Contents

Symbols Used

I THE RECURSIVE ApPROACH

1 Introduction

2 An Overview

2.1 A Deterministic Model of Optimal Growth 92.2 A Stochastic Model of Optimal Growth 162.3 Competitive Equilibrium Growth 222.4 Conclusions and Plans 32

II DETERMINISTIC MODELS

3 Mathematical Preliminaries

3.1 Metric Spaces and Normed Vector Spaces 433.2 The Contraction Mapping Theorem 493.3 The Theorem of the Maximum 55

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4 Dynamic Programming under Certainty 66

4.1 The Principle of Optimality 674.2 Bounded Returns 774.3 Constant Returns to Scale 874.4 Unbounded Returns 924.5 Euler Equations 97

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5 Applications of Dynamic Programming under Certainty 103

5.1 The One-Sector Model of Optimal Growth 1035.2 A "Cake-Eating" Problem 1055.3 Optimal Growth with Linear Utility 1055.4 Growth with Technical Progress 1055.5 A Tree-Cutting Problem 1075.6 Learning by Doing 1075.7 Human Capital Accumulation 1095.8 Growth with Human Capital 1115.9 Investment with Convex Costs 112

5.10 Investment with Constant Returns 1135.11 Recursive Preferences 1145.12 Theory of the Consumer with Recursive

Preferences 1165.13 A Pareto Problem with Recursive Preferences 1175.14 An (5, S) Inventory Problem 1185.15 The Inventory Problem in Continuous Time 1225.16 A Seller with Unknown Demand 1235.17 A Consumption-Savings Problem 126

6 Deterministic Dynamics 131

6.1 One-Dimensional Examples 1336.2 Global Stability: Liapounov Functions 1396.3 Linear Systems and Linear Approximations 1436.4 Euler Equations 1486.5 Applications 157

III STOCHASTIC MODELS

7 Measure Theory and Integration

7.1 Measurable Spaces 1687.2 Measures 1707.3 Measurable Functions 1777.4 Integration 1847.5 Product Spaces 1957.6 The Monotone Class Lemma 1997.7 Conditional Expectation 202

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8 Markov Processes8.1 Transition Functions 2128.2 Probability Measures on Spaces of Sequences 2208.3 Iterated Integrals 2258.4 Transitions Defined by Stochastic Difference

Equations 234

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9 Stochastic Dynamic Programming 239

9.1 The Principle of Optimality 2419.2 Bounded Returns 2599.3 Constant Returns to Scale 2709.4 Unbounded Returns 2739.5 Stochastic Euler Equations 2809.6 Policy Functions and Transition Functions 283

10 Applications of Stochastic Dynamic Programming 288

10.1 The One-Sector Model of Optimal Growth 28810.2 Optimal Growth with Two Capital Goods 29010.3 Optimal Growth with Many Goods 29010.4 Industry Investment under Uncertainty 29210.5 Production and Inventory Accumulation ' 29710.6 Asset Prices in an Exchange Economy 30010.7 A Model of Search Unemployment 30410.8 The Dynamics of the Search Model 30810.9 Variations on the Search Model 310

10.10 A Model of Job Matching 31110.11 Job Matching and Unemployment 314

11 Strong Convergence of Markov Processes 316

11.1 Markov Chains 31911.2 Convergence Concepts for Measures 33411.3 Characterizations of Strong Convergence 33811.4 Sufficient Conditions 344

12 Weak Convergence of Markov Processes 352

12.1 Characterizations of Weak Convergence 35312.2 Distribution Functions 36412.3 Weak Convergence of Distribution Functions 369

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12.4 Monotone Markov Processes 37512.5 Dependence of the Invariant Measure on a

Parameter 38312.6 A Loose End 386

13 Applications of Convergence Results for MarkovProcesses

13.1 A Discrete-Space (5, S) Inventory Problem 38913.2 A Continuous-State (5, S) Process 39013.3 The One-Sector Model of Optimal Growth 39113.4 Industry Investment under Uncertainty 39513.5 Equilibrium in a Pure Currency Economy 39713.6 A Pure Currency Economy with Linear Utility 40113.7 A Pure Credit Economy with Linear Utility 40213.8 An Equilibrium Search Economy 404

14 Laws of Large Numbers

14.1 Definitions and Preliminaries 41614.2 A Strong Law for Markov Processes 425

IV COMPETITIVE EQUILIBRIUM

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Pareto Optima and Competitive Equilibria

15.1 Dual Spaces 44515.2 The First and Second Welfare Theorems 45115.3 Issues in the Choice of a Commodity Space 45815.4 Inner Product Representations of Prices 463

Applications of Equilibrium Theory

16.1 A One-Sector Model of Growth under Certainty16.2 A Many-Sector Model of Stochastic Growth 48116.3 An Economy with Sustained Growth 48516.4 Industry Investment under Uncertainty 48716.5 Truncation: A Generalization 49116.6 A Peculiar Example 49316.7 An Economy with Many Consumers 495

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17 Fixed-Point Arguments

17.1 An Overlapping-Generations Model 50217.2 An Application of the Contraction Mapping

Theorem 508

17.3 The Brouwer Fixed-Point Theorem 51617.4 The Schauder Fixed-Point Theorem 51917.5 Fixed Points of Monotone Operators 52517.6 Partially Observed Shocks 531

18 Equilibria in Systems with Distortions

18.1 An Indirect Approach 54318.2 A Local Approach Based on First-Order

Conditions 54718.3 A Global Approach Based on First-Order

Conditions 554

References

Index of Theorems

General Index

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PA RT I

The Recursive Approach