Stability and chaos of the duopoly model of Kopel: A study based on symbolic computations

TL;DR

This paper provides an analytical study of Kopel's asymmetric duopoly model using symbolic computations, establishing conditions for equilibria stability and proving the existence of chaos through snapback repellers.

Contribution

It offers the first rigorous analytical conditions for equilibrium stability and demonstrates chaos in Kopel's model via symbolic computation and Marotto's theorem.

Findings

Necessary and sufficient conditions for multiple positive equilibria

Rigorous stability conditions for equilibria with adaptive expectations

Proof of chaos existence via snapback repellers

Abstract

Since Kopel's duopoly model was proposed about three decades ago, there are almost no analytical results on the equilibria and their stability in the asymmetric case. The first objective of our study is to fill this gap. This paper analyzes the asymmetric duopoly model of Kopel analytically by using several tools based on symbolic computations. We discuss the possibility of the existence of multiple positive equilibria and establish necessary and sufficient conditions for a given number of positive equilibria to exist. The possible positions of the equilibria in Kopel's model are also explored. Furthermore, in the asymmetric model of Kopel, if the duopolists adopt the best response reactions or homogeneous adaptive expectations, we establish rigorous conditions for the local stability of equilibria for the first time. The occurrence of chaos in Kopel's model seems to be supported by observations through numerical simulations, which, however, is challenging to prove rigorously. The second objective is to prove the existence of snapback repellers in Kopel's map, which implies the existence of chaos in the sense of Li-Yorke according to Marotto's theorem.