Bertrand-Edgeworth oligopoly: Characterization of mixed strategy equilibria when some firms are large and the others are small

Salvadori, Neri and De Francesco, Massimo A. (2020): Bertrand-Edgeworth oligopoly: Characterization of mixed strategy equilibria when some firms are large and the others are small.

Abstract

This paper studies Bertrand-Edgeworth competition among firms producing a homogeneous commodity under efficient rationing and constant (andidentical across firms) marginal cost until full capacity utilization is reached. Our focus is on a subset of the no pure-strategy equilibrium region of the capacity space in which, in a well-defined sense, some firms are large and the others are small. We characterize equilibria for such subset. For each firm, the payoffs are the same at any equilibrium and, for each type of firm, they are proportional to capacity. While there is a single profile of equilibrium distributions for the large firms, there is a continuum of equilibrium distributions for the small firms: what is uniquely determined, for the latter, is the capacity-weighted sum of their equilibrium distributions and hence the union of the supports of their equilibrium strategies.

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

References

Armstrong, M. and J Vickers, Patterns of Competition with Captive Customers, MPRA working paper No. 90362 (2018).

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

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

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

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

De Francesco, M. A., Salvadori N., Bertrand-Edgeworth games under oligopoly with a complete characterization for the triopoly, MPRA working paper No. 10767 (2008) and MPRA working paper No. 24087 (2010).

De Francesco, M. A., Salvadori N., Bertrand-Edgeworth competition in an almost symmetric oligopoly, Journal of Microeconomics, 1(1), 2011, pp. 99-105; reprinted in Studies in Microeconomics, 1(2), 2013, pp. 213-219.

De Francesco, M. A., Salvadori N., Bertrand-Edgeworth games under triopoly: the payoffs, MPRA working paper No. 64638 (2015).

De Francesco, M. A., Salvadori N., Bertrand-Edgeworth games under triopoly: the equilibrium strategies when the payoffs of the two smallest firms are proportional to their capacities, MPRA working paper No. 69999 (2016).

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

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

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

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

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

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

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