On Complex Dynamics of Differentiated Products: Cournot Duopoly Model under Average Profit Maximization

Abstract

In this paper, some important dynamic characteristics such as multistability and synchronization phenomena are investigated for a game of an economic Cournot duopoly whose time evolution is received by the iteration of a noninvertible map in the plane. In the asymmetric case, the equilibrium points of game’s map are calculated, and their stability conditions are obtained. The obtained results show that the Nash equilibrium point loses its stability through flip bifurcation. Under some restrictions, the map’s coordinate axes form an invariant manifold, and hence their dynamics are studied based on a one-dimensional discrete dynamic map. In the symmetric case where both firms are identical, the map has the property of symmetry, and this implies that the diagonal $q_1=q_2$ forms an invariant manifold and therefore synchronization phenomena occur. Global analysis of the behavior of the noninvertible map is carried out through studying critical manifolds of the map that categorize it as $Z_4-Z_2-Z_0$ type. Furthermore, global bifurcation of the basins of attraction is confirmed through contact between the critical curves and the boundaries of escaping domain.