Bayesian–Walrasian equilibria: beyond the rational expectations equilibrium

In general rational expectations equilibrium (REE), as introduced in Radner (Econometrica 47:655–678, 1978) in an Arrow–Debreu–McKenzie setting with uncertainty, does not exist. Moreover, it fails to be fully Pareto optimal and incentive compatible and is also not implementable as a perfect Bayesian equilibrium of an extensive form game (Glycopantis et al. in Econ Theory 26:765–791, 2005). The lack of all the above properties is mainly due to the fact that the agents are supposed to predict the equilibrium market clearing price (as agent’s expected maximized utility is conditioned on the information that equilibrium prices reveal), which leads inevitably to the presumption that agents know all the primitives in the economy, i.e., random initial endowments, random utility functions and private information sets. To get around this problematic equilibrium notion, we introduce a new concept called Bayesian–Walrasian equilibrium (BWE) which has Bayesian features. In particular, agents try to predict the market-clearing prices using Bayesian updating and evaluate their consumption in terms of Bayesian price estimates, which are different for each individual. In this framework agents maximize expected utility conditioned on their own private information about the state of nature, subject to a Bayesian estimated budget constraint. Market clearing is not an intrinsic part of the definition of BWE. However, both in the case of perfect foresight and in the case of symmetric information BWE leads to a statewise market clearing; it then becomes an ex post Walrasian equilibrium allocation. This new BWE exists under standard assumptions, in contrast to the REE. In particular, we show that our new BWE exists in the well-known example in Kreps (J Econ Theory 14:32–43, 1977), where REE fails to exist. This work was done in the Spring of 2005, when EJB was a visiting professor at the University of Illinois.     

A bargaining model in general equilibrium

The goal of the paper is to present a simple model of rational endogenous household formation in a general equilibrium framework in which Pareto optimality at the economy level is not necessarily obtained. The simplest example of household formation is the case in which pairs of individuals engage themselves in a bargaining process on the division of some wealth: if an agreement on the distribution is (not) reached, we can say that the household is (not) formed. The vast majority of existing bargaining models predicts agreements on an efficient outcome. A seminal paper by Crawford (Econometrica 50:607–637, 1982) describes a very simple game with incomplete information in which, even with rational agents, disagreement causes welfare losses. We embed that model in a general equilibrium framework and present some results on equilibria both in the bargaining game and the associated exchange economy. Crawford’s results support Schelling’s intuition on the reasons of disagreement: it may arise if players’ commitments are reversible. Crawford shows that high probabilities of reversibility tend to favor the bargaining impasse, in fact with low probability. We prove that even if those probabilities are arbitrarily close to zero, disagreement is an equilibrium outcome, with high probability. That conclusion seems to be an even stronger support to Schelling’s original viewpoint. In the exchange economy model with that noncooperative bargaining game as a first stage, we present significant examples of economies for which equilibria exist. Because of disagreement, Pareto suboptimal exchange economy equilibria exist for all elements in the utility function and endowment spaces and they may coexist with Pareto optimal equilibria even at the same competitive prices.     

A Note on Market Power in an Emission Permits Market with Banking

In this paper, we investigate the effect of market power on equilibrium path of an emission permits market in which firms can bank current permits for use in later periods. In particular, we study the market equilibrium for a large (potentially dominant) firm and competitive fringe with rational expectations. We characterize the equilibrium solution for different permits allocations and discuss the large firm’s stock-holding constraints needed for credible market manipulation.     

Information at equilibrium

Summary. In a game with rational expectations, individuals simultaneously refine their information with the information revealed by the strategies of other individuals. At a Nash equilibrium of a game with rational expectations, the information of individuals is essentially symmetric: the same profile of strategies is also an equilibrium of a game with symmetric information; and strategies are common knowledge. If each player has a veto act, which yields a minimum payoff that no other profile of strategies attains, then the veto profile is the only Nash equilibrium, and it is is an equilibrium with rational expectations and essentially symmetric information; which accounts for the impossibility of speculation. Received: June 20, 2001; revised version: January 9, 2002 RID="*" ID="*" We wish to thank Pierpaolo Battigalli, Fran?oise Forges, Franco Donzelli, Leonidas Koutsougeras, Aldo Rustichini, Rajiv Vohra and Nicholas Yannelis for their comments. Correspondence to: H. Polemarchakis     

Market crashes,speculation and learning in financial markets

A natural conjecture is that if agents’ beliefs are almost correct then equilibrium prices should be close to rational expectations prices. Sandroni (J Econ Theory 82:1–18, 1998) gives a counterexample in an economy with sunspots and complete markets. We extend Sandroni’s result by showing that the conjecture is generically true for economies with complete markets. We consider a standard General Equilibrium model with large but finite horizon and complete markets. We show that, for almost every such economy, if conditional beliefs eventually become correct along a path of events then equilibrium prices of assets traded along this path converge to rational expectations equilibria in the sup-norm. Moreover, we establish that, generically, there exist along any such path local diffeomorphisms between individual beliefs and equilibrium prices. I would like to thank C. Ewerhart and A. Kirman for their comments, as well as the seminar participants at the University of Minho, the General Equilibrium Workshop 2005 in Zurich, and the 15th Asian General Equilibrium Conference 2007 in Singapore. An anonymous referee also provided very valuable comments.     

Coordination of strategic responses to security threats: Laboratory evidence

Laboratory experiments are used to evaluate the extent to which players in games can coordinate investments that diminish the probability of losses due to security breaches or terrorist attacks. In this environment, economically sensible investments may be foregone if their potential benefits are negated by failures to invest in security at other sites. The result is a coordination game with a desirable high-payoff, high-security equilibrium and an undesirable low-security equilibrium that may result if players do not expect others to invest in security. One unique feature of this coordination situation is that investment in security by one player generates a positive externality such that all other players’ expected payoffs are increased, regardless of those other players’ investment decisions. Coordination failures are pervasive in a baseline experiment with simultaneous decisions, but coordination is improved if players are allowed to move in an endogenously determined sequence. In addition, coordinated security investments are observed more often when the largest single security threat to individuals is preventable by their own decisions to invest in security. The security coordination game is a “potential game,” and the success of coordination on the more secure equilibrium is related to the notion of potential function maximization and basin of attraction.      

Optimal pricing and capacity choice for a public service under risk of interruption

We develop rules for pricing and capacity choice for an interruptible service that recognize the interdependence between consumers’ perceptions of system reliability and their market behavior. Consumers post ex ante demands, based on their expectations on aggregate demand. Posted demands are met if ex post supply capacity is sufficient. However, if supply is inadequate all ex ante demands are proportionally interrupted. Consumers’ expectations of aggregate demand are assumed to be rational. Under reasonable values for the consumer’s degrees of relative risk aversion and prudence, demand is decreasing in supply reliability. We derive operational expressions for the optimal pricing rule and the capacity expansion rule. We show that the optimal price under uncertainty consists of the optimal price under certainty plus a markup that positively depends on the degrees of relative risk aversion, relative prudence and system reliability. We also show that any reliability enhancing investment—though lowering the operating surplus of the public utility—is socially desirable as long as it covers the cost of investment.     

Unemployment and econometric learning

We apply well-known results of the econometric learning literature to the Mortensen and Pissarides real business cycle model. Agents can always learn the unique rational expectations equilibrium (REE), for all possible well-defined sets of parameter values, by using the minimum-state-variable solution to the model and decreasing gain learning. From this perspective the assumption of rational expectations in the model could be seen as reasonable. But using a parametrisation with UK data, simulations show that the speed of convergence to the REE is slow. This type of learning dampens the cyclical response of unemployment to small structural shocks.     

Admissible market data structures: A complete characterization

This paper characterizes the market data which, if observed by traders in a stochastic exchange environment, will permit the general existence of (rational) expectations equilibria. It is proved that if a trader observes and conditions his expectations on some nonconstant market data, he must observe at least the equilibrium price and his own equilibrium trade. Any data which do not satisfy this requirement, such as the price alone, or the price and the volume of trade, will fail to permit the existence of an expectations equilibrium for some otherwise well-behaved stochastic environment.     

Introduction to the stability of rational expectations equilibrium

This paper surveys the literature which examines the stability of the expectations that agents are assumed to have in a rational expectations equilibrium (REE). This issue is more complex than the usual statistical estimation problem because the relationship between observable variables and payoff relevant variables is endogenous. One approach taken in the literature yields convergence to a REE but requires agents to have extensive knowledge about the structure and dynamics of the model that prevails while they learn. A second approach does not assume that agents have correctly specified likelihood functions and finds that REE may not be stable.     

A dynamic model of expectations equilibrium

This paper describes a dynamic information adjustment process which achieves (rational) expectations equilibria for stochastic exchange environments. An informational temporary equilibrium is an exchange equilibrium in which agents' expectations are conditioned on their initial information and market data generated in previous informational temporary equilibria. An equilibrium is a temporary equilibrium which reveals no further information. This process leads to an equilibrium even when the data observed by agents is insufficient to permit the existence of an expectations equilibrium.     

A price competition game under free entry

This study builds a game of Bertrand-like price competition in a market with free entry. Under the assumption of a standard U-shaped average cost curve, it demonstrates that even if the number of sellers is small, a long-run competitive outcome can be supported as a Nash equilibrium. This game provides unifying treatments to the standard Bertrand equilibrium, the long-run competitive outcome, Demsetz’s equilibrium as well as other types of equilibria that have not been known in the existing literature.This paper is dedicated to Professor Mukul Majumdar to celebrate his sixtieth birthday. I am deeply indebted to Mukul for his generosity and kindness towards me throughout my career. I am also grateful to Tapan Mitra and anonymous referees for their very useful comments and suggestions in the revision and to Fumio Dei, Jim Friedman, Koji Ishibashi, Atsushi Kajii, Takashi Kamihigashi, Kunio Kawamata, Kei-ichi Koda, Takashi Negishi, Michihiro Ohyama, Santanu Roy, and Yoshimasa Shirai for useful conversations and correspondence and, in particular, Motonari Kurasawa, who drew to my attention Demsetz’s work.     

Monetary Rules and Deficit Shocks

Modern theory on interest rate rules is based on the representative agent framework with infinite-horizon consumers, thereby ignoring redistributions of the fiscal burden across generations due to deficit shocks. We show how the ‘Taylor principle’ relies on this restrictive assumption. In a dynamic New Keynesian general equilibrium model with overlapping generations, the existence of a unique stable rational expectations equilibrium may also occur under a passive monetary policy. However, active monetary policy is still required to stabilize the economy in response to fiscal shocks. Thanks are due to an anonymous referee, Andrea Costa, Jordi Galí and Giancarlo Marini for very useful comments and discussions. Financial support from CNR and the FIRB project is gratefully acknowledged. The usual disclaimer applies.     

Learning,estimation, and the stability of rational expectations

The stability of the rational expectations equilibrium of a simple asset market model is studied in a situation where a group of traders learn about the relationship between the price and return on the asset using ordinary least squares estimation, and then use their estimates in predicting the return from the price. The model which they estimate is a well-specified model of the rational expectations equilibrium, but a misspecified model of the situation in which the traders are learning. It is shown that for appropriate values of a stability parameter the situation converges almost surely to the rational expectations equilibrium.     

Local coordination and market equilibria

We reformulate the local stability analysis of market equilibria in a competitive market as a local coordination problem in a market game, where the map associating market prices to best-responses of all traders is common knowledge and well-defined both in and out of equilibrium. Initial expectations over market variables differ from their equilibrium values and are not common knowledge. This results in a coordination problem as traders use the structure of the market game to converge back to equilibrium. We analyse a simultaneous move and a sequential move version of the market game and explore the link with local rationalizability.     

A decomposition algorithm for N-player games

An N-player game can be decomposed by adding a coordinator who interacts bilaterally with each player. The coordinator proposes profiles of strategies to the players, and his payoff is maximized when players’ optimal replies agree with his proposal. When the feasible set of proposals is finite, a solution of an associated linear complementarity problem yields an equilibrium of the approximate game and thus an approximate equilibrium of the original game. Computational efficiency is improved by using vertices of a triangulation of the players’ strategy space for the coordinator’s pure strategies. Computational experience is reported.     

Information Structure and the Tragedy of the Commons in Resource Extraction

Synopsis This paper considers the well-known Levhari-Mirman discrete-time model of resource extraction, and investigates the effects of the information structure of the dynamic game – open-loop, Markovian or history-dependent – on the equilibrium consumption path and the overall utility of the agents. Due to the special structure of the model, the open-loop regime yields a Pareto-optimal outcome. The Markovian regime leads to the most pronounced version of the tragedy of the commons. History-dependent behavior yields an outcome set that is intermediate between the other two cases, and that may include the Pareto-optimal outcome in some cases. The level of efficiency of equilibrium behaviour is thus U-shaped as a function of the level of information the agents’ extraction strategies are based on. The analysis suggests that in environments characterized by a dynamic (and no market) externality, forcing agents to commit to open-loop behavior would constitute welfare-improving regulation.     

Strategic managerial delegation and cross-border mergers

The purpose of this paper is to clarify the relationship between the market structure in equilibrium and the most preferred structure with respect to each country’s social welfare and/or total social welfare, when all existing firms can freely merge with each other in an international oligopoly under the segmented market assumption in three cases: the case wherein all the firms are entrepreneurial and the cases wherein they use two different types of managerial delegation contracts. We focus our attention on the coincidence/non-coincidence between the equilibrium market structure (EMS) and the most socially preferred structure with respect to each country’s social welfare and/or total social welfare, as each firm’s production efficiency varies. When each firm’s production efficiency is relatively low, in all the three cases, the EMS coincides with the most socially preferred structure with respect to each country’s social welfare and total social welfare in a large area of the physical trade cost. On the other hand, when each firm’s production efficiency is relatively high, in the cases wherein they use the two different types of managerial delegation contracts, there exists an area of each firm’s production efficiency such that the EMS does not coincide with the most socially preferred structure with respect to each country’s social welfare and total social welfare. Therefore, as each firm’s organizational structure proceeds from entrepreneurial to managerial delegation, a more active merger policy is needed with respect to each country’s social welfare and total social welfare.     

The dynamics of a banking duopoly with capital regulations

This paper analyses the dynamics of a banking duopoly game with heterogeneous and homogeneous players (as regards the type of expectations' formation), to investigate the effects of the capital requirements introduced by international accords (Basel-I in 1988 and more recently Basel-II and Basel-III), in the context of the Monti-Klein model. This analysis reveals that the policy of introducing a capital requirement tends to stabilise the market equilibrium (both with heterogeneous and homogeneous banks). Moreover, it is shown that 1) when the capital standard is reduced the market stability is lost through a flip bifurcation and subsequently a cascade of flip bifurcations may lead to periodic cycles and chaos; 2) when the expectations are heterogeneous even the case of multi-stability may be present.Therefore, although on the one side the capital regulation is harmful for the equilibrium loans' volume and profit, on the other side it is effective in keeping or restoring the stability of the Cournot–Nash equilibrium in the banking duopoly.     

Rationally mispriced assets in equilibrium

We present a simple model of trading in a financial market where agents are asymmetrically informed and information is transmitted through the price system. We characterize the equilibrium for this economy and show that ‘rational mispricing’ of assets occurs if the price system fails to reveal the insider information accurately. It is argued that the communication of wrong information through equilibrium prices is compatible with full rationality on the part of the investors and may explain deviations from the efficient markets hypothesis.