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Direct Proofs of the Existence of Equilibrium, the Gale-Nikaido-Debreu Lemma and the Fixed Point Theorems 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, IPAG Business School, TIMAS - Institute of Mathematics and Applied Science, CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

  • Ngoc-Sang Pham

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

  • Cagri Saglam

    (Universite Bilkent [Ankara] - Bilkent University [Ankara])

Abstract

In this paper we use only Sperner's lemma to prove the existence of general equilibrium for a competitive economy with production or with uncertainty and financial assets. We show that the direct use of Sperner's lemma together with Carathéodory's convexity theorem and basic properties of topology such as partition of unit, finite covering of a compact set allow us to bypass the Kakutani fixed point theorem even in establishing the Gale-Nikaido-Debreu Lemma. We also provide a new proof of the Kakutani fixed point theorem based on Sperner's lemma.

Suggested Citation

  • Thanh Le & Cuong Le Van & Ngoc-Sang Pham & Cagri Saglam, 2021. "Direct Proofs of the Existence of Equilibrium, the Gale-Nikaido-Debreu Lemma and the Fixed Point Theorems using Sperner's Lemma," Working Papers halshs-02993655, HAL.
    • Handle: RePEc:hal:wpaper:halshs-02993655
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-02993655v2
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    References listed on IDEAS

    as
    1. Florenzano, Monique, 2009. "Two lemmas that changed general equilibrium theory," Games and Economic Behavior, Elsevier, vol. 66(2), pages 603-605, July.
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    More about this item

    Keywords

    Gale- Nikaido-Debreu Lemma; Fixed Point Theorem; Subdivision; Simplex; Sperner lemma;
    All these keywords.

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • D5 - Microeconomics - - General Equilibrium and Disequilibrium

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