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

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  • Ceparano, Maria Carmela
  • Quartieri, Federico

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 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 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.

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  • Ceparano, Maria Carmela & Quartieri, Federico, 2019. "A second welfare theorem in a non-convex economy: The case of antichain-convexity," Journal of Mathematical Economics, Elsevier, vol. 81(C), pages 31-47.
    • Handle: RePEc:eee:mateco:v:81:y:2019:i:c:p:31-47
    DOI: 10.1016/j.jmateco.2018.12.007
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    1. Bonnisseau, Jean-Marc & Cornet, Bernard, 1988. "Valuation equilibrium and pareto optimum in non-convex economies," Journal of Mathematical Economics, Elsevier, vol. 17(2-3), pages 293-308, April.
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    8. Khan, M Ali, 1999. "The Mordukhovich Normal Cone and the Foundations of Welfare Economics," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 1(3), pages 309-338.
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    1. Maria Carmela Ceparano & Federico Quartieri, 2020. "On Pareto Dominance in Decomposably Antichain-Convex Sets," Journal of Optimization Theory and Applications, Springer, vol. 186(1), pages 68-85, July.

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    More about this item

    Keywords

    Second theorem of welfare economics; Non-convex economies; Chain-convexity and antichain-convexity; Separation theorem; Convex sum of non-convex sets;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • D51 - Microeconomics - - General Equilibrium and Disequilibrium - - - Exchange and Production Economies
    • D61 - Microeconomics - - Welfare Economics - - - Allocative Efficiency; Cost-Benefit Analysis

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