Summary
This paper characterizes the set of Nash equilibria in a price setting duopoly in which firms have limited capacity, and in which unit costs of production up to capacity may differ. Assuming concave revenue and efficient rationing, we show that the case of different unit costs involves a tractable generalization of the methods used to analyze the case of identical costs. However, the supports of the two firms' equilibrium price distributions need no longer be connected and need not coincide. In addition, the supports of the equilibrium price distributions need no longer be continuous in the underlying parameters of the model.
As an application of our characterization, we examine the Kreps-Scheinkman model of capacity choice followed by Bertrand-Edgeworth price competition and show that, unlike in the case of identical costs, Cournot equilibrium capacity levels need not arise as subgame-perfect equilibria. The low-cost firm has greater incentive to price its rival out of the market than exists under Cournot behavior.
Similar content being viewed by others
References
Allen, B., Deneckere, R., Faith, R., Kovenock, D.: Capacity precommitment as a barrier to entry. Social Systems Research Institute, Discussion Paper No. 9440, University of Wisconsin-Madison, 1994
Allen, B., Hellwig, M.: Bertrand-Edgeworth oligopoly in large markets. Review of Economic Studies53, 175–204 (1986)
Dasgupta, P., Maskin, E.: The existence of equilibrium in discontinuous economic games, I: Theory. Review of Economic Studies53, 1–26 (1986)
Davidson, C., Deneckere, R.: Long-run competition in capacity, short-run competition in price, and the Cournot model. The Rand Journal of Economics17, 404–415 (1986)
Deneckere, R., Kovenock, D.: Capacity-constrained price competition when unit costs differ. CMSEMS Discussion Paper No. 861, Northwestern University, 1989
Deneckere, R., Kovenock, D.: Bertrand-Edgeworth oligopoly. Mimeo, Purdue University 1990
Deneckere, R., Kovenock, D.: Price leadership. Review of Economic Studies59, 143–162 (1992)
Faith, T.: Essays in Bertrand-Edgeworth competition. Ph.D. Thesis, Purdue University, 1992
Gelman, J., Salop, S.: Judo economics: Capacity limitation and coupon competition. Bell Journal of Economics14, 315–325 (1983)
Ghemawat, P.: Short run competition: IO models for cross-sectional analysis. Mimeo, Harvard University, 1988
Glicksberg, I.: A further generalization of the Kakutani fixed point theorem with application to Nash equilibrium points. Proceedings of the AMS3, 170–174 (1952)
Kreps, D., Scheinkman, J.: Quantity precommitment and Bertrand competition yield Cournot outcomes. Bell Journal of Economics14, 326–337 (1983)
Levitan, R., Shubik, M.: Price duopoly and capacity constraints. International Economic Review13, 111–123 (1972)
Osborne, M., Pitchik, C.: Price competition in a capacity constrained duopoly. Columbia University, Department of Economics, Working Paper No. 185, 1983
Osborne, M., Pitchik, C.: Price competition in a capacity-constrained duopoly. Journal of Economic Theory38, 238–260 (1986)
Simon, L.: Games with discontinuous payoffs. Review of Economic Studies54, 569–597 (1987)
Author information
Authors and Affiliations
Additional information
We are grateful to Joseph Harrington, Marie Thursby, Casper de Vries and, especially, William Novshek for helpful discussions and comments. Thomas Faith and Ioannis Tournas provided valuable research assistance. This paper was presented at the Winter Meetings of the Econometric Society in December 1988, the Midwest Mathematical Economics Conference in April 1989, the Sixteenth Annual Congress of the European Association for Research in Industrial Economics in August 1989, the European Meetings of the Econometric Society in September 1989, and in seminars at the Ecole Nationale des Ponts et Chaussées, Erasmus University Rotterdam, Indiana University, INSEAD, Texas A&M University, Tilburg University, the University of Bonn and the University of Florida. Deneckere acknowledges financial support through National Science Foundation Grant SES-8619012 and the Kellogg Graduate School of Management's Beatrice/Esmark Research Chair. Kovenock acknowledges financial support through Erasmus University Rotterdam, the Purdue Research Foundation, the Ford Motor Company Fund, and an Ameritech Foundation Summer Faculty Research Grant.
Rights and permissions
About this article
Cite this article
Deneckere, R.J., Kovenock, D. Bertrand-Edgeworth duopoly with unit cost asymmetry. Econ Theory 8, 1–25 (1996). https://doi.org/10.1007/BF01212009
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01212009