On the Failure O F Core Convergence in Economies with Asymmetric Information

THE AIM OF THIS PAPER is to study core convergence in economies with asymmetric information. We will argue that the positive results of Debreu and Scarf (1963) and Aumann (1964) do not extend to a model with incomplete information in which decisions are made at the interim stage.2 Consider first the case in which decisions are made at the ex-ante stage, i.e., before information is revealed to any agent. A well-known price equilibrium concept for such a model is the notion of an Arrow-Debreu equilibrium in (complete) markets with contingent commodities. The fact that in such markets an Arrow-Debreu equilibrium is identical to a Walrasian equilibrium with an appropriate indexing of commodities by time and state immediately implies the standard relationship between such allocations and a corresponding notion of the ex-ante core: (a) Arrow-Debreu allocations belong to the ex-ante core, and (b) the Debreu-Scarf argument can be applied (with no more than a reinterpretation) to assert that any ex-ante core allocation that survives replication is an Arrow-Debreu allocation. While the Arrow-Debreu model involves incomplete information, it is essentially one of symmetric uncertainty. Asymmetry of information can be incorporated into this ex-ante framework by postulating that consumers differ in their ex-post information. One such approach is the one introduced by Radner (1968), which imposes the requirement that an agent's trades be measurable with respect to her private information. Equilibrium allocations so defined (Radner allocations) bear the standard relationship with an ex-ante core concept that similarly imposes such measurability restrictions (as in Allen (1991) and Yannelis (1991)); see Einy, Moreno, and Shitovitz (2001). Another approach for dealing with ex-post asymmetry of information (based on mechanism design) is to directly impose incentive compatibility on agents' trades. A corresponding price equilibrium notion is the one used in Prescott and Townsend (1984). Here, even in the ex-ante case, matters are no longer so simple. As Forges, Heifetz, and Minelli (2001) show, core equivalence does not generally hold, although a positive result can be established under certain conditions. We shall concentrate on the interim stage, i.e., the stage when agents have received their private information. In this context, too, the existing literature (Goenka and Shell (19971, Kobayashi (19801, and Yannelis (1991)) seems to point to the validity of the convergence principle. However, we will begin by showing that the coarse core of Wilson (1978) does not converge to any set of price equilibrium allocations considered in the literature. To prove our main point we construct a simple example of a replicated sunspot