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Noncooperative oligopoly in markets with a continuum of traders: a limit theorem a la Cournot

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  • Francesca Busetto
  • Giulio Codognato
  • Sayantan Ghosal

Abstract

In this paper, we consider an exchange economy `a la Shitovitz (1973), with atoms and an atomless set. We associate with it a strategic market game of the kind first proposed by Lloyd S. Shapley and known as the Shapley window model. We analyze the relationship between the set of the Cournot-Nash equilibrium allocations of the strategic market game and the Walras equilibrium allocations of the exchange economy with which it is associated. We show, with an example, that even when atoms are countably infinite, any Cournot-Nash equilibrium allocation of the game is not a Walras equilibrium of the underlying exchange economy. Accordingly, in the original spirit of Cournot (1838), we partially replicate the mixed exchange economy by increasing the number of atoms, without affecting the atomless part, and ensuring that the measure space of agents remains finite. We show that any sequence of Cournot-Nash equilibrium allocations of the strategic market games associated with the partially replicated exchange economies approximates a Walras equilibrium allocation of the original exchange economy.

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Paper provided by Business School - Economics, University of Glasgow in its series Working Papers with number 2014_01.

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Handle: RePEc:gla:glaewp:2014_01

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  1. Peck, James & Shell, Karl & Spear, Stephen E., 1992. "The market game: existence and structure of equilibrium," Journal of Mathematical Economics, Elsevier, vol. 21(3), pages 271-299.
  2. D'ASPREMONT, Claude & DOS SANTOS, R. & GÉRARD-VARET, Louis-André, 1992. "General equilibrium concepts under imperfect competition: a Cournotion approach," CORE Discussion Papers 1992057, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  3. Busetto, Francesca & Codognato, Giulio & Ghosal, Sayantan, 2011. "Noncooperative oligopoly in markets with a continuum of traders," Games and Economic Behavior, Elsevier, vol. 72(1), pages 38-45, May.
  4. Codognato, Giulio & Ghosal, Sayantan, 2000. "Cournot-Nash equilibria in limit exchange economies with complete markets and consistent prices," Journal of Mathematical Economics, Elsevier, vol. 34(1), pages 39-53, August.
  5. Ruffin, R J, 1971. "Cournot Oligopoly and Competitive Behaviour," Review of Economic Studies, Wiley Blackwell, vol. 38(116), pages 493-502, October.
  6. Novshek, William & Sonnenschein, Hugo, 1983. "Walrasian equilibria as limits of noncooperative equilibria. Part II: Pure strategies," Journal of Economic Theory, Elsevier, vol. 30(1), pages 171-187, June.
  7. Benyamin Shitovitz, 1997. "A comparison between the core and the monopoly solutions in a mixed exchange economy," Economic Theory, Springer, vol. 10(3), pages 559-563.
  8. Codognato, Giulio & Gabszewicz, Jean J, 1993. "Cournot-Walras Equilibria in Markets with a Continuum of Traders," Economic Theory, Springer, vol. 3(3), pages 453-64, July.
  9. Novshek, William & Sonnenschein, Hugo, 1987. "General Equilibrium with Free Entry: A Synthetic Approach to the Theory of Perfect Competition," Journal of Economic Literature, American Economic Association, vol. 25(3), pages 1281-1306, September.
  10. Busetto, Francesca & Codognato, Giulio & Ghosal, Sayantan, 2008. "Cournot-Walras Equilibrium as a Subgame Perfect Equilibrium," The Warwick Economics Research Paper Series (TWERPS) 837, University of Warwick, Department of Economics.
  11. Novshek, William, 1980. "Cournot Equilibrium with Free Entry," Review of Economic Studies, Wiley Blackwell, vol. 47(3), pages 473-86, April.
  12. Shitovitz, Benyamin, 1973. "Oligopoly in Markets with a Continuum of Traders," Econometrica, Econometric Society, vol. 41(3), pages 467-501, May.
  13. Roberts, Kevin, 1980. "The limit points of monopolistic competition," Journal of Economic Theory, Elsevier, vol. 22(2), pages 256-278, April.
  14. Mas-Colell, Andreu, 1983. "Walrasian equilibria as limits of noncooperative equilibria. Part I: Mixed strategies," Journal of Economic Theory, Elsevier, vol. 30(1), pages 153-170, June.
  15. Sahi, Siddhartha & Yao, Shuntian, 1989. "The non-cooperative equilibria of a trading economy with complete markets and consistent prices," Journal of Mathematical Economics, Elsevier, vol. 18(4), pages 325-346, September.
  16. Masahiro Okuno, 1977. "Oligopoly and Competition in Large Markets," Discussion Papers 305, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  17. Amir, Rabah & Sahi, Siddharta & Shubik, Martin & Yao, Shuntian, 1990. "A strategic market game with complete markets," Journal of Economic Theory, Elsevier, vol. 51(1), pages 126-143, June.
  18. Artstein, Zvi, 1979. "A note on fatou's lemma in several dimensions," Journal of Mathematical Economics, Elsevier, vol. 6(3), pages 277-282, December.
  19. Shapley, Lloyd S & Shubik, Martin, 1977. "Trade Using One Commodity as a Means of Payment," Journal of Political Economy, University of Chicago Press, vol. 85(5), pages 937-68, October.
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