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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

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  • Le, Thanh
  • Le Van, Cuong
  • Pham, Ngoc-Sang
  • Sağlam, Çağrı

Abstract

In this paper, we use Sperner’s lemma to prove the existence of general equilibrium for a competitive economy with production or with uncertainty and financial assets. We then 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

  • Le, Thanh & Le Van, Cuong & Pham, Ngoc-Sang & Sağlam, Çağrı, 2020. "Direct Proofs of the Existence of Equilibrium, the Gale-Nikaido-Debreu Lemma and the Fixed Point Theorems using Sperner’s Lemma," MPRA Paper 110933, University Library of Munich, Germany, revised 28 Oct 2020.
    • Handle: RePEc:pra:mprapa:110933
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    References listed on IDEAS

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    More about this item

    Keywords

    Sperner lemma; Simplex; Subdivision; Fixed Point Theorem; Gale-Nikaido-Debreu Lemma; General Equilibrium.;
    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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