Pricing and matching under duopoly with imperfect buyer mobility

Recent contributions have explored how lack of buyer mobility affects pricing. For example, Burdett, Shi, and Wright (2001) envisage a two-stage game where, once prices are set by the firms, the buyers play a static game by choosing independently which firm to visit. We incorporate imperfect mobility in a duopolistic pricing game where the buyers are involved into a multi-stage game. The firms are shown to have an incentive to give service priority to loyal customers. Under this rationing rule, equilibrium prices converge to their value under perfect buyer mobility as the number of stages of the buyer game increasesBertrand competition, matching, imperfect mobility, sequential equilibrium, buyerloyalty

Existence of pure strategy equilibrium in Bertrand-Edgeworth games with imperfect divisibility of money

This paper incorporates imperfect divisibility of money in a price game where a given number of identical firms produce a homogeneous product at constant unit cost up to capacity. We find necessary and sufficient conditions for the existence of a pure strategy equilibrium. Unlike in the continuous action space case, with discrete pricing there may be a range of symmetric pure strategy equilibria - which we fully characterize - a range which may or may not include the competitive price. Also, we determine the maximum number of such equilibria when competitive pricing is itself an equilibrium.Bertrand-Edgeworth competition

On stability of Bertrand-Nash equilibrium in a simple model of the labour market

We examine a Bertrand-Edgeworth model of competition in a labour market where the workers simultaneously set wages disregarding any influence their current decision may have on opponents' future decisions. The iterated best response process is shown to converge in finite time to a Bertrand-Nash solution, where wages are set at the market-clearing level. This convergence result is also shown to hold when the assumption of static expectations is replaced by milder restrictions on beliefs about opponents'' wages.

A dynamic entry and price game with capacity indivisibility

Strategic market interaction is here modelled as a two-stage game in which potential entrants choose capacities and active firms compete in prices. Due to capital indivisibility, the capacity choice is made from a finite grid and there are substantial economies of scale. In the simplest version of the model assuming a single production technique, the equilibrium of the game is shown to depend on the market size - namely, on total demand at a price equal to the minimum average cost -relative to the firm minimum efficient scale: if the market is sufficiently large, then the competitive price (the minimum of average cost) emerges at a subgame-perfect equilibrium of the game; if the market is not that large, then the firms randomize in prices on the equilibrium path of the game. The role of the market size for the competitive outcome is even more important for the case of two production techniquesBertrand-Edgeworth, oligopoly, price game, mixed strategy equilibrium, capacity indivisibility

Endogenous entry under Bertrand-Edgeworth and Cournot competition with capacity indivisibility

Strategic market interaction is modelled as a two-stage game where potential entrants choose capacities and active firms compete in prices or quantities. Due to capital indivisibility, the capacity choice is made from a finite grid. In either strategic setting, the equilibrium of the game depends on the size of total demand at a price equal to the minimum average cost. With a sufficiently large market, the long-run competitive price emerges at a subgame-perfect equilibrium of either game. Failing the large market condition, equilibrium outcomes are quite different in the two games (in contrast to Kreps and Scheinkman), and neither game reproduces the competitive equilibrium.Entry, Bertrand-Edgeworth, Cournot, capacity indivisibility

On a property of mixed strategy equilibria of the pricing game

Before solving a two-stage capacity and pricing game for oligopoly, Boccard and Wauthy (2000) argue that, as under duopoly, at a mixed strategy equilibrium of the pricing game the largest firm's expected profit is the profit accruing to it as a Stackelberg follower when the rivals supply their entire capacity. We point to a serious mistake in their argument and then we see how this important property can be satisfactorily established.Bertrand competition

Existence of pure strategy equilibria in Bertrand-Edgeworth games with imperfect divisibility of money

This paper incorporates imperfect divisibility of money in a price game where a given number of identical firms produce a homogeneous product at constant unit cost up to capacity. We find necessary and sufficient conditions for the existence of a pure strategy equilibrium. Unlike in the continuous action space case, under discrete pricing there may be a range of symmetric pure strategy equilibria - which we fully characterize - a range which may or may not include the competitive price. Also, we determine the maximum number of such equilibria when competitive pricing is itself an equilibrium.Bertrand-Edgeworth competition; Price game; Oligopoly; Pure strategy equilibrium; Discrete pricing

Bertrand-Edgeworth competition in an almost symmetric oligopoly

We analyze a Bertrand-Edgeworth game in homogeneous product industry, under effcient rationing, constant marginal cost until full capacity utilization, and identical technology across firms. We solve for the equilibrium and establish its uniqueness for capacity configurations in the mixed strategy region of the capacity space such that the capacities of the largest and smallest firm are sufficiently close.Bertrand-Edgeworth competition; mixed strategy equilibrium; almost symmetric oligopoly; Mixed strategy equilibrium

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

The paper extends the analysis of price competition among capacity-constrained sellers beyond the cases of duopoly and symmetric oligopoly. We first provide some general results for the oligopoly and then focus on the triopoly, providing a complete characterization of the mixed strategy equilibrium of the price game. The region of the capacity space where the equilibrium is mixed is partitioned according to the features of the mixed strategy equilibrium arising in each subregion. Then computing the mixed strategy equilibrium becomes a quite simple task. The analysis reveals features of the mixed strategy equilibrium which do not arise in the duopoly (some of them have also been discovered by Hirata (2008)).Bertrand-Edgeworth; Price game; Oligopoly; Triopoly; Mixed strategy equilibrium