Stability and Chaos of the Duopoly Model of Kopel: A Study Based on Symbolic Computations

Abstract

Since Kopel’s duopoly model was proposed about 3 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 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 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 of this paper 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.