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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 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.

Suggested Citation

  • Ceparano, Maria Carmela & Quartieri, Federico, 2018. "A Second Welfare Theorem in a Non-convex Economy: The Case of Antichain-convexity," MPRA Paper 87531, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:87531
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    References listed on IDEAS

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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.
    2. M. Ali Khan, 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, July.
    3. Monique Florenzano & Pascal Gourdel & Alejandro Jofré, 2006. "Supporting weakly Pareto optimal allocations in infinite dimensional nonconvex economies," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 29(3), pages 549-564, November.
    4. M. Ali Khan & Rajiv Vohra, 1987. "An Extension of the Second Welfare Theorem to Economies with Nonconvexities and Public Goods," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 102(2), pages 223-241.
    5. Ceparano, Maria Carmela & Quartieri, Federico, 2017. "Nash equilibrium uniqueness in nice games with isotone best replies," Journal of Mathematical Economics, Elsevier, vol. 70(C), pages 154-165.
    6. Flam, S.D. & Jourani, A., 2000. "Prices and Pareto Optima," Norway; Department of Economics, University of Bergen 0800, Department of Economics, University of Bergen.
    7. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680.
    8. repec:bla:jpbect:v:1:y:1999:i:3:p:309-38 is not listed on IDEAS
    9. Guesnerie, Roger, 1975. "Pareto Optimality in Non-Convex Economies," Econometrica, Econometric Society, vol. 43(1), pages 1-29, January.
    10. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, January.
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    Cited by:

    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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