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Differentiability of Equilibria for Linear Exchange Economies

Author

Listed:
  • Jean-Marc Bonnisseau

    (CERMSEM - CEntre de Recherche en Mathématiques, Statistique et Économie Mathématique - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

  • Michael Florig

    (CERMSEM - CEntre de Recherche en Mathématiques, Statistique et Économie Mathématique - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

  • Alejandro Jofré

    (CMM - Centre de modélisation mathématique / Centro de Modelamiento Matemático [Santiago] - UCHILE - Universidad de Chile = University of Chile [Santiago] - CNRS - Centre National de la Recherche Scientifique)

Abstract

The purpose of this paper is to study the differentiability properties of equilibrium prices and allocations in a linear exchange economy when the initial endowments and utility vectors vary. We characterize an open dense subset of full measure of the initial endowment and utility vector space on which the equilibrium price vector is a real analytic function, hence infinitely differentiable function. We provide an explicit formula to compute the equilibrium price and allocation around a point where it is known. We also show that the equilibrium price is a locally Lipschitzian mapping on the whole space. Finally, using the notion of the Clarke generalized gradient, we prove that linear exchange economies satisfy a property of gross substitution.

Suggested Citation

  • Jean-Marc Bonnisseau & Michael Florig & Alejandro Jofré, 2001. "Differentiability of Equilibria for Linear Exchange Economies," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00265685, HAL.
    • Handle: RePEc:hal:cesptp:halshs-00265685
    DOI: 10.1023/A:1017558204399
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    References listed on IDEAS

    as
    1. J. M. Bonnisseau & M. Florig & A. Jofré, 2001. "Continuity and Uniqueness of Equilibria for Linear Exchange Economies," Journal of Optimization Theory and Applications, Springer, vol. 109(2), pages 237-263, May.
    2. repec:dau:papers:123456789/5639 is not listed on IDEAS
    3. Mas-Colell,Andreu, 1990. "The Theory of General Economic Equilibrium," Cambridge Books, Cambridge University Press, number 9780521388702.
    4. Jouini, Elyes, 1993. "The graph of the Walras correspondence : The production economies case," Journal of Mathematical Economics, Elsevier, vol. 22(2), pages 139-147.
    5. Gale, David, 1976. "The linear exchange model," Journal of Mathematical Economics, Elsevier, vol. 3(2), pages 205-209, July.
    6. Cheng, Hsueh-Cheng, 1979. "Linear economies are "gross substitute" systems," Journal of Economic Theory, Elsevier, vol. 20(1), pages 110-117, February.
    7. Eaves, B. Curtis, 1976. "A finite algorithm for the linear exchange model," Journal of Mathematical Economics, Elsevier, vol. 3(2), pages 197-203, July.
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    Cited by:

    1. Gaël Giraud & Dimitrios Tsomocos, 2010. "Nominal uniqueness and money non-neutrality in the limit-price exchange process," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 45(1), pages 303-348, October.
    2. J. M. Bonnisseau & M. Florig & A. Jofré, 2001. "Continuity and Uniqueness of Equilibria for Linear Exchange Economies," Journal of Optimization Theory and Applications, Springer, vol. 109(2), pages 237-263, May.
    3. M. Florig, 2004. "Equilibrium Correspondence of Linear Exchange Economies," Journal of Optimization Theory and Applications, Springer, vol. 120(1), pages 97-109, January.

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