Analysis of the stability and the bifurcations of two heterogeneous triopoly games with an isoelastic demand

In this paper, we explore two heterogeneous triopoly games where the market demand function is isoelastic. The local stability and the bifurcations of these games are systematically analyzed using a symbolic approach, proposed by the author, of counting real solutions of a parametric system. The novelty of our study is twofold. On one hand, we introduce into the study of oligopoly games several methods of symbolic computation, which can establish analytical results and are different from the existing methods in the literature based on numerical simulations. In particular, we obtain the analytical conditions of the local stability and prove the existence of double routes to chaos through the period-doubling bifurcation and the Neimark-Sacker bifurcation. On the other hand, in the special case of the involved firms having identical marginal costs, we acquire the analytical conditions of the local stability for the two models. By further analyzing these conditions, it seems that the presence of the local monopolistic approximation (LMA) mechanism has a stabilizing effect for heterogeneous triopoly games with the isoelastic demand.