Analytic Solutions of Nonlinear Cournot Duopoly Game

The main aim of this study is to construct an explicit form of an analytic solution of a multidimensional difference equation. To this end, we present a nonlinear Cournot duopoly model with unimodal reaction functions, show that its dynamical system is double logistic when the production externalities involved get stronger, and then give a constructive proof of the existence of an analytic solution of the dynamical system. It is well known that there is an existence theorem for nonlinear differential equations, which guarantees the existence of an analytic solution. It is also well known that there is no general existence theorem for a nonlinear difference equation. In consequence, we check whether a discrete-time system to be considered has a solution or not, and then we examine its dynamical characteristic. In spite of this inconvenience, a discrete-time dynamical system is useful for analyzing economic phenomena observed in the real economic world such as business fluctuations that occur in real time. As far as global dynamics are concerned, the usual procedure is to perform numerical simulations to visualize how a dynamical process evolves over time, even if analytical treatment is insufficient. However, it is still possible that numerical simulations do not approximate the true behavior of a discrete-time system, mainly due to computer’s storage limitations; truncations errors and the binary representation of numbers. To examine exact phenomena generated by a discrete-time system, it is therefore natural to attempt to provide its analytical solutions. This study is organized as follows. Section 2 constructs a nonlinear duopoly model with unimodal reaction functions. The first half of Section 3 considers the existence of particular solutions. The second half, the main part of this study, seeks general analytic solutions and demonstrates their explicit forms.