Optimization and Chaos PDF
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Studies in Economic Theory Editors Charalambos D. Aliprantis Purdue University Department of Economics West Lafayette, IN 47907-1310 USA Nicholas C. Yannelis Department of Economics University of Illinois Champaign, IL 61820 USA Springer-Verlag Berlin Heidelberg GmbH Titles in the Series M. Ali Khan and Nicholas C. Yannelis (Eds.) Equilibrium Theory in Infinite Dimensional Spaces Charalambos D. Aliprantis, Kim C. Border and Wilhelmus A.J. Luxemburg (Eds.) Positive Operators, Riesz Spaces, and Economics Donald G. Saari Geometry of Voting Charalambos D. Aliprantis and Kim C. Border Infinite Dimensional Analysis Jean-Pierre Aubin Dynamic Economic Theory Mordecai Kurz (Ed.) Endogenous Economic Fluctuations Jean-Franyois Laslier Tournament Solutions and Majority Voting Ahmet Alkan, Charalambos D. Aliprantis, Nicholas C. Yannelis (Eds.) Theory and Applications James C. Moore Mathematical Methods for Economic Theory 1 James C. Moore Mathematical Methods for Economic Theory 2 Mukul Majumdar, Tapan Mitra, Kazuo Nishimura Optimization and Chaos Mukul Majumdar Tapan Mitra Kazuo Nishimura Optimization and Chaos With 13 Figures i Springer Prof. Mukul Majumdar Prof. Tapan Mitra Cornell University Department of Economics 418 Uris Hall Ithaca, NY 14853-7601 USA [email protected] Prof. Kazuo Nishimura Kyoto University Institute for Economic Research Kyoto 606 Japan ISSN 1431-8849 ISBN 978-3-642-08636-6 ISBN 978-3-662-04060-7 (eBook) DOI 10.1007/978-3-662-04060-7 Library of Congress Cataloging-in-Publication Data Die Deutsche Bibliothek - CIP-Einheitsaufnahme Optimization and chaos I Mukul Majumdar, Tapan Mitra, Kazuo Nishimura. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singa pore; Tokyo: Springer, 2000 (Studies in economic theory; 11) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 2000 Originally published by Springer-Verlag Berlin Heidelberg New York in 2000. Softcover reprint ofthe hardcover Ist edition 2000 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Hardcover design: Erich Kirchner, Heidelberg SPIN 10733605 42/2202-5 4 3 2 1 0 - Printed on acid-free paper Preface This collection of essays brings together some articles on dynamic optimization models that exhibit chaotic behavior. Chapters 3, 4, 5, 6, 7, and 9 appeared in a Symposium on Chaotic Dynamical Systems in Economic Theory (Volume 4, Number 5, 1994). Also, Chapters 10,11, and 12 appeared in the Journal of Economic The ory. We would like to thank the authors, and Academic Press for permission to reprint. We are grateful to Professor C.D. Aliprantis for suggesting the idea of a book structured around the Economic Theory Symposium, and without the support and patience of Dr. Mueller this project could not have been completed. We would like to thank Ms. Amy Gowan who cheerfully per formed the arduous task of typing the manuscript. Thanks are also due to Xiao Qing Yu, Tridip Ray and Malabika Majumdar for their help at various stages in the preparation of the manuscript. For a course on dynamic optimization addressed to students with a good background in economic theory and real analysis, one can assign Chapter 2 as a partial introduction to the basic tech niques. Chapters 3 and 4 can be assigned to provide examples of simple optmization models generating complicated behavior. The role of discounting in generating complexity can be taken up through Chapters 10-12. Many of the major problems of intertem poral allocation theory cannot be discussed in the framework of Chapter 2: short run shocks, indivisibilities, increasing returns, structural breaks for the "late-comers in development" , export-Ied strategies of develoment, sustained growth through human capital accumulation, just to indicate a few. Appropriate readings can be assigned for some of these topics. The course can be made more challenging by introducing the standard model of discounted dy namic programming under uncertainty with a single decision mak l, ing agent [using any of the standard texts and then extending the coverage to models involving a small number of interacting deci- VI Preface sion makers. The excellent review of Dutta and Sundaram (1998) is particuarly useful in the last context. Parts of the book can also be used for a course on chaotic mod els in economic theory. In such a course, Chapters 1 and 2 will provide some of the core material. The instructor can move on to Chapters 3-6, to provide examples of bifurcation, cycling and chaos. The question of "robustness" can be dealt with by drawing from Chapters 3,7 and 9. Finally, parts of the last four chapters can be used to clarify the role of discounting. Progress of a science, we are told, is made through funerals. It is our hope that some of the concepts and techniques will continue to be useful in the analysis of models that will engage researchers in the future. List of Contributors Robert Becker, Indiana University Ciprian Foias, Indiana University Mukul Majumdar, Cornell University Tapan Mitra, Cornell University Kazuo Nishimura, Kyoto University Gerhard Sorger, University of London Makoto Yano, Keio University Contents Preface V List of Contributors VII 1 Dynamical Systems: A Tutorial 1 Mukul Majumdar and Tapan Mitra 2 Introduction to Dynamic Optimization Theory 31 Tapan Mitra 3 Periodic and Chaotic Prograrns of Optimal Intertemporal Allocation in an Aggregative Model with Wealth Effects 109 Mukul Majumdar and Tapan Mitra 4 Optimal Chaos, Nonlinearity, and Feasibility Conditions 149 K azuo Nishimura and M akoto Yano 5 The Local Bifurcation of Ramsey Equilibrium 173 Roben A. Becker and Ciprian Foias 6 On the Structure of Ramsey Equilibirum: Cycles, Indeterminacy and Sunspots 209 Gerhard Sorger 7 Robust Ergodic Chaos in Discounted Dynamic Optimization Models 240 Mukul Majumdar and Tapan Mitra X Contents 8 Non-Linear Dynamics and Chaos in Optimal Growth: A Constructive Exposition 258 K azuo Nishimura and M akoto Yano 9 Ergodic Chaos in Optimal Growth Models with Low Discount Rates 296 Kazuo Nishimura, Gerhard Sorger and Makoto Yano 10 On the Minimum Rate of Impatience for Complicated Optimal Growth Paths 315 Gerhard Sorger 11 An Exact Discount Factor Restriction for Period-Three Cycles in Dynamic Optimization Models 339 Tapan Mitra 12 On the Least Upper Bound ofDiscount Factors that are Compatible with Optimal Period-Three Cycles 368 Kazuo Nishimura and Makoto Yano 13 On the Relationship between Discounting and Entropy of Dynamic Optimization Models 403 Tapan Mitra References 434
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