Recursive Macroeconomic Theory Second edition Recursive Macroeconomic Theory Second edition Lars Ljungqvist Stockholm School of Economics Thomas J Sargent New York University and Hoover Institution c 2000 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 Cataloging-in-Publication Data Ljungqvist, Lars Recursive macroeconomic theory / Lars Ljungqvist, Thomas J Sargent p cm Includes bibliographical references and index ISBN 0-262-19451-1 Macroeconomics Recursive functions Statics and dynamics (Social sciences) I Sargent, Thomas J II Title HB172.5 L59 2000 339’.01’51135–dc21 00-056067 Contents Acknowledgements xvii Preface xviii Part I: The imperialism of recursive methods Overview 1.1 Warning 1.2 A common ancestor 1.3 The savings problem 1.3.1 Linear-quadratic permanent income theory 1.3.2 Precautionary savings 1.3.3 Complete markets, insurance, and the distribution of wealth 1.3.4 Bewley models 1.3.5 History dependence in standard consumption models 1.3.6 Growth theory 1.3.7 Limiting results from dynamic optimal taxation 1.3.8 Asset pricing 1.3.9 Multiple assets 1.4 Recursive methods 1.4.1 Methodology: dynamic programming issues a challenge 1.4.2 Dynamic programming challenged 1.4.3 Imperialistic response of dynamic programming 1.4.4 History dependence and ‘dynamic programming squared’ 1.4.5 Dynamic principal-agent problems 1.4.6 More applications Part II: Tools Time series 26 2.1 Two workhorses 2.2 Markov chains 2.2.1 Stationary distributions 2.2.2 Asymptotic stationarity 2.2.3 Expectations 2.2.4 Forecasting functions 2.2.5 Invariant functions and ergodicity 2.2.6 Simulating a Markov chain 2.2.7 The likelihood function 2.3 Continuous state Markov chain 2.4 Stochastic linear difference equations 2.4.1 First and second moments 2.4.2 Impulse response function 2.4.3 Prediction and discounting 2.4.4 Geometric sums of quadratic forms 2.5 –v– vi Contents Population regression 2.5.1 The spectrum 2.5.2 Examples 2.6 Example: the LQ permanent income model 2.6.1 Invariant subspace approach 2.7 The term structure of interest rates 2.7.1 A stochastic discount factor 2.7.2 The log normal bond pricing model 2.7.3 Slope of yield curve depends on serial correlation of log mt+1 2.7.4 Backus and Zin’s stochastic discount factor 2.7.5 Reverse engineering a stochastic discount factor 2.8 Estimation 2.9 Concluding remarks 2.10 Exercises A A linear difference equation Dynamic Programming 82 3.1 Sequential problems 3.1.1 Three computational methods 3.1.2 Cobb-Douglas transition, logarithmic preferences 3.1.3 Euler equations 3.1.4 A sample Euler equation 3.2 Stochastic control problems 3.3 Concluding remarks 3.4 Exercise Practical Dynamic Programming 93 4.1 The curse of dimensionality 4.2 Discretization of state space 4.3 Discrete-state dynamic programming 4.4 Application of Howard improvement algorithm 4.5 Numerical implementation 4.5.1 Modified policy iteration 4.6 Sample Bellman equations 4.6.1 Example 1: calculating expected utility 4.6.2 Example 2: risk-sensitive preferences 4.6.3 Example 3: costs of business cycles 4.7 Polynomial approximations 4.7.1 Recommended computational strategy 4.7.2 Chebyshev polynomials 4.7.3 Algorithm: summary 4.7.4 Shape preserving splines 4.8 Concluding remarks Linear Quadratic Dynamic Programming 5.1 Introduction 5.2 The optimal linear regulator problem 5.2.1 Value function iteration 5.2.2 Discounted linear regulator problem 5.2.3 Policy improvement algorithm 5.3 The stochastic optimal linear regulator problem 5.3.1 Discussion of certainty equivalence 5.4 Shadow prices in the linear regulator 5.4.1 Stability 5.5 A Lagrangian formulation 5.6 The Kalman filter 5.6.1 Muth’s example 5.6.2 Jovanovic’s example 5.7 Concluding remarks A Matrix formulas B Linear-quadratic approximations 5.B.1 An example: the stochastic growth model 5.B.2 Kydland and Prescott’s method 5.B.3 Determination of z 5.B.4 Log linear approximation 5.B.5 Trend removal ¯ 5.10 Exercises 107 Contents Search, Matching, and Unemployment vii 137 6.1 Introduction 6.2 Preliminaries 6.2.1 Nonnegative random variables 6.2.2 Mean-preserving spreads 6.3 McCall’s model of intertemporal job search 6.3.1 Effects of mean preserving spreads 6.3.2 Allowing quits 6.3.3 Waiting times 6.3.4 Firing 6.4 A lake model 6.5 A model of career choice 6.6 A simple version of Jovanovic’s matching model 6.7 A longer horizon version of Jovanovic’s model 6.7.1 The Bellman equations 6.8 Concluding remarks A More numerical dynamic programming 6.A.1 Example 4: Search 6.A.2 Example 5: A Jovanovic model 6.A.3 Wage distributions 6.A.4 Separation probabilities 6.A.5 Numerical examples 6.10 Exercises Part III: Competitive equilibria and applications Recursive (Partial) Equilibrium 186 7.1 An equilibrium concept 7.2 Example: adjustment costs 7.2.1 A planning problem 7.3 Recursive competitive equilibrium 7.4 Markov perfect equilibrium 7.4.1 Computation 7.5 Linear Markov perfect equilibria 7.5.1 An example 7.6 Concluding remarks 7.7 Exercises Equilibrium with Complete Markets 8.1 Time- versus sequential trading 8.2 The physical setting: preferences and endowments 8.3 Alternative trading arrangements 8.3.1 History dependence 8.4 Pareto problem 8.4.1 Time invariance of Pareto weights 8.5 Time-0 trading: Arrow-Debreu securities 8.5.1 Equilibrium pricing function 8.5.2 Optimality of equilibrium allocation 8.5.3 Equilibrium computation 8.5.4 Interpretation of trading arrangement 8.6 Examples 8.6.1 Example 1: Risk sharing 8.6.2 Example 2: No aggregate uncertainty 8.6.3 Example 3: Periodic endowment processes 8.7 Primer on asset pricing 8.7.1 Pricing redundant assets 8.7.2 Riskless consol 8.7.3 Riskless strips 8.7.4 Tail assets 8.7.5 Pricing one period returns 8.8 Sequential trading: Arrow securities 8.8.1 Arrow securities 8.8.2 Insight: wealth as an endogenous state variable 8.8.3 Debt limits 8.8.4 Sequential trading 8.8.5 Equivalence of allocations 8.9 Recursive competitive equilibrium 8.9.1 Endowments governed by a Markov process 8.9.2 Equilibrium outcomes inherit the Markov property 8.9.3 Recursive formulation of optimization and equilibrium 8.10 j -step pricing kernel 203 viii Contents 8.10.1 Arbitrage free pricing 8.11 Consumption strips and the cost of business cycles 8.11.1 Consumption strips 8.11.2 Link to business cycle costs 8.12 Gaussian asset pricing model 8.13 Recursive version of Pareto problem 8.14 Static models of trade 8.15 Closed economy model 8.15.1 Two countries under autarky 8.15.2 Welfare measures 8.16 Two countries under free trade 8.16.1 Small country assumption 8.17 A tariff 8.17.1 Nash tariff 8.18 Concluding remarks 8.19 Exercises Overlapping Generations Models 258 9.1 Endowments and preferences 9.2 Time- trading 9.2.1 Example equilibrium 9.2.2 Relation to the welfare theorems 9.2.3 Nonstationary equilibria 9.2.4 Computing equilibria 9.3 Sequential trading 9.4 Money 9.4.1 Computing more equilibria 9.4.2 Equivalence of equilibria 9.5 Deficit finance 9.5.1 Steady states and the Laffer curve 9.6 Equivalent setups 9.6.1 The economy 9.6.2 Growth 9.7 Optimality and the existence of monetary equilibria 9.7.1 Balasko-Shell criterion for optimality 9.8 Within generation heterogeneity 9.8.1 Nonmonetary equilibrium 9.8.2 Monetary equilibrium 9.8.3 Nonstationary equilibria 9.8.4 The real bills doctrine 9.9 Gift giving equilibrium 9.10 Concluding remarks 9.11 Exercises 10 Ricardian Equivalence 306 10.1 Borrowing limits and Ricardian equivalence 10.2 Infinitely lived– agent economy 10.2.1 Solution to consumption/savings decision 10.3 Government 10.3.1 Effect on household 10.4 Linked generations interpretation 10.5 Concluding remarks 11 Fiscal policies in the nonstochastic growth model 11.1 Introduction 11.2 Economy 11.2.1 Preferences, technology, information 11.2.2 Components of a competitive equilibrium 11.2.3 Competitive equilibria with distorting taxes 11.2.4 The household: no arbitrage and asset pricing formulas 11.2.5 User cost of capital formula 11.2.6 Firm 11.3 Computing equilibria 11.3.1 Inelastic labor supply 11.3.2 The equilibrium steady state 11.3.3 Computing the equilibrium path with the shooting algorithm 11.3.4 Other equilibrium quantities 11.3.5 Steady state R and s/q 11.3.6 Lump sum taxes available 11.3.7 No lump sum taxes available 11.4 A digression on ‘back-solving’ 11.5 Effects of taxes on equilibrium allocations 317 Contents ix and prices 11.6 Transition experiments 11.7 Linear approximation 11.7.1 Relationship between the λi ’s 11.7.2 Once and for all jumps 11.7.3 Simplification of formulas 11.7.4 A one-time pulse 11.7.5 Convergence rates and anticipation rates 11.8 Elastic labor supply 11.8.1 Steady state calculations 11.8.2 A digression on accuracy: Euler equation errors 11.9 Growth 11.10 Concluding remarks A Log linear approximations 11.12 Exercises 12 Recursive competitive equilibria 360 12.1 Endogenous aggregate state variable 12.2 The growth model 12.3 Lagrangian formulation of the planning problem 12.4 Time- trading: Arrow-Debreu securities 12.4.1 Household 12.4.2 Firm of type I 12.4.3 Firm of type II 12.4.4 Equilibrium prices and quantities 12.4.5 Implied wealth dynamics 12.5 Sequential trading: Arrow securities 12.5.1 Household 12.5.2 Firm of type I 12.5.3 Firm of type II 12.5.4 Equilibrium prices and quantities 12.5.5 Financing a type II firm 12.6 Recursive formulation 12.6.1 Technology is governed by a Markov process 12.6.2 Aggregate state of the economy 12.7 Recursive formulation of the planning problem 12.8 Recursive formulation of sequential trading 12.8.1 The ‘Big K , little k ’ trick 12.8.2 Price system 12.8.3 Household problem 12.8.4 Firm of type I 12.8.5 Firm of type II 12.9 Recursive competitive equilibrium 12.9.1 Equilibrium restrictions across decision rules 12.9.2 Using the planning problem 12.10 Concluding remarks 13 Asset Pricing 13.1 Introduction 13.2 Asset Euler equations 13.3 Martingale theories of consumption and stock prices 13.4 Equivalent martingale measure 13.5 Equilibrium asset pricing 13.6 Stock prices without bubbles 13.7 Computing asset prices 13.7.1 Example 1: Logarithmic preferences 13.7.2 Example 2: A finite-state version 13.7.3 Example 3: Asset pricing with growth 13.8 The term structure of interest rates 13.9 State-contingent prices 13.9.1 Insurance premium 13.9.2 Man-made uncertainty 13.9.3 The Modigliani-Miller theorem 13.10 Government debt 13.10.1 The Ricardian proposition 13.10.2 No Ponzi schemes 13.11 Interpretation of risk-aversion parameter 13.12 The equity premium puzzle 13.13 Market price of risk 13.14 HansenJagannathan bounds 13.14.1 Inner product representation of the pricing kernel 13.14.2 Classes of stochastic discount factors 13.14.3 A Hansen-Jagannathan bound 13.14.4 The Mehra-Prescott data 13.15 386 x Contents Factor models 13.16 Heterogeneity and incomplete markets 13.17 Concluding remarks 13.18 Exercises 14 Economic Growth 443 14.1 Introduction 14.2 The economy 14.2.1 Balanced growth path 14.3 Exogenous growth 14.4 Externality from spillovers 14.5 All factors reproducible 14.5.1 One-sector model 14.5.2 Two-sector model 14.6 Research and monopolistic competition 14.6.1 Monopolistic competition outcome 14.6.2 Planner solution 14.7 Growth in spite of nonreproducible factors 14.7.1 “Core” of capital goods produced without nonreproducible inputs 14.7.2 Research labor enjoying an externality 14.8 Concluding comments 14.9 Exercises 15 Optimal Taxation with Commitment 15.1 Introduction 15.2 A nonstochastic economy 15.2.1 Government 15.2.2 Households 15.2.3 Firms 15.3 The Ramsey problem 15.3.1 Definitions 15.4 Zero capital tax 15.5 Limits to redistribution 15.6 Primal approach to the Ramsey problem 15.6.1 Constructing the Ramsey plan 15.6.2 Revisiting a zero capital tax 15.7 Taxation of initial capital 15.8 Nonzero capital tax due to incomplete taxation 15.9 A stochastic economy 15.9.1 Government 15.9.2 Households 15.9.3 Firms 15.10 Indeterminacy of state-contingent debt and capital taxes 15.11 The Ramsey plan under uncertainty 15.12 Ex ante capital tax varies around zero 15.12.1 Sketch of the proof of Proposition 15.13 Examples of labor tax smoothing 15.13.1 Example 1: gt = g for all t ≥ 15.13.2 Example 2: gt = for t = T , and gT > 15.13.3 Example 3: gt = for t = T , and gT is stochastic 15.13.4 Lessons for optimal debt policy 15.14 Taxation without state-contingent government debt 15.14.1 Future values of {gt } become deterministic 15.14.2 Stochastic {gt } but special preferences 15.14.3 Example revisited: gt = for t = T , and gT is stochastic 15.15 Zero tax on human capital 15.16 Should all taxes be zero? 15.17 Concluding remarks 15.18 Exercises 473 Contents xi Part IV: The savings problem and Bewley models 16 Self-Insurance 540 16.1 Introduction 16.2 The consumer’s environment 16.3 Nonstochastic endowment 16.3.1 An ad hoc borrowing constraint: nonnegative assets 16.3.2 Example: Periodic endowment process 16.4 Quadratic preferences 16.5 Stochastic endowment process: i.i.d case 16.6 Stochastic endowment process: general case 16.7 Economic intuition 16.8 Concluding remarks A Supermartingale convergence theorem 16.10 Exercises 17 Incomplete Markets Models 17.1 Introduction 17.2 A savings problem 17.2.1 Wealth-employment distributions 17.2.2 Reinterpretation of the distribution λ 17.2.3 Example 1: A pure credit model 17.2.4 Equilibrium computation 17.2.5 Example 2: A model with capital 17.2.6 Computation of equilibrium 17.3 Unification and further analysis 17.4 Digression: the nonstochastic savings problem 17.5 Borrowing limits: “natural” and “ad hoc” 17.5.1 A candidate for a single state variable 17.5.2 Supermartingale convergence again 17.6 Average assets as function of r 17.7 Computed examples 17.8 Several Bewley models 17.8.1 Optimal stationary allocation 17.9 A model with capital and private IOUs 17.10 Private IOUs only 17.10.1 Limitation of what credit can achieve 17.10.2 Proximity of r to ρ 17.10.3 Inside money or ‘free banking’ interpretation 17.10.4 Bewley’s basic model of fiat money 17.11 A model of seigniorage 17.12 Exchange rate indeterminacy 17.12.1 Interest on currency 17.12.2 Explicit interest 17.12.3 The upper bound on M p 17.12.4 A very special case 17.12.5 Implicit interest through inflation 17.13 Precautionary savings 17.14 Models with fluctuating aggregate variables 17.14.1 Aiyagari’s model again 17.14.2 Krusell and Smith’s extension 17.15 Concluding remarks 17.16 Exercises 561 References 1061 Wallace, Neil 1983 “A Legal Restrictions Theory of the Demand for ‘Money’ and the Role of Monetary Policy.” Quarterly Review, Federal Reserve Bank of Minneapolis, Vol 7(1), pp 1–7 Wallace, Neil 1989 “Some Alternative Monetary Models and Their Implications for the Role of Open-Market Policy.” In Robert J Barro (ed.), Modern Business Cycle Theory Cambridge, MA: Harvard University Press, pp 306–328 Walsh, Carl E 1998 Monetary Theory and Policy Cambridge: MIT Press Wang, Cheng, and Stephen D Williamson 1996 “Unemployment Insurance with Moral Hazard in a Dynamic Economy.” Carnegie-Rochester Conference Series on Public Policy, Vol 44(0), pp 1–41 Wang, Neng 2003 “Caballero meets Bewley: The permanent-income hypothesis in general equilbrium” American Economic Review, Vol 93, No 3, pp 927–936 Watanabe, Shinichi 1984 “Search Unemployment, the Business Cycle, and Stochastic Growth.” Mimeo Ph.D dissertation, University of Minnesota Weil, Philippe 1989 “The Equity Premium Puzzle and the Risk-Free Rate Puzzle.” Journal of Monetary Economics, Vol 24(2), pp 401–421 Weil, Philippe 1990 “Nonexpected Utility in Macroeconomics.” Quarterly Journal of Economics, Vol 105, pp 29–42 Weil, Philippe 1993 “Precautionary Savings and the Permanent Income Hypothesis.” Review of Economic Studies, Vol 60(2), pp 367–383 Werning, Ivan 2002 “Optimal unemployment insurance with hidden savings” Mimeo University of Chicago working paper Wen, Y 1998 “Can a Real Business Cycle Pass the Watson Test” Journal of Monetary Economics, Vol 42, pp 185-202 Whiteman, Charles H 1983 Linear Rational Expectations Models: A Users Guide Minneapolis: University of Minnesota Press Whittle, Peter 1963 Prediction and Regulation by Linear Least-Square Methods Princeton, NJ: Van Nostrand-Reinhold Whittle, Peter 1990 Risk-Sensitive Optimal Control New York: Wiley Wilcox, David W 1989 “The Sustainability of Government Deficits: Implications of the Present-Value Borrowing Constraint.” Journal of Money, Credit, and Banking, Vol 21(3), pp 291–306 Woodford, Michael 1994 “Monetary Policy and Price Level Determinacy in a Cashin-Advance Economy.” Economic Theory, Vol 4, pp 345–380 Woodford, Michael 1995 “Price-Level Determinacy without Control of a Monetary Aggregate.” Carnegie-Rochester Conference Series on Public Policy, Vol 43(0), pp 1–46 Woodford, Michael 2003 “Optimal Interest-Rate Smoothing” Review of Economic Studies, Vol 70(4), pp 861-886 Woodford, Michael 20003 “Interest and Prices.” Mimeo Princeton University Wright, Randall 1986 “Job Search and Cyclical Unemployment.” Journal of Political Economy, Vol 94(1), pp 38–55 Young, Alwyn 1998 “Growth without Scale Effects.” Journal of Political Economy, Vol 106, pp 41–63 1062 References Zeira, Joseph 1999 “Informational Overshooting, Booms, and Crashes.” Journal of Monetary Economics, Vol 43, No 1, pp 237–257 Zeldes, Stephen P 1989 “Optimal Consumption with Stochastic Income: Deviations from Certainty Equivalence.” Quarterly Journal of Economics, Vol 104(2), pp 275–298 Zhang, Harold 1997 “Endogenous Borrowing Constraints with Incomplete Markets.” Journal of Finance, Vol 52, No 5.pp 2187–2209 Zhao, Rui 2001 “The Optimal Unemployment Insurance Contract: Why a Replacement Ratio?” Mimeo University of Illinois, Champagne-Urbana Zhu, Xiaodong 1992 “Optimal Fiscal Policy in a Stochastic Growth Model.” Journal of Economic Theory, Vol 58, pp 250–289 Index endogenous, 736 natural, 306 adaptive expectations, 122, 124 affine factor model, 434 amnesia, 724 of risk-sharing contract, 640, 698 ante capital tax, 498 approximation higher order, 353 arbitrage, 231, 232, 477 free pricing theory, 231 no-arbitrage principle, 427 pricing theory, 232 Arrow securities, 203, 218, 219, 735, 820 asymmetric information, 657 hidden income and hidden storage, 668 autarky value, 636 autocovariogram, 42 back-solving, 329 balanced growth path, 446 Bellman equation, 82, 88, 400, 780, 823, 1007 dynamic games, 202 stacked, 194 Benveniste-Scheinkman formula, 88, 188, 229, 549, 550, 575 best response, 974 Bewley model, 150 Bewley models, 561, see euilibrium, incomplete markets565 Black-Scholes formula, 393 borrowing constraint, 387 call option European, 393 capital overaccumulation, 335 cash-in-advance constraint, 881 Cauchy-Schwarz inequality, 424 certainty equivalence, 549 certainty equivalence principle, 113 certainty equivalence principle: proof, 112 Chebyshev polynomial, 104 coefficient of relative risk aversion, 418 coinsurance, 663 commitment one-sided lack of, 635 technology, 768 two-sided, 657, 668 two-sided lack of, 692 competitive equilibrium, 771, 900, 912 sequential-trading, 222 complete markets, 203, 820, 899 no role for money, 852 conditional covariance, 390 conditional expectation, 30 constant absolute risk-aversion, 596 constant relative risk-aversion, 596 contract design, 693 dynamic program, 695 costate vector, 117 covariance stationary, 40 credible government policies, 797 credit and currency, 897 – 1063 – 1064 cross-equation restrictions, 343 curse of dimensionality, 92 debt limit natural, 220 deficit finance, 267 as cause of inflation, 861, 910 deposit insurance, 605 discretization of state space, 93 distorted transition density, 391 distribution Gaussian, 68 invariant, 29 log-normal, 421 multinomial, 35 stationary, 28, 29, 49, 564 double coincidence of wants, 971 doubling algorithms, 111 duality, 116 of control and filtering, 1018 dynamic optimization, 82 dynamic programming, 46, 82, 768 discrete state space, 94 squared, 21 eigenvalue decomposition, 118 eigenvalues, 114 employment lottery, 936, 955 layoff taxes, 959 endowment stream, 632 equilibrium, 857, 920, 923 incomplete markets, 561 multiple, 260 rational expectations, 191 recursive competitive, 191 stationary, 566, 568 Index equity premium, 102 equivalent martingale measure, 256 Euler equation, 116 Euler’s theorem, 445 European unemployment, 983 exchange rate, 918 determinacy, 871 indeterminacy, 589, 870, 922 expectations disappearance of, 192 without promises, 269 expectations theory of the term structure, 62 fanning out of wealth distribution, 650 filter linear, 120 linear, 69 nonlinear, 1029 fiscal theory of inflation, 867 folk theorem, 768 Fourier inversion formula, 50 Friedman rule, 865, 873, 885, 907, 910 Bewley’s model, 593 credibility, 886 functional equation, 83 gift giving game, 907 with overlapping generations, 291 Gram-Schmidt orthogonalization, 1013 growth exogenous, 447 externality, 448, 462 nonreproducible factors, 460 reproducible factors, 450 Index research and monopolistic competition, 455, 462 Growth model stochastic, 361 guess-and-verify method, 44, 85, 87, 260 hazard rate, 149 history dependence, 210, see reputation lack of, 226 of consumption stream, 635 of contracts, 636 of strategies, 798 Howard improvement algorithm, 46 human capital, 453, 523 hyperbolic absolute risk-aversion, 596 impulse response function, 43 Inada condition, 362, 445 incentive-compatibility constraint, 658, 682 incomplete markets, 512 indeterminacy of equilibrium, 619 innovation in time series representation, 121 interest rate peg, 868 intertemporal elasticity of substitution, 447 inverse optimal prediction, 122 island model, 935, 936, 979 layoff taxes, 962 Italy and Brazil inflation and measured deficits, 890 Jensen’s inequality, 406 1065 Kalman filter, 120, 121, 1012 Kalman gain, 121 Kronecker product, 171 Laffer curve, 274, 862, 913 Lagrange multiplier, 113 law of iterated expectations, 30, 31 Law of large numbers, 48 law of one price, 426 layoff taxes, 957 employment lottery, 959 island model, 962 matching model, 966 least squares projection, 48 legal restrictions, 865, 916 Terror, 866 lending with moral hazard, 817 likelihood function, 35, 48 Gaussian, 68 multinomial, 35 linear rational expectations models, 68 linear-quadratic approximations, 127 linear-quadratic dynamic games, 194 logarithmic preferences, 86 lotteries man made, 404 Lucas tree in overlapping generations model, 265 Lyapunov equation, 110, 111 discrete, 44, 237 market price of risk, 424 Markov chain, 26, 564 hidden, 1029 Markov perfect equilibrium, 195 1066 Index fish and fishers, 194 linear, 194 prices and inventory example, 197 self-control, 202 martingale, 38, 392 convergence theorem, 550, 665, 667 difference sequence, 38 equivalant measure, 392 equivalent measure, 256, 390 matching model, 935, 941 dispersion of match values, 984 heterogeneous jobs, 948 layoff taxes, 966 match surplus, 935, 943, 946 matching function, 942 separate markets, 952 skill-biased technological change, 983 wage announcements, 954 maximum likelihood, 35, 69 measurement equation, 1030 method of undetermined coefficients, 87 money commodity, 922 demand function, 856 inside, 586 outside, 586 search model, 970 monopolistic competition, 455, 879 moving average representation, 43 Nash bargaining, 943 Nash equilibrium, 772, 971 infinite repetition, 787 natural debt limit, 220, 307 no-trade result, 217 observer equation, 1031 open market operation, 857 another definition, 863 in private securities, 290 one big one, 866 one definition, 863 optimal growth, 87 optimal inflation tax, 873 optimal linear regulator, 106, 116, 127, 195, 1006 dynamic game, 194 optimal linear regulator: stochastic, 111 optimal quantity of money, 864 Friedman rule, 873 optimal saving problem, 93, 540 optimal taxation commitment, 473 ex ante capital tax varies around zero, 498 human capital, 523 incomplete taxation, 490 indeterminacy of state-contingent debt and capital taxes, 495 initial capital, 489 labor tax smoothing, 502 redistribution, 483 zero capital tax, 480, 488 Pareto optimal allocation, 900 Pareto problem, 899 participation constraint, 636, 638, 640 permanent income theory, 897 Index policy function, 82 policy improvement algorithm, 85, 96–98, 110, 794 modified, 98 Ponzi schemes ruling out, 220 precautionary savings, 595 predetermined wage, 880 price system, 857 primal approach, 485 primary surplus, 411 promise-keeping constraint, 638 promised value as state variable, 631, 637 public policies credible, 769 sustainable, 769 pulse, 335 pure credit model, 566 pure exchange economy, 566 quadratic approximation, 130 quantity theory of money, 860, 883 Ramsey outcome, 772 Ramsey plan, 473, 772, 876, 884 Ramsey problem, 473, 480, 772, 876 primal approach, 485 uncertainty, 497 randomization, 683 real bills doctrine, 290, 933 reciprocal pairs of eigenvalues, 118 recursive, 783 competitive equilibrium, 189, 191 recursive competitive equilibrium, 230 recursive contracts, 631 redistribution, 483, 531 1067 redundant assets, see arbitrage pricing theory regression equation, 48 relaxation method, 569 parameter, 569 renegotiation proof, 739 repeated principal-agent problem, 681 representative agent, 191 reputation, 768, 769 Ricardian doctrine, 897 Ricardian proposition, 901 Riccati equation, 111, 120 matrix difference, 1008 stacked, 194 Riccati equation: algebraic, 108 Riesz representation theorem, 426 risk sharing mechanisms, 561 scale effects, 458 Schur decomposition, 118 search model, 169 business cycles, 980 money, 970 shocks to human capital, 984 second-moment restrictions, 37 self-control Markov perfect equilibrium, 202 self-enforcing equilibrium, 795 self-generation, 793, 794 self-insurance, 541, 561, 667 sequential, see recursive shadow price, 113 shooting, 317 shooting algorithm, 325 shopping technology, 854 single-crossing property, 139 1068 Index social welfare function, 483 spectral density, 49 spline shape preserving, 848 spline: shape preserving, 106 stability properties, 114 stabilizable, 115 stable matrix, 114 state, 26, 82 state-contingent policies, 768 stochastic discount factor, 45, 218, 392, 424–426 stochastic linear difference equation, 26 stochastic matrix, 27 stochastic process, 26, 40 strips, 216 subgame perfect equilibrium, 785, 795, 886 supermartingale convergence theorem, 550, 574 sustainable contract, 636 sustainable plans, see credible government policies Sylvester equation discrete Lyapunov equation, 127 symplectic matrix, 118 term structure expectations theory, 401 factor models, 432 slope, 401 tightness of labor market, 942 time consistency, 503, 768, 772, 853, 878 transition matrix, 27 transversality condition, 188, 412 trigger strategy, 788 turnpike Townsend’s, 905 twisted transition measure, see martingale, 392 typical spectral shape, 51 unemployment compensation, 746 unpleasant monetarist arithmetic, 863 vacancy, 941 value function, 45, see Bellman equation, 83 iteration, 85 vector autoregression, 39 voluntary unemployment, 168 wealth-employment distributions, 563 white noise, 38 Yule-Walker equations, 42 zero inflation policy, 862 Author Index Abel, Andrew, 292, 396 Abreu, Dilip, xxxiv, 631, 769, 790, 800, 802 Acemoglu, Daron, 948, 979 Adda, Jerome, xx Aghion, Philippe, xx, 465 Aiyagari, Rao, 308, 475, 509, 531, 561, 562, 567 Albarran, Pedro, 733 Altug, Sumru, xx Alvarez, Fernando, 14, 102, 232, 591, 736 Anant, T.C.A., 465 Anderson, Evan, 111, 115, 120 Ang, Andrew, 13, 61 Apostol, Tom, 655 Arrow, Kenneth J., 203, 218, 443, 449 Atkeson, Andrew, xxiii, xxxiv, 817, 848 Attanasio, Orazio, 293, 438, 733 Azariadis, Costas, xx 779, 512, 233, 219, 666, Backus, David K., 13, 60, 64 Bagwell, Kyle, 315 Balasko, Yves, 285 Barro, Robert J., 315, 452, 508, 768, 800, 878 Barsky, Robert, 316 Bassetto, Marco, 316, 811 Baumol, William J., 854 Bellman, Richard, xviii, 82 Benassy, Jean-Pascal, 465 Benoit, Jean-Pierre, 779 Benveniste, Lawrence, 83, 713 Bernheim, B Douglas, 315 Bertola, Giuseppe, 948 Bertsekas, Dimitri, 82, 83, 109 Bewley, Truman, 561, 897 Black, Fischer, 393 Blackwell, David, 162 Blanchard, Olivier J., xx, 116, 120, 315 Blundell, Richard, 3, 70 Bohn, Henning, 413 Bond, Eric, 817, 827, 833 Bond, Eric W., xxiii, 817, 827, 833 Breeden, Douglas T., 11, 393 Brock, William A., 127, 262, 265, 293, 353, 361, 891 Browning, Martin, xxvi Brumelle, Shelby, 96 Bruno, Michael, 274, 862 Bryant, John, 866, 916 Buiter, Willem, 869 Burdett, Kenneth, 978 Burnside, Craig, 69 Caballero, Ricardo J., 596, 948 Campbell, John Y., 13, 57, 386 Carroll, Christopher D., 596 Cass, David, 285 Chamberlain, Gary, 540, 542, 554 Chamley, Christophe, 10, 473, 480, 482, 483 Champ, Bruce, 293 Chang, Roberto, 769, 811 Chari, V.V., 474, 492, 500, 531, 769, 810, 873 Chatterjee, Satyajit, 897 – 1069 – 1070 Author Index Chen, Ren-Raw, 434 Cho, In-Koo, 810 Chow, Gregory, 82, 111, 116 Christiano, Lawrence J., 69, 130, 353, 474, 492, 500, 531, 810, 873 Clower, Robert, 881 Cochrane, John H., 213, 733 Cogley, Timothy, 436, 438 Cole, Harold, xxii, 23, 632 Cole, Harold L., 668, 764 Constantinides, George, 438 Constantinides, George M., 330 Cooper, Russell, xx Corbae, Dean, 897 Correia, Isabel H., 473, 490, 873 Evans, George, 619 Fackler, Paul L., xix Faig, Miguel, 873 Farmer, Roger, xx Fischer, Stanley, xx, 274, 862 Fisher, Irving, 287 Fisher, Jonas, 353 Fitzgerald, T., 591 Flavin, Marjorie A., 418 Freeman, Scott, 293 Friedman, Milton, 864, 907 Fudenberg, Drew, 810 Dai, Qiang, 434 Daniel, James W., 1018 Davies, James B., Davis, Steven J., 948 Debreu, Gerard, 203, 259, 261 Den Haan, Wouter, 347 Diamond, Peter A., 292, 315, 935, 941 Diaz-Gim´nez, J., 6, 591 e Dinopoulos, Elias, 465 Dixit, Avinash K., 455 Domeij, David, 531 Doob, Joseph L., 665 Duffie, Darrell, 330, 386, 438 Gale, David, 262, 263 Gali, Jordi, 418 Geanakoplos, John, 598 Genicot, Garance, 739 Gilles, Saint-Paul, 967 Gittins, J.C., 183 Golosov, Mikhail, 531 Gomes, Joao, 979, 980 Gong, Frank F., 434 Gordon, David B., 768, 800, 878 Granger, C.W.J., xxxiii, 57 Green, Edward J., xxxiv, 633 Green, Jerry R., 191, 211, 261 Greenwood, Jeremy, 353, 979, 980 Grossman, Gene M., 465 Guidotti, Pablo E., 873 Gul, Faruk, 202 Eichenbaum, Martin, 69, 810 Eichenbaum, Martin , 353 Engle, Robert F., 57 Epple, Dennis, xxxiii, 116, 653 Ethier, Wilfred J., 455 Evans, Charles, 353 Hall, George, 94 Hall, Robert E., 320, 353 Hamilton, James D., 418 Hansen, Gary D., 94, 562, 936, 955 Hansen, Lars P., xxvi, xxxiii, 13, 68, 111, 115, 116, 120, 234, 387, Author Index 418, 421, 425, 653 Hansen, Lars, Peter, 102 Heathcote, Jonathan, 531 Heckman, James J., xxvi Helpman, Elhanan, 465 Hercowitz, Zvi, 353 Holmstrăm , Bengt, 635 o Honkapohja, Seppo, 619 Hopenhayn, Hugo A., 102, 746, 958 Hosios, Arthur J., 945, 952 Howitt, Peter, xx, 465 Huggett, Mark, 561, 565, 595 Imrohoro˘lu, Ay¸e, 562 g s Imrohoro˘lu, Selahattin, 94, 562 g Ingram, Beth Fisher, 353 Ireland, Peter N., 878, 897 Jagannathan, Ravi, 425 Jermann, Urban, 14, 102, 232, 736 Joines, Douglas, 562 Jones, Charles I., 466 Jones, Larry, 10 Jones, Larry E., 452, 460, 475, 485, 491, 523, 530 Jorgenson, Dale, 320 Jovanovic, Boyan, xxxiii, 125, 155, 169 Judd, Kenneth L., xix, 10, 197, 455, 473, 482, 483, 529 233, 480, 137, 347, Kahn, Charles, 116, 120, 291 Kalman, Rudolf, xviii Kandori, Michihiro, 291 Kareken, John, xxxiii, 293, 588, 870, 922 Kehoe, Patrick J., 474, 492, 500, 531, 1071 656, 769, 873 Kehoe, Timothy, 735, 736 Keynes, John Maynard, 347, 916 Kim, J., 353 Kim, S., 353 Kimball, Miles S., 316, 596 Kimbrough, Kent P., 873 King, Robert G., 127, 879 Kiyotaki, Nobuhiro, 246, 936, 970 Kletzer, Kenneth M., 739 Kocherlakota, Narayana R., xxii, 23, 292, 316, 353, 531, 631, 632, 668, 693, 707, 764, 838, 907 Koeppl, Thorsten V., 713 Kreps, David M., 427, 768, 810 Krishna, Vijay, 779 Krueger, Dirk, 6, 718, 733 Krusell, Per, 597 Kwakernaak, Huibert, 50, 111, 115 Kydland, Finn E., xxxiii, 127, 653, 768, 769, 878 Labadie, Pamela, xx Lagos, Ricardo, 978 Laibson, David I., 202 Leland, Hayne E., 595 LeRoy, Stephen, 393 LeRoy, Steven, 425 Levhari, David, 194 Levine, David, 593, 735, 736 Ligon, Ethan, Ljungqvist, Lars, 958, 983 Lo, Andrew W., 386 Lucas, Robert E., Jr., xx, xxvi, xxxiii, 11, 82, 83, 90, 186, 191, 246, 353, 393, 404, 443, 453, 474, 485, 502, 509, 666, 866, 881, 1072 Author Index 935, 936, 966, 979 Ludvigson, Sydney, 57 Lustig, Hanno, 232, 233, 738 Mace, Barbara, 213, 733 MacKinlay, A Craig, 386 Mankiw, Gregory, 14, 292, 316, 386, 396, 435–437 Manuelli, Rodolfo, 10, 452, 460, 475, 480, 485, 491, 523, 530, 897 Marcet, Albert, 274, 347, 475, 509, 512, 598, 653, 656, 862 Marimon, Ramon, 274, 653, 656, 862, 869, 983 Mas-Colell, Andreu, 191, 211, 261, 598 McCall, B.P., 151 McCall, John, 137 McCallum, Bennett T., 852, 869 McCandless, George T., 293 McGrattan, Ellen R., 111, 115, 120 McLennan, Andrew, 598 Mehra, Rajnish, 186, 191, 246, 397, 421 Miao, Jianjun, 598 Miller, Bruce L., 596 Miller, Merton, 386, 404 Miranda, Mario J., xix Mirman, Leonard J., 127, 194, 353, 361 Modigliani, Franco, 386, 404 Moen, Espen R., 954 Montgomery, James D., 978 Mortensen, Dale T., 935, 941, 969, 983 Murphy, Kevin, 186 Muth, John F., xxxiii Neal, Derek, 151 Nicolini, Juan Pablo, 102, 274, 746 Nobel, Ben, 1018 Nyarko, Yaw, xxxiii Ospina, Sandra, 595 Paal, Beatrix, 866 Park, Jee-Haejwong, xxiii Park, Jee-Hyeong, 817, 827, 833 Pearce, David, xxxiv, 631, 769, 779, 790, 800 Perri, Fabrizio, 6, 656, 718, 733 Persson, Mats, 503 Persson, Torsten, 503 Pesendorfer, Wolfgang, 202 Peters, Michael, 978 Phelan, Christopher, xxxiv, 631, 683, 769, 811 Phelps, Edmund S., 202, 935 Piazzesi, Monika, 13, 61, 434 Pissarides, Christopher, 935, 941, 946, 969, 983 Plosser, Charles I., 127 Pollak, Robert A., 202 Pratt, John, 419 Prescott, Edward C., xx, xxxiii, 82, 83, 90, 127, 186, 191, 246, 353, 397, 421, 591, 653, 768, 769, 878, 935, 936, 966, 979 Preston, Ian, 3, 70 Putterman, Martin L., 96 Quadrini, V., Quah, Danny, 70 Rıios-Rull, J.V., Ray, Debraj, 739 Author Index Rebelo, Sergio, 69, 127, 444, 460, 979, 980 Remolona, Eli M., 434 Roberds, William, xxxiii, 116, 291, 418, 653 Rodriquez, S.B., Rogerson, Richard, 936, 955, 958 Rogoff, Kenneth, 768, 769 Romer, David, xx Romer, Paul M., 443, 449, 455, 462, 465, 466 Rosen, Sherwin, 186 Rossi, Peter, 10 Rossi, Peter E., 452, 475, 480, 485, 491, 523, 530 Rotemberg, Julio J., 879 Rubinstein, Mark, 435 Runkle, David, 14 Sala-i-Martin, Xavier, 452 Samuelson, Paul, 257, 269 Sandmo, Agnar, 595 Sargent, Thomas J., 50, 109, 115, 274, 475, 509, 512, 862, 869, 983 Savin, N.E., 353 Scheinkman, Jose, 83, 186, 713 Schmitt-Grohe, Stephanie, 353, 522 Scholes, Myron, 393 Scott, Louis, 434 Seater, John, J, 315 Segerstrom, Paul S., 465, 466 Seppălă, Juha, 475, 509, 512 aa Shavell, Stephen, xxxiv Shavell, Steven, xxii, 746 Shell, Karl, 261, 285 Shi, Shouyong, 972, 978 1073 Shiller, Robert, 57, 425 Shimer, Robert, 979 Shin, M.C., 96 Shorrocks, Anthony F., Shreve, Steven, 82 Sibley, David S., 596 Sims, Christopher A., 330, 591, 868 Singleton, Kenneth J., 387, 421, 434, 598 Siu, Henry, 522 Sivan, Raphael, 50, 111, 115 Smith, Adam, 290 Smith, Anthony, 597 Smith, Bruce, 922 Smith, Bruce D., 286, 330 Smith, Lones, 291 Sotomayor, Marilda A de Oliveira, 542 Spear, Stephen E., xxxiv, 631, 682 Srivastava, Sanjay, 631, 682 Stacchetti, Ennio, xxxiv, 631, 769, 779, 790, 800, 811 Stigler, George, 137 Stiglitz, Joseph E., 455, 491 Stokey, Nancy L., xx, 82, 83, 90, 474, 485, 502, 509, 769, 800 Storesletten, Kjetil, 438 Summers, Lawrence, 292, 396 Sunder, Shyam, 274, 862 Svensson, Lars E.O., 503, 879 Tallarini, Thomas, 102, 234 Teles, Pedro, 873 Telmer, Chris, 438 Thomas, Jonathan, xxii, xxxiv, 631, 657, 693, 838 1074 Author Index Tobin, James, 854 Topel, Robert, 186 Townsend, Robert M., xxxiii, xxxiv, 213, 246, 631, 683, 733, 897, 905 Trejos, Alberto, 972 Tsyvinski, Aleh, 531 Uribe, Martin, 353, 522 Uzawa, Hirofumi, 443, 453 Vegh, Carlos A., 873 Velde, Fran¸ois, 330, 866, 922 c Villamil, Anne, 916 Wald, Abraham, xviii Wallace, Neil, xxxiii, 278, 286, 291, 293, 588, 866, 869, 870, 916, 922 Walsh, Carl, xx Wang, Neng, 595 Wash, Carl, xx Weber, Gugliemo, 438 Wei, Chao, 94 Weiss, Laurence, xxii, xxxiv, 746 Werning, Ivan, 764 Whinston, Michael D., 191, 211, 261 Whiteman, Charles, 116, 120 Wilcox, David W., 418 Wilson, Charles, 540, 542, 554 Wolman, Alexander L., 879 Woodford, Michael, xx, 120, 868, 879 Worrall, Tim, xxii, xxxiv, 631, 657, 693, 838 Wright, Brian D., 739 Wright, Randall, 246, 936, 970, 972, 978 Yaron, Amir, 438 Young, Alwyn, 466 Zame, William, 593 Zeckhauser, Richard, 292, 396 Zeira, Joseph, 441 Zeldes, Stephen P., 14, 316, 596 Zhang, Harold, 14, 738 Zhao, Rui, xxii, 746 Zhu, Xiaodong, 474, 492, 495, 500 Zilibotti, Fabrizio, 983 Zin, Stanley E., 13, 60, 64 Matlab Index Matlab Index hugofoc1.m, 752 impulse.m, 43 judd.m, 198 aiyagari2.m, 579 kfilter.m, 122 bewley99.m, 579 bewley99v2.m, 579 bewleyplot.m, 579 bewleyplot2.m, 579 bigshow2.m, 51 dimpulse.m, 43 dlyap.m, 44, 111, 237 doublej.m, 42, 44, 111, 237 fsolve.m, 129 hjbnd1.m, 431 hjbnd2.m, 431 hugo.m, 752 hugo1a.m, 752 markov.m, 35, 565 markovapprox.m, 579 nash.m, 198, 202 neal2.m, 152 nnash.m, 198 olipololy5.m, 624 olrp.m, 110 policyi.m, 110 schumaker.m, 105 schurg.m, 118 valhugo.m, 752 1075 ... Cataloging-in-Publication Data Ljungqvist, Lars Recursive macroeconomic theory / Lars Ljungqvist, Thomas J Sargent p cm Includes bibliographical references and index ISBN 0-262-19451-1 Macroeconomics Recursive functions.. .Recursive Macroeconomic Theory Second edition Lars Ljungqvist Stockholm School of Economics Thomas J Sargent New York University and Hoover Institution c 2000 Massachusetts... Lucas, and Prescott (1989) and Bertsekas (1976) remain standard references for recursive methods in macroeconomics Chapters and appendix A in this book revise material appearing in Chapter of Sargent