Nonlinear Cournot Duopoly Model : Under Cubic Cost Functions with a Negative Externality and Differences in Production Costs

Abstract

We consider the Cournot duopoly model under the assumption that cost functions are a cubic function of own and competitors' outputs. Therefore, we incorporated a negative external effect into cost functions because cost functions lead to non-monotonic, in particular unimodal reaction functions. The shape of the reaction curve is the same as the Puu [5] model. The number and the stability of the Nash-equilibrium depend on the degree of external effect and the differences in the coefficients of the cost functions. If there exists no differences in the coefficients, the three Nash-equilibria increase the degree of external effects, and the adjustment fails to converge to any equilibria. If there exists differences in the coefficients, the number of equilibria decrease in some cases, but do not always mean stabilization of the Nash-equilibrium. The Nash-equilibrium becomes unstable again when the differences expand. If there exist differences in the coefficients, a general cubic equation must be solved to result in the Nash-equilibrium. The shape of the solutions is algebraically complex, because they have a cube root. The stability of the Nash-equilibrium is explained by graphical methods. In the graphical methods, the boundaries that result in the the discriminant solutions becoming zero to provide some information.JEL Classification: C63, D21, D43