Endogenous sunspots, pseudo-bubbles, and beliefs about beliefs

Abstract

We analyze a simple sunspot model that represents a standard securities market without sidebets on an exogenous sunspot event. The multiple self–fulfilling equilibria that arise in the model are based on investors' uncertainty about other investors' beliefs. Hence, these multiple equilibria are `endogenous sunspots'. We show that endogenous sunspots can arise even with complete markets to which all investors have access and homogeneous beliefs, provided the homogeneity is not common knowledge. We also show that endogenous sunspots can produce `pseudo–bubbles' in which the risky asset price is higher (or lower) at all dates than in a no–sunspot equilibrium.

Introduction

Large and rapid movements in securities prices, such as those of October 19–20, 1987, are dramatic events for which several possible explanations have been offered. One explanation is that these market movements, in fact all market price movements, reflect the arrival of new information about earnings prospects and other fundamental factors affecting the dividend stream. There is little question that such information plays a major role in much market movement but many market observers question the notion that this explanation covers cases such as the October 1987 crash, for example. At the other extreme is the explanation that fads and fashions are an important factor underlying securities prices and that large price movements may result from major changes in fads.

The many times that stock prices have changed by large amounts without any accompanying news that might explain the price move has led many to believe that stock markets are inherently emotional. Keynes in his classic writings expressed this intuition with his notion of animal spirits or market psychology affecting asset prices. More recently, financial economists have investigated more rigorously the nature of stock market volatility. The paper by Shiller (1981) for example has led to a growing literature trying to determine whether stock prices are too volatile relative to the volatility of their fundamental value. The related idea that trading creates volatility was given empirical support in French and Roll (1986) who found that prices are more volatile when markets are open for trading.

Economists starting with Cass and Shell have investigated asset pricing models that contain a role for market psychology or more generally extrinsic uncertainty referred to as `sunspots'. Although the seminal article on sunspot equilibria by Cass and Shell (1983) begins with reference to the stock market, sunspot equilibria have not played a significant role in models of securities markets in the financial economics literature. Rational expectations models incorporating sunspot equilibria have either not featured financial securities other than money or, in the case of the Cass–Shell model, involved securities whose payoffs are contingent on the occurrence of the exogenous event called `sunspots'. In other words, the financial securities available in the Cass–Shell sunspot equilibrium are sidebets about the occurrence of sunspots. Such sidebets on extraneous events are not, of course, a significant factor in actual securities markets.

In this paper we consider a simple `sunspot' model that represents standard financial assets where the role of market psychology or extrinsic uncertainty is played by beliefs about beliefs. We show that beliefs about beliefs are capable of explaining how trading by rational investors can create volatility that is unrelated to changes in fundamental value. The beliefs about beliefs equlibrium we consider in this paper fulfils the intuition of market psychology because the equilibrium in a sense pulls itself up by its own bootstraps. Investors in the model must believe that the equilibrium exists for it to exist. If even some investors believe that it might exist then it can exist.

One of the purposes of this paper is to explore the existence of multiple self-fulfilling equilibria in securities markets when the securities' payoffs are not contingent on some extraneous event unrelated to the fundamentals of the economy (i.e., preferences or endowments). That is, the securities in our model include the sorts of risky and riskless assets that figure in usual securities market models but exclude explicit sidebets on extraneous events, i.e., on sunspots. Consequently, the multiple self-fulfilling equilibria that arise in our model are based on factors endogenous to the market; hence, we call them `endogenous sunspots'. The factor that gives rise to endogenous sunspots in our model is uncertainty about the beliefs of other market participants.

In our model, markets are complete and it is common knowledge that individuals share common beliefs about payoff-relevant events. Individuals' beliefs about the endogenous-sunspot event are not common knowledge, however. We show that endogenous-sunspot equilibria can occur when there is uncertainty concerning aggregate beliefs about the sunspot event. In this situation, securities prices may be affected by beliefs about the endogenous-sunspot event when individuals actually share common beliefs about this event but the fact of common beliefs is not common knowledge.

Our pure exchange economy has three dates, of which the first two are times when securities are traded and the final date is when all security payoffs and consumption occur. All investors act as price takers in the model. There is a riskless security that serves as numeraire and a risky security. The item of particular interest in the model is the price of the risky security at the second trading date, especially how it is affected by investors' beliefs about it. The state of beliefs about this price acts as the event that generates the possibility of endogenous-sunspot equilibria. These equilibria exist because of investors' uncertainty about the beliefs of other investors about beliefs about the future price of the risky asset.

Because we wish to abstract from the effects of private information about payoff-relevant events, we assume that it is known that everyone has the same information about the payoff of the risky security. This immediately raises the question of why investors would care about the beliefs of other investors in choosing their portfolios. If we considered a one-period model in which returns depended only on final security payoffs, the answer is that they wouldn't.

Our model has two trading dates before security payoffs occur, so the return from holding the risky security at the first date depends on its price at the second date. Investors base their initial trading date portfolio choices on their beliefs about the price at the second trading date. The model contains investors with different risk preferences, so the conditions for aggregation in Brennan and Kraus (1978) and Rubinstein (1974) are not present. As a result, the price at the second trading date depends on the portfolio choices investors make at the initial trading date. However, the only factor that could lead to one price as opposed to another at the second trading date is differences in beliefs about that price, which means differences in beliefs about beliefs since nothing else is unknown.

Here is what we envisage. Each investor, being atomistic, has a belief about the probability distribution of the second trading date price. This can be thought of as a reduced form of that investor's beliefs about the portfolio choices other investors make based on their own beliefs. Different investors' beliefs may be similar or they may not; the only information an investor gets that reveals anything about others' beliefs is the initial trading date price. Investors choose initial trading date portfolios based on their beliefs about the second trading date price, which in turn is dependent on their initial trading date portfolios.

One set of beliefs and prices that satisfies this scenario is for every investor to believe that the only possible price at the second trading date is the initial trading date price. If everyone believes that the price will not change, then everyone trades to his or her final portfolio at the initial trading date and nothing changes at the final trading date. This is analogous to the `no trade theorem' in Milgrom and Stokey (1982). This no-sunspot equilibrium is one possible equilibrium, but not the only one. More interesting equilibria also exist that involve situations in which the price does change at the second trading date and further trading occurs. These endogenous-sunspot equilibria exist because investors are uncertain about what other investors are thinking – in other words, because of beliefs about beliefs.

If investors are uncertain about the future price, it is because they are uncertain about the beliefs of other investors. Since investors are not homogeneous in risk preference, the second trading date price is affected by the distribution of securities across investors resulting from the portfolios chosen at the initial trading date. Since the latter depend on investors' beliefs about the second trading date price, an investor's uncertainty about the beliefs of other investors implies uncertainty about the future price. Our model presents equilibria in which this uncertainty is not resolved by observing the current price. In these endogenous-sunspot equilibria, investors' beliefs about beliefs are the sole cause of the price movement between the two trading dates.

The way the endogenous-sunspot equilibrium appears to individual investor i is as follows. Investor i believes that the beliefs of investors in the economy have one of two configurations and investor i attaches some probability to each configuration. One of the configurations of beliefs is consistent with a price P_a at the second trading date and the second configuration is associated with a price P_b. Therefore, these possible prices are used by investor i in choosing a portfolio at the initial trading date.

The implication of the rational beliefs character of the endogenous-sunspot equilibrium is that if the portfolio choices of all investors are made as described above, with each investor believing in the same two configurations of beliefs but not necessarily attaching the same probabilities to the likelihood of each configuration, and if beliefs are actually those of the first configuration, then when investors choose optimal portfolios at the final trading date the stock price will indeed be P_a. Similarly, if beliefs are actually those of the second configuration, then the final trading date stock price will indeed be P_b. In other words, beliefs are rational because they are self-fulfilling.

Two aspects of the endogenous-sunspot equilibria are particularly interesting. The first concerns how the possible second trading date prices relate to the price in the no-sunspot equilibrium. One possibility is for initial trading date beliefs to reflect a probability distribution of the future price with support on both sides of the no-sunspot price, i.e., in the endogenous-sunspot equilibrium investors believe that the future price may be higher or may be lower than the no-sunspot price.

Surprisingly, it is also possible for an endogenous-sunspot equilibrium to involve current and future prices which are higher with probability one in the beliefs of all investors than in the no-sunspot equilibrium, or for it to involve prices that are lower with probability one in the beliefs of all investors than in the no-sunspot equilibrium. In the former case, for example, the price at the initial trading date is higher than in the no-sunspot equilibrium and the price at the second trading date, although investors believe that it may be above or below the initial trading date price, is certain to be above the no-sunspot equilibrium price. Thus, the existence of an endogenous-sunspot equilibrium can lead to a raising (or a lowering) of security prices at all dates, relative to the absence of an endogenous-sunspot equilibrium. We call this a `pseudo-bubble'. It should be noted that this effect does not represent a true bubble, however, since the price remains higher through the second trading date and ultimate security payoffs are not affected.

The second interesting aspect of endogenous-sunspot equilibria concerns a welfare comparison with the no-sunspot equilibrium. We show by numerical example that an endogenous-sunspot equilibrium may involve all investors having higher expected utilities, using their own beliefs, than they have under the no-sunspot equilibrium. In other words, self-supporting conjectures about future prices, in a market where investors can take positions to bet on their conjectures, may make everyone feel better off ex ante. A more typical result, however, is that replacing the no-sunspot equilibrium by an endogenous-sunspot equilibrium makes one group of investors better off and makes the other group worse off.

Our model can be characterized as a noisy rational expectations equilibrium. However, our model is sharply distinguished from other noisy rational expectations models in the fact that no investor in our model has information about security payoffs or endowments that other investors do not have. In other words, in our model you are interested in my belief but learning it would not change your belief about the final payoff. In previous models, it would. Previous noisy rational expectations models include those of Grossman–Stiglitz (1980), Hellwig (1980), Diamond–Verrecchia (1981), Admati (1985), Kyle (1989), Grundy–McNichols (1989), Allen et al. (1993) and Romer (1993).

Our endogenous-sunspot model is related, in name and substance, to the seminal idea of sunspot equilibrium introduced in Cass–Shell (1983). However, there are significant conceptual differences between our model and the economy analyzed in Cass–Shell. In their model, the existence of a sunspot equilibrium rests on one of three conditions. Either markets are incomplete, or not all individuals are allowed to trade in all markets (in the Cass–Shell overlapping generations model some individuals cannot trade in the securities market for sunspot sidebets because they are born after the sunsport event occurs), or individuals' beliefs about the sunspot event are heterogeneous and a sidebet is traded.

We show that an endogenous-sunspot equilibrium can occur even in the absence of the above conditions, (i.e. with complete markets, all individuals having access to all markets, and homogeneous beliefs.) The basic source of the difference is that the sunspot event in Cass–Shell is an exogenous event that will occur or not regardless of what happens in the market. In our model, on the other hand, the endogenous sunspot event is the revelation at the second trading date of the true state of investors' beliefs about other investors' beliefs. This difference has significant implications.

First, as noted earlier, the financial securities in the Cass–Shell model are cash and a sidebet on the occurrence of the exogenous-sunspot event. The sunspot sidebet is in zero net supply and its payoff does not depend on what happens in the economy. Rather, sidebet payoffs, assuming investors take nonzero positions in the sidebet, are what cause the equilibrium at the second trading date (the date when commodities are traded in the Cass–Shell model) to depend on the occurrence of the sunspot event. In our model, the financial securities are cash and an asset in positive net supply whose risky payoff at the final date determines the aggregate consumption in the economy. The market value of the risky asset at the second trading date, which plays a role in our model analogous to the payoff on the sunspot sidebet in the Cass–Shell model, clearly does depend on the nature of the equilibrium in the economy.

Second, the role of heterogeneous beliefs is different in our model compared to Cass–Shell. In the Cass–Shell model with complete markets, homogeneous beliefs prevent a sunspot equilibrium (i.e., an equilibrium in which prices are affected by the occurrence of sunspots) from occurring, regardless of whether the fact of homogeneous beliefs is or is not common knowledge. This is because, with homogenous beliefs, the market for the sunspot sidebet clears only when all investors hold zero positions in the sidebet, so the exogenous sunspot event has no effect on equilibrium at the second trading date.

In our model, as we demonstrate later, it is possible to have an endogenous-sunspot equilibrium with homogenous beliefs, provided the fact of homogeneous beliefs is not common knowledge. In other words, what is crucial in the endogenous sunspot model is not the actual degree of heterogeneity of beliefs but the degree of investors' uncertainty about the state of beliefs. An endogenous-sunspot equilibrium can exist when beliefs are homogeneous but investors do not know that this is the case and investors believe that other investors' beliefs may differ from their own. On the other hand, if homogeneous beliefs are common knowledge then these beliefs are degenerate (since they are beliefs about beliefs) and only the no-sunspot equilibrium exists.

The organization of the remainder of the paper is as follows. Section 2contains the basic assumptions and notation. Section 3deals with the equilibrium at the second trading date. Section 4describes the initial trading date equilibrium. Section 5presents numerical examples to illustrate the no-sunspot and endogenous-sunspot equilibria. Section 6gives conclusions.