IDEAS home Printed from https://ideas.repec.org/a/eee/mateco/v75y2018icp84-92.html
   My bibliography  Save this article

Power method tâtonnements for Cobb–Douglas economies

Author

Listed:
  • Shikhman, V.
  • Nesterov, Yu.
  • Ginsburgh, V.

Abstract

We consider an economy with consumers maximizing Cobb–Douglas utilities from the algorithmic perspective. It is known that in this case finding equilibrium prices reduces to the eigenvalue problem for a particularly structured stochastic matrix. We show that the power method for solving this eigenvalue problem can be naturally interpreted as a tâtonnement executed by an auctioneer. Its linear rate of convergence is established under the reasonable assumption of pairwise connectivity w.r.t. commodities within submarkets. We show that the pairwise connectivity remains valid under sufficiently small perturbations of consumers’ tastes and endowments. Moreover, the property of pairwise connectivity holds for almost all Cobb–Douglas economies.

Suggested Citation

  • Shikhman, V. & Nesterov, Yu. & Ginsburgh, V., 2018. "Power method tâtonnements for Cobb–Douglas economies," Journal of Mathematical Economics, Elsevier, vol. 75(C), pages 84-92.
    • Handle: RePEc:eee:mateco:v:75:y:2018:i:c:p:84-92
    DOI: 10.1016/j.jmateco.2017.12.010
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S030440681830003X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmateco.2017.12.010?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to look for a different version below or search for a different version of it.

    Other versions of this item:

    References listed on IDEAS

    as
    1. Du, Ye & Lehrer, Ehud & Pauzner, Ady, 2015. "Competitive economy as a ranking device over networks," Games and Economic Behavior, Elsevier, vol. 91(C), pages 1-13.
    2. Bonnisseau, Jean-Marc, 2003. "Regular economies with non-ordered preferences," Journal of Mathematical Economics, Elsevier, vol. 39(3-4), pages 153-174, June.
    3. J. M. Bonnisseau & M. Florig & A. Jofré, 2001. "Continuity and Uniqueness of Equilibria for Linear Exchange Economies," Journal of Optimization Theory and Applications, Springer, vol. 109(2), pages 237-263, May.
    4. Debreu, Gerard, 1970. "Economies with a Finite Set of Equilibria," Econometrica, Econometric Society, vol. 38(3), pages 387-392, May.
    5. Herbert E. Scarf, 1967. "The Approximation of Fixed Points of a Continuous Mapping," Cowles Foundation Discussion Papers 216R, Cowles Foundation for Research in Economics, Yale University.
    6. Nesterov, Yurii & Nemirovski, Arkadi, 2015. "Finding the stationary states of Markov chains by iterative methods," Applied Mathematics and Computation, Elsevier, vol. 255(C), pages 58-65.
    7. Eaves, B. Curtis, 1976. "A finite algorithm for the linear exchange model," Journal of Mathematical Economics, Elsevier, vol. 3(2), pages 197-203, July.
    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. Kovalenko, Olga & Bokiy, Olena & Rybak, Yaroslav & Lysenko, Hanna & Voznesenska, Nataliia, 2023. "Assessment of export potential and state of foreign food and agriculture trade in the world," Agricultural and Resource Economics: International Scientific E-Journal, Agricultural and Resource Economics: International Scientific E-Journal, vol. 9(3), September.
    2. Wang, Jingfan & Tchapmi, Lyne P. & Ravikumar, Arvind P. & McGuire, Mike & Bell, Clay S. & Zimmerle, Daniel & Savarese, Silvio & Brandt, Adam R., 2020. "Machine vision for natural gas methane emissions detection using an infrared camera," Applied Energy, Elsevier, vol. 257(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. M. Florig, 2004. "Equilibrium Correspondence of Linear Exchange Economies," Journal of Optimization Theory and Applications, Springer, vol. 120(1), pages 97-109, January.
    2. Yves Balasko & Mich Tvede, 2010. "General equilibrium without utility functions: how far to go?," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 45(1), pages 201-225, October.
    3. Nguyen, Van-Quy, 2021. "Endowment-regarding preferences," Journal of Mathematical Economics, Elsevier, vol. 94(C).
    4. Pascal Gauthier & Timothy J. Kehoe & Erwan Quintin, 2022. "Constructing pure-exchange economies with many equilibria," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 73(2), pages 541-564, April.
    5. Michael S. Harr'e & Adam Harris & Scott McCallum, 2019. "Singularities and Catastrophes in Economics: Historical Perspectives and Future Directions," Papers 1907.05582, arXiv.org.
    6. J. M. Bonnisseau & M. Florig & A. Jofré, 2001. "Differentiability of Equilibria for Linear Exchange Economies," Journal of Optimization Theory and Applications, Springer, vol. 109(2), pages 265-288, May.
    7. Van Quy Nguyen, 2020. "Endowments-regarding preferences," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-02966848, HAL.
    8. Jugal Garg & Ruta Mehta & Vijay V. Vaziranic, 2018. "Substitution with Satiation: A New Class of Utility Functions and a Complementary Pivot Algorithm," Mathematics of Operations Research, INFORMS, vol. 43(3), pages 996-1024, August.
    9. Van Quy Nguyen, 2020. "Endowments-regarding preferences," Post-Print halshs-02966848, HAL.
    10. Van Quy Nguyen, 2020. "Endowments-regarding preferences," Documents de travail du Centre d'Economie de la Sorbonne 20017, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    11. Yang Zhan & Peixuan Li & Chuangyin Dang, 2020. "A differentiable path-following algorithm for computing perfect stationary points," Computational Optimization and Applications, Springer, vol. 76(2), pages 571-588, June.
    12. Michael Zierhut, 2021. "Generic regularity of differentiated product oligopolies," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(1), pages 341-374, February.
    13. Dolf Talman & Zaifu Yang, 2012. "On a Parameterized System of Nonlinear Equations with Economic Applications," Journal of Optimization Theory and Applications, Springer, vol. 154(2), pages 644-671, August.
    14. R. M. Harstad & R. Selten, 2014. "Bounded-rationality models:tasks to become intellectually competitive," Voprosy Ekonomiki, NP Voprosy Ekonomiki, issue 5.
    15. Herings, P.J.J. & Talman, A.J.J. & Yang, Z.F., 1999. "Variational Inequality Problems With a Continuum of Solutions : Existence and Computation," Other publications TiSEM 73e2f01b-ad4d-4447-95ba-a, Tilburg University, School of Economics and Management.
    16. Bottazzi, Jean-Marc & De Meyer, Bernard, 2003. "A market game for assets and taxed investors," Journal of Mathematical Economics, Elsevier, vol. 39(5-6), pages 657-675, July.
    17. John Geanakoplos, 2008. "Overlapping Generations Models of General Equilibrium," Cowles Foundation Discussion Papers 1663, Cowles Foundation for Research in Economics, Yale University.
    18. J. M. Bonnisseau & M. Florig & A. Jofré, 2001. "Continuity and Uniqueness of Equilibria for Linear Exchange Economies," Journal of Optimization Theory and Applications, Springer, vol. 109(2), pages 237-263, May.
    19. Bernard Dumas & Andrew Lyasoff, 2012. "Incomplete-Market Equilibria Solved Recursively on an Event Tree," Journal of Finance, American Finance Association, vol. 67(5), pages 1897-1941, October.
    20. Jean-Marc Bonnisseau & Elena Mercato, 2010. "Externalities, consumption constraints and regular economies," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 44(1), pages 123-147, July.

    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:eee:mateco:v:75:y:2018:i:c:p:84-92. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/jmateco .

    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.