Chaos in Oligopoly Models

Abstract

In this article, the authors investigate the dynamics of two oligopoly games. In the first game, they consider a nonlinear Cournot-type duopoly game with homogeneous goods and same rational expectations. The authors investigate the case, where managers have a variety of attitudes toward relative performance that are indexed by their type. In the second game they consider a Cournot-Bertrand duopoly game with linear demand, quadratic cost function and differentiated goods. In the two games they suppose a linear demand and a quadratic cost function. The games are modeled with a system of two difference equations. Existence and stability of equilibria of the systems are studied. The authors show that the models gives more complex, chaotic and unpredictable trajectories, as a consequence of change in the parameter k of speed of the player's adjustment (in the first game) and in the parameter d of the horizontal product differentiation (in the second game). The authors prove that the variation of the parameter k (resp. d) destabilizes the Nash equilibrium via a period doubling bifurcation (resp. through a Neimark-Sacker bifurcation). The chaotic features are justified numerically via computing Lyapunov numbers and sensitive dependence on initial conditions. In the second game they show that in the case of a quadratic cost there are stable trajectories and a higher or lower degree of product differentiation does not tend to destabilize the economy. They verify these results through numerical simulations. Finally, the authors control the chaotic behavior of the games introducing a new parameter. For some values of this parameter, the Nash equilibrium is stable for every value of the main parameter k or d.