Rational belief structures and rational belief equilibria

Summary The paper introduces some simplifying tools and methods for studying Rational Beliefs and for proving existence of Rational Belief Equilibria. We identify a set of stable non-stationary stochastic processes, named SIDS processes. Furthermore we introduce the concept of a Rational Belief Structure, which formulates the Rational Beliefs of the agents as beliefs about the distribution of exogenous variables and the beliefs of other agents. The use of the developed apparatus is demonstrated by showing existence of a set of Rational Belief Equilibria in an Overlapping Generations Model with money and one commodity.The results presented in this paper are taken from my Ph.D. thesis from Stanford University ([13]). The dissertation is devoted to the study of the theory of Rational Beliefs as developed by Mordecai Kurz. I gratefully acknowledge the inspiration obtained from innumerable discussions with him over the years about this subject. His suggestions were instrumental for writing the thesis and this paper. The paper benefited much from a thorough reading by Trinidad Casasus, Mordecai Kurz and Ho-Mou Wu. Financial support from The Academy for Research (Forskerakademiet), Aarhus, Denmark and The University of Copenhagen, during my time as a Ph.D. student and from Danish Social Science Research Council thereafter and in addition from Fondazione Eni Enrico Mattei of Milan, Italy is gratefully acknowledged.     

On rational belief equilibria

Summary We study equilibria in which agent's belief are rational in the sense of Kurz [1994]. The market is formulated by specifying a stochastic demand function and a continuum of producers, each with a quadratic cost function who must select their output before knowing prices. Holding Rational Beliefs about future prices, producers maximize expected profits. In a Rational Belief Equilibrium (RBE) agents select diverse forecast functions but each one is rational in the sense that it is based on a theory which cannot be rejected by the data. It is shown that there exists a continuum of RBE's and they could entail very different patterns of time series for the economy and consequently different aggregate levels of longterm volatility. Since the model contains exogenously specified random variables, the difference in the level of long-term volatility of prices among the different RBE's arises endogenously as an amplification of the volatility of exogenous variables. The paper derives exact bounds on the possible levels of such amplification.This research was supported by NSF Grant IRI-8814954 to Stanford University. The author is thankful to Steven N. Durlauf and Carsten K. Nielsen for useful comments on an earlier draft.     

On the uniqueness of fully informative rational expectations equilibria

Summary. This paper analyzes two equivalent equilibrium notions under asymmetric information: risk neutral rational expectations equilibria (rn-REE), and common knowledge equilibria. We show that the set of fully informative rn-REE is a singleton, and we provide necessary and sufficient conditions for the existence of partially informative rn-REE. In a companion paper (DeMarzo and Skiadas (1996)) we show that equilibrium prices for the larger class of quasi-complete economies can be characterized as rn-REE. Examples of quasi-complete economies include the type of economies for which demand aggregation in the sense of Gorman is possible (with or without asymmetric information), the setting of the Milgrom and Stokey no-trade theorem, an economy giving rise to the CAPM with asymmetric information but no normality assumptions, the simple exponential-normal model of Grossman (1976), and a case of no aggregate endowment risk. In the common-knowledge context, we provide necessary and sufficient conditions for a common knowledge posterior estimate, given common priors, to coincide with the full communication posterior estimate. Received: May 29, 1997; revised version: July 18, 1997     

Construction of a state space for interrelated securities with an application to temporary equilibrium theory

Summary We construct an endogenous state space in an exchange economy with possibly infinite horizon. Every period agents trade securities whose payoffs depend on future dividends and asset prices. We reject the perfect foresight assumption on the ground that agents have not only limited knowledge of other individuals' endowments and preferences, but also limited capacity to compute equilibria. We choose instead absence of arbitrage as the principle which allows agents to determine if a system of future prices is possible. We give an alogrithm to compute the set of nonarbitrage prices every period, with both finite and infinite horizon. We then apply this endogenous structure of uncertainty to an infinite horizon temporary equilibrium model.I would like to thank Professor Donald Brown for his constant help and guidance. I have also greatly benefited from helpful discussions with Professors Jacques Drèze, Bernard Dumas, Mordecai Kurz, Carsten Nielsen, Jan Werner, and Ho-Mou Wu.     

Optimal tranching with diverse beliefs

In this paper, we consider the optimal selling strategy for an asset securitization originator (the issuer) when the potential buyers hold diverse beliefs. We find that the tranching process is beneficial to the issuer because it increases the total sale price. We also consider the optimal tranching strategy that maximizes the revenue of the issuer. This helps explain the widespread use of the tranching technique in the asset securitization practice. Our findings are also enlightening to understand the price bubble problem when the junior tranche is interpreted as a collateral.     

Internal rationality, imperfect market knowledge and asset prices

We present a decision theoretic framework in which agents are learning about market behavior and that provides microfoundations for models of adaptive learning. Agents are ‘internally rational’, i.e., maximize discounted expected utility under uncertainty given dynamically consistent subjective beliefs about the future, but agents may not be ‘externally rational’, i.e., may not know the true stochastic process for payoff relevant variables beyond their control. This includes future market outcomes and fundamentals. We apply this approach to a simple asset pricing model and show that the equilibrium stock price is then determined by investors? expectations of the price and dividend in the next period, rather than by expectations of the discounted sum of dividends. As a result, learning about price behavior affects market outcomes, while learning about the discounted sum of dividends is irrelevant for equilibrium prices. Stock prices equal the discounted sum of dividends only after making very strong assumptions about agents? market knowledge.     

Coordination risk and the price of debt

Creditors of a distressed borrower face a coordination problem. Even if the fundamentals are sound, fear of premature foreclosure by others may lead to pre-emptive action, undermining the project. Recognition of this problem lies behind corporate bankruptcy provisions across the world, and it has been identified as a culprit in international financial crises, but has received scant attention from the literature on debt pricing. Without common knowledge of fundamentals, the incidence of failure is uniquely determined provided that private information is precise enough. This affords a way to price the coordination failure. Comparative statics on the unique equilibrium provides several insights on the role of information and the incidence of inefficient liquidation.     

Endogenous uncertainty in a general equilibrium model with price contingent contracts

Summary This paper views uncertainty and economic fluctuations as being primarily endogenous and internally propagated phenomena. The most important Endogenous Uncertainty examined in this paper is price uncertainty which arises when agents do not have structural knowledge and are complelled to make decisions on the basis of their beliefs. We assume that agents adopt Rational Beliefs as in Kurz [1994a]. The trading of endogenous uncertainty is accomplished by using Price Contingent Contracts (PCC) rather than the Arrow-Debreu state contingent contracts. The paper provides a full construction of the price state space which requires the expansion of the exogenous state space to include the state of beliefs. This construction is central to the analysis of equilibrium with endogenous uncertainty and the paper provides an existence theorem for a Rational Belief Equilibrium with PCC. It shows how the PCC completes the markets for trading endogenous uncertainty and lead to an allocation which is Pareto optimal. This paper also demonstrates that endogenous uncertainty is generically present in this new equilibrium.This research was supported in part by the Fondazione Eni Enrico Mattei of Milan, Italy, and by the National Science Council of Taiwan. The authors thank Carsten K. Nielsen for valuable suggestions.     

Sunspots and predictable asset returns

This paper uses a stylised asset-pricing model to show that sunspots may cause asset returns to be predictable, a widely documented feature of many speculative markets. This result parallels and extends previous works showing that sunspots render asset prices excessively volatile.     

Overlapping risk adjusted sets of priors and the existence of efficient allocations and equilibria with short-selling

The theory of existence of equilibrium with short-selling is reconsidered under risk and ambiguity modelled by risk averse variational preferences. No-arbitrage conditions are given in terms of risk adjusted priors. A sufficient condition for existence of efficient allocations is the overlapping of the interiors of the risk adjusted sets of priors or the inexistence of mutually compatible trades, with non-negative expectation with respect to any risk adjusted prior. These conditions are necessary when agents are not risk neutral at extreme levels of wealths. It is shown that the more uncertainty averse or risk averse the agents, the more likely are efficient allocations and equilibria to exist.     

The dynamics of efficient asset trading with heterogeneous beliefs

This paper analyzes the dynamic properties of portfolios that sustain dynamically complete markets equilibrium when agents have heterogeneous priors. We argue that the conventional wisdom that belief heterogeneity generates continuous trade and significant fluctuations in individual portfolios may be correct but it needs some qualifications. We consider an infinite horizon stochastic endowment economy populated by many Bayesian agents with heterogeneous priors over the stochastic process of the states of nature. Our approach hinges on studying the portfolios that decentralize Pareto optimal allocations. Since these allocations are typically history dependent, we propose a methodology to provide a complete recursive characterization when agents believe that the process of states of nature is i.i.d. but disagree about the probability of the states. We show that even though heterogeneous priors within that class can indeed generate genuine changes in the portfolios of any dynamically complete markets equilibrium, these changes vanish with probability one if the true process consists of i.i.d. draws from a common distribution and the support of some agent's prior belief contains the true distribution. Finally, we provide examples in which asset trading does not vanish because either (i) no agent learns the true conditional probability of the states or (ii) some agent does not know the true process generating the data is i.i.d.     

Art market inefficiency

Art is often used as an investment vehicle. Given the importance of market efficiency in finance, we use a large auction-based index to test whether the art market is weakly efficient. Evidence reveals that returns on artworks exhibit high positive auto-correlation. We attribute this result to price truncation resulting from unobservable reserve prices in auctions. We conclude that the art market is not efficient, mainly because price formation is opaque to outsiders who lack information on unsold artworks.     

Markov rational expectations equilibria in an overlapping generations model

In this paper, we analyze rational expectations equilibrium paths in a stochastic overlapping generations model. The work presented here builds on results of S. E. Spear (J. Econ. Theory 35 (1985), 251–275), where is is shown that in a model with multiple goods and time non-separable preferences, a stochastic steady state equilibrium will generically fail to exist. A stochastic steady state is defined as an equilibrium in which the stochastic process of endogenously determined variables is measure isomorphic to the exogenous process driving the model. In this paper, we establish the existence of non-steady state equilibria and provide a characterization of their stochastic properties.     

Market for Information and Identity Disclosure in an Experimental Open Limit Order Book

This paper analyses the effects of pre‐trade transparency on market quality in an experimental open limit order book preceded by a market for information. The design of the trading game is akin to the system in use in an increasing number of financial markets. We find that the disclosure of traders' identities reduces the incentive to acquire information, liquidity and volatility. We also show that a positive relation exists between the proportion of traders buying information and liquidity. The results are consistent with a standard model of price formation where the number of informed traders is endogenous .     

Asymmetric information in a competitive market game: Reexamining the implications of rational expectations

Summary. We examine price formation in a simple static model with asymmetric information, an infinite number of risk neutral traders and no noise traders. Here we re-examine four results associated with rational expectations models relating to the existence of fully revealing equilibrium prices, the advantage of becoming informed, the costly acquisition of information, and the impossibility of having equilibrium prices with higher volatility than the underlying fundamentals. Received: August 27, 1997; revised version: February 11, 1998     

Tournament incentives and asset price bubbles: Evidence from a field experiment

In this note we show at the example of an experimental stock market, conducted on the occasion of the World Soccer Championship 2010 in South Africa, how tournament incentives might contribute to the formation of asset price bubbles.     

Competitive equilibrium with unawareness in economies with production

This paper investigates an extension of the GEI-unawareness framework by Modica et al. (Econ. Theory 12 (1998) 259) to economies with entrepreneurial production. Existence of equilibrium is guaranteed given decreasing returns to scale. Firm's value and investment decision in equilibrium are characterized. An example of commodity innovation shows that the effect of different degrees of awareness on investment decisions depend upon the degree of risk aversion. In the case of log preferences unawareness may not matter for commodity innovation, although this depends on other preference features.     

Robustness of sunspot equilibria

Summary. A sunspot equilibrium (SSE) is based on some extrinsic randomizing device (RD). We analyze the robustness of SSE. (1) We say that an SSE allocation is robust to refinements if it is also an SSE allocation based on any refinement of its RD. (2) We introduce two core concepts for analyzing the robustness of SSE in the face of cooperative-coalition formation. In the first, the blocking allocations are based on the RD that defines the SSE. In the second (stronger) core concept, coalitions select their own RDs. For the convex economy with restricted market participation, SSE allocations are robust under each of the definitions and the cores converge on replication of the economy to the set of SSE allocations. For the economy with an indivisible good, SSE allocations are not always robust. We provide examples of each of the following: (i) an SSE allocation that is not robust to refinement, (ii) an SSE allocation that is in neither core, (iii) an SSE allocation that is in the first core, but not in the second, and (iv) a core that does not converge upon replication to the set of SSE allocations. Received: July 31, 1995; revised version August 30, 1996     

Measuring subjective expectations in developing countries: A critical review and new evidence

The majority of economic decisions are forward-looking and thus involve expectations of future outcomes. Understanding the expectations that individuals have is thus of crucial importance to designing and evaluating policies in health, education, finance, migration, social protection, and many other areas. However, the majority of developing country surveys are static in nature and many do not elicit subjective expectations of individuals. Possible reasons given for not collecting this information include fears that poor, illiterate individuals do not understand probability concepts, that it takes far too much time to ask such questions, or that the answers add little value. This paper provides a critical review and new analysis of subjective expectations data from developing countries and refutes each of these concerns. We find that people in developing countries can generally understand and answer probabilistic questions, such questions are not prohibitive in time to ask, and the expectations are useful predictors of future behavior and economic decisions. The paper discusses the different methods used for eliciting such information, the key methodological issues involved, and the open research questions. The available evidence suggests that collecting expectations data is both feasible and valuable, suggesting that it should be incorporated into more developing country surveys.     

Bayesian learning and convergence to Nash equilibria without common priors

Summary. Consider an infinitely repeated game where each player is characterized by a “type” which may be unknown to the other players in the game. Suppose further that each player's belief about others is independent of that player's type. Impose an absolute continuity condition on the ex ante beliefs of players (weaker than mutual absolute continuity). Then any limit point of beliefs of players about the future of the game conditional on the past lies in the set of Nash or Subjective equilibria. Our assumption does not require common priors so is weaker than Jordan (1991); however our conclusion is weaker, we obtain convergence to subjective and not necessarily Nash equilibria. Our model is a generalization of the Kalai and Lehrer (1993) model. Our assumption is weaker than theirs. However, our conclusion is also weaker, and shows that limit points of beliefs, and not actual play, are subjective equilibria. Received: March 3, 1995; revised version: February 17, 1997