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Linking Behavioral Economics, Axiomatic Decision Theory

and General Equilibrium Theory

A Dissertation

Presented to the Faculty of the Graduate School

of

Yale University

in Candidacy for the Degree of

Doctor of Philosophy

by

Katsutoshi Wakai

Dissertation Director: Professor Stephen Morris

May 2002

c 2002 by Katsutoshi Wakai

All rights reserved.

Abstract

Linking Behavioral Economics, Axiomatic Decision Theory

and General Equilibrium Theory

Katsutoshi Wakai

2002

My dissertation links behavioral economics, axiomatic decision theory and general equi-

librium theory to analyze issues in nancial economics. I investigate two behavioral con-

cepts: time-variability aversion, i.e., the aversion to volatility (uctuation in payo¤s over

time) and uncertainty aversion, i.e., the aversion to uncertainty of state realizations. Chap-

ter 1 develops a new intertemporal choice theory by endogenizing discount factors based on

time-variability aversion, and shows that the new model can explain widely noted stylized

facts in nance. I nd that (1) time-variability aversion can be represented by time-varying

discount factors based on very parsimonious axioms; (2) under the assumption of dynamic

consistency, time-variability aversion implies gain/loss asymmetry in discount factors (3)

the gain/loss asymmetry boosts e¤ective risk aversion over states by extreme dislike of

losses while maintaining positive average time-discounting. This intertemporal substitution

mechanism explains why the risk premium of equity needs to be very high relative to the

risk-free rate.

Chapter 2 provides the conditions under which the no-trade theorem of Milgrom &

Stokey (1982) holds for an economy of agents whose preferences follow uncertainty aversion

as captured by the multiple prior model of Gilboa and Schmeidler (1989). First, I prove

that given the agentsknowledge of the ltration, dynamic consistency and consequentialism

imply that a set of ex-ante priors must satisfy the recursive structure. Next, I show that with

perfect anticipation of ex-post knowledge, the no-trade theorem holds under the economy

such that agents follow dynamically consistent multiple prior preferences.

Chapter 3 examines risk-sharing among agents who are uncertainty averse. The main

objective is to provide conditions in the exchange economy such that agentse¤ective priors

(and equilibrium consumptions) will be comonotonic and their marginal rates of substitution

(weighted by these priors) will be equalized when agents have heterogeneous multiple prior

sets. One set of su¢ cient conditions is for each agents multiple prior set to be symmetric

(or to be dened by a convex capacity) around the center of the simplex.

Acknowledgments:

I thank my committee, Stephen Morris (chairman), Benjamin Polak, and John Geanako-

plos for their valuable suggestions. I beneted greatly from their advice and encouragement,

which helped me to complete this dissertation. I am also grateful to Itzhak Gilboa for pro-

viding invaluable advice regarding Chapter 2.

I also appreciated comments from Giuseppe Moscarini, Robert Shiller, Leeat Yariv, and

especially those from Larry Epstein for the work of Chapter 2, and from Larry Blume for

the work in Chapters 3 and 4.

Finally, I owe Max Schanzenbach for his help in proof reading. All errors are strictly

my own responsibility.

v

Table of Contents:

Acknowledgments ... v

Chapter 1 - Introduction ... 1

1.1 Introduction ... 2

Chapter 2 - A Model of Consumption Smoothing ... 6

with an Application to Asset Pricing

2.1 Introduction ... 7

2.2 Time-Variability vs. Atemporal Risk ... 12

2.3 Multiple Discount Factors under Certainty ... 14

2.3.1 Multiple Discount Factors: Examples ... 14

2.3.2 Representation of Intertemporal Preferences ... 16

2.3.3 Interpretation of Discount Factors ... 24

2.3.4 Application of (2.3.2) ... 25

2.4 Multiple Discount Factors under Uncertainty ... 27

2.4.1 Representation of Intertemporal Preferences ... 27

2.4.2 Interpretation of Discount Factors ... 31

2.5 Implications for Asset Pricing under Multiple Discount Factors ... 33

2.5.1 Asset Pricing Equation ... 33

vi

2.5.2 Calibration: Equity-Premium and Risk-Free-Rate Puzzles ... 36

2.5.3 Estimation: Simple Test for UK Data ... 43

2.6 Comparison with Other Intertemporal Utility Functions ... 50

2.6.1 Recursive Utility, Gilboa (1989) and Shalev (1997) ... 50

2.6.2 Loss Aversion and Habit Formation ... 52

2.6.3 Comparison of Empirical Implications ... 55

2.7 Derivation of the Representation of (2.4.1) ... 57

2.8 Conclusions and Extensions ... 64

Appendices 2.A - 2.F ... 66

References ... 92

Chapter 3 - Conditions for Dynamic Consistency ... 97

and No-Trade Theorem under Multiple Priors

3.1 Introduction ... 98

3.2 Consistency for Individual Preference ... 101

3.3 Ex-ante and Ex-post Knowledge ... 117

3.4 Consistency under Equilibrium ... 125

3.5 Conclusion ... 131

Appendices 3.A - 3.C ... 132

References ... 143

vii

Chapter 4 - Aggregation of Agents with Multiple Priors ... 145

and Homogeneous Equilibrium Behavior

4.1 Introduction ... 146

4.2 Stochastic Exchange Economy with Uncertainty Aversion ... 150

4.2.1 Intertemporal Utility Functions and Structure of Beliefs ... 150

4.2.2 The Structure of Economy ... 154

4.2.3 Special Case ... 157

4.2.4 Utility Supergradients and Asset Prices ... 159

4.3 Single Agent Economy ... 161

4.3.1 Background ... 161

4.3.2 General Order Property of Utility Process ... 162

4.3.3 Su¢ cient Conditions for the Order Property ... 165

4.3.4 Time and State Heterogeneous Prior Set ... 169

4.4 Multiple Agents Economy with the Identical MP Sets ... 173

4.4.1 Background ... 173

4.4.2 Denition of the Representative Agent ... 175

4.4.3 Single Period Economy ... 177

4.4.4 Dynamic Setting ... 184

4.4.5 Su¢ cient Conditions for the Representative Agent ... 192

4.5 Multiple Agents Economy with the Heterogeneous MP Sets ... 195

4.5.1 Background ... 195

viii

4.5.2 Denition of Commonality ... 198

4.5.3 Single Period Economy ... 201

4.5.4 Dynamic Setting ... 209

4.6 Continuum of Equilibrium Prices ... 217

4.6.1 Single Agent Economy ... 217

4.6.2 Multiple Agents Economy ... 219

4.7 Conclusion ... 222

4.8 Extension ... 223

Appendices 4.A - 4.N ... 224

References ... 255

ix

Chapter 1

Introduction

1

1.1 Introduction

My dissertation links behavioral economics, axiomatic decision theory and general equilib-

rium theory to analyze issues in nancial economics. The behavioral issues I investigate are

time-variability aversion and uncertainty aversion. The analysis develops new theories and

combines them with estimation and calibration.

Chapter 1 develops a new behavioral notion, time-variability aversion, and then applies

this idea to a consumption-saving problem to derive implications for asset pricing. Con-

ventionally, risk aversion is regarded as dislike of variations in payo¤s of random variables

within a period. By contrast, time-variability is variation in payo¤s over time. In princi-

ple, an agent could be averse to such variation even in the absence of risk. For example,

Loewenstein & Prelec (1993) show that, in experiments, agents prefer smooth allocations

over time even under certainty, and their preferences for smoothing cannot be explained by

a time-separable discounted utility representation.

I dene time-variability aversion to mean that an agent is averse to mean-preserving

spreads of utility over time. To capture this idea, I provide a representation, adapting a

method developed in a di¤erent context by Gilboa & Schmeidler (1989). In this represen-

tation, risk aversion is captured by the concavity of a von Neumann-Morgenstern utility

function. Time-variation aversion is captured by the agent selecting a sequence of (normal-

ized) discount factors (from a given set) that minimizes the present discounted value of a

given payo¤ stream. I provide an axiomatization for this representation. More formally, the

assignment of discount factors is determined recursively. At each time t, the agent compares

2

present consumption with the discounted present value of future consumption from t+1 on-

ward and then selects the time-t discount factor to minimize the weighted sum of these

two values. These recursive preferences are non-time-separable and dynamically consistent

by construction (but they di¤er in form and implication from those used by Epstein & Zin

(1989)). Intuitively, this representation exhibits time-variability aversion by allocating a

high discount factor when tomorrows consumpt