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Market Equilibrium for CES Exchange Economies: Existence, Multiplicity, and Computation

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FSTTCS 2005: Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3821))

Abstract

We consider exchange economies where the traders’ preferences are expressed in terms of the extensively used constant elasticity of substitution (CES) utility functions. We show that for any such economy it is possible to say in polynomial time whether an equilibrium exists.

We then describe a convex formulation of the equilibrium conditions, which leads to polynomial time algorithms for a wide range of the parameter defining the CES utility functions. This range includes instances that do not satisfy weak gross substitutability. As a byproduct of our work, we prove the uniqueness of equilibrium in an interesting setting where such a result was not known.

The range for which we do not obtain polynomial-time algorithms coincides with the range for which the economies admit multiple disconnected equilibria.

Part of the first author’s work was done when visiting the Toyota Technological Institute at Chicago. The work by the last three authors was in part supported by NSF CAREER award CCR-0237431. This paper is an expanded and revised version of an unpublished paper that was presented at a DIMACS Workshop on Large Scale Games in April, 2005.

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References

  1. Arrow, K.J., Debreu, G.: Existence of an Equilibrium for a Competitive Economy. Econometrica 22(3), 265–290 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arrow, K.J., Chenery, H.B., Minhas, B.S., Solow, R.M.: Capital-Labor Substitution and Economic Efficiency. The Review of Economics and Statistics 43(3), 225–250 (1961)

    Article  Google Scholar 

  3. Chen, N., Deng, X., Sun, X., Yao, A.: Fisher Equilibrium Price with a Class of Concave Utility Functions. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 169–179. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  4. Codenotti, B., McCune, B., Varadarajan, K.: Market Equilibrium via the Excess Demand Function. In: Andersson, S.I. (ed.) Summer University of Southern Stockholm 1993. LNCS, vol. 888, Springer, Heidelberg (1995)

    Google Scholar 

  5. Codenotti, B., Pemmaraju, S., Varadarajan, K.: Algorithms Column: The Computation of Market Equilibria. SIGACT News 35(4) (December 2004)

    Google Scholar 

  6. Codenotti, B., Pemmaraju, S., Varadarajan, K.: On the Polynomial Time Computation of Equilibria for certain Exchange Economies. SODA (2005)

    Google Scholar 

  7. Codenotti, B., Saberi, A., Varadarajan, K., Ye, Y.: Leontief Economies EncodeNonzero Sum Two-Player G ames. Electronic Colloquium on Computational Complexity, Report TR05-055. To appear in SODA 06

    Google Scholar 

  8. Codenotti, B., Varadarajan, K.: Efficient Computation of Equilibrium Prices mfor Markets with Leontief Utilities. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 371–382. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  9. Debreu, G.: Economies with a Finite Set of Equilibria. Econometrica 38(3), 387–392 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  10. Deng, X., Papadimitriou, C.H., Safra, M.: On the Complexity of Equilibria. In: Andersson, S.I. (ed.) Summer University of Southern Stockholm 1993. LNCS, vol. 888, Springer, Heidelberg (1995)

    Google Scholar 

  11. Devanur, N.R., Papadimitriou, C.H., Saberi, A., Vazirani, V.V.: Market Equilibrium via a Primal-Dual-Type Algorithm. FOCS 2002, 389–395 (2002)

    Google Scholar 

  12. Eaves, B.C.: Finite Solution of Pure Trade Markets with Cobb-Douglas Utilities. Mathematical Programming Study 23, 226–239 (1985)

    MATH  MathSciNet  Google Scholar 

  13. Gale, D.: The Linear Exchange Model. Journal of Mathematical Economics 3, 205–209 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  14. Garg, R., Kapoor, S.: Auction Algorithms for Market Equilibrium. In: Proc. STOC (2004)

    Google Scholar 

  15. Garg, R., Kapoor, S., Vazirani, V.V.: An Auction-Based Market Equilbrium Algorithm for the Separable Gross Substitutibility Case. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 128–138. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  16. Gjerstad, S.: Multiple Equilibria in Exchange Economies with Homothetic, Nearly Identical Preference, University of Minnesota, Center for Economic Research, Discussion Paper 288 (1996)

    Google Scholar 

  17. Jain, K.: A polynomial time algorithm for computing the Arrow-Debreu market equilibrium for linear utilities. In: FOCS 2004 (2004)

    Google Scholar 

  18. Jain, K., Mahdian, M., Saberi, A.: Approximating Market Equilibria. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 98–108. Springer, Heidelberg (2003)

    Google Scholar 

  19. Jain, K., Varadarajan, K.: Equilibria for Economies with Production: Constant-Returns Technologies and Production Planning Constraints. To appear in SODA 2006 (2006)

    Google Scholar 

  20. Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill, New York (1969)

    MATH  Google Scholar 

  21. Mas-Colell, A.: On the Uniqueness of Equilibrium Once Again. In: Barnett, W., Cornet, B., DAspremont, C., Gabszewicz, J., Mas-Colell, A. (eds.) Equilibrium Theory and Applications, Cambridge University Press, Cambridge (1991)

    Google Scholar 

  22. Mas-Colell, A., Whinston, M.D., Green, J.R.: Microeconomic Theory. Oxford University Press, Oxford (1995)

    Google Scholar 

  23. Maxfield, R.R.: General Equilibrium and the Theory of Directed Graphs. Journal of Mathematical Economics 27, 23–51 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  24. Nenakov, E.I., Primak, M.E.: One algorithm for finding solutions of the Arrow-Debreu model, Kibernetica, vol. 3, pp. 127–128 (1983) (in Russian)

    Google Scholar 

  25. Papadimitriou, C.H.: On the Complexity of the Parity Argument and otherInefficient Proofs of Existence. Journal of Computer and System Sciences 48, 498–532 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  26. Shoven, J.B., Whalley, J.: Applying General Equilibrium. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  27. Solov, R.: A Contribution to the Theory of Economic Growth. Quarterly Journal of Economics 70, 65–94 (1956)

    Article  Google Scholar 

  28. Varian, H.: Microeconomic Analysis. W.W. Norton, New York (1992)

    Google Scholar 

  29. Ye, Y.: A Path to the Arrow-Debreu Competitive Market Equilibrium. To appear in Mathematical Programming

    Google Scholar 

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Author information

Authors and Affiliations

  1. IIT-CNR, Pisa, Italy

    Bruno Codenotti

  2. Department of Computer Science, The University of Iowa, Iowa City, IA, 52242, USA

    Benton McCune & Kasturi Varadarajan

  3. Department of Mathematics, Arizona State University, Tempe, AZ, 85287, USA

    Sriram Penumatcha

Authors
  1. Bruno Codenotti
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  2. Benton McCune
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  3. Sriram Penumatcha
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  4. Kasturi Varadarajan
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Editor information

Editors and Affiliations

  1. National Institute of Advanced Studies, Indian Institute of Science Campus, 560012, Bangalore, India

    Sundar Sarukkai

  2. Department of Comp. Sc. & Engg., Indian Institute of Technology Delhi, 110016, New Delhi, India

    Sandeep Sen

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Codenotti, B., McCune, B., Penumatcha, S., Varadarajan, K. (2005). Market Equilibrium for CES Exchange Economies: Existence, Multiplicity, and Computation. In: Sarukkai, S., Sen, S. (eds) FSTTCS 2005: Foundations of Software Technology and Theoretical Computer Science. FSTTCS 2005. Lecture Notes in Computer Science, vol 3821. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11590156_41

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  • DOI: https://doi.org/10.1007/11590156_41

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-30495-1

  • Online ISBN: 978-3-540-32419-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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