Efficient allocations and equilibria with short-selling and incomplete preferences

This article reconsiders the theory of existence of efficient allocations and equilibria when consumption sets are unbounded below under the assumption that agents have incomplete preferences. It is motivated by an example in the theory of assets with short-selling where there is risk and ambiguity. Agents have Bewley’s incomplete preferences. As an inertia principle is assumed in markets, equilibria are individually rational. It is shown that a necessary and sufficient for existence of an individually rational efficient allocation or of an equilibrium is that the relative interiors of the risk adjusted sets of probabilities intersect. The more risk averse, the more ambiguity averse the agents, the more likely is an equilibrium to exist. The paper then turns to incomplete preferences with concave multi-utility representations. Several definitions of efficiency and of equilibrium with inertia are considered. Sufficient conditions and necessary and sufficient conditions are given for existence of efficient allocations and equilibria with inertia.