Stability of Equilibrium Prices in a Dynamic Duopoly Bertrand Game with Asymmetric Information and Cluster Spillovers

Abstract

Bounded rationality, asymmetric information and spillover effects are widespread in the economic market, and had been studied extensively in oligopoly games, but few references discussed incomplete information in a duopoly market with rationality expectations. Considering the positive externalities brought by the spillover effect between enterprises in a cluster, a duopoly Bertrand game with bounded rationality and asymmetric information is proposed in this paper. In our model, a firm with private information, high or low marginal cost, is introduced. Interestingly, our theoretical analysis reveals that: (1) In a dynamic duopoly Bertrand game with perfect rationality and asymmetric information, the equilibrium price is positively correlated with product substitution rate and the probability of a high marginal cost, while it is negatively correlated with the cluster spillover. (2) In a dynamic duopoly Bertrand game with asymmetric information and adaptive expectation adopted by both firms, the Nash equilibrium prices are always asymptotically stable. (3) In a dynamic duopoly Bertrand game with heterogenous expectation and asymmetric information, where two firms use adaptive expectation and boundedly rational expectation respectively, the Nash equilibrium prices are locally stable under certain conditions. Furthermore, results indicate that, high product substitution rate or large probability of high marginal cost for firm 2 with private information may make the market price unstable, bifurcating or even falling into chaos, while high technology spillover is conducive to stabilize the market by contrast. It is also shown that the chaos can be controlled by a hybrid control strategy with the state variables feedback and parameter variation. Our research has an important theoretical and practical significance to the price competition in oligopoly markets.