IDEAS home Printed from https://ideas.repec.org/a/eee/mateco/v81y2019icp31-47.html
   My bibliography  Save this article

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

Author

Listed:
  • 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.

Suggested Citation

  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304406819300011
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.jmateco.2018.12.007?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to look for a different version below or search for a different version of it.

    Other versions of this item:

    References listed on IDEAS

    as
    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, December.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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.
    2. Paul Oslington, 2012. "General Equilibrium: Theory and Evidence," The Economic Record, The Economic Society of Australia, vol. 88(282), pages 446-448, September.
    3. W D A Bryant, 2009. "General Equilibrium:Theory and Evidence," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 6875, August.
    4. Flam, S.D. & Jourani, A., 2000. "Prices and Pareto Optima," Norway; Department of Economics, University of Bergen 0800, Department of Economics, University of Bergen.
    5. Jean-Marc Bonnisseau & Matias Fuentes, 2022. "Increasing returns, externalities and equilibrium in Riesz spaces," Documents de travail du Centre d'Economie de la Sorbonne 22025, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    6. Sushama Murty, 2014. "Necessary and sufficient conditions for an environmental Kuznets curve with some illustrative examples," Discussion Papers 1407, University of Exeter, Department of Economics.
    7. Jean-Marc Bonnisseau, 2000. "The Marginal Pricing Rule in Economies with Infinitely Many Commodities," Econometric Society World Congress 2000 Contributed Papers 0262, Econometric Society.
    8. 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.
    9. Jean-Marc Bonnisseau & Matías Fuentes, 2020. "Market Failures and Equilibria in Banach Lattices: New Tangent and Normal Cones," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 338-367, February.
    10. Jean Paul Chavas, 2015. "Coase Revisited: Economic Efficiency under Externalities, Transaction Costs, and Nonconvexity," Journal of Institutional and Theoretical Economics (JITE), Mohr Siebeck, Tübingen, vol. 171(4), pages 709-734, December.
    11. Antonio Villar, 1994. "Existence and efficiency of equilibrium in economics with increasing returns to scale: an exposition," Investigaciones Economicas, Fundación SEPI, vol. 18(2), pages 205-243, May.
    12. Jean-Marc Bonnisseau & Matías Fuentes, 2022. "Increasing returns, externalities and equilibrium in Riesz spaces," Working Papers halshs-03908326, HAL.
    13. Paweł Dziewulski, 2011. "On Time-to-Build Economies with Multiple-Stage Investments," Gospodarka Narodowa. The Polish Journal of Economics, Warsaw School of Economics, issue 9, pages 23-49.
    14. Torregrosa, Ramon. J., 2008. "Preference for Income Taxation with Several Heterogeneous Consumers," MPRA Paper 14291, University Library of Munich, Germany.
    15. Bonnisseau, J.-M. & Cornet, B., 2008. "Existence of equilibria with a tight marginal pricing rule," Journal of Mathematical Economics, Elsevier, vol. 44(7-8), pages 613-624, July.
    16. Brown, Donald J. & Heller, Walter P. & Starr, Ross M., 1992. "Two-part marginal cost pricing equilibria: Existence and efficiency," Journal of Economic Theory, Elsevier, vol. 57(1), pages 52-72.
    17. Sudhir A. Shah, 2016. "The Generalized Arrow-Pratt Coefficient," Working papers 254, Centre for Development Economics, Delhi School of Economics.
    18. Jean-Marc Bonnisseau & Bernard Cornet & Marc-Olivier Czarnecki, 2007. "The marginal pricing rule revisited," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 33(3), pages 579-589, December.
    19. Doraszelski, Ulrich & Escobar, Juan F., 2019. "Protocol invariance and the timing of decisions in dynamic games," Theoretical Economics, Econometric Society, vol. 14(2), May.
    20. Murty, Sushama, 2010. "Externalities and fundamental nonconvexities: A reconciliation of approaches to general equilibrium externality modeling and implications for decentralization," Journal of Economic Theory, Elsevier, vol. 145(1), pages 331-353, January.

    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

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:mateco:v:81:y:2019:i:c:p:31-47. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/jmateco .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.