A Second Welfare Theorem in a Non-convex Economy: The Case of Antichain-convexity

We introduce the notion of an antichain-convex set to extend Debreu (1954)'s version of the second welfare theorem to economies where either the aggregate production set or preference relations are not convex. We show that (possibly after some redistribution of individuals' wealth) the Pareto optima of some economies which are marked by certain types of non-convexities can be spontaneously obtained as valuation quasi-equilibria and equilibria: both equilibrium notions are to be understood in Debreu (1954)'s sense. From a purely structural point of view, the mathematical contribution of this work is the study of the conditions that guarantee the convexity of the Minkowski sum of finitely many possibly non-convex sets. Such a study allows us to obtain a version of the Minkowski\Hahn-Banach separation theorem which dispenses with the convexity of the sets to be separated and which can be naturally applied in standard proofs of the second welfare theorem; in addition (and equally importantly) the study allows to get a deeper understanding of the conditions on the single production sets of an economy that guarantee the convexity of their aggregate.