Equilibrium theory with asymmetric information and with infinitely many commodities

The traditional deterministic general equilibrium theory with infinitely many commodities cannot cover economies with private information constraints on the consumption sets. We bring the level of asymmetric information equilibrium theory at par with that of the deterministic one. In particular, we establish results on equilibrium existence for exchange economies with asymmetric (differential) information and with an infinite dimensional commodity space. Our new equilibrium existence theorems include, as a special case, classical results, e.g. Bewley [Existence of equilibria in economies with infinitely many commodities, J. Econ. Theory 4 (1972) 514-540] or Mas-Colell [The price equilibrium existence problem in topological vector lattices, Econometrica 54 (1986) 1039-1053].     

A unified treatment of finite and infinite economies: limited arbitrage is necessary and sufficient for the existence of equilibrium and the core

Summary. A single condition, limited arbitrage, is shown to be necessary and sufficient for the existence of a competitive equilibrium and the core in economies with any number of markets, finite or infinite, with or without short sales. This extends earlier results of Chichilnisky [8] for finite economies. This unification of finite and infinite economies is achieved by proving that in Hilbert spaces limited arbitrage is necessary and sufficient for the compactness of the Pareto frontier. Limited arbitrage has also been shown to be necessary and sufficient for a resolution of the social choice paradox [9], [10], [12], [13], [14]. Received: August 4, 1995; revised version: April 11, 1997     

Economies with observable types

We study economies of asymmetric information with observable types. Trade takes place in lotteries. Individuals face a standard budget constraint, while the incentive compatibility constraints are imposed on the production set of the intermediaries. This formalization encompasses moral hazard, as in [Jerez, 2003] and [Jerez, 2005], and private information economies. Equilibrium allocations are constrained efficient, but, contrary to what stated for example in Jerez (2005), the set of equilibrium allocations may be empty and the Second Welfare Theorem may fail. This happens for two reasons. First, constrained efficient allocations may violate the necessary and sufficient conditions of price supportability for the individuals. Second, even when constrained efficient allocation are price supportable, they may fail to be a profit maximizing choice of the firm at the individual supporting prices. To restore existence of an equilibrium the firm has to be restricted to supply allocations with support in the set of incentive compatible contracts.     

Trade and revelation of information

We consider an economy with asymmetric information and two types of agents, fully informed and uninformed. Uninformed agents update their information observing equilibrium prices and the equilibrium levels of other agents’ excess demand. We show that, for a generic set of economies, there are rational expectations equilibria which are partially revealing on an open, dense set of signals of positive Lebesgue measure, provided that the dimension of the signal space is sufficiently larger than the dimension of the commodity space.     

A complete characterization of Pareto optimality for general OLG economies

In the context of general pure exchange OLG economies where agents can have heterogeneous longevities, we provide both sufficient and necessary conditions for Pareto optimality of competitive equilibria. For the case in which all agents live for the same number of periods, we find that these conditions are equivalent. We also find this equivalence when agents can have different lifetimes, but in this case we need to impose particular restrictions on relative equilibrium prices. Moreover, we show that without these conditions on prices the equivalence, and hence a full characterization, is not necessarily obtained.     

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.     

The type-agent core for exchange economies with asymmetric information

The type-agent core is a new solution concept for exchange economies with asymmetric information. It coincides with the set of subgame-perfect equilibrium outcomes of a simple competitive screening game. Uninformed intermediaries help the agents to cooperate in an attempt to make some profit. The paper extends the work of Perez-Castrillo [Cooperative outcomes through non-cooperative games, Games Econ. Behav. 7 (1994) 428-440] to exchange economies with non-transferable utility and asymmetric information. The type-agent core is a subset of Wilson's coarse core [Wilson, Information, efficiency, and the core of an economy, Econometrica 46 (1978) 807-816]. It is never empty, even though it may be a strict subset of Wilson's fine core. In addition, it converges towards the set of constrained market equilibria as the economy is replicated.     

Full revelation of information in Sender-Receiver games of persuasion

We provide necessary and sufficient conditions on both players' preferences and information that can be certified for a Sender-Receiver game to possess a separating equilibrium, as well as sufficient conditions for every equilibrium of such a game to be separating. Accordingly, we generalize Seidmann and Winter's [D.J. Seidmann, E. Winter, Strategic information transmission with verifiable messages, Econometrica 65 (1997) 163-170] results to games with partial provability.     

Trade,Growth, and Increasing Returns to Infrastructure: The Role of the Sophisticated Monopolist*

Equilibrium in international trade with increasing returns in infrastructure depends on whether the infrastructure provider is “naïve” or sophisticated. A monopolist produces infrastructure under decreasing cost using fixed equipment. Unlike similar work, we derive a unique closed‐economy equilibrium. In a small open economy, with “naïve” infrastructure provider(s), multiple equilibria obtain. The industrial export potential of the economy depends on unexhausted economies of scale, and equilibria are possible where manufactures are exported despite an autarky price higher than the world price. With a sophisticated infrastructure provider, even an open economy has a unique equilibrium, which, at least as long as economies of scale are unexhausted, also involves more industrialization than the “naïve” equilibria. Access to the unlimited world market is necessary for significant industrialization but is not sufficient: one may also require “Schumpeterian” entrepreneurs, monopolists with a panoramic vision of the economy and of their catalytic role in it.     

The communication requirements of social choice rules and supporting budget sets

The paper examines the communication requirements of social choice rules when the (sincere) agents privately know their preferences. It shows that for a large class of choice rules, any minimally informative way to verify that a given alternative is in the choice rule is by verifying a “budget equilibrium”, i.e., that the alternative is optimal to each agent within a “budget set” given to him. Therefore, any communication mechanism realizing the choice rule must find a supporting budget equilibrium. We characterize the class of choice rules that have this property. Furthermore, for any rule from the class, we characterize the minimally informative messages (budget equilibria) verifying it. This characterization is used to identify the amount of communication needed to realize a choice rule, measured with the number of transmitted bits or real variables. Applications include efficiency in convex economies, exact or approximate surplus maximization in combinatorial auctions, the core in indivisible-good economies, and stable many-to-one matchings.     

Broadcasting Opinions with an Overconfident Sender*

We analyze a model where an altruistic, but possibly overconfident sender broadcasts one of a finite set of messages to rational receivers. If broadcasting is costless and the sender is rational, there is an informationally efficient equilibrium, but multiple equilibria may arise, and asymmetric equilibria might be more informative than the symmetric equilibrium even if the prior is symmetric. Although overconfidence on the part of the sender reduces informativeness in some cases, it may also eliminate less informative equilibria and lead to better information transmission. Overconfidence can also improve the informativeness of the message when broadcasting is costly.     

Wage–vacancy contracts and multiplicity of equilibria in a directed search model of the labour market

This paper studies a directed search model of the labour market, which is standard in all aspects except two. First, we allow firms to post wage–vacancy contracts advertising the number of workers they would pay as well as the payment all will receive. Second, we consider two cases: one where workers are risk neutral and one where workers are risk averse, both in finite and large economies. Our paper shows that when firms post wage–vacancy contracts, whether workers are modelled as risk neutral or risk averse matters: the types of symmetric equilibria and the nature of multiplicity of equilibria are different. Somewhat surprisingly, when there are finite numbers of risk‐neutral workers and firms, we obtain a finite number of symmetric equilibria, but when workers are risk averse, we obtain a continuum of equilibria. Furthermore, our paper sounds a cautionary note on using large economies as an approximation of finite economies: when workers are risk neutral, the nature of equilibrium is preserved going from a finite to a large economy, but the nature of equilibrium is different when workers are risk averse.     

Intertemporal equilibria with Knightian uncertainty

We study a dynamic and infinite-dimensional model with incomplete multiple prior preferences. In interior efficient allocations, agents share a common risk-adjusted prior and subjective interest rate. Interior efficient allocations and equilibria coincide with those of economies with subjective expected utility and priors from the agents? multiple prior sets. A specific model with neither risk nor uncertainty at the aggregate level is considered. Risk is always fully insured. For small levels of ambiguity, there exists an equilibrium with inertia where agents also insure fully against Knightian uncertainty. When the level of ambiguity exceeds a critical threshold, full insurance no longer prevails and there exist equilibria with inertia where agents do not insure against uncertainty at all. We also show that equilibria with inertia are indeterminate.     

Generalized potentials and robust sets of equilibria

This paper introduces generalized potential functions of complete information games and studies the robustness of sets of equilibria to incomplete information. A set of equilibria of a complete information game is robust if every incomplete information game where payoffs are almost always given by the complete information game has an equilibrium which generates behavior close to some equilibrium in the set. This paper provides sufficient conditions for the robustness of sets of equilibria in terms of argmax sets of generalized potential functions. These sufficient conditions unify and generalize existing sufficient conditions. Our generalization of potential games is useful in other game theoretic problems where potential methods have been applied.     

Lotteries, Sunspots, and Incentive Constraints

We study a prototypical class of exchange economies with private information and indivisibilities. We establish an equivalence between lottery equilibria and sunspot equilibria and show that the welfare and existence theorems hold. To establish these results, we introduce the concept of the stand-in consumer economy, which is a standard, convex, finite consumer, finite good, pure exchange economy. With decreasing absolute risk aversion and no indivisibilities, we prove that no lotteries are actually used in equilibrium. We provide a simple numerical example with increasing absolute risk aversion in which lotteries are necessarily used in equilibrium. We also show how the equilibrium allocation in this example can be implemented in a sunspot equilibrium. Journal of Economic Literature Classification Numbers: D11, D50, D82.     

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     

Risk aversion in symmetric and asymmetric contests

We analyze existence, uniqueness and properties of equilibria in incompletely discriminating Tullock contests with logistic contest success functions, when contestants are risk averse. We prove that a Nash equilibrium for such a contest exists, but give an example of a symmetric contest with both symmetric and asymmetric equilibria, showing that risk aversion may lead to multiple equilibria. Symmetric contests have unique symmetric equilibria but additional conditions are necessary for general uniqueness. We also study the effects on incumbents of additional competitors entering the contest under these conditions and examine the effects of risk aversion on rent dissipation in symmetric and asymmetric contests.     

A two-stage core with applications to asset market and differential information economies

Summary. We introduce a new core concept, called the two-stage core, which is appropriate for economies with sequential trade. We prove a general existence theorem and present two applications of the two-stage core: (i) In asset markets economies where we extend our existence proof to the case of consumption sets with no lower bound, in order to capture the case of arbitrary short sales of assets. Further, we show that the two-stage core is non empty in the Hart (1975) example where a rational expectations equilibrium fails to exist. (ii) In differential information economies where we provide sufficient conditions for the incentive compatibility of trades. Namely, that no coalition of agents can misreport the true state and provide improvements to all its members, even by redistributing the benefits from misreporting. Received: December 20, 1995; revised version: December 6, 1996     

General equilibrium without utility functions: how far to go?

How far can we go in weakening the assumptions of the general equilibrium model? Existence of equilibrium, structural stability and finiteness of equilibria of regular economies, genericity of regular economies and an index formula for the equilibria of regular economies have been known not to require transitivity and completeness of consumers’ preferences. We show in this paper that if consumers’ non-ordered preferences satisfy a mild version of convexity already considered in the literature, then the following properties are also satisfied: (1) the smooth manifold structure and the diffeomorphism of the equilibrium manifold with a Euclidean space; (2) the diffeomorphism of the set of no-trade equilibria with a Euclidean space; (3) the openness and genericity of the set of regular equilibria as a subset of the equilibrium manifold; (4) for small trade vectors, the uniqueness, regularity and stability of equilibrium for two version of tatonnement; (5) the pathconnectedness of the sets of stable equilibria.     

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.