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A Direct Proof of the Gale–Nikaido–Debreu Lemma Using Sperner’s Lemma

Author

Listed:
  • Thanh Le

    (University of Wollongong [Australia])

  • Cuong Le Van

    (PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - ENPC - École des Ponts ParisTech - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement, CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, IPAG Business School, TIMAS - Institute of Mathematics and Applied Science)

  • Ngoc-Sang Pham

    (Métis Lab EM Normandie - EM Normandie - École de Management de Normandie)

  • Cagri Saglam

    (Bilkent University [Ankara])

Abstract

The Gale–Nikaido–Debreu lemma plays an important role in establishing the existence of competitive equilibrium. In this paper, we use Sperner's lemma and basic elements of topology to prove the Gale–Nikaido–Debreu lemma.

Suggested Citation

  • Thanh Le & Cuong Le Van & Ngoc-Sang Pham & Cagri Saglam, 2022. "A Direct Proof of the Gale–Nikaido–Debreu Lemma Using Sperner’s Lemma," PSE-Ecole d'économie de Paris (Postprint) hal-03960256, HAL.
    • Handle: RePEc:hal:pseptp:hal-03960256
    DOI: 10.1007/s10957-022-02067-2
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    References listed on IDEAS

    as
    1. Gale, D. & Mas-Colell, A., 1975. "An equilibrium existence theorem for a general model without ordered preferences," Journal of Mathematical Economics, Elsevier, vol. 2(1), pages 9-15, March.
    2. Florenzano, Monique, 2009. "Two lemmas that changed general equilibrium theory," Games and Economic Behavior, Elsevier, vol. 66(2), pages 603-605, July.
    3. Till Düppe & E. Roy Weintraub, 2014. "Finding Equilibrium: Arrow, Debreu, McKenzie and the Problem of Scientific Credit," Economics Books, Princeton University Press, edition 1, number 10206.
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    Cited by:

    1. Pascal Gourdel & Cuong Le Van & Ngoc-Sang Pham & Cuong Tran Viet, 2023. "Hartman-Stampacchia theorem, Gale-Nikaido-Debreu lemma, and Brouwer and Kakutani fixed-point theorems," Working Papers hal-04008394, HAL.
    2. Le, Thanh & Le Van, Cuong & Pham, Ngoc-Sang & Sağlam, Çağrı, 2024. "Sperner’s lemma and competitive equilibrium with incomplete financial markets," Economics Letters, Elsevier, vol. 238(C).

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