Growth and business cycles with equilibrium indeterminacy

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


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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
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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
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23
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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
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56
62
64
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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 . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

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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 . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

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Contents

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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
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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 . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

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172
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178

7 New Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.1 Microfoundations of Keynesian Economics . . . . . . .. . . . . . . . . . . . . . . . . . . .

7.2 Financial Frictions and Bubbles .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.3 Search Frictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.4 Agent Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

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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  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  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 a ¤ 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 . //
ΠC
. / Q0 .R . //

R . / :
satisfies

The steady state rate of inflation denoted by
R

D

C

;

(1.24)