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Advances in Japanese Business and Economics 13

Kazuo Mino

Growth and

Business Cycles

with Equilibrium

Indeterminacy

Advances in Japanese Business and Economics

Volume 13

Editor in Chief

RYUZO SATO

C.V. Starr Professor Emeritus of Economics, Stern School of Business,

New York University

Senior Editor

KAZUO MINO

Professor Emeritus, Kyoto University

Managing Editors

HAJIME HORI

Professor Emeritus, Tohoku University

HIROSHI YOSHIKAWA

Professor, Rissho University; Professor Emeritus, The University of Tokyo

KUNIO ITO

Professor Emeritus, Hitotsubashi University

Editorial Board Members

TAKAHIRO FUJIMOTO

Professor, The University of Tokyo

YUZO HONDA

Professor Emeritus, Osaka University; Professor, Kansai University

TOSHIHIRO IHORI

Professor Emeritus, The University of Tokyo; Professor, National Graduate Institute for Policy Studies

(GRIPS)

TAKENORI INOKI

Professor Emeritus, Osaka University; Special University Professor, Aoyama Gakuin University

JOTA ISHIKAWA

Professor, Hitotsubashi University

KATSUHITO IWAI

Professor Emeritus, The University of Tokyo; Visiting Professor, International Christian University

MASAHIRO MATSUSHITA

Professor Emeritus, Aoyama Gakuin University

TAKASHI NEGISHI

Professor Emeritus, The University of Tokyo; Fellow, The Japan Academy

KIYOHIKO NISHIMURA

Professor, The University of Tokyo

TETSUJI OKAZAKI

Professor, The University of Tokyo

YOSHIYASU ONO

Professor, Osaka University

JUNJIRO SHINTAKU

Professor, The University of Tokyo

KOTARO SUZUMURA

Professor Emeritus, Hitotsubashi University; Fellow, The Japan Academy

Advances in Japanese Business and Economics showcases the research of Japanese

scholars. Published in English, the series highlights for a global readership the

unique perspectives of Japan’s most distinguished and emerging scholars of business

and economics. It covers research of either theoretical or empirical nature, in both

authored and edited volumes, regardless of the sub-discipline or geographical coverage, including, but not limited to, such topics as macroeconomics, microeconomics,

industrial relations, innovation, regional development, entrepreneurship, international trade, globalization, financial markets, technology management, and business

strategy. At the same time, as a series of volumes written by Japanese scholars,

it includes research on the issues of the Japanese economy, industry, management

practice and policy, such as the economic policies and business innovations before

and after the Japanese “bubble” burst in the 1990s.

Overseen by a panel of renowned scholars led by Editor-in-Chief Professor

Ryuzo Sato, the series endeavors to overcome a historical deficit in the

dissemination of Japanese economic theory, research methodology, and analysis.

The volumes in the series contribute not only to a deeper understanding of Japanese

business and economics but to revealing underlying universal principles.

More information about this series at http://www.springer.com/series/11682

Kazuo Mino

Growth and Business Cycles

with Equilibrium

Indeterminacy

123

Kazuo Mino

Kyoto University Institute of Economic

Research

Kyoto

Kyoto, Japan

ISSN 2197-8859

ISSN 2197-8867 (electronic)

Advances in Japanese Business and Economics

ISBN 978-4-431-55608-4

ISBN 978-4-431-55609-1 (eBook)

DOI 10.1007/978-4-431-55609-1

Library of Congress Control Number: 2017943342

© Springer Japan KK 2017

This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of

the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,

broadcasting, reproduction on microfilms or in any other physical way, and transmission or information

storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology

now known or hereafter developed.

The use of general descriptive names, registered names, trademarks, service marks, 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.

The publisher, the authors and the editors are safe to assume that the advice and information in this book

are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or

the editors give a warranty, express or implied, with respect to the material contained herein or for any

errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional

claims in published maps and institutional affiliations.

Printed on acid-free paper

This Springer imprint is published by Springer Nature

The registered company is Springer Japan KK

The registered company address is: Chiyoda First Bldg. East, 3-8-1 Nishi-Kanda, Chiyoda-ku, Tokyo

101-0065, Japan

Preface

Why do macroeconomic variables of an economy fluctuate, even though no fundamental shock hits the economy? Why do countries with similar initial conditions

sometimes display very different patterns of growth and development? To answer

these questions, it is often helpful to use growth and business cycle models that give

rise to multiple equilibria. In these models, the equilibrium path of an economy is

indeterminate without specifying agents’ expectations. Therefore, in the presence

of equilibrium indeterminacy, extrinsic uncertainty that only affects expectations of

agents may alter patterns of business cycles and long-run growth. Over the last two

decades, the issue of equilibrium indeterminacy has been a well-explored research

theme in macroeconomics. The central concern of this book is to elucidate various

topics discussed in this line of research.

Chapter 1 provides the readers with basic concepts and analytical methods

used in the literature on macroeconomic models with equilibrium indeterminacy.

After presenting a brief historical review, we consider two simple examples: a

univariable rational expectations model and a monetary dynamic model of an

exchange economy. When analyzing both models, we classify the models into three

cases: the steady-state equilibrium of the model economy is (i) unique, (ii) multiple,

and (iii) a continuum. Those classifications apply to the growth and business cycle

models examined in the subsequent chapters.

Chapters 2 and 3 explore baseline models of growth and business cycles that

hold equilibrium indeterminacy. Chapter 2 focuses on the real business cycle

models with external increasing returns and clarifies the conditions under which

equilibrium indeterminacy emerges. This chapter also examines related studies

that extend the baseline model into various directions. In Chap. 3, we study

equilibrium indeterminacy in endogenous growth models. We treat the basic models

of endogenous growth and reveal the similarities and differences in indeterminacy

conditions between the real business cycle models and the endogenous growth

models.

Chapter 4 considers growth models that involve multiple steady states. We examine a neoclassical growth model with threshold externalities and an endogenous

growth model with global indeterminacy.

v

vi

Preface

Chapters 5 and 6 discuss applied topics. Chapter 5 investigates how fiscal and

monetary policy rules give rise to equilibrium indeterminacy in both real business

cycle models and endogenous growth models. Chapter 6 considers equilibrium

indeterminacy in open-economy models. We discuss indeterminacy conditions in

small open-economy models as well as in two-country models. When inspecting

both types of models, we consider both exogenous and endogenous growth settings.

The final short chapter (Chap. 7) refers to a sample of recent studies that intended to

pursue new directions.

Although this book is not a mere collection of my publications, the main content

of the book is based on my foregoing research on macroeconomic models with

equilibrium indeterminacy. First of all, I would like to thank my coauthors, Daisuke

Amano, Been-Lon Chen, Koichi Futagami, Seiya Fujisaki, Yu-Shan Hsu, Yunfang

Hu, Jun-ichi Itaya, Yasuhiro Nakamoto, Kazuo Nishimura, Akihisa Shibata, (late)

Koji Shimomura, and Ping Wang, for their productive collaboration. At various

stages of my research, many people provided useful comments. Among others, I

particularly thank Shin-ichi Fukuda, Jang-Ting Guo, Makoto Saito, Danyang Xie,

and Chon-Ki Yip for their constructive comments and suggestions on my papers on

which this book partially depends.

Professor Ryuzo Sato, chief editor of the Advances in Japanese Business and

Economics series, encouraged me to publish this book. I am grateful for his

continuing support, since I learned economics under his guidance as a graduate

student at Brown University in the early 1980s. I also thank Juno Kawakami of

Springer Japan for her helpful editorial assistance. Finally, I thank my wife, Yoko

Hayami, for her understanding and constant support.

Kyoto, Japan

March 2017

Kazuo Mino

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1.1 A Brief Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1.2 A Univariable Rational Expectations Model . . . . . .. . . . . . . . . . . . . . . . . . . .

1.2.1 Base Model.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1.2.2 Fundamental Disturbances . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1.3 General Equilibrium Models of the Monetary Economy . . . . . . . . . . . . .

1.3.1 Base Model.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1.3.2 The Case with a Unique Steady State. . . . . .. . . . . . . . . . . . . . . . . . . .

1.3.3 The Case with Multiple Steady States . . . . .. . . . . . . . . . . . . . . . . . . .

1.3.4 A Model with a Continuum of Steady States .. . . . . . . . . . . . . . . . .

1.4 References and Related Studies . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1

1

3

3

7

8

8

10

11

15

18

2 Indeterminacy in Real Business Cycle Models . . . . . . .. . . . . . . . . . . . . . . . . . . .

2.1 One-Sector Growth Models with Fixed Labor Supply . . . . . . . . . . . . . . . .

2.1.1 A Model with Production Externalities .. . .. . . . . . . . . . . . . . . . . . . .

2.1.2 A Model with Productive Consumption . . .. . . . . . . . . . . . . . . . . . . .

2.2 The Benhabib-Farmer-Guo Approach . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

2.2.1 Base Model.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

2.2.2 Dynamic System . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

2.2.3 Indeterminacy Conditions . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

2.2.4 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

2.3 The Source of Indeterminacy .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

2.3.1 Strategic Complementarity .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

2.3.2 Intuitive Implication of Indeterminacy Conditions.. . . . . . . . . . .

2.4 Related Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

2.4.1 Indeterminacy Under Mild Increasing Returns .. . . . . . . . . . . . . . .

2.4.2 Preference Structure . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

2.4.3 Consumption Externalities . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

2.4.4 News Versus Sunspots .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

2.4.5 Local Versus Global Indeterminacy.. . . . . . .. . . . . . . . . . . . . . . . . . . .

2.5 References and Related Studies . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

19

19

19

22

23

23

25

27

28

31

31

33

36

36

41

45

50

52

54

vii

viii

Contents

3 Indeterminacy in Endogenous Growth Models . . . . . . .. . . . . . . . . . . . . . . . . . . .

3.1 A One-Sector Model with Social Increasing Returns . . . . . . . . . . . . . . . . .

3.1.1 Separable Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

3.1.2 Non-separable Utility . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

3.2 A Two-Sector Model with Intersectoral Externalities .. . . . . . . . . . . . . . . .

3.2.1 Model.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

3.2.2 Dynamic System . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

3.2.3 Indeterminacy Conditions . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

3.3 A Two-Sector Model with Flexible Labor Supply . . . . . . . . . . . . . . . . . . . .

3.3.1 Model.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

3.3.2 Dynamic System . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

3.3.3 Conditions for Indeterminacy . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

3.3.4 An Alternative Formulation . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

3.4 Indeterminacy Under Social Constant Returns . . . .. . . . . . . . . . . . . . . . . . . .

3.4.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

3.4.2 The Dynamic System . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

3.4.3 Local Dynamics.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

3.4.4 Conditions for Local Indeterminacy . . . . . . .. . . . . . . . . . . . . . . . . . . .

3.4.5 Intuitive Implication . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

3.4.6 General Technology and Factor Income Taxation .. . . . . . . . . . . .

3.5 References and Related Studies . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

55

56

56

62

64

64

66

68

69

69

71

73

75

76

77

78

81

83

84

88

91

4 Growth Models with Multiple Steady States .. . . . . . . . .. . . . . . . . . . . . . . . . . . . .

4.1 History Versus Expectations .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

4.2 A Neoclassical Growth Model with Threshold Externalities . . . . . . . . .

4.2.1 Optimal Growth Under a Concave-Convex

Production Function . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

4.2.2 A Model with Threshold Externalities .. . . .. . . . . . . . . . . . . . . . . . . .

4.2.3 Model.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

4.2.4 Steady State Equilibria and Local Dynamics .. . . . . . . . . . . . . . . . .

4.2.5 Patterns of Global Dynamics .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

4.3 Global Indeterminacy in an Endogenous Growth .. . . . . . . . . . . . . . . . . . . .

4.3.1 Model.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

4.3.2 Market Equilibrium Conditions .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

4.3.3 Growth Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

4.3.4 A Simplified System .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

4.3.5 Local Dynamics.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

4.3.6 Global Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

4.3.7 Implications .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

4.4 References and Related Studies . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

93

93

97

97

99

101

102

103

105

106

108

109

111

113

116

118

119

5 Stabilization Effects of Policy Rules . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

5.1 Fiscal Policy Rules in Real Business Cycle Models . . . . . . . . . . . . . . . . . .

5.1.1 Balanced Budget Rule .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

5.1.2 Nonlinear Taxation . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

121

121

121

127

Contents

ix

5.2 Interaction Between Fiscal and Monetary Policies . . . . . . . . . . . . . . . . . . . .

5.2.1 Model.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

5.2.2 Policy Rules and Macroeconomic Stability . . . . . . . . . . . . . . . . . . .

5.2.3 Discussion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

5.3 Policy Rules in Endogenous Growth Models .. . . . .. . . . . . . . . . . . . . . . . . . .

5.3.1 Nonlinear Taxation Under Endogenous Growth . . . . . . . . . . . . . .

5.3.2 Interest-Rate Control Rules Under Endogenous Growth . . . . .

5.4 References and Related Studies . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

131

131

135

142

143

144

148

157

6 Indeterminacy in Open Economies . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

6.1 A One-Sector Model of Small Open Economy .. . .. . . . . . . . . . . . . . . . . . . .

6.1.1 Baseline Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

6.1.2 Endogenous Growth . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

6.2 A Two-Sector Model of Small Open Economy.. . .. . . . . . . . . . . . . . . . . . . .

6.2.1 Production .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

6.2.2 Households .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

6.2.3 Equilibrium (In)determinacy .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

6.2.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

6.3 A Two-Country Model with Free Trade of Commodities .. . . . . . . . . . . .

6.3.1 Baseline Setting .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

6.3.2 Global Equilibrium Conditions . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

6.3.3 Equilibrium Indeterminacy and Patterns of Trade . . . . . . . . . . . .

6.4 A Two-Country Model with Financial Transactions . . . . . . . . . . . . . . . . . .

6.4.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

6.4.2 Market Equilibrium Conditions and Aggregate

Dynamics.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

6.4.3 Steady State of the World Economy . . . . . . .. . . . . . . . . . . . . . . . . . . .

6.4.4 Indeterminacy Conditions . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

6.4.5 Long-Run Wealth Distribution .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

6.4.6 Non-tradable Consumption Goods .. . . . . . . .. . . . . . . . . . . . . . . . . . . .

6.4.7 Implication of the Indeterminacy Conditions . . . . . . . . . . . . . . . . .

6.4.8 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

6.5 A Two-Country Model with Variable Labor Supply . . . . . . . . . . . . . . . . . .

6.5.1 Model.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

6.5.2 Equilibrium Dynamics . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

6.5.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

6.5.4 Endogenous Growth . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

6.6 References and Related Studies . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

159

159

159

163

165

165

167

169

171

172

172

174

175

178

178

7 New Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

7.1 Microfoundations of Keynesian Economics . . . . . . .. . . . . . . . . . . . . . . . . . . .

7.2 Financial Frictions and Bubbles .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

7.3 Search Frictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

7.4 Agent Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

207

207

208

209

211

179

182

183

184

185

187

188

189

189

193

195

197

201

x

Contents

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 213

Author Index.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 223

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 227

About the Author

Kazuo Mino is a professor of economics at Doshisha University and a professor

emeritus of Kyoto University. He is the former president of the Japanese Economic

Association and the former editor of the Japanese Economic Review. Prior to

joining Doshisha University, he worked at Hiroshima, Tohoku, Kobe, and Osaka

Universities as well as at the Kyoto Institute of Economic Research at Kyoto

University. Mino has published extensively on various topics in macroeconomic

theory including growth and business cycle models, monetary and fiscal policies,

and open-economy macroeconomics.

xi

Chapter 1

Introduction

This chapter reviews the issue of equilibrium indeterminacy in macroeconomics.

Instead of providing a broad literature survey, we consider two simple examples.

One is a univariable rational expectations model of asset price determination. The

other is a general equilibrium model of monetary economy. When discussing both

examples, we classify the models into three categories: the steady state of the model

economy is (i) unique, (ii) multiple, and (iii) a continuum. The majority of foregoing

studies have treated models with a unique steady state. However, there are some

interesting situations in which multiple steady state equilibria exist or the steady

state of the economy constitutes a continuum. The main parts of the subsequent

chapters in this book also treat case (i). Chapters 2, 3 and 5 focus on the models that

have a unique interior steady state. Most of Chaps. 5 and 6 also discuss this case.

On the other hand, Chap. 4 examines the models with multiple steady states, while

Chap. 6 refers to the models that yield a continuum of steady states.

The models treated in this chapter are much simpler than the growth and business

cycle models explored in the main body of this book. However, they are helpful for

clarifying the key concepts and analytical methods used in the subsequent chapters.

1.1 A Brief Overview

If the equilibrium path of a dynamic macroeconomic model is not uniquely determined under rational expectations, which path is realized depends on a specification

of expectations of agents. In this situation, non-fundamental shocks that only

affect expectations of economic agents fluctuate economic activities. Therefore,

in the presence of equilibrium indeterminacy, extrinsic uncertainty is a driving

force of business cycles. Furthermore, if the equilibrium path of an economy is

© Springer Japan KK 2017

K. Mino, Growth and Business Cycles with Equilibrium Indeterminacy,

Advances in Japanese Business and Economics 13,

DOI 10.1007/978-4-431-55609-1_1

1

2

1 Introduction

indeterminate, the long-run growth and development process of the economy would

be affected by extrinsic uncertainty.1

Early studies on rational expectations models in the 1970s found that the rational

expectations equilibrium may be multiple without imposing ad hoc restrictions.2

Since most of the early rational expectations models lacked microfoundations, it was

expected that the indeterminacy problem can be resolved, if one constructs models

in which rational agents solve their dynamic optimization problems. However,

as revealed by Brock (1974) and Calvo (1979), monetary dynamic models with

optimizing agents easily exhibit equilibrium indeterminacy. Hence, constructing

microfounded models cannot resolve the indeterminacy problem.

While the presence of equilibrium indeterminacy poses a difficult question for

policy makers, it can give an alternative source of business fluctuations. This

idea led to a line of research that focuses on the role of extrinsic uncertainty in

macroeconomic models. Using a two-period model of general equilibrium, Cass

and Shell (1983) revealed that if some agents cannot participate insurance contracts,

extrinsic uncertainty has real effects even in the presence of complete financial

markets. Cass and Shell (1983) called extrinsic uncertainty “sunspots.”3 Azariadis

(1981) examined a two-period-lived overlapping generations model and found that

extrinsic uncertainty, which is called “self-fulfilling prophecies,” may generate

cyclical behavior of the aggregate economy. Since then, extrinsic uncertainty has

also been called “animal spirits,” “sentiments,” or “market psychology”.

Although the sunspot-driven business cycles theory developed in the 1980s made

an important theoretical contribution, it had little impact on the empirical research

on business cycles. This is because in the two-period lived overlapping generations

economy, the length of one period is about 30 years, so that fluctuations in such

an environment are not suitable for describing business cycles in the conventional

sense. A special issue of the Journal of Economic Theory published in 1994

substantially changed the situation. The articles in this issue explored equilibrium

indeterminacy in infinite horizon models of growth and business cycles. Among

others, Benhabib and Farmer (1994) introduced external increasing returns into

an otherwise standard real business cycle model and revealed that there exists a

continuum of equilibrium paths that converge to the steady state if the degree

1

Cass and Shell (1983) distinguished extrinsic uncertainty from intrinsic uncertainty. The former

has no effect on the fundamentals of an economy such as preferences and technologies, whereas

the latter affects the fundamentals.

2

“Multiple equilibria” and “equilibrium indeterminacy” are sometimes used as interchangeable

terms. Precisely speaking, the presence of multiple equilibria in macrodynamic models is

necessary but not sufficient for equilibrium indeterminacy In the literature, if a model economy

involves multiple paths under rational expectations (perfect foresight in the case of deterministic

environment), then the equilibrium path of the economy is called indeterminate.

3

As is well known, Jevons (1884) claimed that solar activities could generate business cycles,

because they could affect weather condition for agriculture. Hence, as opposed to Cass and

Shell (1983), Jevons consided that sunspots represent intrinsic uncertainty that directly affects the

agricultural production condition.

1.2 A Univariable Rational Expectations Model

3

of increasing returns is sufficiently strong. Moreover, Farmer and Guo (1994)

examined a calibrated version of the Benhabib and Farmer model. They found that if

indeterminacy holds, the model economy exhibits empirically plausible fluctuations

even in the absence of fundamental technological shocks. The Benhabib-FarmerGuo line of research attracted a considerable attention and spawned a large body of

literature in the last 20 years. The main concern of this book is to elucidate relevant

issues discussed in this class of studies.

Before examining growth and business cycle models in the subsequent chapters,

the rest of this chapter considers two simple examples that do not involve capital

and investment.

1.2 A Univariable Rational Expectations Model

1.2.1 Base Model

In this section we focus on a univariable dynamic system given by

pt D f .Et ptC1 / ;

(1.1)

where pt denotes the price of some asset whose initial value is not historically

specified. This equation means that the price in period t is determined by the

conditional expected price in period tC1: If the system does not involve uncertainty,

then Et ptC1 D ptC1 for all t

0; so that perfect foresight prevails. To avoid

unnecessary classification of patterns of dynamics, we assume that function f .:/

is monotonically increasing. Additionally, we assume that agents anticipate that pt

will converge neither to C1 nor to zero. Therefore, we exclude asset price bubbles.

1.2.1.1 The Case with a Unique Steady State

We first specify (1.1) as a linear system in such a way that

pt D ˛Et ptC1 C b; a > 0; a ¤ 1:

(1.2)

Here, we assume that a and b are deterministic parameters and that fundamental

shocks do not hit this dynamic system. However, there may exist non-fundamental

shocks that only affect expectations of economic agents. If there is no extrinsic

uncertainty that gives rise to non-fundamental shocks, perfect foresight holds and

the dynamic system becomes

pt D aptC1 C b:

(1.3)

4

1 Introduction

An obvious solution of (1.2) is the stationary one given by

pt D p D

b

1

a

for all t

0:

(1.4)

When this condition holds, we can set Et ptC1 D p :

Now assume that there is extrinsic uncertainty that only affects agents’ expectations. For example, suppose that agents believe that pt D pH if the state of

period t is H; while pt D pL if the state of period t is L: We also assume that

pL < p < pH : Furthermore, the transition of two states follows a stationary Markov

chain whose transition matrix is given by

Ä

QD

1

q

1

s

q

s

: 0 < s; q < 1:

Thus, for example,

Pr fstate of period t C 1 D H; state of period t D Hg D q;

Pr fstate of period t C 1 D L; state of period t D Hg D 1

q:

Given the above assumptions, it holds that

Et ptC1 D qpH C .1

Et ptC1 D .1

q/ pL if the state in period t is H;

s/ pH C spL if the state in period t is L:

First, suppose that 0 < a < 1 and b > 0: Then, pL < Et ptC1 < pH ; which means

that

pH > ˛Et ptC1 C b > pL :

In this case, it is impossible to find q; s 2 .0; 1/ that support

pH D a ŒqpH C .1 q/ pL C b;

pL D a Œ.1 s/ pH C spL C b:

(1.5)

As a result, agents’ predictions under which pt may be either pH or pL cannot be selffulfilled. The only equilibrium price that will not continue diverging is the stationary

price, so that under 1 < a < 1; it holds that pt D p D 1 b a for all t 0: In this

sense, the equilibrium path of pt is determinate, and non-fundamental shocks fail to

affect the equilibrium price levels.

Conversely, suppose that a > 1 and b < 0: Then we see that

pH < apH C b;

pL > apL C b:

1.2 A Univariable Rational Expectations Model

5

Since pL < Et ptC1 < pH , it is possible to find q; s 2 .0; 1/ that establish (1.5) for

any levels of pH and pL satisfying pL < p < pH : Namely, the system supports nonfundamental equilibrium prices pt D pH and pt D PL as well as the fundamental

equilibrium, pt D c= .1 a/ :

Note that the assumption of a two-state, stationary Markov chain is made only

for simplicity of discussion. We may find various forms of sunspots. For example, if

pO t satisfies (1.2), then pt D pO t C "t is also its solution, where "t is white noise. In this

case the, stochastic disturbance, "t , represents a sunspot shock that hits the agents’

expectations in period t:

If there is no extrinsic uncertainty, then Et ptC1 D ptC1 : In this case (1.2) can be

written as

ptC1 D

1

pt

a

b

:

a

(1.6)

Since the characteristic root of the above system is 1=a; the dynamic system has

a stable root if a > 1; while it has an unstable root if 0 < a < 1: Note that

this system does not involve non-jump state variables. Thus, if 0 < a < 1; the

number of stable roots equals the number of non-jump variables, which is zero in

this example. In contrast, if a > 1; the number of stable roots, which is one in our

model, exceeds the number of non-jump state variables. In other words, if a > 1 and

there is no uncertainty, pt may converge to p from any initial level of p0 ; implying

that p0 is indeterminate so that the subsequent path of fpt g1

tD0 converging to p is

indeterminate as well.

The above discussion can be applied to the original nonlinear system (1.1).

If pt D f . ptC1 / has a stationary solution satisfying f . p / D p ; the linear

approximation system at pt D p is expressed as

ptC1 D

1

. pt

f0 .p /

p /Cp :

Hence, setting 1=f 0 . p / D a and p .1 1=f 0 . p //

p D b; we see that

system (2.1) is locally determinate (indeterminate) around the steady state if and

only if 0 < f 0 . p / < 1 . f 0 . p / > 1/ :

To sum up, local determinacy/indeterminacy around the interior steady state

can be shown by checking the following conditions. Namely, the necessary and

sufficient conditions for local determinacy is:

number of non-jump variables D number of stable roots.

On the other hand, the necessary and sufficient conditions for local indeterminacy

is:

number of non-jump variables < number of stable roots.

These criteria have been used frequently in the literature.

6

1 Introduction

pt+1

p t+1

(a)

pt +1 = f −1 ( pt)

450

(b)

450

p t +1 = f

0

p*

p**

pt

0

p*

p**

−1

( p t)

pt

Fig. 1.1 (a) f(.) is strictly concave. (b) f(.) is strictly convex

1.2.1.2 The Case with Multiple Steady States

Next, assume that f .:/ in (1.1) is either a strictly convex function with f .0/ > 0 or a

strictly concave function with f .0/ < 0: Since f .:/ is assumed to be invertible, the

dynamic system under perfect foresight is written as

ptC1 D f

1

. pt / :

(1.7)

Figure 1.1a, b depict the relation between ptC1 and pt given by (1.7). The figures

show that the system has dual interior steady states. According to the criteria

mentioned above, p is locally determinate and p is locally indeterminate in

Fig. 1.1a, while the opposite results hold in Fig. 1.1b. The initial level of pt can

be selected from Œ0; p in case (a), while it can be chosen from Œ p ; C1/ in case

(b). In both cases, global indeterminacy is established.

1.2.1.3 The Case with a Continuum of Steady States

Again, we use the linear system (1.2) and set a D 1 and b D 0; which leads to

pt D Et ptC1 :

(1.8)

The corresponding deterministic system is pt D ptC1 and, hence, pt stays constant:

pt D p for all t

0:

However, in this case, the dynamic system fails to pin down the level of p :

Therefore, any feasible price level can be a stationary solution, and sunspot shocks

1.2 A Univariable Rational Expectations Model

7

may affect the selection of p : Furthermore, even if we select a particular level of

p as an equilibrium solution, p C "tC1 with Et "tC1 D 0 also fulfills (1.8). Namely,

even after the deterministic system selects the stationary level of pt ; the asset price

may fluctuate due to the presence of non-fundamental shocks.

1.2.2 Fundamental Disturbances

So far, we have assumed that there are no fundamental shocks. To check whether

the baseline results shown above will not change in the presence of fundamental

shocks, let us consider the following model:

pt D aEt ptC1 C bt ;

Â/ bN C Âbt

bt D .1

1

(1.9)

0 < Â < 1; bN > 0:

C "t ;

(1.10)

In this model, bt is not stationary and is disturbed by an exogenous shock, "t ; in

each period. Here, "t is represents an independent and identically distributed (i.i.d)

stochastic variable. We seek non-divergent solutions.

First, suppose that 0 < a < 1: In this case, iterative substitution in (1.9) up to

t D T > 0 presents

p t D Et

T

X

a j btCj C bt C atCT Et ptCT :

jD1

Hence, in view of 0 < a < 1; when T goes to infinity, we obtain

p t D Et

1

X

a btCj C bt D Et

j

1

X

jD1

a Â bt C .1

j

j

Â/ bN t

jD0

1

X

a j;

(1.11)

jD1

which yields

pt D

1

1

aÂ

bt C

a.1 Â/ N

b:

1 a

(1.12)

As a result, pt is uniquely determined. Note that if there is no fundamental

uncertainty ."t D 0 for all t 0/ and bt is fixed at bN .so Â D 0/, then the above

N .1 a/ : This is the steady state solution of the deterministic

reduces to pt D b=

system.

Next, consider the case of a > 1: We assume that aÂ ¤ 1: As a possible solution,

we try pt D bt C ; where and are unknown constants. Then, it holds that

bt C D a EbtC1 C a C bt ; which leads to

bt C

D a Âbt C a .1

Â/ bN C a C bt :

8

1 Introduction

Thus, we find

D

1

1

aÂ

;

D

a .1

1

Â/ bN

;

a

so that we again obtain (1.12).

Note that (1.12) is derived by letting T ! 1 in (1.11). Therefore, if 0 < a < 1;

then (1.12) is a unique, non-diverging solution. In the case of a > 1; let us define a

fundamental solution as

pO t D

1

1

aÂ

bt C

a .1

1

Â/ bN

:

a

Then it is obvious that the following also fulfills pt D aEt ptC1 C bt W

pt D pO t C

t;

where t is a white noise with Et "tCj D 0 for all j 0: Consequently, the necessary

condition for the presence of sunspot equilibrium is a > 1; which ensures the

local indeterminacy condition for the corresponding system without fundamental

disturbances.

1.3 General Equilibrium Models of the Monetary Economy

The simple model examined in the previous section lacks microfoundation. In this

section, we reconsider indeterminacy and sunspots in general equilibrium models

of monetary economies in which agents’ optimization behaviors are explicitly

formulated. As shown in the previous section, the key condition for the presence

of sunspot-driven fluctuations in a stochastic model is that the corresponding

deterministic models with perfect foresight display equilibrium indeterminacy. For

expositional convenience, in this section we focus on continuous-time, deterministic

models of monetary economies.

1.3.1 Base Model

Consider a money-in-the-utility function model of an exchange economy. There is

an infinitely lived representative household that maximizes a discounted sum of

utilities

Ã

Â

Z 1

M

t

dt;

>0

UD

e u c;

p

0

1.3 General Equilibrium Models of the Monetary Economy

9

subject to the flow budget constraint:

P D RB C p . y C

BP C M

c/ :

Here, c is consumption, y is the real income, M is the nominal money stock, B is the

stock of private bond, p is the price level, R is the nominal interest rate, and denotes

a real transfer from the government. The initial holdings of nominal stocks of money

and bond, M0 and B0 ; are exogenously specified. For simplicity, we assume that the

real income y is an exogenously given endowment that is a positive constant.

Let us define A D B C M; a D A=p; m D M=p and D pP =p: Then, we find that

the flow budget constraint given above is expressed as

aP D .R

/aCyC

c

Rm:

The instantaneous utility function, u .c; m/, is assumed to be monotonically increasing and strictly concave with respect to consumption, c; and real money balances,

M=p:

Denoting the implicit price of total asset, a; by q; the household’s optimization

conditions include the following:

uc .c; m/ D q;

(1.13)

um .c; m/ D Rq:

(1.14)

Conditions (1.13) and (1.14) yield

um .c; m/

D R;

uc .c; m/

(1.15)

which means that the marginal rate of substitution between consumption and real

balances equals the nominal interest rate. The implicit price of asset, q; changes

according to

qP D q . C

R/ :

(1.16)

Additionally, the implicit value of asset, qa; should fulfill the transversality condition, limt!1 e t qt at D 0:

The market equilibrium condition for final goods is

c D y:

(1.17)

Since there is no outstanding bond, the equilibrium condition for the financial

market is

b D 0:

(1.18)

10

1 Introduction

Finally, we assume that the seigniorage revenue of the government is distributed

back the households as a lump-sum transfer. Thus, the government’s budget

constraint is

P Dp :

M

(1.19)

1.3.2 The Case with a Unique Steady State

We now assume that the monetary authority keeps the growth rate of nominal money

stock at a constant rate of : Hence, the government’s budget (1.19) is expressed as

D m: Equations (1.13) and (1.15), together with (1.17), present

uc . y; m/

m

P

D

m

ucm . y; m/

Ä

C

um . y; m/

:

uc . y; m/

Eliminating from the above by use of

D

m=m;

P

we obtain a complete

dynamic system with respect to the real money balances as follows:

uc . y; m/ m

m

P D

ucm . y; m/ C uc . y; m/

Ä

C

um . y; m/

:

uc . y; m/

(1.20)

Now assume that there is a stationary solution of m that fulfills

C

D

um . y; m /

:

uc . y; m /

Since mt D Mt =pt is a jump variable, local determinacy holds if

ˇ

dm

P t ˇˇ

D

dmt ˇmt Dm

Ä

d um . y; m /

uc . y; m / m

> 0:

ucm . y; m / C uc . y; m / dm uc . y; m /

Since the above shows that the linearized system has an unstable root and

mt .D Mt =pt / is a jump variable, the economy always stays in the steady state

so that local indeterminacy cannot emerge.

Conversely, if the following condition holds, the steady state of the monetary

economy exhibits local indeterminacy:

ˇ

dm

P t ˇˇ

D

dmt ˇmt Dm

Ä

d um . y; m /

uc . y; m / m

< 0:

ucm . y; m / C uc . y; m / dm uc . y; m /

Under the above condition, the linearized system has a stable root so that any m0

around m can lead the economy to the steady state equilibrium.

1.3 General Equilibrium Models of the Monetary Economy

11

It is to be noted that non-separability of the utility function is a key condition for

holding local intermediacy. To see this, suppose that the utility function is additively

separable in such a way that

u .c; m/ D v .v/ C x .m/ ;

where v .c/ and x .m/ satisfy strict concavity. Given this specification, (1.20)

becomes

Ä

x0 .m/

:

(1.21)

m

P Dm C

v 0 . y/

It is easy to confirm that this system gives a unique steady value of mt and that

ˇ

dm

P t ˇˇ

D

dmt ˇmt Dm

m

x00 .m /

> 0;

v 0 . y/

where m is the steady state level of real money balances.

1.3.3 The Case with Multiple Steady States

Monetary economies often involve multiple steady states. In the following, we

examine two typical examples.

1.3.3.1 Hyper Inflation

Brock (1974) is the first study on the perfect-foresight competitive equilibrium of

money in the utility function model (the Sidrauski model). He pointed out that

the hyper-deflationary path on which real money balances go to infinity can be

eliminated by the transversality condition on the household’s optimization behavior.

At the same time, Brock (1974) also reveals that the hyper inflationary path on

which real money balances converge to zero may be supported as a perfect-foresight

competitive equilibrium.

Obstfeld and Rogoff (1983) present a comprehensive discussion on the presence

of hyper-inflationary equilibrium. According to their analysis, when the utility

function is additively separable, the phase diagram of (1.21) has three alternative

patterns as depicted by Paths A, B and C in Fig. 1.2. We see that each path satisfies

12

1 Introduction

&

m

Fig. 1.2 Alternativve paths

⎡

x ' ( m )⎤

& = m ⎢ρ + μ −

m

⎥

'(m)⎦

v

⎣

0

m*

A

m

B

C

the following conditions:

Path A W lim

mt !0

Path B W

mt v 0 .mt / D 0;

1 < lim

mt !0

Path C W lim

mt !0

mt v 0 .mt / < 0;

mt v 0 .mt / D 1:

Path A has two steady states, that is, an interior steady state wherein mt D m and

a non-monetary steady state wherein mt D 0: As claimed by Brock (1974), since

the non-monetary steady state satisfies the transversality condition, it fulfills all the

conditions for perfect-foresight competitive equilibrium. On the other hand, if the

equilibrium path is either Path B or Path C; the transversailty condition is violated,

so that the hyperinflationary path cannot be in competitive equilibrium. However,

Obstfeld and Rogoff (1983) prove that to realize Paths B and C; the utility function

should satisfy

lim v .mt / D 1;

mt !0

implying that the household’s utility becomes minus infinity when its real balance

holding conveyers to zero. This is obviously an extreme assumption as to the utility

of holding money. As a result, the feasible equilibrium is Path A alone. This means

1.3 General Equilibrium Models of the Monetary Economy

13

that the equilibrium of the economy is either the interior steady state, mt D m ;

or a path that converges to mt D 0: In this sense, the economy exhibits global

indeterminacy.

1.3.3.2 Taylor Rule

In the previous example, one of the dual steady states is a boundary point .mt D 0/ :

We now consider the case of dual interior steady states. Suppose that the monetary

authority adjusts nominal interest rate in response to the rate of inflation in such a

way that

R D R . / ; R0 . / > 1:

(1.22)

That is, the monetary authority follows the Taylor principle under which a rise in the

rate of inflation increases the real interest rate, r D R

: Notice that in this policy

regime, the nominal money stock is adjusted in order to support the interest-rate

control rule mentioned above.

In this example, we use a non-separable utility function in which the consumption

and real money balances are Edgeworth complements to each other so that

ucm .c; m/ > 0: First, condition (1.13) and the market equilibrium condition, y D c;

give uc . y; m/ D q: Thus, due to the assumption of ucm > 0; the relation between

m and q is expressed as m D m .q/ with m0 .m/ > 0: Then, (1.14) leads to

um . y; m .q// D qR: As a result, the relation between q and R is given by

q D Q .R/ ; Q0 .R/ D

umm m0 q um q

< 0:

q2

(1.23)

From (1.22) and (1.23), we obtain

qP

Q0 .R/ P

Q0 .R/ 0

D

R P:

RD

q

Q .R/

Q .R/

By use of (1.16), the above equation yields a complete dynamic system of the rate

of inflation in such a way that

P D

R0

Q .R . //

Œ C

. / Q0 .R . //

R . / :

satisfies

The steady state rate of inflation denoted by

R

D

C

;

(1.24)