Sharing Risk Efficiently Under Suboptimal Punishments for Defection

I study the efficient 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 efficient with respect to some punishment convention. The conditions rationalizing such efficiency are very weak; they are (i) resource exhaustion, (ii) satisfaction of individual rationality constraints at each continuation, and (iii) finiteness of the value of the allocation under the implicit decentralizing price system. I show how efficient allocations may be decentralized, and I state versions of the Welfare Theorems for these economies