A second welfare theorem in a non-convex economy: The case of antichain-convexity

Abstract

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 …nitely many possibly non-convex sets. Such a study allows us to obtain a version of the MinkowskinHahn-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.

On $C$-Pareto dominance in decomposably $C$-antichain-convex sets

“…, X n are K-antichain-convex. Part 1 of Lemma 4 in [9] and part 1 of Theorem 1 guarantee that X is K-upward. As X is K-upward and K contains the zero vector, part 3 of Lemma 5 in [9] implies X + K = X and hence…”

“…Put C 0 = C ∪ {0}. Part 1 of Lemma 1 in [9] guarantees that C 0 is a cone in V . We are done if we show that C 0 + C 0 ⊆ C 0 .…”

“…The following facts are two general results of some importance for the sequel of this work. Theorem 1 is essentially new while Theorem 2 uses-but does not directly follow from-a previous result shown in [9].…”

“…possibly non-quasiconcave function). The first type of results can be used, for instance, in welfare economics in order to obtain variants of the classical "Second Welfare Theorem": within the mathematical strand of literature motivated by the influential paper [15], we mention [21,22,4,20,24,3,12,13,18,16,9]. The second type of results can be used, for instance, in demand theory where the convexity of the solutions of a consumer's constrained maximization problem is a property with useful consequences for the theory of general equilibrium: see, e.g., [5,Chapter 18]-as well as the literature cited therein-for illustrations of the use of convex-valued (excess) demand correspondences in the proofs of the existence of an equilibrium price.…”

“…The notion of C-antichain-convexity stipulates the usual definition of convexity only for pairs of vectors that are unrelated through R C . Such a notion has been introduced 1 in [9], where it is shown that the Minkowski sum of two Cantichain-convex sets is convex if one of the two summands is C-upward (a set is C-upward whenever its Minkowski sum with the cone C is included in the set itself). This mathematical fact-which properly generalizes the assertion that the sum of two convex sets is convex-is used in that article to prove a separation theorem for possibly non-convex sets.…”