Ceparano, Maria Carmela and Quartieri, Federico (2018): A Second Welfare Theorem in a Nonconvex Economy: The Case of Antichainconvexity.

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Abstract
We introduce the notion of an antichainconvex 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 nonconvexities can be spontaneously obtained as valuation quasiequilibria 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 nonconvex sets. Such a study allows us to obtain a version of the Minkowski\HahnBanach 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.
Item Type:  MPRA Paper 

Original Title:  A Second Welfare Theorem in a Nonconvex Economy: The Case of Antichainconvexity 
Language:  English 
Keywords:  Second Theorem of Welfare Economics; Nonconvex Economies; Chainconvexity and Antichainconvexity; Separation Theorem; Convex Sum of Nonconvex Sets. 
Subjects:  C  Mathematical and Quantitative Methods > C0  General > C02  Mathematical Methods C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C60  General D  Microeconomics > D5  General Equilibrium and Disequilibrium > D51  Exchange and Production Economies D  Microeconomics > D6  Welfare Economics > D61  Allocative Efficiency ; CostBenefit Analysis 
Item ID:  87531 
Depositing User:  Federico Quartieri 
Date Deposited:  24 Jun 2018 16:36 
Last Modified:  24 Jun 2018 16:36 
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URI:  https://mpra.ub.unimuenchen.de/id/eprint/87531 