slp solutions manual

Solutions M an ual for R ecursiv e M ethods in Eco n o mic Dy n amics Solutions M anual for Recursive Methods in E conomic D ynamics Claud io Irig oyen Esteban Rossi-Hansberg Ma r k L. J. Wr i g h t Harvard University Press Cambridge, Massachusetts, and London, England 2002 Copyright c ° 2002 by the President and Fellows of Harvard College All rights reserved Printed in the United States of America Library of Congress C ataloging-in-Publication Data To Marta, Santiago, and Federico —CI To Maria Jose —ERH To Christine —MLJW Contents 1 Introduction 1 2 An Overview 3 3 Mathematical Preliminaries 20 4 Dynamic Programming under Certainty 43 5 Applications of Dynamic Programming under Certainty 56 6 Deterministic Dynamics 85 7 Measure Theory and Integration 102 8 Markov Processes 137 9 Stochastic Dynamic Programming 154 10 Applications of Stochastic Dynamic Programming 179 11 Strong Convergence of Markov P rocesses 199 12 Weak Convergence of Markov Processes 208 13 Applications of Convergence Results for Markov Processes 223 14 Laws of Large Numbers 246 15 Pareto Optima and Competitive Equilibria 252 vi vii 16 Applications of Equilibrium Theory 266 17 Fixed-Point Arguments 284 18 Equilibria in Systems with Distortions 298 Foreword Over the years we have received many requests for an answer book for the exercises in Recursive Methods in Economic Dynamics.These requests have come not from inept teac hers or lazy students, but from serious readers who have wanted to make sure their time was being well spent. For a student trying to master the material in Recursive Methods, the exercises are critical, and some of them are quite hard. Thus it is useful for the reader to be reassured along the way that he or she is on the right track, and to have misconceptions corrected quickly when they occur. In addition, some of the problems need more specific guidelines or sharper formulations, and a few (not too many, we like to think) contain errors — com m ands to prove assertions that, under the stated assumptions, are just not true. Consequently, when three of our best graduate students proposed to write a Solutions Manual, we were delighted. While we firmly believe in the value of working out problems for oneself, in learning by doing, it is clear that the present book will be an invaluable aid for students engaged in this enterprise. The exercises in Recursive Methods are of two types, reflecting the organization of the book. Some c hapters in the book are self- contained expositions of theoretical tools that are e ssential to mod- ern practitioners of dynamic stochastic economics. These “core” chapters contain dozens of problems that are basically mathemat- ical: exercises to help a reader make sure that an abstract definition or theorem has been grasped, or to provide a proof (some of them quiteimportant)thatwasomittedfromthetext. ThisSolutions Manual contains solutions for most of the exercises of this sort. In viii ix particular, proofs are provided for results that are fundamental in the subsequent development of the theory. Other chapters of Recursive Methods contain applications of those theoretical tools, organized by the kind of mathematics they require. The exercises in these chapters are quite different in character. Many of them guide the reader through classic papers drawn from various substantive areas of economics: growth, macroeconomics, monetary theory, labor, information economics, and so on. These papers, which appeared in leading j ournals over the last couple of decades, repre- sented the cutting edge, both technically and substantively. Turning a paper of this sort into an exercise meant providing enough struc- ture to keep the reader on course, while leaving enough undone to challenge even the best students. The present book provides answers for only a modest proportion of these problems. (Of course, for many of the rest the journal article on which the problem is based provides a solution!) We hope that readers will think of this Solutions Manual as a trio of especially helpful classmates . Claudio, Esteban, and Mark are people you might look for in the library when you a re stuck on a problem and need some help, or with whom you want to compare notes when you have hit on an especially clever argument. This is the way a generation of University of Chicago students have thought of them, and we hope that this book will let many more students, in a wide variety of places, benefit from their company as well. Nancy L. Stokey Robert E. Lucas Solutions M anual for R ecursiv e M eth ods in Econom ic Dynam ics [...]... readers can post corrections, comments and alternative answers This website is currently hosted at: http://home.uchicago.edu/~mwright2/SLPSolutions.html In the process of completing this project we have incurred various debts A number of people provided us with their own solutions to problems in the text, including Xavier Gine and Rui Zhao Others, including Vadym Lepetyuk, Joon Hyuk Song and Ivan Werning,... equation represented in Figure 2.1 Insert Figure 2.1 About Here As can be seen in the gure, the rst-order dierence equation has two steady states (that is, zs such that zt+1 = zt = z), which are the two solutions to the characteristic equation z 2 (1 + )z + = 0 These are given by z = 1 and b zT as Using the boundary condition zT +1 = 0 we can solve for zT = 1 + Substituting recursively into (??) . or to provide a proof (some of them quiteimportant)thatwasomittedfromthetext. ThisSolutions Manual contains solutions for most of the exercises of this sort. In viii ix particular, proofs are. at: http://home.uchicago.edu/~m wright2/SLPSolutions.html In the process of completing this project we have incurred vari- ous debts. A number of people provided us with their own solutions to problems in the. Solutions M an ual for R ecursiv e M ethods in Eco n o mic Dy n amics Solutions M anual for Recursive Methods in E conomic D ynamics Claud