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Power Method Tâtonnements for Cobb-Douglas Economies

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  • V. Shikhman
  • Yu Nesterov
  • Victor Ginsburgh

Abstract

We consider an economy with consumers maximizing Cobb-Douglas utilities from the algorithmic perspective. It is known that in this case nding 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 t^atonnement 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 suciently small perturbations of consumers' tastes and endowments. Moreover, the property of pairwise connectivity holds for almost all Cobb-Douglas economies.

Suggested Citation

  • V. Shikhman & Yu Nesterov & Victor Ginsburgh, 2017. "Power Method Tâtonnements for Cobb-Douglas Economies," Working Papers ECARES ECARES 2017-09, ULB -- Universite Libre de Bruxelles.
  • Handle: RePEc:eca:wpaper:2013/248466
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    References listed on IDEAS

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    1. 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.
    2. Debreu, Gerard, 1970. "Economies with a Finite Set of Equilibria," Econometrica, Econometric Society, vol. 38(3), pages 387-392, May.
    3. Bonnisseau, Jean-Marc, 2003. "Regular economies with non-ordered preferences," Journal of Mathematical Economics, Elsevier, vol. 39(3-4), pages 153-174, June.
    4. repec:eee:apmaco:v:255:y:2015:i:c:p:58-65 is not listed on IDEAS
    5. 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.
    6. 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.
    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.
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    Keywords

    exchange economy; Cobb-Douglas utility; tâtonnement; power method; regular economy; stochastic matrix;

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