I. Introduction

Growing evidence casts doubt on the view that asset prices are determined by investors’ unbiased expectations of fundamental values, ruling out persistent over- and undervaluation of assets.² To explain the occurrence of mispricing, behavioral finance has analyzed various systematic psychological phenomena that might interfere with efficient asset pricing. However, one of the most influential theories in social psychology—namely, cognitive dissonance—has not been applied to asset pricing. This paper attempts to fill that gap by allowing for the possibility that investors might alter their beliefs in response to conflicting cognitions: when information about the assets they hold is sufficiently unfavorable, investors might discard the information because it conflicts with their self-image of being smart investors. We show how this type of behavior can be modeled as the outcome of an individual optimization problem, how it can lead to over- and undervaluation of assets, and how it can cause sometimes too much and sometimes too little asset price volatility.

A broad range of evidence suggests that asset prices may not always be priced efficiently. In addition to the well-known equity premium puzzle (Mehra and Prescott, 1985) and indications of excess volatility (Shiller, 1981), recent studies have found evidence for underreaction and overreaction of stock prices.³ At short horizons, stock returns tend to exhibit positive autocorrelation, indicating that stock prices underreact to news as new information is only slowly incorporated into the prices.⁴ By contrast, over horizons of three to five years, the autocorrelation in stock returns seems to be slightly negative, suggesting that stock prices overreact to extreme news, be it good or bad.⁵ Moreover, stock market crashes are typically difficult to reconcile with market efficiency, in part because the new information revealed just prior to stock market crashes does not usually warrant the huge drop in stock prices. All of these deviations from the efficient markets model appear to be systematic and remain largely unresolved within models where rational investors maximize their utility and/or material wealth and regard their information sets as exogenous (De Bondt and Thaler, 1995).⁶

To explain these phenomena, behavioral finance has integrated concepts developed by psychologists into asset pricing models (see Barberis and Thaler, 2002; Daniel and others, 2002; and Hirshleifer, 2001, for reviews).⁷ Psychologists find that agents are not purely rational in processing and interpreting information. Research indicates that people tend to systematically underweight some types of information and overweight other types (for surveys, see Rabin, 1998; and Shiller, 1998). Three closely related biases, in particular, have been identified, and their implications for asset pricing have been analyzed: (1) overconfidence—people tend to overstate their abilities and the precision of their knowledge (Daniel and others, 1998; and Odean, 1998); (2) availability/representativeness heuristic—agents tend to overweight memorable and vivid evidence, and they tend to underweight overall prior odds (Tversky and Kahneman, 1974); and (3) confirmatory bias—decision makers tend to overweight information that is consistent with their beliefs, and underweight information that is at odds (Fiske and Taylor, 1991; Rabin and Schrag, 1999).

In this paper, we introduce cognitive dissonance—one of the most influential theories of social psychology—into an asset pricing model.⁸ Festinger (1957) postulated that if individuals are presented with conflicting cognitions (pieces of knowledge), they suffer distress and feel compelled to change one of these cognitions. They either avoid information that is likely to increase dissonance or select information that is consistent with their a priori beliefs. In particular, individuals tend to have a positive self-image and do not want to feel incompetent. Thus, information that conflicts with the positive self-image tends to be ignored, rejected, or misinterpreted, or the conflict is resolved by changes in other beliefs (Harmon-Jones and others, 1999). For example, persons who have made decisions often tend to discard information that would suggest that the decisions were made in error. In general, the potential for cognitive dissonance can lead to the avoidance of information that could be inconsistent with one’s beliefs and attitudes.

Biases in decision making—including the effects of cognitive dissonance—may be particularly prevalent in financial market settings, since investors are confronted with complex and ambiguous evidence, and they have to selectively collect and interpret information. When faced with complex and ambiguous evidence, agents tend to overstate the strength of confirming information and to downplay the reliability of conflicting evidence (Rabin, 1998). Particularly when processing available information is costly so that some selection of information will have to be made, an investor might selectively ignore information about his asset portfolio if heeding that information reduces (expected) welfare.⁹ Moreover, these biases are particularly prevalent when the complexity and ambiguity of the situation requires the use of prior hypotheses to interpret data and to decide which data to scrutinize—another key feature of the decision making environment in financial markets.

In this paper, we construct a general equilibrium model where the quasirational behavior associated with cognitive dissonance emerges endogenously; that is, individuals choose to ignore unfavorable or contradictory information.¹⁰ In our model investors make investment decisions in the initial period based on prior beliefs about asset payoffs. If the information that arrives in the next period conflicts too much with their ex ante beliefs that they made the right portfolio choice, they choose to ignore the information. The thresholds that separate information that is heeded from information that is ignored are determined within the model and not exogenously set. Specifically, we model cognitive dissonance behavior by assuming that if the expected utility in the second period conditional on the signal exceeds the unconditional expected utility without the signal, the investor will heed the information; otherwise, he will ignore it.

The way we model cognitive dissonance may appear ad hoc, but we consider it a tractable approach with some advantages. A possible alternative approach (similar to the one adopted by Akerlof and Dickens, 1982) would arbitrarily assume psychological costs associated with deviations from the cognition of a positive self-image. The investor’s choice over the information set would then be modeled as a tradeoff between the loss from a suboptimal portfolio choice if the information is discarded and the costs of psychological discomfort due to cognitive dissonance when the information is heeded. We do not follow this route; instead we want to stay as close to the rational model as possible to facilitate direct comparisons of equilibrium outcomes based on rational decision making with those involving quasirational decision making.

Unlike in other models where quasirational behavior is assumed throughout, in our model, what looks like rational behavior emerges endogenously under some circumstances and quasirational behavior under other circumstances. In fact, and this is a relatively novel feature of our approach, even though agents have identical preferences and endowments, they do not necessarily behave in the same way as a result of general equilibrium effects: in certain equilibria some agents choose to act quasirationally, while others act rationally. Since investors choose their information sets by deciding whether to heed or to ignore some information, our model may provide micro foundation for some nonstandard behavioral assumptions.

Our key results describe how the asset price responds to the information that is revealed in the second period. We show how the equity price can both under- and overreact to a publicly observable signal—unlike in Daniel and others (1998) who found overreaction to the private signal and underreaction to the public signal.¹¹ Equity prices overreact to moderately negative information, while prices underreact to extremely positive and extremely negative information. For slightly worse information, the price becomes totally insensitive to the news. Since the deviation from rational behavior is endogenous, the mispricing is bounded—if information is sufficiently negative, it is advantageous for some investors to revert to rational behavior because these informed investors can earn abnormal profits based on their superior information that compensate them for the discomfort associated with cognitive dissonance. As a result, equity prices become again sensitive to information that is extremely unfavorable. However this arbitrage mechanism is not perfect, and price distortions remain in equilibrium. Moreover, excessive price volatility and excessive price stability arise endogenously. We also describe how the distortions of the pricing process are related to the precision of the information. In our model, the price mechanism remains distorted even if the signal becomes asymptotically uninformative—a finding that might contribute to an understanding of the emergence of sunspot equilibria.

The paper is organized as follows. The next section describes the model and outlines the two benchmark cases when all investors heed the information and when all investors ignore the information. Section III derives the equilibrium of the model with cognitive dissonance. Section IV explains how different realizations of the information signal affect the equilibrium asset price, and Section V explores the effects of changing the information content of the signal. Concluding remarks and a discussion of further research are presented in Section VI.

II. The Asset Pricing Model

The economy lasts for three periods and is populated by a continuum of identical agents, I = [0,1]. Agents consume only at time 3. Preferences are described by the utility function U (c) = -exp(-βc), where β is the constant degree of absolute risk aversion; c is the final wealth to be consumed by the agent at time 3. Two assets are available: a risky asset (shares) and a safe asset, with a price that is normalized to one (numeraire). At time 1, each agent is endowed with one unit of the risky asset and e > 0 units of the safe asset. The safe asset pays a fixed rate of return, r, in periods 2 and 3, and is in perfectly elastic supply. The second asset pays a random dividend, $\tilde{d}=\tilde{\gamma }+\tilde{\epsilon }$, in period 3. The variables $\tilde{\gamma }$ and $\tilde{\epsilon }$ are stochastically independent random variables with $E\{\tilde{\gamma }\}>0$ and $E\{\tilde{\epsilon }\}=0$. The realization of $\tilde{\epsilon }$ can be observed at time 2, while $\tilde{\gamma }$ will not be revealed prior to time 3. Thus some information about the risky asset’s payoff is revealed before the dividend, $\tilde{d}$, actually materializes. We denote the realizations of the random variables $\tilde{\gamma }$ and $\tilde{\epsilon }$ by γ and ε. In light of the new information that is received at time 2, investors may reconsider their portfolio decisions made at time 1.

The sequence of events can be summarized as follows:

• Period 1: investors choose their portfolios and can trade on the asset markets.

• Period 2: the investors observe the signal ε, and then decide whether to heed the information and, if necessary, to reshape their portfolios by trading on the asset markets, or whether to ignore the new information to avoid internal conflict (such as cognitive dissonance).

• Period 3: the investors consume the asset payoffs.

In period 1, each agent can invest his endowment of e units of the safe asset and one unit of the risky asset into a portfolio consisting of both assets. The budget constraint in period 1 is

where p_t is the real price of a share at time t, while the price of the safe asset is normalized to 1; s_t denotes the units of the safe asset held at the end of period t; and λ_t denotes the number of shares held at the end of time t.

In period 2, the portfolio can be changed subject to the constraint

Wealth (i.e. consumption) in period 3 is

At the beginning of period 2, when the signal ε is revealed, investors may find themselves facing an internal psychological conflict akin to cognitive dissonance. On the one hand, agents want to consider themselves to be smart investors; on the other hand, a realization of the information ε that differs significantly from the agents’ ex ante expectations would indicate that, in hindsight, they might have chosen the wrong portfolio at time 1. To resolve this internal conflict, investors might be prompted to ignore the signal ε. We will model the choice of heeding or ignoring the information as follows: the investor will ignore the new information if his unconditional expected utility (without the information of the signal) is at least as large as the expected utility conditional on the new information. Since the (conditional) expected utility depends, inter alia, on the portfolio choice at time 1, this approach captures the key aspect of cognitive dissonance: information that conflicts too much with the portfolio decision taken earlier will be ignored. To derive the effects of this cognitive bias on asset prices, we will first consider two benchmark cases before we turn to a detailed description of the equilibrium with cognitive dissonance: (1) all investors heed the information, and (2) all investors ignore the information.

III. Asset Pricing with Cognitive Dissonance

We now turn to modeling how each investor individually decides whether or not to heed the information signal—and thus whether to act rationally or quasirationally. This decision is modeled as follows: if the investor achieves a higher expected utility at time 2 by ignoring the signal, he will choose to do so; otherwise he will rationally use the information provided by the signal.

Suppose an investor heeds the information ε and observes the asset price p at time 2.¹⁴ In equilibrium, all investors hold the same asset portfolio at date 1 consisting of e bonds and one share, i.e. s₁ = e, λ₁ = 1. Let the full-information value function at time 2, Ψ(p, ε), be defined by

and denote the solution to the problem by ${\lambda }_2^H(p,\epsilon )$; it is the per-capita demand for the risky asset when the information ε is heeded.

By contrast, suppose the investor ignores the information ε but observes the asset price p. In this case, let Φ(p) denote the no-information value function, which is defined as

It is the maximum expected utility at price p when the signal is ignored. The demand for the risky asset in period 2 is in this case denoted by ${\lambda }_2^D(p)$.

Formally, for a given signal ε, an investor faces cognitive dissonance and will choose to ignore the information ε if and only if

We can now define the Nash equilibria in this model.

Definition 1 (equilibria in pure or mixed strategies): Consider a realization of the signal ε′.

1. $\left\{{\lambda }_2^H\left(p^{*}(\epsilon \text{'}),\epsilon \text{'}\right),p^{*}(\epsilon \text{'})\right\}$ is a pure-strategy full-information equilibrium at ε′ if

   1. p^*(ε′) is such that ${\lambda }_2^H\left(p^{*}(\epsilon \text{'}),\epsilon \text{'}\right)=1$;

   2. Ψ(p^*(ε′),ε′) ≥ Φ(p^*(ε′)).

2. $\left\{{\lambda }_2^D(\overline{p}),\overline{p}\right\}$ is a pure-strategy no-information equilibrium at ε′ if

   1. $\overline{p}$ is such that ${\lambda }_2^D(\overline{p})=1$;

   2. ${\Phi }(\overline{p})\ge {\Psi }(\overline{p},\epsilon \text{'})$.

3. $\{{\lambda }_2^H\left(\widehat{p}(\epsilon \text{'}),\epsilon \text{'}\right),{\lambda }_2^D\left(\widehat{p}(\epsilon \text{'})\right),\widehat{p}(\epsilon \text{'})\}$ is a mixed-strategy equilibrium at ε′ if

   1. $\exists \mu \in (0,1)$ such that $\mu {\lambda }_2^H\left(\widehat{p}(\epsilon \text{'}),\epsilon \text{'}\right)+(1-\mu ){\lambda }_2^D\left(\widehat{p}(\epsilon \text{'})\right)=1$

   2. ${\Psi }\left(\widehat{p}(\epsilon \text{'}),\epsilon \text{'}\right)={\Phi }\left(\widehat{p}(\epsilon \text{'})\right)$.

The conditions (ii) in parts (a), (b), and (c) of Definition 1 ensure that, in equilibrium, an agent will not ignore (heed) the signal if he can achieve a higher welfare by heeding (ignoring) the information.

Before we can characterize the equilibria, a few useful properties of the value functions need to be derived (see Figure 1).

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Value Functions of the Investors’ Optimization Problem

Citation: IMF Working Papers 2005, 009; 10.5089/9781451860283.001.A001

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Step 1: ${\lambda }_2^H$ satisfies the first-order condition

from which we derive

since ${\lambda }_2^H=1$ holds in equilibrium. In fact, the above inequality is valid for any p that satisfies ${\lambda }_2^H(p,\epsilon )=1$. Thus we conclude

Step 2: Differentiation of the full-information value function with respect to p yields

Combining (13) and (14), we get

which proves the claim. QED.

To simplify the characterization of the equilibrium equity price function at time 2, we now assume that $\tilde{\gamma }$ and $\tilde{\epsilon }$ have a multivariate normal distribution. Recall that individual preferences are U (c) = -exp(-βc) and that in equilibrium all investors hold the same asset portfolio at time 1, (s₁, λ₁) = (e, 1). Thus, if an investor observes the signal ε, his expected utility at time 2 is

where ${\sigma }_{\gamma }:=\mathrm{var}(\tilde{\gamma })$ and $\overline{e}:={(1+r)}^{2}e$. If the agent does not heed the signal ε, his perceived expected utility at time 2 is

where ${\sigma }_{\epsilon }:=\mathrm{var}(\tilde{\epsilon })$. The first-order necessary conditions are

and

Thus, desired equity holdings as a function of the equity price are

when the information is heeded, and

when the information is ignored.

IV. How Information Affects Asset Prices

Having derived investors’ asset demand functions, we can now investigate how equity prices respond to new information. The key results are presented by an equity price function that describes how the price depends on the realization of the signal ε. The shape of this function will differ for various regions of the realization of the signal. Each region, it turns out, is characterized by a specific price formation process resulting from the way in which investors treat the new information (Figure 2).

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The Equilibrium Pricing Function

Citation: IMF Working Papers 2005, 009; 10.5089/9781451860283.001.A001

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