Analysis of stability and bifurcation for two heterogeneous triopoly games with the isoelastic demand

TL;DR

This paper provides an analytical study of stability and bifurcation phenomena in two heterogeneous triopoly models with isoelastic demand, revealing routes to chaos and conditions for stability, with implications for market dynamics.

Contribution

It introduces a rigorous analytical approach to stability and bifurcation analysis in triopoly games, including conditions for chaos and the stabilizing effect of the LMA mechanism.

Findings

Existence of double routes to chaos via bifurcations.

Necessary and sufficient conditions for local stability.

LMA mechanism may stabilize heterogeneous triopoly games.

Abstract

In this paper, we investigate two heterogeneous triopoly games where the demand function of the market is isoelastic. The local stability and the bifurcation of these games are systematically analyzed using the symbolic approach proposed by the author. The novelty of the present work is twofold. On one hand, the results of this paper are analytical, which are different from the existing results in the literature based on observations through numerical simulations. In particular, we rigorously prove the existence of double routes to chaos through the period-doubling bifurcation and through the Neimark-Sacker bifurcation. On the other hand, for the special case of the involved firms having identical marginal costs, we acquire the necessary and sufficient conditions of the local stability for both models. By further analyzing these conditions, it seems that that the presence of the local monopolistic approximation (LMA) mechanism might have a stabilizing effect for heterogeneous triopoly games with the isoelastic demand.