The set of sunspot equilibria in economies with incomplete financial markets: variable asset prices

Summary. I consider the set of equilibria of two-period economies with S extrinsic states of nature in the second period and I assets with linearly independent nominal payoffs. Asset prices are variable. If the number of agents is greater than (S-I), the payoff matrix is in general position and S 2I, the set of equilibrium allocations generically (in utility function space) contains a smooth manifold of dimension (S-1). Moreover, the map from states o f nature to equilibrium allocations (restricted to this manifold) is one-to-one at each equilibrium. Received: February 23, 1998; revised version: June 1, 2000     

On the existence of rational expectations equilibrium

We prove a theorem on the existence of rational expectations general economic equilibrium when agents condition on prices as well as on private information, and maximize the expectation of a state-dependent utility function. The key to the result is a new idealization of what it means for a set of empirical distributions to support agents' expectations. This idealization depends on the notion that agents compare their expectations with continuous versions of the random empirical distributions that are generated by the workings of the economy. The existence theorem covers all strictly concave utility functions, arbitrary distributions of the state variable, and situations in which the dimension of the state variable is large relative to the number of commodities.     

The generic existence of rational expectations equilibrium in the higher dimensional case

The study of the generic existence of rational expectations equilibrium has concentrated on three cases, according to whether the space of “states of private information” has dimension less than, equal to, or greater than the price space. Previous results have shown that in the first case fully revealing equilibria exist generically and in the second case there are open sets of environments with no equilibrium. This paper shows that in the third case there is a residual set of environments which permits equilibria which are arbitrarily close to fully revealing.     

Risk sharing and retrading in incomplete markets

At a competitive equilibrium of an incomplete-markets economy agents’ marginal valuations for the tradable assets are equalized ex-ante. We characterize the finest partition of the state space conditional on which this equality holds for any economy. This leads naturally to a necessary and sufficient condition on information that would induce agents to retrade, if such information was to become publicly available after the initial round of trade.     

How equilibrium prices reveal information in a time series model with disparately informed, competitive traders

Accommodating asymmetric information in a dynamic asset pricing model is technically challenging due to the problems associated with higher-order expectations. That is, rational investors are forced into a situation where they must forecast the forecasts of other agents. In a dynamic setting, this problem telescopes into the infinite future and the dimension of the relevant state space approaches infinity. By using the frequency domain approach of Whiteman [Linear Rational Expectations Models: A User's Guide, University of Minnesota Press, Minneapolis, 1983] and Kasa [Forecasting the forecasts of others in the frequency domain, Rev. Econ. Dynam. 3 (2000) 726-756], this paper demonstrates how information structures previously believed to preserve asymmetric information in equilibrium, converge to a symmetric information, rational expectations equilibrium. The revealing aspect of the price process lies in the invertibility of the observed state space, which makes it possible for agents to infer the economically fundamental shocks and thus eliminating the need to forecast the forecasts of others.     

Strongly robust equilibrium and competing-mechanism games

This paper formulates the notion of a strongly robust equilibrium relative to a set of mechanisms specified in any competing-mechanism game of complete information with multiple principals and multiple agents. It shows that when agents’ efforts are contractible, any strongly robust pure-strategy equilibrium relative to single-incentive contracts persists, regardless of the continuation equilibrium that agents play upon any principal's deviation to any complex mechanism.     

Real indeterminacy in incomplete financial market economies without aggregate risk

Summary We analyze an exchange economy with incomplete financial markets and assets whose returns are fixed in units of account. Moreover, we assume absence of aggregate risk, i.e., that individual preferences and total resources are constrained to be invariant across different states of the world. In this framework we show that the set of (commodity) price-endowment equilibria is diffeomorphic to a Euclidean space. We then exploit this global parameterization to prove that the set of equilibrium allocations associated with each endowment in a generic set contains a smooth manifold, whose dimension is equal to the number of missing assets.     

General equilibrium and the new neoclassical synthesis

We present a general equilibrium model of the new neoclassical synthesis that has the same level of generality as the Arrow–Debreu model. This involves a stochastic multi-period economy with a monetary sector and sticky commodity prices. We formulate the notion of a sticky price equilibrium where all agents form rational expectations on prices for commodities and assets, interest rates, and rationing. We present a general result showing that monetary policy imposes no restrictions whatsoever on nominal equilibrium price levels and that the set of sticky price equilibria has a dimension equal to the number of terminal date-events. Stickiness of prices implies that this indeterminacy is real.     

Private Value Allocations in Large Economies with Differential Information

We consider a Radner-type (e.g., Radner, 1968, Econometrica36, 31–58) pure exchange economy with differential information and a continuum of agents. We show that under appropriate assumptions the set of Aumann–Shapley private value allocations in such an economy coincides with the set of Radner competitive equilibrium allocations. Journal of Economic Literature Classification Numbers: C71, D51, D82.     

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.     

On the set of Pareto efficient allocations in economies with public goods

Summary We examine the set of Pareto-efficient allocations in economies with public goods. We show that even if preferences are continuous and strongly monotonic, it need not coincide with the set of weakly efficient allocations. We then study topological properties of the Pareto set. We show that it is neither connected nor closed in allocation space. Furthermore, if the public goods are local, the image of the Pareto set in utility space need not be closed or connected. We provide two independent sufficient conditions for the closedness of the Pareto set. The results are directly applicable to private goods economies with joint production. Our results should be of interest for general equilibrium and mechanism design theory; where for example, the properties of the efficient set are important for proving the existence of an equilibrium and for the study of the properties of monotone-path social choice correspondences.We thank Hideo Konishi, Tomoichi Shinotsuka, Nicholas Yannelis and an anonymous referee for valuable comments on previous drafts of this paper.     

Existence of a continuum of equilibria in a monetary random-matching model

In this paper, we study a monetary random-matching model where both goods and money are perfectly divisible, production is costly, and there is no exogenous upper bound on agents' money holdings, information on which is private to the agent. We show that there is a continuum of stationary equilibria where agents have either no money or a set amount, and buyers spend all their money. As in the previous studies, the equilibrium value function is step-like, which emerges as a self-fulfilling prophecy. The endogenous upper bound on agents' money holdings is the result of private information on agents' money holdings. Buyers post an offer that is accepted only by sellers without money, who set a higher value on money.     

A Characterization of Robust Sunspot Equilibria

In nonconvex environments, a sunspot equilibrium can sometimes be destroyed by the introduction of new extrinsic information. We provide a simple test for determining whether or not a particular equilibrium survives, or is robust to, all possible refinements of the state space. We use this test to provide a characterization of the set of robust sunspot-equilibrium allocations of a given economy; it is equivalent to the set of equilibrium allocations of the associated lottery economy. Journal of Economic Literature Classification Numbers: D51, D84, E32.     

Representing equilibrium aggregates in aggregate games with applications to common agency

An aggregate game is a normal-form game with the property that each playerʼs payoff is a function of only his own strategy and an aggregate of the strategy profile of all players. Such games possess properties that can often yield simple characterizations of equilibrium aggregates without requiring that one solves for the equilibrium strategy profile. When payoffs have a quasi-linear structure and a degree of symmetry, we construct a self-generating maximization program over the space of aggregates with the property that the solution set corresponds to the set of equilibrium aggregates of the original n-player game. We illustrate the value of this approach in common-agency games where the playersʼ strategy space is an infinite-dimensional space of nonlinear contracts. We derive equilibrium existence and characterization theorems for both the adverse selection and moral hazard versions of these games.     

How common are common priors?

To answer the question in the title we vary agents? beliefs against the background of a fixed knowledge space, that is, a state space with a partition for each agent. Beliefs are the posterior probabilities of agents, which we call type profiles. We then ask what is the topological size of the set of consistent type profiles, those that are derived from a common prior (or a common improper prior in the case of an infinite state space). The answer depends on what we term the tightness of the partition profile. A partition profile is tight if in some state it is common knowledge that any increase of any single agent?s knowledge results in an increase in common knowledge. We show that for partition profiles that are tight the set of consistent type profiles is topologically large, while for partition profiles that are not tight this set is topologically small.     

Equilibrium price formation in markets with differentially informed agents

We consider a pure exchange economy with a finite set of types of agents which have incomplete and asymmetric information on the states of nature. Our aim was to describe the equilibrium price formation and analyze how the lack of information may affect the allocation of resources. To do so, we adapt to an asymmetric information scenario a variant of the Shapley–Shubik game introduced by Dubey and Geanakoplos (J Math Econ 39:391–400, 2003 ).     

The bargaining set of a large economy with differential information

Summary. We study the Mas-Colell bargaining set of an exchange economy with differential information and a continuum of traders. We established the equivalence of the private bargaining set and the set of Radner competitive equilibrium allocations. As for the weak fine bargaining set, we show that it contains the set of competitive equilibrium allocations of an associated symmetric information economy in which each trader has the “joint information” of all the traders in the original economy, but unlike the weak fine core and the set of fine value allocations, it may also contain allocations which are not competitive in the associated economy. Received: February 15, 1999; revised version: August 9, 1999     

Non-cooperative bargaining and the incomplete informational core

In an exchange economy with incomplete information, the signaling core is defined by the set of state-contingent allocations to which no coalitions object under informational leakage through proposals by informed agents. An objection underlying the signaling core is supported by a sequential equilibrium of an ultimatum bargaining game with an informed proposer. We prove that a stationary sequential equilibrium allocation in a Rubinstein-type sequential bargaining game with a restart rule belongs to the signaling core if the belief of players satisfies a self-selection property.     

Growth in economies with non convexities: sunspots and lottery equilibria

Summary. We investigate the relation between lotteries and sunspot allocations in a dynamic economy where the utility functions are not concave. In an intertemporal competitive economy, the household consumption set is identified with the set of lotteries, while in the intertemporal sunspot economy it is the set of measurable allocations in the given probability space of sunspots. Sunspot intertemporal equilibria whenever they exist are efficient, independently of the sunspot space specification. If feasibility is, at each point in time, a restriction over the average value of the lotteries, competitive equilibrium prices are linear in basic commodities and intertemporal sunspot and competitive equilibria are equivalent. Two models have this feature: Large economies and economies with semi-linear technologies. We provide examples showing that in general, intertemporal competitive equilibrium prices are non-linear in basic commodities and, hence, intertemporal sunspot equilibria do not exist. The competitive static equilibrium allocations are stationary, intertemporal equilibrium allocations, but the static sunspot equilibria need not to be stationary, intertemporal sunspot equilibria. We construct examples of non-convex economies with indeterminate and Pareto ranked static sunspot equilibrium allocations associated to distinct specifications of the sunspot probability space.Received: 25 August 2003, Revised: 16 March 2004, JEL Classification Numbers: D84, D90.Correspondence to: Paolo SiconolfiWe thank Herakles Polemarchakis for helpful conversations on the topic. The research of Aldo Rustichini was supported by the NSF grant NSF/SES-0136556.     

Selecting equilibria in common agency games

We characterize equilibrium payoffs of a delegated common agency game in a public good context where principals use smooth contribution schedules. We prove that under complete information, payoff vectors of equilibria with truthful schedules coincide with the set of smooth equilibrium payoffs, including non-truthful schedules. We next consider whether the presence of arbitrarily small amounts of asymmetric information is enough to refine this payoff set. Providing that the extensions of the equilibrium schedules beyond the equilibrium point are flatter than truthful schedules, the set of equilibrium payoffs is strictly smaller than the set of smooth (equivalently, truthful) equilibrium payoffs. Interestingly, some forms of asymmetric information do not sufficiently constrain the slopes of the extensions and fail to refine the payoff set. In the case of a uniform distribution of types and arbitrary out-of-equilibrium contributions, the refinement has no bite. If, however, one restricts out-of-equilibrium behavior in a natural way, the refinement is effective. Alternatively, we may consider an exponential distribution with unbounded support (and hence no out-of-equilibrium choices) and we find that the refinement selects a unique equilibrium payoff vector equal to Lindahl prices.As a separate contribution, equilibria with forcing contracts are also considered both under complete and asymmetric information.