Optimality of equilibria in differential information economies with restricted coalitions

Abstract

It is well known that the set of unblocked allocations, the core, and the set of Walras allocations coincide in an atomless economy when all measurable coalitions are allowed to form. Clearly, if only a subset of the set of all coalitions is allowed to form, the set of unblocked allocations enlarges, and generally we can merely say that this larger set contains the set of Walrasian allocations. In reality the lack of communication restricts the set of coalitions that can be formed. The purpose of our work is investigate the Core-Walras equivalence by imposing to the set of all coalitions some restrictions. The notion of the core is based on the premise that any group of agents can cooperate and agree upon a coordinated set of actions which can then be enforced. In the context of a differential information economy, an allocation should be seen as a state-contingent allocation satisfying physical resource constraints in each information state. A central role is played by the information set of each agent. Agents which enter into a coalition contract at the ex-ante stage, i.e. before any agents receives private information, or at the interim, i.e. after each agent has received her private information. It is well known that the ex-post stage, i.e. decisions are made after the information state is known, is no different from a model with complete information. An appropriate notion of the core must take into account of whether the coalition decisions stage is ex-ante or interim. In our work we specially deal with the private core which is an ex-ante concept. It has some interesting properties: it exists under standard continuity and concavity assumptions on utility functions, it is coalitionally incentive, i.e. there is truthful revelation of information in each coalition, and it takes into account the information superiority of traders. From the non cooperative side, we deal with two main equilibrium concepts: • the rational expectations equilibrium which is an interim concept in which prices are referred to as signals reflecting and transferring information; • the competitive private equilibrium, which is closer to the Walrasian equilibrium notion in the deterministic case: this non-cooperative solution concept presumes that agents maximize their ex-ante expected utility subject to their budget constraint in which information constraints, besides the classical ones, are considered. We will consider a differential information exchange economy obtained by introducing in the classical Arrow-Debreu model both uncertainty and asymmetries in information. In these model, uncertainty is exogenous and is represented by a measure space (Ω,F) where Ω denotes the finite set of all possible states of nature and F is the set of all possible events. We will assume that agents make coalitional decisions at the ex-ante stage but each agent receives private information which is not publicly verifiable before consumption takes place. In particular, agents trades with the anonymous market rather than with other agents directly and it becomes very natural to require that agents' trade be measurable with respect to their private information. We characterize in terms of decentralizing prices several notions of core allocations resulting from different possible restrictions imposed to the set of blocking coalitions. Reciprocal relations among cores are also studied. Another issue we have investigated in our work, starting from the reality restrictions of coalitions due by various rules imposed over the society (i.e. information, transportation, legal and institutional constraints), is the number and composition of the set of blocking coalitions.