PRICE DYNAMICS IN EQUILBRIUM MODELS

PRICE DYNAMICS IN EQUILBRIUM MODELS

Advances in Computational Economics VOLUME 16 SERIES EDITORS Hans Amman, University of Amsterdam, Amsterdam, The Netherlands Anna Nagumey, University of Massachusetts at Amherst, USA EDITORIAL BOARD Anantha K. Duraiappah, European University Institute John Geweke, University of Minnesota Manfred GilIi, University of Geneva Kenneth L. Judd, Stanford University David Kendrick, University of Texas at Austin Daniel MeFadden, University of California at Berkeley Ellen MeGrattan, Duke University Reinhard Neck, University of Klagenfurt Adrian R. Pagan, Australian National University John Rust, University ofwisconsin Bere Rustem, University of London Hal R. Varian, University of Michigan

Price Dynamics in Equilibrium Models The Search for Equilibrium and the Emergence of Endogenous Fluctuations by J an Tuinstra University of Amsterdam ~. " Springer Science+Business Media, LLC

Library of Congress Cataloging-in-Publication Data Tuinstra, Jan. Price dynamics in equilibrium models: the search for equilibrium and the emergence of endogenous fluctuations / by Jan Tuinstra. p.cm. -- (Advances in computational economics ; v. 16) Includes bibliographical references and index. ISBN 978-1-4613-5665-3 ISBN 978-1-4615-1661-3 (ebook) DOI 10.1007/978-1-4615-1661-3 1. Prices--Mathematical models. 2. Equilibrium (Economics)--Mathematical models. 3. Statics and dynamics (Social sciences) 4. Non1inear theories. 1. TitIe. II. Series. HB221. TI7 2000 338.5'2--dc21 00-051986 Copyright c 2001 by Springer Science+Business Media New York Originally published by Kluwer Academic Publishers in 2001 Softcover reprint orthe hardcover lst edition 2001 AII rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photo-copying, recording, or otherwise, without the prior written permission of the publisher, Springer Science+Business Media, LLC. Printed on acid-free pa per.

Contents 1 Introduction and Outline 1.1 Bounded Rationality............. 1.2Theory of Nonlinear Dynamics.... 1.2.1 Nonlinear economic dynamics: spective... 1.2.2 An example. 1.3 Outline. a historical per- 1 2 4 5 7 12 2 A TAtonnement Process 2.1 Introduction........ 2.2Price adjustment models.. 2.3 The Model.......... 2.3.1 Price normalization 2.3.2 Model specifications 2.3.3 Some typical numerical simulations. 2.4Symmetry...................... 34 2.5Local bifurcation analysis............. 39 2.5.1 Symmetry and the Jacobian matrix 40 2.5.2 Rotational symmetry and the Hopf bifurcation 43 2.5.3 Reflection symmetry and the period-doubling bifurcation... 43 2.5.4 D3 symmetry and the Equivariant Branching Lemma 45 2.6Global dynamics......................... 48 2.6.1 Symmetry-breaking and -increasing bifurcations 48 2.6.2 Rotational symmetry..... 50 2.6.3 Reflection symmetry..... 53 2.7 An asymmetric price adjustment process. 57 2.8Multiplicity of equilibria.......... 60 17 17 18 28 28 30 32

vi CONTENTS 2.8.1 Transcritical and saddle-node bifurcations: a case study... 60 2.8.2 The continuous tatonnement process 62 2.9Summary and conclusions.... 63 3 Perfect Foresight Cycles in Overlapping Generations Mode~ 65 3.1 The overlapping generations model.... 65 3.2 Equivalence................. 71 3.2.1 A cyclical exchange economy 71 3.2.2 An overlapping generations model 74 3.2.3 Equivalence of cycles and asymmetric equilibria. 75 3.2.4 A special case: the two generations overlapping generations model and Sarkovskii's theorem 78 3.2.5 Extensions... 80 3.3 Examples 3.3.1... 83 A three generations overlapping generations model with CES utility functions.............. 83 3.3.2 A two generations overlapping generations model with CARA utility functions... 85 3.3.3 An example with two generations and capital 88 4 Learning in Overlapping Generations Models 91 4.1 Perfect foresight versus learning.. 91 4.2The overlapping generations model 94 4.3 Static expectations......... 99 4.4 A regression on price levels... 105 4.5A regression on inflation rates - part I 110 4.6A regression on inflation rates - part II. 117 4.7Summary................. 123 5 An Evolutionary Model of Cournot Competition 125 5.lIntroduction...................... 125 5.2The Model....................... 128 5.2.1 Traditional Cournot Duopoly Analysis 128 5.2.2 Quantity Dynamics.... 131 5.2.3 Population Dynamics.. 133 5.2.4 Local Instability Results. 139 5.3Best-Reply versus Rational Players 143 5.4Local Bifurcation Analysis....... 145

CONTENTS Vll 5.4.1 Discrete Choice Dynamics. 146 5.4.2 Replicator Dynamics.... 153 5.5Global Bifurcation Analysis....... 158 5.5.1 Homoclinic Bifurcation Theory 159 5.6Imitators versus Best-Reply Players. 171 5.6.1 Discrete Choice Model 172 5.6.2 Replicator Dynamics. 175 5.7 Concluding Remarks........ 176 5.8 Appendix.............. 178 5.8.1 Derivation of Profit Functions. 178 5.8.2 Equivalence of the Cournot and Cobweb Model. 179 5.8.3 The Jacobian matrix.. 181 5.8.4 Some Important Curves 183 5.8.5 Diffeomorphisms.... 184 6 Price Adjustment in Monopolistic Competition 187 6.lIntroduction............ 187 6.2 A partial equilibrium model... 189 6.2.1 Best-reply dynamics 191 6.2.2 Gradient systems.. 197 6.3A short memory learning procedure. 199 6.3.1 Examples: convergence 202 6.3.2 Example: nonconvergence 204 6.4A long memory learning procedure 210 6.5 Concluding remarks... 217 References 219 Index 231

List of Figures 1.1 Long run behaviour for the tatonnement process in the state space. On the horizontal axis PIt and on the vertical axis P2t. a) Period two orbit created in period doubling bifurcation. ct = i and.a = 0.85. b) Quasi periodic cycle created in Hopf bifurcation. ct = -i and.a = 0.95..... 8 1.2 Time series of first 200 values of PIt for a) ct = i and.a = 0.85, and b) ct = -i and.a = 0.95............ 9 1.3 Strange attractors for the tatonnement process. a) ct = i and.a = 1.20, b) ct = -~ and.a = 1.11............ 10 1.4 a) Two time series (for ct = ~ and.a = 1.20) of first 200 values of Pit for (plo,p20) = (1,0.99) (solid line) and for qit with (qlo,q20) = (1,0.99001) (dotted line). b) Time series of difference Pit - qit.................. 11 2.1 Attractors of the tatonnement process with rotational symmetry. a) ct12 = ct23 = ct31 = 1, ct13 = ct21 = ct32 =!, (J' =! and.a = 4.95. b) ct12 = ct23 = ct31 = 1, ct13 = ct21 = ct32 =!, (J' =! and.a = 8.45.............. 32 2.2 Attractors of the tatonnement process with reflectional symmetry. a) ct12 = ct21 = i, ct13 = ct23 = ct31 = ct32 =!, (J' = lo and.a = 32. b) ct12 = ct21 = ~, ct13 = ct23 = ct31 = ct32 = ~, (J' = 1~ and.a = 31.................. 33 2.3 Strange attractors of the tatonnement process with cyclical and reflectional symmetry. a) ctij = 1, all i,j, (J' = /0 and.a = 26. b) ct12 = ct13 = ct21 = ct23 = ct31 = ct32 = 10' (J' = 1~ and.a = 95....................... 34

x LIST OF FIGURES 2.4 Primary bifurcations for the tatonnement process with different symmetry groups. a) Invariant closed curve created in Hopf bifurcation for the tatonnement process with cyclical symmetry, 0 = 1, {3 =!, (J' =! and A = 4.5. b) Period two orbit off the symmetry axis created in period doubling bifurcation for the tatonnement process with reflectional symmetry, 0 = i, {3 = 1 =!, (J' = 1~ and A = 25. c) Period two orbit on symmetry axis created in period doubling bifurcation for the tatonnement process with reflectional symmetry, 0 = :1, (3 = 1 =!, a = 1~ and A = 25. d) Six period two orbits created in symmetry breaking period doubling bifurcation in tatonnement process with cyclical and reflectional symmetry, 0 = 1, a = l~o and A = 220............................. 46 2.5 Bifurcation scenario for tatonnement process with D3 symmetry, 0 = 1, a = 1~' a) A = 220, b) A = 258, c) A = 260, d) A = 261........................... 49 2.6 Bifurcation curves for tatonnement process with cyclical symmetry, 0 = 1 and a = :1.................. 51 2.7 Bifurcation scenario for tatonnement process with cyclical symmetry, 0 = 1, (3 =!, a =!. a) A = 4.85, b) A = 4.90, c) A = 4.925 and d) A = 4.93................. 52 2.8 Period doubling bifurcation curves for the tatonnement process with reflectional symmetry, {3 =!, 1 =! and (J' = lo'... 55 2.9 Bifurcation scenario for tatonnement process with reflectional symmetry 0 = 1, {3 =!, 1 =!, (J' = 1~' a) A = 25, b) A = 27, c) A = 28 and d) A = 28.35... " 56 2.10 a) Stable period two curves for tatonnement process with D3 symmetry (Oij = 1, a = 1~) created through symmetry breaking period doubling bifurcation. b) Period two curves in a tatonnement process close to D3 symmetry (Oij = 1, 012 = 0.95, 013 = 0.9, a = I~O)'... 57 2.11 Attractors for asymmetric tatonnement process. a)-c) three different coexisting attractors for A = 259. d) unique attractor for A = 265....................... 58 2.12 Aggregate demand of good 1 on Lt. a) Saddle-node bifurcation. b) Transcritical bifurcation............ " 60

LIST OF FIGURES Xl 3.1 Equilibria in the cyclical exchange economy with four commodities and CARA utility functions as a function of f3.. 87 4.1 Savings functions. Upper diagram: CES savings function, lower diagram: complicated non-monotonic savings function 96 4.2 The perfect foresight mapping. The lower equilibrium corresponds to the monetary steady state () and the higer equilibrium corresponds to the autarkic steady state 7r a. 98 4.3 Attractors for the OG model with naive expectations. a) Invariant circle created through Hopf bifurcation for OG with CES savings function, p = i and () = 0.85. b) Period two cycle created through flip bifurcation for the complicated savings function, p = i and () = 0.36. c) and d) Strange attractors for OG with complicated savings function, p = i and () = 0.7926 and () = 2.14, respectively... 102 4.4 Attractors for the OG model with least squares learning on price levels. a) CES savings function, p = i and () = 1.40. b) CES savings function, p = i and () = 1.41. c) Complicated savings function, p = i and () = 1.5. d) Complicated savings function, p = ~ and () = 1.8...... 108 4.5 Attractors of (4.20) and examples of beliefs-equilibra. a) BE for CES savings function with p = i and () = 0.85. b) BE from a) with small noise. c) and d) Nonexistence of BE for complicated savings function with p = i and p = ~, respectively and () = 1.... 116 4.6 Time series of 7rt and f3 t for (4.20) with CES savings function and p = i and () = 0.85.... 117 4.7 Time series of 7rt and f3t for (4.20) with complicated savings function and p = i and () = 1.............. 118 4.8 Attractors of (4.26) and examples of beliefs-equilibria with complicated savings function. a) BE corresponding to steady state, p = i, () = 1, ao = 5 and f30 = -to b) Nonexistence of BE, p = i, () = 1, ao = 5 and f30 = -!. c) BE corresponding to period two cycle, p = i, () = 1, ao = 5 and f3 0 = -0.9. d) BE corresponding to four cycle, p = i, () = 1.5, ao = 5 and f30 = -!... 122

xii LIST OF FIGURES 5.1 Attractors for the model with best-reply versus rational players and discrete choice dynamics with a = 17, b = 1, c = 10, d = 1.25, T = 1 and a) (3 = 8.1, b) (3 = 8.2, c) (3 = 8.5 and d) (3 = 25... 152 5.2 Time series of quantities and fraction of rarional players for the model with best-reply versus rational players and discrete choice dynamics with a = 17, b = 1, c = 10, d = 1.25, T = 1 and (3 = 25.................. 153 5.3 Attractors for the model with best-reply versus rational players and replicator dynamics with a = 17, b = 1, c = 10, d = 1.1, T = 1 and a) 8 = 0.00080, b) 8 = 0.00066, c) 8 = 0.00064 and d) 8 = 0.00060................ 156 5.4 Time series of quantities and fraction of rational players for the model with best-reply versus rational players and replicator dynamics with a = 17, b = 1, c = 10, d = 1.1, T = 1 and 8 = 0.00060..... 157 5.5 Shape of stable and unstable manifolds (WS (p) and W U (p)) if there is a homo clinic intersection point q......... 159 5.6 Homoclinic bifurcation. a) Stable and unstable manifolds before the bifurcation (a > ao), b) Stable and unstable manifolds at the homo clinic bifurcation (a = ad) and c) Stable and unstable manifolds after the homo clinic bifurcation (a < ad)..... 160 5.7 Stable and unstable manifolds for the evolutionary model of Cournot competition with replicator dynamics with a = 17, b = 1, c = 10, d = 1.1, T = 1. a) Three parts of the stable manifold for 8 = 0.1, b) Stable and unstable manifolds for 8 = 0.005, c) Stable and unstable manifolds for 8 = O.OOland d) Stable and unstable manifolds for 8 = 0.0005752.... 163 5.8 Homoclinic bifurcation: a) Stable and unstable manifolds for 8 = 0.00070, b) Stable and unstable manifolds for 8 = 0.0005752, c) Stable and unstable manifolds for 8 = 0.00040 and d) Part of the strange attractor for {j = 0.0005752............................ 164 5.9 Horseshoe in the evolutionary model of Cournot competition: the rectangular box is mapped into the horseshoelike figure in 12 iterations..... 166

LIST OF FIGURES xiii 5.10 Some important curves: Lb and L r are the curves along which net profits for the best-reply and rational rule are zero, respectively. L n is the curve along which the denominator of 9 (X, n) vanishes, DIet is the curve along which the determinant of the Jacobian matrix vanishes and LIII is the third component of the stable manifold of the equilibrium............................. 168 5.11 Graphical sketch of the proof of homo clinic intersection.. 170 5.12 Attractors for the model with imitators versus best-reply players and discrete choice dynamics with a = 17, b = 1, c = 10, d = 1.1, T = 1 and a) (3 = 2.2, b) (3 = 2.6, c) (3 = 2.8 and d) (3 = 3.3..... 174 5.13 Attractors for the model with imitators versus best-reply players and replicator dynamics with a = 17, b = 1, c = 10, d = 1.1, T = 1 and a) 6 = 0.0175, b) 6 = 0.0130, c) 6 = 0.0114 and d) 6 = 0.0111................. 176 6.1 Attractors of the short memory learning model with (311 = (322 = -1, (312 = (321 = 1, 111 = 122 = 2, a1 = a2 = 3 and 112 = -'21 = 6, where we have: a) 6 = 3, b) 6 = 4, c) 6 = 41 and d) 6 = 5..... 206 6.2 Attractors of the short memory learning model with (311 = (322 = -1, (312 = (321 = 1, 111 = 122 = 2, a1 = a2 = 3 and 112 = 121 = 6, where we have: a) 6 = 1, b) 6 = 21, c) 6 = 2.70 and d) 6 = 2.78... 208 6.3 Attractors of the short memory learning model with (311 = (322 = -1, (312 = (321 = 1, 111 = 122 = 2, a1 = a2 = 3 and 112 = 121 = 6, where we have: a) 6 = -1, b) 6 = -2, c) 6 = -2! and d) 6 = -5..... 209

Preface A long-standing unsolved problem in economic theory is how economic equilibria are attained. This book considers a number of adjustment processes in different economic models and investigates their dynamical behaviour. Two important themes arising in this context are "bounded rationality" and "nonlinear dynamics". The results presented here indicate that endogenous fluctuations are the rule rather than the exception in the search for equilibrium. The book is written for anyone with an interest in economic theory in general and bounded rationality and endogenous fluctuations in particular. It is entirely self-contained and accessible to readers with only a limited knowledge of economic theory. Parts of the work presented here are based upon research projects with Edward Droste, Cars Hommes and Claus Weddepohl, whom I thank for allowing me to use the results of our joint work. Chapter 3 is based upon an article published by the Journal of Economics-Zeitschrijt fur Nationalokonomie. I thankfully acknowledge Springer-Verlag for permission of reproducing these results here. Furthermore, I benefitted greatly from vivid discussions with Maarten-Pieter Schinkel and Michel van de Velden and I am thankful to Cees Diks and Henk van de Velden for assistance with practical issues. I am grateful to Allard Winterink and his associates at Kluwer Academic Publishers for providing me with the opportunity to write this book. Amsterdam, September 2000