On a Bertrand Duopoly Game With Heterogeneous Expectations and Emissions Costs

Abstract

This study was based on the dynamics of a nonlinear Bertrand-type duopoly game with differentiated goods, linear demand and a cost function that included emissions costs. This duopoly game was modeled with a system of two difference equations. Existence and stability of equilibria of this system were studied. It was shown that the model gave more complex chaotic and unpredictable trajectories as a consequence of changes in the speed of adjustment parameter and horizontal product differentiation parameter. Numerical simulations showed that a higher value of the speed of adjustment and a higher or lower (negative) value of product differentiation (weaker or fiercer competition) can destabilize the economy. The chaotic features were justified numerically via computing Lyapunov numbers and sensitive dependence on initial conditions. Also, it was shown that in this case of a duopoly game, there were stable trajectories, and a higher (lower) degree of product differentiation did not tend to destabilize the economy. Keywords: Bertrand duopoly game, discrete dynamical system, heterogeneous expectations, stability, chaotic behavior