Contents

Chapter 1: The Recursive Utility Approach

  1. Introduction...1
  2. What Is A Recursive Utility Function?...2
  3. Why Study Recursive Utility?...9
    1. The Long-Run Incidence of Capital Taxation...9
      • The Tax Model
      • Tax Incidence with TAS Utility
      • Tax Incidence with Epstein-Hynes Utility
    2. The Impatience Problem...16
      • The Impatience Problem with Epstein-Hynes Utility
  4. Recursive Utility and Commodity Spaces...18
    1. Diminishing Returns and Bounded Growth...19
    2. Nondecreasing Returns and Sustained Growth...21
      • Growth and Exogenous Technical Progress
      • Endogenous Growth Models
    3. Order Structures...25
      • Weak Separability of the Future from the Present
      • Partial Orders on the Commodity Space
  5. Conclusion...27

Chapter 2: Commodity and Price Spaces

  1. Introduction...29
  2. Commodity Spaces...29
    1. Order Properties...31
      • Free Disposal
    2. Topological Properties...33
      • Metric Spaces
      • Continuity
      • Compactness and Product Spaces
      • Connectedness
    3. Linear Topologies...42
      • Order Convergence
      • Semicontinuity
      • Contraction Mapping Theorems
  3. Commodity Price Dualities...49
    1. Duals and Hyperplanes...52
    2. Hahn-Banach Theorems...53
    3. Dual Pairs and Weak Topologies...55
    4. Order Duals...58
    5. The Dual of l∞...60
  4. Conclusion...62

Chapter 3: Representation of Recursive Preferences

  1. Introduction...63
  2. Preference Orders and Utility Theory...64
  3. Recursive Utility: The Koopmans Axioms...71
    1. The Axioms...72
    2. Biconvergence...75
    3. Recursive Preferences and Additivity...78
  4. Impatience, Discounting and Myopia...84
    1. Impatience and Time Perspective...85
    2. Myopia and the Continuity Axiom...86
    3. The Norm of Marginal Impatience Conditions...92
  5. Recursive Utility: The Aggregator...96
    1. Basic Properties of the Aggregator...97
    2. The Existence of Recursive Utility...100
    3. Aggregators Bounded From Below...101
    4. Unbounded Aggregators...103
  6. Conclusion...108

Chapter 4: Existence and Characterization of Optimal Paths

  1. Introduction...110
  2. Fundamentals of Existence Theory...110
    1. A Simple Capital Accumulation Model...112
    2. The Weierstrass Theorem...114
    3. One-Sector TAS Existence Theory...115
    4. Extended Utilitarianism...118
  3. Multisector Capital Accumulation Models...121
    1. The von Neumann and Malinvaud Models...123
    2. The Feasible Correspondence...127
  4. The Existence and Sensitivity of Optimal Paths...128
    1. The Maximum Theorem...129
    2. Optimal Paths...130
  5. Recursive Dynamic Programming...135
    1. Dynamic Programming with TAS Utility...135
    2. Recursive Utility and Multisector Models...138
    3. Dynamic Programming and Extended Utilitarianism...141
  6. Characterization of Optimal Paths...143
    1. No-Arbitrage Conditions...143
    2. Complete Characterization of Optimal Paths...149
  7. Conclusion...154

Chapter 5: Statics and Dynamics of Optimal Paths

  1. Introduction...155
  2. One-Sector Models...157
    1. The Inada Conditions...157
    2. Stationary States in One-Sector Models...159
    3. Monotonicity and Turnpikes in TAS Models...162
      • Differential Approach
      • Nonclassical Models
    4. Monotonicity and Turnpikes in Recursive Models...167
    5. Growing Economies...170
  3. Steady States in Multisectoral Models...173
    1. Stationary Optimal Programs for Additive Utility...173
    2. Stationary Optimal Programs for Recursive Utility...176
  4. Stability of Multisectoral Models...186
    1. The Undiscounted Model...186
    2. The Visit Lemma...190
    3. Uniqueness of Steady States...191
    4. Local Analysis of Steady States...194
    5. Local and Global Stability...198
  5. Cycles and Chaos in Optimal Growth...202
    1. The Existence of Cycles...203
    2. Chaotic Dynamics...208
  6. Conclusion...211

Chapter 6: Equivalence Principles and Dynamic Equilibria

  1. Introduction...213
  2. Equivalence Principles for One-Sector Models...216
    1. The Perfect Foresight Equivalence Theoremj...217
      • Perfect Foresight Competitive Equilibrium
      • The PFCE Equivalence Principle
    2. The Fisher Equivalence Theorem...220
    3. The Equivalence Theorem and Transversality...222
    4. Recursive Competitive Equilibrium and Equivalence...225
  3. Multisector Equivalence Principles...228
    1. The Portfolio Equilibrium Condition...228
    2. 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
    3. Dynamics and the Two-Sector Equivalence Theorem...236
  4. Transversality and the Hahn Problem...237
  5. Transversality and Decentralization...241

Chapter 7: Comparative Dynamics

  1. Introduction...242
  2. The Reduced-Form TAS Model...246
    1. Comparative Dynamics for Monotonic Programs...247
    2. Comparative Dynamics for Oscillating Programs...254
    3. Comparative Dynamics and Capital Income Tax Reform...257
  3. A Primer of Lattice Programming...260
    1. More About Lattices...261
    2. An Introduction to Monotone Comparative Statics...264
    3. Topkis's Theorems...266
  4. Lattice Programming and the TAS Model...271
    1. Monotonicity of Optimal Capital Policy Functions...272
    2. The Capital Deepening Theorem...274
  5. Recursive Utility Models...277
    1. Recursive Utility, Monotonicity and Lattice Programming...277
    2. Increasing Impatience and Recursive Utility...277
    3. Capital Deepening and Recursive Utility...279
  6. Conclusion...284

Chapter 8: Dynamic Competitive Equilibrium

  1. Introduction...285
  2. Dynamic Economies...286
    1. Infinite Horizon Economies...288
    2. Existence of Pareto Optima...297
  3. The Core and Edgeworth Equilibria...298
    1. Existence of Core Allocations...298
    2. Replicas and Edgeworth Equilibria...299
  4. The Core and Competitive Equilibrium...301
    1. Core Equivalence...301
    2. The Welfare Theorems...308
    3. Representation of Equilibrium as Welfare Maximum...309
  5. Models with Very Heterogeneous Discounting...311
  6. Conclusion...315