Bertrand-Edgeworth games under oligopoly with a complete characterization for the triopoly

De Francesco, Massimo A. and Salvadori, Neri (2009): Bertrand-Edgeworth games under oligopoly with a complete characterization for the triopoly.

Abstract

The paper extends the analysis of price competition among capacity-constrained sellers beyond the cases of duopoly and symmetric oligopoly. We rst provide some general results for the oligopoly, highlighting features of a duopolistic mixed strategy equilibrium that generalize to oligopoly. Unlike in the duopoly, however, there can be in nitely many equilibria when the capacity of a subset of rms is so large that no strategic interaction among smaller rms exists. Then we focus on the triopoly, providing a complete characterization of the mixed strategy equilibrium of the Bertrand-Edgeworth game. The mixed-strategy region of the capacity space is partitioned according to key equilibrium features. We also prove the possibility of a disconnected support of an equilibrium strategy and show how gaps are then determined. Computing the mixed strategy equilibrium then becomes quite a simple task.

• Item Type: MPRA Paper
• Language: English
• Keywords: Bertrand-Edgeworth; Price game; Oligopoly; Triopoly; Mixed strategy equilibrium
• Subjects: L - Industrial Organization > L1 - Market Structure, Firm Strategy, and Market Performance > L13 - Oligopoly and Other Imperfect Markets; D - Microeconomics > D4 - Market Structure, Pricing, and Design > D43 - Oligopoly and Other Forms of Market Imperfection; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C72 - Noncooperative Games

References

1. Baik, K. H., Horizontal mergers of price-setting rms with sunk capacity costs, The Quarterly Journal of Economics and Finance, Vol. 35, No. 3, Fall, 245-256 (1995).

2. Boccard, N., Wauthy, X., Bertrand competition and Cournot outcomes: further results, Economics Letters, 68, 279-285 (2000).

3. Dasgupta, P., E. Maskin, The existence of equilibria in discontinuous economic games I: Theory, Review of Economic Studies, 53, 1-26 (1986).

4. Davidson, C., R. J. Deneckere, Horizontal mergers and collusive behavior, International Journal of Industrial Organization, vol. 2, Iss. 2, 117-32 (1984).

5. Davidson, C., R. J. Deneckere, Long-run competition in capacity, short-run competition in price, and the Cournot model, Rand Journal of Economics, Vol. 17, No. 3, Autumn, 404-415 (1986).

6. De Francesco, M. A., On a property of mixed strategy equilibria of the pricing game, Economics Bulletin, 4, 1-7 (2003).

7. De Francesco, M. A., Salvadori N., Bertrand-Edgeworth games under oligopoly with a complete characterization for the triopoly, MPRA working paper No. 8634 (2008).

8. Deneckere, R. J., D. Kovenock, Price leadership, Review of Economic Studies, 59, 143-162 (1992).

9. Deneckere, R. J., D. Kovenock, Bertrand-Edgeworth duopoly with unit cost asymmetry, Economic Theory, vol. 8, Iss. 1, 1-25 (1996).

10. Hirata, D., Bertrand-Edgeworth equilibrium in oligopoly, MPRA working paper No. 7997 (2008).

11. Hirata, D., Asymmetric Bertrand-Edgeworth oligopoly and mergers, The B.E. Journal of Theoretical Economics. Topics, volume 9, Issue 1, 1-23 (2009).

12. Kreps, D., J. Sheinkman, Quantity precommitment and Bertrand competition yields Cournot outcomes, Bell Journal of Economics, 14, 326-337 (1983).

13. Levitan, R., M. Shubik, Price duopoly and capacity constraints, International Economic Review, 13, 111-122 (1972).

14. Loertscher, S., Market making oligopoly, The Journal of Industrial Economics, 56, 263-289 (2008).

15. Osborne, M. J., C. Pitchik, Price competition in a capacity-constrained duopoly, Journal of Economic Theory, 38, 238-260 (1986).

16. Ubeda, L., Capacity and market design: discriminatory versus uniform auctions, Dep. Fundamentos del Analisis Economico, Universidad de Alicante, January, 2007.

17. Vives, X., Rationing rules and Bertrand-Edgeworth equilibria in large markets, Economics Letters 21, 113-116 (1986).