On the Failure of Core Convergence in Economies with Asymmetric Information

In interim economies with asymmetric information, we show a robust failure of the core convergence theorem. We 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 economy with strictly convex and monotonic preferences. We also show that the coarse core may not satisfy the equal treatment property. Moreover, coarse core allocations satisfying the equal treatment property may not converge to price equilibrium allocations. The result applies to any notion of price equilibrium (including rational expectations equilibria, Radner equilibria, sunspot equilibria) in which fully informed consumers receive a consumption plan maximizing ex-post utility over the ex-post budget set. The non-convergence result holds for several different notions of the interim core proposed in the literature, including the fine core (Wilson (1978)), the private core (Yannelis (1991)), the core in the pooling case (Ichiishi and Idzik (1996), Goenka and Shell (1997)), the incentive compatible core (Vohra (1999)), the coarse+ core (Lee and Volij (forthcoming)), the core with endogenous communication (Volij (forthcoming)). This failure of core convergence is a basic consequence of asymmetric information, which at the interim stage, imposes important restrictions on coalition formation. (Informational constraints may rule out many coalitions of types of consumers that would be present in the standard Debreu and Scarf (1963) complete information framework.) In particular, our negative results do not rely on the lack of `informational smallness' of agents (at least in the sense in which this term has recently been formalized in McLean and Postlewaite (1999)).