ISBN: 3540323082
TITLE: Handbook of Optimal Growth 1
AUTHOR: Dana et al.
TOC: 

1. Optimal Growth Without Discounting
Rose-Anne Dana and Cuong Le Van 1
1.1 Introduction 1
1.2 The Model 2
1.3 Good Programmes 3
1.3.1 Good Programmes 3
1.3.2 Convergence Properties of Good Programmes 5
1.3.3 Existence of Good Programmes from x_0 7
1.4 Optimal and Weakly Optimal Programmes 8
1.4.1 Definition and First Properties 8
1.4.2 Definition and Characterisation 8
1.4.3 Existence of Optimal Programmes 9
1.4.4 Euler Equation 11
1.5 Bellman's Equation, Optimal Policy 12
1.5.1 Bellman's Equation 12
1.5.2 Optimal Policy 14
1.5.3 Examples 15
2. Optimal Growth Models with Discounted Return
Cuong Le Van 19
2.1 Bounded from Below Utility 21
2.1.1 The General Case 21
2.1.2 The Case of a Concave Return Function and a Convex Technology 32
2.1.3 Examples 37
2.2 Unbounded from Below Utility 45
3. Duality Theory in Infinite Horizon Optimization Models
Tapan Mitra 55
3.1 Introduction 55
3.2 A General Intertemporal Allocation Model 57
3.3 Characterization of Optimal Programs in Terms of Dual Variables 58
3.3.1 When Are Competitive Programs Optimal? 59
3.3.2 When Are Optimal Programs Competitive? 60
3.3.3 An Example 63
3.4 Duality Theory for Stationary Optimal Programs 65
3.4.1 Existence of a Stationary Optimal Stock via Duality Theory 65
3.4.2 Quasi-Stationary Price Support for Stationary Optimal Programs 67
3.5 Replacing the Transversality Condition by a Period-by-Period Condition 68
3.6 Are Competitive Programs Optimal? 71
3.7 Duality Theory in the Consumption Model 74
3.7.1 The Model 74
3.7.2 Conversion to the Format of the General Model 76
3.7.3 Characterization of Optimal Plans in Terms of Dual Variables 76
3.7.4 Existence of a Stationary Optimal Output 78
3.8 Weitzman's Theorem an the NNP 79
3.9 Bibliographic Notes 81
4. Rationalizability in Optimal Growth Theory
Gerhard Sorger 85
4.1 Introduction 85
4.2 Problem Formulation 87
4.3 Optimal Policy Functions 90
4.4 Discount Factor Restrictions 99
4.5 Extensions .106
5. On Stationary Optimal Stocks in Optimal Growth Theory: Existence and Uniqueness Results
Tapan Mitra and Kazuo Nishimura 115
5.1 Introduction 115
5.2 Preliminaries .117
5.2.1 Notation 117
5.2.2 The Model 118
5.2.3 Existence of Optimal Programs and the Principle of Optimality 118
5.3 Equivalence of Discounted Golden-Rule and Stationary Optimal Stocks 119
5.4 Existence of Discounted Golden-Rule and Stationary Optimal Stocks 121
5.5 Uniqueness of Non-trivial SOS 124
5.5.1 Description of the Framework 125
5.5.2 A Uniqueness Result Under Normality 126
5.5.3 An Example of Non-uniqueness of SOS 129
5.6 Uniqueness of Interior SOS for Smooth Economies 132
5.7 Bibliographic Remarks 137
6. Optimal Cycles and -Chaos
Tapan Mitra, Kazuo Nishimura, and Gerhard Sorger 141
6.1 Introduction 141
6.2 Basic Definitions and Results 142
6.2.1 Dynamical Systems 143
6.2.2 Optimal Growth Models 145
6.3 Optimal Cycles 147
6 3.1 Monotonic Policy Functions 148
6.3.2 The Role of Discounting 150
6.3.3 Variations on Weitzman's Example 153
6.3.4 Variations on Sutherland's Example 156
6.4 Optimal Chaos 156
6.4.1 Sources of Optimal Chaos 158
6.4.2 Optimal Chaos Under Weak Impatience 163
7. Intertemporal Allocation with a Non-convex Technology
Mukul Majumdar 171
7.1 Introduction 171
7.2 Optimal Allocation in a Closed Economy 174
7.2.1 Production 174
7.2.2 Programs 175
7.2.3 Evaluation Criteria 175
7.3 Characterization of Inefficiency 177
7.4 The Ramsey Problem: Undiscounted Utilities 178
7.4.1 Stationary Programs: The Golden Rule Equilibrium 179
7.4.2 Non-stationary Programs 181
7.5 Mild Discounting: Comparative Dynamics and Stability 185
7.5.1 The Modified Golden Rule 185
7.5.2 Non-stationary Programs 186
7.6 Discounting: The Linear Utility Function 190
7.6.1 An Alternative Interpretation: A Competitive Fishery 190
7.6.2 Characterization of Optimal Programs 191
7.7 A Multi-sector Non-convex Economy: The Undiscounted Case 192
7.8 Optimal Allocation in a Small Open Economy 195
7.8.1 A Small Open Economy with Non-convexity 195
7.8.2 -Interpretation 196
7.8.3 Escape from the Poverty Trap 197
7.9 Some Concluding Comments 198
B. Isotone Recursive Methods: The Case of Homogeneous Agents
Manjira Datta and Kevin L. Reffett 203
8.1 Introduction 203
8.2 Preliminaries 205
8 2.1 Ordered Spaces 205
8.2.2 Mappings 207
8.3 Fixed Point Theory in Ordered Spates 208
8.3.1 Existence 208
8.3.2 Computational Fixed Point Theory 212
8.3.3 Monotone Selections and the Equilibrium Correspondence 214
8.4 An Economy with Classical Technology 215
8.4.1 The Existence of MEDPs 218
8.4.2 The Uniqueness of MEDPs 220
8.4.3 Monotone Comparison Theorems Using Euler Equation Methods 222
8.5 An Economy with Nonclassical Technology 223
8.5.1 The Parameter Spate and Household Decision Problems 225
8.5.2 The Existence of MEDPs 231
8.5.3 Monotone Comparison Theorems via Lattice Programming Methods 231
8.6 An Economy with Elastic Labor Supply 234
8.6.1 The Household Decision and Equilibrium 235
8.6.2 The Existence of Equilibrium 238
8.6.3 The Uniqueness of Equilibrium 241
8.7 Concluding Remarks 244
9. Discrete-Time Recursive Utility
John H. Boyd III 251
9.1 Why Recursive Utility? 251
9.2 Recursive Utility and Aggregators 252
9.2.1 Construction of Recursive Utility from an Aggregator 253
9.3 Existence of Optimal Paths 259
9.4 One-Sector Model with Recursive Utility 260
9.4.1 Dynamic Programming 261
9.5 Optimal Paths in the One-Sector Model 264
9.5.1 The Inada Conditions 265
9.5.2 Monotonicity 266
9.6 Homogeneous Recursive Utility and Sustained Growth 268
10. Indeterminacy in Discrete-Time Infinite-Horizon Models
Kazuo Nishimura and Wain Venditti 273
10.1 Introduction 273
10.2 One-Sector Models 274
10.3 Two-Sector Models with Cobb-Douglas Technologies 277
10.4 Two-Sector Models with CES Technologies 281
10.4.1 Symmetric Elasticities of Substitution 282
10.4.2 Asymmetric Elasticities of Substitution 283
10.5 Extensions with Cobb-Douglas Technologies 286
10.5.1 Partial Depreciation 286
10.5.2 Intersectoral Externalities 289
10.6 Other Formulations 291
10.6.1 Variable Capital Utilization 291
10.6.2 Two-Sector Models with General Technologies 292
11. Theory of Stochastic Optimal Economic Growth
Lars J. Olson and Santanu Roy 297
11.1 Introduction 297
11.2 The Classical Framework 298
11.2.1 The One Sector Classical Model: Basic Properties 298
11.2.2 Stochastic Steady States and Convergence Properties in the One Sector Classical Model 304
11.2.3 Stochastic Steady States and Convergence Properties in the Multisector Classical Model 310
11.3 Extensions of the Classical Framework 312
11.3.1 Sustained (Long Run) Growth 312
11.3.2 Stochastic Growth with Irreversible Investment 313
11.3.3 Stochastic Growth with Experimentation and Learning 314
11.4 Non-Classical Models of Optimal Stochastic Growth 316
11.4.1 Stochastic Growth with Non-convex Technology 316
11.4.2 Stochastic Growth with Stock-Dependent Utility 320
11.5 Comparative Dynamics 322
11.6 Solving the Stochastic Growth Model 324
11.7 Conclusion .325
12. The von Neumann-Gale Growth Model and Its Stochastic Generalization
Igor V. Evstigneev and Klaus R. Schenk-Hoppe 337
12.1 Introduction 337
12.2 The Model and the Main Concepts 339
12.2.1 Basic Definitions 339
12.2.2 Nonlinear von Neumann Models 342
12.2.3 Modeling Financial Markets 344
12.2.4 Stationary Models 346
12.3 Assumptions and Results 348
12.3.1 Assumptions 348
12.3.2 Finite Rapid Paths 349
12.3.3 Infinite Rapid Paths: Existence and Quasi-Optimality 351
12.3.4 Turnpike Theorems 352
12.3.5 The Stationary Case: von Neumann Ray and von Neumann Equilibrium 354
12.3.6 Duality and Reachability 356
12.4 Model Description 357
12.4.1 General (Non-stationary) Model 357
12.4.2 Stationary Model 358
12.4.3 Rapid Paths 360
12.5 Key Assumptions and Results in the Non-stationary Case 361
12.5.1 Assumptions 361
12.5.2 Finite Rapid Paths 363
12.5.3 Quasi-Optimality of Infinite Rapid Paths 364
12.5.4 Turnpike Theorems and Infinite Rapid Paths 365
12.6 Stationary Models: von Neumann Equilibrium 366
12.6.1 Von Neumann Equilibrium 366
12.6.2 The Existence Problem for a von Neumann Equilibrium 366
12.6.3 Randomization 367
12.6.4 Von Neumann Path, Lob Optimal Investments and the Numeraire Portfolio 369
12.7 Stochastic Version of the Perron-Frobenius Theorem and Its Applications: 370
12.7.1 Stochastic Perron-Frobenius Theorem 370
12.7.2 Von Neumann-Gale Systems Defined by Positive Random Matrices 371
12.7.3 Volatility-Induced Financial Growth 372
12.8 Asset Pricing and Hedging 373
12.8.1 Model Description 373
12.8.2 Hedging Problem and Duality 375
12.8.3 An Example: A Currency Market Without Short Sales 376
A The Kuhn-Tucker Theorem 378
B Cones and Separation Theorems 378
C Positive Matrices 379
13. Equilibrium Dynamics with Many Agents
Robert A. Becker 385
13.1 Introduction: Ramsey's Steady State Conjecture 385
13.2 Impatience and the Distribution of Wealth 386
13.2.1 Rae's and Fisher's Time Preference Theories 387
13.2.2 The Solow-Stiglitz Convergence Hypothesis 388
13.2.3 General vs. Temporary Equilibrium 390
13.2.4 Mankiw's Savers-Spenders Model 392
13.3 The Ramsey Equilibrium Model 394
13.3.1 The Basic Model and Blanket Assumptions 394
13.3.2 The Households' Problems 396
13.3.3 The Production Sector's Objective 398
13.3.4 The Ramsey Economy 398
13.3.5 The Equilibrium Concept 399
13.4 Stationary Ramsey Equilibrium Models 400
13.4.1 Heterogeneous Households and Differing Rates of Impatience 400
13.4.2 Stationary Strategic Ramsey Equilibria 406
13.4.3 Heterogeneous Households and Identical Rates of Impatience 407
13.4.4 Stationary Ramsey Equilibria with Flexible Time Preference 408
13.4.5 Uncertainty and Stationary Equilibrium 410
13.4.6 Comments on Ramsey's Conjecture 417
13.5 Ramsey Equilibrium Dynamics 418
13.5.1 Existence of Ramsey Equilibrium and Sufficient Conditions for a Ramsey Equilibrium Path 420
13.5.2 The Recurrence and Turnpike Properties 424
13.5.3 Equilibrium Dynamics with Capital Income Monotonicity 431
13.5.4 Special Ramsey Equilibria 433
13.6 Conclusion .436
14. Dynamic Games in Economics
Rabah Amir 443
14.1 Introduction 443
14.1.1 Purpose and Scope of the Survey 444
14.1.2 Special Features of Economic Applications of Dynamic Games 444
14.1.3 Organization of the Survey 445
14.2 Open-Loop vs Markovian Equilibrium 446
14.2.1 Open-Loop Strategies in Deterministic Dynamic Games 446
14.2.2 Open-Loop Equilibrium in Economic Models 447
14.2.3 Markovian Equilibrium 448
14.2.4 On Open-Loop Versus Markovian Equilibria 449
14.3 Special Classes of Dynamic Games 450
14.3.1 Linear-Quadratic Dynamic Gaines 450
14.3.2 Dynamic Games with Myopic Equilibrium 452
14.4 Common-Property Productive Assets 453
14.4.1 The Beginnings of this Literature 454
14.4.2 General Functional Forms 455
14.5 General Existence Results 456
14.5.1 Existence of Mixed-Strategy Markov Equilibrium 457
14.5.2 Existence of Pure-Strategy Markov Equilibrium 457
14.6 Dynamic Games in Industrial Organization 461
14.6.1 Dynamic Competition with a Fixed Number of Firms 462
14.6.2 Dynamic Competition with Entry and Exit 462
14.6.3 Empirical and Computational Work an Industry Dynamics 463
14.7 Dynamic Games of Perfect Information 464
14.7.1 Games of Strategic Bequests 464
14.7.2 A Class of Games with Alternating Moves 465
14.7.3 General Existence Results 466
14.8 Dynamic Games with a Continuum of Players 466
14.9 Computational Methods 467
14.10 Experimental Research on Dynamic Games 468
14.11 Appendix 469
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