Amsterdam «Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo North-Holland is an imprint of Elsevier

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1 HANDBOOK OF COMPUTATIONAL ECONOMICS KARL SCHMEDDERS KENNETH L. JUDD ELSEVIER Amsterdam «Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo North-Holland is an imprint of Elsevier

2 CONTENTS Contributors Acknowledgments Introduction to the Series Introduction for Volume 3 ofthe Handbook ofcomputational Economics ix xi xiii xv 1. Learning About Learning in Dynamic Economic Models 1 David A. Kendrick, Hans M. Amman, and Marco P. Tucci 1. Introduction 2 2. The Framework 3 3. What We Have Learned 6 4. What We Hope to Learn Algorithms and Codes A Showcase on Active Learning Learning with Forward Looking Variables Other Applications of Active Learning Summary 31 References On the Numerical Solution of Equilibria in Auction Models with Asymmetries within the Private Values Paradigm 37 Timothy P. Hubbard and Harry J. Paarsch 1. Motivation and Introduction Theoretical Model Primer on Relevant Numerical Strategie: Previous Research Concerning Numerical Solutions Some Examples Comparisons of Relative Performance and Potential Improvements Summary and Conclusions 112 Acknowledgments 112 References Analyzing Fiscal Policies in a Heterogeneous-Agent Overlapping-Generations Economy 117 Shinichi Nishiyama and Kent Smetters 1. Introduction Existing Literature 119 v

3 vi Contents 3. Stylized Model Economy 4. Computational Algorithm Calibration to the US Economy ^8 6. Policy Experiments Concluding Remarks 156 References On Formulating and Solving Portfolio Decision and Asset Pricing Problems 161 Yu Chen, Thomas F. Cosimano, and Alex A. Himonas 1. Introduction Discrete Time Portfolio Decision Making Discrete Time Asset Pricing Continuous Time Portfolio Decision Problem Continuous Time Asset Pricing Conclusion 218 Acknowledgments 219 References Computational Methods for Derivatives with Early Exercise Features 225 Carl Chiarella, Boda Kang, Gunter Meyer, and Andrew Ziogas 1. General Introduction The Problem Statement In the Gase of Stochastic Volatility and Poisson Jump Dynamics American Call Options Under Jump-Diffusion Processes American Call Options under Jump-Diffusion and Stochastic Volatility Processes Conclusion 273 References Solving and Simulating Models with Heterogeneous Agents and Aggregate Uncertainty 277 Yann Algan, Olivier Allais, Wouter J. Den Haan, and Pontus Rendahl 1. Introduction Example Economy Algorithms Overview Models with Nontrivial Market Clearing Approximate Aggregation Simulation with a Continuum of Agents Accuracy ^ 8. Comparison ^3 9. Other Types of Heterogeneity Concluding Comments,,0

4 Contents vii A. Explicit Aggregation and Perturbation Techniques 319 Acknowledgments 322 References Numerical Methods for Large-Scale Dynamic Economic Models 325 Lilia Mailar and Serguei Mailar 1. Introduction Literature Review The Chapter at a Glance Nonproduct Approaches to Representing, Approximating, and Interpolating Functions Approximation of Integrals Derivative-Free Optimization Methods Dynamic Programming Methods for High-Dimensional Problems Precomputation Techniques Local (Perturbation) Methods Parallel Computation Numerical Analysis ofa High-Dimensional Model Numerical Results for the Multicountry Model Conclusion 469 Acknowledgments 469 References Advances in Numerical Dynamic Programming and New Applications 479 Yongyang Cai and Kenneth L. Judd 1. Introduction Theoretical Challenges Numerical Methods for Dynamic Programming Tools from Numerical Analysis Shape-Preserving Dynamic Programming Parallelization Dynamic Portfolio Optimization with Transaction Costs Dynamic Stochastic Integration of Climate and Economy Conclusions 514 Acknowledgments 515 References Analysis of Numerical Errors 517 Adrian Peralta-Alva and Manuel S. Santos 1. Introduction Dynamic Stochastic Economies Numerical Solution of Simple Markov Equilibria 525

5 viii Contents 4. Recursive Methods for Non-Optimal Economies Numerical Experiments Concluding Remarks 553 References GPU Computing in Economics 557 Eric M. Aldrich 1. Introduction Basics ofgpgpu Computing A Simple GPGPU Example Example: Value Function Iteration Example: A General Equilibrium Asset Pricing Model with Fleterogeneous Beliefs The Road Ahead Conclusion 596 References Computing All Solutions to Polynomial Equations in Economics 599 Felix Kubler, Philipp Renner, and Karl Schmedders 1. Introduction Gröbner Bases and Polynomial Equations Applying Gröbner Bases to Economic Models All-Solution Homotopy Methods Applying Flomotopy Methods Conclusion 651 Acknowledgments 651 References 651 Wey 653