Sharing Risk E¢ ciently under Suboptimal Punishments for Defection

I study the e¢ cient risk-sharing in an endowments economy when enforcement is achieved by the threat of reversion to punishments that may be less severe than autarkic consumption. I characterize (up to a technical condition) the set of allocations that may be interpreted as e¢ cient with respect to some punishment convention. The conditions rationalizing such e¢ ciency are very weak; they are (i) resource exhaustion, (ii) satisfaction of individual rationality constraints at each continuation, and (iii) …niteness of the value of the allocation under the implicit decentralizing price system. I show how e¢ cient allocations may be decentralized, and I state versions of the Welfare Theorems for these economies.