Price Competition with Decreasing Returns-to-Scale: A General Model of Bertrand-Edgeworth Duopoly

We present a novel approach to analyzing models of price competition. By realizing price competition as a class of all-pay contests, we are able to generalize the models in which pricing behavior can be characterized, accommodating convex (possibly asymmetric) cost structures and general demand rationing schemes. Using this approach, we identify necessary and sufficient conditions for a pure strategy equilibrium and use them to demonstrate the fragility of deterministic outcomes in pricing games. Consequently, we characterize bounds on equilibrium pricing and profits of all mixed strategy equilibria and examine the effect of demand and supply shifts on those bounds. Our focus on bounds can be motivated by the potential for multiple non-payoff equivalent equilibria, as we identify two types of equilibrium strategies through a derivation of sufficient conditions for uniqueness of equilibrium.