Pareto-optimal assignments by hierarchical exchange

A version of the Second Fundamental Theorem of Welfare Economics that applies to a money-free environment, in which a set of indivisible goods needs to be matched to some set of agents, is established. In such environments, 'trade' can be identified with the set of hierarchical exchange mechanisms de ned by Papai (2000). Papai (2000)'s result - that any such mechanism yields Pareto-optimal allocations - can be interpreted as a version of the First Fundamental Theorem of Welfare Economics for the given environment. In this note, I show that for any Pareto-optimal allocation and any hierarchical exchange mechanism one can add an initial allocation of ownership rights, such that the given Pareto-optimal allocation arises as a result of trade.