Exchange Market Equilibria with Leontief’s Utility: Freedom of Pricing Leads to Rationality

  • Yinyu Ye
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3828)


This paper studies the equilibrium property and algorithmic complexity of the exchange market equilibrium problem with more general utility functions: piece-wise linear functions, which include Leontief’s utility functions. We show that the Fisher model again reduces to the general analytic center problem, and the same linear programming complexity bound applies to approximating its equilibrium. However, the story for the Arrow-Debreu model with Leontief’s utility becomes quite different. We show that, for the first time, that solving this class of Leontief exchange economies is equivalent to solving a known linear complementarity problem whose algorithmic complexity status remains open.


Utility Function Equilibrium Problem Linear Complementarity Problem Price Vector Initial Endowment 
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  1. 1.
    Arrow, K.J., Debreu, G.: Existence of an equilibrium for competitive economy. Econometrica 22, 265–290 (1954)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Brainard, W.C., Scarf, H.E.: How to compute equilibrium prices in 1891. Cowles Foundation Discussion Paper 1270 (August 2000)Google Scholar
  3. 3.
    Codenotti, B., Varadarajan, K.: Efficient computation of equilibrium prices for market with Leontief utilities. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 371–382. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    Codenotti, B., Pemmaraju, S., Varadarajan, K.: On the polynomial time computation of equilibria for certain exchange economies. In: The Proceedings of SODA 2005 (2005)Google Scholar
  5. 5.
    Cottle, R., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, Boston (1992)zbMATHGoogle Scholar
  6. 6.
    Deng, X., Huang, S., Ye, Y.: Computation of the Arrow-Debreu equilibrium for a class of non-homogenous utility functions. Technical Report, State Key Laboratory of Intelligent Technology and Systems, Dept. of Computer Science and Technology, Tsinghua Univ., Beijing, 10084, ChinaGoogle Scholar
  7. 7.
    Deng, X., Papadimitriou, C.H., Safra, S.: On the complexity of price equilibria. Journal of Computer and System Sciences 67, 311–324 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Devanur, N.R., Papadimitriou, C.H., Saberi, A., Vazirani, V.V.: Market equilibria via a primal-dual-type algorithm. In: The Proceedings of FOCS 2002 (2002)Google Scholar
  9. 9.
    Eaves, B.C.: A finite algorithm for the linear exchange model. J. of Mathematical Economics 3, 197–203 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Eaves, B.C.: Finite solution for pure trade markets with Cobb-Douglas utilities. Mathematical Programming Study 23, 226–239 (1985)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Eisenberg, E.: Aggregation of utility functions. Management Sciences 7, 337–350 (1961)CrossRefGoogle Scholar
  12. 12.
    Eisenberg, E., Gale, D.: Consensus of subjective probabilities: The pari-mutuel method. Annals of Mathematical Statistics 30, 165–168 (1959)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Gale, D.: The Theory of Linear Economic Models. McGraw-Hill, N.Y (1960)Google Scholar
  14. 14.
    Güler, O.: Existence of interior points and interior paths in nonlinear monotone complementarity problems. Mathematics of Operations Research 18, 128–147 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Jain, K.: A polynomial time algorithm for computing the Arrow-Debreu market equilibrium for linear utilities. In: The Proceedings of FOCS 2004 (2004)Google Scholar
  16. 16.
    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
  17. 17.
    Jain, K., Vazirani, V.V., Ye, Y.: Market equilibria for homothetic, quasi-concave utilities and economies of scale in production. In: The Proceedings of SODA 2005 (2005)Google Scholar
  18. 18.
    Kojima, M., Megiddo, N., Noma, T., Yoshise, A.: A Unified Approach for Interior-Point Algorithms for Linear Complementarity Problems. In: Kojima, M., Noma, T., Megiddo, N., Yoshise, A. (eds.) A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems. LNCS, vol. 538. Springer, Heidelberg (1991)Google Scholar
  19. 19.
    Megiddo, N.: Pathways to the optimal set in linear programming. In: Megiddo, N. (ed.) Progress in Mathematical Programming : Interior Point and Related Methods, pp. 131–158. Springer, New York (1989)Google Scholar
  20. 20.
    Nenakhov, E., Primak, M.: About one algorithm for finding the solution of the Arrow-Debreu model. Kibernetica 3, 127–128 (1983)Google Scholar
  21. 21.
    Primak, M.: A converging algorithm for a linear exchange model. J. of Mathematical Economics 22, 181–187 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Scarf, H.E.: The Computation of Economic Equilibria. In: With collaboration of T. Hansen, Cowles Foundation Monograph No. 24. Yale University Press, New Haven (1973)Google Scholar
  23. 23.
    Todd, M.J.: The Computation of Fixed Points and Applications. LNEMS, vol. 124. Springer, Berlin (1976)zbMATHGoogle Scholar
  24. 24.
    Walras, L.: Elements of Pure Economics, or the Theory of Social Wealth (1874); (1899, 4th ed.; 1926, rev ed., 1954, Engl. Transl.)Google Scholar
  25. 25.
    Yang, Z.: Computing Equalibria and Fixed Points: The Solution of Nonlinear Inequalities. Kluwer Academic Publishers, Boston (1999)Google Scholar
  26. 26.
    Ye, Y.: A path to the Arrow-Debreu competitive market equilibrium. Working Paper, Department of Management Science and Engineering, Stanford University, Stanford, CA 94305 (2004); To appear in Mathematical ProgrammingGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Yinyu Ye
    • 1
  1. 1.Stanford UniversityStanfordUSA

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