IDEAS home Printed from https://ideas.repec.org/p/hal/pseptp/hal-03960256.html
   My bibliography  Save this paper

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
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a search for a similarly titled item that would be available.

    Other versions of this item:

    References listed on IDEAS

    as
    1. 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.
    2. Florenzano, Monique, 2009. "Two lemmas that changed general equilibrium theory," Games and Economic Behavior, Elsevier, vol. 66(2), pages 603-605, July.
    3. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Gourdel, Pascal & Le Van, Cuong & Pham, Ngoc-Sang & Tran Viet, Cuong, 2023. "Hartman-Stampacchia theorem, Gale-Nikaido-Debreu lemma, and Brouwer and Kakutani fixed-point theorems," MPRA Paper 116541, University Library of Munich, Germany.
    3. Le, Thanh & Le Van, Cuong & Pham, Ngoc-Sang & Sağlam, Çağrı, 2020. "Sperner Lemma, Fixed Point Theorems, and the Existence of Equilibrium," MPRA Paper 100084, University Library of Munich, Germany.
    4. Lionel DE BOISDEFFRE, 2015. "Existence of financial equilibrium with differential information: the no-arbitrage characterization," Working Papers 2015-2016_5, CATT - UPPA - Université de Pau et des Pays de l'Adour, revised Nov 2015.
    5. Kajii, Atsushi, 1996. "How to discard non-satiation and free-disposal with paper money," Journal of Mathematical Economics, Elsevier, vol. 25(1), pages 75-84.
    6. Philippe Bich & Rida Laraki, 2017. "Externalities in Economies with Endogenous Sharing Rules," PSE-Ecole d'économie de Paris (Postprint) halshs-01437507, HAL.
    7. Nuno Gouveia & Abdelkrim Seghir, 2009. "Endogenous Default Penalties in Nominal Incomplete Markets," Annals of Economics and Finance, Society for AEF, vol. 10(2), pages 391-418, November.
    8. Móczár, József, 2017. "Ergodic Versus Uncertain Financial Processes – Part II: Neoclassical and Institutional Economics," Public Finance Quarterly, Corvinus University of Budapest, vol. 62(4), pages 478-501.
    9. Lionel De Boisdeffre, 2015. "Existence of financial equilibrium with differential information: the no-arbitrage characterization," Working Papers hal-02945462, HAL.
    10. Robert M. Anderson & Haosui Duanmu & M. Ali Khan & Metin Uyanik, 2022. "Walrasian equilibrium theory with and without free-disposal: theorems and counterexamples in an infinite-agent context," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 73(2), pages 387-412, April.
    11. Joshua S Gans & Fiona Murray, 2023. "Markets for Scientific Attribution," The Journal of Law, Economics, and Organization, Oxford University Press, vol. 39(3), pages 828-846.
    12. Charalambos Aliprantis & Kim Border & Owen Burkinshaw, 1996. "Market economies with many commodities," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 19(1), pages 113-185, March.
    13. Florenzano, Monique & Gourdel, Pascal & Pascoa, Mario Rui, 2001. "Overlapping generations models with incomplete markets," Journal of Mathematical Economics, Elsevier, vol. 36(3), pages 201-218, December.
    14. Robert Becker & Stefano Bosi & Cuong Van & Thomas Seegmuller, 2015. "On existence and bubbles of Ramsey equilibrium with borrowing constraints," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 58(2), pages 329-353, February.
    15. Aouani, Zaier & Cornet, Bernard, 2009. "Existence of financial equilibria with restricted participation," Journal of Mathematical Economics, Elsevier, vol. 45(12), pages 772-786, December.
    16. Lionel de Boisdeffre, 2021. "Dropping the Cass Trick and Extending Cass' Theorem to Asymmetric Information and Restricted Participation," Post-Print halshs-03196923, HAL.
    17. Bonnisseau, Jean-Marc, 2003. "Regular economies with non-ordered preferences," Journal of Mathematical Economics, Elsevier, vol. 39(3-4), pages 153-174, June.
    18. repec:ipg:wpaper:16 is not listed on IDEAS
    19. Jean-Marc Bonnisseau & Lalaina Rakotonindrainy, 2015. "A Note on the Characterization of Optimal Allocations in OLG Models with Multiple Goods," PSE-Ecole d'économie de Paris (Postprint) halshs-01199162, HAL.
    20. Vladiir Yefimov, 2015. "Two Disputes Of Methods, Three Constructivisms, And Three Liberalisms. Part I," Economy of region, Centre for Economic Security, Institute of Economics of Ural Branch of Russian Academy of Sciences, vol. 1(1), pages 29-38.
    21. Philippe Bich & Rida Laraki, 2017. "Externalities in Economies with Endogenous Sharing Rules," Post-Print halshs-01437507, HAL.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:hal:pseptp:hal-03960256. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Caroline Bauer (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.