A Bertrand duopoly game with differentiated products reconsidered

TL;DR

This paper analyzes a dynamic Bertrand duopoly with differentiated products, revealing how product substitutability impacts equilibrium stability and contrasting these effects with Cournot models, supported by symbolic computation and numerical simulations.

Contribution

It provides the first rigorous stability conditions for this model and highlights the destabilizing effect of increased substitutability, contrasting with Cournot model findings.

Findings

Increasing substitutability destabilizes equilibrium.

Lower product differentiation leads to lower prices and profits.

Complex dynamics like chaos are observed in simulations.

Abstract

In this paper, we explore a dynamic Bertrand duopoly game with differentiated products, where firms are boundedly rational and consumers are assumed to possess an underlying CES utility function. We mainly focus on two distinct degrees of product substitutability. Several tools based on symbolic computations such as the triangular decomposition method and the PCAD method are employed in the analytical investigation of the model. The uniqueness of the non-vanishing equilibrium is proved and rigorous conditions for the local stability of this equilibrium are established for the first time. Most importantly, we find that increasing the substitutability degree or decreasing the product differentiation has an effect of destabilization for our Bertrand model, which is in contrast with the relative conclusions for the Cournot models. This finding could be conducive to the revelation of the essential difference between dynamic Cournot and Bertrand oligopolies with differentiated goods. In the special case of identical marginal costs, we derive that lower degrees of product differentiation mean lower prices, higher supplies, lower profits, and lower social welfare. Furthermore, complex dynamics such as periodic orbits and chaos are reported through our numerical simulations.