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LARS LJUNGQVIST

THOMAS J. SARGENT

recursive

macroeconomic theory

SECOND EDITION

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Recursive Macroeconomic Theory

Second edition

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To our parents, Zabrina, and Carolyn

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Recursive Macroeconomic Theory

Second edition

Lars Ljungqvist

Stockholm School of Economics

Thomas J. Sargent

New York University

and

Hoover Institution

The MIT Press

Cambridge, Massachusetts

London, England

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c 2004 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. – 2nd ed.

p. cm.

Includes bibliographical references and index.

ISBN 0-262-12274-X

1. Macroeconomics. 2. Recursive functions. 3. Statics and dynamics

(Social sciences)

I. Sargent, Thomas J. II. Title.

HB172.5 .L59 2004

339’.01’51135–dc22

2004054688

10 9 8 7 6 5 4 3 2 1

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Contents

Acknowledgements

xvii

Preface to the second edition

xviii

Part I: The imperialism of recursive methods

1. Overview

3

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 saving. 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.

–v–

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Part II: Tools

2. Time Series

29

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. Continuousstate 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.

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.

A. A linear diﬀerence equation. 2.11. Exercises.

3. Dynamic Programming

85

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.

4. Practical Dynamic Programming

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.

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5. Linear Quadratic Dynamic Programming

vii

109

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.

6. Search, Matching, and Unemployment

139

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. 6.6.1. Recursive formulation and solution. 6.6.2. Endogenous

statistics. 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. 6.10. Exercises.

Part III: Competitive equilibria and applications

7. Recursive (Partial) Equilibrium

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.

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8. Equilibrium with Complete Markets

208

8.1. Time 0 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. 8.10.1. Arbitrage-free pricing. 8.11. Consumption strips and the

cost of business cycles. 8.11.1. 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.

9. Overlapping Generations Models

9.1. Endowments and preferences. 9.2. Time 0 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.

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10. Ricardian Equivalence

ix

312

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 Growth Model

323

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

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

12.1. Endogenous aggregate state variable. 12.2. The stochastic growth

model. 12.3. Lagrangian formulation of the planning problem. 12.4.

Time 0 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. A “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.

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12.9.1. Equilibrium restrictions across decision rules. 12.9.2. Using the

planning problem. 12.10. Concluding remarks.

13. Asset Pricing

392

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.

Factor models. 13.16. Heterogeneity and incomplete markets. 13.17.

Concluding remarks. 13.18. Exercises.

14. Economic Growth

449

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.

15.4. 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.

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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 2. 15.13. Examples of labor tax smoothing . 15.13.1. Example 1: gt = g for all t ≥ 0 .

15.13.2. Example 2: gt = 0 for t = T , and gT > 0 . 15.13.3. Example 3:

gt = 0 for t = T , and gT is stochastic. 15.14. Lessons for optimal debt

policy. 15.15. Taxation without state-contingent debt. 15.15.1. Future

values of {gt } become deterministic. 15.15.2. Stochastic {gt } but special preferences. 15.15.3. Example 3 revisited: gt = 0 for t = T , and

gT is stochastic. 15.16. Zero tax on human capital. 15.17. Should all

taxes be zero?. 15.18. Concluding remarks. 15.19. Exercises.

Part IV: The savings problem and Bewley models

16. Self-Insurance

545

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 a 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

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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 deﬂation.

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.

Part V: Recursive contracts

18. Dynamic Stackelberg Problems

615

18.1. History dependence. 18.2. The Stackelberg problem. 18.3. Solving the Stackelberg problem. 18.3.1. Step 1: solve an optimal linear

regulator. 18.3.2. Step 2: use the stabilizing properties of shadow

price P yt . 18.3.3. Stabilizing solution. 18.3.4. Step 3: convert implementation multipliers. 18.3.5. History-dependent representation of

decision rule. 18.3.6. Digression on determinacy of equilibrium. 18.4.

A large ﬁrm with a competitive fringe. 18.4.1. The competitive fringe.

18.4.2. The monopolist’s problem. 18.4.3. Equilibrium representation.

18.4.4. Numerical example. 18.5. Concluding remarks. A. The stabilizing µt = P yt . B. Matrix linear diﬀerence equations. C. Forecasting

formulas. 18.9. Exercises.

19. Insurance Versus Incentives

19.1. Insurance with recursive contracts. 19.2. Basic environment. 19.3.

One-sided no commitment. 19.3.1. Self-enforcing contract. 19.3.2. Recursive formulation and solution. 19.3.3. Recursive computation of contract . 19.3.4. Proﬁts. 19.3.5. P (v) is strictly concave and continuously diﬀerentiable. 19.3.6. Many households. 19.3.7. An example.

19.4. A Lagrangian method. 19.5. Insurance with asymmetric information. 19.5.1. Eﬃciency implies bs−1 ≥ bs , ws−1 ≤ ws . 19.5.2. Local

upward and downward constraints are enough. 19.5.3. Concavity of

P . 19.5.4. Local downward constraints always bind. 19.5.5. Coinsurance. 19.5.6. P (v) is a martingale. 19.5.7. Comparison to model with

commitment problem. 19.5.8. Spreading continuation values. 19.5.9.

Martingale convergence and poverty. 19.5.10. Extension to general

equilibrium. 19.5.11. Comparison with self-insurance. 19.6. Insurance

with unobservable storage. 19.6.1. Feasibility. 19.6.2. Incentive compatibility. 19.6.3. Eﬃcient allocation. 19.6.4. The case of two periods

(T = 2 ). 19.6.5. Role of the planner. 19.6.6. Decentralization in a

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closed economy. 19.7. Concluding remarks. A. Historical development.

19.A.1. Spear and Srivastava. 19.A.2. Timing. 19.A.3. Use of lotteries.

19.9. Exercises.

20. Equilibrium without Commitment

697

20.1. Two-sided lack of commitment. 20.2. A closed system. 20.3.

Recursive formulation. 20.4. Equilibrium consumption. 20.4.1. Consumption dynamics. 20.4.2. Consumption intervals cannot contain each

other. 20.4.3. Endowments are contained in the consumption intervals.

20.4.4. All consumption intervals are nondegenerate (unless autarky is

the only sustainable allocation). 20.5. Pareto frontier and ex ante division of the gains. 20.6. Consumption distribution. 20.6.1. Asymptotic

distribution. 20.6.2. Temporary imperfect risk sharing. 20.6.3. Permanent imperfect risk sharing. 20.7. Alternative recursive formulation.

20.8. Pareto frontier revisited. 20.8.1. Values are continuous in implicit consumption. 20.8.2. Diﬀerentiability of the Pareto frontier. 20.9.

Continuation values a` la Kocherlakota. 20.9.1. Asymptotic distribution

is nondegenerate for imperfect risk sharing (except for when S = 2 ).

20.9.2. Continuation values do not always respond to binding participation constraints. 20.10. A two-state example: amnesia overwhelms

memory. 20.10.1. Pareto frontier. 20.10.2. Interpretation. 20.11. A

three-state example. 20.11.1. Perturbation of parameter values. 20.11.2.

Pareto frontier. 20.12. Empirical motivation. 20.13. Generalization

. 20.14. Decentralization. 20.15. Endogenous borrowing constraints.

20.16. Concluding remarks. 20.17. Exercises.

21. Optimal Unemployment Insurance

21.1. History-dependent unemployment insurance. 21.2. A one-spell

model. 21.2.1. The autarky problem. 21.2.2. Unemployment insurance

with full information. 21.2.3. The incentive problem. 21.2.4. Unemployment insurance with asymmetric information. 21.2.5. Computed

example. 21.2.6. Computational details. 21.2.7. Interpretations. 21.2.8.

Extension: an on-the-job tax. 21.2.9. Extension: intermittent unemployment spells. 21.3. A multiple-spell model with lifetime contracts.

21.3.1. The setup. 21.3.2. A recursive lifetime contract. 21.3.3. Compensation dynamics when unemployed. 21.3.4. Compensation dynamics

while employed. 21.3.5. Summary. 21.4. Concluding remarks. 21.5. Exercises.

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22. Credible Government Policies

773

22.1. Introduction. 22.2. Dynamic programming squared: synopsis.

22.3. The one-period economy. 22.3.1. Competitive equilibrium. 22.3.2.

The Ramsey problem. 22.3.3. Nash equilibrium. 22.4. Examples of

economies. 22.4.1. Taxation example. 22.4.2. Black-box example with

discrete-choice sets. 22.5. Reputational mechanisms: General idea.

22.5.1. Dynamic programming squared. 22.6. The inﬁnitely repeated

economy. 22.6.1. A strategy proﬁle implies a history and a value. 22.6.2.

Recursive formulation. 22.7. Subgame perfect equilibrium (SPE). 22.8.

Examples of SPE. 22.8.1. Inﬁnite repetition of one-period Nash equilibrium. 22.8.2. Supporting better outcomes with trigger strategies.

22.8.3. When reversion to Nash is not bad enough. 22.9. Values of all

SPEs. 22.9.1. The basic idea of dynamic programming squared. 22.10.

Self-enforcing SPE. 22.10.1. The quest for something worse than repetition of Nash outcome. 22.11. Recursive strategies. 22.12. Examples

of SPE with recursive strategies. 22.12.1. Inﬁnite repetition of Nash

outcome. 22.12.2. Inﬁnite repetition of a better-than-Nash outcome.

22.12.3. Something worse: a stick-and-carrot strategy. 22.13. The best

and the worst SPE values. 22.13.1. When v1 is outside the candidate

set. 22.14. Examples: alternative ways to achieve the worst. 22.14.1.

Attaining the worst, method 1. 22.14.2. Attaining the worst, method 2.

22.14.3. Attaining the worst, method 3. 22.14.4. Numerical example.

22.15. Interpretations. 22.16. Concluding remarks. 22.17. Exercises.

23. Two Topics in International Trade

23.1. Two dynamic contracting problems. 23.2. Lending with moral

hazard and diﬃcult enforcement. 23.2.1. Autarky. 23.2.2. Investment

with full insurance. 23.2.3. Limited commitment and unobserved investment. 23.2.4. Optimal capital outﬂows under distress. 23.3. Gradualism in trade policy. 23.3.1. Closed-economy model. 23.3.2. A Ricardian

model of two countries under free trade. 23.3.3. Trade with a tariﬀ.

23.3.4. Welfare and Nash tariﬀ. 23.3.5. Trade concessions. 23.3.6. A

repeated tariﬀ game. 23.3.7. Time-invariant transfers. 23.3.8. Gradualism: time-varying trade policies. 23.3.9. Baseline policies. 23.3.10.

Multiplicity of payoﬀs and continuation values. 23.4. Concluding remarks. A. Computations for Atkeson’s model. 23.6. Exercises.

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Part VI: Classical monetary economics and search

24. Fiscal-Monetary Theories of Inﬂation

857

24.1. The issues. 24.2. A shopping time monetary economy. 24.2.1.

Households. 24.2.2. Government. 24.2.3. Equilibrium. 24.2.4. “Short

run” versus “long run”. 24.2.5. Stationary equilibrium. 24.2.6. Initial

date (time 0). 24.2.7. Equilibrium determination. 24.3. Ten monetary doctrines. 24.3.1. Quantity theory of money. 24.3.2. Sustained

deﬁcits cause inﬂation. 24.3.3. Fiscal prerequisites of zero inﬂation

policy. 24.3.4. Unpleasant monetarist arithmetic. 24.3.5. An “open

market” operation delivering neutrality. 24.3.6. The “optimum quantity” of money. 24.3.7. Legal restrictions to boost demand for currency.

24.3.8. One big open market operation. 24.3.9. A ﬁscal theory of the

price level. 24.3.10. Exchange rate indeterminacy . 24.3.11. Determinacy of the exchange rate retrieved . 24.4. An example of exchange rate

(in)determinacy. 24.4.1. Trading before sunspot realization. 24.4.2. Fiscal theory of the price level. 24.5. Optimal inﬂation tax: the Friedman

rule. 24.5.1. Economic environment. 24.5.2. Household’s optimization

problem. 24.5.3. Ramsey plan. 24.6. Time consistency of monetary

policy. 24.6.1. Model with monopolistically competitive wage setting.

24.6.2. Perfect foresight equilibrium. 24.6.3. Ramsey plan. 24.6.4. Credibility of the Friedman rule. 24.7. Concluding discussion. 24.8. Exercises.

25. Credit and Currency

25.1. Credit and currency with long-lived agents. 25.2. Preferences

and endowments. 25.3. Complete markets. 25.3.1. A Pareto problem.

25.3.2. A complete markets equilibrium. 25.3.3. Ricardian proposition.

25.3.4. Loan market interpretation. 25.4. A monetary economy. 25.5.

Townsend’s “turnpike” interpretation. 25.6. The Friedman rule. 25.6.1.

Welfare. 25.7. Inﬂationary ﬁnance. 25.8. Legal restrictions. 25.9. A

two-money model. 25.10. A model of commodity money. 25.10.1. Equilibrium. 25.10.2. Virtue of ﬁat money. 25.11. Concluding remarks.

25.12. Exercises.

904

xvi

Contents

26. Equilibrium Search and Matching

940

26.1. Introduction. 26.2. An island model. 26.2.1. A single market (island). 26.2.2. The aggregate economy. 26.3. A matching model. 26.3.1.

A steady state. 26.3.2. Welfare analysis. 26.3.3. Size of the match surplus. 26.4. Matching model with heterogeneous jobs. 26.4.1. A steady

state. 26.4.2. Welfare analysis. 26.4.3. The allocating role of wages

I: separate markets. 26.4.4. The allocating role of wages II: wage announcements. 26.5. Model of employment lotteries . 26.6. Lotteries for

households versus lotteries for ﬁrms. 26.6.1. An aggregate production

function. 26.6.2. Time-varying capacity utilization. 26.7. Employment

eﬀects of layoﬀ taxes. 26.7.1. A model of employment lotteries with layoﬀ taxes. 26.7.2. An island model with layoﬀ taxes. 26.7.3. A matching

model with layoﬀ taxes. 26.8. Kiyotaki-Wright search model of money.

26.8.1. Monetary equilibria. 26.8.2. Welfare. 26.9. Concluding comments. 26.10. Exercises.

Part VII: Technical appendixes

A. Functional Analysis

1005

A.1. Metric spaces and operators. A.2. Discounted dynamic programming. A.2.1. Policy improvement algorithm. A.2.2. A search problem.

B. Control and Filtering

1017

B.1. Introduction. B.2. The optimal linear regulator control problem.

B.3. Converting a problem with cross products in states and controls to

one with no such cross products. B.4. An example. B.5. The Kalman

ﬁlter. B.6. Duality. B.7. Examples of Kalman ﬁltering. B.8. Linear

projections. B.9. Hidden Markov models. B.9.1. Optimal ﬁltering.

1. References

1044

2. Index

1072

3. Author Index

1077

4. Matlab Index

1082

Acknowledgments

We wrote this book during the 1990s and early 2000s 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

DeNardi, William Dupor, William Fuchs, George Hall, Cristobal Huneeus, Sagiri Kitao, Hanno Lustig, Sergei Morozov, Eva Nagypal, Monika Piazzesi, Navin

Kartik, Martin Schneider, Juha Sepp¨

al¨

a, Yongseok Shin, Christopher Sleet, Stijn

Van Nieuwerburgh, Laura Veldkamp, Neng Wang, Chao Wei, Mark Wright,

Sevin Yeltekin, Bei Zhang, and Lei Zhang. Each of these people made substantial suggestions 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 Jesus Fernandez-Villaverde, Gary Hansen,

Jonathan Heathcote, Berthold Herrendorf, Mark Huggett, Charles Jones, Narayana Kocherlakota, Dirk Krueger, Per Krusell, Francesco Lippi, Rodolfo Manuelli,

Beatrix Paal, Adina Popescu, Jonathan Thomas, and Nicola Tosini.

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 25 are versions of ones in Sargent (1987b).

Fran¸cois Velde provided substantial help with the TEX and Unix macros that

produced this book. Maria Bharwada helped typeset it. We thank P.M. Gordon

Associates for copyediting.

For providing good environments to work on this book, Ljungqvist thanks

the Stockholm School of Economics and Sargent thanks the Hoover Institution

and the departments of economics at the University of Chicago, Stanford University, and New York University.

– xvii –

Preface to the second edition

Recursive Methods

Much of this book is about how to use recursive methods to study macroeconomics. 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 the mutual covariation of 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 sequence of problems, each one posing 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 of

the model today into the state tomorrow, and another function mapping the

state into the other endogenous variables of the model. The state is a vector

of variables that characterizes the system’s current position. Time series are

generated from these objects by iterating the transition law.

– xviii –

Preface to the second edition

xix

Recursive approach

Recursive methods constitute a powerful approach to dynamic economics due

to their described focus on a tradeoﬀ between the current period’s utility and a

continuation value for utility in all future periods. As mentioned, the simpliﬁcation arises from dealing with the evolution of state variables that capture the

consequences of today’s actions and events for all future periods, and in the case

of uncertainty, for all possible realizations in those future periods. This is not

only a powerful approach to characterizing and solving complicated problems,

but it also helps us to develop intuition, conceptualize, and think about dynamic economics. Students often ﬁnd that half of the job in understanding how

a complex economic model works is done once they understand what the set of

state variables is. Thereafter, the students are soon on their way to formulating

optimization problems and transition equations. Only experience from solving

practical problems fully conveys the power of the recursive approach. This book

provides many applications.

Still another reason for learning about the recursive approach is the increased importance of numerical simulations in macroeconomics, and most computational algorithms rely on recursive methods. When such numerical simulations are called for in this book, we give some suggestions for how to proceed

but without saying too much on numerical methods. 1

Philosophy

This book mixes tools and sample applications. Our philosophy is to present the

tools with 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 has become 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, there are several good recent books covering parts of the ﬁeld that

we neglect, for example, Aghion and Howitt (1998), Barro and Sala-i-Martin

(1995), Blanchard and Fischer (1989), Cooley (1995), Farmer (1993), Azariadis

1 Judd (1998) and Miranda and Fackler (2002) provide good treatments of numerical

methods in economics.

www.ebook3000.com

xx

Preface to the second edition

(1993), Romer (1996), Altug and Labadie (1994), Walsh (1998), Cooper (1999),

Adda and Cooper (2003), Pissarides (1990), and Woodford (2000). Stokey, Lucas, and Prescott (1989) and Bertsekas (1976) remain standard references for

recursive methods in macroeconomics. Chapters 6 and Appendix A in this book

revise material appearing in chapter 2 of Sargent (1987b).

Changes in the second edition

This edition contains seven new chapters and substantial revisions of important

parts of about half of the original chapters. New to this edition are chapters 1,

11, 12, 18, 20, 21, and 23. The new chapters and the revisions cover exciting

new topics. They widen and deepen the message that recursive methods are

pervasive and powerful.

New chapters

Chapter 1 is an overview that discusses themes that unite many of the apparently diverse topics treated in this book. Because it ties together ideas that can

be fully appreciated only after working through the material in the subsequent

chapters, we were ambivalent about whether this chapter should be ﬁrst or last.

We have chosen to put this last chapter ﬁrst because it tells our destination. The

chapter emphasizes two ideas: (1) a consumption Euler equation that underlies

many results in the literatures on consumption, asset pricing, and taxation; and

(2) a set of recursive ways to represent contracts and decision rules that are

history-dependent. These two ideas come together in the several chapters on

recursive contracts that form Part V of this edition. In these chapters, contracts or government policies cope with enforcement and information problems

by tampering with continuation utilities in ways that compromise the consumption Euler equation. How the designers of these contracts choose to disrupt the

consumption Euler equation depends on detailed aspects of the environment

that prevent the consumer from reallocating consumption across time in the

way that the basic permanent income model takes for granted. These chapters

on recursive contracts convey results that can help to formulate novel theories

of consumption, investment, asset pricing, wealth dynamics, and taxation.

Preface to the second edition

xxi

Our ﬁrst edition lacked a self-contained account of the simple optimal

growth model and some of its elementary uses in macroeconomics and public ﬁnance. Chapter 11 corrects that deﬁciency. It builds on Hall’s 1971 paper

by using the standard nonstochastic growth model to analyze the eﬀects on equilibrium outcomes of alternative paths of ﬂat rate taxes on consumption, income

from capital, income from labor, and investment. The chapter provides many

examples designed to familiarize the reader with the covariation of endogenous

variables that are induced by both the transient (feedback) and anticipatory

(feedforward) dynamics that are embedded in the growth model. To expose the

structure of those dynamics, this chapter also describes alternative numerical

methods for approximating equilibria of the growth model with distorting taxes

and for evaluating the accuracy of the approximations.

Chapter 12 uses a stochastic version of the optimal growth model as a vehicle for describing how to construct a recursive competitive equilibrium when

there are endogenous state variables. This chapter echoes a theme that recurs

throughout this edition even more than it did in the ﬁrst edition, namely, that

discovering a convenient state variable is an art. This chapter extends an idea

of chapter 8, itself an extensively revised version of chapter 7 of the ﬁrst edition, namely, that a measure of household wealth is a key state variable both

for achieving a recursive representation of an Arrow-Debreu equilibrium price

system, and also for constructing a sequential equilibrium with trading each

period in one-period Arrow securities. The reader who masters this chapter will

know how to use the concept of a recursive competitive equilibrium and how to

represent Arrow securities when there are endogenous state variables.

Chapter 18 reaps rewards from the powerful computational methods for linear quadratic dynamic programming that are discussed in chapter 5, a revision

of chapter 4 of the ﬁrst edition. Our new chapter 18 shows how to formulate and

compute what are known as Stackelberg or Ramsey plans in linear economies.

Ramsey plans assume a timing protocol that allows a Ramsey planner (or government) to commit, i.e., to choose once-and-for-all a complete state contingent

plan of actions. Having the ability to commit allows the Ramsey planner to

exploit the eﬀects of its time t actions on time t + τ actions of private agents

for all τ ≥ 0 , where each of the private agents chooses sequentially. At one time,

it was thought that problems of this type were not amenable recursive methods

because they have the Ramsey planner choosing a history-dependent strategy.

Indeed, one of the ﬁrst rigorous accounts of the time inconsistency of a Ramsey

xxii

Preface to the second edition

plan focused on the failure of the Ramsey planner’s problem to be recursive in

the natural state variables (i.e., capital stocks and information variables). However, it turns out that the Ramsey planner’s problem is recursive when the state

is augmented by costate variables whose laws of motion are the Euler equations

of private agents (or followers). In linear quadratic environments, this insight

leads to computations that are minor but ingenious modiﬁcations of the classic

linear-quadratic dynamic program that we present in chapter 5.

In addition to substantial new material, chapters 19 and 20 contain comprehensive revisions and reorganizations of material that had been in chapter

15 of the ﬁrst edition. Chapter 19 describes three versions of a model in which a

large number of villagers acquire imperfect insurance from a planner or money

lender. The three environments diﬀer in whether there is an enforcement problem or some type of information problem (unobserved endowments or perhaps

both an unobserved endowments and an unobserved stock of saving). Important new material appears throughout this chapter, including an account of a

version of Cole and Kocherlakota’s (2001) model of unobserved private storage.

In this model, the consumer’s access to a private storage technology means that

his consumption Euler inequality is among the implementability constraints that

the contract design must respect. That Euler inequality severely limits the planner’s ability to manipulate continuation values as a way to manage incentives.

This chapter contains much new material that allows the reader to get inside

the money-lender villager model and to compute optimal recursive contracts by

hand in some cases.

Chapter 20 contains an account of a model that blends aspects of models

of Thomas and Worrall (1988) and Kocherlakota (1996b). Chapter 15 of our

ﬁrst edition had an account of this model that followed Kocherlakota’s account

closely. In this edition, we have chosen instead to build on Thomas and Worrall’s

work because doing so allows us to avoid some technical diﬃculties attending

Kocherlakota’s formulation. Chapter 21 uses the theory of recursive contracts to

describe two models of optimal experience-rated unemployment compensation.

After presenting a version of Shavell and Weiss’s (1979) model that was in

chapter 15 of the ﬁrst edition, it describes a version of Zhao’s (2001) model

of a “lifetime” incentive-insurance arrangement that imparts to unemployment

compensation a feature like a “replacement ratio.”

Chapter 23 contains two applications of recursive contracts to two topics in

international trade. After presenting a revised version of an account of Atkeson’s

Preface to the second edition

xxiii

(1991) model of international lending with both information and enforcement

problems, it describes a version of Bond and Park’s (2002) model of gradualism

in trade agreements.

Revisions of other chapters

We have added signiﬁcant amounts of material to a number of chapters, including chapters 2, 8, 15, and 16. Chapter 2 has a better treatment of laws of large

numbers and two extended economic examples (a permanent income model of

consumption and an arbitrage-free model of the term structure) that illustrate

some of the time series techniques introduced in the chapter. Chapter 8 says

much more about how to ﬁnd a recursive structure within an Arrow-Debreu

pure exchange economy than did its successor. Chapter 16 has an improved

account of the supermartingale convergence theorem and how it underlies precautionary saving results. Chapter 15 adds an extended treatment of an optimal

taxation problem in an economy in which there are incomplete markets. The

supermartingale convergence theorem plays an important role in the analysis

of this model. Finally, chapter 26 contains additional discussion of models in

which lotteries are used to smooth nonconvexities facing a household and how

such models compare with ones without lotteries.

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 alternative 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

one-period 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 let ﬁat money in.

xxiv

Preface to the second edition

2. A class of incomplete markets models with heterogeneous agents: The 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 family’s 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 government’s intertemporal budget constraint and the demand for ﬁat money,

aspects that transcend many models. We also use Samuelson’s overlapping

generations model, Bewley’s incomplete markets model, and Townsend’s

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 government’s

budget constraint. Other important doctrines do too. These doctrines,

known as Modigliani-Miller and Ricardian equivalence theorems, have a

common structure. They embody an equivalence class of government policies that produce the same allocations. We display the structure of such

theorems with an eye to ﬁnding the features whose absence causes them to

fail, letting particular policies matter.

5. Ramsey taxation problem: 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 the choice variables are allocations

rather than tax rates. Permissible allocations are those that satisfy resource

constraints and implementability constraints, where the latter are budget

constraints in which the consumer and ﬁrm ﬁrst-order conditions are used

to substitute out for 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 must balance providing insurance against providing proper incentives. Applications include the provision of unemployment