of θ 1066 Credible Government Policies II 2533 Finding set Ω The preceding calculations assume that we know the set Ω We describe Changs method for constructing Ω later in this chapter but also provide a brief sketch here Chang uses backward induction in the style of Abreu Pearce and Stac chetti Thus Chang starts by guessing a compact set Ω0 of candidate continua tion θs For a reason to be explained momentarily it is important for Chang to start with guess Ω0 that contains Ω He defines a threetuple x0 h0 m0 E and a θ1 Ω0 that together satisfy time 0 versions of restrictions 2524 and 2528 as admissible with respect to Ω0 Notice that if the guess Ω0 happens to equal Ω then being admissible with respect to Ω0 means that x0 h0 m0 would be time t 0 variables for a competitive equilibrium that is associated with a continuation competitive equilibrium marginal utility of money θ1 be cause θ1 is in Ω But because Ω0 might be bigger than Ω Chang cant be sure that that a fourtuple x0 h0 m0 θ1 that is admissible with respect to Ω0 rep resents a competitive equilibrium Therefore he proceeds as follows He starts by seeking all fourtuples x0 h0 m0 that are admissible with respect to Ω0 For each admissible fourtuple he calculates a value θ0 ufx0m0 x0 He then constructs a set call it Ω1 of all such θ0 s associated with fourtuples that are admissible with respect to Ω0 In this way Chang constructs an op erator D that maps a set Ω0 of candidate continuation θ1s into a set Ω1 of implied time 0 θ0s Call this operator D Chang constructs Ω by iterating to convergence on D Chang shows that if he starts with a big enough set Ω0 this algorithm converges to Ω Changs proof strategy relies on verifying that D is monotone 2534 An example Figures 2531 2532 2533 and 2534 describe policies that attain Bellman equation 25332534 for fundamentals uc logc vm 1 2000m m 5m25 fx 180 4x2 and m 30 β 9 with h is confined to the interval 8 13 The domain for each of the functions reported in these figures is the set Ω of marginal utilities of money affiliated with competitive equilibria which we computed by the method described in subsection 2533 Figure 2531 shows θ0 the maximizer of value function Jθ of a continuation Ramsey plan ner it also shows the limit point θ limt θt that the marginal utility Recursive approach to Ramsey problem 1067 010 012 014 016 018 020 θ 5226 5228 5230 5232 5234 Jθ θ0 θ Figure 2531 Value function Jθ for continuation Ramsey planner of money under a Ramsey plan approaches as t Figure 2531 shows θ as a function of θ The policy functions in figure 2533 and the time series that they imply displayed in figure 2534 show how the Ramsey planner grad ually raises the tax rate xt and the inverse money growth rate ht measures that cause real balances mt gradually to rise The Ramsey planner wants to push up real balances m but dislikes the distorting taxes x required to make the inverse money growth rate bigger than 1 The Ramsey planner also under stands that the households forward looking behavior makes its demand for real balances depend inversely on future rates of inflation and therefore on future inverse money growth rates4 Therefore a Ramsey planner who plans to set high inverse money growth rates at dates t 1 reaps benefits in terms of higher real balances at time 0 By setting time 0 inverse money growth and distorting taxes to be high the Ramsey planner reaps no such benefits from higher real balances at date t 0 The different structures of payoffs to the Ramsey plan ner from settings of xt ht at different dates accounts for the Ramsey planners decision gradually to raise inverse money growth rates and distorting taxes 4 See the analysis of a demand function for money that highlights this channel in exercise 191 of chapter19 1068 Credible Government Policies II 010 012 014 016 018 020 θ 010 012 014 016 018 020 θ θ0 θ Figure 2532 θt1 as function of θt together with initial con dition θ0 and fixed point θ under Ramsey plan 010 012 014 016 018 020 θ 010 012 014 016 018 θ 010 012 014 016 018 020 θ 16 18 20 22 24 26 28 30 m 010 012 014 016 018 020 θ 090 095 100 105 110 115 120 h 010 012 014 016 018 020 θ 2 1 0 1 2 3 4 5 6 x Figure 2533 Policy functions showing θ m h and x as func tions of θ 254 Changs formulation This section describes Changs 1998 way of formulating competitive equilib ria the set Ω of marginal utilities associated with continuation competitive equilibria and a Ramsey plan Inventory of key objects 1069 0 20 40 60 80 100 t 0115 0120 0125 0130 0135 0140 θ 0 20 40 60 80 100 t 210 215 220 225 230 235 m 0 20 40 60 80 100 t 100 101 102 103 104 105 h 0 20 40 60 80 100 t 00 02 04 06 08 10 12 x Figure 2534 Time series of θ m h x under Ramsey plan 2541 Competitive equilibrium Definition A government policy is a pair of sequences h x where ht Π t 0 A price system is a nonnegative value of money sequence q An allocation is a triple of nonnegative sequences c m y It is required that time t components mt xt ht E Definition Given M1 a government policy h x price system q and allocation c m y are said to be a competitive equilibrium if i mt qtMt and yt fxt ii The government budget constraint 2524 is satisfied iii Given q x y c m solves the households problem 255 Inventory of key objects Chang constructs the following objects 1 A set Ω of initial marginal utilities of money θ0 Let Ω denote the set of initial promised marginal utilities of money θ0 asso ciated with competitive equilibria Chang exploits the fact that a competitive equilibrium consists of a first period outcome h0 m0 x0 and a continuation competitive equilibrium with marginal utility of money θ1 Ω 1074 Credible Government Policies II Note that the proposition relies on knowing the set Ω To find Ω Chang uses a method reminiscent of chapter 24s APS iteration to convergence on an operator B that maps continuation values into values We want an operator that maps a continuation θ into a current θ Chang lets Q be a nonempty bounded subset of IR Elements of the set Q are candidates for continuation marginal utilities Chang defines an operator BQ θ IR there is m x h θ E Q such that 2563 2564 and 2565 hold Thus BQ is the set of first period θs attainable with m x h E and some θ Q Proposition i Q BQ implies BQ Ω selfgeneration ii Ω BΩ factorization The proposition characterizes Ω as the largest fixed point of B It is easy to establish that BQ is a monotone operator This property allows Chang to compute Ω as the limit of iterations on B provided that iterations begin from a sufficiently large initial set 2561 Notation Let ht h0 h1 ht denote a history of inverse money creation rates with time t component ht Π A government strategy σ σt t0 is a σ0 Π and for t 1 a sequence of functions σt Πt1 Π Chang restricts the governments choice of strategies to the following space CEπ h Π there is some m x such that m xh CE In words CEπ is the set of money growth sequences consistent with the ex istence of competitive equilibria Chang observes that CEπ is nonempty and compact Definition σ is said to be admissible if for all t 1 and after any history ht1 the continuation ht implied by σ belongs to CEπ Analysis 1075 Admissibility of σ means that anticipated policy choices associated with σ are consistent with the existence of competitive equilibria after each possible subsequent history After any history ht1 admissibility restricts the govern ments choice in period t to the set CE0 π h Π there is h CEπ with h h0 In words CE0 π is the set of all first period money growth rates h h0 each of which is consistent with the existence of a sequence of money growth rates h starting from h0 in the initial period and for which a competitive equilibrium exists Remark CE0 π h Π there is m θ 0 m Ω such that mufh 1m vm βθ with equality if m m Definition An allocation rule is a sequence of functions α αt t0 such that αt Πt 0 m X Thus the time t component of αtht is a pair of functions mtht xtht Definition Given an admissible government strategy σ an allocation rule α is called competitive if given any history ht1 and ht CE0 π the continuations of σ and α after ht1 ht induce a competitive equilibrium sequence 2562 An operator At this point it is convenient to introduce an operator D that can be used to compute a Ramsey plan For computing a Ramsey plan this operator is wasteful because it works with a state vector that is bigger than necessary We introduce operator D because it helps to prepare the way for Changs operator DZ that we shall define in section 257 It is also useful because a fixed point of the operator DZ is a good guess for an initial set from which to initiate iterations on Changs settoset operator DZ Let S be the set of all pairs w θ of competitive equilibrium values and associated initial marginal utilities Let W be a bounded set of values in IR Analysis 1077 7420 7425 7430 7435 7440 7445 7450 w 000 001 002 003 004 005 006 θ R Figure 2561 Sets of w θ pairs associated with competitive equilibria the larger set and with sustainable plans the smaller set for β 3 The Ramsey plan is associated with the w θ pair denoted R which among points in the larger set maximizes w Attaining R requires an initial θ equal to the projection of R onto the vertical axis The larger sets in figures 2561 and 2562 report sets of w θ pairs asso ciated with competitive equilibria for two parameterizations We will discuss the smaller sets in the next section about sustainable plans In both figures uc logc vm 1 2000m m 5m25 fx 180 4x2 and m 30 In figure 2561 β 3 and h is confined to the interval 9 2 In figure 2562 β 8 and h is confined to the interval 9 18 In both figures the Ramsey outcome is associated with the w θ pair denoted R which among points in the larger set maximizes w To find the initial θ associated with the Ramsey plan project R onto the vertical axis9 The value of the Ramsey plan is the projection of the point R onto the horizontal axis 9 We thank Sebastian Graves for computing these sets and also the smaller ones to be described below Graves used the outer approximation method of Judd Yeltekin and Conklin 2003 to compute this set A public randomization device is introduced to convexify the set of equilibrium values 1078 Credible Government Policies II 2590 2595 2600 2605 2610 2615 2620 w 000 005 010 015 020 025 θ R Figure 2562 Sets of w θ pairs associated with competitive equilibria the larger set and with sustainable plans the smaller set for β 8 The Ramsey plan is associated with the w θ pair denoted R which among points in the larger set maximizes w Attaining R requires an initial θ equal to the projection of R onto the vertical axis 257 Sustainable plans Definition A government strategy σ and an allocation rule α constitute a sustainable plan SP if i σ is admissible ii Given σ α is competitive iii After any history ht1 the continuation of σ is optimal for the government ie the sequence ht induced by σ after ht1 maximizes 2521 over CEπ given α Remark Given any history ht1 the continuation of a sustainable plan is a sustainable plan Definition Let Θ m xh CE there is an SP whose outcome is m xh Sustainable outcomes are elements of Θ Concluding remarks 1081 a w θ pair associated with the Ramsey plan For the low β figure 2561 economy the Ramsey outcome is not sustainable while for the high β figure 2562 economy it is This structure delivers the following recursive representation of a sustainable plan 1 choose an initial w0 θ0 S 2 generate outcomes recursively by iterating on 2552 which we repeat here for convenience ˆht hwt θt mt mht wt θt xt xht wt θt wt1 χht wt θt θt1 Ψht wt θt 258 Concluding remarks This chapter has studied how Roberto Chang 1998 encodes two sources of history dependence each with its own forward looking state variable One state variable indexes a continuation competitive equilibrium while the other is a discounted present value that an earlier government decision maker had promised that subsequent government decision makers would deliver Chang represents credible government plans recursively in terms of these two state variables The need to assure that government plans are credible impelled Chang to expand the dimension of the state beyond those used in either chapters 19 and 20 or in chapter 24 Chapter 26 Two Topics in International Trade 261 Two dynamic contracting problems This chapter studies two models in which recursive contracts are used to over come incentive problems commonly thought to occur in international trade The first is Andrew Atkesons model of lending in the context of a dynamic setting that contains both a moral hazard problem due to asymmetric information and an enforcement problem due to borrowers option to disregard the contract It is a considerable technical achievement that Atkeson managed to include both of these elements in his contract design problem But this substantial technical accomplishment is not just showing off As we shall see both the moral hazard and the selfenforcement requirement for the contract are required in order to explain the feature of observed repayments that Atkeson was after that the occurrence of especially low output realizations prompt the contract to call for net repayments from the borrower to the lender exactly the occasions when an unhampered insurance scheme would have lenders extend credit to borrowers The second is Bond and Parks model of a recursive contract that induces two countries to adopt free trade when they begin with a pair of promised values that implicitly determine the distribution of eventual welfare gains from trade liberalization The new policy is accomplished by a gradual relaxation of tariffs accompanied by trade concessions Bond and Parks model of gradualism is all about the dynamics of promised values that are used optimally to manage participation constraints 1083 1094 Two Topics in International Trade contract manages to encode all history dependence in an extremely economical fashion In the end there is no need as occurred in the problems that we studied in chapter 21 to add a promised value as an independent state variable 263 Gradualism in trade policy We now describe a version of Bond and Parks 2002 analysis of gradualism in bilateral agreements to liberalize international trade Bond and Park cite examples in which a large country extracts a possibly rising sequence of transfers from a small country in exchange for a gradual lowering of tariffs in the large country Bond and Park interpret gradualism in terms of the historydependent policies that vary the continuation value of the large country in a way that induces it gradually to reduce its distortions from tariffs while still gaining from a move toward free trade They interpret the transfers as trade concessions6 We begin by laying out a simple general equilibrium model of trade between two countries7 The outcome of this theorizing will be a pair of indirect utility functions rL and rS that give the welfare of a large and small country respec tively both as functions of a tariff tL that the large country imposes on the small country and a transfer eS that the small country voluntarily offers to the large country 6 Bond and Park say that in practice the trade concessions take the form of reforms of policies in the small country about protecting intellectual property protecting rights of foreign investors and managing the domestic economy They do not claim explicitly to model these features 7 Bond and Park 2002 work in terms of a partial equilibrium model that differs in details but shares the spirit of our model 1096 Two Topics in International Trade Substituting for χγ from 2634 gives 2 γ 1γ 1ℓ which can be rearranged to become ℓ Lγ 1 γ 1 γ 2635 It follows that per capita the equilibrium quantity of each intermediate good is given by x1 x2 χγ χγ1 Lγ 2 γ2 1 γ2 2636 Two countries under autarky Suppose that there are two countries named L and S denoting large and small Country L consists of N 1 identical consumers while country S consists of one household All households have the same preferences 2631 but technologies differ across countries Specifically country L has production parameter γ 1 while country S has γ γS 1 Under no trade or autarky each country is a closed economy whose alloca tions are given by 2635 2636 and 2633 Evaluating these expressions we obtain ℓL n1L n2L cL 0 05 05 y 1 ℓS n1S n2S cS LγS χγS χγSγS y 2 χγS The relative price between the two intermediate goods is 1 in country L while for country S intermediate good 2 trades at a price γ1 S in terms of intermediate good 1 The difference in relative prices across countries implies gains from trade Gradualism in trade policy 1097 2632 A Ricardian model of two countries under free trade Under free trade country L is large enough to meet both countries demands for intermediate good 2 at a relative price of 1 and hence country S will specialize in the production of intermediate good 1 with n1S 1 To find the time n1L that a worker in country L devotes to the production of intermediate good 1 note that the world demand at a relative price of 1 is equal to 05N 1 and after imposing market clearing that N n1L 1 05 N 1 n1L N 1 2N The freetrade allocation becomes ℓL n1L n2L cL 0 N 12N N 12N y 1 ℓS n1S n2S cS 0 1 0 y 1 Notice that the welfare of a household in country L is the same as under autarky because we have ℓL 0 cL y 1 The invariance of country Ls allocation to opening trade is an immediate implication of the fact that the equilibrium prices under free trade are the same as those in country L under autarky Only country S stands to gain from free trade 2633 Trade with a tariff Although country L has nothing to gain from free trade it can gain from trade if it is accompanied by a distortion to the terms of trade that is implemented through a tariff on country Ls imports Thus assume that country L imposes a tariff of tL 0 on all imports into L For any quantity of intermediate or final goods imported into country L country L collects a fraction tL of those goods by levying the tariff A necessary condition for the existence of an equilibrium with trade is that the tariff does not exceed 1γS because otherwise country S would choose to produce intermediate good 2 rather than import it from country L Given that tL 1 γS we can find the equilibrium with trade as follows From the perspective of country S 1 tL acts like the production parameter 1100 Two Topics in International Trade Measure world welfare by uW tL uStLuLtL This measure of world welfare satisfies d uW tL d tL 2 tL 2 tL3 0 26312a and d 2uW tL d t2 L 4 1 tL 2 tL4 0 26312b We summarize our findings Proposition 1 World welfare uW tL is strictly concave is decreasing in tL 0 and is maximized by setting tL 0 But uLtL is strictly concave in tL and is maximized at tN L 0 Therefore uLtN L uL0 A consequence of this proposition is that country L prefers the Nash equilibrium to free trade but country S prefers free trade To induce country L to accept free trade country S will have to transfer resources to it We now study how country S can do that efficiently in an intertemporal version of the model 2635 Trade concessions To get a model in the spirit of Bond and Park 2002 we now assume that the two countries can make trade concessions that take the form of a direct transfer of the consumption good between them We augment utility functions uL uS of the form 2631 with these transfers to obtain the payoff functions rLtL eS uLtL eS 26313a rStL eS uStL eS 26313b where tL 0 is a tariff on the imports of country L eS 0 is a transfer from country S to country L These definitions make sense because the indirect utility functions 2639 are linear in consumption of the final consumption good so that by transferring the final consumption good the small country transfers utility The transfers eS are to be voluntary and must be nonnegative ie the country cannot extract transfers from the large country We have already seen that uLtL is strictly concave and twice continuously differentiable with u L0 0 and that uW tL uStL uLtL is strictly concave and 1102 Two Topics in International Trade 2637 Timeinvariant transfers We first study circumstances under which there exists a timeinvariant transfer eS 0 that will induce country L to move to free trade Let vN i uitN L 1β be the present discounted value of country i when the static Nash equilibrium is repeated forever If both countries are to prefer free trade with a timeinvariant transfer level eS 0 the following two participation constraints must hold vL uL0 eS 1 β uLtN L eS βvN L 26315 vS uS0 eS 1 β uS0 βvN S 26316 The timing here articulates what it means for L and S to choose simultane ously when L defects from 0 eS L retains the transfer eS for that period Symmetrically if S defects it enjoys the zero tariff for that one period These temporary gains provide the temptations to defect Inequalities 26315 and 26316 say that countries L and S both get higher continuation values from remaining in free trade with the transfer eS than they get in the repeated static Nash equilibrium Inequalities 26315 and 26316 invite us to study strate gies that have each country respond to any departure from what it had expected the other country to do this period by forever after choosing the Nash equilib rium actions tL tN L for country L and eS 0 for country S Thus the response to any deviation from anticipated behavior is to revert to the repeated static Nash equilibrium itself a subgame perfect equilibrium8 Inequality 26315 the participation constraint for L and the definition of vN L can be rearranged to get eS uLtN L uL0 β 26317 Timeinvariant transfers eS that satisfy inequality 26317 are sufficient to in duce L to abandon the Nash equilibrium and set its tariff to zero The minimum timeinvariant transfer that will induce L to accept free trade is then eSmin uLtN L uL0 β 26318 8 In chapter 24 we study the consequences of reverting to a subgame perfect equilibrium that gives worse payoffs to both S and L and how the worst subgame perfect equilibrium payoffs and strategies can be constructed Gradualism in trade policy 1105 Here y is the continuation value for L meaning next periods value of vL Constraint 26327a is the promisekeeping constraint while 26327b and 26327c are the participation constraints for countries L and S respectively The constraint set is convex and the objective is concave so PvL is concave though not strictly concave an important qualification as we shall see As with our study of Thomas and Worralls and Kocherlakotas model we place nonnegative multipliers θ on 26327a and µL µS on 26327b and 26327c respectively form a Lagrangian and obtain the following firstorder necessary conditions for a saddlepoint tL u StL θ µLu LtL 0 0 if tL 0 26328a y P y1 µS θ µL 0 26328b eS 1 θ µS 0 0 if eS 0 26328c We analyze the consequences of these firstorder conditions for the optimal con tract in three regions delineated by the continuation values v L v L We break our analysis into two parts We begin by displaying particular policies that attain initial values on the constrained Pareto frontier Later we show that there can be many additional policies that attain the same values which as we shall see is a consequence of a flat interval in the constrained Pareto frontier 2639 Baseline policies Region I vL v L v L neither PC binds When the initial value is in this interval the continuation value stays in this interval From the envelope property P vL θ If vL v L v L neither participation constraint binds and we have µS µL 0 Then 26328b implies P y P vL This can be satisfied by setting y vL Then y vL and the always binding promisekeeping constraint in 26327a imply that vL y uLtL eS 1 β v L vN L uLtN L 1 β 26329 1106 Two Topics in International Trade where the weak inequality states that vL trivially satisfies the lower bound of region I which in turn is strictly greater than the Nash value vN L according to expression 26325 Because uLt is maximized at tN L the strict inequality in expression 26329 holds only if eS 0 Then inequality 26328c and eS 0 imply that θ 1 Rewrite 26328a as u W tL 0 0 if tL 0 By Proposition 1 this implies that tL 0 We can solve for eS from vL uL0 eS 1 β 26330 and then obtain PvL from uS0eS 1β Before turning to region II with vL v L we shall first establish that there indeed exist such high continuation values for the large country which cannot be sustained by a timeinvariant transfer scheme This is done by showing that Pv L vN S That is there is scope for further increasing the continuation value of the large country beyond v L before the associated continuation value of the small country is reduced to vN S The argument goes as follows Pv L uW 0 1 β v L uL0 uS0 1 β uL0 βuS0 uStN L 1 β uS0 β uStN L 1 β uStN L β uStN L 1 β vN S 26331 where the first equality uses the fact that the continuation value v L lies on the unconstrained Pareto frontier whose slope is 1 and the second equality invokes expression 26322 It then follows that Pv L vN S Region II vL v L PCS binds We shall verify that in region II there is a solution to the firstperiod first order necessary conditions with µS 0 and eS 0 When vL v L µS 0 and µL 0 When µS 0 inequality 26328c and eS 0 imply θ 1 µS 1 26332 1108 Two Topics in International Trade compared to the transfer e S that the small country pays in period t 1 and forever afterwards we notice that e S is also subject to a participation constraint 26327c with the very same continuation value Py but where uSt L uS0 Hence we can express 26327c for all periods t 1 given a time invariant continuation value Py determined by 26335 as e S βPy vN S eS where the equality sign follows from 26335c We conclude that the transfer is nonincreasing over time for our solution to an initial continuation value in region II Thus in region II tL 0 in period 0 followed by t L 0 thereafter Moreover the initial promised value to the large country vL v L is followed by a lower timeinvariant continuation value y v L Subtracting 26335b from 26334 gives y vL uL0 uLtL e S eS The contract sets the continuation value y vL by making tL 0 thereby making uL0 uLtL 0 and also possibly letting e S eS 0 so that transfers can fall between periods 0 and 1 In region II country L induces S to accept free trade by a twostage lowering of the tariff from the Nash level so that 0 tL tN L in period 0 with t L 0 for t 1 in return it gets period 0 transfers of eS 0 and constant transfers e S 0 thereafter Region III vL vN L v L PCL binds The analysis of region III is subtle9 It is natural to expect that µS 0 µL 0 in this region However assuming that µL 0 can be shown to lead to a contradiction implying that the pair vL PvL both is and is not on the unconstrained Pareto frontier10 We can avoid the contradiction by assuming that µL 0 so that the partic ipation constraint for country L is barely binding We shall construct a solution to 26328 and 26327 with period 0 transfer eS 0 Note that 26328c with eS 0 implies θ 1 which from the envelope property P vL θ 9 The findings of this section reproduce ones summarized in Bond and Parks 2002 corollary to their Proposition 2 10 Please show this in exercise 262 Gradualism in trade policy 1109 implies that vL PvL is actually on the unconstrained Pareto frontier a reflection of the participation constraint for country L barely binding With θ 1 and µL 0 26328a implies that tL 0 which confirms vL PvL being on the Pareto frontier We can then solve the following equations for PvL eS y Py PvL vL uW 0 1 β 26336a vL uL0 eS βy 26336b uL0 eS βy uLtN L eS βvN L 26336c PvL uS0 eS βPy 26336d Py y uW 0 1 β 26336e We shall soon see that these constitute only four linearly independent equa tions Equations 26336a and 26336e impose that both vL PvL and y Py lie on the unconstrained Pareto frontier We can solve these equa tions recursively First solve for y from 26336c Then solve for Py from 26336e Next solve for PvL from 26336a Get eS from 26336b Finally equations 26336a 26336b and 26336d imply that equation 26336e holds which establishes the reduced rank of the system of equations We can use 26334 to compute e S the transfer from period 1 onward In particular e S satisfies y uL0 e S βy Subtracting 26336b from this equation gives y vL e S eS 0 Thus when vL v L country S induces country L immediately to reduce its tariff to zero by paying transfers that rise between period 0 and period 1 and that thereafter remain constant That the initial tariff is zero means that we are immediately on the unconstrained Pareto frontier It just takes timevarying transfers to put us there Interpretations For values of vL within regions II and III timeinvariant transfers eS from country S to country L are not capable of sustaining immediate and enduring free trade But patterns of timevarying transfers and tariff reductions are able to induce both countries to move permanently to free trade after a oneperiod 1110 Two Topics in International Trade v v v Pv L L L L N Figure 2631 The constrained Pareto frontier vS PvL in the BondPark model transition There is an asymmetry between regions II and III revealed in Figure 2631 and in our finding that tL 0 in region III so that the move to free trade is immediate The asymmetry emerges from a difference in the quality of instruments that the unconstrained country L in region II S in region III has to induce the constrained country eventually to accept free trade by moving those instruments over time appropriately to manipulate the continuation values of the constrained country to gain its assent In region II where S is constrained all that L can do is manipulate the time path of tL a relatively inefficient instrument because it is a distorting tax By lowering tL gradually L succeeds in raising the continuation values of S gradually but at the cost of imposing a distorting tax thereby keeping vL PvL inside the Pareto frontier In region III where L is constrained S has at its disposal a nondistorting instrument for raising country Ls continuation value by increasing the transfer eS after period 0 The basic principle at work is to make the continuation value rise for the country whose participation constraint is binding Gradualism in trade policy 1111 26310 Multiplicity of payoffs and continuation values We now find more equilibrium policies that support values in our three regions The unconstrained Pareto frontier is a straight line in the space vL vS with a slope of 1 vL vS uW 0 1 β W This reflects the fact that utility is perfectly transferable between the two countries As a result there is a continuum of ways to pick current payoffs ri i L S and continuation values v i i L S that deliver the promised values vL and vS to country L and S respectively For example each country could receive a current payoff equal to the annuity value of its promised value ri 1 βvi and retain its promised value as a continuation value v i vi That would clearly deliver the promised value to each country ri βv i 1 βvi βvi vi Another example would reduce the prescribed current payoff to country S by S 0 and increase the prescribed payoff to country L by the same amount Continuation values v S v L would then have to be set such that 1 βvS S βv S vS 1 βvL S βv L vL Solving from these equations we get S βv S vS βv L vL Here country S is compensated for the reduction in current payoff by an equiv alent increase in the discounted continuation value while country L receives corresponding changes of opposite signs Since the constrained Pareto frontier coincides with the unconstrained Pareto frontier in regions I and III we would expect that the tariff games would also be characterized by multiplicities of payoffs and continuation values We will now examine how the participation constraints shape the range of admissible equilibrium values Another model 1115 264 Another model Fuchs and Lippi 2006 are motivated by a vision about the nature of mone tary unions that was not well captured by work in the previous literature In particular earlier work 1 assumed away commitment problems between mem bers that would occur within an ongoing currency union and 2 modeled the consequences of abandoning a currency as reversion to a worst case outcome of a repeated game played by independent monetaryfiscal authorities The FuchsLippi paper repairs both of these deficiencies by 1 imposing participa tion constraints each period for each member within a currency union and 2 assuming that the consequence of a breakup is to move to the best outcome of the game played by independent monetaryfiscal authorities Here is the setup Two countries have ideal levels of a policy setting eg an interest rate that are each hit by countryspecific idiosyncratic shocks The history of these shocks is common knowledge When not in a union the countries play a repeated game The best equilibrium outcome is the point to which the countries revert after a breakup When in a union the two countries play another repeated game The authors model the benefit of being in the union as making it harder to effect a surprise change than it is outside it thereby making it easier to abstain from opportunistic monetary policy that eg exploits the Phillips curve to get short run benefits in exchange for longrun costs The authors use dynamic programming squared to express equilibrium strategies within the currency union game in terms of the current observed shock vector and continuation values The union chooses a public good namely the common policy each period It is a weighted average of the ideal points for the two individual countries with the weights being tilted a country whose partici pation constraint is binding that period The authors show that there are three possible cases 1 the shocks and initial continuation values are such that only country As constraint is binding in which case the policy tilts toward country As ideal point 2 only country Bs participation constraint is binding in which case the policy tilts toward country Bs ideal point 3 the continuation values and shocks are such that both countries participation constraints are binding In case 3 the currency union breaks up 1116 Two Topics in International Trade Depending on specifications of functional forms preferences and the joint distribution of shocks case 3 may or may not be possible When it is one can use the model to calculate waiting times to breakup of a union Many currency unions have broken up in the past an observation that could be used to help reverse engineer parameter values something that the authors dont do It is interesting to compare this model with an earlier risksharing model of Thomas and Worrall and Kocherlakota that we studied in chapter 21 In that model there was no case 3 and the analogue of the union the relationship be tween the firm and the worker in Thomas and Worrall or between two consumers in Kocherlakota lasts forever One never observes defaults along the equilibrium path What is the source of the different outcome in the FuchsLippi model The answer hinges on the part of the payoff structure of the FuchsLippi model that captures the public good aspect of the monetary union policy choice Both countries have to live with the same setting of a policy instrument and what one gains the other does not necessarily lose In the chapter 21 model each period when one person gets more the other necessarily gets less creating a symmetry in the participation constraints that prevents them from binding simultaneously 265 Concluding remarks Although substantive details differ mechanically the models of this chapter work much like models that we studied in chapters 19 21 and 24 The key idea is to cope with binding incentive constraints in this case participation constraints partly by changing the continuation values for those agents whose incentive constraints are binding For example that creates intertemporal tieins that Bond and Park interpret as gradualism Exercises 1119 Exercises Exercise 261 Consider a version of Bond and Parks model with γS 4 and payoff functions 26313a and 26313b with uLtL 5tL 52 uW tL 5t2 L where uW tL uLtL uStL a Compute the cutoff value βc from 26323 For β βc 1 compute v L v L b Compute the constrained Pareto frontier Hint In region II use 26335 for a grid of values vL satisfying vL v L c For a given vL vN L v compute eS e S y Exercise 262 Consider the BondPark model analyzed above Assume that in region III µL 0 µS 0 Show that this leads to a contradiction Part VI Classical Monetary and Labor Economics 1128 FiscalMonetary Theories of Inflation 2723 Equilibrium We use the following definitions Definition A price system is a pair of positive sequences Rt pt t0 Definition We take as exogenous sequences gt τt t0 We also take B0 b0 and M0 m0 0 as given An equilibrium is a price system a consumption sequence ct t0 a sequence for government indebtedness Bt t1 and a posi tive sequence for the money supply Mt t1 for which the following statements are true a given the price system and taxes the households optimum problem is solved with bt Bt and mt Mt b the governments budget constraint is satisfied for all t 0 and c ct gt y 2724 Short run versus long run We shall study government policies designed to ascribe a definite meaning to a distinction between outcomes in the short run initial date and the long run stationary equilibrium We assume gt g t 0 τt τ t 1 Bt B t 1 27219 We permit τ0 τ and B0 B These settings of policy variables are designed to let us study circumstances in which the economy is in a stationary equilibrium for t 1 but starts from some other position at t 0 We have enough free policy variables to discuss two alternative meanings that the theoretical literature has attached to the phrase open market operations A shopping time monetary economy 1129 2725 Stationary equilibrium We seek an equilibrium for which ptpt1 Rm t 0 Rt R t 0 ct c t 0 st s t 0 27220 Substituting equations 27220 into equations 27214 and 27217 yields R β1 mt1 pt fRm 27221 where we define fRm Fc RmR and we have suppressed the constants c and R in the money demand function fRm in a stationary equilibrium Notice that f Rm 0 an inequality that plays an important role below Substituting equations 27219 27220 and 27221 into the govern ment budget constraint 27218 using the equilibrium condition Mt mt and rearranging gives g τ BR 1R fRm1 Rm t 1 27222 Given the policy variables g τ B equation 27222 determines the station ary rate of return on currency Rm In 27222 g τ is the net of interest deficit sometimes called the operational deficit g τ BR1R is the gross of interest government deficit and fRm1 Rm is the rate of seigniorage revenues from printing currency5 The inflation tax rate is 1 Rm and the quantity of real balances fRm is the base of the inflation tax 5 The stationary value of seigniorage per period is given by Mt1 Mt pt Mt1 pt Mt pt1 pt1 pt fRm1 Rm 1130 FiscalMonetary Theories of Inflation 2726 Initial date time 0 Because M1p0 fRm the government budget constraint at t 0 can be written M0p0 fRm g B0 τ0 BR 27223 2727 Equilibrium determination Given the policy parameters g τ τ0 B the initial stocks B0 and M0 and the equilibrium gross real interest rate R β1 equations 27222 and 27223 determine Rm p0 The two equations are recursive equation 27222 de termines Rm then equation 27223 determines p0 0 02 04 06 08 1 2 0 2 4 6 8 10 Rm g τ BR1R fRm 1Rm Figure 2721 The stationary rate of return on currency Rm is determined by the intersection between the stationary gross of interest deficit g τ BR 1R and the stationary seigniorage fRm1 Rm A shopping time monetary economy 1131 0 02 04 06 08 1 0 20 40 60 80 100 M0 p0 Rm fRm g B0 τ0 BR Figure 2722 Given Rm the real value of initial money balances M0p0 is determined by fRm g B0 τ0 BR Thus the price level p0 is determined because M0 is given It is useful to illustrate the determination of an equilibrium with a parametric example Let the utility function and the transaction technology be given by uct lt c1δ t 1 δ l1α t 1 α Hct mt1pt ct 1 mt1pt where the latter is a modified version of equation 2725 so that transactions can be carried out even in the absence of money For parameter values β δ α c 096 07 05 04 Figure 2721 displays the stationary gross of interest deficit g τ BR 1R and the stationary seigniorage fRm1Rm6 Figure 2722 shows fRmgB0τ0BR Stationary equilibrium is determined as follows name constant values g τ B which imply a stationary gross of interest deficit g τ BR 1R then read an associated stationary value Rm from Figure 2721 that satisfies equation 6 For our parameterization in Figure 2721 households choose to hold zero money balances for Rm 015 so at these rates there is no seigniorage collected Seigniorage turns negative for Rm 1 because the government is then continuously withdrawing money from circulation to raise the real return on money above 1 1132 FiscalMonetary Theories of Inflation 27222 for this value of Rm find the value of fRm g B0 τ0 BR in Figure 2722 which is equal to M0p0 by equation 27223 Thus the initial price level p0 is determined because M0 is given in period 0 273 Ten monetary doctrines We now use equations 27222 and 27223 to explain some important doc trines about money and government finance 2731 Quantity theory of money The classic quantity theory of money experiment is to increase M0 by some factor λ 1 a helicopter drop of money leaving all of the other parameters of the model fixed including the fiscal policy parameters τ0 τ g B The effect is to multiply the initial equilibrium price and money supply sequences by λ and to leave all other variables unaltered 2732 Sustained deficits cause inflation The parameterization in Figures 2721 and 2722 shows that there can be mul tiple values of Rm that solve equation 27222 As can be seen in Figure 2721 some values of the grossofinterest deficit g τ BR 1R can be financed with either a low or high rate of return on money The tax rate on real money balances is 1 Rm in a stationary equilibrium so the higher Rm that solves equation 27222 is on the good side of a Laffer curve in the inflation tax rate If there are multiple values of Rm that solve equation 27222 we shall always select the highest one for the purposes of doing our comparative dynamic exercises7 The stationary equilibrium with the higher rate of return on currency 7 In chapter 9 we studied the perfectforesight dynamics of a closely related system and saw that the stationary equilibrium selected here was not the limit point of those dynamics Our selection of the higher rate of return equilibrium can be defended by appealing to various forms of adaptive nonrational dynamics See Bruno and Fischer 1990 Marcet and Sargent 1989 and Marimon and Sunder 1993 Also see exercise 272 1134 FiscalMonetary Theories of Inflation price level p0 can go either way depending on the slope of the revenue curve fRm1 Rm the decrease in Rm reduces the righthand side of equation 27223 fRm g B0 τ0 BR while the increase in B raises the value Thus the upward shift of the curve in Figure 2722 due to the higher value of B and the downward movement along that new curve due to the lower equilibrium value of Rm can cause M0p0 to move up or down that is a decrease or an increase in the initial price level p0 The effect of a decrease in the money supply M1 accomplished through such an open market operation is at best temporarily to drive the price level downward at the cost of causing the inflation rate to be permanently higher Sargent and Wallace 1981 called this unpleasant monetarist arithmetic 2735 An open market operation delivering neutrality We now alter the definition of open market operations for the purpose of dis arming unpleasant monetarist arithmetic We supplement the fiscal powers of the monetary authority in a way that lets open market operations have effects like those in the quantity theory experiment Let there be an initial equilibrium with policy values denoted by bars over variables Consider an open market sale or purchase defined as a decrease in M1 and simultaneous increases in B and τ sufficient to satisfy 1 1R ˆB B ˆτ τ 2731 where variables with hats denote the new values of the corresponding variables We assume that ˆτ0 τ0 As long as the tax rate from time 1 on is adjusted according to equation 2731 equation 27222 will be satisfied at the initial value of Rm Equation 2731 imposes a requirement that the lumpsum tax τ be adjusted by just enough to service whatever additional interest payments are associated with the alteration in B resulting from the exchange of M1 for B8 Under this definition of an open market operation reductions in M1 achieved by increases in B and the taxes needed to service B cause proportionate decreases in the paths of the money supply and the price level and leave Rm unaltered In this way we have salvaged a version of the pure quantity theory of money 8 This definition of an open market operation imputes unrealistic power to a monetary authority on earth central banks dont set tax rates Ten monetary doctrines 1135 2736 The optimum quantity of money Friedmans 1969 ideas about the optimum quantity of money can be repre sented in Figures 2721 and 2722 Friedman noted that given the stationary levels of g B the representative household prefers stationary equilibria with higher rates of return on currency In particular the higher the stationary level of real balances the better the household likes it By running a sufficiently large grossofinterest surplus that is a negative value of g τ BR 1R the government can attain any value of Rm 1 β1 Given g B and the target value of Rm in this interval a tax rate τ can be chosen to assure the required surplus The proceeds of the tax are used to retire currency from circulation thereby generating a deflation that makes the rate of return on currency equal to the target value of Rm According to Friedman the optimal policy is to satiate the system with real balances insofar as it is possible to do so The social value of real money balances in our model is that they reduce households shopping time The optimum quantity of money is the one that minimizes the time allocated to shopping For the sake of argument suppose there is a satiation point in real balances ψc for any consumption level c that is Hmpc mt1pt 0 for mt1pt ψc According to condition 27215 the government can attain this optimal allocation only by choosing Rm R since λt µt 0 Utility is assumed to be strictly increasing in both consumption and leisure Thus welfare is at a maximum when the economy is satiated with real balances For the transaction technology given by equation 2725 the Friedman rule can be attained only approximately because money demand is insatiable 2737 Legal restrictions to boost demand for currency If the government can somehow force households to increase their real money balances to fRm fRm it can finance a given stationary gross of interest deficit g τ BR 1R at a higher stationary rate of return on currency Rm The increased demand for money balances shifts the seigniorage curve in Figure 2721 upward to fRm1 Rm thereby increasing the higher of the two intersections of the curve fRm1 Rm with the grossofinterest deficit line in Figure 2721 By increasing the base of the inflation tax the rate 1 Rm of inflation taxation can be diminished Examples of legal restrictions 1136 FiscalMonetary Theories of Inflation to increase the demand for government issued currency include a restrictions on the rights of banks and other intermediaries to issue bank notes or other close substitutes for government issued currency9 b arbitrary limitations on trading other assets that are close substitutes with currency and c reserve requirements Governments intent on raising revenues through the inflation tax have fre quently resorted to legal restrictions and threats designed to promote the de mand for its currency In chapter 28 we shall study a version of Bryant and Wallaces 1984 theory of some of those restrictions Sargent and Velde 1995 describe the sharp tools used to enforce such restrictions during the Terror during the French Revolution To assess the welfare effects of policies forcing households to hold higher real balances we must go beyond the incompletely articulated transaction process underlying equation 2724 We need an explicit model of how money facili tates transactions and how the government interferes with markets to increase the demand for real balances In such a model there would be opposing effects on social welfare On the one hand our discussion of the optimum quantity of money says that a higher real return on money Rm tends to improve wel fare On the other hand the imposition of legal restrictions aimed at forcing households to hold higher real balances might elicit socially wasteful activities devoted to evading those restrictions 2738 One big open market operation Lucas 1986 and Wallace 1989 describe a large open market purchase of pri vate indebtedness at time 0 The purpose of the operation is to provide the government with a portfolio of interestearning claims on the private sector one that is sufficient to permit it to run a grossofinterest surplus The government uses the surplus to reduce the money supply each period thereby engineering a deflation that raises the gross rate of return on money above 1 That is the government uses its own lending to reduce the gap in rates of return between its money and higheryield bonds As we know from our discussion of the optimum 9 In the US Civil War the US Congress taxed out of existence the notes that state chartered banks had issued which before the war had been the countrys paper currency 1138 FiscalMonetary Theories of Inflation holdings almost worthless the private sectors real balances at the end of period 0 M1p0 come almost entirely from that periods openmarket operation The government injects that money stock into the economy in exchange for interest earning claims on the private sector BR M1p0 In future periods the government keeps those bond holdings constant while using the net interest earnings to reduce the money supply in each future period The government passes the interest earnings on to money holders by engineering a deflation that yields a return on money equal to Rm R 2739 A fiscal theory of the price level The preceding sections have illustrated what might be called a fiscal theory of inflation This theory assumes that at time t 0 the government commits to a specific sequence of exogenous variables ranging over t 0 In particular the government sets g τ0 τ and B while B0 and M0 are inherited from the past The model then determines Rm and p0 via equations 27222 and 27223 This system of equations determining equilibrium values is recursive given g τ and B equation 27222 determines the rate of return on currency Rm and therefore in light of equation 2728 inflation then given g τ B and Rm equation 27223 determines p0 After p0 is determined M1 is determined from M1p0 fRm In this setting the government commits to a longrun grossofinterest government deficit g τ BR 1R and then the market determines p0 Rm Woodford 1995 and Sims 1994 have converted a version of the same model into a fiscal theory of the price level by altering assumptions about the variables that the government sets Rather than assuming that the government sets B and thereby the grossofinterest government deficit Woodford assumes that B is endogenous and that instead the government sets in advance a present value of seigniorage fRm1RmR1 This assumption is equivalent to saying that the government commits to fix either the nominal interest rate or the gross rate of inflation R1 m the nominal interest rate and Rm are locked together by equation 2728 Woodford emphasizes that in the present setting such a nominal interest rate peg leaves the equilibrium price level process determinate11 To 11 Woodford 1995 interprets this finding against the background of a literature that oc casionally asserted a different result namely that interest rate pegging led to price level 1140 FiscalMonetary Theories of Inflation Several commentators have remarked that the SimsWoodford use of these equations puts the government on a different setting than the private agents12 Private agents demand curves are constructed by requiring their budget con straints to hold for all hypothetical price processes not just the equilibrium one However under Woodfords assumptions about what the government has already chosen regardless of the p0 Rm it faces the only way an equilibrium can exist is if p0 adjusts to make equation 2734 satisfied The government budget constraint would not be satisfied unless p0 adjusts to satisfy 2734 By way of contrast in the fiscal theory of inflation described by Sargent and Wallace 1981 and Sargent 2013 embodied in our description of unpleasant monetarist arithmetic the focus is on how the one tax rate that is assumed to be free to adjust the inflation tax responds to fiscal conditions that the government inherits Sims and Woodford forbid the inflation tax from adjusting having set it once and all for by pegging the nominal interest rate They thereby force other aspects of fiscal policy and the price system to adjust 27310 Exchange rate indeterminacy Kareken and Wallaces 1981 exchange rate indeterminacy result provides a good laboratory for putting the fiscal theory of the price level to work First we will describe a version of Kareken and Wallaces result Then we will show how it can be overturned by changing the assumptions about policy to ones like Woodfords To describe the theory of exchange rate indeterminacy we change the pre ceding model so that there are two countries with identical technologies and preferences Let yi and gi be the endowment of the good and government pur chases for country i 1 2 where y1 y2 y and g1 g2 g Under the assumption of complete markets equilibrium consumption ci in country i is constant over time and c1 c2 c Each country issues currency The government of country i has Mit1 units of its currency outstanding at the end of period t The price level in terms of currency i is pit and the exchange rate et satisfies the purchasing power parity condition p1t etp2t The household is indifferent about which currency to use so long as both currencies bear the same rate of return and will not hold one 12 See Buiter 2002 and McCallum 2001 Ten monetary doctrines 1141 with an inferior rate of return This fact implies that p1tp1t1 p2tp2t1 which in turn implies that et1 et e Thus the exchange rate is constant in a nonstochastic equilibrium with two currencies being valued We let Mt1 M1t1 eM2t1 For simplicity we assume that the money demand function is linear in the transaction volume Fc RmR c ˆFRmR It then follows that the equilibrium condition in the world money market is Mt1 p1t fRm 2736 In order to study stationary equilibria where all real variables remain con stant over time we restrict attention to identical monetary growth rates in the two countries Mit1Mit 1ǫ for i 1 2 We let τi and Bi denote constant steadystate values for lumpsum taxes and real government indebtedness for government i The budget constraint of government i is τi gi Bi 1 R R Mit1 Mit pit 2737 Here is a version of Kareken and Wallaces exchange rate indeterminacy result Assume that the governments of each country set gi Bi and Mit1 1 ǫMit planning to adjust the lumpsum tax τi to raise whatever revenues are needed to finance their budgets Then the constant monetary growth rate implies Rm 1ǫ1 and equation 2736 determines the worldwide demand for real balances But the exchange rate is not determined under these policies Specifically the market clearing condition for the money market at time 0 holds for any positive e with a price level p10 given by M11 eM21 p10 fRm 2738 For any such pair e p10 that satisfies equation 2738 with an associated value for p20 p10e governments budgets are financed by setting lumpsum taxes according to 2737 Kareken and Wallace conclude that under such settings for government policy variables something more is needed to deter mine the exchange rate With policy as specified here the exchange rate is indeterminate13 13 See Sargent and Velde 1990 for an application of this theory to events surrounding German monetary unification 1142 FiscalMonetary Theories of Inflation 27311 Determinacy of the exchange rate retrieved A version of Woodfords assumptions about the variables that governments choose can render the exchange rate determinate Thus suppose that each government sets a real level of seigniorage xi Mit1 Mitpit for all t 1 The budget constraint of government i is then τi gi Bi 1 R R xi 2739 In order to study stationary equilibria where all real variables remain constant over time we allow for three cases with respect to x1 and x2 they are both strictly positive strictly negative or equal to zero To retrieve exchange rate determinacy we assume that the governments of each country set gi Bi xi and τi so that budgets are financed according to 2739 Hence the endogenous inflation rate is pegged to deliver the targeted levels of seigniorage x1 x2 fRm1 Rm 27310 The implied return on money Rm determines the endogenous monetary growth rates in a stationary equilibrium R1 m Mit1 Mit 1 ǫ for i 1 2 27311 That is nominal supplies of both monies grow at the rate of inflation so that real money supplies remain constant over time The levels of those real money supplies satisfy the equilibrium condition that the real value of net monetary growth is equal to the real seigniorage chosen by the government ǫMit pit xi for i 1 2 27312 Equations 27312 determine the price levels in the two countries so long as the chosen amounts of seigniorage are not equal to zero which in turn determine a unique exchange rate e p1t p2t M1t M2t x2 x1 1 ǫtM10 1 ǫtM20 x2 x1 M10 M20 x2 x1 Thus with this SimsWoodford structure of government commitments ie set ting of exogenous variables the exchange rate is determinate It is only the An example of exchange rate indeterminacy 1143 third case of stationary equilibria with x1 and x2 equal to zero where the exchange rate is indeterminate because then there is no relative measure of seigniorage levels that is needed to pin down the denomination of the world real money supply for the purpose of financing governments budgets 274 An example of exchange rate indeterminacy As an illustration of the KarekenWallace exchange rate indeterminacy and the SimsWoodford fiscal theory of the price level consider the following version of the twocountry environment in section 27310 y1 y2 y2 2741a g1 g2 0 2741b B1 B2 0 2741c M10 M20 2741d M1t1 M1t M2t1 M2t 1 ǫ 1 t 0 2741e The governments in the two countries have no purchases to finance and no bond holdings The seigniorage raised by printing money is handed over as lumpsum transfers to the households in each country respectively The budget constraint of government i is τi Mit1 Mit pit xi 2742 where the negative lumpsum tax τi is equal to the real value of the countrys seigniorage xi To operationalize the concept of exchange rate indeterminacy we assume that there is a sunspot variable that can take on three values at the start of the economy14 Each realization of the sunspot variable is associated with a particular belief about the equilibrium value of the exchange rate e 0 1 that will prevail in period 0 and forever thereafter That is depending on the sunspot realization all households will coordinate on one of the following three beliefs about the equilibrium outcome in the world money market 14 Sunspots were introduced by Cass and Shell 1983 to explain excess market volatility Sunspots represent extrinsic uncertainty not related to the fundamentals of the economy 1144 FiscalMonetary Theories of Inflation 1 the currency of country 2 is worthless e 0 and p2t t 0 2 the two currencies are traded one for one e 1 and p1t p2t t 0 3 the currency of country 1 is worthless e and p1t t 0 We assume that all households share the same belief about the sunspot process and that each sunspot realization is perceived to occur with the same probability equal to 13 We also postulate that all households are riskaverse with identical prefer ences and as stated in 2741a that they have the same constant endowment stream As initial conditions the representative household in country i owns the beginningofperiod money stock Mi0 of its country 2741 Trading before sunspot realization The equilibrium allocation in this economy will depend on whether or not house holds can trade before observing the sunspot realization In chapter 8 we as sumed that all trade took place after any uncertainty had been resolved in the first period In our current setting this would translate into households trading after the sunspot realization ie after the agents have seen the sunspot and therefore after the coordination of beliefs about the equilibrium value of the ex change rate In cases i and iii this implies that the households in the country with a valued currency will be better off because their initial money holdings are valuable and they will receive lumpsum transfers equal to their governments revenue from seigniorage in each period In case ii all households are equally well off in the world economy because of identical budget constraints Alternatively we can assume that households can trade in markets before the sunspot realization In a complete market world agents would be able to trade in contingent claims with payoffs conditional on the sunspot realization Given the symmetries in the environment with respect to preferences endowment and expected assettransfer outcomes associated with the sunspot process the equi librium allocation will be one of perfect pooling with each household consuming y2 in every period15 Hence the households will use security markets to pool the risks associated with the sunspot process Given the ex ante symmetry in 15 See Lucas 1982 for a perfect pooling equilibrium in a twocountry world with two curren cies However Lucas considers only intrinsic uncertainty arising from stochastic endowment streams 1146 FiscalMonetary Theories of Inflation theory of the price level is at its core a device for selecting equilibria from the continuum which can exist in monetary models Kocherlakota and Phelan 1999 are skeptical about this recommendation for selecting an equilibrium The fiscal theory proposes to rule out other equi libria by specifying government policies in such a way that government budget constraints hold only for one particular exchange rate But what would happen if the sunspot realization signals case i or case iii to the households so that they actually abandon one currency making it worthless The fiscal theory formulated by Sims and Woodford contains no answer to this question Critics of the fiscal theory of the price level instead prefer to specify government poli cies so that a governments budget constraint is satisfied for all hypothetical outcomes including e 0 For example a government that finds itself issuing a worthless currency could surrender its aspiration to make lumpsum transfers with strictly positive value to its citizens while the other government would accept that the value of the transfer of newly printed money to its citi zens has doubled in real terms But of course this remedy to the puzzle would refute the fiscal theory of the price level and once again render the exchange rate indeterminate 2743 A game theoretic view of the fiscal theory of the price level Bassetto 2002 agrees with criticisms of the fiscal theory of the price level that question how the government can adopt a fiscal policy without being concerned about outcomes that could make the policy infeasible Bassetto reformulates the fiscal theory of the price level in terms of a game The essence of his argument is that in order to select an equilibrium a government must specify strategies for all arbitrary outcomes so that its desired outcome is the only one that can be supported as an equilibrium outcome merely on the basis of individual rationality of private actors Bassetto 2002 studies a government that seeks to finance occasional deficits by issuing debt in a model with trading posts In such a trading environment it might happen that not all government debt can be sold because private agents fail to submit enough bids What would the equilibrium outcome be then The fiscal theory formulated by Sims and Woodford contains no answer since it presupposes that the government budget constraint will be satisfied for a given An example of exchange rate indeterminacy 1147 fiscal policy Bassetto provides an answer by arguing that the government should formulate a strategy for that and all other arbitrary outcomes Specifically the following government strategy supports the desired fiscal policy as a unique equilibrium outcome If some debt cannot be sold the government responds by increasing taxes to make up for the present shortfall but without altering future taxes Thus the onset of a debt crisis would be accompanied by an increase in the amount of resources that are offered in repayment of debt and hence an increase in the rate of return of government debt As a consequence any rational household would respond to a debt crisis by lending the government more rather than less which ensures that no such crisis can occur in an equilibrium16 Because Bassettos argument works equally well in a real economy the pre ceding paragraph did not mention money or nominal prices Moreover our omission of money seems appropriate since Bassetto studies a cashless economy where the relative price of goods and nominal bonds merely determines the value of the unit of account the dollar Atkeson et al 2010 extend the analysis to a monetary economy and follow the same approach to multiplicity of equilibria that we took in the cashinadvance model in section 16173 While theirs is a newKeynesian model they analyze sunspot equilibria that satisfy a constraint similar to ours when we imposed an unchanged value for the denominator of equation 16167 without any constraint on each individual nextperiod price level In the analysis of Atkeson et al 2010 the corresponding restriction on sunspot equilibria is that the expected inflation is unchanged when perturbing the sunspotdriven uncertainty in next periods price level Note that different versions of the fiscal theory of the price level share the same key assumption that a government can fully commit to its policy or strat egy In chapters 24 and 25 we study credible government policies policies that a government would like to enact under all circumstances 16 A similar strategy would establish Bassettos version of the fiscal theory of the price level in section 2742 For example suppose that each government promises to increase taxation in order to purchase its currency if it turns worthless say at the price level that would have prevailed in case ii Such strategies can effectively rule out cases i and iii as equilibrium outcomes and make exchange rate e 1 the only possible equilibrium Optimal inflation tax the Friedman rule 1151 The Ramsey problem is to maximize expression 2722 subject to equation 27512 and a feasibility constraint that combines equations 2751 through 2753 1 ℓt Hct ˆmt1 ct gt 0 27513 Let Φ and θt t0 be a Lagrange multiplier on equation 27512 and a se quence of Lagrange multipliers on equation 27513 respectively Firstorder conditions for this problem are ct uct Φ ucctct uct uℓct 1 ℓt 1 νHct ˆmt1 1 νuℓtHct θt Hct 1 0 27514a ℓt uℓt Φ ucℓtct uℓt uℓℓt 1 ℓt 1 νHct ˆmt1 θt 27514b ˆmt1 H ˆmt Φ1 νuℓt θt 0 27514c The firstorder condition for real money balances 27514c is satisfied when either H ˆmt 0 or θt Φ1 νuℓt 27515 We now show that equation 27515 cannot be a solution of the problem Notice that when ν 1 equation 27515 implies that the multipliers Φ and θt will either be zero or have opposite signs Such a solution is excluded because Φ is nonnegative while the insatiable utility function implies that θt is strictly positive When ν 1 a strictly positive θt also excludes equation 27515 as a solution To reject equation 27515 for ν 0 1 we substitute equation 27515 into equation 27514b uℓt Φ ucℓtct νuℓt uℓℓt 1 ℓt 1 νHct ˆmt1 0 which is a contradiction because the left side is strictly positive given our as sumption that ucℓt 0 We conclude that equation 27515 cannot charac terize the solution of the Ramsey problem when the transaction technology is homogeneous of degree ν 0 so the solution has to be H ˆmt 0 In other words the social planner follows the Friedman rule and satiates the economy with real balances According to condition 2758c this aim can be accom plished with a monetary policy that sustains a zero net nominal interest rate 1152 FiscalMonetary Theories of Inflation As an illustration of how the Ramsey plan is implemented suppose that gt g in all periods Example 1 of chapter 16 presents the Ramsey plan for this case if there were no transaction technology and no money in the model The optimal outcome is characterized by a constant allocation ˆc ˆn and a constant tax rate ˆτ that supports a balanced government budget We conjecture that the Ramsey solution to the present monetary economy shares that real allocation But how can it do so in the present economy with its additional constraint in the form of a transaction technology First notice that the preceding Ramsey solution calls for satiating the economy with real balances so there will be no time allocated to shopping in the Ramsey outcome Second the real balances needed to satiate the economy are constant over time and equal to Mt1 pt ψˆc t 0 27516 and the real return on money is equal to the constant real interest rate pt pt1 R t 0 27517 Third the real balances in equation 27516 also equal the real value of assets acquired by the government in period 0 from selling the money supply M1 to the households These government assets earn a net real return in each future period equal to R 1ψˆc R Mt pt1 Mt1 pt pt1 pt Mt pt1 Mt1 pt Mt Mt1 pt where we have invoked equations 27516 and 27517 to show that the in terest earnings just equal the funds for retiring currency from circulation in all future periods needed to sustain an equilibrium in the money market with a zero net nominal interest rate It is straightforward to verify that households would be happy to incur the indebtedness of the initial period They use the borrowed funds to acquire money balances and meet future interest payments by surrendering some of these money balances Yet their real money balances are unchanged over time because of the falling price level In this way money holdings are costless to the households and their optimal decisions with respect to consumption and labor are the same as in the nonmonetary version of this economy Time consistency of monetary policy 1153 276 Time consistency of monetary policy The optimality of the Friedman rule was derived in the previous section under the assumption that the government can commit to a plan for its future actions The Ramsey plan is not time consistent and requires that the government have a technology to bind itself to it In each period along the Ramsey plan the govern ment is tempted to levy an unannounced inflation tax in order to reduce future distortionary labor taxes Rather than examine this time consistency problem due to distortionary taxation we now turn to another time consistency problem arising from a situation where surprise inflation can reduce unemployment Kydland and Prescott 1977 and Barro and Gordon 1983a 1983b study the time consistency problem and credible monetary policies in reducedform models with a tradeoff between surprise inflation and unemployment In their spirit Ireland 1997 proposes a model with microeconomic foundations that gives rise to such a tradeoff because monopolistically competitive firms set nominal goods prices before the government sets monetary policy17 The gov ernment is here tempted to create surprise inflation that erodes firms markups and stimulates employment above a suboptimally low level But any anticipated inflation has negative welfare effects that arise as a result of a postulated cash inadvance constraint More specifically anticipated inflation reduces the real value of nominal labor income that can be spent or invested first in the next period thereby distorting incentives to work The following setup modifies Irelands model and assumes that each house hold has some market power with respect to its labor supply while a single good is produced by perfectly competitive firms 17 Irelands model takes most of its structure from those developed by Svensson 1986 and Rotemberg 1987 See Rotemberg and Woodford 1997 and King and Wolman 1999 for empirical implementations of related models Time consistency of monetary policy 1155 2 The nominal wage for labor of type i at time t is chosen by household i at the very beginning of period t Given the nominal wage wit household i is obliged to deliver any amount of labor nit that is demanded in the economy The governments only task is to increase or decrease the money supply by making lumpsum transfers xt 1Mt to the households where Mt is the per capita money supply at the beginning of period t and xt is the gross growth rate of money in period t Mt1 xtMt 2764 Following Ireland 1997 we assume that xt β x These bounds on money growth ensure the existence of a monetary equilibrium The lower bound will be shown to yield a zero net nominal interest rate in a stationary equilibrium whereas the upper bound x guarantees that households never abandon the use of money altogether During each period t events unfold as follows for household i The house hold starts period t with money mit and real private bonds bit and the house hold sets the nominal wage wit for its type of labor After the wage is deter mined the government chooses a nominal transfer xt 1Mt to be handed over to the household Thereafter the household enters the asset market to settle maturing bonds bit and to pick a new portfolio composition with money and real bonds bit1 After the asset market has closed the household splits into a shopper and a worker20 During period t the shopper purchases cit units of the single good subject to the cashinadvance constraint mit pt xt 1Mt pt bit bit1 Rt cit 2765 where pt and Rt are the price level and the real interest rate respectively Given the households predetermined nominal wage wit the worker supplies all the labor nit demanded by firms At the end of period t when the goods market has closed the shopper and the worker reunite and the households money holdings mit1 now equal the workers labor income witnit plus any unspent cash from the shopping round Thus the budget constraint of the 20 The interpretation that the household splits into a shopper and a worker follows Lucass 1980b cashinadvance framework It embodies the constraint on transactions recommended by Clower 1967 Time consistency of monetary policy 1157 The Lagrange multiplier λit is the shadow value of relaxing the budget con straint in period t by one unit measured in utils at time t Since preferences 2761 are linear in the disutility of labor λ1 it is the value of leisure in period t in terms of the units of the budget constraint at time t Equation 2768 is then the familiar expression that the monopoly price ˆwyt nit should be set as a markup above marginal cost λ1 it and the markup is inversely related to the absolute value of the demand elasticity of labor type i ǫit Firstorder conditions 2767c and 2767d for asset decisions can be used to solve for rates of return pt pt1 λit β λit1 µit1 2769a Rt λit µit β λit1 µit1 2769b Whenever the Lagrange multiplier µit on the cashinadvance constraint is strictly positive money has a lower rate of return than bonds or equivalently the net nominal interest rate is strictly positive as shown in equation 2728 Given initial conditions mi0 M0 and bi0 0 we now turn to character izing an equilibrium under the additional assumption that the cashinadvance constraint 2765 holds with equality even when it does not bind Since all households are perfectly symmetric they will make identical consumption and labor decisions cit ct and nit nt so by goods market clearing and the constantreturnstoscale technology 2762 we have ct yt nt 27610a and from the expression for the marginal product of labor in equation 2763 ˆwyt nt 1 27610b Equilibrium asset holdings satisfy mit1 Mt1 and bit1 0 The substi tution of equilibrium quantities into the cashinadvance constraint 2765 at equality yields Mt1 pt ct 27610c where a version of the quantity theory of money determines the price level pt Mt1ct We now substitute this expression and conditions 2767a and Time consistency of monetary policy 1159 The Ramsey plan then follows directly from inspecting the oneperiod return of the Ramsey optimization problem cγ t γ ct 27613 which is strictly concave and reaches a maximum at c 1 Thus the Ramsey solution calls for xt1 β for t 0 in order to support ct c for t 0 Notice that the Ramsey outcome can be supported by any initial money growth x0 It is only future money growth rates that must be equal to β in order to eliminate labor supply distortions that would otherwise arise from the cashin advance constraint if the return on money were to fall short of the return on bonds The Ramsey outcome equalizes the returns on money and bonds that is it implements the Friedman rule with a zero net nominal interest rate It is instructive to highlight the inability of the Ramsey monetary policy to remove the distortions coming from monopolistic wage setting Using the fact that the equilibrium real wage is unity we solve for λit from equation 2768 and substitute into equation 2767a cγ1 it µit 1 α 1 α 1 27614 The left side of equation 27614 is the marginal utility of consumption Since technology 2762 is linear in labor the marginal utility of consumption should equal the marginal utility of leisure in a firstbest allocation But the right side of equation 27614 exceeds unity which is the marginal utility of leisure given preferences 2761 While the Ramsey monetary policy succeeds in removing distortions from the cashinadvance constraint by setting the Lagrange multi plier µit equal to zero the policy cannot undo the distortion of monopolistic wage setting manifested in the markup 1 α1 α23 Notice that the Ramsey solution converges to the firstbest allocation when the parameter α goes to zero that is when households market power goes to zero To illustrate the time consistency problem we now solve for the Ramsey plan when the initial nominal wages are taken as given wi0 w0 βM0 xM0 First setting the initial period 0 aside it is straightforward to show that the solution for t 1 is the same as before That is the optimal policy calls for 23 The government would need to use fiscal instruments that is subsidies and taxation to correct the distortion from monopolistically competitive wage setting 1160 FiscalMonetary Theories of Inflation xt1 β for t 1 in order to support ct c for t 1 Second given w0 the firstbest outcome c0 1 can be attained in the initial period by choosing x0 w0M0 The resulting money supply M1 w0 will then serve to transact c0 1 at the equilibrium price p0 w0 Specifically firms are happy to hire any number of workers at the wage w0 when the price of the good is p0 w0 At the price p0 w0 the goods market clears at full employment since shoppers seek to spend their real balances M1p0 1 The labor market also clears because workers are obliged to deliver the demanded n0 1 Finally money growth x1 can be chosen freely and does not affect the real allocation of the Ramsey solution The reason is that because of the preset wage w0 there cannot be any labor supply distortions at time 0 arising from a low return on money holdings between periods 0 and 1 2764 Credibility of the Friedman rule Our comparison of the Ramsey equilibria with or without a preset initial wage w0 hints at the governments temptation to create positive monetary surprises that will increase employment We now ask if the Friedman rule is credible when the government lacks the commitment technology implicit in the Ramsey optimization problem Can the Friedman rule be supported with a trigger strat egy where a government deviation causes the economy to revert to the worst possible subgame perfect equilibrium Using the concepts and notation of chapter 24 we specify the objects of a strategy profile and state the definition of a subgame perfect equilibrium SPE Even though households possess market power with respect to their labor type they remain atomistic visavis the government We therefore stay within the framework of chapter 24 where the government behaves strategically and the households behavior can now be summarized as a monopolistically competitive equilibrium that responds nonstrategically to the governments choices At every date t for all possible histories a strategy of the households σh and a strategy of the government σg specify actions wt W and xt X β x respectively where wt wt Mt and xt Mt1 Mt Time consistency of monetary policy 1161 That is the actions multiplied by the beginningofperiod money supply Mt produce a nominal wage and a nominal money supply This scaling of nominal variables is used by Ireland 1997 throughout his analysis since the size of the nominal money supply at the beginning of a period has no significance per se Definition A strategy profile σ σh σg is a subgame perfect equilibrium if for each t 0 and each history wt1 xt1 W t Xt 1 Given the trajectory of money growth rates xt1j xσ wt1xt1j j1 the wagesetting outcome wt σh t wt1 xt1 constitutes a monopolistically competitive equilibrium 2 The government cannot strictly improve the households welfare by deviating from xt σg t wt1 xt1 that is by choosing some other money growth rate η X with the implied continuation strategy profile σ wtxt1η Besides changing to a monopolistically competitive equilibrium the main dif ference from Definition 6 of chapter 24 lies in requirement 1 The equilibrium in period t can no longer be stated in terms of an isolated government action at time t but requires the trajectory of the current and all future money growth rates generated by the strategy profile σ wt1xt1 The monopolistically com petitive equilibrium in requirement 1 is understood to be the perfect foresight equilibrium described previously When the government is contemplating a de viation in requirement 2 the equilibrium is constructed as follows In period t when the deviation takes place equilibrium consumption ct is a function of η and wt as implied by the cashinadvance constraint at equality ct ηMt pt ηMt wt η wt 27615 where we use the equilibrium condition pt wt Starting in period t 1 the deviation has triggered a switch to a new perfect foresight equilibrium with a trajectory of money growth rates given by xtj xσ wtxt1ηj j1 We conjecture that the worst SPE has ct c for all periods and the candi date strategy profile ˆσ is ˆσh t x c t wt1 xt1 ˆσg t x t wt1 xt1 The strategy profile instructs the government to choose the highest permissible money growth rate x for all periods and for all histories Similarly the house holds are instructed to set the nominal wages that would constitute a perfect Concluding remarks 1163 less incentive to deviate when households are patient and put a high weight on future outcomes Moreover the Friedman rule is credible for a sufficiently small value of α which is equivalent to households having little market power The associated small distortion from monopolistically competitive wage setting means that the potential welfare gain of a monetary surprise is also small so the government is less tempted to deviate from the Friedman rule 277 Concluding remarks Besides shedding light on a number of monetary doctrines this chapter has brought out the special importance of the initial date t 0 in the analysis This point is especially pronounced in Woodfords 1995 model where the initial interestbearing government debt B0 is not indexed but rather denominated in nominal terms So although the construction of a perfect foresight equilibrium ensures that all future issues of nominal bonds will ex post yield the real rates of return that are needed to entice the households to hold these bonds the realized real return on the initial nominal bonds can be anything depending on the price level p0 Activities at the initial date were also important when we considered dynamic optimal taxation in chapter 16 Monetary issues are also discussed in other chapters of the book Chapters 9 and 18 study money in overlapping generations models and Bewley mod els respectively Chapters 28 and 29 present other explicit environments that give rise to a positive value of fiat money Townsends turnpike model and the KiyotakiWright search model Exercises 1167 Let the rate of return on money be Rmt ptpt1 Let the nominal interest rate at time t be 1 it Rtpt1pt Rtπt a Derive the demand for money and show that it decreases with the nominal interest rate b Suppose that the government policy is such that gt g Bt B and τt τ Prove that the real interest rate R is constant and equal to the inverse of the discount factor c Define the deficit as d where d g BRR1τ What is the highest possible deficit that can be financed in this economy An economist claims that increases in d which leave g unchanged will result in increases in the inflation rate Discuss this view d Suppose that the economy is open to international capital flows and that the world interest rate is R β1 Assume that d 0 and that Mt M At t T the government increases the money supply to M 1 µM This increase in the money supply is used to purchase government bonds This of course results in a smaller deficit at t T In this case it will result in a surplus However the government also announces its intention to cut taxes starting at T 1 to bring the deficit back to zero Argue that this open market operation will have the effect of increasing prices at t T by µ p 1 µp where p is the price level from t 0 to t T 1 e Consider the same setting as in d Suppose now that the open market operation is announced at t 0 it still takes place at t T Argue that prices will increase at t 0 and in particular that the rate of inflation between T 1 and T will be less than 1 µ Exercise 275 Interest elasticity of the demand for money donated by Rodolfo Manuelli Consider an economy in which the demand for money satisfies mt1pt Fct RmtRt where Rmt ptpt1 and Rt is the oneperiod interest rate Consider the following open market operation At t 0 the government sells bonds and destroys the money it receives in exchange for those bonds No other real variables eg government spending or taxes are changed Find conditions on 1168 FiscalMonetary Theories of Inflation the income elasticity of the demand for money such that the decrease in money balances at t 0 results in an increase in the price level at t 0 Exercise 276 Dollarization donated by Rodolfo Manuelli In recent years several countries eg Argentina and countries hit by the Asian crisis have considered the possibility of giving up their currencies in favor of the US dollar Consider a country say A with deficit d and inflation rate π 1Rm Output and consumption are constant and hence the real interest rate is fixed with R β1 The grossofinterestpayments deficit is d with d g τ BRR 1 Let the demand for money be mt1pt Fct RmtRt and assume that ct y g Thus the steadystate government budget constraint is d Fy g βRm1 Rm 0 Assume that the country is considering at t 0 the retirement of its money in exchange for dollars The government promises to give to each person who brings a peso to the Central Bank 1e dollars where e is the exchange rate in pesos per dollar between the countrys currency and the US dollar Assume that the US inflation rate before and after the switch is given and equal to π 1R m π and that the country is on the good part of the Laffer curve a If you are advising the government of A how much would you say that it should demand from the US government to make the switch Why b After the dollarization takes place the government understands that it needs to raise taxes Economist 1 argues that the increase in taxes on a per period basis will equal the loss of revenue from inflation Fy g βRm1 Rm while Economist 2 claims that this is an overestimate More precisely he or she claims that if the government is a good negotiator visavis the US government taxes need only increase by Fy g βRm1 Rm Fy g βR m1 R m per period Discuss these two views Exercise 277 Currency boards donated by Rodolfo Manuelli In the last few years several countries eg Argentina 1991 Estonia 1992 Lithuania 1994 Bosnia 1997 and Bulgaria 1997 have adopted the currency Exercises 1169 board model of monetary policy In a nutshell a currency board is a commitment on the part of the country to fully back its domestic currency with foreign denominated assets For simplicity assume that the foreign asset is the US dollar The governments budget constraint is given by gt Bt B t1eRpt τt Bt1R B t ept Mt1 Mtpt where B t is the stock of oneperiod bonds denominated in dollars held by this country e is the exchange rate pesos per dollar and 1R is the price of oneperiod bonds both domestic and dollar denominated Note that the budget constraint equates the real value of income and liabilities in units of consumption goods The currency board contract requires that the money supply be fully backed One interpretation of this rule is that the domestic money supply is Mt eB t Thus the right side is the local currency value of foreign reserves in bonds held by the government while the left side is the stock of money Finally let the law of one price hold pt ep t where p t is the foreign US price level a Assume that Bt B and that foreign inflation is zero p t p Show that even in this case the properties of the demand for money which you may take to be given by Fy g βRm are important in determining total revenue In particular explain how a permanent increase in y income per capita allows the government to lower taxes permanently b Assume that Bt B Let foreign inflation be positive that is π 1 In this case the price in dollars of a oneperiod dollardenominated bond is 1Rπ Go as far as you can describing the impact of foreign inflation on domestic inflation and on per capita taxes τ c Assume that Bt B Go as far as you can describing the effects of a once andforall surprise devaluation ie an unexpected and permanent increase in e on the level of per capita taxes 1170 FiscalMonetary Theories of Inflation Exercise 278 Growth and inflation donated by Rodolfo Manuelli Consider an economy populated by identical individuals with instantaneous util ity function given by uc ℓ cϕℓ1ϕ1σ1 σ Assume that shopping time is given by st ψctmt1pt Assume that in this economy income grows exogenously at the rate γ 1 Thus at time t yt γty Assume that government spending also grows at the same rate gt γtg Finally ct yt gt a Show that for this specification if the demand for money at t is x mt1pt then the demand at t 1 is γx Thus the demand for money grows at the same rate as the economy b Show that the real rate of interest depends on the growth rate You may assume that ℓ is constant for this calculation c Argue that even for monetary policies that keep the price level constant that is pt p for all t the government raises positive amounts of revenue from printing money Explain d Use your finding in c to discuss why following monetary reforms that generate big growth spurts many countries manage to monetize their economies this is just jargon for increases in the money supply without generating inflation Chapter 28 Credit and Currency 281 Credit and currency with longlived agents This chapter describes Townsends 1980 turnpike model of money and puts it to work The model uses a particular pattern of heterogeneity of endowments and locations to create a demand for currency The model is more primitive than the shopping time model of chapter 27 As with the overlapping generations model the turnpike model starts from a setting in which diverse intertemporal endowment patterns across agents prompt borrowing and lending If something prevents loan markets from operating it is possible that an unbacked currency can play a role in helping agents smooth their consumption over time Following Townsend we shall eventually appeal to locational heterogeneity as the force that causes loan markets to fail in this way The turnpike model can be viewed as a simplified version of the stochastic model proposed by Truman Bewley 1980 We use the model to study a number of interrelated issues and theories including 1 a permanent income model of consumption 2 a Ricardian doctrine that government borrowing and taxes have equivalent economic effects 3 some restrictions on the operation of private loan markets needed in order that unbacked currency be valued 4 a theory of inflationary finance 5 a theory of the optimal inflation rate and the optimal behavior of the currency stock over time 6 a legal restrictions theory of inflationary finance and 7 a theory of exchange rate indeterminacy1 1 Some of the analysis in this chapter follows Manuelli and Sargent 2010 Also see Chatterjee and Corbae 1996 and Ireland 1994 for analyses of policies within a turnpike environment 1171 1174 Credit and Currency where µ is a nonnegative Lagrange multiplier The firstorder conditions for the households problem are βtuct µq0 t if ct 0 Definition 1 A competitive equilibrium is a price sequence qo t t0 and an allocation co t ce t t0 that have the property that a given the price sequence the allocation solves the optimum problem of households of both types and b co t ce t 1 for all t 0 To find an equilibrium we have to produce an allocation and a price system for which we can verify that the firstorder conditions of both households are satisfied We start with a guess inspired by the constantconsumption property of the Pareto optimal allocation We guess that co t co ce t ce t where ce co 1 This guess and the firstorder condition for the odd agents imply q0 t βtuco µo or q0 t q0 0βt 2834 where we are free to normalize by setting q0 0 1 For odd agents the right side of the budget constraint evaluated at the prices given in equation 2834 is then 1 1 β2 and for even households it is β 1 β2 The left side of the budget constraint evaluated at these prices is ci 1 β i o e For both of the budget constraints to be satisfied with equality we evidently require that co 1 β 1 ce β β 1 2835 1178 Credit and Currency Definition 3 A competitive equilibrium is an allocation co t ce t t0 nonneg ative money holdings mo t me t t1 and a nonnegative price level sequence pt t0 such that a given the price level sequence and mo 1 me 1 the al location solves the optimum problems of both types of households and b co t ce t 1 mo t1 me t1 MN for all t 0 The periodic nature of the endowment sequences prompts us to guess the following twoparameter form of stationary equilibrium co t t0 c0 1 c0 c0 1 c0 ce t t0 1 c0 c0 1 c0 c0 2843 and pt p for all t 0 To determine the two undetermined parameters c0 p we use the firstorder conditions and budget constraint of the odd agent at time 0 His endowment sequence for periods 0 and 1 yo 0 yo 1 1 0 and the Inada condition 2821 ensure that both of his firstorder conditions at time 0 will hold with equality That is his desire to set co 0 0 can be met by consuming some of the endowment yo 0 and the only way for him to secure consumption in the following period 1 is to hold strictly positive money holdings mo 0 0 From his firstorder conditions at equality we obtain βu1 c0 p uc0 p which implies that c0 is to be determined as the root of β uc0 u1 c0 0 2844 Because β 1 it follows that c0 5 1 To determine the price level we use the odd agents budget constraint at t 0 evaluated at mo 1 0 and mo 0 MN to get pc0 MN p 1 or p M N1 c0 2845 See Figure 2841 for a graphical determination of c0 From equation 2844 it follows that for β 1 c0 05 and 1 c0 05 Thus both types of agents experience fluctuations in their consumption sequences in this monetary equilibrium Because Pareto optimal allocations have constant consumption sequences for each type of agent this equilibrium allocation is not Pareto optimal 1184 Credit and Currency which by 2862 is the zeroinflation equilibrium τ 0 For the even agents the preferred allocation given by U ec0 0 implies c0 05 and can there fore not be implemented as a monetary equilibrium above Hence the even agents preferred stationary monetary equilibrium is the one with the smallest permissible c0 ie c0 05 According to 2862 this allocation can be supported by choosing money growth rate 1 τ β which is then also the equilibrium gross rate of deflation Notice that all agents both odd and even are in agreement that they prefer no inflation to positive inflation that is they prefer c0 determined by 2864 to any higher value of c0 To abstract from the described conflict of interest between odd and even agents suppose that the agents must pick their preferred monetary policy under a veil of ignorance before knowing their true identity Since there are equal numbers of each type of agent an individual faces a fiftyfifty chance of her identity being an odd or an even agent Hence prior to knowing ones identity the expected lifetime utility of an agent is Uc0 1 2U oc0 1 2U ec0 uc0 u1 c0 21 β The ex ante preferred allocation c0 is determined by the firstorder condition U c0 0 which has the solution c0 05 Collecting equations 2861 2862 and 2863 this preferred policy is characterized by pt pt1 1 1 τ uco t βuco t1 uce t βuce t1 1 β t 0 where ci j 05 for all j 0 and i o e Thus the real return on money ptpt1 equals a common marginal rate of intertemporal substitution β1 and this return would therefore also constitute the real interest rate if there were a credit market Moreover since the gross real return on money is the inverse of the gross inflation rate it follows that the gross real interest rate β1 multiplied by the gross rate of inflation is unity or the net nominal interest rate is zero In other words all agents are ex ante in favor of Friedmans rule Figure 2861 shows the utility possibility frontier associated with this econ omy Except for the allocation associated with Friedmans rule the allocations associated with stationary monetary equilibria lie inside the utility possibility frontier Inflationary finance 1185 o U e U Friedmans Rule Arrow Debreu Zero Inflation Monetary Equilibrium A B C Figure 2861 Utility possibility frontier on the Townsend turn pike The locus of points ABC denotes allocations attainable in stationary monetary equilibria Point B is the allocation asso ciated with the zeroinflation monetary equilibrium Point A is associated with Friedmans rule while points between B and C correspond to stationary monetary equilibria with inflation 287 Inflationary finance The government prints new currency in total amount MtMt1 in period t and uses it to purchase a constant amount G of goods in period t The governments time t budget constraint is Mt Mt1 ptG t 0 2871 1186 Credit and Currency Preferences and endowment patterns of odd and even agents are as specified previously We now use the following definition Definition 4 A competitive equilibrium is a price level sequence pt t0 a money supply process Mt t1 an allocation co t ce t Gt t0 and nonnega tive money holdings mo t me t t1 such that a given the price sequence and mo 1 me 1 the allocation solves the optimum problems of households of both types b the governments budget constraint is satisfied for all t 0 and c Nco t ce t Gt N for all t 0 and mo t me t MtN for all t 1 For t 1 write the governments budget constraint as Mt Npt pt1 pt Mt1 Npt1 G N or mt Rt1 mt1 g 2872 where g GN mt MtNpt is peroddperson real balances and Rt1 pt1pt is the rate of return on currency from t 1 to t To compute an equilibrium we guess an allocation of the periodic form co t t0 c0 1 c0 g c0 1 c0 g ce t t0 1 c0 g c0 1 c0 g c0 2873 We guess that Rt R for all t 0 and again guess a quantity theory outcome mt m t 0 Evaluating the odd households time 0 firstorder condition for currency at equality gives βR uc0 u1 c0 g 2874 With our guess real balances held by each odd agent at the end of period 0 mo 0p0 equal 1c0 and time 1 consumption which also is R times the value of these real balances held from 0 to 1 is 1c0g Thus 1c0R 1c0g which implies that R 1 c0 g 1 c0 2875 Equations 2874 and 2875 are two simultaneous equations that we want to solve for c0 R Inflationary finance 1187 0 01 02 03 04 05 06 07 08 09 0 01 02 03 04 05 06 07 08 09 1 Figure 2871 Revenue from inflation tax mR1 R and deficit for β 95 δ 2 g 2 The gross rate of return on currency is on the xaxis g and the revenue from inflation are on the yaxis Use equation 2875 to eliminate 1c0 g from equation 2874 to get βR uc0 uR1 c0 Recalling that 1 c0 m0 this can be written βR u1 m0 uRm0 2876 For the power utility function uc c1δ 1δ this equation can be solved for m0 to get the demand function for currency m0 mR βR1δ1δ 1 βR1δ1δ 2877 Substituting this into the government budget constraint 2872 gives mR1 R g 2878 This equation equates the revenue from the inflation tax namely mR1 R to the government deficit g The revenue from the inflation tax is the product of real balances and the inflation tax rate 1 R The equilibrium value of R solves equation 2878 1188 Credit and Currency 0 005 01 015 02 025 03 0 01 02 03 04 05 06 07 08 09 1 Figure 2872 Revenue from inflation tax mR1 R and deficit for β 95 δ 7 g 2 The rate of return on currency is on the xaxis g and the revenue from inflation are on the yaxis Here there is a Laffer curve Figures 2871 and 2872 depict the determination of the stationary equilib rium value of R for two sets of parameter values For the case δ 2 shown in Figure 2871 there is a unique equilibrium R there is a unique equilibrium for every δ 1 For δ 1 the demand function for currency slopes upward as a function of R as for the example in Figure 2873 For δ 1 there can occur multiple stationary equilibria as for the example in Figure 2872 In such cases there is a Laffer curve in the revenue from the inflation tax Notice that the demand for real balances is downward sloping as a function of R when δ 1 The initial price level is determined by the time 0 budget constraint of the government evaluated at equilibrium time 0 real balances In particular the time 0 government budget constraint can be written M0 Np0 M1 Np0 g or m g M1 Np0 Equating m to its equilibrium value 1 c0 and solving for p0 gives p0 M1 N1 c0 g Legal restrictions 1189 01 015 02 025 03 035 04 045 05 0 01 02 03 04 05 06 07 08 09 1 Figure 2873 Demand for real balances on the yaxis as a func tion of the gross rate of return on currency on the xaxis when β 95 δ 2 288 Legal restrictions This section adapts ideas of Bryant and Wallace 1984 and Villamil 1988 to the turnpike environment Those authors analyzed situations in which the government could make all savers better off by introducing a price discrimination scheme for marketing its debt The analysis formalizes some ideas mentioned by John Maynard Keynes 1940 Figure 2881 depicts the terms on which an odd agent at t 0 can transfer consumption between 0 and 1 in an equilibrium with inflationary finance The agent is endowed at the point 1 0 The monetary mechanism allows him to transfer consumption between periods on the terms c1 R1 c0 depicted by the budget line connecting 1 on the ctaxis with the point B on the ct1axis The government insists on raising revenues in the amount g for each pair of an odd and an even agent which means that R must be set so that the tangency between the agents indifference curve and the budget line c1 R1c0 occurs 1190 Credit and Currency ct1 ct 1g 1g 1 1 c R1c 0 1 I A B 1F H D Figure 2881 The budget line starting at 1 0 and ending at the point B describes an odd agents time 0 opportunities in an equilibrium with inflationary finance Because this equilibrium has the private consumption feasibility menu intersecting the odd agents indifference curve a forced saving legal restriction can be used to put the odd agent onto a higher indifference curve than I while leaving even agents better off and the government with revenue g If the individual is confronted with a minimum denomination F at the rate of return associated with the budget line ending at H he would choose to consume 1 F at the intersection of the budget line and the straight line connecting 1 g on the ct axis with the point 1 g on the ct1axis At this point the marginal rate of substitution for odd agents is uc0 βu1 c0 g R Legal restrictions 1191 ct1 ct 1g 1g 1 1 c R1c 0 1 I A B 1F H D I E Figure 2882 The minimum denomination F and the return on money can be lowered visavis their setting associated with line DH in Figure 2881 to make the odd household better off raise the same revenues for the government and leave even households better off as compared to no government intervention The lower value of F puts the odd household at E which leaves him at the higher indifference curve I The minimum denomination F and the return on money can be lowered visavis their setting asso ciated with line DH in Figure 2881 to make the odd household better off raise the same revenues for the government and leave even households better off as compared to no government inter vention The lower value of F puts the odd household at E which leaves him at the higher indifference curve I because currency holdings are positive For even agents the marginal rate of substitution is u1 c0 g βuc0 1 β2R 1 1192 Credit and Currency where the inequality follows from the fact that R 1 under inflationary finance The fact that the odd agents indifference curve intersects the solid line con necting 1 g on the two axes indicates that the government could improve the welfare of the odd agent by offering him a higher rate of return subject to a minimal real balance constraint The higher rate of return is used to send the line c1 1 Rc0 into the lensshaped area in Figure 2881 onto a higher indifference curve The minimal real balance constraint is designed to force the agent onto the postgovernment share feasibility line connecting the points 1 g on the two axes Thus notice that in Figure 2881 the government can raise the same rev enue by offering odd agents the higher rate of return associated with the line connecting 1 on the ct axis with the point H on the ct1 axis provided that the agent is required to save at least F if he saves at all This minimum saving requirement would make the households budget set the point 1 0 together with the heavy segment DH With the setting of F R associated with the line DH in Figure 2881 odd households have the same twoperiod utility as with out this scheme Points D and A lie on the same indifference curve However it is apparent that there is room to lower F and lower R a bit and thereby move the odd household into the lensshaped area See Figure 2882 The marginal rates of substitution that we computed earlier indicate that this scheme makes both odd and even agents better off relative to the original equilibrium The odd agents are better off because they move into the lens shaped area in Figure 2881 The even agents are better off because relative to the original equilibrium they are being permitted to borrow at a gross rate of interest of 1 Since their marginal rate of substitution at the original equilibrium is 1β2R 1 this ability to borrow makes them better off 1194 Credit and Currency We use the following definition Definition 5 A competitive equilibrium with two valued fiat currencies is an al location co t ce t G1t G2t t0 nonnegative money holdings mo 1t me 1t mo 2t me 2t t1 a pair of finite price level sequences p1t p2t t0 and currency supply sequences M1t M2t t1 such that a given the price level sequences and mo 11 me 11 mo 21 me 21 the allocation solves the households problems b the budget constraints of the governments are satisfied for all t 0 and c Nco t ce t G1t G2t N for all t 0 and mo jt me jt MjtN for j 1 2 and all t 1 In the case of constant government expenditures G1t G2t Ng1 Ng2 for all t 0 we guess an equilibrium allocation of the form 2873 where we reinterpret g to be g g1 g2 We also guess an equilibrium with a constant real value of the world money supply that is m M1t Np1t M2t Np2t and a constant exchange rate so that we impose condition 2892 We let R p1tp1t1 p2tp2t1 be the constant common value of the rate of return on the two currencies With these guesses the sum of the two countries budget constraints for t 1 and the conjectured form of the equilibrium allocation imply an equation of the form 2878 where now mR M1t p1tN M2t p2tN Equation 2878 can be solved for R in the fashion described earlier Once R has been determined so has the constant real value of the world currency supply m To determine the time t price levels we add the time 0 budget constraints of the two governments to get M10 Np10 M20 Np20 M11 eM21 Np10 g1 g2 or m g M11 eM21 Np10 A twomoney model 1195 In the conjectured allocation m 1 c0 so this equation becomes M11 eM21 Np10 1 c0 g 2893 which given any e 0 has a positive solution for the initial country 1 price level Given the solution p10 and any e 0 the price level sequences for the two countries are determined by the constant rate of return on currency R To determine the values of the nominal currency stocks of the two countries we use the government budget constraints 2891 Our findings are a special case of the following remarkable proposition Proposition Exchange Rate Indeterminacy Given the initial stocks of currencies M11 M21 that are equally distributed among the even agents at time 0 if there is an equilibrium for one constant exchange rate e 0 then there exists an equilibrium for any ˆe 0 with the same consumption allocation but different currency supply sequences Proof Let p10 be the country 1 price level at time zero in the equilibrium that is assumed to exist with exchange rate e For the conjectured equilibrium with exchange rate ˆe we guess that the corresponding price level is ˆp10 p10 M11 ˆeM21 M11 eM21 After substituting this expression into 2893 we can verify that the real value at time 0 of the initial world money supply is the same across equi libria Next we guess that the conjectured equilibrium shares the same rate of return on currency R and constant endofperiod real value of the world money supply m as the the original equilibrium By construction from the original equilibrium we know that this setting of the world money supply pro cess guarantees that the consolidated budget constraint of the two governments is satisfied in each period To determine the values of each countrys prices and nominal money supplies we proceed as above That is given ˆp10 and ˆe the price level sequences for the two countries are determined by the constant rate of return on currency R The evolution of the nominal money stocks of the two countries is governed by government budget constraints 2891 Versions of this proposition were stated by Kareken and Wallace 1980 See chapter 27 for a discussion of a possible way to alter assumptions to make the exchange rate determinate A model of commodity money 1197 C1 eC2 e C0 C1 e e C0 C1 e e 0 45o Ct Ct1 Expansion path at MRS 1 1 1 C0 C1 eC2 e 1C 0 Figure 28101 Determination of equilibrium when uvS βuc0 For as long as it is feasible the even agent sets uce t1uce t β by running down his silver holdings This implies that ce t1 ce t during the rundown period Eventually the even agent runs out of silver so that the tail of his allocation is c0 1 c0 c0 1 c0 determined as before The figure depicts how the spending of silver pushes the agent onto lower twoperiod budget sets 28101 Equilibrium Definition 6 A competitive equilibrium is an allocation co t ce t t0 and non negative asset holdings mo t me t t1 such that given mo 1 me 1 the allocation solves each agents optimum problem Adding the budget constraints of the two types of agents with equality at time t gives co t ce t 1 vSt1 St 28101 where St mo t me t is the total per odd person stock of silver in the country at time t Equation 28101 asserts that total domestic consumption at time t is the sum of the countrys endowment plus its imports of goods where the latter equals its exports of silver vSt1 St Given the opportunity to choose nonnegative asset holdings with a gross rate of return equal to 1 the equilibrium allocation to the odd agent is co t t0 A model of commodity money 1199 to the even agents is determined by gluing this initial piece with declining consumption onto a tail of the allocation assigned to even agents in the original model starting on an odd date ct tT 1 c0 1 c0 c0 1 c0 9 28102 Virtue of fiat money This is a model with an exogenous price level and an endogenous stock of cur rency The model can be used to express a version of Friedmans and Keyness condemnation of commodity money systems the equilibrium allocation can be Pareto dominated by the allocation in a fiat money equilibrium in which in addition to the stock of silver at time 0 the even agents are endowed with M units of an unbacked fiat currency We can then show that there exists a monetary equilibrium with a constant price level p satisfying 2845 p M N1 c0 In effect the time 0 endowment of the even agents is increased by 1c0 units of consumption good Fiat money creates wealth by removing commodity money from circulation which instead can be transformed into consumption Since the initial horizon T satisfied 28102 and 28103 with nonnegative savings it follows that so must also horizon T 1 Therefore the largest horizon T must occur on an even date 9 Is the equilibrium with uvS βuc0 a stylized model of Spain in the sixteenth century At the beginning of the sixteenth century Spain suddenly received a large claim on silver and gold from the New World During the century Spain exported gold and silver to the rest of Europe to finance government and private purchases Exercises 1201 a Define a competitive equilibrium with oneperiod consumption loans b Compute a competitive equilibrium with oneperiod consumption loans c Is the equilibrium allocation Pareto optimal Compare the equilibrium allocation with that for the corresponding ArrowDebreu equilibrium for an economy with identical endowment and preference structure Exercise 283 Stock market Consider a stock market version of an economy with endowment and prefer ence structure identical to the one in the previous economy Now odd and even agents begin life owning one of two types of trees Odd agents own the odd tree which is a perpetual claim to a dividend sequence yo t t0 1 0 1 0 while even agents initially own the even tree which entitles them to a per petual claim on dividend sequence ye t t0 0 1 0 1 Each period there is a stock market in which people can trade the two types of trees These are the only two markets open each period The time t price of type j trees is aj t j o e The time t budget constraint of agent h is ch t ao tsho t ae tshe t ao t yo t sho t1 ae t ye t she t1 where shj t is the number of shares of stock in tree j held by agent h from t to t 1 We assume that soo 1 1 see 1 1 sjk 1 0 for j k a Define an equilibrium of the stock market economy b Compute an equilibrium of the stock market economy c Compare the allocation of the stock market economy with that of the corre sponding ArrowDebreu economy Exercise 284 Inflation Consider a Townsend turnpike model in which there are N odd agents and N even agents who have endowment sequences respectively of yo t t0 1 0 1 0 ye t t0 0 1 0 1 Exercises 1205 household of type h chooses to carry over mh t 0 of currency from time t to t 1 We start households out with these debts or assets at time 0 to support a stationary equilibrium Each period t 0 households can issue indexed oneperiod debt in amount bt promising to pay off btRt at t1 subject to the constraint that bt FRt where F 0 is a parameter characterizing the borrowing constraint and Rt is the rate of return on these loans between time t and t 1 When F 0 we get the BewleyTownsend model A households period t budget constraint is ct mtpt bt yt mt1pt bt1Rt1 where Rt1 is the gross real rate of return on indexed debt between time t 1 and t If bt 0 the household is borrowing at t and if bt 0 the household is lending at t a Define a competitive equilibrium in which valued fiat currency and private loans coexist b Argue that in the equilibrium defined in part a the real rates of return on currency and indexed debt must be equal c Assume that 0 F 1co2 where co is the solution of equation 2844 Show that there exists a stationary equilibrium with a constant price level and that the allocation equals that associated with the stationary equilibrium of the F 0 version of the model How does F affect the price level Explain d Suppose that F 1 co2 Show that there is a stationary equilibrium with private loans but that fiat currency is valueless in that equilibrium e Suppose that F β 1β For a stationary equilibrium find an equilibrium allocation and interest rate f Suppose that F 1 co2 β 1β Argue that there is a stationary equi librium without valued currency in which the real rate of return on debt is R 1 β1 Exercise 2810 Initial conditions and inside money Consider a version of the preceding model in which each odd person is initially endowed with no currency and no IOUs and each even person is initially en dowed with MN units of currency but no IOUs At every time t 0 each 1206 Credit and Currency agent can issue oneperiod IOUs promising to pay off FRt units of consump tion in period t 1 where Rt is the gross real rate of return on currency or IOUs between periods t and t1 The parameter F obeys the same restrictions imposed in exercise 289 a Find an equilibrium with valued fiat currency in which the tail of the alloca tion for t 1 and the tail of the price level sequence respectively are identical with that found in exercise 289 b Find the price level the allocation and the rate of return on currency and consumption loans at period 0 Exercise 2811 Real bills experiment Consider a version of exercise 289 The initial conditions and restrictions on borrowing are as described in exercise 289 However now the government augments the currency stock by an open market operation as follows In period 0 the government issues M M units per each odd agent for the purpose of purchasing units of IOUs issued at time 0 by the even agents Assume that 0 F At each time t 1 the government uses any net real interest payments from its stock IOUs from the private sector to decrease the outstanding stock of currency Thus the governments budget constraint sequence is M M p0 t 0 Mt Mt1 pt Rt1 1 t 1 HereRt1 is the gross rate of return on consumption loans from t 1 to t and Mt is the total stock of currency outstanding at the end of time t a Verify that there exists a stationary equilibrium with valued fiat currency in which the allocation has the form 2843 where c0 solves equation 2844 b Find a formula for the price level in this stationary equilibrium Describe how the price level varies with the value of c Does the quantity theory of money hold in this example Chapter 29 Equilibrium Search Matching and Lotteries 291 Introduction This chapter presents various equilibrium models of the labor market We de scribe 1 Lucas and Prescotts version of search in an island model 2 some matching models in the style of Mortensen Pissarides and Diamond and 3 a model of employment lotteries as formulated by Rogerson and Hansen Chapter 6 studied the optimization problem of a single unemployed worker who searches for a job by drawing from an exogenous wage offer distribution We now turn to a model with a continuum of workers who interact across a large number of spatially separated labor markets Phelps 1970 introductory chapter recommended such an island economy as a good model of labor market frictions We present an analysis in the spirit of Lucas and Prescotts 1974 version of such an economy Workers on an island can choose to work at the marketclearing wage in their own labor market or seek their fortune by moving to another island and its labor market In an equilibrium agents tend to move to islands that experience good productivity shocks while an island with bad productivity may see some of its labor force depart Frictional unemployment arises because moves between labor markets take time A distinct approach to modeling unemployment is the matching framework described by Diamond 1982 Mortensen 1982 and Pissarides 1990 This framework postulates a matching function that maps measures of unemployment and vacancies into a measure of matches A match pairs a worker and a firm who then have to bargain about how to share the match surplus that is the value that will be lost if the two parties cannot agree and break the match In contrast to the island model with pricetaking behavior and no externalities the decentralized outcome in the matching framework is in general not efficient Unless parameter values satisfy a knifeedge restriction there will be either too many or too few vacancies posted in an equilibrium The efficiency problem is further exacerbated if it is assumed that heterogeneous jobs must be created via 1207 A matching model 1213 xv u x vu x Figure 2921 The curve maps an economys average labor force per market x into the stationary equilibrium value to search vu 293 A matching model Another model of unemployment is the matching framework as described by Diamond 1982 Mortensen 1982 and Pissarides 1990 The basic model is as follows Let there be a continuum of identical workers with measure normalized to 1 The workers are infinitely lived and risk neutral The objective of each worker is to maximize the expected discounted value of leisure and labor income The leisure enjoyed by an unemployed worker is denoted z while the current utility of an employed worker is given by the wage rate w The workers discount factor is β 1 r1 The production technology is constant returns to scale with labor as the only input Each employed worker produces y units of output Without loss of generality suppose each firm employs at most one worker A firm entering the economy incurs a vacancy cost c in each period when looking for a worker and in a subsequent match the firms perperiod earnings are yw All matches are exogenously destroyed with perperiod probability s Free entry implies that the expected discounted stream of a new firms vacancy costs and earnings is equal to zero The firms have the same discount factor as the workers who would be the owners in a closed economy 1218 Equilibrium Search Matching and Lotteries and equation 29314 both curves are negatively sloped and convex to the origin y z r sα φ φ θ qθ 1 φqθ c 29320 When we also require that the point of tangency satisfy the equilibrium condition 29314 it can be seen that φ α maximizes the value of being unemployed in a decentralized equilibrium The solution is the same as the social optimum be cause the social planner and an unemployed worker both prefer an optimal rate of investment in vacancies one that takes matching externalities into account 2933 Size of the match surplus The size of the match surplus depends naturally on the output y produced by the worker which is lost if the match breaks up and the firm is left to look for another worker In principle this loss includes any returns to production factors used by the worker that cannot be adjusted immediately It might then seem puzzling that a common assumption in the matching literature is to exclude payments to physical capital when determining the size of the match surplus see eg Pissarides 1990 Unless capital can be moved without friction in the economy this exclusion of payments to physical capital must rest on some implicit assumption of outside financing from a third party that is removed from the wage bargain between the firm and the worker For example suppose the firms capital is financed by a financial intermediary that demands specific rental payments in order not to ask for the firms bankruptcy As long as the financial intermediary can credibly distance itself from the firms and workers bargaining it would be rational for the two latter parties to subtract the rental payments from the firms gross earnings and bargain over the remainder In our basic matching model there is no physical capital but there is invest ment in vacancies Let us consider the possibility that a financial intermediary provides a single firm funding for this investment The simplest contract would be that the intermediary hand over funds c to a firm with a vacancy in ex change for a promise that the firm pay ǫ in every future period of operation If the firm cannot find a worker in the next period it fails and the intermediary writes off the loan and otherwise the intermediary receives the stipulated inter est payment ǫ so long as a successful match stays in business This agreement Employment lotteries 1227 optimal steady state are such that firms would like freely to announce them and to participate in the corresponding markets without any wage bargaining The equal value of an unemployed worker across markets ensures the participation of workers who now also act as wage takers 295 Employment lotteries Consider a labor market without search and matching frictions but where labor is indivisible An individual can supply either one unit of labor or no labor at all as assumed by Hansen 1985 and Rogerson 1988 In such a setting em ployment lotteries can be welfare enhancing The argument is best understood in Rogersons static model but with physical capital and its implication of diminishing marginal product of labor removed from the analysis We assume that a single good can be produced with labor n as the sole input in a constant returns to scale technology fn γn where γ 0 2951 In a competitive equilibrium the equilibrium wage is then equal to γ Follow ing Hansen and Rogerson the preferences of an individual are assumed to be additively separable in consumption c and labor uc vn The standard assumptions are that both u and v are twice continuously differ entiable and increasing but while u is strictly concave v is convex However as pointed out by Rogerson the precise properties of the function v are not essential because of the indivisibility of labor The only values of vn that matter are v0 and v1 Let v0 0 and v1 A 0 An individual who can supply one unit of labor in exchange for γ units of goods would then choose to do so if uγ A u0 and otherwise the individual would choose not to work The proposed allocation might be improved upon by introducing employment lotteries That is each individual chooses a probability of working ψ 0 1 1228 Equilibrium Search Matching and Lotteries and he trades his stochastic labor earnings in contingency markets We assume a continuum of agents so that the idiosyncratic risks associated with employment lotteries do not pose any aggregate risk and the contingency prices are then determined by the probabilities of events occurring See chapters 8 13 and 14 Let c1 and c2 be the individuals choice of consumption when working and not working respectively The optimization problem becomes max c1c2ψ ψ uc1 A 1 ψ uc2 subject to ψc1 1 ψc2 ψγ c1 c2 0 ψ 0 1 At an interior solution for ψ the firstorder conditions for consumption imply that c1 c2 ψ uc1 ψ λ 1 ψ uc2 1 ψ λ where λ is the multiplier on the budget constraint Since there is no harm in also setting c1 c2 when ψ 0 or ψ 1 the individuals maximization problem can be simplified to read max cψ uc ψ A subject to c ψγ c 0 ψ 0 1 2952 The welfareenhancing potential of employment lotteries is implicit in the re laxation of the earlier constraint that ψ could only take on two values 0 or 1 With employment lotteries the marginal rate of transformation between leisure and consumption is equal to γ The solution to expression 2952 can be characterized by considering three possible cases Case 1 Au0 γ Case 2 Au0 γ Auγ Case 3 Auγ γ The introduction of employment lotteries will only affect individuals behavior in the second case In the first case if Au0 γ it will under all circum stances be optimal not to work ψ 0 since the marginal value of leisure in terms of consumption exceeds the marginal rate of transformation even at a zero Employment lotteries 1229 consumption level In the third case if Auγ γ it will always be optimal to work ψ 1 since the marginal value of leisure falls short of the marginal rate of transformation when evaluated at the highest feasible consumption per worker The second case implies that expression 2952 has an interior solution with respect to ψ and that employment lotteries are welfare enhancing The optimal value ψ is then given by the firstorder condition A uγψ γ An example of the second case is shown in Figure 2951 The situation here is such that the individual would choose to work in the absence of employment lotteries because the curve uγnu0 is above the curve vn when evaluated at n 1 After the introduction of employment lotteries the individual chooses the probability ψ of working and his welfare increases by ψ γ u n u 0 ψ n 1 A Utils v n ψ Figure 2951 The optimal employment lottery is given by prob ability ψ of working which increases expected welfare by ψ as compared to working fulltime n 1 1230 Equilibrium Search Matching and Lotteries 296 Lotteries for households versus lotteries for firms Prescott 2005b focuses on the role of nonconvexities at the level of individual households and production units in the study of business cycles On the house hold side he envisions indivisibilities in labor supply like those in the previous section while on the firm side he uses capacity constraints as an example In spite of these nonconvexities at the micro level where all units are assumed to be infinitesimal Prescott points out that the aggregate economy is convex when there are lotteries for households and lotteries for firms that serve to smooth the nonconvexities and that thereby deliver both a standin household and a standin firm Prescott thus recommends an aggregation theory to rationalize a standin household that is analogous to betterknown aggregation results that underlie the standin firm and the aggregate production function He emphasizes the formal similarities associated with smoothing out nonconvexities by aggregating over firms on the one hand and aggregating over households on the other Here we shall argue that the economic interpretations that attach to these two types of aggregation make the two aggregation theories very different3 Perhaps this explains why this aggregation method has been applied more to firms than to households4 Before turning to a critical comparison of the two aggregation theories we first describe a simple technology that will capture the essence of Prescotts example of nonconvexities on the firm side while leaving intact most of our analysis in section 295 3 Our argument is based on Ljungqvist and Sargents 2005 comment on Prescott 2005b 4 Sherwin Rosen often used a lottery model for the household Instead of analyzing why a particular individual chose higher education Rosen modeled a family with a continuum of members that allocates fractions of its members to distinct educational choices that involve different numbers of years of schooling See Ryoo and Rosen 2003 Lotteries for households versus lotteries for firms 1233 the equilibrium wage to w g1 γ That is firms are then not a scarce input in production and therefore earn no rents Given the equilibrium wage w γ the first inequality in 2963 states that the standin households firstorder conditions would be violated if N 0 Second we can reject an equilibrium outcome with N 1 as follows At full employment all firms are operating and aggregate output is given by Z g1Z which is also equal to per capita output since the measure of households is normalized to one Moreover according to section 295 households will trade in contingent claims prior to the outcome of the employment lottery so that each households consumption is also given by c Z g1Z The equilibrium wage at full employment is given by w g1Z ie the marginal product of labor in an individual firm that employs the same amount of labor as all other firms Given the consumption outcome and wage rate when N 1 we can ask if the standin household would indeed choose the probability of working equal to one that would be required in order for this allocation to constitute an equilibrium According to the second inequality in 2963 the answer is no because the standin household would then value a marginal increase in leisure more than the loss of wage income Thus we can conclude that parameter restrictions 2963 guarantee an interior solution with respect to the probability of working when A A In contrast when the preference shock is A 0 the standin household will inelastically supply one unit of labor since there is no disutility of working The economy will then be operating at full employment with no idle firms Hence different realizations of the preference shock A 0 A will trigger changes in unemployment and potentially changes in capacity utilization where the latter depends on the size of the given measure of firms Everything else being equal a higher Z makes it more likely that the preference shock A entails idle firms in an equilibrium The households and firms that are designated to be unemployed and idle respectively are determined by the outcome of lotteries among households and lotteries among firms Prescotts assertion that the aggregation theory for households is the analogue of the aggregation theory for firms seems to be accurate So what is the difference between these two aggregation theories An important distinction between firms and households is that firms have no independent preferences They serve only as vehicles for generating rental pay ments for employed factors and profits for their owners When a firm becomes inactive the firm itself does not care whether it continues or ceases to exist 1234 Equilibrium Search Matching and Lotteries Our example of a nonconvex production technology that generates timevarying capacity utilization illustrates this point very well The firms that do not find any workers stay idle that is just as well for those idle firms because the firms in operation earn zero rents In short whether individual firms operate or remain idle is the end of the story in the aggregation theory behind the aggregate pro duction function in 2962 But in the aggregation theory behind the standin households utility function in 2952 it is really just the beginning Individual households do have preferences and care about alternative states of the world So the aggregation theory behind the standin household has an additional as pect that is not present in the theory that aggregates over firms namely it says how consumption and leisure are smoothed across households with the help of an extensive set of contingent claim markets This market arrangement and randomization device stands at the center of the employment lottery model To us it seems that they make the aggregation theory behind the standin house hold fundamentally different than the wellknown aggregation theory for the firm side 297 Employment effects of layoff taxes The models of employment determination in this chapter can be used to address the question how do layoff taxes affect an economys employment Hopen hayn and Rogerson 1993 apply the model of employment lotteries to this very question and conclude that a layoff tax would reduce the level of employment Mortensen and Pissarides 1999b reach the opposite conclusion in a matching model We will here examine these results by scrutinizing the economic forces at work in different frameworks The purpose is both to gain further insights into the workings of our theoretical models and to learn about possible effects of layoff taxes7 Common features of many analyses of layoff taxes are as follows The pro ductivity of a job evolves according to a Markov process and a bad enough realization triggers a layoff The government imposes a tax τ on each layoff 7 The analysis is based on Ljungqvists 2002 study of layoff taxes in different models of employment determination 1238 Equilibrium Search Matching and Lotteries to population ratio equal to 06 which leads us to choose A 16 Figures 29712975 show how equilibrium outcomes vary with the layoff tax The curves labelled L pertain to the model of employment lotteries As derived in equation 2975 the reservation productivity in Figure 2971 falls when it becomes more costly to lay off workers Figure 2972 shows how the decrease in number of layoffs is outweighed by the higher tax per layoff so total layoff taxes as a fraction of GNP increase over almost the whole range Figure 2973 reveals changing job prospects where the probability of working falls with a higher layoff tax which is equivalent to falling employment in a model of employment lotteries The welfare loss associated with a layoff tax is depicted in Figure 2974 as the amount of consumption that an agent would be willing to give up in exchange for a steady state with no layoff tax and the willingness to pay is expressed as a fraction of per capita consumption at a zero layoff tax Figure 2975 reproduces Hopenhayn and Rogersons 1993 result that em ployment falls with a higher layoff tax except at the highest layoff taxes In tuitively from a private perspective a higher layoff tax is like a deterioration in the production technology the optimal change in the agents employment lotteries will therefore depend on the strength of the substitution effect versus the income effect The income effect is largely mitigated by the governments lumpsum transfer of the tax revenues back to the private economy Thus lay off taxes in models of employment lotteries have strong negative employment implications that are caused by substituting leisure for work Formally the loga rithmic preference specification gives rise to an optimal choice of the probability of working which is equal to the employment outcome as given by ψ 1 A T Π w 29712 The precise employment effect here is driven by profit flows from firms gross of layoff taxes expressed in terms of the wage rate Since these profits are to a large extent generated in order to pay for firms future layoff taxes a higher layoff tax tends to increase the accumulation of such funds with a corresponding negative effect on the optimal choice of employment Negative employment effect of layoff taxes when evaluated at τ 0 Under the assumption that p0 1 ie the initial productivity of a new job is equal to the upper support of the uniform distribution Gp on the unit interval 0 1 we will show that the derivative of equilibrium employment is strictly negative with respect to the layoff tax when evaluated at τ 0 1240 Equilibrium Search Matching and Lotteries 0 2 4 6 8 10 12 14 0 005 01 015 02 025 03 035 04 045 05 RESERVATION PRODUCTIVITY LAYOFF TAX S Ma L Mb Figure 2971 Reservation productivity for different values of the layoff tax 0 2 4 6 8 10 12 14 0 001 002 003 004 005 006 007 008 009 01 FRACTION OF GNP LAYOFF TAX S Ma L Mb Figure 2972 Total layoff taxes as a fraction of GNP for different values of the layoff tax Employment effects of layoff taxes 1241 0 2 4 6 8 10 12 14 0 01 02 03 04 05 06 07 PROBABILITY LAYOFF TAX S Ma L Mb Figure 2973 Probability of working in the model with employ ment lotteries and probability of finding a job within 10 weeks in the other models for different values of the layoff tax 0 2 4 6 8 10 12 14 002 0 002 004 006 008 01 012 014 FRACTION OF CONSUMPTION LAYOFF TAX S Ma L Mb Figure 2974 A job finders welfare loss due to the presence of a layoff tax computed as a fraction of per capita consumption at a zero layoff tax 1246 Equilibrium Search Matching and Lotteries The equilibrium conditions that firms post vacancies until the expected profits are driven down to zero become 1 φSap0 c βqθ 29729 1 φSbp0 φτ c βqθ 29730 for Nash products 29723 and 29724 respectively In the calibration we choose a matching function Mu v 001u05v05 a workers bargaining strength φ 05 and the same value of leisure as in the island model z 025 Qualitatively the results in Figures 2971 through 2974 are the same across all the models considered here The curve labeled Ma pertains to the matching model in which the workers relative share of the match surplus is constant while the curve Mb refers to the model in which the share is positively related to the layoff tax However matching model Mb does stand out Its reservation productivity plummets in response to the layoff tax in Figure 2971 and is close to zero at τ 11 A zero reservation productivity means that labor reallocation comes to a halt and the economys tax revenues fall to zero in Figure 2972 The more dramatic outcomes under Mb have to do with layoff taxes increasing workers relative share of the match surplus The equilibrium condition 29730 requiring that firms finance incurred vacancy costs with retained earnings from the matches becomes exceedingly difficult to satisfy when a higher layoff tax erodes the fraction of match surpluses going to firms Firms can break even only if the expected time to fill a vacancy is cut dramatically that is there has to be a large number of unemployed workers for each posted vacancy This equilibrium outcome is reflected in the sharply falling probability of a worker finding a job within 10 weeks in Figure 2973 As a result there are larger welfare costs in model Mb as shown by the welfare loss of a job finder in Figure 2974 The welfare loss of an unemployed agent is even larger in model Mb whereas the differences between employed and unemployed agents in the three other model specifications are negligible not shown in any figure In Figure 2975 matching model Ma looks very much like the island model with increasing employment and matching model Mb displays initially falling employment similar to the model of employment lotteries The later sharp reversal of the employment effect in the Mb model is driven by our choice of a KiyotakiWright search model of money 1249 production of goods is the agents own prior consumption After consuming one of his consumption goods an agent produces next period a new good drawn randomly from the set of all commodities We assume that agents can consume neither their own output nor their initial endowment so for consumption and production to take place there must be exchange In each period an agent meets one other agent with probability θ 0 1 he meets no other agent with probability 1 θ Two agents who meet trade if there is a mutually agreeable transaction Any transaction must be quid pro quo because private credit arrangements are ruled out by the assumptions of a random matching technology and a continuum of agents We also assume that there is a transaction cost ǫ 0 U in terms of disutility which is incurred whenever accepting a commodity in trade Thus a trader who is indifferent between holding two goods will never trade one for the other Agents choose trading strategies in order to maximize their expected dis counted utility from consumption net of transaction costs taking as given the strategies of other traders Following Kiyotaki and Wright 1993 we restrict our attention to symmetric Nash equilibria where all agents follow the same strate gies and all goods are treated the same and to steady states where strategies and aggregate variables are constant over time In a symmetric equilibrium an agent will trade only if he is offered a com modity that belongs to his set of consumption goods and then consumes it im mediately Accepting a commodity that is not ones consumption good would only give rise to a transaction cost ǫ without affecting expected future trading opportunities This statement is true because no commodities are treated as special in a symmetric equilibrium and therefore the probability of a commod ity being accepted by the next agent one meets is independent of the type of commodity one has11 It follows that x is the probability that a trader lo cated at random is willing to accept any given commodity and x2 becomes the probability that two traders consummate a barter in a situation of double coincidence of wants At the beginning of a period before the realization of the matching process the value of an agents optimization problem becomes V n c θ x2 U ǫ βV n c 11 Kiyotaki and Wright 1989 analyze commodity money in a related model with nonsym metric equilibria where some goods become media of exchange 1250 Equilibrium Search Matching and Lotteries where β 0 1 is the discount factor The superscript and subscript of V n c denote a nonmonetary equilibrium and a commodity trader respectively to set the stage for our next exploration of the role for money in this economy How will fiat money affect welfare Keep the benchmark of a barter economy in mind V n c θ x2 U ǫ 1 β 2981 2981 Monetary equilibria At the beginning of time suppose a fraction M 0 1 of all agents are each offered one unit of fiat money The money is indivisible and an agent can store at most one unit of money or one commodity at a time That is fiat money will enter into circulation only if some agents accept money and discard their endowment of goods These decisions must be based solely on agents beliefs about other traders willingness to accept money in future transactions because fiat money is by definition unbacked and intrinsically worthless To determine whether or not fiat money will initially be accepted we will therefore first have to characterize monetary equilibria12 Fiat money adds two state variables in a symmetric steady state the prob ability that a commodity trader accepts money Π 0 1 and the amount of money circulating M 0 M which is also the fraction of all agents carrying money An equilibrium pair Π M must be such that an individuals choice of probability of accepting money when being a commodity trader π coincides with the economywide Π and the amount of money M is consistent with the decisions of those agents who are initially free to replace their commodity endowment with fiat money In a monetary equilibrium agents can be divided into two types of traders An agent brings either a commodity or a unit of fiat money to the trading process that is he is either a commodity trader or a money trader At the beginning of a period the values associated with being a commodity trader and 12 If money is valued in an equilibrium the relative price of goods and money is trivially equal to 1 since both objects are indivisible and each agent can carry at most one unit of the objects Shi 1995 and Trejos and Wright 1995 endogenize the price level by relaxing the assumption that goods are indivisible 1252 Equilibrium Search Matching and Lotteries could accept money with any probability Based on these results the individuals bestresponse correspondence is as shown in Figure 2981 and there are exactly three values consistent with Π π Π 0 Π 1 and Π x 45o 1 x π Π Figure 2981 The bestresponse correspondence We can now answer our first question namely how many of the agents who are initially free to exchange their commodity endowment for fiat money will choose to do so The answer is implicit in our discussion of the bestresponse correspondence Thus we have the following three types of symmetric equilibria 1 A nonmonetary equilibrium with Π 0 and M 0 which is identical to the barter outcome in the previous section Agents expect that money will be valueless so they never accept it and this expectation is selffulfilling All agents become commodity traders associated with a value of V n c as given by equation 2981 2 A pure monetary equilibrium with Π 1 and M M Agents expect that money will be universally acceptable From our previous discussion we know that agents will then prefer to bring money rather than commodities to the trading process It is therefore a dominant strategy to accept money whenever possible that is expectation is selffulfilling Another implication is that the fraction M of agents who are initially free to exchange their commodity Concluding remarks 1255 Thus according to equation 2987 money can cannot increase welfare if x 5 x 5 Intuitively speaking when x 5 each agent is willing to consume and therefore accept at least half of all commodities so barter is not very difficult The introduction of money would here only reduce welfare by diverting real resources from the economy When x 5 barter is sufficiently difficult so that the introduction of some fiat money improves welfare The optimum quantity of money is then found by setting equation 2987 equal to zero M 1 2x2 2x That is M varies negatively with x and the optimum quantity of money increases when x shrinks and the problem of double coincidence of wants becomes more difficult In particular M converges to 5 when x goes to zero 299 Concluding remarks The frameworks of search and matching present various ways of departing from the frictionless ArrowDebreu economy where all agents meet in a complete set of markets This chapter has mainly focused on labor markets as a central application of these theories The presented models have the concept of frictions in common but there are also differences The island economy has frictional unemployment without any externalities An unemployed worker does not inflict any injury on other job seekers other than what a seller of a good imposes on his competitors The equilibrium value to search vu serves the function of any other equilibrium price of signaling to suppliers the correct social return from an additional unit supplied In contrast the matching model with its matching function is associated with externalities Workers and firms impose congestion effects when they enter as unemployed in the matching function or add another vacancy in the matching function To arrive at an efficient allocation in the economy it is necessary that the bilaterally bargained wage be exactly right In a labor market with homogeneous firms and workers efficiency prevails only if the workers bargaining strength φ is exactly equal to the elasticity of the matching function with respect to the measure of unemployment α In the case of heterogeneous jobs in the same labor market with a single matching function we established the impossibility of efficiency without government intervention 1256 Equilibrium Search Matching and Lotteries The matching model unarguably offers a richer analysis through its extra in teraction effects but it comes at the cost of the models microeconomic structure In an explicit economic environment feasible actions can be clearly envisioned for any population size even if there is only one Robinson Crusoe The island economy is an example of such a model with its microeconomic assumptions such as the time it takes to move from one island to another In contrast the matching model with its matching function imposes relationships between ag gregate outcomes It is therefore not obvious how the matching function arises when gradually increasing the population from one Robinson Crusoe to an econ omy with more agents Similarly it is an open question what determines when heterogeneous firms and labor have to be matched through a common matching function and when they have access to separate matching functions Peters 1991 and Montgomery 1991 suggest some microeconomic under pinnings to labor market frictions which are further pursued by Burdett Shi and Wright 2001 Firms post vacancies with announced wages and unem ployed workers can apply to only one firm at a time If the values of filled jobs differ across firms firms with more valued jobs will have an incentive to post higher wages to attract job applicants In an equilibrium workers will be indifferent between applying to different jobs and they are assumed to use identical mixed strategies in making their applications In this way vacancies may remain unfilled because some firms do not receive any applicants and some workers may find themselves second in line for a job and therefore remain un employed When assuming a large number of firms that take market tightness as given for each posted wage Montgomery finds that the decentralized equi librium does maximize welfare for reasons similar to Moens 1997 identical finding that was discussed earlier in this chapter Lagos 2000 derives a matching function from a model without any exoge nous frictions at all He studies a dynamic market for taxicab rides in which taxicabs seek potential passengers on a spatial grid and the fares are regu lated exogenously In each location the shorter side determines the number of matches It is shown that a matching function exists for this model but this matching function is an equilibrium object that changes with policy experi ments Lagos sounds a warning that assuming an exogenous matching function when doing policy analysis might be misleading Exercises 1257 Throughout our discussion of search and matching models we have assumed riskneutral agents Acemoglu and Shimer 1999 and Gomes Greenwood and Rebelo 2001 analyze a matching model and a search model respectively where agents are risk averse and hold precautionary savings because of imperfect insurance against unemployment As a work horse for frictional unemployment in macro labor research we continue to explore the mechanics of the matching framework in chapter 30 In chapter 31 we study how time averaging has replaced employment lotteries as a theoretical foundation of aggregate labor supply Exercises Exercise 291 An island economy Lucas and Prescott 1974 Let the island economy in this chapter have a productivity shock that takes on two possible values θL θH with 0 θL θH An islands productivity remains constant from one period to another with probability π 5 1 and its productivity changes to the other possible value with probability 1 π These symmetric transition probabilities imply a stationary distribution where half of the islands experience a given θ at any point in time Let ˆx be the economys labor supply as an average per market a If there exists a stationary equilibrium with labor movements argue that an islands labor force has two possible values x1 x2 with 0 x1 x2 b In a stationary equilibrium with labor movements construct a matrix Γ with the transition probabilities between states θ x and explain what the employment level is in different states c In a stationary equilibrium with labor movements we observe only four values of the value function vθ x where θ θL θH and x x1 x2 Argue that the value function takes on the same value for two of these four states d Show that the condition for the existence of a stationary equilibrium with labor movements is β2π 1θH θL 1 Exercises 1259 b How would an increase in µ affect an unemployed workers behavior Part II Equilibrium unemployment rate The economy is populated with a continuum of the workers just described There is an exogenous rate of new workers entering the labor market equal to µ which equals the death rate New entrants are unemployed and must draw a new wage c Find an expression for the economys unemployment rate in terms of exoge nous parameters and the endogenous reservation wage Discuss the determinants of the unemployment rate We now change the technology so that the economy fluctuates between booms B and recessions R In a boom all employed workers are paid an extra z 0 That is the income of a worker with wage w is It w z in a boom and It w in a recession Let whether the economy is in a boom or a recession define the state of the economy Assume that the state of the economy is iid and that booms and recessions have the same probabilities of 5 The state of the economy is publicly known at the beginning of a period before any decisions are made d Describe the optimal behavior of employed and unemployed workers When if ever might workers choose to quit e Let wB and wR be the reservation wages in booms and recessions respec tively Assume that wB wR Let Gt be the fraction of workers employed at wages w wB wR in period t Let Ut be the fraction of workers unem ployed in period t Derive difference equations for Gt and Ut in terms of the parameters of the model and the reservation wages F µ wB wR f Figure 291 contains a simulated time series from the solution of the model with booms and recessions Interpret the time series in terms of the model Exercise 293 Business cycles and search again The economy is either in a boom B or recession R with probability 5 The state of the economy R or B is iid through time At the beginning of each period workers know the state of the economy for that period At the beginning of each period a previously employed worker can choose to work at her last periods wage or draw a new wage If she draws a new wage the old wage is lost b is received this period and she can start working at the new Exercises 1261 Exercises 294296 European unemployment The following three exercises are based on work by Ljungqvist and Sargent 1998 Marimon and Zilibotti 1999 and Mortensen and Pissarides 1999b who calibrate versions of search and matching models to explain high European unemployment Even though the specific mechanisms differ they all attribute the rise in unemployment to generous benefits in times of more dispersed labor market outcomes for job seekers Exercise 294 Skillbiased technological change Mortensen and Pis sarides 1999b Consider a matching model in discrete time with infinitely lived and riskneutral workers who are endowed with different skill levels A worker of skill type i produces hi goods in each period that she is matched to a firm where i 1 2 N and hi1 hi Each skill type has its own but identical matching function Mui vi Auα i v1α i where ui and vi are the measures of unemployed workers and vacancies in skill market i Firms incur a vacancy cost c hi in every period that a vacancy is posted in skill market i that is the vacancy cost is proportional to the workers productivity All matches are exogenously destroyed with probability s 0 1 at the beginning of a period An unemployed worker receives unemployment compensation b Wages are de termined in Nash bargaining between matched firms and workers Let φ 0 1 denote the workers bargaining weight in the Nash product and we adopt the standard assumption that φ α a Show analytically how the unemployment rate in a skill market varies with the skill level hi b Assume an even distribution of workers across skill levels For different ben efit levels b study numerically how the aggregate steadystate unemployment rate is affected by meanpreserving spreads in the distribution of skill levels c Explain how the results would change if unemployment benefits are propor tional to a workers productivity Exercise 295 Dispersion of match values Marimon and Zilibotti 1999 We retain the matching framework of exercise 294 but assume that all workers have the same innate ability h h and any earnings differentials are purely match specific In particular we assume that the meeting of a firm and a worker 1262 Equilibrium Search Matching and Lotteries is associated with a random draw of a matchspecific productivity p from an exogenous distribution Gp If the worker and firm agree to stay together the output of the match is then p h in every period as long as the match is not exogenously destroyed as in exercise 294 We also keep the assumptions of a constant unemployment compensation b and Nash bargaining over wages a Characterize the equilibrium of the model b For different benefit levels b study numerically how the steadystate unem ployment rate is affected by meanpreserving spreads in the exogenous distribu tion Gp Exercise 296 Idiosyncratic shocks to human capital Ljungqvist and Sargent 1998 We retain the assumption of exercise 295 that a workers output is the product of his human capital h and a jobspecific component which we now denote w but we replace the matching framework with a search model In each period of unemployment a worker draws a value w from an exogenous wage offer distri bution Gw and if the worker accepts the wage w he starts working in the following period The wage w remains constant throughout the employment spell that ends either because the worker quits or the job is exogenously de stroyed with probability s at the beginning of each period Thus in a given job with wage w a workers earnings wh can only vary over time because of changes in human capital h For simplicity we assume that there are only two levels of human capital h1 and h2 where 0 h1 h2 At the beginning of each period of employment a workers human capital is unchanged from last period with probability πe and is equal to h2 with probability 1 πe Losses of human capital are only triggered by exogenous job destruction In the period of an exogenous job loss the laid off workers human capital is unchanged from last period with probability πu and is equal to h1 with probability 1 πu All unemployed workers receive unemployment compensation and the benefits are equal to a replacement ratio γ 0 1 times a workers last job earnings a Characterize the equilibrium of the model b For different replacement ratios γ study numerically how the steadystate unemployment rate is affected by changes in h1 Exercises 1265 where M is concave increasing in each argument and homogeneous of degree 1 In this setting ut is interpreted as the total number of unemployed workers and vt is the total number of vacancies Let θ vu and let qθ Mu vv be the probability that a vacant job or firm will meet a worker Similarly let θqθ Mu vu be the probability that an unemployed worker is matched with a vacant job Jobs are exogenously destroyed with probability s In order to create a vacancy a firm must pay a cost c 0 per period in which the vacancy is posted ie unfilled There is a large number of potential firms or jobs and this guarantees that the expected value of a vacant job V is zero Finally assume that when a worker and a vacant job meet they bargain according to the Nash bargaining solution with the workers share equal to ϕ Assume that yt y for all t a Show that the zeroprofit condition implies that w y r scqθ b Show that if workers and firms negotiate wages according to the Nash bar gaining solution with workers share equal to ϕ wages must also satisfy w z ϕy z θc c Describe the determination of the equilibrium level of market tightness θ d Suppose that at t 0 the economy is at its steady state At this point there is a onceandforall increase in productivity The new value of y is y y Show how the new steadystate value of θ θ compares with the previous value Argue that the economy jumps to the new value right away Explain why there are no transitional dynamics for the level of market tightness θ e Let ut be the unemployment rate at time t Assume that at time 0 the economy is at the steadystate unemployment rate corresponding to θ the old market tightness and display this rate Denote this rate as u0 Let θ0 θ Note that change in unemployment rate is equal to the difference between job destruction at t JDt and job creation at t JCt It follows that JDt 1 uts JCt θtqθtut ut1 ut JDt JCt 1268 Equilibrium Search Matching and Lotteries Exercise 2911 Financial wealth heterogeneity and unemployment donated by Rodolfo Manuelli Consider the behavior of a riskneutral worker who seeks to maximize the ex pected present discounted value of wage income Assume that the discount factor is fixed and equal to β with 0 β 1 The interest rate is also con stant and satisfies 1 r β1 In this economy jobs last forever Once the worker has accepted a job he or she never quits and the job is never destroyed Even though preferences are linear a worker needs to consume a minimum of a units of consumption per period Wages are drawn from a distribution with sup port on a b Thus any employed individual can have a feasible consumption level There is no unemployment compensation Individuals of type i are born with wealth ai i 0 1 2 where a0 0 a1 a a2 a1 β Moreover in the period that they are born all individuals are unemployed Population Nt grows at the constant rate 1 n Thus Nt1 1 nNt It follows that at the beginning of period t at least nNt1 individuals those born in that period will be unemployed Of the nNt1 individuals born at time t ϕ0 are of type 0 ϕ1 of type 1 and the rest 1 ϕ0 ϕ1 are of type 2 Assume that the mean of the offer distribution the mean offered not necessarily accepted wage is greater than aβ a Consider the situation of an unemployed worker who has a0 0 Argue that this worker will have a reservation wage w0 a Explain b Let wi be the reservation wage of an individual with wealth i Argue that w2 w1 w0 What does this say about the crosssectional relationship between financial wealth and employment probability Discuss the economic reasons underlying this result c Let the unemployment rate be the number of unemployed individuals at t Ut relative to the population at t Nt Thus ut UtNt Argue that in this economy the unemployment rate is constant d Consider a policy that redistributes wealth in the form of changes in the fraction of the population that is born with wealth ai Describe as completely as you can the effect upon the unemployment rate of changes in ϕi Explain your results Extra credit Go as far as you can describing the distribution of the random variable number of periods unemployed for an individual of type 2 Chapter 30 Matching Models Mechanics 301 Introduction We reserve the term search models to denote ones in the spirit of McCall 1970 like the searchisland model of Lucas and Prescott 1974 described in section 292 What are now widely called matching models have matching functions that are designed to represent congestion externalities concisely12 This chapter explores some of the mechanics of matching models especially those governing the responses of labor market outcomes to productivity shocks To get big responses of unemployment to movements in productivity match ing models require a high elasticity of market tightness with respect to productiv ity Shimer 2005 pointed out that for common calibrations of what was then a standard matching model the elasticity of market tightness is too low to explain business cycle fluctuations To increase that elasticity researchers reconfigured matching models in various ways by elevating the utility of leisure by mak ing wages sticky by assuming alternatingoffer wage bargaining by introducing costly acquisition of credit or by assuming fixed matching costs Ljungqvist and Sargent 2017 showed that beneath this apparent diversity there resides an essential unity all of these redesigned matching models increase responses of unemployment to movements in productivity by diminishing what Ljungqvist and Sargent called the fundamental surplus fraction a name they gave to an upper bound on the fraction of a jobs output that the invisible hand can allo cate to vacancy creation Business cycle and welfare state dynamics of an entire class of reconfigured matching models operate through this common channel Across a variety of matching models the fundamental surplus fraction is the single intermediate channel through which economic forces generating a 1 We encountered these earlier in section 293 In chapter 6 the word matching described Jovanovics 1979a analysis of a process in which workers and firms gradually learn about match quality In macro labor the term matching models has come instead to mean models that postulate matching functions 2 Petrongolo and Pissarides 2001 call the matching function a black box because it de scribes outcomes of labor market frictions without explicitly modeling them 1269 1270 Matching Models Mechanics high elasticity of market tightness with respect to productivity must operate Differences in the fundamental surplus explain why unemployment responds sensitively to movements in productivity in some matching models but not in others The role of the fundamental surplus in generating that response sensi tivity transcends diverse matching models having very different outcomes along other dimensions that include the elasticity of wages with respect to productivity and whether or not outside values affect bargaining outcomes For any model with a matching function to arrive at the fundamental sur plus take the output of a job then deduct the sum of the value of leisure the annuitized values of layoff costs and training costs and a workers ability to exploit a firms cost of delay under alternatingoffer wage bargaining and any other items that must be set aside The fundamental surplus is an upper bound on what the invisible hand could allocate to vacancy creation If that funda mental surplus constitutes a small fraction of a jobs output it means that a given change in productivity translates into a much larger percentage change in the fundamental surplus Because such large movements in the amount of resources that could potentially be used for vacancy creation cannot be offset by the invisible hand significant variations in market tightness ensue causing large movements in unemployment In contrast to search models matching models with inputs of unemployed workers and vacancies in matching functions are typically plagued by external ities What types of workers perhaps differentiated by education skill age and what types of jobs perhaps differentiated by required skills and strengths does the analyst make sit within the same matching functions Broadly speak ing matching analyses can be divided into those that focus on congestion exter nalities and those that seek to eliminate such externalities in order to facilitate analytical tractability For example we describe a way of proliferating matching functions and assigning workers to them that can be interpreted as expressing directed search and that succeeds in arresting congestion externalities and im proving analytic tractability along some dimensions To illustrate the two types of matching analyses that either emphasize or eliminate externalities we turn to aging as one key source of heterogeneity Cheron Hairault and Langot 2013 and Menzio Telyukova and Visschers 2016 study overlapping generations models in which unemployed workers ei ther enter a single matching function or are assigned to typespecific matching functions In a version of the model of Cheron Hairault and Langot we show Fundamental surplus 1271 that it is optimal to subsidize the continuing employment of old workers and to tax that of young workers in order properly to rearrange the age composition of unemployed workers sitting inside a single matching function The agespecific matching functions of Menzio Telyukova and Visschers make those externali ties vanish and unleash market forces that make job finding rates decrease with age Equilibrium computation turns out to be block recursive because agents value and policy functions depend on realizations of exogenous shocks but not on the distribution of agents across employment and unemployment states This makes it easy to compute outofsteadystate dynamics as well as equilibria with aggregate shocks 302 Fundamental surplus With exogenous separation a comparative steady state analysis decomposes the elasticity of market tightness with respect to productivity into two multiplicative factors both of which are bounded from below by unity In a matching model of variety j let ηj θy be the elasticity of market tightness θ with respect to productivity y ηj θy d θ d y y θ Υj y y xj 3021 The first factor Υj has an upper bound coming from a consensus about values of the elasticity of matching with respect to unemployment The second factor yy xj is the inverse of what we define to be the fundamental surplus fraction The fundamental surplus y xj equals a quantity that deducts from productivity y a value xj that the invisible hand cannot allocate to vacancy creation a quantity whose economic interpretation differs across models Unlike Υj the fraction yy xj has no widely agreed upon upper bound To get a high elasticity of market tightness requires that yy xj must be large ie that what we call the fundamental surplus fraction must be small3 Across reconfigured matching models many details differ but what ultimately matters is the fundamental surplus 3 We call y x the fundamental surplus and yx y the fundamental surplus fraction Fundamental surplus 1273 where the second equality is obtained after using equation 3022 to rearrange the numerator while in the denominator we invoke the constant elasticity of matching with respect to unemployment the third equality follows from multi plying and dividing by θ qθ The elasticity of market tightness with respect to productivity is then given by ηθy r s φ θ qθ αr s φ θ qθ y y z ΥNash y y z 3024 This multiplicative decomposition of the elasticity of market tightness is central to our analysis Similar decompositions prevail in all of the reconfigured match ing models to be described below and those in Ljungqvist and Sargent 2017 The first factor ΥNash in expression 3024 has counterparts in other setups A consensus about reasonable parameter values bounds its contribution to the elasticity of market tightness Hence the magnitude of the elasticity of market tightness depends mostly on the second factor in expression 3024 ie the inverse of what we define to be the fundamental surplus fraction In the standard matching model with Nash bargaining the fundamental surplus is simply what remains after deducting the workers value of leisure from productivity x z in expression 3021 To induce them to work workers have to receive at least the value of leisure so the invisible hand cannot allocate that value to vacancy creation 3023 Shimers critique Shimer 2005 observed that the average job finding rate θ qθ is large relative to the observed value of the sum of the net interest rate and the separation rate r s When combined with reasonable parameter values for a workers bargaining power φ and the elasticity of matching with respect to unemployment α this implies that the first factor ΥNash in expression 3024 is close to its lower bound of unity More generally the first factor in 3024 is bounded from above by 1α Because reasonable values of the elasticity α imply an upper bound on the first factor the second factor yy z in expression 3024 becomes critical for generating movements in market tightness For values of leisure within a commonly assumed range well below productivity the second factor is not large enough to generate the high volatility of market tightness associated with observed business cycles This is Shimers critique Fundamental surplus 1275 and key to our new perspective iii the parts of fundamental surpluses from future employment matches that are not allocated to match surpluses Ψextra u θqθ r s Ψmsurplus u 3027 which can be deduced from equation 3025 after replacing Ψmsurplus u with expression 3026 We can use decomposition 3025 of a workers outside value U to shed light on the activities of the invisible hand that make the elasticity of mar ket tightness with respect to productivity be low for common calibrations of matching models Those parameter settings entail a value of leisure z well be low productivity and a significant share φ of match surpluses being awarded to workers which together with a high job finding probability θqθ imply that the sum Ψmsurplus u Ψextra u in equation 3025 forms a substantial part of a workers outside value Furthermore Ψextra u is the much larger term in that sum which follows from expression 3027 and the assumption that θqθ is large relative to rs That big term Ψextra u makes it easy for the invisible hand to realign a workers outside value in a way that leaves the match surplus almost unchanged when productivity changes Offsetting changes in Ψextra u can absorb the impact of productivity shocks so that resources devoted to vacancy creation can remain almost unchanged which in turn explains why unemployment does not respond sensitively to productivity But in Hagedorn and Manovskiis 2008 calibration with a high value of leisure the fundamentalsurplus components of a workers outside value are so small that there is little room for the invisible hand to realign things as we have described making the equilibrium amount of resources allocated to vacancy cre ation respond sensitively to variations in productivity That results in a high elasticity of market tightness with respect to productivity Put differently since the fundamental surplus is a part of productivity it follows that a given change in productivity translates into a greater percentage change in the fundamental surplus by a factor of yyz ie the inverse of the fundamental surplus frac tion Thus the small fundamental surplus fraction in calibrations like Hagedorn and Manovskiis having high values of leisure imply large percentage changes in the fundamental surplus Such large changes in the amount of resources that could potentially be used for vacancy creation cannot be offset by the invisible hand and hence variations in productivity lead to large variations in vacancy 1276 Matching Models Mechanics creation resulting in a high elasticity of market tightness with respect to pro ductivity5 3025 Relationship to match surplus How does the fundamental surplus relate to the match surplus The fundamen tal surplus is an upper bound on resources that the invisible hand can allocate to vacancy creation Its magnitude as a fraction of output is the prime determinant of the elasticity of market tightness with respect to productivity6 In contrast although it is directly connected to resources that are devoted to vacancy cre ation match surplus that is small relative to output has no direct bearing on the elasticity of market tightness Recall that in the standard matching model the zeroprofit condition for vacancy creation implies that the expected present value of a firms share of match surpluses equals the average cost of filling a vacancy Since common calibrations award firms a significant share of match surpluses and since vacancy cost expenditures are calibrated to be relatively small it fol lows that equilibrium match surpluses must form small parts of output across various matching models regardless of the elasticity of market tightness in any particular model Fundamental surpluses yield match surpluses which in turn include firms profits A small fundamental surplus fraction necessarily implies small match surpluses and small firms profits But small match surpluses and small firms profits dont necessarily imply small fundamental surpluses Therefore the size 5 It is instructive to consider a single perturbation φ 0 to common calibrations of the standard matching model for which a workers outside value in expression 3025 solely equals the capitalized value of leisure and the worker receives no part of fundamental sur pluses Ψmsurplus u Ψextra u 0 What explains that the elasticity of market tightness with respect to productivity remains low for such perturbed parameter settings in which large fun damental surpluses end up affecting only firms profits that in equilibrium are all used for vacancy creation The answer lies precisely in the outcome that firms profits would then be truly large therefore even though variations in productivity then affect firms profits directly the percentage wise impact of productivity shocks on such huge profits is negligible so mar ket tightness and unemployment hardly changes This shows that decomposition 3025 of a workers outside value can only go so far to shed light on the sensitivity of market tight ness to changes in productivity because what ultimately matters is evidently the size of the fundamental surplus fraction in expression 3024 6 We express the fundamental surplus as a flow value while the match surplus is typically a capitalized value Fundamental surplus 1279 3027 Sticky wages The standard assumption of Nash bargaining in matching models is one way to determine a wage but not the only one Matching frictions create a range of wages that a firm and worker both prefer to breaking a match Hall noted that a constant wage can be consistent with no private inefficiencies in contractual arrangements within a matching model That motivated Hall 2005 to assume sticky wages in the form of a constant wage in his main analysis as a way of responding to the Shimer critique Hall posited a wage norm ˆw inside the Nash bargaining set that must be paid to workers Here we show that an appropriately defined fundamental surplus fraction determines how does such a constant wage affects the elasticity of market tightness with respect to productivity Given a constant wage w ˆw an equilibrium is characterized by the zero profit condition for vacancy creation in expression 2936 of the standard matching model ˆw y r s qθ c 30215 There exists an equilibrium for any constant wage ˆw z y r sc The lower bound is a workers utility of leisure and the upper bound is determined by the zeroprofit condition for vacancy creation evaluated at the point where the probability of a firm filling a vacancy is at its maximum value of qθ 1 After implicitly differentiating 30215 we can compute the elasticity of market tightness as ηθy 1 α y y ˆw Υsticky y y ˆw 30216 This equation resembles the earlier one for ηθy in 3024 Not surprisingly if the constant wage equals the value of leisure ˆw z then the elasticity 30216 is equal to that earlier elasticity of market tightness in the standard matching model with Nash bargaining when the worker has a zero bargaining weight φ 0 With such lopsided bargaining power the equilibrium wage would indeed be the constant value z of leisure This outcome reminds us that the first factor in expression 3024 can play only a limited role in magnifying the elasticity ηθy because it is bounded from above by the inverse of the elasticity of matching with respect to unemployment α In 30216 the upper bound is attained So again it is the second factor the inverse of the fundamental surplus fraction that tells whether the elasticity of market tightness is high or low The pertinent definition of the fundamental 1280 Matching Models Mechanics surplus is now the difference between productivity and the stipulated constant wage In Halls 2005 model all of the fundamental surplus goes to vacancy cre ation as also occurs in the standard matching model with Nash bargaining when the workers bargaining weight is zero A given percentage change in productivity is multiplied by a factor yy ˆw to become a larger percentage change in the fundamental surplus Because all of the fundamental surplus now goes to vacancy creation there is a correspondingly magnified impact on un employment Numerical simulations of economies with aggregate productivity shocks in section 3031 reaffirm this interpretation 3028 Alternatingoffer wage bargaining Hall and Milgrom 2008 proposed yet another response to the Shimer critique Instead of Nash bargaining a firm and a worker take turns making wage offers The threat is not to break up and receive outside values but instead to continue to bargain because that choice has a strictly higher payoff than accepting the outside option After each unsuccessful bargaining round the firm incurs a cost of delay γ 0 while the worker enjoys the value of leisure z There is also a probability δ that the job opportunity is exogenously destroyed between bargaining rounds sending the worker to the unemployment pool It is optimal for both bargaining parties to make barely acceptable offers The firm always offers wf and the worker always offers ww Consequently in an equilibrium the first wage offer is accepted Hall and Milgrom assume that firms make the first wage offer Hall and Milgrom 2008 p 1673 chose to emphasize that the limited in fluence of unemployment the outside value of workers on the wage results in large fluctuations in unemployment under plausible movements in productiv ity It is more accurate to emphasize that the key force is actually that an appropriately defined fundamental surplus fraction has to be calibrated to be small Without a small fundamental surplus fraction it matters little that the outside value has been prevented from influencing bargaining To illustrate this we compute the elasticity of market tightness with respect to productivity and look under the hood Business cycle simulations 1283 303 Business cycle simulations To illustrate that a small fundamental surplus fraction is essential for generating ample unemployment volatility over the business cycle in matching models we use Halls 2005 specification with discrete time and a random productivity process The monthly discount factor β corresponds to a 5percent annual rate and the value of leisure is z 040 The elasticity of matching with respect to unemployment is α 0235 and the exogenous monthly separation rate is s 0034 Aggregate productivity takes on five values ys uniformly spaced around a mean of one on the interval 09935 100565 and is governed by a monthly transition probability matrix Π with probabilities that are zero except as follows π12 π45 21 ρ π23 π34 31 ρ with the upper triangle of the transition matrix symmetrical to the lower triangle and the diagonal elements equal to one minus the sums of the nondiagonal elements The resulting serial correlation of y is ρ which is parameterized to be ρ 09899 To facilitate the sensitivity analysis following Ljungqvist and Sargent 2017 we alter Halls model period from one month to one day 3031 Halls sticky wage Following Hall 2005 we posit a fixed wage ˆw 09657 which equals the flex ible wage that would prevail at the median productivity level under standard Nash bargaining with equal bargaining weights φ 05 Figure 3031 repro duces Halls figures 2 and 4 for those two models The solid line and the upper dotted line depict unemployment rates at different productivities for the sticky wage model and the standard Nashbargaining model respectively9 Unemploy ment is almost invariant to productivity under Nash bargaining but responds sensitively under the sticky wage These outcomes are explained by differences in jobfinding rates as shown by the dashed line and the lower dotted line for the stickywage model and the standard Nashbargaining model respectively 9 Unemployment is a state variable that is not just a function of the current productivity as are all of the other variables but depends on the history of the economy But high persistence of productivity and the high jobfinding rates make the unemployment rate that is observed at a given productivity level be well approximated by expression 2932 evaluated at the market tightness θ prevailing at that productivity see Hall 2005 p 59 1284 Matching Models Mechanics 0994 0996 0998 1 1002 1004 1006 1 2 3 4 5 6 7 8 9 Productivity Unemployment rate Job finding rate Figure 3031 Stickywage model Unemployment rates and daily jobfinding rates at different productivities given a fixed wage ˆw 09657 where the dotted lines with almost no slopes are counterparts from a standard Nashbargaining model expressed at our daily frequency10 Under the sticky wage high productivities cause firms to post many vacancies making it easy for unemployed workers to find jobs while the opposite is true when productivity is low We conduct a sensitivity analysis of the choice of the fixed wage The solid line in Figure 3032 shows how the average unemployment rate varies with the fixed wage ˆw A small set of wages spans outcomes ranging from very low to very high average unemployment rates Small variations in a fixed wage close to pro ductivity generate large changes in the fundamental surplus fraction y ˆwy Free entry of firms makes that map directly into the amount of resources devoted to vacancy creation The dashed line in Figure 3032 delineates implications for the volatility of unemployment The standard deviation of unemployment is nearly zero at the left end of the graph where the jobfinding probability is almost one for all productivity levels Unemployment volatility then increases for higher constant wages until outside of the graph at the right end vacancy creation becomes so unprofitable that average unemployment converges to its maximum of 100 percent causing there to be no more fluctuations 10 Our daily jobfinding rates are roughly 130 of the monthly rates in Hall 2005 figures 2 and 4 confirming our conversion from a monthly to a daily frequency Business cycle simulations 1285 095 0955 096 0965 097 0975 098 0 2 4 6 8 10 Fixed wage meanu stdu Halls Fixed Wage Figure 3032 Stickywage model Average unemployment rate and standard deviation of unemployment for different postulated values of the fixed wage At Halls fixed wage ˆw 09657 Figure 3032 shows a standard deviation of unemployment equal to 180 percentage points which is close to the target of 154 to which Hall 2005 calibrated his model 3032 Hagedorn and Manovskiis high value of leisure It turns out that by elevating the value of leisure the standard Nashbargaining model can attain the same volatility of unemployment as does the sticky wage model of the previous subsection To illustrate this we use Halls 2005 param eterized environment but now simply assume standard Nash wage bargaining in order to study Hagedorn and Manovskiis 2008 analysis of the consequences of positing a high value z 0960 of leisure and a low bargaining power of workers φ 00135 These parameter values imply a high standard deviation of 14 percentage points for unemployment Figure 3033 which depicts outcomes for different constellations of z 04 99 and φ 0001 05 sheds light on the sensitivity of outcomes to the choice of parameters To construct the figure for each pair z φ we adjusted the efficiency parameter A of the matching function to make the average unemployment rate stay at 55 percent Because 1286 Matching Models Mechanics it implies a a small fundamental surplus fraction a high value of leisure is es sential for obtaining large variations in market tightness and a high volatility of unemployment To match the elasticity of wages with respect to productivity Hagedorn and Manovskii 2008 require a low bargaining power for workers Given the above parameterization with z φ 0960 00135 we obtain a wage elasticity of 044 which is approximately the value that Hagedorn and Manovskii had targeted To conduct a sensitivity analysis to variations in z and φ Figure 3034 employs the same computational approach underlying Figure 3033 The figure confirms that a low φ is required to obtain a low wage elasticity11 Taken together Figures 3033 and 3034 seem to settle a difference of opin ions in favor of Hagedorn and Manovskii 2008 p 1696 who argued that the volatility of labor market tightness is almost independent of φ and is deter mined only by the level of z Rogerson and Shimer 2011 p 660 apparently disagreed when they instead emphasized that wages are rigid under the cali bration of Hagedorn and Manovskii 2008 although it is worth noting that the authors do not interpret their paper as one with wage rigidities They cali brate a small value for the workers bargaining power φ This significantly amplifies productivity shocks But Figures 3033 and 3034 indicate that the low wage elasticity of Hagedorn and Manovskii 2008 is incidental to and neither necessary nor sufficient to obtain a high volatility of unemployment We suggest that instead of stressing the importance of a rigid wage as Rogerson and Shimer did what should be concluded is the general principle that the fundamental surplus fraction must be small in order to amplify business cycle responses to productivity changes 11 Note that the axes in Figure 3034 are rotated relative to Figure 3033 for easy viewing of the relationship Business cycle simulations 1287 0 02 04 04 06 08 1 0 05 1 15 2 25 Flow value of leisure Bargaining power stdu in Hagedorn Manovskii Figure 3033 Nashbargaining model Standard deviation of unemployment in percentage points for different constellations of the value of leisure z and the bargaining power of workers φ 0 02 04 04 06 08 1 0 02 04 06 08 1 Flow value of leisure Bargaining power Elasticity Hagedorn Manovskii Figure 3034 Nashbargaining model Wage elasticity with re spect to productivity for different constellations of the value of leisure z and the bargaining power of workers φ Note that the axes are rotated relative to Figure 3033 1288 Matching Models Mechanics 3033 Hall and Milgroms alternatingoffer bargaining Hall and Milgroms 2008 model of alternatingoffer wage bargaining is another way to increase unemployment volatility Except for the wage formation pro cess their environment is Halls 2005 But Hall and Milgrom parameterize it differently One difference between Hall and Milgroms parameterization and Halls 2005 plays an especially important role in setting the fundamental sur plus Hall and Milgrom raised the value of leisure to z 071 from Halls value of z 040 Section 3028 taught us that the values of leisure and of the firms cost of delay in bargaining γ are likely to be critical determinants of the elasticity of market tightness with respect to productivity and hence of the volatility of unemployment But that is not what Hall and Milgrom 2008 chose to emphasize Instead they stressed how much the outside value of unemployment is suppressed in alternatingoffer wage bargaining since disagreement no longer leads to unem ployment but instead to another round of bargaining So from Hall and Mil groms perspective a key parameter is the exogenous rate δ at which parties break up between bargaining rounds Figure 3035 shows how different con stellations of γ δ affect the standard deviation of unemployment For each pair γ δ we adjust the efficiency parameter A of the matching function to make the average unemployment rate stay at 55 percent Because Hall and Milgrom 2008 assumed that productivity shocks are not the sole source of unemployment fluctuations leading them to lower their target standard devia tion of unemployment to 068 percentage points a target attained with their parameterization γ δ 027 00055 and reproduced in Figure 3035 Figure 3035 supports our earlier finding that the cost of delay γ together with the value of leisure z are the keys to generating higher volatility of unem ployment Without a cost of delay sufficiently high to reduce the fundamental surplus fraction the exogenous separation rate between bargaining rounds mat ters little12 Although Hall and Milgrom 2008 p 1670 notice that their sum of z and γ is not very different from the value of z by itself in Hagedorn and Manovskiis calibration as studied in our section 3032 they demphasized 12 To be specific our formula 30226 for the steadystate comparative statics is an ap proximation of the elasticity of market tightness at the rear end of Figure 3035 where the exogenous rate δ at which parties break up between bargaining rounds is equal to Hall and Milgroms 2008 assumed job destruction rate of 00014 per day Business cycle simulations 1289 024 026 028 03 032 2 3 4 5 x 10 3 05 1 15 2 25 Sep while bargaining Cost of delay stdu in Hall and Milgrom Figure 3035 Alternatingoffer bargaining model Standard de viation of unemployment in percentage points for different constel lations of firms cost of delay γ in bargaining and the exogenous separation rate δ while bargaining this similarity and instead emphasized differences in mechanisms across Hage dorn and Manovskiis model and theirs Focusing on the fundamental surplus tells us that it is their similarity that should be stressed Hall and Milgroms and Hagedorn and Manovskiis models are united in requiring a small fundamen tal surplus fraction to generate high unemployment volatility over the business cycle 3034 Matching and bargaining protocols in a DSGE model Christiano Eichenbaum and Trabandt 2016 compare consequences of assum ing alternativeoffer bargaining AOB and Nash bargaining in a dynamic stochas tic general equilibrium DSGE model with a matching function They find that if they adjust structural parameters across the two models to fit the data models parameters estimated under the two alternative assumptions are able to account for the data equally well That includes comparable performance in generating observed unemployment volatility The solid lines in Figure 3036 de pict responses of unemployment to a neutral technology shock that are virtually identical across the two models But beneath those nearly identical responses there resides a substantial difference in estimates of a key parameter under the 1290 Matching Models Mechanics 0 2 4 6 8 10 12 14 02 015 01 005 0 005 01 015 Time in quarters Percentage points Figure 3036 Impulse response of unemployment to a neutral technology shock in the DSGE analyses The solid lines refer to estimates of AOB and Nash bargaining models respectively The dashed lines refer to perturbed models where parameter values for the replacement ratio and in the AOB model for a firms cost to make a counteroffer are cut in half The two solid dashed lines are almost indistinguishable except for the Nash bargaining model being slightly below the AOB model two assumptions namely the replacement rate from unemployment insurance a parameter that corresponds to our value of leisure z They estimate a value of 037 under the AOB model versus 088 with the Nash bargaining model Christiano et al 2016 pp 15511552 remark that their high estimate of the value of leisure in the Nash bargaining model is reminiscent of Hage dorn and Manovskiis 2008 argument that a high replacement ratio has the potential to boost the volatility of unemployment13 To elaborate Christiano et al demonstrate that if they restrict the replacement rate in the Nash bar gaining model to be the same as that of the AOB model and then recalculate the impulse response functions then there occurs a dramatic deterioration in the performance of the Nash bargaining model Thus the dashed line in Fig ure 3036 show how unemployment becomes much less responsive to a neutral technology shock under that perturbation in the replacement rate 13 See section 3032 above Overlapping generations in one matching function 1291 Christiano et al 2016 p 1547 proceed to interpret their low estimate of the value of leisure in the AOB model as meaning that the replacement ratio does not play a critical role in the AOB models ability to account for the data Their account conceals that the fundamental surplus is really at work once again Christiano et al generously conducted for us a perturbation of the AOB model that can be regarded as the analogue to their perturbation of the Nash bargaining model namely a cutting in half of both the replacement rate 037 and a firms cost of delay in bargaining where the latter in their model is a firms cost of making a counteroffer calibrated to 06 of a firms daily revenue per worker14 As sections 3028 and 3033 lead us to expect this perturbation of the AOB model also brings a dramatic deterioration in performance one as bad as that of the perturbed Nash bargaining model the dashed lines depicting a dampened impulse response of unemployment to a neutral technology shock in Figure 3036 are almost the same across the two perturbed models We conclude from this exercise that contrary to what Christiano et al say the replacement ratio is critical in the AOB model too and that what is needed to make the fundamental surplus fraction small in that model is a combination of very high values of the replacement rate and a firms cost of delay in bargaining 304 Overlapping generations in one matching function To emphasize the important role of congestion externalities it is useful to study a matching model in which workers are heterogeneous along one or more di mensions for example age Cheron Hairault and Langot 2013 and Menzio Telyukova and Visschers 2016 study overlapping generations models under alternative arrangements in which unemployed workers either enter a single matching function or are assigned to typespecific matching functions In this section we adopt a framework of Cheron Hairault and Langot They assume a single matching function and an exogenous retirement age T 1 Each period a retiring generation is replaced by a new generation of the same size normalized to unity All newborn workers enter the labor market being unemployed 14 Christiano et al 2016 assume that it takes one day for a wage offer to be extended with a firm and a worker alternating in making an offer 1302 Matching Models Mechanics at a sufficiently young age the subsidy becomes negative and turns into a tax on employment of young workers when ˆκi1 1 Note that except for one caveat ˆκi1 as defined in 30425 is the expected nextperiod surplus for an employed worker of age i relative to a weighted average across employed workers of all ages where the weights are agespecific unemployment ui as a fraction of total unemployment u The caveat is that these weights sum to less than one because unemployment of the youngest generation u1 1 is included in u while there are no employed workers in that generation However this caveat just serves to emphasize that there is a critical cutoff age i at which ˆκj1 1 for all j i since the expected nextperiod surplus of such a young employed worker which tends to be greater than an economywide weighted average is compared to something less than a weighted average of expected nextperiod surpluses of all employed workers The justification for the subsidy δT c θ to employed workers in the last period before retirement is that if one of them joins the ranks of the unemployed the economy incurs a vacancy cost per unemployed equal to c θ with no poten tial gain in terms of future matches So long as this cost exceeds a workers value of leisure when unemployed net of the output in the present job c θ z ǫ it is socially optimal for the worker to remain employed the subsidy accomplishes this by lowering the reservation productivity to RT z c θ Similarly em ployed workers further from retirement are also subsidized but by less in order to ameliorate congestion in the matching function Interestingly the argument is reversed for sufficiently young workers whose employment should instead be taxed because otherwise they would fail to internalize the positive externality that they exert in the matching function Directed search agespecific matching functions 1303 305 Directed search agespecific matching functions Following Menzio Telyukova and Visschers 2016 we now assume agespecific matching functions Within a particular submarket and matching function firms post vacancies for a particular age of unemployed workers only workers of that age are allowed to sit in that matching function Such a directed search setting leads to a block recursive structure in which agents value and policy functions and measures of market tightness are independent of the distribution of workers across states of employment and unemployment Two important fea tures are 1 computation of equilibria simplifies 2 the congestion externali ties of section 304 vanish because there is no longer a mixture of heterogeneous workers sitting inside a matching function To facilitate a transparent presentation we shut down differences in pro ductivity each employed worker produces y and let age be the only source of heterogeneity Matches break up exogenously with probability s 3051 Value functions and market tightness A key difference from the section 304 setting is that there are now agespecific measures of market tightness θi and values of vacancy creation Vi Correspond ing to value functions 30423045 we have Ji y wi β1 sJi1 3051 Vi c βqθiJi1 3052 Ei wi β 1 sEi1 sUi1 3053 Ui z βθiqθiEi1 β 1 θiqθi Ui1 z βUi1 βθiqθi Ei1 Ui1 3054 where we have already imposed the zeroprofit condition in vacancy creation on the right sides of these equations After also imposing Vi 0 on the left side of 3052 a zeroprofit condition becomes qθi c βJi1 3055 The agespecific match surplus is given by Si Ji Ei Ui 3056 Directed search agespecific matching functions 1307 matching model with infinitelylived workers as given by equation 29314 The incentives of those firms and young workers to engage in job creation are practically identical with those in a standard matching model with infinitely lived workers because the matches into which they enter will almost certainly break before workers retire and at most of those future separations the worker will still be much younger than retirement age T 1 Therefore in the setting of this section young workers experiences are similar to those of infinitelylived workers Because workers have finite lives equilibrium values of θi have to be less than market tightness in the standard matching model with infinitelylived workers Subject to the risk of workers retiring before an exogenous job destruction shock the invisible hand compensates firms that create vacancies with a higher prob ability of filling vacancies ie a lower equilibrium value of market tightness By the last equality in 30515 it follows that Si Si1 is strictly positive so match surpluses are decreasing in age Si Si1 By substituting 3057 into 3055 qθi cβ1 φSi1 we can confirm that market tightness also declines in age and hence a workers job finding rate decreases with age The declining job finding rate becomes especially pronounced towards the end of a workers labor market career more so with a low exogenous separation rate s ie when jobs are expected to last long in the absence of retirement As an illustration Figure 3051 reports a numerical example where the job finding index is a workers job finding probability relative to that of the youngest worker with the highest job finding probability19 The axis labeled mean job duration calculated as 1s identifies different economies defined by their exogenous job destruction rate s For each such economy the agespecific job finding index is shown for older workers defined by their times to retirement 19 As in common parameterizations of matching models the elasticity α of a CobbDouglas matching function with respect to unemployment and a workers bargaining power φ are set equal to each other and near the middle of the unit interval α φ 05 and the replacement ratio in unemployment is around half of worker productivity zy 06 The annual discount rate is 4 percent and a workers labor market career lasts 45 years The model period is set to be one day For different economies indexed by their exogenous job destruction rates s the vacancy cost c and a multiplicative efficiency parameter A in the matching function are chosen to yield an unemployment rate of 5 percent in a corresponding matching model with infinitelylived workers When there is no calibration target for vacancies fixing either c or A amounts to a normalization with the other parameter then being used to target an unemployment rate 1308 Matching Models Mechanics 0 5 10 15 0 1 2 3 0 02 04 06 08 1 Years before retirement Mean job duration Job finding index Figure 3051 Job finding index of older workers defined by their time left until retirement in different economies defined by their mean job duration The index is a workers job finding probability relative to that of the youngest worker in respective economy 3053 Block recursive equilibrium computation The value and policy functions and measures of market tightness described in the previous subsections are independent of the distributions of workers across age and employment and unemployment states It is analytically convenient to be able to derive those quantities before computing distributions of workers in a steady state or along transition paths or in response to aggregate shocks For an example of business cycle analysis in a matching model with directed search see Menzio and Shi 2011 who bring out the benefits of such block recursive structures They show how agents value and policy functions depend on the aggregate state of the economy through realization of aggregate shocks only and not through endogenous distributions of workers across employment and unemployment states In our present framework it is easy to compute a steady state Under the assumption of a stationary population in which the number of new labor market entrants of age 1 equals the number of retiring workers of age T 1 the age specific unemployment rates in a steady state are computed as follows At age 1 all new entrants are unemployed since it takes at least one period to be matched with a vacancy u1 1 The unemployment rates for subsequent ages 1312 Matching Models Mechanics 306 Concluding remarks Ljungqvist and Sargent 2017 showed that in a variety of matching models pro ductivity changes affect both business cycle and welfare state dynamics through a single intermediate channel called the fundamental surplus Thus in studying welfare state dynamics Mortensen and Pissarides 1999b and Ljungqvist and Sargent 2007 attribute the outbreak of European unemployment after the late 1970s to changes in the economic environment in conjunction with the generous unemployment benefits offered by European government ie a higher z in the formulation of matching models in this chapter In a matching model with di rected search by workers with permanently different productivities Mortensen and Pissarides 1999b model skillbiased technology shocks in terms of a mean preserving spread of the distribution of productivities There is a convex inverse relationship between the unemployment rate and worker productivity across sub markets so moving workers to a lower range of productivities causes a larger increase in unemployment than a decrease that would caused by moving workers to higher productivities Because the relationship becomes more convex for a higher value of z unemployment increases more in high z Europe than in low z America In a matching model with skill accumulation and unemployment benefits that are paid as a fixed replacement rate of a workers past earnings Ljungqvist and Sargent 2007 study how European unemployment erupts in turbulent times modeled as an increased risk of skill loss at layoff events both under random search in a single matching function but more so under directed search They conclude that the cost of posting vacancies is the lynchpin or to use a less kind metaphor the tail that wags the dog of matching models Then how is it that vacancy costs that are commonly calibrated to be small relative to aggregate output turn out to wag the dog in some matching models but not in others The answer is that it all depends on whether the fundamen tal surplus fraction is small Here it helps to remember that the fundamental surplus fraction serves as an upper bound on the fraction of a jobs output that the invisible hand can allocate to vacancy creation Concluding remarks 1313 As mentioned in the introduction of this chapter as well as in the concluding remarks of chapter 29 an important difference between matching models and search models is whether there are congestion externalities It is helpful to recall how Lucas and Prescott 1974 and Pissarides 1992 summarized these distinct frameworks In their searchisland economy Lucas and Prescott remarked that the injury a searching worker imposes on his fellows is of exactly the same type as the injury a seller of any good imposes on his fellow sellers the equilibrium expected return from job search serves the function of any other equilibrium price of signalling to suppliers the correct social return from an additional unit supplied Things are very different in Pissaridess 1992 matching model with twoperiodlived overlapping generations in which workers who remain unem ployed in the first period of life lose skills Because all unemployed workers congest the same matching function a temporary shock to employment can persist for a long time outlasting the maximum duration of any workers un employment The key mechanism is a thin market externality that reduces the supply of jobs when the duration of unemployment increases persistence and multiple equilibria are possible even with constant returns production and matching technologies Directed search in matching models disarms congestion externalities because heterogeneous workers andor heterogeneous jobs no longer sit in the same matching function Directed search simultaneously simplifies equilibrium com putation and eradicates congestion externalities21 In some matching models with directed search all types of heterogeneous workers prefer to sit in their assigned matching functions but there are other models populated by some workers who would like to sneak into another matching function For exam ple an older worker in section 305 would prefer to sit in a matching function with younger workers and thereby enjoy a higher job finding probability A firm encountering such a deviant job applicant would be disappointed with that workers type but nevertheless to recover some of its sunk vacancyposting cost would engage in Nash bargaining with the older worker and form a match be cause the match surplus is positive 21 Scope for beneficial government interventions remain in matching models with directed search whenever the elasticity of a matching function with respect to unemployment does not equal a workers Nash bargaining power φ ie whenever the Hosios efficiency condition is violated Chapter 31 Foundations of Aggregate Labor Supply 311 Introduction The section 295 employment lotteries model for years served as the foundation of the high aggregate labor supply elasticity that generates big employment fluc tuations in real business cycle models In the original version of his Nobel prize lecture Prescott 2005a highlighted the central role of employment lotteries for real business cycle models when he asserted that Rogersons aggregation result is every bit as important as the one giving rise to the aggregate produc tion function But Prescotts enthusiasm for employment lotteries has not been shared universally especially by researchers who have studied labor market ex periences of individual workers For example Browning Hansen and Heckman 1999 expressed doubts about the employment lotteries model when they as serted that the employment allocation mechanism strains credibility and is at odds with the micro evidence on individual employment histories This chap ter takes such criticisms of the employment lotteries to heart by investigating how the aggregate labor supply elasticity would be affected were we to replace employment lotteries and complete markets for consumption insurance with the incomplete markets arrangements that seem more natural to labor economists This change reorients attention away from the fraction of its members that a representative family chooses to send to work at any moment to career lengths chosen by individual workers who selfinsure by saving and dissaving We find that abandoning the employment lotteries coupled with complete consumption insurance claims trading assumed within many real business cycle models and re placing them with individual workers who selfinsure by trading a riskfree bond does not by itself imperil that high aggregate labor supply elasticity championed by Prescott The labor supply elasticity depends on whether shocks and gov ernment financed social security retirement schemes leave most workers on or off corners with respect to their retirement decisions in a model of indivisible labor 1315 1316 Foundations of Aggregate Labor Supply During the last half decade macroeconomists have mostly abandoned em ployment lotteries in favor of timeaveraging and incomplete markets as an aggregation theory for aggregate labor supply This is undoubtedly a positive development because now researchers who may differ about the size of the ag gregate labor supply elasticity can at least talk in terms of a common framework and can focus on their disagreements about the proper quantitative settings for a commonly agreed on set of parameters and constraints To convey these ideas we build on an analysis of Ljungqvist and Sargent 2007 who in a particular continuous time model showed that the very same aggregate allocation and individual expected utilities that emerge from a Rogersonstyle completemarket economy with employment lotteries are also attained in an incompletemarket economy without lotteries In the Ljungqvist Sargent setting instead of trading probabilities of working at any point in time agents choose fractions of their lifetimes to devote to work and use a credit mar ket to smooth consumption across episodes of work and times of retirement1 This chapter studies how two camps of researchers namely those who cham pion high and low labor supply elasticities respectively both came to adopt the same theoretical framework2The first part of the chapter revisits equivalence results between an employment lotteries model and a timeaveraging model 1 Larry Jones and Casey Mulligan anticipated aspects of this equivalence result In the context of indivisible consumption goods in the original 1988 version of his paper Jones 2008 showed how timing could replace lotteries when there is no discounting In the 2008 published version of his paper he extended the analysis to cover the case of discounting In comparing an indivisiblelabor completemarket model and a representativeagent model with divisible labor Mulligan 2001 suggested that the elimination of employment lotteries and complete markets for consumption claims from the former model might not make much of a quantitative difference The smallest labor supply decision has an infinitesimal effect on lifetime consumption and the marginal utility of wealth in the divisiblelabor model and a smallbutlargerthaninfinitesimal effect on the marginal utility of wealth in the indivisible labor model as long as the effect on lifetime consumption is a small fraction of lifetime income or the marginal utility of wealth does not diminish too rapidly However as we shall learn later in this chapter these qualifications vanish when time is continuous as well as for infinitelylived agents in discrete time As a discussant of Ljungqvist and Sargent 2007 Prescott 2007 endorsed their incomplete markets career length model as a model of aggregate labor supply In addition he reduced his previous stress on the employment lotteries model by adding a new section The life cycle and labor indivisibility to the final version of his Nobel lecture published in America Prescott 2006 2 This is the theme of Ljungqvist and Sargent 2011 1318 Foundations of Aggregate Labor Supply 3121 Choosing career length At each point in time an agent can work at a wage rate w and can save or dissave at an interest rate r An agents asset holdings at time t are denoted by at and its time derivative by at Initial assets are assumed to be zero a0 0 and the budget constraint at time t is at rat wnt ct 3122 with a terminal condition a1 0 This is a noPonzi scheme condition To solve the agents optimization problem we formulate the currentvalue Hamiltonian Ht uct Bnt λt rat wnt ct 3123 where λt is the multiplier on constraint 3122 It is called the costate variable associated with the state variable at Firstorder conditions with respect to ct and nt respectively are uct λt 0 3124a B λtw 0 if nt 0 0 if indifferent to nt 0 1 0 if nt 1 3124b Furthermore the costate variable obeys the differential equation λt λtρ Ht at λtρ r 3125 When r ρ Ljungqvist and Sargent 2007 show that the solution to this optimization problem yields the same lifetime utility as if the agent had access to employment lotteries and complete insurance markets including consumption claims that are contingent on lottery outcomes First we note from equation 3125 that when r ρ the costate variable is constant over time and hence by equation 3124a the optimal consumption stream is constant over time ct c Then after invoking optimality condition 3124b there are three possible cases with respect to the agents lifetime labor supply B ucw 0 Case 1 nt 0 for all t 0 Case 2 indifference to nt 0 1 at any particular instance in time 0 Case 3 nt 1 for all t 3126 1322 Foundations of Aggregate Labor Supply 313 Taxation and social security We study taxation and social security in a continuoustime overlapping genera tions model At each instance in time there is a constant measure of newborn ex ante identical agents like those in section 3121 entering the economy Thus the economys population and age structure stay constant over time Our focus is not on the determination of intertemporal prices in this overlapping genera tions environment with its possible dynamic inefficiencies see chapter 9 so we retain our small open economy assumption of an exogenously given interest rate which also implies a given wage rate if the economys production technology is constant returns to scale in labor and capital4 We assume that utility is logarithmic in consumption uc logc and that there is no discounting r ρ 0 The assumption of no discounting is inessential for most of our results and where it matters we will take note The analytical convenience is that the optimal career length is uniquely determined and does not depend on the timing of an agents lifetime labor supply as shown in expressions 31210 and 31211 As emphasized by Prescott 2005 if labor income is taxed and tax revenues are handed back lump sum to agents a model with indivisible labor and em ployment lotteries exhibits a large labor supply elasticity Under the equivalence result in section 312 we follow Ljungqvist and Sargent 2007 and demonstrate that the same high labor supply elasticity arises in the incompletemarket model where career lengths rather than the odds of working in employment lotteries are shortened in response to such a tax system In the spirit of Ljungqvist and Sargent 2012 we offer a qualification to the high labor supply elasticity in a model of lifetime labor supply When a government program such as social security is associated with a large implicit tax on working beyond an official retirement age there might not be much of an effect of taxation on career length for those agents who could be at a corner solution strictly preferring to retire at the official retirement age 4 In the case of a constantreturnstoscale CobbDouglas production function equation 3185b shows how the interest rate in international capital markets determines the capital labor ratio in a small open economy which in turn determines the wage rate in 3185a Taxation and social security 1327 i if R T τ then TR τ T τ retire after the official retirement age ii if R T τ then TR τ T τ retire before the official retirement age iii otherwise TR τ R retire at the official retirement age Given R 06 the solid curve in Figure 3131 displays equilibrium career length as a function of τ Within a range of tax rates between 1640 percent equilibrium career length does not respond to changes in the tax rate because agents are at a corner solution and strictly prefer to retire at the official retire ment age R Away from that corner career length is highly sensitive to the social security tax rate τ in Figure 3131 When an equilibrium has agents retiring before the official retirement age R T T τ equilibrium career length 31310 is identical to outcome 3135 under the Prescott tax system The reasons are that a under our assumption that average earnings alone determine the replacement rate without regard to career length agents regard their social security contributions purely 0 02 04 06 08 1 0 02 04 06 08 1 Tax rate Career length T τ T τ Figure 3131 Social security Solid curve depicts equilibrium career length as a function of a social security tax rate τ given an official retirement age R 06 At low high tax rates τ 016 τ 040 an agent retires after before the official retirement age where the actual retirement age lies along the curve T τ T τ given a disutility of work B 1 Earningsexperience profiles 1331 case when work experience does not affect earnings the aggregate labor supplies as well as the expected lifetime utilities are exactly the same across the two economies as asserted in the equivalence result of section 312 3143 Prescott tax and transfer scheme It is instructive to revisit Prescotts tax analysis in section 3131 for the present environment with earningsexperience profiles We invite the readers to verify that the equilibrium career length in the time averaging economy is then T 1 τφ 1 B 3149 and the employmentpopulation ratio in the employment lotteries economy is ψ 1 τ B 31410 While the labor supplies in 3149 and 31410 differ we note that the elas ticity of the supply with respect to the netoftax rate 1 τ is the same and equal to one This equality is another reflection of broad similarities that typically prevail across incompletemarket and completemarket economies with indivisible labor We shall encounter another example in section 318 when we compare the aggregate labor supply in a Bewley incomplete markets economy with its completemarket counterpart 3144 No discounting now matters Recall that under a flat earningsexperience profile φ 0 in section 3121 an agent is indifferent about the multitude of labor supply paths that yield the same presentvalue of labor income in budget constraint 3129 The reason is that two alternative labor supply paths with the same presentvalue of labor income imply the same lifetime disutility of work when ρ r Note that for strictly positive discounting ρ r 0 a labor supply path that is tilted toward the future means that an agent will have to work for a longer period of time to generate the same presentvalue of labor income as compared to a labor supply path that is tilted toward the present But that is acceptable to the agent since 1332 Foundations of Aggregate Labor Supply future disutilities of work are discounted at the same rate as labor earnings when the subjective discount rate is equal to the market discount rate But if there is an upwardsloping earningsexperience profile φ 0 an agent is no longer indifferent to the described variation in career length associ ated with the timing of lifetime labor supply In particular when ρ r 0 an agent strictly prefers to shift his labor supply to the end of life because at a given lifetime disutility of work working later in life would mean spending more total time working That would push the worker further up the experienceearnings profile and thereby increase the present value of lifetime earnings Features not present in our model would attenuate such a desire to postpone labor supply to the end of life eg borrowing constraints that force an agent to finance consumption with current labor earnings incomplete insurance markets that compel an agent to resolve career uncertainties earlier and forecast declines in dexterity with advances in age 315 Intensive margin Prescott et al 2009 extend the analysis of Ljungqvist and Sargent 2007 in section 312 by introducing an intensive margin in labor supply ie nt 0 1 is now a continuous rather than a discrete choice variable However to retain the central force of indivisible labor they postulate a nonlinear mapping from nt to effective labor services in particular an increasing mapping that is first convex and then concave For expositional simplicity we let the effective labor services associated with nt be nt n where n 0 1 As noted by Prescott et al 2009 such a mapping can reflect costs associated with getting set up in a job learning about coworkers and so on The preferences are the same as those of Ljungqvist and Sargent 2007 in 3121 but now with no discounting ρ r 0 Under the present assumption that nt is a continuous choice variable we need to make additional assumptions about the function v The instantaneous disutility function over work vn is strictly increasing strictly convex and twice continuously differentiable 1336 Foundations of Aggregate Labor Supply Along with Prescott et al 2009 we conclude that the effects of taxation are the same as in Ljungqvist and Sargent 2007 ie all the adjustment of labor supply takes place along the extensive margin and the elasticity of aggregate labor supply with respect to the netoftax rate 1 τ is equal to one The reason that none of the adjustment takes place along the intensive mar gin is that any changes in labor when already working occur along an increasing marginal disutility of work while adjustment along the extensive margin are made at a constant disutility of work by varying the fraction of ones lifetime devoted to work The constancy of the latter terms of trade between working and not working was the essential ingredient of the famous or depending on your viewpoint infamous high labor supply elasticity in models of employment lotteries when labor is indivisible Rogerson and Wallenius 2009 break the constancy of the terms of trade between working and not working by adding a life cycle earnings profile to the present framework but in contrast to section 314 they take that earnings pro file as exogenously given rather than having it be determined as a function of an agents past work experience In the Rogerson and Wallenius setup two results follow immediately a agents choose to work when their life cycle earnings profile is highest namely when it exceeds an optimally chosen reservation level and b labor supply nt at a point in time varies positively with the exogenous earnings level Taxation in this augmented framework affects labor supply along both the intensive and extensive margins While an increasing marginal disutil ity of work continues to frustrate adjustment along the intensive margin there is now decreasing earnings when extending the career beyond the heights of the exogenous life cycle earnings profile which then also frustrates adjustment along the extensive margin The assumed curvatures of the disutility of work at the intensive margin and that of the exogenous lifecycle earnings profile determine how much adjustment occurs along the intensive and extensive margins BenPorath human capital 1337 316 BenPorath human capital We return to the assumption that labor is strictly indivisible nt 0 1 and add a BenPorath human capital accumulation technology to the framework of section 312 We take note of BenPoraths 1967 p 361 observation that if the technology were to exhibit exact constant returns to scale the marginal cost of additional units of human capital would be constant until all of the agents current human capital is devoted to the effort of accumulating human capital and hence the optimal rate of investment at any point in time would be either full specialization or no investment at all Under our simplifying assumption of no depreciation of human capital it follows that an agent would specialize and make all of his investment in human capital upfront Acquiring human capital can be thought of as formal education before starting to work To represent the notion of specializing in human capital investments in a simple way we assume that an agent has access to a technology that can instan taneously determine his human capital through the investment of m 0 units of goods in himself which produces a human capital level h mγ γ 0 1 3161 and there is no depreciation of human capital It follows trivially that it will be optimal for an agent to use that technology once and for all before starting to work Under our assumption of a perfect credit market an agent chooses in vestment goods m that maximize his present value labor income in conjunction with his choice of an optimal career length T Papers by Guvenen et al 2011 and Manuelli et al 2012 that incorporate BenPorath human capital technologies in life cycle models inspire our analysis Those papers mainly focus on tax dynamics driven not by the force in the Prescott tax system in section 3131 but instead by wedges that distort an agents investment in human capital Guvenen et al 2011 postulate progressive labor income taxation while Manuelli et al 2012 assume that investments in human capital are not fully taxdeductible In both cases the central force is that the tax rate on returns to human capital is higher than the rate applied to labor earnings foregone while investing in human capital or the rate at which goods input to human capital can be deducted from an agents tax liabilities Following Manuelli et al 2012 we assume a flatrate tax τ 0 1 on labor income and that only a fraction ǫ 0 1 of goods input to human capital 1346 Foundations of Aggregate Labor Supply 3171 Interpretation of wealth and substitution effects For an agent with positive asset holdings at ˆt a negative wage shock means that returns to working fall relative to the marginal value of his wealth That induces the agent to enjoy more leisure because doing that has now become relatively less expensive But with negative asset holdings at ˆt a negative wage shock compels the agent to supply more labor both to pay off time ˆt debt and to moderate the adverse effect of the shock on his future consumption With a positive wage shock leisure becomes more expensive causing the agent to substitute away from leisure and toward consumption This force makes lifetime labor supply increase for an agent with positive wealth But why does a positive wage shock lead to a reduction in lifetime labor supply when time ˆt assets are negative In the case of a positive wage shock and negative time ˆt assets consider a hypothetical asset path that would have prevailed if the agent had enjoyed the higher wage rate ˆw from the beginning starting at t 0 Along that hypothetical path the agent would have been even further in debt at ˆt since the optimal constant consumption level would have been equal to ˆwB as given by 3128 So at ˆt the agent actually finds himself richer at ˆt than he would have in our hypothetical scenario Because there is less debt to be repaid at ˆt the agent chooses to supply less labor than he would have in the hypothetical scenario In other words it is not optimal to make up for what would have been past underconsumption relative to our hypothetical path so the agent chooses instead to enjoy more leisure because he has relatively less debt at ˆt than he would along the hypothetical path Time averaging in a Bewley model 1351 3183 Simulations of Prescott taxation We adopt the calibration of Chang and Kim 2007 except that we shut down aggregate productivity shocks To highlight differences and similarities across our incomplete and completemarket versions of the economy we compute equi librium outcomes under Prescotts tax and transfer scheme in section 3131 0 20 40 60 80 100 0 05 1 15 2 25 3 35 Asset holdings Productivity τI 070 τI 030 τC 070 τC 030 Figure 3181 Reservation productivity as a function of asset holdings in the economy with incomplete markets solid curves and complete markets dashed curves respectively where the lower upper curve refers to tax rate 030 070 For labor tax rates of 030 and 070 respectively reservation productivities as functions of asset holdings are displayed in Figure 3181 In the incomplete market economy solid curves an agents reservation productivity increases in his asset holdings A high asset level means that everything else equal an agent is poised to enjoy one of his intermittent spells of leisure which will result in asset decumulation and his ultimate return to work For an agent with high assets to postpone such a desired spell of leisure the agent must experience a relatively high productivity to be willing to continue to work for a while As one would expect the reservation productivities for the higher tax rate 070 lie well above those for the lower tax rate 030 since Prescotts tax and transfer scheme is very potent in suppressing agents labor supply and causing them to 1352 Foundations of Aggregate Labor Supply choose more leisure In the completemarket economy the single productivity cutoff dashed curve is indicative of a privately efficient allocation It is the most productive agents who work at any point in time 0 1 2 3 4 5 0 005 01 Productivity Density Population distribution τC7 τI3 τI7 τC3 Figure 3182 Productivity distribution The upper solid curve is the population productivity distribution while the other two in descending order show the agents thereof who are employed in the incompletemarket economy given tax rate 030 and 070 re spectively The corresponding masses of employed agents in the completemarket economy are the halves of the population distri bution to the right of a vertical dashed line where the left right dashed line refers to tax rate 030 070 The top solid curve in Figure 3182 depicts the stationary distribution of productivities in the population A dashed vertical line is the productivity cutoff in the completemarket economy where the left right one refers to tax rate 030 070 ie the same reservation productivity as the corresponding dashed line in Figure 3181 All agents with productivities to the right of the dashed line work in the completemarket economy and hence the area under that portion of the population distribution equals the employmentpopulation ratio In the incompletemarket economy the endogenous stationary distribution of agents across both productivities and asset holdings Ja z together with the decision rule for whether or not to work na z determine how many agents Time averaging in a Bewley model 1353 are at work at different productivity levels Those employed workers in the incompletemarket economy are depicted by the solid curves that lie weakly below the top population curve which in descending order refer to tax rates 030 and 070 respectively As in the completemarket economy virtually all agents with high productivities are working in the incompletemarket economy But over a midrange of productivities there are significant differences between the two economies On the one hand some agents in the incompletemarket do not work but would have been working in the completemarket economy The reason is that because their asset holdings are relatively high their shadow value of additional wealth falls below the utility of leisure On the other hand other agents in the incompletemarket economy work but would not have worked in the completemarket economy These agents have low asset holdings and so feel compelled to work despite their low productivities The work and asset decisions of individual agents in the incompletemarket economy determine the distribution of asset holdings and the capital stock For labor tax rates 030 and 070 respectively the solid curves in Figure 3183 depict the cumulative distribution function for asset holdings in the incomplete market economy At the high tax rate 070 upper solid curve asset holdings become concentrated at lower levels As in the case of the elevated reservation productivities in Figure 3181 taxation suppresses market activity in favor of leisure In the completemarket economy tax rate 070 is associated with a similar large decline in per capita asset holdings as depicted by the vertical dashed lines in Figure 3183 where the left right one refers to tax rate 070 030 From a production perspective what matters is the capital stock relative to the aggregate supply of efficiency units of labor In the completemarket econ omy that capitallabor ratio is determined by steadystate relationship 3189 which does not depend on the labor tax rate but would have depended on any intertemporal tax wedge such as a tax on capital income Since the wage rate is a function of the capitallabor ratio in 3185a it follows in Figure 3184 that the wage rate in the completemarket economy dashed curve is invariant to the labor tax rate In contrast the wage rate in the incompletemarket economy solid curve falls with the labor tax rate and lies above the wage rate of the completemarket economy To understand the latter outcome we recall that in a Bewley model like ours with infinitelylived agents the interest rate must fall below the subjective rate of discounting β1 which is the steadystate 1354 Foundations of Aggregate Labor Supply 0 10 20 30 40 50 0 02 04 06 08 1 Asset holdings CDF τC 3 τC 7 τI 070 τI 030 Figure 3183 Asset distribution The lower upper solid curve is the cumulative distribution function for asset holdings in the incompletemarket economy when the tax rate is 030 070 The right left vertical dashed line is the per capita asset holdings in the completemarket economy when the tax rate is 030 070 interest rate in the completemarket economy Since the equilibrium interest rate is inversely related to the capitallabor ratio in expression 3185b it fol lows immediately that the capitallabor ratio is higher in the incompletemarket economy and therefore by expression 3185a so is the wage rate Figure 3185 shows that the fraction of the population employed is higher in the incompletemarket economy than in the completemarket economy As seen in Figure 3182 there are those agents who work and those who do not work in the incompletemarket economy but who would have done the opposite if they instead had lived in the completemarket economy Evidently the group of agents who work in the incompletemarket economy but would not have worked in the completemarket economy is larger With no insurance markets agents on average work more in order to accumulate precautionary savings in the event of low productivity in the future9 9 Marcet et al 2007 conduct an analysis similar to that of Chang and Kim 2007 but where labor is divisible nt 0 1 and the idiosyncratic productivity shock takes on only two values zt 0 1 In addition to the precautionary savings effect that tends to increase Time averaging in a Bewley model 1355 0 01 02 03 04 05 06 07 2 205 21 215 22 225 23 235 24 Wage rate Labor tax rate Incomplete markets Complete markets Figure 3184 Wage rate per efficiency unit of labor in the econ omy with incomplete markets solid curve and complete markets dashed curve as a function of the labor tax rate 0 01 02 03 04 05 06 07 0 01 02 03 04 05 06 07 08 Labor tax rate Employment Incomplete markets Complete markets Figure 3185 Employmentpopulation ratio in the economy with incomplete markets solid line and complete markets dashed line as a function of the labor tax rate The dotted line represents the former economy with a less persistent productivity process the capital stock under uncertainty they identify an ex post wealth effect on labor supply that 1356 Foundations of Aggregate Labor Supply What makes the employmentpopulation ratio to converge across the two economies at higher tax rates in Figure 3185 A key reason is that Prescotts tax and transfer scheme effectively insures the agents by collecting tax revenues and then returning them lump sum as equal amounts to all agents To explore how precautionary savings drive the employment wedge between the incomplete and completemarket economies at low tax rates consider the following per turbation of the idiosyncratic productivity process Specifically suppose that agents face a transition probability distribution function πzz λ 1 λπzz λGz 31813 where λ 0 1 For λ 0 the productivity process is the same as that of Chang and Kim 2007 while for λ 1 productivities are independent and identically distributed across agents and time with realizations governed by the stationary unconditional distribution of Chang and Kims process Such per turbations do not affect equilibrium outcomes in the completemarket economy because they do not affect the constraints of the representative family But agents in the incompletemarket are now ex ante relieved when they do not have to bear as much of the risk associated with the persistence of Chang and Kims productivity process The dotted line in Figure 3185 shows equilibrium outcomes in the incompletemarket economy for λ 01 where employment is now closer to that of the completemarket economy A striking feature of Figure 3185 is the high elasticity of aggregate labor supply to taxation in the completemarket as well as in the incompletemarket economy This message is shared with the first part of this chapter when agents were finitely lived and at interior solutions with respect to their choices of career length can depress the aggregate hours of work as well as the capital stock in an incompletemarket economy See section 178 1358 Foundations of Aggregate Labor Supply Thus any asset accumulation or decumulation by an agent can only be mo tivated by that agents desire to engage in time averaging with respect to his labor supply For an agent with assets in some range a a we shall show that time averaging is indeed optimal because it enables him to finance an optimal constant consumption level c wB But first we discuss our guess of the value function outside of this asset range If an agent has too little too much assets he will choose to work forever to never work and to consume the highest affordable constant consumption level associated with that labor supply plan Consider an agent whose beginning ofperiod assets a a are so low that if he works forever and consumes the highest affordable constant consumption w ra that consumption level will be less than or equal to c wB We can verify later that such a poor agent will indeed choose to work forever and to consume w ra in each period After invoking r 1β 1 the critical asset limit a is w 1 β β a w B a βw 1 β B1 1 3192 If nt 1 and ct w ra for all t preference specification 3181 yields lifetime utility given by the conjectured value function 3191 when a a11 Next consider an agent whose beginningofperiod assets a a are so high that if he never works and consumes the highest affordable constant consump tion ra that consumption level will be greater than or equal to c wB We can later verify that such a rich agent will indeed choose never to work and to consume ra in each period After invoking r 1β 1 the critical asset limit a is 1 β β a w B a βw 1 β B1 3193 shifting consumption from periods of high to periods of low consumption An agents employ ment status does not matter since preference specification 3181 is additively separable in consumption and leisure 11 Under the implicit but necessary parameter restriction for an equilibrium with time aver aging B 1 note that asset limit a in 3192 is negative ie only agents who are initially indebted a 0 could conceivably want to choose to work forever with constant consumption equal to w ra 1360 Foundations of Aggregate Labor Supply Note that the conjectured value function 3191 is weakly concave so that the two inner optimization problems one for working another for not working on the right side of 3194 are both concave programming problems Moreover since the conjectured value function is continuous and differentiable everywhere we can solve each optimization problem for working and for not working one by one using firstorder conditions and compare the values Let Wa 1 and Wa 0 denote the value of working and not working respectively and hence V a maxWa 1 Wa 0 We start by verifying the conjectured value function for a a a when time averaging should be an optimal policy First conditional on working take a firstorder condition with respect to a in the first inner optimization problem on the right side of 3194 1 1 ra w a βV a a 1 β a w w B 3195 Here we have invoked the conjectured steadystate interest rate 1 r β1 and proceeded as if a also falls in the range a a where the conjectured value function 3191 has derivative V a Bβw Since a exceeds a it follows that a must fall below some upper bound a a in order for a a a where that upper bound a is given by 12 a a βw a 3196 Given the optimal choice of a in expression 3195 we can compute from the budget constraint that the implied consumption level is c wB With 12 Using expression 3195 for a the upper bound a on asset level a that ensures a a can be solved from 1 β a w w B a Multiplying both sides by β and subtracting and adding a on the right side yield a βw βw B βa a a After invoking expression 3193 for a we find that the last term on the left side is equal to the first two terms on the right side and hence we have arrived at the equality in 3196 L and S equivalence meets C and Ks agents 1363 3193 Equivalence of time averaging and lotteries Krusell et al 2008 argue that there exists a stationary equilibrium for the incompletemarket economy where all agents engage in time averaging with assets in the range a a and the aggregate values of K and L are the same as in a corresponding completemarket economy with employment lotteries We have already studied equilibrium outcomes in a more general version of the completemarket economy in section 3182 Under our present assumption that all agents have a constant productivity level that is normalized to one equa tion 31810 shows that the optimal consumption level is c wB and the aggregate labor supply is given by the appropriate version of equation 31812 K L 1 β K L w w BL 3199 where once again the capitallabor ratio KL and the wage w are determined by equations 3189 and 3185a Hence we can solve for the aggregate labor supply L from equation 3199 In the stationary equilibrium of the incompletemarket economy with time averaging agents are indifferent to alternative lifetime labor supply paths that yield equal present values of labor income In a competitive equilibrium an invisible hand arranges agents labor and savings decisions so that at every point in time the aggregate labor supply and aggregate asset holdings equal the same constant aggregates L and K as those in the completemarkets economy An equilibrium interest equal to 1 r 1β makes a constant consumption c wB be the optimal choice for the workerconsumer 1364 Foundations of Aggregate Labor Supply 3110 Two pillars for high elasticity at extensive margin The high labor supply elasticity at an interior solution for career length rests on two pillars indivisibilities in labor supply and time separable preferences Labor supply indivisibilities cause workers to divide their lifetimes into parts working and not working Timeseparable preferences make the choice between those two parts occur at a constant perperiod disutility of work generating that high elasticity of labor supply at an interior solution for career length ie at an extensive margin The laborsupplyindivisibility pillar is typically justified by the observation that workers hours of work are mostly bunched at a few common values with the fulltime value predominating Alternative assumptions about technologies and preferences can generate that outcome Simple examples include a setup cost at work and a fixed disutility of work The timeseparablepreferences pillar is typically justified as doing a good job of approximating workers wishes to rest and refresh between periods 3111 No pillars at intensive margin No pillars have yet been discovered that would imply a high labor supply elas ticity at an intensive margin Examples of utility functions that generate a high elasticity at an intensive margin have been constructed but they are very spe cial and seem to rest on no general principles about preferences We illustrate the absence of pillars by studying reasoning behind a claim by Rogerson and Wallenius 2013 hereafter RW that based on existing estimates of the size of nonconvexities and measures of fulltime work prior to retirement it is hard to rationalize values of the IES intertemporal elasticity of substitution for labor at the intensive margin that are less than 075 at the fulltime work option No pillars at intensive margin 1367 0 01 02 03 04 05 0 05 1 15 Hours worked Disutility of work np nf Figure 31111 Disutility of work where the solid line depicts vRW with IES for labor equal to 077 at nf and the dashed lines represent alternative parameterizations of vLS with an IES at nf of around 01 02 03 04 05 06 and 07 respectively when moving from left to right at the top of the figure 31112 Fragility of the special example RWs finding that the IES for labor must be high to match observations depends sensitively on assuming utility function 31111 or 31112 Ljungqvist and Sargent 2018 establish that sensitivity by blending those two utility functions to get an alternative one vLSn µ1vRWn µ2vPmax0 n np 31114 The utility function 31114 augments the RW disutility of work vRWn with extra disutility for hours of work above np measured by the Prescott et al 2009 disutility vPmax0 n np Specifically to parameterize the dashedline utility functions in Figure 31111 we can proceed as follows We set a common preference parameter γ in vRW to be larger than the value deduced in the preceding subsection and select a weight µ1 that assures that vLSnp 05075 Each dashed line in the figure is constructed using a different 1368 Foundations of Aggregate Labor Supply value of φ in vP and an appropriately adjusted weight µ2 that assures that vLSnf 116 The utility functions in Figure 31111 are normalized to be one at nf by construction they attain the value 05 075 at np Hence a worker with any of those preferences would be indifferent between working full time or part time at an interior solution to career length RW use that indifference to establish a lower bound on the IES for labor given utility function vRW Without relying on any other nonconvexity than the one used by RW our alternative utility functions vLS in Figure 31111 demonstrate fragility of RWs conclu sion that it is hard to rationalize values of the IES that are less than 075 To the contrary Ljungqvist and Sargent 2018 showed that it is easy by sim ply blending preferences that were actually used by RW themselves in closely related contexts17 While utility functions 31111 and 31112 are often used to estimate labor supply elasticities at intensive margins the parametric specifications are best thought of as local approximations of the curvature at some observed hours 16 By setting γ and µ1 as described in the text we are assured that µ1vRWnf 1 which leaves room for a positive quantity µ2vPnf np 0 to be part of vLSnf 1 The weights µ1 and µ2 then satisfy µ1 05 075 vRWnp and µ2 1 µ1vRWnf vPnf np For the record but without any particular significance the dashed lines in Figure 31111 are drawn for a parameter γ that is twice as large as the one for the solid line 096 instead of 048 and the values of the parameter φ are set equal to 014 024 033 043 053 063 and 073 respectively when moving across the dashed lines from left to right at the top of the figure 17 The reasoning behind our parameterization of the two components in vLS is as follows Regarding 31111 with a constant IES γ for leisure we note that the IES for work as given by γ 1 nn asymptotes to infinity when hours of work goes to zero Therefore for some initial range of hours of work the IES for labor will necessarily be high and that heightened willingness to substitute intertemporally means that the disutility of work increases almost linearly over an initial range of hours of work Thus with a relatively high γ our utility function 31114 relies on an initial extended nearly linear segment of vRW that serves to suppress the attraction of the parttime work option In specification 31112 with a constant IES φ for work a lower value of φ means an accelerated growth in disutility because at any level of supplied hours less willingness to substitute intertemporally necessarily shows up as a relatively larger increase in the disutility of work ie in a more convex shape Thus with a low value of φ our utility function 31114 unleashes that strong growth component vP closer to nf eventually coming to dominate the curvature of our utility function so the IES for labor at the fulltime work option is close to the assumed small value of φ Concluding remarks 1369 of work Instead RW chose to draw dramatic conclusions from the assumption of a globally constant IES for leisure They would have drawn even more dramatic ones if they had assumed a globally constant IES for labor In our judgement drawing sweeping conclusions about the labor supply elasticity at the intensive margin from such a shaky extrapolation from a reasonable local property of a utility function to a global one falls far short of providing the general economic forces provided by the two pillars that support a high elasticity at the extensive margin 3112 Concluding remarks A high aggregate labor supply elasticity hinges on a substantial fraction of agents being at an interior solution with respect to their lifetime labor supplies This finding emerges from models with finitelylived agents who choose career length and also in Chang and Kims 2007 model of infinitelylived agents who engage in time averaging across periods of work and leisure When agents are finitely lived two forces can lower the labor supply elas ticity 1 government financed social security retirement schemes that leave agents at a corner solution with respect to their choices of career lengths and 2 large adverse labor market shocks towards the end of working lives that prematurely terminate careers by pushing the shadow value of additional labor earnings below the utility of leisure in early retirement It is an occasion to celebrate that two camps of researchers namely those who have championed high and low labor supply elasticities have come together in adopting the same theoretical framework Nevertheless the serious division between the two camps about quantitative magnitudes of labor supply elastic ities persists But we see the emergence of agreement over a basic theoretical framework as genuine progress relative to the earlier stalemate when proponents of employment lotteries used macroeconomic observations to build support for their aggregation theory while opponents brought a different set of microeco nomic observations to refute the employment lotteries allocation mechanism18 18 It would be a mistake to regard the abandonment of a standin household with its employ ment lotteries as unconditional surrender to the other tradition in macroeconomics of over lapping generations models that has commonly postulated incomplete markets The reason is that earlier work in the overlapping generations tradition has often postulated an exogenous 1370 Foundations of Aggregate Labor Supply To illustrate how far we have come we revisit our own section 296 reason ing where we are concerned about an asymmetry between idle firms and idle workers in a particular model While idle firms are truly indifferent about their operating status because operating firms are just breaking even without mak ing any profits the aggregation theory behind the standin household has an additional aspect that is not present in the theory that aggregates over firms namely it says how consumption and leisure are smoothed across households with the help of an extensive set of contingent claim markets This market arrangement and randomization device stands at the center of the employment lottery model To us it seems that they make the aggregation theory behind the standin household fundamentally different than the wellknown aggregation theory for the firm side Well we now also can assert that this difference is not important for those households who being at an interior solution for lifetime labor supply are about to choose whether to supply more of their indivisible labor by extending their careers before retiring Having a diverse group of researchers focus on a common set of observations on lifetime labor supply within a common theoretical framework bodes well for the eventual arrival of what we hope will be the labor supply elasticity accord foretold by Ljungqvist and Sargent 2011 retirement age shutting down the key choice focused on in timeaveraging models of career choice It is the possibility of interior solutions to lifetime labor supply in combination with indivisible labor that have led real business cycle researchers like Prescott 2006 to embrace lifecycle models of labor supply 1376 Functional Analysis These examples illustrate the fact that whether a given sequence is Cauchy depends on the metric space within which it is embedded in particular on the metric that is being used The sequence tn is Cauchy in C0 1 d2 and more generally in C0 1 dp for 1 p The sequence tn however is not Cauchy in the metric space C0 1 d The first example also illustrates the fact that a Cauchy sequence in X d need not converge to a limit point x0 belonging to the metric space The property that Cauchy sequences converge to points lying in the metric space is desirable in many applications We give this property a name Definition A14 A metric space X d is said to be complete if each Cauchy sequence in X d is a convergent sequence in X d That is in a complete metric space each Cauchy sequence converges to a point belonging to the metric space The following metric spaces are complete lp0 dp 1 p l0 d C0 T d The following metric spaces are not complete C0 T dp 1 p Proofs that lp0 dp for 1 p and C0 T d are complete are contained in Naylor and Sell 1982 chap 3 In effect we have already shown by counterexample that the space C0 1 d2 is not complete because we displayed a Cauchy sequence that did not converge to a point in the metric space A definition may now be stated Definition A15 A function f mapping a metric space X d into itself is called an operator We need a notion of continuity of an operator Definition A16 Let f X X be an operator on a metric space X d The operator f is said to be continuous at a point x0 X if for every ǫ 0 Metric spaces and operators 1377 there exists a δ 0 such that dfx fx0 ǫ whenever dx x0 δ The operator f is said to be continuous if it is continuous at each point x X We shall be studying an operator with a particular property the application of which to any two distinct points x y X brings them closer together Definition A17 Let X d be a metric space and let f X X We say that f is a contraction or contraction mapping if there is a real number k 0 k 1 such that dfx fy kdx y for all x y X It follows directly from the definition that a contraction mapping is a continuous operator We now state the following theorem Theorem A11 Contraction Mapping Let X d be a complete metric space and let f X X be a contraction Then there is a unique point x0 X such that fx0 x0 Furthermore if x is any point in X and xn is defined inductively according to x1 fx x2 fx1 xn1 fxn then xn converges to x0 Proof Let x be any point in X Define x1 fx x2 fx1 Express this as xn f nx To show that the sequence xn is Cauchy first assume that n m Then dxn xm df nx f mx df mxnm f mx kdf m1xnm f m1x By induction we get dxn xm kmdxnm x When we repeatedly use the triangle inequality the preceding inequality implies that dxn xm kmdxnm xnm1 dx2 x1 dx1 x Applying gives dxn xm kmknm1 k 1dx1 x Discounted dynamic programming 1379 all points in the domain of definition of the functions in X For any positive real c and every x X T xc T xβc for some β satisfying 0 β 1 Then T is a contraction mapping with modulus β Proof For all x y X x y dx y Applying properties a and b to this inequality gives T x T y dx y T y βdx y Exchanging the roles of x and y and using the same logic implies T y T x βdx y Combining these two inequalities gives T x T y βdx y or dT x T y βdx y A2 Discounted dynamic programming We study the functional equation associated with a discounted dynamic pro gramming problem vx max uRkrx u βvx x gx u 0 β 1 A21 We assume that rx u is real valued continuous concave and bounded and that the set x x u x gx u u Rk is convex and compact We define the operator T v max uRkrx u βvx x gx u x X We work with the space of continuous bounded functions mapping X into the real line We use the metric dv w supxX vxwx This metric space is complete The operator T maps a continuous bounded function v into a continuous bounded function T v For a proof see Stokey and Lucas with Prescott 19893 3 The assertions in the preceding two paragraphs are the most difficult pieces of the argument to prove 1380 Functional Analysis We now establish that T is a contraction by verifying Blackwells pair of sufficient conditions First suppose that vx wx for all x X Then T v max uRkrx u βvx x gx u max uRkrx u βwx x gx u T w Thus T is monotone Next notice that for any positive constant c T v c max uRkrx u βvx c x gx u max uRkrx u βvx βc x gx u T v βc Thus T discounts Therefore T satisfies both of Blackwells conditions It follows that T is a contraction on a complete metric space Therefore the functional equation A21 which can be expressed as v T v has a unique fixed point in the space of bounded continuous functions This fixed point is approached in the limit in the d metric by iterations vn T nv0 starting from any bounded and continuous v0 Convergence in the d metric implies uniform convergence of the functions vn Stokey and Lucas with Prescott 1989 show that T maps concave functions into concave functions It follows that the solution of v T v is a concave function A21 Policy improvement algorithm For ease of exposition in this section we shall assume that the constraint x gx u holds with equality For the purposes of describing an alternative way to solve dynamic programming problems we introduce a new operator We use one step of iterating on the Bellman equation to define the new operator Tµ as follows Tµv T v or Tµv rx µx βvgx µx Discounted dynamic programming 1381 where µx is the policy function that attains T vx For a fixed µx Tµ is an operator that maps bounded continuous functions into bounded continuous functions Denote by C the space of bounded continuous functions mapping X into X For any admissible policy function µx the operator Tµ is a contraction mapping This fact can be established by verifying Blackwells pair of sufficient conditions 1 Tµ is monotone Suppose that vx wx Then Tµv rx µx βvgx µx rx µx βwgx µx Tµw 2 Tµ discounts For any positive constant c Tµv c rx µ β vgx µx c Tµv βc Because Tµ is a contraction operator the functional equation vµx Tµvµx has a unique solution in the space of bounded continuous functions This solu tion can be computed as a limit of iterations on Tµ starting from any bounded continuous function v0x C vµx lim k T k µv0 x The function vµx is the value of the objective function that would be attained by using the stationary policy µx each period The following proposition describes the policy iteration or Howard improve ment algorithm Theorem A21 Let vµx Tµvµx Define a new policy µ and an associated operator Tµ by Tµvµx T vµx that is µ is the policy that solves a oneperiod problem with vµx as the terminal value function Compute the fixed point vµx Tµvµx Then vµx vµx If µx is not optimal then vµx vµx for at least one x X 1382 Functional Analysis Proof From the definition of µ and Tµ we have Tµvµx rx µx βvµgx µx T vµx rx µx βvµgx µx Tµvµx vµx or Tµvµx vµx Apply Tµ repeatedly to this inequality and use the monotonicity of Tµ to con clude vµx lim n T n µ vµx vµx This establishes the asserted inequality vµx vµx If vµx vµx for all x X then vµx Tµvµx T vµx where the first equality follows because Tµvµx vµx and the second equality follows from the definitions of Tµ and µ Because vµx T vµx the Bellman equation is satisfied by vµx The policy improvement algorithm starts from an arbitrary feasible policy and iterates to convergence on the two following steps4 Step 1 For a feasible policy µx compute vµ Tµvµ Step 2 Find µ by computing T vµ Use µ as the policy in step 1 In many applications this algorithm proves to be much faster than iterating on the Bellman equation 4 A policy µx is said to be unimprovable if it is optimal to follow it for the first period given a terminal value function vx In effect the policy improvement algorithm starts with an arbitrary value function then by solving a oneperiod problem it generates an improved policy and an improved value function The proposition states that optimality is characterized by the features first that there is no incentive to deviate from the policy during the first period and second that the terminal value function is the one associated with continuing the policy Hidden Markov models 1387 B2 Hidden Markov models This section gives a brief introduction to hidden Markov models a tool that is useful to study a variety of nonlinear filtering problems in finance and economics We display a solution to a nonlinear filtering problem that a reader might want to compare to the linear filtering problem 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uncertainty 582 amnesia 965 of risksharing contract 881 939 approximation higher order 448 arbitrage 281 282 666 noarbitrage principle 281 282 559 Arrow securities 249 266 976 1086 asymmetric information 897 hidden income and hidden storage 909 autarky value 876 autocovariogram 47 backsolving 409 588 817 balanced growth path 634 Bayes Law with search model 178 beliefs distorted 597 Bellman equation 105 111 517 787 823 824 860 861 866 923 936 948 977 980 989 990 995 1000 1007 1022 1023 1064 1086 1087 1089 1209 1235 1242 1251 1274 1348 1359 1380 dynamic games 247 stacked 240 BenvenisteScheinkman formula 107 111 227 278 769 770 800 Bertrand competition 187 best response 1252 beta distribution 179 Bewley models 172 785 789 Big K little k 226 470 489 BlackScholes formula 510 block recursive equilibrium 1308 borrowing constraint 504 endogenous 976 natural 379 bubbles and volume 538 leverage constraints 538 short sales constraints 538 call option European 510 capacity utilization 1232 capital tax 684 carrot and stick 996 cashinadvance constraint 1155 CauchySchwarz inequality 556 certainty equivalence 135 768 proof 133 certainty equivalent 566 Chebyshev polynomial 126 coefficient of relative risk aversion 550 coinsurance 903 cointegration 78 768 commitment onesided lack of 875 technology 1011 twosided 897 909 twosided lack of 933 communism rational expectations 290 competitive equilibrium 1014 1174 1186 sequential trading 271 complete markets 249 1086 1173 no role for money 1123 conditional covariance 507 conditional expectation 34 consol 399 constant absolute risk aversion 821 constant relative risk aversion 821 continuation of a sequence 846 of a Stackelberg plan 846 contract design 934 dynamic program 936 costate vector 138 covariance stationary 45 credible government policies 1040 1425 1426 Subject Index credit and currency 1171 crossequation restrictions 423 curse of dimensionality 114 debt limit natural 268 deficit finance 340 as cause of inflation 1132 1184 deposit insurance 829 detection error probability 582 deterministic stochastic process 585 directed search 1302 discount factor 394 discretization of state space 115 distorted transition density 508 distorting tax 410 distribution Gaussian 55 invariant 32 log normal 553 multinomial 40 stationary 31 32 69 788 double coincidence of wants 1249 doubling algorithms 132 DSGE model 1290 duality 137 dynamic optimization 105 dynamic programming 53 105 1011 discrete state space 116 linear quadratic 129 squared 23 1022 dynamic programming squared 861 etymology 1025 eigenvalue 136 decomposition 140 employment lottery 1208 1227 layoff taxes 1235 endowment stream 872 entropy 292 577 detection errors 582 rlative 577 equilibrium 1128 1194 1196 incomplete markets 785 multiple 333 rational expectations 230 recursive competitive 230 stationary 790 793 equilibrium price distribution BurdettJudd model 183 equity premium 124 equity premium puzzle 552 565 equivalent martingale measure 314 Euler equation 137 Eulers theorem 633 exchange rate 1192 determinacy 1141 indeterminacy 813 1140 1195 expectations disappearance of 232 theory of the term structure 395 604 without promises 342 family one big happy 1230 fanning out of wealth distribution 890 filter linear 56 59 142 nonlinear 1386 fiscal theory of inflation 1138 fiscal theory of price level 717 folk theorem 1011 Fourier inversion formula 69 Friedman rule 717 1135 1147 1159 1181 1184 and free banking 818 Bewleys model 818 credibility 1160 functional equation 106 fundamental surplus 1271 giftgiving game 1181 with overlapping generations 364 Golden rule augmented 405 government deficit gross of interest 397 net of interest 397 operational 397 primary 397 GramSchmidt process 59 Subject Index 1427 growth exogenous 635 externality 636 650 nonreproducible factors 648 reproducible factors 638 research and monopolistic competi tion 643 650 growth model stochastic 472 guessandverify method 50 108 110 333 Hamiltonian 1318 hazard rate 171 hidden Markov model 34 60 63 Kalman filter 57 history dependence 256 see reputation lack of 275 of consumption stream 876 of contracts 876 of strategies 1041 holding period yields 395 Howard policy improvement algorithm 53 104 human capital 641 732 hyperbolic absolute risk aversion 821 impulse response function 48 Inada condition 473 633 incentivecompatibility constraint 899 923 incomplete markets 704 indirect utility function 579 innovation in time series representation 143 innovations representation 61 63 interest rate peg 1138 intertemporal elasticity of substitution 635 inverse generalized 284 inverse optimal prediction 64 inverse probability integral transform 834 island model 1207 1208 1257 layoff taxes 1242 Italy and Brazil inflation and measured deficits 1164 Jensens inequality 74 523 Kalman filter 55 59 142 and optimal linear regulator 129 and vector autoregressions 64 crossproducts between measurement and state noise 97 dual to linear regulator 143 Kalman gain 143 Kronecker product 203 lHopitals rule 579 Laffer curve 347 812 1132 1188 Lagrange multiplier 135 law of iterated expectations 34 law of large numbers 54 law of one price 558 layoff taxes 1229 employment lottery 1235 island model 1242 matching model 1244 leastsquares projection 54 284 legal restrictions 1135 1189 Terror 1136 lending with moral hazard 1083 likelihood function 40 54 Gaussian 55 multinomial 40 likelihood ratio 571 martingale 576 linear quadratic dynamic games 240 dynamic programming 129 linear rational expectations models 55 logarithmic preferences 109 long run risk 618 lotteries firms 1230 households 1227 manmade 521 Lucas tree in overlapping generations model 338 Lyapunov equation 132 discrete 50 543 M1 362 market price of model uncertainty 581 market price of risk 555 Markov chain 29 788 as difference equation 33 1428 Subject Index hidden 1386 Markov perfect equilibrium 240 fish and fishers 239 linear 239 prices and inventory example 242 selfcontrol 247 martingale 43 510 convergence theorem 178 770 868 906 908 difference sequence 43 equivalant measure 509 equivalent measure 315 507 likelihood ratio 576 martingale difference 1387 matching model 1207 1213 dispersion of match values 1261 heterogeneous jobs 1220 layoff taxes 1244 match surplus 1207 1215 1218 matching function 1214 separate markets 1224 skillbiased technological change 1261 wage announcements 1225 maximum likelihood 40 56 meanpreserving spread 160 measurability constraints AMSS model 866 measurement equation 1387 method of undetermined coefficients 110 moment generating function 563 money commodity 1195 demand function 1127 inside 811 outside 811 search model 1248 monopolistic competition 643 1153 moving average representation 48 multiplier preferences 579 ambiguity 614 618 Nash bargaining 1215 Nash equilibrium 1015 1249 infinite repetition 1031 natural debt limit 269 380 notrade result 268 noisy search Burdett and Judd 184 nonexpected utility 566 nondistorting tax 410 observational equivalence risk sensitivity and robustness 579 580 observer equation 1388 occupational choice 234 oneperiod deviations 1071 returns 395 open market operation 1128 another definition 1134 in private securities 363 one big one 1136 one definition 1134 optimal growth 109 optimal inflation tax 1147 optimal linear regulator 128 132 138 241 dynamic game 239 optimal quantity of money 1134 Friedman rule 1147 optimal savings problem 115 759 optimal taxation ex ante capital tax varies around zero 684 commitment 661 857 human capital 732 incomplete taxation 676 indeterminacy of statecontingent debt and capital taxes 681 initial capital 675 labor tax smoothing 688 zero capital tax 669 674 Pareto optimal allocation 1174 Pareto problem 1173 Pareto weights time invariance 254 participation constraint 876 878 881 permanent income model 1171 general equilibrium version 496 policy function 105 policy improvement algorithm 108 118 120 132 1037 Subject Index 1429 modified 120 Ponzi schemes ruling out 268 population regression 53 Kalman filter 57 precautionary savings 74 565 820 predetermined wage 1155 price system 1128 primal approach 670 primary surplus 528 promisekeeping constraint 878 promised value as state variable 871 877 public policies credible 1012 sustainable 1012 pulse 417 pure consumption loans economy 790 pure credit model 790 puzzle definition xxv quantity theory of money 1132 1157 quasilinear utility 713 Ramsey outcome 1014 Ramsey plan 661 1014 1150 1158 Ramsey problem 661 668 1014 1151 primal approach 670 uncertainty 683 randomization 924 rational expectations 290 291 595 communism 290 real bills doctrine 363 835 1206 real business cycle model 472 reciprocal pairs of eigenvalues 140 recursive 1027 competitive equilibrium 228 230 279 contracts 871 redistribution 740 redundant assets see arbitrage pricing the ory regression population 53 regression equation 54 relative entropy 292 577 relaxation method 794 parameter 794 renegotiation proof 979 repeated principalagent problem 922 representative agent 231 reputation 1011 resolvent operator 35 695 reverse engineer 587 Ricardian proposition 410 1171 1175 Riccati equation 132 141 algebraic 130 matrix difference 60 stacked 240 Riesz representation theorem 559 risk aversion versus robustness 575 risk exposure 593 risk neutral probabilities 595 risk prices 593 affine 593 risk sensitivity 568 579 versus robustness 575 riskfree rate puzzle 555 565 risksensitivity 579 risksharing mechanisms 785 Rosen schooling model 232 scale effects 646 Schur decomposition 140 search model 201 business cycles 1258 money 1248 shocks to human capital 1262 secondmoment restrictions 42 selfcontrol Markov perfect equilibrium 247 selfenforcing contract 875 selfenforcing equilibrium 1038 selfgeneration 1036 1037 selfinsurance 760 785 908 sequential see recursive shadow price 135 Sharpe ratio 556 shooting algorithm 392 406 shopping technology 1124 1430 Subject Index short sales 400 singlecrossing property 159 160 Smirnov transform 834 spectral density 69 spline shape preserving 128 1117 stability properties 135 stabilizable 137 stable matrix 136 state 29 105 statecontingent policies 1011 stick and carrot 1043 1045 sticky wage 1279 stochastic linear difference equations 29 matrix 30 process 29 45 stochastic differential utility 569 stochastic discount factor 51 266 509 549 555 557 559 570 exponential affine 615 stochastic volatility 619 strips 264 subgame perfect equilibrium 1028 1038 1160 submartingale 74 sunspots 1143 supermartingale convergence theorem 770 799 sustainable contract 876 sustainable plans see credible government policies Sylvester equation discrete Lyapunov equation 144 symplectic matrix 139 term structure affine yield model 592 expectations theory 518 of interest rates 394 slope 518 term structure of interest rates 394 tightness of labor market 1214 time consistency 839 1011 1015 1124 1152 time inconsistency of Ramsey plan 696 time to build Rosen schooling model 233 transition matrix 30 transversality condition 227 529 trigger strategy 1031 turnpike Townsends 1179 twisted transition measure see martingale 510 typical spectral shape 72 uncertainty measuring 582 unemployment compensation 987 European 1261 1312 voluntary 200 unpleasant monetarist arithmetic 1133 vacancy 1213 value function 51 105 see Bellman equa tion iteration 108 vector autoregression 44 63 wealthemployment distributions 788 white noise 43 yield to maturity 283 zero coupon bonds 395 zero inflation policy 1133 zero lower bound 720 Author Index Abel Andrew 365 513 Abreu Dilip xxxvii 871 1012 1022 1033 1043 1045 1066 Acemoglu Daron 1220 1257 Adda Jerome xxiv Aghion Philippe xxiv 653 Aiyagari Rao 380 663 701 704 740 785 786 792 857 864 Albarran Pedro 974 Allen Franklin 872 909 Altug Sumru xxiv 511 Alvarez Fernando 16 124 584 689 816 976 Anant TCA 653 Anderson Evan W 132 136 142 577 580 583 Ang Andrew 15 595 Angeletos GeorgeMarios 398 Apostol Tom 895 Arrow Kenneth J 249 266 631 637 Atkeson Andrew xxxvii 907 1083 1117 1147 Attanasio Orazio 366 587 974 Azariadis Costas xxiv Backus David K 15 436 595 602 606 Bagwell Kyle 388 Balasko Yves 357 Barillas Francisco 406 582 Barro Robert J 388 640 1011 1043 1153 Barseghyan Levon 575 Barsky Robert 389 Bassetto Marco 389 732 1026 1054 1059 1146 Baumol William J 1125 Beker Pablo 295 Bellman Richard xxiii 105 BenPorath Yoram 1328 1337 Benassy JeanPascal xxiv 653 Benoit JeanPierre 1020 Benveniste Lawrence 106 954 Bernheim B Douglas 388 Bertola Giuseppe 1220 Bertsekas Dimitri 105 106 131 Bewley Truman 785 1171 Bhandari Anmol 406 Bigio Saki 406 Black Fischer 510 Blackwell David 195 Blanchard Olivier J xxiv 138 142 388 Blume Lawrence 250 293 295 Blundell Richard 5 82 Bohn Henning 530 Bond Eric W 1083 1094 1100 Breeden Douglas T 13 510 Brock William A 335 338 366 448 472 511 1165 Browning Martin xxx 1315 Brumelle Shelby 118 Bruno Michael 348 1133 Bryant John 1136 1189 Buera Francisco 398 Buiter Willem H 1139 Burdett Kenneth 182 183 1256 Burnside Craig 68 Caballero Ricardo J 821 1220 Cagan Phillip 64 Calvo Guillermo A 1060 Campbell John Y 15 78 503 Canova Fabio xxiv 56 68 Carroll Christopher D 821 Cass David 357 1143 Cheron Arnaud 1270 1291 Chamberlain Gary 759 761 777 Chamley Christophe 12 661 669 670 Champ Bruce 366 Chang Roberto 859 1012 1054 1059 Chang Yongsung 1317 1347 1351 1357 Chari VV 662 663 678 686 717 727 740 1012 1053 1147 1148 Chatterjee Satyajit 1171 Chattopadhyay Subir 295 Chen RenRaw 595 Chow Gregory 105 133 137 Christensen Bent Jasper xxiv 68 Christiano Lawrence J 68 448 662 663 1431 1432 Author Index 678 686 717 727 740 1053 1148 1290 Clower Robert 1155 Cochrane John H 259 727 974 Cogley Timothy 295 591 Colacito Riccardo 406 Cole Harold L 25 872 909 1005 Conklin James 1077 Constantinides George M 409 587 590 591 786 Cooper Russell xxiv Corbae Dean 1171 Correia Isabel H 661 676 1148 Currie DA 19 Dai Qiang 595 Dave Chetan xxiv 56 68 Davies James B 8 Davig Troy 448 Davis Steven J 1220 De Santis Massimiliano 587 590 Debreu Gerard 249 290 332 334 DeJong David xxiv 56 68 Den Haan Wouter 431 Diamond Peter A 365 388 1207 1213 DiazGimenez J 8 816 Dinopoulos Elias 653 Dixit Avinash K 643 980 Dolmas Jim 566 Domeij David 740 Doob Joseph L 906 Duffie Darrell xxxii 409 503 569 587 590 591 786 Easley David 250 293 295 Eichenbaum Martin 68 448 1053 1290 Ellsberg Daniel 579 Engle Robert F 78 Epple Dennis xxxvi 138 893 Epstein Larry G 566 569 Ethier Wilfred J 643 Evans Charles 448 Evans David 851 Evans George W 229 825 Fackler Paul L xxiv Faig Miguel 1148 Farmer Roger xxiv Fernandes Ana 898 Fischer Stanley xxiv 348 1133 Fisher Irving 360 Fisher Jonas 448 Fitzgerald T 816 Flavin Marjorie A 535 Foley Duncan K 774 Freeman Scott 366 Friedman Milton 64 292 1134 1181 Fuchs William 1115 Fudenberg Drew 1053 Gale David 335 336 Gali Jordi xxiv 535 Geanakoplos John 822 Genicot Garance 979 Gittins JC 215 Golosov Mikhail 740 Gomes Joao 1257 1258 Gordon David B 1011 1043 1153 Granger CWJ xxxvi 78 Green Edward J xxxvii 873 Green Jerry R 230 257 334 Greenwood Jeremy 448 777 1257 1258 Grossman Gene M 653 980 Guidotti Pablo E 1148 Gul Faruk 247 980 Guvenen Fatih 1337 Hagedorn Marcus 1275 1285 Hairault JeanOlivier 1270 1291 Hall George 116 Hall Robert E 399 448 1279 1280 1283 1288 Hamilton James D 56 535 Hansen Gary D 116 786 1208 1227 Hansen Lars P xxiv xxx xxxii xxxvi 15 19 55 124 132 136138 142 143 504 535 553 557 577 579 580 582 583 893 1315 Harrison J Michael 535 Heathcote Jonathan 590 740 786 Heckman James J xxx 1315 Hellwig Martin F 774 Helpman Elhanan 653 Hercowitz Zvi 448 777 Author Index 1433 Holmstrom Bengt 875 Honkapohja Seppo 229 825 Hopenhayn Hugo A 124 987 1234 Hosios Arthur J 1217 1224 Howitt Peter xxiv 653 Huffman Gregory W 777 Huggett Mark 785 789 820 Imrohoroglu Ayse 786 Imrohoroglu Selahattin 116 786 Ingram Beth Fisher 448 Ireland Peter N 1153 1171 Jagannathan Ravi 557 Jermann Urban 16 124 584 976 Joines Douglas 786 Jones Charles I 654 Jones Larry E 12 640 648 663 671 677 732 739 Jorgenson Dale 399 Jovanovic Boyan xxxvi 67 157 179 201 1269 Judd Kenneth L xxiv 12 183 242 431 643 661 670 1077 Juillard Michel 406 Kahn Charles 138 142 364 Kalman Rudolf xxiii Kandori Michihiro 364 Kaplan Greg 786 Kareken John xxxvi 366 813 1140 1195 Kehoe Patrick J 436 662 663 678 686 689 717 727 740 897 1012 1147 1148 Kehoe Timothy 975 976 Keynes John Maynard 431 1189 Kiefer Nicholas M xxiv 68 Kim ChangJin 64 Kim J 448 Kim S 448 Kim SunBin 1317 1347 1351 1357 Kimball Miles S 389 821 Kimbrough Kent P 1148 King Robert G 392 1153 Kitao Sagiri 406 Kiyotaki Nobuhiro 288 1208 1248 Kletzer Kenneth M 979 Kocherlakota Narayana R 25 365 389 448 565 740 871 872 909 934 948 1005 1104 1146 1181 Koeppl Thorsten V 954 Kreps David M 535 559 566 569 1011 1053 Krishna Vijay 1020 Krueger Dirk 8 786 959 974 Krusell Per 822 897 1357 Kuruscu Burhanettin 1337 Kwakernaak Huibert 69 133 136 Kydland Finn 861 Kydland Finn E xxxvii 19 436 472 839 851 893 1011 1012 1153 Labadie Pamela xxiv Lagos Ricardo 1256 Laibson David I 247 Langot Francois 1270 1291 Leeper Eric M 448 Leland Hayne E 820 LeRoy Stephen 510 557 Lettau Martin 78 Levhari David 239 Levine David K 818 975 976 1053 Levine PL 19 Ligon Ethan 5 Lippi Francesco 1115 Ljungqvist Lars 1230 1234 1261 1269 1312 1317 1332 1343 Lo Andrew W 503 Lucas Robert E Jr xxiv xxix xxxvi 13 105 106 113 225 230 288 338 398 448 496 510 521 573 631 641 662 671 688 701 717 857 907 1136 1144 1155 1207 1208 1244 1257 1269 1312 Ludvigson Sydney 78 Lustig Hanno 979 Mace Barbara 259 974 MacKinlay A Craig 503 Majumdar Mukul xxiv Mankiw Gregory 15 365 389 513 587 Manovskii Iourii 1275 1285 Manuelli Rodolfo xxxiv 12 640 648 663 671 677 732 739 1171 1337 1434 Author Index Marcet Albert 229 348 431 663 701 704 774 822 825 851 857 864 893 897 1133 1354 Marimon Ramon 348 851 893 897 1133 1261 MasColell Andreu 230 257 334 822 Matthes Christian 406 McCall B P 173 McCall John 157 1269 McCallum Bennett T 1123 1139 McCandless George T 366 McGrattan Ellen R 132 136 142 McLennan Andrew 822 Mehra Rajnish 225 230 306 496 514 549 Mendoza Enrique G 436 Menzio Guido 1270 1291 1302 1308 Mertens Karel 436 Miao Jianjun 822 Milgrom Paul R 1280 1288 Miller Bruce L 821 Miller Marcus 19 Miller Merton 503 521 Miranda Mario J xxiv Mirman Leonard J 239 448 472 Modigliani Franco 503 521 Moen Espen R 1226 Molinari Francesca 575 Montgomery James D 1256 Mortensen Dale T 182 1207 1213 1247 1261 1312 Mukoyama Toshihiko 1357 Murphy Kevin 225 Muth John F xxxvi 64 Neal Derek 157 173 Nelson Charles R 64 Neumeyer Pablo Andres 689 Nicolini Juan Pablo 124 348 398 987 Niepelt Dirk 727 Nyarko Yaw xxxvi ODonoghue Ted 575 ObiolsHoms Francesc 774 1354 Obstfeld Maurice 566 Ospina Sandra 820 Ozkan Serdar 1337 Paal Beatrix 1136 Park JeeHyeong 1083 1094 1100 Pavan Ronni 177 Pavoni Nicola 1005 Pearce David xxxvii 871 1012 1022 1033 1043 1066 Pearlman J G 19 Perri Fabrizio 786 897 959 974 PerriFabrizio 8 Pesendorfer Wolfgang 247 Peters Michael 1256 Petrongolo Barbara 1269 Phelan Christopher xxxvii 871 898 924 1012 1054 1059 1146 Phelps Edmund S 247 1207 Piazzesi Monika 15 595 596 Pissarides Christopher 1207 1213 1218 1247 1261 1269 1277 1312 Pistaferri Luigi 786 Plosser Charles I 392 Pollak Robert A 247 Porteus Evan 566 569 Pratt John 550 Prescott Edward C xxiv xxxvii 19 105 106 113 225 230 306 434 448 472 496 514 549 816 839 851 861 893 1011 1012 1153 1207 1208 1230 1244 1257 1269 1312 1315 1316 1331 1332 Preston Ian 5 82 Putterman Martin L 118 Quadrini V 8 Quah Danny 82 RıosRull JV 8 Ravn Morten 0 436 Ray Debraj 979 Rebelo Sergio 68 392 632 648 1257 1258 Remolona Eli M 595 Roberds William xxxvi 19 138 364 535 893 Rodriquez SB 8 Rogerson Richard 1208 1227 1234 1286 1332 1357 1364 Author Index 1435 Rogoff Kenneth 1011 1012 Romer David xxiv Romer Paul M 631 637 643 650 653 654 Rosen Sherwin 225 232 234 237 1230 Rossi Peter E 12 640 663 671 677 732 739 Rotemberg Julio J 1153 Rothschild Michael 182 Runkle David 15 Ryoo Jaewoo 1230 Sahin Aysgul 1357 SaintPaul Gilles 1245 SalaiMartin Xavier 640 Salmon Mark 19 Salomao Juliana 596 Samuelson Paul 330 342 Sandmo Agnar 820 Santos Manuel S 107 Sargent Thomas J 69 131 136 143 229 295 348 406 577 579 580 582 583 663 701 704 730 825 851 857 864 1053 1133 1140 1171 1230 1261 1269 1312 1317 1332 1343 Savin NE 448 Scheinkman Jose 106 225 541 954 SchmittGrohe Stephanie 448 727 Schneider Martin 595 596 Scholes Myron 510 Scott Louis 595 Seater John J 388 Segerstrom Paul S 653 654 Seppala Juha 663 701 704 857 864 Shavell Stephen xxxvii 19 987 Shell Karl 334 357 1143 Shi Shouyong 1250 1256 1308 Shiller Robert 78 557 Shimer Robert xxiv 1257 1269 1273 1286 Shin MC 118 Shin Yongseok 398 406 1337 Shorrocks Anthony F 8 Shreve Steven 105 Sibley David S 821 Sims Christopher A 409 816 1138 Singleton Kenneth J 504 553 595 822 Siow A 234 237 Siu Henry 727 Sivan Raphael 69 133 136 Smith Adam 363 Smith Anthony 822 897 Smith Bruce D 359 409 1195 Smith Lones 364 Sotomayor Marilda A de Oliveira 761 Spear Stephen E xxxvii 871 922 Srivastava Sanjay 871 922 Stacchetti Ennio xxxvii 871 1012 1022 1033 1043 1054 1059 1066 Stachurski John xxiv Stigler George 157 Stiglitz Joseph E 643 677 Stokey Nancy L xxiv 105 106 113 398 662 671 688 701 717 857 1012 1043 Storesletten Kjetil 590 591 786 Straub Ludwig 661 Summers Lawrence 365 513 Sunder Shyam 348 1133 Svensson Lars EO 1153 Tallarini Thomas 124 Tallarini Thomas D 566 573 Teitelbaum Joshua C 575 Teles Pedro 1148 Telmer Chris 591 Telyukova Irina A 1270 1291 1302 Tesar Linda L 436 Thomas Jonathan xxxvii 871 898 934 937 1104 Tobin James 1125 Topel Robert 225 Townsend Robert M xxxvi xxxvii 259 288 871 924 974 1171 1179 Trabandt Mathias 1290 Trejos Alberto 1250 Tsyrennikov Viktor 295 Tsyvinski Aleh 740 Uribe Martin 448 727 Uzawa Hirofumi 631 641 1436 Author Index Vegh Carlos A 1148 Velde Francois 409 1136 1195 Villamil Anne 1189 Violante Giovanni L 590 786 1005 Visschers Ludo 1270 1291 1302 Wald Abraham xxiii Walker Todd B 448 Wallace Neil xxxvi 351 359 364 366 730 813 1136 1140 1189 1195 Wallenius Johanna 1332 1364 Walsh Carl xxiv Wang Neng 580 820 Weber Gugliemo 587 Wei Chao 116 Weil Philippe 565 566 573 774 1354 Weiss Laurence xxxvii 19 987 Werning Ivan 661 Werning Ivan 704 1005 Whinston Michael D 230 257 334 Whiteman Charles 138 142 Wilcox David W 535 Williams Noah 348 Wilson Charles 759 761 777 Wolman Alexander L 1153 Woodford Michael xxiv 142 1138 1153 Worrall Tim xxxvii 871 898 934 937 1104 Wright Brian D 979 Wright Randall 288 1208 1248 1250 1256 Yaron Amir 591 Yeltekin Sevin 1077 Young Alwyn 654 Zame William 818 Zeckhauser Richard 365 513 Zeira Joseph 612 Zeldes Stephen P 15 389 821 Zha Tao 348 Zhang Harold 16 979 Zhao Rui 987 Zhu Shenghao 774 1354 Zhu Xiaodong 662 678 681 685 Zilibotti Fabrizio 1261 Zin Stanley E 15 563 566 595 602 606 Matlab Index aiyagari2m 804 bewley99m 804 bewley99v2m 804 bewleyplotm 804 bewleyplot2m 804 bigshow3m 70 dimpulsem 49 dlyapm 50 132 543 doublejm 47 50 132 543 hugom 993 hugo1am 993 hugofoc1m 993 impulsem 49 juddm 243 kfilterm 62 102 143 markovm 40 789 markovapproxm 804 nashm 243 247 neal2m 174 nnashm 243 olipololy5m 851 olrpm 132 policyim 132 schumakerm 127 schurgm 140 search learn beta 2m 179 search learn francisco 3m 179 valhugom 993 1437