On Bertrand duopoly game with differentiated goods

• Utility function of the form U(q1,q2)=q1α+q2α is introduced.
• New dynamical Bertrand games are derived.
• Stability conditions of equilibrium points are discussed.
• Chaos and bifurcation behavior are studied.

The paper investigates a dynamic Bertrand duopoly with differentiated goods in which boundedly rational firms apply a gradient adjustment mechanism to update their price in each period. The demand functions are derived from an underlying CES utility function. We investigate numerically the dynamical properties of the model. We consider two specific parameterizations for the CES function and study the Nash equilibrium and its local stability in the models. The general finding is that the Nash equilibrium becomes unstable as the speed of adjustment increases. The Nash equilibrium loses stability through a period-doubling bifurcation and the system eventually becomes chaotic either through a series of period-doubling bifurcations or after a Neimark–Sacker bifurcation.