A simple model of production and trade in an oligopolistic market: back to basics

We provide a two good model of oligopolistic production and trade with one good being commodity money. There is the usual demand function of the consumers for the produced good that producer-sellers face. Each seller is a budget constrained preference maximizer and derives utility (or satisfaction) from consuming bundles comprising commodity money and the produced good. We define a competitive equilibrium strategy profile and a Cournotian equilibrium and show that under our assumptions both exist. We further show that at a competitive equilibrium strategy profile, each seller maximizes profits given his own consumption of the produced good and the price of the produced good, the latter being determined by the inverse demand function. Similarly we show that at a Cournotian the sellers are at a Cournot equilibrium given their own consumption of the produced good. Assuming sufficient differentiability of the cost functions we show that at a competitive equilibrium each seller either sets price equal to marginal cost or exhausts his capacity of production; at a Cournotian equilibrium each seller either sets marginal revenue equal to marginal cost or exhausts his capacity of production. We also study the evolution of Cournotian strategies as the sellers and buyers are replicated. As the number of buyers and sellers go to infinity any sequence of interior symmetric Cournotian equilibrium strategies admits a convergent subsequence, which converges to an interior symmetric competitive equilibrium strategy. In a final section we discuss the Bertrand Edgeworth price setting game and show that a Bertrand Edgeworth equilibrium must be a derived from a competitive equilibrium price. Here we show that if at a symmetric competitive equilibrium, the sellers consume positive quantities of the produced good then the competitive equilibrium cannot be a Bertrand Edgeworth equilibrium. Thus, if at all symmetric competitive equilibria the sellers consume positive amounts of the produced good, then a Bertrand Edgeworth equilibrium simply does not exist.