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

Xiaoliang Li; Xiaoliang Li

doi:10.3934/math.20221065

AIMS Mathematics

2022, Volume 7, Issue 10: 19388-19414. doi: 10.3934/math.20221065

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Research article

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

Xiaoliang Li ^,

School of Digital Economics, Dongguan City University, Wenchang Road, Liaobu, Dongguan 523419, China

Received: 04 July 2022 Revised: 23 August 2022 Accepted: 30 August 2022 Published: 01 September 2022
MSC : 37B25, 37M20, 37N40

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.
- triopoly games,
- heterogeneous firms,
- dynamics,
- stability,
- bifurcations,
- symbolic computation
Citation: Xiaoliang Li. Analysis of the stability and the bifurcations of two heterogeneous triopoly games with an isoelastic demand[J]. AIMS Mathematics, 2022, 7(10): 19388-19414. doi: 10.3934/math.20221065

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Abstract

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.

References

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