On a master-slave Bertrand game model

A master-slave Bertrand game model is proposed for upstream and downstream monopolies owned by different parties, in which the upstream monopolist's output is used as the main factor of production by the downstream monopolist who is a small purchaser of the upstream monopolist's output. The bifurcation of the Bertrand-Nash equilibrium is analyzed with Schwarzian derivative. Numerical simulations are employed to show the model's complex dynamics by means of the largest Lyapunov exponents (LLEs), bifurcation, time series diagrams and phase portraits. With the modified straight-line stabilization method, chaos control is used to improve the aggregate profits of the two oligopolists. Lastly the welfare impacts of price fluctuations and chaos controls are briefly discussed.