Table Of ContentStudies 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
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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
