Dynamics of a discrete-time mixed oligopoly Cournot-type model with three time delays

TL;DR

This paper studies the dynamic behavior of a discrete-time mixed oligopoly model with three types of delays, analyzing stability, bifurcations, and complex dynamics involving one public and multiple private firms.

Contribution

It introduces a novel discrete-time Cournot model with multiple delays and thoroughly analyzes the stability and bifurcation phenomena in this complex setting.

Findings

Boundary equilibrium is a saddle point.

Stability regions depend on delay parameters.

Numerical examples show complex dynamic behaviors.

Abstract

The paper analyzes the interactions among one public firm and $n$ private firms on the market, in the framework of a discrete-time Cournot game with time delay. The production of the public firm is influenced by previous output levels of private firms. The productions of private companies are influenced by the past productions of the public company, as well as by the previous productions of the other private companies. The associated nonlinear system admits two equilibrium points: the positive one and the boundary equilibrium. After the stability analysis, we obtained that the boundary equilibrium point is a saddle point. If there is no delay, for the positive equilibrium point we have determined the stability region. Then, for different particular cases of delays, we found the conditions for which the positive equilibrium is asymptotically stable. The flip and Neimark-Sacker bifurcations are investigated. In addition, numerous numerical examples are performed to reveal the complex dynamic behavior of the system.