Bertrand–Edgeworth equilibrium with a large number of firms

Abstract

We examine a model of price competition with strictly convex costs where the firms simultaneously decide on both price and quantity, are free to supply less than the quantity demanded, and there is discrete pricing. If firms are symmetric then, for a large class of residual demand functions, there is a unique equilibrium in pure strategies whenever, for a fixed grid size, the number of firms is sufficiently large. Moreover, this equilibrium price is within a grid-unit of the competitive price. The results go through to a large extent when the firms are asymmetric, or they are symmetric but play a two stage game and the tie-breaking rule is 'weakly manipulable'. JEL Classification Number: D43, D41, L13.

Information, Bertrand–Edgeworth competition and the law of one price

“…Alternative methods of delivering pure strategy equilibria include modifying the timing of the game (Deneckere and Kovenock, 1992;Deneckere and Peck, 2012;Dudey, 1992), allowing sellers to choose list prices and subsequent discount prices (Garcìa Díaz et al, 2009;Myatt and Ronayne, 2019), requiring integer pricing (Chowdhury, 2008), imposing a cost on firms that turn customers away (Dixon, 1990) and introducing a public social-surplus maximising seller (Rácz and Tasnádi, 2016).…”

An Empirical Study of Pricing Strategies in an Online Market with High-Frequency Price Information

“…Tirole (1988) provides a general discussion Chowdhury (2008). shows that the unique equilibrium of a repeated game with convex costs (related to Edgeworth's capacity constraints) in which firms can choose price and quantity is a grid point above marginal cost if the number of firms is large enough.…”