Table Of Content

Economic Dynamics Theory and Computation Second Edition 1 Economic Dynamics Theory and Computation Second Edition John Stachurski The MIT Press Cambridge, Massachusetts London, England 2 © 2022 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. This book was typeset in LATEX by the author. Library of Congress Cataloging-in-Publication Data. Names: Stachurski, John, 1969-author. Title: Economic dynamics : theory and computation / John Stachurski. Description: Second Edition. | Cambridge, Massachusetts : The MIT Press, [2022] | Revised edition of the author's Economic dynamics, c2009. | Includes bibliographical references and index. Identifiers: LCCN 2021061989 (print) | LCCN 2021061990 (ebook) | ISBN 9780262544771 (paperback) | ISBN 9780262372442 (epub) | ISBN 9780262372459 (pdf) Subjects: LCSH: Statics and dynamics (Social sciences)–Mathematical models. | Economics– Mathematical models. Classification: LCC HB145 .S73 2022 (print) | LCC HB145 (ebook) | DDC 330.1/519– dc23/eng/20220214 LC record available at https://lccn.loc.gov/2021061989 LC ebook record available at https://lccn.loc.gov/2021061990 10 9 8 7 6 5 4 3 2 1 d_r0 3 To Cleo 4 Contents Preface Common Symbols Part I: Introduction to Dynamics Chapter 1: Introduction Chapter 2: Programming Chapter 3: Analysis in Metric Space Chapter 4: Introduction to Dynamics Chapter 5: Further Topics for Finite MCs Chapter 6: Infinite State Space Part II: Advanced Techniques Chapter 7: Integration Chapter 8: Density Markov Chains Chapter 9: Measure-Theoretic Probability Chapter 10: Stochastic Dynamic Programming Chapter 11: Stochastic Dynamics Chapter 12: More Stochastic Dynamic Programming Part III: Appendixes Appendix A: Real Analysis Appendix B: Chapter Appendixes Bibliography Index List of figures Chapter 1 Figure 1.1 A simple Markov model Figure 1.2 Distribution of population after 100 periods Figure 1.3 No route out of poverty Figure 1.4 Fraction of time spent in each state by a single household Figure 1.5 A spin configuration in the plane (light = −1 and dark = +1) Figure 1.6 An initial configuration of households Figure 1.7 An absorbing state with significant segregation Figure 1.8 Another initial configuration of households Figure 1.9 The result of iteration with low-level mixing Figure 1.10 Sequence of marginal distributions Figure 1.11 Stationary distribution Figure 1.12 Time series 5 Figure 1.13 Sample mean of time series Chapter 2 Figure 2.1 Partition (I(x))x∈S created by ϕ Chapter 3 Figure 3.1 Limit of a sequence Figure 3.2 An ϵ-ball for d∞ Figure 3.3 Fixed points in one dimension Chapter 4 Figure 4.1 The result of mapping x ↦ h(x) for a grid of x values Figure 4.2 Global stability Figure 4.3 45 degree diagram Figure 4.4 Threshold externalities Figure 4.5 Trajectory of the quadratic map Figure 4.6 Quadratic maps, r ∈ [0, 4] Figure 4.7 Bifurcation diagram Figure 4.8 The unit simplex with N = 3 Figure 4.9 Finite Markov chain Figure 4.10 Simulation of the Hamilton Markov chain Figure 4.11 Top: X0 = 0. Bottom: X0 = 4 Figure 4.12 Top: X0 = 1. Bottom: X0 = 4 Figure 4.13 Trajectories of(P(S), MH) Chapter 5 Figure 5.1 The value function computed via value function iteration Figure 5.2 Stationary wealth distribution ψ* Figure 5.3 Optimal policy as computed by policy iteration Figure 5.4 Best response dynamics Figure 5.5 Fraction of time spent at N as ϵ → 0 Chapter 6 Figure 6.1 Time series plot Figure 6.2 The map g provides smooth transition between two affine functions Figure 6.3 Sampling from the marginal distribution Figure 6.4 Expected steady state consumption as a function of the savings rate Figure 6.5 Empirical distribution functions for the Solow model Figure 6.6 Persistence in time series Figure 6.7 Persistence in time series Figure 6.8 Nonparametric kernel density estimates of normal sample Figure 6.9 Sequence of marginal densities for the STAR model Figure 6.10 Stationary look-ahead estimator, nonconvex growth model Figure 6.11 The map := L ◦ T Figure 6.12 Approximation via linear interpolation 6 Figure 6.13 FVI algorithm iterates Figure 6.14 Approximate optimal policy Figure 6.15 Densities of the asset process Figure 6.16 The trajectory Tn P and an (approximate) fixed pointp* Chapter 7 Figure 7.1 The measure of a rectangle I ⊂ ℝ2 Figure 7.2 A covering of A by elements of ℐ Figure 7.3 A simple function on the plane Figure 7.4 A measurable function Figure 7.5 Image measure Figure 7.6 Equivalence classes for (ℝ2, ρ) Chapter 8 Figure 8.1 The stochastic kernel p(x, y)dy = N(ax + b, 1) Figure 8.2 Time series generated by the Gaussian stochastic kernel Figure 8.3 Two time series Figure 8.4 Divergence to +∞ Figure 8.5 Nondiverging sequence Figure 8.6 Multiple invariant sets Figure 8.7 Dobrushin coefficient is zero Chapter 9 Figure 9.1 Correlated shocks, α = −0.9, ρ = −0.9, Wt ≡ 0 Figure 9.2 Correlated shocks, α = 0.9, ρ = 0.9, Wt ∼ N(0, 0.25) Figure 9.3 Correlated shocks, α = 0.9, ρ = 0.0, Wt ∼ N(0, 0.25) Chapter 10 Figure 10.1 Stochastic dynamic programming Chapter 11 Figure 11.1 The function θ(w) Figure 11.2 Determination of kt+1 Figure 11.3 Estimates of F* for different λ Figure 11.4 Coupling of(Xt)t≥0 and at t = 5 Figure 11.5 The process (XL)t≥0 7 Preface Aims and Scope The aim of this book is to teach foundational topics in stochastic dynamics such as stability, ergodicity, and dynamic programming, with applications from economics and finance. As we travel down this path, we will delve into a variety of related fields, including simulation and numerical methods, fixed point theory, stochastic process theory, function approximation, and coupling. In writing the book I had two main goals. First, I wanted to show that sound understanding of relevant mathematical concepts leads to effective algorithms for solving real world problems. Second, I wanted the book to be enjoyable to read, with an emphasis on building intuition. Hence the material is driven by examples—I believe the fastest way to grasp a new concept is through studying examples—and makes extensive use of programming to illustrate ideas. Running simulations and computing equilibria helps bring abstract concepts to life. The primary intended audience is advanced undergraduate and, especially, beginning graduate students in economics. However, the techniques discussed in the second half of the book add some shiny new toys to the standard tool kit used for economic modeling, and as such they should be of interest to advanced graduate students and researchers. The book is as self-contained as possible, given space constraints. Part I of the book covers material that all well-rounded graduate students should know. The style is relatively mathematical, and those who find the going hard might start by working through the exercises in appendix A. Part II is significantly more challenging. In designing the text it was not my intention that all of those who read part I should go on to read part II. Rather, part II is written for researchers and graduate students with a particular interest in technical problems. Those who do read the majority of part II will gain a very strong understanding of infinite-horizon dynamic programming and (nonlinear) stochastic models. How does this book differ from other textbooks? There are several books on computational macroeconomics and macrodynamics that treat related topics. In comparison, this book is not specific to macroeconomics. It should be of interest to (at least some) people working in microeconomics, operations research, and finance. Second, computation and theory are tightly integrated. When numerical methods are discussed, I have tried to emphasize mathematical analysis of the algorithms. Readers will acquire a strong knowledge of the probabilistic and function-analytic framework that underlies proposed solutions. Like any text containing a significant amount of mathematics, the notation piles up thick and fast. To aid readers I have worked hard to keep notation minimal and consistent. Uppercase symbols such as A and B usually refer to sets, while lowercase symbols such as x and y are elements of these sets. Functions use uppercase and lowercase symbols such as f , g, F, and G. Calligraphic letters such as and ℬ represent sets of sets or, occasionally, sets of functions. Proofs end with the symbol □. I provide a table of common symbols on page xvii. Furthermore, the index begins with an extensive list of symbols, along with the number of the page on which they are defined. Solutions, Code, and Online Resources Solutions to exercises, code, and online resources for the textbook can be found at https://johnstachurski.net/edtc.html Solutions to most of the exercises are collected in a PDF that's freely available to all readers. You will also find an online code book that accompanies this text, created using Jupyter Book. 8 The code book contains Python code that generates the figures and runs computations discussed herein. Solutions to exercises involving computation are also included in the code book.1 Additional related code can be found at https://quantecon.org. The code there includes Python and Julia implementations of algorithms discussed in this text, such as routines for simulation of Markov chains and solution of Markov decision processes. Acknowledgements In the process of writing this book I received invaluable comments and suggestions from my colleagues. In particular, I wish to thank Martin Barbie, Robert Becker, Toni Braun, Roger Farmer, Onésimo Hernández-Lerma, Timothy Kam, Takashi Kamihigashi, Noritaka Kudoh, Vance Martin, Sean Meyn, Len Mirman, Tomoyuki Nakajima, Kevin Reffett, Ricard Torres, and Yiannis Vailakis. Many graduate students also contributed to the end product, including Real Arai, Yu Awaya, Yiyong Cai, Chase Coleman, Rosanna Chan, Katsuhiko Hori, Shu Hu, Saya Ikegaway, Spencer Lyon, Qingyin Ma, Dimitris Mavridis, Murali Neettukattil, Flint O’Neil, Jenö Pál, Guanlong Ren, Natasha Watkins, Katsunori Yamada and Junnan Zhang. Regarding the second edition, I particularly want to thank Saya Ikegawa, who read the manuscript extremely carefully and made excellent comments and suggestions regarding exposition, code, and models. I have benefited greatly from the generous help of my intellectual elders and betters. Special thanks go to John Creedy, Cuong Le Van, Kazuo Nishimura, Kevin Reffett, and Thomas Sargent. Extra special thanks go to Kazuo and Tom, who have helped me at every step of my personal random walk. I have been inspired by reading the work of many brilliant mathematicians. Three favorites who come to mind are Wolfgang Doeblin, Torgny Lindvall, and Andrzej Lasota. I had the pleasure of meeting the last two, albeit briefly. Wolfgang Doeblin died far too young but his beautiful ideas still shine brightly. The editorial team at MIT Press has been first rate. I deeply appreciate their enthusiasm and their professionalism, as well as their gentle and thoughtful criticism: Nothing is more valuable to the author who believes he is always right—especially when he isn't. I am grateful to the Department of Economics at Melbourne University, the Center for Operations Research and Econometrics at Université Catholique de Louvain, and the Institute for Economic Research at Kyoto University for providing me with the time, space, and facilities to complete this text. While completing the second edition, I benefited from funding generously donated by Schmidt Futures. I thank my parents for their love and support, and Andrij, Nic, and Roman for the same. Thanks also go to my extended family of Aaron, Kirdan, Merric, Murdoch, and Simon, who helped me weed out all those unproductive brain cells according to the principles set down by Charles Darwin. Dad gets an extra thanks for his long-running interest in this project, and for providing the gentle push that is always necessary for a task of this size. Finally, I thank my beautiful wife, Cleo, for suffering with patience and good humor the absent-minded husband, the midnight tapping at the computer, the highs and the lows, and then some more lows, until the job was finally done. This book is dedicated to you. 1 As an aside, I have played a small role in the development of Jupyter Book, as one of the founders of Executable Book Project (EBP), which manages Jupyter Book. My co-founders are Chris Holdgraf from Berkeley and Greg Caporaso from Northern Arizona University. Jupyter Book is open source and promotes open, reproducible science. EBP is generously supported by the Alfred P. Sloan Foundation. 9