Abstract
This paper considers a Cournot–Bertrand game model based on the relative profit maximization with bounded rational players. The existence and stability of the Nash equilibrium of the dynamic model are investigated. The influence of product differentiation degree and the adjustment speed on the stability of the dynamic system is discussed. Furthermore, some complex properties and global stability of the dynamic system are explored. The results find that the higher degree of product differentiation enlarges the stable range of the dynamic system, while the higher unit product cost decreases the stable range of price adjustment and increases the one of output adjustment; period cycles and aperiodic oscillation (quasi-period and chaos) occur via period-doubling or Neimark–Sacker bifurcation, and the attraction domain shrinks with the increase of adjustment speed values. By selecting appropriate control parameters, the chaotic system can return to the stable state. The research of this paper is of great significance to the decision-makers’ price decision and quantity decision.
Highlights
Oligarchic market is one type of market structures controlled by several players who produce homogeneous products and take actions on the basis of their rival reactions for seeking profit maximization
Our theoretical contribution is as follows. e first contribution is to construct a dynamic Cournot–Bertrand game model, in which one firm adopts price as his decision variable and the other adopts quantity as his decision variable based on the relative profit maximization. e second contribution is to study the influence of parameters changing on the stability and profitability of the dynamic Cournot–Bertrand game model and to obtain some interesting phenomenon, such as existing period-doubling bifurcation and Neimark–Sacker bifurcation
Firm 1 regards the quantity of the products as the strategy variable, while firm 2 considers the price of the products as the strategy variable. at is to say, firm 1 competes in product quantity (q1) and firm 2 competes in product price (p2). Both firms will take actions based on the information from their rival and market for seeking relative profit maximization. is paper mainly focuses on contributing to construct a dynamic Cournot–Bertrand game model based on relative profit maximization and investigate the dynamic behaviors of players and the complex properties of the system
Summary
Oligarchic market is one type of market structures controlled by several players who produce homogeneous products and take actions on the basis of their rival reactions for seeking profit maximization. E first contribution is to construct a dynamic Cournot–Bertrand game model, in which one firm adopts price as his decision variable and the other adopts quantity as his decision variable based on the relative profit maximization. Is paper mainly focuses on contributing to construct a dynamic Cournot–Bertrand game model based on relative profit maximization and investigate the dynamic behaviors of players and the complex properties of the system. When the values of the adjustment speeds of two firms escape from the stable region (red region), the Nash equilibrium point will become unstable, perioddoubling cycles or Neimark–Sacker bifurcations occur and even evolve into chaos . Two firms should adjust the parameters according to the actual market conditions so that the dynamic system (12) is in a stable state
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