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An Algebraic Index Theorem for Non-smooth Economies

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  • Gaël Giraud

    (BETA - Bureau d'Économie Théorique et Appliquée - INRA - Institut National de la Recherche Agronomique - UNISTRA - Université de Strasbourg - UL - Université de Lorraine - CNRS - Centre National de la Recherche Scientifique)

Abstract

In this paper, we prove an existence theorem for equilibria in production economies with increasing returns, which generalizes the classic results on this topic. In particular, we eliminate both the free-disposal assumptions and any smoothness requirements on the boundary of the production sets. For this purpose, we propose a new definition of the topological degree for non-convex-valued correspondences defined on non-smooth topological manifolds.

Suggested Citation

  • Gaël Giraud, 2001. "An Algebraic Index Theorem for Non-smooth Economies," Post-Print hal-00460314, HAL.
    • Handle: RePEc:hal:journl:hal-00460314
    DOI: 10.1016/S0304-4068(01)00074-X
    Note: View the original document on HAL open archive server: https://hal.science/hal-00460314
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    References listed on IDEAS

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    1. Cornet, Bernard, 1988. "General equilibrium theory and increasing returns : Presentation," Journal of Mathematical Economics, Elsevier, vol. 17(2-3), pages 103-118, April.
    2. Debreu, Gerard, 1970. "Economies with a Finite Set of Equilibria," Econometrica, Econometric Society, vol. 38(3), pages 387-392, May.
    3. Jouini, Elyes, 1992. "Existence of equilibria in nonconvex economies without free disposal," Economics Letters, Elsevier, vol. 38(1), pages 37-42, January.
    4. Jouini, Elyes, 1992. "An index theorem for nonconvex production economies," Journal of Economic Theory, Elsevier, vol. 57(1), pages 176-196.
    5. Geanakoplos, John & Shafer, Wayne, 1990. "Solving systems of simultaneous equations in economics," Journal of Mathematical Economics, Elsevier, vol. 19(1-2), pages 69-93.
    6. repec:dau:papers:123456789/5638 is not listed on IDEAS
    7. Bonnisseau, Jean-Marc & Cornet, Bernard, 1988. "Existence of equilibria when firms follow bounded losses pricing rules," Journal of Mathematical Economics, Elsevier, vol. 17(2-3), pages 119-147, April.
    8. McLennan, Andrew, 1991. "Approximation of contractible valued correspondences by functions," Journal of Mathematical Economics, Elsevier, vol. 20(6), pages 591-598.
    9. Kamiya, Kazuya, 1988. "Existence and uniqueness of equilibria with increasing returns," Journal of Mathematical Economics, Elsevier, vol. 17(2-3), pages 149-178, April.
    10. Hamano, Tadashi, 1994. "On the existence of equilibria without convexity or free disposal on production technologies," Journal of Mathematical Economics, Elsevier, vol. 23(6), pages 565-583, November.
    11. Beato, Paulina & Mas-Colell, Andreu, 1985. "On marginal cost pricing with given tax-subsidy rules," Journal of Economic Theory, Elsevier, vol. 37(2), pages 356-365, December.
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    Citations

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    Cited by:

    1. Covarrubias, Enrique, 2013. "The number of equilibria of smooth infinite economies," Journal of Mathematical Economics, Elsevier, vol. 49(4), pages 263-265.
    2. Gaël Giraud & Céline Rochon, 2007. "Natural rate of unemployment and efficiency: a dynamic analysis with flexible prices and increasing returns," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00155739, HAL.
    3. Hans-Jürgen Salchow, 2005. "Non-existence of equilibria with free elimination," Cahiers de la Maison des Sciences Economiques b05057, Université Panthéon-Sorbonne (Paris 1).
    4. Acurio Vásconez, Verónica & Giraud, Gaël & Mc Isaac, Florent & Pham, Ngoc-Sang, 2015. "The effects of oil price shocks in a new-Keynesian framework with capital accumulation," Energy Policy, Elsevier, vol. 86(C), pages 844-854.
    5. Antoine Mandel, 2007. "Changes in the firms behavior after the opening of an allowance market," Documents de travail du Centre d'Economie de la Sorbonne b07027, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    6. Gaël Giraud & Cécile Renouard, 2011. "In search of an alternative to shareholder value maximization," Documents de travail du Centre d'Economie de la Sorbonne 11038, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    7. Antoine Mandel, 2009. "Changes in the firms behavior after the opening of markets of allowances," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 40(1), pages 1-25, July.
    8. Mandel, Antoine, 2008. "An index formula for production economies with externalities," Journal of Mathematical Economics, Elsevier, vol. 44(12), pages 1385-1397, December.
    9. Salchow, Hans-Jurgen, 2006. "An essay on the state of economic science," Journal of Mathematical Economics, Elsevier, vol. 42(6), pages 653-660, September.

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    More about this item

    Keywords

    non-convex production; topological degree; local homology; increasing returns; Non-smooth production; Production non-lisse; production non-convexe; rendements croissants; degré topologique; homologie locale;
    All these keywords.

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • D51 - Microeconomics - - General Equilibrium and Disequilibrium - - - Exchange and Production Economies

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