Stability analysis in a Bertrand duopoly with different product quality and heterogeneous expectations

We study the local stability properties of a duopoly game with price competition, different product quality and heterogeneous expectations. We show that the Nash equilibrium can loose stability through a flip bifurcation when the consumer's type range increases. This result occurs irrespective of whether the high(low)-quality firm has either bounded rational (naive) or naive (bounded rational) expectations about the price that should be set in the future by the rival to maximise profits. Therefore, although, on the one hand, an increase in the consumer's types range increases profits, on the other hand, it contributes to reduce the parametric stability region of the unique interior equilibrium. Moreover, we show that the stability region is larger when the high-quality firm has naive expectations and the low-quality firm has bounded rational expectations. This implies that when the expectations formation mechanism of the high-quality firm becomes more complicated than the naive one, and, in particular, it follows the mechanism proposed by Dixit (1986), the stability of the Nash equilibrium in a duopoly market with price competition becomes under increasing strain.