Abstract
Price competition has become a universal commercial phenomenon nowadays. This paper considers a dynamic Bertrand price game model, in which enterprises have heterogeneous expectations. By the stability theory of the dynamic behavior of the Bertrand price game model, the instability of the boundary equilibrium point and the stability condition of the internal equilibrium point are obtained. Furthermore, bifurcation diagram, basin of attraction, and critical curve are introduced to investigate the dynamic behavior of this game. Numerical analysis shows that the change of model parameters in a dynamic system has a significant impact on the stability of the system and can even lead to complex dynamic behaviors in the evolution of the entire economic system. This kind of complex dynamic behavior will cause certain damage to the stability of the whole economic system, causing the market to fall into a chaotic state, which is manifested as a kind of market disorder competition, which is very unfavorable to the stability of the economic system. Therefore, the chaotic behavior of the dynamical system is controlled by time-delay feedback control and the numerical analysis shows that the effective control of the dynamical system can be unstable behavior and the rapid recovery of the market can be stable and orderly.
Highlights
Price competition has become a universal commercial phenomenon nowadays. is paper considers a dynamic Bertrand price game model, in which enterprises have heterogeneous expectations
Under price competition, the behavior of gradient firms may constitute a source of instability. is phenomenon occurs when participants are more sensitive to market information and make their adjustments too fast. is result confirms the conclusions presented in the literature
The adaptive expectations of enterprises have an impact on the stability or instability of the game model, because if enterprises are more inclined to change the number of previous periods, Nash equilibrium will lose local stability. is result is consistent with the conclusions in [14, 43] in the absence of product differentiation
Summary
The equilibrium points of the nonlinear discrete system (12) are calculated. en, through qualitative en, the stability characteristic equilibrium point of system erefore, system (12) can be converted into the following (12) is analyzed qualitatively. Proof: To prove this result, we calculate the Jacobian matrix (14) at the boundary equilibrium point E1 as follows: J. Erefore, according to Lemma 1, the discrete complex dynamic system (12) Nash equilibrium stability determination condition can be known that E1 is an unstable boundary equilibrium. We can conclude that the Nash equilibrium is locally asymptotically stable by adjusting the threshold of the following speeds: α1 αB (25). This monotonic relationship does not appear in the substitute product In this case, the threshold for adjusting the speed is the largest and has a certain degree of substitutability (for the fixed β); if the degree of difference between the products is increased, the stability of the Nash equilibrium is difficult to be guaranteed (note: d is true when the value approaches 0 on the right side). This result is related to the complementary intertemporal strategy of the product
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