On the dynamics of Cournot duopoly game with private firms: Investigations and analysis

Abstract

In this paper, we introduce a Cournot duopoly game whose players’ inverse prices are isoelastic functions and are derived from a simple quadratic utility function. The proposed game consists of two competing firms seeking the optimum of their productions by maximizing weighted objectives that are profit and social welfare under the same marginal costs. Based on those objectives, the bounded rationality is adopted to construct the model describing the game's evolution. A unique fixed point which is Nash equilibrium point is obtained for the game. The stability conditions for this point are discussed and shown that the equilibrium point can be unstable through flip bifurcation. In our discussion, we show that the model's map is noninvertible and belongs to Z3−Z1 type. Numerical experiments for the dynamics of the map are performed and shown that the map possesses several stable attractors. Furthermore, the shapes of some attractive basins for the map are analyzed providing the existence of the so-called lobes.