## Contents

- List of Examples...xi
- Preface...xv
- Chapter 1: Introduction: The Recursive Utility Approach...1
- Chapter 2: Commodity and Price Spaces...29
- Chapter 3: The Representation of Recursive Preferences...63
- Chapter 4: Optimal Growth: Existence and Characterization...110
- Chapter 5: The Statics and Dynamics of Optimal Paths...155
- Chapter 6: Equivalence Principles for Dynamic Equilibrium...213
- Chapter 7: Comparative Dynamics...242
- Chapter 8: Dynamic Competitive Equilibrium...285
- References...316
- Index...334

### Chapter 1: The Recursive Utility Approach

- Introduction...1
- What Is A Recursive Utility Function?...2
- Why Study Recursive Utility?...9
- The Long-Run Incidence of Capital Taxation...9
- The Tax Model
- Tax Incidence with TAS Utility
- Tax Incidence with Epstein-Hynes Utility

- The Impatience Problem...16
- The Impatience Problem with Epstein-Hynes Utility

- The Long-Run Incidence of Capital Taxation...9
- Recursive Utility and Commodity Spaces...18
- Diminishing Returns and Bounded Growth...19
- Nondecreasing Returns and Sustained Growth...21
- Growth and Exogenous Technical Progress
- Endogenous Growth Models

- Order Structures...25
- Weak Separability of the Future from the Present
- Partial Orders on the Commodity Space

- Conclusion...27

### Chapter 2: Commodity and Price Spaces

- Introduction...29
- Commodity Spaces...29
- Order Properties...31
- Free Disposal

- Topological Properties...33
- Metric Spaces
- Continuity
- Compactness and Product Spaces
- Connectedness

- Linear Topologies...42
- Order Convergence
- Semicontinuity
- Contraction Mapping Theorems

- Order Properties...31
- Commodity Price Dualities...49
- Duals and Hyperplanes...52
- Hahn-Banach Theorems...53
- Dual Pairs and Weak Topologies...55
- Order Duals...58
- The Dual of
*l*^{∞}...60

- Conclusion...62

### Chapter 3: Representation of Recursive Preferences

- Introduction...63
- Preference Orders and Utility Theory...64
- Recursive Utility: The Koopmans Axioms...71
- The Axioms...72
- Biconvergence...75
- Recursive Preferences and Additivity...78

- Impatience, Discounting and Myopia...84
- Impatience and Time Perspective...85
- Myopia and the Continuity Axiom...86
- The Norm of Marginal Impatience Conditions...92

- Recursive Utility: The Aggregator...96
- Basic Properties of the Aggregator...97
- The Existence of Recursive Utility...100
- Aggregators Bounded From Below...101
- Unbounded Aggregators...103

- Conclusion...108

### Chapter 4: Existence and Characterization of Optimal Paths

- Introduction...110
- Fundamentals of Existence Theory...110
- A Simple Capital Accumulation Model...112
- The Weierstrass Theorem...114
- One-Sector TAS Existence Theory...115
- Extended Utilitarianism...118

- Multisector Capital Accumulation Models...121
- The von Neumann and Malinvaud Models...123
- The Feasible Correspondence...127

- The Existence and Sensitivity of Optimal Paths...128
- The Maximum Theorem...129
- Optimal Paths...130

- Recursive Dynamic Programming...135
- Dynamic Programming with TAS Utility...135
- Recursive Utility and Multisector Models...138
- Dynamic Programming and Extended Utilitarianism...141

- Characterization of Optimal Paths...143
- No-Arbitrage Conditions...143
- Complete Characterization of Optimal Paths...149

- Conclusion...154

### Chapter 5: Statics and Dynamics of Optimal Paths

- Introduction...155
- One-Sector Models...157
- The Inada Conditions...157
- Stationary States in One-Sector Models...159
- Monotonicity and Turnpikes in TAS Models...162
- Differential Approach
- Nonclassical Models

- Monotonicity and Turnpikes in Recursive Models...167
- Growing Economies...170

- Steady States in Multisectoral Models...173
- Stationary Optimal Programs for Additive Utility...173
- Stationary Optimal Programs for Recursive Utility...176

- Stability of Multisectoral Models...186
- The Undiscounted Model...186
- The Visit Lemma...190
- Uniqueness of Steady States...191
- Local Analysis of Steady States...194
- Local and Global Stability...198

- Cycles and Chaos in Optimal Growth...202
- The Existence of Cycles...203
- Chaotic Dynamics...208

- Conclusion...211

### Chapter 6: Equivalence Principles and Dynamic Equilibria

- Introduction...213
- Equivalence Principles for One-Sector Models...216
- The Perfect Foresight Equivalence Theoremj...217
- Perfect Foresight Competitive Equilibrium
- The PFCE Equivalence Principle

- The Fisher Equivalence Theorem...220
- The Equivalence Theorem and Transversality...222
- Recursive Competitive Equilibrium and Equivalence...225

- The Perfect Foresight Equivalence Theoremj...217
- Multisector Equivalence Principles...228
- The Portfolio Equilibrium Condition...228
- The Two-Sector Equivalence Theorem...229
- The Household Sector
- The Production Sector
- The Transformation Function
- Perfect Foresight Equilibrium
- The Optimal Growth Problem
- The Equivalence Theorem

- Dynamics and the Two-Sector Equivalence Theorem...236

- Transversality and the Hahn Problem...237
- Transversality and Decentralization...241

### Chapter 7: Comparative Dynamics

- Introduction...242
- The Reduced-Form TAS Model...246
- Comparative Dynamics for Monotonic Programs...247
- Comparative Dynamics for Oscillating Programs...254
- Comparative Dynamics and Capital Income Tax Reform...257

- A Primer of Lattice Programming...260
- More About Lattices...261
- An Introduction to Monotone Comparative Statics...264
- Topkis's Theorems...266

- Lattice Programming and the TAS Model...271
- Monotonicity of Optimal Capital Policy Functions...272
- The Capital Deepening Theorem...274

- Recursive Utility Models...277
- Recursive Utility, Monotonicity and Lattice Programming...277
- Increasing Impatience and Recursive Utility...277
- Capital Deepening and Recursive Utility...279

- Conclusion...284

### Chapter 8: Dynamic Competitive Equilibrium

- Introduction...285
- Dynamic Economies...286
- Infinite Horizon Economies...288
- Existence of Pareto Optima...297

- The Core and Edgeworth Equilibria...298
- Existence of Core Allocations...298
- Replicas and Edgeworth Equilibria...299

- The Core and Competitive Equilibrium...301
- Core Equivalence...301
- The Welfare Theorems...308
- Representation of Equilibrium as Welfare Maximum...309

- Models with Very Heterogeneous Discounting...311
- Conclusion...315