Advertisement

On Iterative Methods for Searching Equilibrium in Pure Exchange Economy with Multiplicative Utilities of Its Agents

  • Leonid D. PopovEmail author
Conference paper
  • 142 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12095)

Abstract

We consider the classical Arrow–Debreu model for a pure exchange economy with multiplicative utilities of its agents. To calculate its equilibrium prices, we present a new iterative algorithm that simulates the simplest intuitive forms of the economic behavior of market agents. It converges under very weak assumptions. The algorithm relies on increasing prices for scarce products only. Moderate inflation, accompanying the computational process, plays a positive role in establishing an equilibrium between commodity supply and demand. Schemes have a meaningful economic interpretation. The convergence theorems are proved, and the results of numerical experiments are presented, including other types of economies.

Keywords

Arrow–Debreu model Cobb–Douglas utility Economic equilibrium Tåtonnement 

References

  1. 1.
    Walras, L.: Elements d’economie politique pure. Corbaz, Lausanne (1874)zbMATHGoogle Scholar
  2. 2.
    Samuelson, P.A.: The stability of equilibrium: comparative statics and dynamics. Econometrica 9(2), 97–120 (1941)CrossRefGoogle Scholar
  3. 3.
    Arrow, K.J., Debreu, G.: Existence of equilibrium for a competitive economy. Econometrica 25, 265–290 (1954)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Arrow, K.J., Hurwicz, L.: On the stability of the competitive equilibrium. Econometrica 26, 522–552 (1958)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Uzawa, H.: Walras’ tatonnenment in the theory of exchange. Rev. Econ. Stud. 27, 182–194 (1960)CrossRefGoogle Scholar
  6. 6.
    Arrow, K.J., Hahn, F.H.: General Competitive Analysis. North-Holland, Amsterdam (1971)zbMATHGoogle Scholar
  7. 7.
    Nicaido, H.: Convex Structures and Economic Theory. Academic Press, New York (1968)Google Scholar
  8. 8.
    Shafer, W.J., Sonnenschein, H.F.: Some theorems on the existence of competitive equilibrium. J. Econ.Theory 11, 83–93 (1975)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Eaves, B.C.: Finite solution of pure trade markets with Cobb – Douglas utilities. Math. Program. Study 23, 226–239 (1985)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Scarf, H.: Some examples of global instability of the competitive equilibrium. Internat Econ. Rev. 1, 157–172 (1960)CrossRefGoogle Scholar
  11. 11.
    Bala, V., Majumdar, M.: Chaotic Tatonnement. Econ. Theory 2(4), 437–445 (1992)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mukherji, A.: A simple example of complex dynamics. Econ. Theory 14, 741–749 (1999)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Tuinstra, J.: A discrete and symmetric price adjustment process on the simplex. J. Econ. Dyn. Control 24(5–7), 881–907 (2000)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Antipin, A.S.: Extra-proximal approach to calculating equilibriums in pure exchange models. Comput. Math. Math. Phys. 46(10), 1687–1698 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Cole, R., Fleischer, I.: Fast-converging tatonnement algorithms for one-time and ongoing market problems. In: Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, pp. 315–324. Association for Computing Machinery, New York (2008)Google Scholar
  16. 16.
    Kitti, M.: Convergence of iterative tatonnement without price normalization. J. Econ. Dyn. Control 34, 1077–1091 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Shikhman, V., Nesterov, Y., Ginsburg, V.: Power method tatonnements for Cobb-Douglas economies. J. Math. Econ. 75, 84–92 (2018)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Krasovskii Institute of Mathematics and Mechanics UB RASYekaterinburgRussia
  2. 2.Ural Federal UniversityYekaterinburgRussia

Personalised recommendations