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On the existence of approximated equilibria in discontinuous economies

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  • Philippe Bich

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

Abstract

In this paper, we prove an existence theorem for approximated equilibria in a class of discontinuous economies. The existence result is a direct consequence of a discontinuous extension of Brouwer's fixed point Theorem (1912), and is a refinement of several classical results in the standard General Equilibrium with Incomplete markets (GEI) model (e.g., Bottazzi (1995), Duffie and Shafer (1985), Husseini et al. (1990), Geanakoplos and Shafer (1990), Magill and Shafer (1991)). As a by-product, we get the first existence proof of an approximated equilibrium in the GEI model, without perturbing the asset structure nor the endowments. Our main theorem rests on a new topological structure result for the asset equilibrium space and may be of interest by itself.

Suggested Citation

  • Philippe Bich, 2005. "On the existence of approximated equilibria in discontinuous economies," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00287685, HAL.
    • Handle: RePEc:hal:cesptp:halshs-00287685
    DOI: 10.1016/j.jmateco.2004.12.003
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-00287685
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    References listed on IDEAS

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    1. Geanakoplos, John & Shafer, Wayne, 1990. "Solving systems of simultaneous equations in economics," Journal of Mathematical Economics, Elsevier, vol. 19(1-2), pages 69-93.
    2. Brown, Donald J & DeMarzo, Peter M & Eaves, B Curtis, 1996. "Computing Equilibria When Asset Markets Are Incomplete," Econometrica, Econometric Society, vol. 64(1), pages 1-27, January.
    3. Magill, Michael & Shafer, Wayne, 1991. "Incomplete markets," Handbook of Mathematical Economics, in: W. Hildenbrand & H. Sonnenschein (ed.), Handbook of Mathematical Economics, edition 1, volume 4, chapter 30, pages 1523-1614, Elsevier.
    4. Hirsch, M. D. & Magill, M. & Mas-Colell, A., 1990. "A geometric approach to a class of equilibrium existence theorems," Journal of Mathematical Economics, Elsevier, vol. 19(1-2), pages 95-106.
    5. Bottazzi, Jean-Marc, 1995. "Existence of equilibria with incomplete markets: The case of smooth returns," Journal of Mathematical Economics, Elsevier, vol. 24(1), pages 59-72.
    6. Husseini, S. Y. & Lasry, J. -M. & Magill, M. J. P., 1990. "Existence of equilibrium with incomplete markets," Journal of Mathematical Economics, Elsevier, vol. 19(1-2), pages 39-67.
    7. Hart, Oliver D., 1975. "On the optimality of equilibrium when the market structure is incomplete," Journal of Economic Theory, Elsevier, vol. 11(3), pages 418-443, December.
    8. Duffie, Darrell & Shafer, Wayne, 1985. "Equilibrium in incomplete markets: I : A basic model of generic existence," Journal of Mathematical Economics, Elsevier, vol. 14(3), pages 285-300, June.
    9. Chichilnisky, Graciela & Heal, Geoffrey, 1996. "On the existence and the structure of the pseudo-equilibrium manifold," Journal of Mathematical Economics, Elsevier, vol. 26(2), pages 171-186.
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    Cited by:

    1. Philippe Bich, 2006. "On the orientability of the asset equilibrium manifold," Post-Print halshs-00287677, HAL.
    2. Bich, Philippe, 2006. "On the orientability of the asset equilibrium manifold," Journal of Mathematical Economics, Elsevier, vol. 42(4-5), pages 452-470, August.
    3. repec:dau:papers:123456789/6111 is not listed on IDEAS
    4. Vincenzo Scalzo, 2011. "Existence of Equilibrium Prices for Discontinuous Excess Demand Correspondences," Czech Economic Review, Charles University Prague, Faculty of Social Sciences, Institute of Economic Studies, vol. 5(2), pages 203-210, August.

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