HANDBOOK OF COMPUTATIONAL ECONOMICS

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HANDBOOK OF COMPUTATIONAL

ECONOMICS

VOLUME 1

Edited by

HANS M. AMMAN University of Amsterdam

DAVID A. KENDRICK

University of Texas

and

JOHN RUST University of Wisconsin

NORTH HOLLAND

Amsterdam • Boston • Heidelberg • London • New York • Oxford

Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo

CONTENTS OF VOLUME I

Introduction to the Series v

Preface to the Handbook vii

PART 1: ECONOMIC TOPICS

Chapter I

Computable General Equilibrium Modelling for Policy Analysis and Forecasting PETER B. DIXON and B.R. PARMENTER 3

1. Introduction 4 1.1. Definition 5

1.2. Brief history 6

2. Solving a CGE model 9 2.1. The programming approach 10

2.2. The derivative approach: The Johansen/Euler method 12

2.3. Solving a multi-period model 24

3. An illustrative CGE model 36 3.1. Input-output database 37

3.2. Equations 39

3.3. Coefficients, parameters, zero problems and initial Solution 48

3.4. Closure of the illustrative model 53

3.5. Simulations 55

4. Concluding remarks: Success, partial success and potential of CGE modelling 67 4.1. Success: Quantifying linkages between different parts of the economy 67

4.2. Partial success: Analysis of welfare effects 69

4.3. Potential: Disaggregated forecasting 76

References 79

Chapter 2

Computation of Equilibria in Finite Games RICHARD D. McKELVEY and ANDREW McLENNAN 87

1. Introduction 88 2. Notation and problem Statement 90

3. Computing a sample equilibrium 3.1. Two-person games: The Lemke-Howson algorithm

3.2. ./V-person games: Simplicial subdivision

3.3. Non-globally convergent methods

4. Extensive form games 4.1. Notation

4.2. Extensive versus normal form

4.3. Computing sequential equilibria

5. Equilibrium refinements 5.1. Two-person games

5.2. /V-person games

6. Finding all equilibria 6.1. Feasibility

6.2. Exemplary algorithms for semi-algebraic sets

6.3. Complexity of finding game theoretic equilibria

7. Practical computational issues 7.1. Software

7.2. Computational complexity

References

Chapter 3

Computational Methods for Macroeconometric Models Ray C. FAIR

1. Introduction 2. Notation 3. Two stage least Squares 4. 3SLS and FIML 5. Two stage least absolute deviations 6. The Gauss-Seidel technique 7. Stochastic Simulation

7.1. Numerical procedures for drawing values

8. Optimal control 8.1. Stochastic Simulation Option

9. Asymptotic distribution accuracy 10. Solution and FIML estimation of RE modeis

10.1. Introduction

10.2. The Solution method

10.3. Computational costs

10.4. FIML estimation

10.5. Stochastic Simulation

10.6. Conclusion

References

Contents of Volume I xiii

Chapter 4

Mechanics of Forming and Estimating Dynamic Linear Economies EVAN W. ANDERSON, LARS PETER HANSEN, ELLEN R. McGRATTAN and THOMAS J. SARGENT 171

1. Introduction 173 2. Control problems 173

2.1. Deterministic regulator problem 174

2.2. Augmented regulator problem 176

2.3. Discounted stochastic regulator problem 177

2.4. A class of linear-quadratic economies 180

3. Solving the deterministic linear regulator problem 182 3.1. Nonsingular Ayy 184

3.2. Singular Ayy 187

3.3. Continuous-üme Systems 189

4. Computational techniques for solving Riccati equations 192 4.1. Schur algorithm 192

4.2. Doubling algorithm 194

4.3. Matrix sign algorithm 200

5. Solving the augmented regulator problem 202 6. Computational techniques for solving Sylvester equations 205

6.1. The Hessenberg-Schur algorithm 205

6.2. Doubling algorithm 207

7. Distorted economies 208 8. Example economies 210

8.1. A model of permanent income with habit persistence 2 1 0

8.2. A model of education 212

8.3. A model of cattle cycles 215

9. Numerical comparisons 218 9.1. Solutions to Riccati equations 219

9.2. Solutions to Sylvester equations 223

10. Innovat ions representa t ions 224

10.1. Wold and autoregressive representations 226

11. The likelihood function 227 12. Estimating the cattle cycles model 228 Appendix A. Computing dL/d6 and 3Lt/3c9 for a state-space model 232

A.l. The formula for 3L/36 232

A.2. Derivation of the formula 235

A.3. Standard errors 242

Appendix B. Differentiating the state-space model with respect to economic parameters 242 B. 1. A linear-quadratic economy without distortions 242

XIV Contents of Volume I

B.2. A nonlinear economy without distortions 244

B.3. A linear-quadratic economy with distortions 2 4 6

References 250

Chapter 5

Nonlinear Pricing and Mechanism Design ROBERT WILSON 253

0. Introduction 255 1. Mirrlees' formulation 255

1.1. Statement of the nonlinear pricing problem 255

1.2. The incomplete problem 2 5 7

1.3. Auxiliary constraints 258

1.4. Statement of the incomplete problem 258

1.5. Necessary conditions for a Solution 259

1.6. Examples 261

1.7. Lessons from a discrete-types formulation 265

2. Computational methods 266 2.1. Direct optimization 267

2.2. Approximation via Fourier series 268

2.3. Introduction to finite-difference methods 269

2.4. Relaxation combined with Newton's method 270

2.5. The pure relaxation algorithm 2 7 7

2.6. Other boundary shapes 278

2.7. Higher dimensions 279

2.8. Nonlinear equations 2 8 0

2.9. Construction of the price schedules and the tariff 281

2.10. An alternative Version 283

3. The complete problem 284 4. A mechanism design formulation 285 5. Summary and conclusions 288 Appendix A. Pseudo-codes 290 Appendix B. APL programs 292 References 292

Chapter 6

Sectoral Economics DAVID A. KENDRICK 295

1. Introduction 296 2. Methods 297

2.1. Activity analysis 298

2.2. Location 304

2.3. Economies of scale 307

Contents of Volume I

3. Software 4. Process industries

4.1. Small static

4.2. Large static

4.3. Small dynamic

4.4. Examples of sectoral modeis

5. The Computer industry 6. Energy 7. Environment 8. Agriculture 9. Linkages to computable general equilibrium and growth modeis 10. Limitations 11. Conclusions References

310 314 314 316 317 318 320 322 324 326 327 327 328 328

PART 2: COMPUTER SCIENCE TOPICS

Chapter 7

Parallel Computation ANNA NAGURNEY 335

1. Introduction 336 2. Technology for parallel computation 341

2.1. Parallel architectures 341

2.2. Parallel programming languages and Compilers 348

2.3. Computer science issues in parallel algorithm development 350

3. Fundamental problem classes and numerical methods 354 3.1. Problem classes 355

3.2. Algorithms 361

4. Applications and numerical results 375 4.1. Nonlinear equations 375

4.2. Optimization problems 376

4.3. Parallel computation of variational inequality problems 384

4.4. Parallel computation of dynamical Systems 393

Acknowledgements 400 References 401

Chapter 8

Artificial Intelligence in Economics and Finance: A State of the Art - 1994 The real estate price and assets and liability analysis case L.F. PAU and TAN, PAN YONG 405

1. Introduction 407

xvi Contents of Volume I

1.1. Historical perspective 407

1.2. Plan of the chapter 408

2. Motivations for the use of AI in economies and finance 409 3. Real estate pricing and lending 411

3.1. Problem analysis and adaptation of AI techniques to decision making goals 411

3.2. Short definitions of main generic approaches 4 1 3

4. Generic tasks 415 5. Conventional AI approaches 416

5.1. Knowledge-based Systems 4 1 7

5.2. Natural language processing 4 1 8

5.3. Qualitative Simulation 419

6. Machine learning approaches 420 6.1. Introduction to alternative machine learning approaches 420

6.2. Tree induetion: ID3 422

6.3. Unsupervized learning: Conceptual clustering 424

6.4. Neural processing 425

7. Case-based reasoning in economies and finance 427 7.1. Introduction 4 2 7

7.2. Implementation of case-based reasoning in economies 428

7.3. Potential applications of CBR in economies and finance 4 3 2

8. Conclusions: Areas for further research 433 Terminology 434 References 437

Chapter 9

Neural Networks for Encoding and Adapting in Dynamic Economies I.-KOO CHO and THOMAS J. SARGENT 441

1. Introduction 443 2. Discriminant funetions 444 3. The pereeptron 445 4. Feedforward neural networks with hidden Units 446 5. Classifier Systems 448

5.1. The brain as an aecounting System 448

5.2. Generality versus discrimination 4 5 0

6. Economic applications 450 6.1. Arthur's artificial bandit 451

6.2. Prisoner's dilemma 452

6.3. Representation 454

6.4. More general games 455

6.5. Capital and moral hazard without discounting 457

6.6. A model 458

6.7. Linear strategies 4 6 0

Contents of Volume I xvii

7. Conclus ions and open quest ions 4 6 2

8. Append ix 463

8.1. Analysis of Theorem 2 4 6 3

References 469

Chapter 10

Modeling Languages in Computational Economics: GAMS STAVROS A. ZENIOS 471

1. Introduction 472 2. Overview of the GAMS modeling language 473

2.1. Domains of parameters and variables 474

2.2. The GAMS arithmetic Operations 475

2.3. The GAMS summation and product Operators 475

2.4. Data entry and manipulations 475

2.5. The GAMS relational Operators 477

2.6. Declaration and definition of equations 477

2.7. Exception handling capabilities 478

2.8. GAMS solvers 478

2.9. The GAMS libraries of economic and financial modeis 479

3. Example applications 479 3.1. A simple transportation model 480

3.2. Asset allocation model 482

3.3. The SAMBAL System: Estimating Social Accounting Matrices 484

References 488

489

490 490 490 491 491 492 493 493 495 496 496 496 496 498

Chapter u Mathematica for Economists HALR

1. 2. 3. 4.

5. 6. 7.

VARIAN

Introduction Design of Mathematica The front end Programming 4.1. An example

4.2. Defining functions

4.3. Programming constructs

4.4. Pattern matching

4.5. Expressions

Packages MathSource Applications in economics 7.1. Comparative statics

7.2. Dynamic programming

7.3.

7.4.

7.5.

7.6.

Nash equilibria

Econometrics and statistics

Graphics

Teaching

8. Summary References

Contents of Volume 1

499 499 502 504 505 505

PART 3: NUMERICAL METHODS

Chapter 12

Approximation, Perturbation, and Projection Methods in Economic Analysis KENNETH L. JUDD 509

1. Introduction 511 2. The uses of approximation ideas: An overview 513 3. The mathematical foundations of regulär perturbation methods 515

3.1. The meaning of "approximation" 515

3.2. Taylor series approximation 516

3.3. Rational approximation 516

3.4. Implicit function theorem 5 1 7

3.5. Generalizations to function Spaces 517

4. Applications of regulär perturbation methods to economics 519 4.1. Comparative statics: A simple rule of thumb in tax theory 520

4.2. Comparative dynamics: A canonical problem 520

4.3. Perturbing dynamic equilibria 521

4.4. The stable manifold theorem and applications to economic theory 524

4.5. Perturbing functional equations from recursive equilibrium analyses 5 2 7

5. Bifurcation methods 540 5.1. Applications of the Hopf bifurcation to dynamic economic theory 541

5.2. Gauge functions 542

5.3. Bifurcation applications to stochastic modelling 542

6. Asymptotic expansions of integrals 545 6.1. Econometric applications of asymptotic methods 547

6.2. Theoretical applications of Laplace's method 5 4 7

7. The mathematics of Lp approximations 548 7.1. Orthogonal polynomials 548

7.2. Least-squares orthogonal polynomial approximation 549

7.3. Interpolation 551

7.4. Approximation through interpolation 552

7.5. Approximation through regression 554

7.6. Piecewise polynomial interpolation 554

Contents of Volume I xix

7.7. Shape-preserving Interpolation 556

7.8. Multidimensional approximation 557

8. Applications of approximation to dynamic programming 560 8.1. Discretization methods 562

8.2. Multilinear approximation 563

8.3. Polynomial approximations 563

9. Projection methods 563 9.1. General projection algorithm 565

10. Applications of projection methods to rational expectations modeis 569 10.1. Discrete-time deterministic optimal growth 569

10.2. Stochastic optimal growth 572

10.3. Problems with inequality constraints 574

10.4. Dynamic games 574

10.5. Continuous time problems 575

10.6. Models with asymmetric information 575

10.7. Convergence properties and accuracy of projection methods 577

11. Hybrid perturbation-projeetion method 578 12. Conclusions 580 References 581

Chapter 13

Numerical Methods for Linear-Quadratic Models HANS AMMAN 587

1. Introduction 588 2. Deterministic optimization 588

2.1. Linear quadratic control: Discrete time 588

2.2. Linear quadratic control: Continuous time 602

3. Stochastic control 604 3.1. Stochastic linear quadratic control: Discrete time 604

3.2. Stochastic linear quadratic control: Continuous time 613

4. Summary 614 A note on Software 615 References 615

Chapter 14

Numerical Dynamic Programming in Economics JOHN RUST 619

1. Introduction 620 2. MDPs and the theory of dynamic programming: A brief review 632

2.1. Definitions of MDPs, DDP's and CDP's 632

2.2. Bellman's equation, contraction mappings, and Blackwell's theorem 633

xx Contents of Volume I

2.3. Examples of analytic Solutions to Bellman's equation

for specific "Test Problems" 636

3. Computational complexity and optimal algorithms 639 3.1. Discrete computational complexity 640

3.2. Continuous computational complexity 641

4. Numerical Solution methods for general MDPs 648 4.1. Discrete finite horizon MDPs 6 4 9

4.2. Discrete infinite horizon MDPs 652

4.3. Continuous finite horizon MDPs 669

4.4. Continuous infinite horizon MDPs 697

5. Conclusion 717 References 722

Chapter 15

Monte Carlo Simulation and Numerical Integration JOHNGEWEKE 731

1. Introduction 733 2. Deterministic methods of integration 734

2.1. Unidimensional quadrature 734

2.2. Multidimensional quadrature 735

2.3. Low discrepancy methods 738

2.4. Other deterministic methods 741

3. Pseudorandom number generation 742 3.1. Uniform pseudorandom number generation 743

3.2. General methods for nonuniform distributions 745

3.3. Selected univariate distributions 752

3.4. Selected multivariate distributions 754

4. Independence Monte Carlo 756 4.1. Simple Monte Carlo 757

4.2. Acceptance methods 759

4.3. Importance sampling 761

4.4. A note on the choice of method 764

5. Variance reduction 769 5.1. Antithetic Monte Carlo 770

5.2. Systematic sampling 772

5.3. The use of conditional expectations 773

5.4. Control variables 774

6. Markov chain Monte Carlo methods 775 6.1. Two Markov chain Monte Carlo algorithms 777

6.2. Mathematical background 779

6.3. Convergence of the Gibbs sampler 781

Contents of Volume I xxi

6.4. Convergence of the Metropolis-Hastings algorithm 784

6.5. Assessing convergence and numerical accuracy 785

7. Some examples 788 7.1. Stochastic volatility 788

7.2. Integration and optimization 793

References 797

Index 801