DO MARKETS PROVE PESSIMISTS RIGHT?

• Author: Jürgen Eichberger,Ani Guerdjikova
• Published date: 01 November 2018
• Date: 01 November 2018
• DOI: http://doi.org/10.1111/iere.12336

INTERNATIONAL ECONOMIC REVIEW

Vol. 59, No. 4, November 2018 DOI: 10.1111/iere.12336

DO MARKETS PROVE PESSIMISTS RIGHT?∗

BYJ¨

URGEN EICHBERGER AND ANI GUERDJIKOVA1

University of Heidelberg, Germany; University of Grenoble Alpes, France

We study how ambiguity and ambiguity attitudes affect asset prices when consumers form their expectations

based on past observations. In an overlapping generations economy with risk-neutral yet ambiguity-sensitive

consumers, we describe limiting asset prices depending on the proportion of investor types. We then study the

evolution of consumer-type shares. With long memory, the market does not select for ambiguity neutrality.

Whenever perceived ambiguity is sufficiently small, but positive, only pessimists survive and determine prices in

the limit. With one-period memory, equilibrium prices are determined by Bayesians. Yet, the average price of

the risky asset is lower than its fundamental value.

1. INTRODUCTION

There is a widespread view that optimism and pessimism may cause excessive procyclical

buying or selling in financial markets. In consequence, asset prices may substantially and per-

sistently deviate from their fundamental values. In an article on the financial crisis, Shefrin and

Statman (2012) quote Keynes on the psychology of financial booms and crises:

The later stages of the boom are characterized by optimistic expectations as to the future yield of capital

goods . .. of speculators who are more concerned with forecasting the next shift of market sentiment

than with a reasonable estimate of the future yield of capital assets, that when disillusion falls upon

an over-optimistic and over-bought market, it should fall with sudden and even catastrophic force.

Moreover, the dismay and uncertainty as to the future which accompanies a collapse in the marginal

efficiency of capital naturally precipitates a sharp increase in liquidity preference. .. . It is not so easy

to revive the marginal efficiency of capital, determined as it is by the uncontrollable and disobedient

psychology of the business world. It is the return of confidence, to speak in ordinary language, which is

so insusceptible to control in an economy of individualistic capitalism Keynes (1936, pp. 315–7).

In this view, investors’ trades in asset markets rely on forecasts not only about the unknown

real returns but also about prices that are endogenously generated by demand and supply.

Investors’ preferences and beliefs determine the equilibrium prices that, in turn, feed back into

the updates of these beliefs. Depending on whether investors hold identical or heterogeneous

subjective beliefs, this feedback process may lead to upward- or downward-biased price predic-

tions and, in consequence, to converging or cyclical price processes. Most of the literature (see

Section 1.2) on the dynamics of financial markets assumes that investors are subjective expected

utility (SEU) maximizers who update their beliefs according to Bayes’ law. Combined with the

assumption of rational expectations, embodied in an equilibrium price functional, procyclical

price movements enter these models via trading frictions and constraints. The “uncontrollable

∗Manuscript received July 2016; revised May 2017.

1This research was supported by grant ANR 2011 CHEX 006 01 of the Agence Nationale de Recherche, by IUF,

and by Labex MME-DII. The hospitality of HIM, Bonn, is gratefully acknowledged. We are thankful to Pablo Beker,

Eva Loecherbach, Marius Ochea, Joerg Oechsler, John Quiggin, Sean Tweedie, and the participants of the Workshop

on Ambiguity (Theory & Applications) at Essex and the Workshop on Heterogeneity and Dynamics in Markets and

Games at Cergy, as well as the seminar participants at the University of Cergy and the Australian National University for

helpful comments and suggestions. Please address correspondence to: Ani Guerdjikova, University of Grenoble Alpes,

1241 rue des R´

esidences 38400 Saint Martin d’H`

eres, France. Phone: +33456528578. E-mail: ani.guerdjikova@univ-

grenoble-alpes.fr.

2259

C

(2018) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

2260 EICHBERGER AND GUERDJIKOVA

and disobedient psychology of the business world” is controlled by a rational expectations

equilibrium price function.

In Eichberger and Guerdjikova (2013), we provide an α-max-min expected utility (α-MEU)

representation of preferences and beliefs, when information is in the form of data sets as in

the case-based decision theory initiated by Gilboa and Schmeidler (2001). The beliefs are given

by a set of probability distributions. They depend not only on the frequency of observed cases

but also on the degree of perceived ambiguity that itself is a function of the type and num-

ber of observations. Attitudes toward uncertainty make investors bias their decision-relevant

beliefs, upward for optimists, and downward for pessimists, whenever beliefs are ambiguous.

If an investor feels no ambiguity, either as an SEU-maximizer or because ambiguity vanishes,

possibly with large amounts of consistent data, then the individual’s attitude becomes irrele-

vant. Thus, over- and underweighting of predictions will depend both on attitudes and on the

endogenously generated price data. We feel that this type of preferences and beliefs can capture

some of the “uncontrollable and disobedient psychology of the business world” as mentioned by

Keynes.

The dynamics of asset prices in a model with case-based investors facing ambiguity will

depend on the distribution of ambiguity attitudes in the population. In turn, as the quote

from Keynes suggests, the general level of optimism and pessimism in the population might

itself be driven by the market. Results in social psychology show that optimism and pessimism

may be intergenerationally transmitted. Zuckerman (2001, p. 184) finds that these attitudes

may be adaptive traits “selected in evolution for our species” characterized by “a low . .. but

significant heritability. . . . Optimism is also influenced by shared familial factors and nonshared

life events, but pessimism seems to be primarily learned by events outside of the shared family

environment.”

Evidence also suggests that macroeconomic events may have an impact on investors’ am-

biguity attitudes and, thus, on their market behavior. Malmendier and Nagel (2011) find that

individuals who have experienced low stock market returns during the Great Depression show

lower willingness to take financial risk, are less likely to participate in the stock market, invest

a lower fraction of their liquid assets in stocks, and are more pessimistic about future stock re-

turns. Such experience need not be “personal”—individual preferences may adjust in response

to the observed performance of others.

In this article, we propose a model of financial markets where (i) investors form ambiguous

beliefs about future asset prices and dividends based on endogenously generated financial data,

(ii) the population of investors consists of optimists, pessimists, and Bayesian SEU maximiz-

ers, and (iii) the shares of these three types of investors evolve depending on their market

performance. Although data about prices are generated endogenously, we treat the amount of

data, that is, the length of memory, on which investors base their predictions as an exogenous

variable. We focus on two cases: short memory, containing the μmost recent observations, and

long memory, containing all past observations. We also treat as exogenous the function relating

the degree of ambiguity to the amount of available data. We discuss the case where ambiguity

vanishes, that is, the set of probability distributions shrinks to a singleton, as the amount of data

grows and the case where ambiguity is persistent, that is, the set of probability distributions

does not converge to a singleton, no matter how much evidence is available.

1.1. Framework and Results. We consider an overlapping generations (OLG) model with

a risky asset and a riskless bond. Investors’ preferences and beliefs are described by the case-

based α-MEU representation developed in Eichberger and Guerdjikova (2013). Investors live

for two periods. They observe data from an exogenous i.i.d. dividend process and endogenously

determined past asset prices. Although each generation plans for just two periods, investors

have access to data sets containing observations of previous generations. Based on these ob-

servations, they form expectations about asset returns. Ambiguity arises because investors

perceive uncertainty about the precision of their predictions depending on quantity and quality

DO MARKETS PROVE PESSIMISTS RIGHT? 2261

of data. We distinguish three types of investors as to their attitude toward ambiguity: optimists,

pessimists, and ambiguity-neutral Bayesians.

With no ambiguity and infinitely many observations, predicted asset prices converge toward

their fundamental values, and in the limit investors hold rational expectations. In general,

however, predictions of future returns depend on the investors’ (limited) memory, on their

perceived ambiguity, and on their attitude toward this ambiguity. Hence, asset prices reflect

these characteristics as well as the shares of investor types in the market.

For fixed shares of types in the population, if memory includes all past cases, consumers learn

the dividend process, and prices converge. However, if ambiguity is persistent, the limit price

exceeds (falls below) the fundamental value for a high share of optimists (pessimists). Market

prices are thus biased by the ambiguity attitudes in the population.

When the memory is short, investors cannot learn the dividend process, and the asset price

does not converge. The price dynamics can be described by an irreducible recurrent Markov

process. The support of the invariant distribution of this process depends on the shares of types

in the economy: It is shifted up (down) if optimists (pessimists) dominate the market. In general,

however, short memory produces cycles of optimism, Bayesianism, and pessimism, each of the

regimes occurring with strictly positive probability.

To capture the idea that attitudes toward ambiguity in the population may evolve, as sug-

gested by Zuckerman (2001) and Malmendier and Nagel (2011), we next assume, in the spirit of

the indirect evolutionary approach2initiated by G ¨

uth and Yaari (1992), that the proportions of

investor types adjust to imitate the more successful types in previous generations. We capture

this process by a replicator dynamics.3Equilibrium prices and population shares are now de-

termined by the learning dynamics for a given memory length and by the replicator dynamics.

When the price of the asset is constant and equals its fundamental value, the replicator dynamics

favors the more cautious pessimistic investors. Hence, the state in which the asset prices are

set by Bayesian investors with rational expectations and coincide with the fundamental values

is not stable. With infinite memory, only pessimists survive in the unique stable steady state of

the economy. Furthermore, with small but persistent ambiguity, the economy converges to this

pessimistic steady state almost surely and in expectations.

In an economy with one-period memory and without Bayesian investors, cycles emerge: A

sequence of high (low) dividend realizations leads to an “optimistic (pessimistic) market,” in

which the share of optimists (pessimists) is relatively high and the equilibrium price equals the

optimists’ (pessimists’) reservation price.

With all three types of consumers and one-period memory, no cycles occur. Almost surely,

after a finite number of periods, the economy reaches a state in which the equilibrium price

in each period is set by the Bayesian investors. However, the average price lies below the

fundamental value. Moreover, although the equilibrium price is determined by a “Bayesian”

regime, optimists or pessimists need not disappear.

1.2. Related Literature. The model presented in this article combines elements of (i) tem-

porary equilibrium theory with OLGs, (ii) case-based decision theory for agents learning from

data with ambiguous beliefs, and (iii) the theory of market selection. The discussion of the

literature focuses on these three aspects.

1.2.1. Rational expectations in stochastic dynamic general equilibrium and temporary equilib-

rium. The literature on dynamic economies with asset markets comprises two approaches

that model price expectations in different ways. The temporary equilibrium approach (with or

without OLG) assumes that expectations about future prices and endowments are based on

2In contrast to a large literature, which will be discussed in more detail in Section 1.2, in our model, evolutionary

pressure does not select for “investment strategies,” or beliefs, but for a feature of the preferences, specifically the

attitude toward ambiguity.

3Hofbauer and Schlag (2000) show that the replicator dynamics can be interpreted as an imitation process.