The dynamics of a Bertrand duopoly with differentiated products: Synchronization, intermittency and global dynamics

Fanti, Luciano; Gori, Luca; Mammana, Cristiana; Michetti, Elisabetta

doi:10.1016/j.chaos.2013.04.002

Cited by 56 publications

(65 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Generally demand functions that are log-convex such as iso-elastic ones, can cause such unusual output in price [6]. Duopoly has been extensively studied not only for the existence of absolutely continuous ergodic measures but also for its dynamics characteristics [28][29][30][31][32].…”

Section: Introductionmentioning

confidence: 99%

Dynamic Cournot duopoly games with nonlinear demand function

Askar

Alshamrani

Alnowibet

2015

Applied Mathematics and Computation

View full text Add to dashboard Cite

Section: Introductionmentioning

confidence: 99%

Dynamic Cournot duopoly games with nonlinear demand function

Askar

Alshamrani

Alnowibet

2015

Applied Mathematics and Computation

View full text Add to dashboard Cite

“…Askar [5] has shown some important results about Cournot duopoly game that was formed by a concave demand function. Other investigations on those complex characteristics can be found in [6][7][8][9][10][11][12]. It has been discussed in [13][14][15] that quantities may have a degree of competition that is low if commodities are substitutes.…”

Section: Introductionmentioning

confidence: 99%

“…Those expectations may be naive, rational, or adaptive expectations, or, alternatively, some weighted sum of previous decision to set the quantities at time + 1. On the other hand, if a player does not have a complete knowledge of the profit function, he/she can use some local estimation of the marginal profit in order to follow the steepest slope of the profit function [6]. Such limited information makes players unable to completely solve the optimization problem max 1 , 2 , +1 ( 1, +1 , 2, +1 ) by considering expectations about the quantity that the competitor will choose for the next period, but they are able to get a correct estimate of their own local slope, that is, the partial derivatives of the profit computed at the current state of production.…”

Section: The Dynamic Modelmentioning

confidence: 99%

Analysis of Nonlinear Duopoly Games with Product Differentiation: Stability, Global Dynamics, and Control

Askar

Al-khedhairi

2017

Discrete Dynamics in Nature and Society

View full text Add to dashboard Cite

Many researchers have used quadratic utility function to study its influences on economic games with product differentiation. Such games include Cournot, Bertrand, and a mixed-type game called Cournot-Bertrand. Within this paper, a cubic utility function that is derived from a constant elasticity of substitution production function (CES) is introduced. This cubic function is more desirable than the quadratic one besides its amenability to efficiency analysis. Based on that utility a two-dimensional Cournot duopoly game with horizontal product differentiation is modeled using a discrete time scale. Two different types of games are studied in this paper. In the first game, firms are updating their output production using the traditional bounded rationality approach. In the second game, firms adopt Puu's mechanism to update their productions. Puu's mechanism does not require any information about the profit function; instead it needs both firms to know their production and their profits in the past time periods. In both scenarios, an explicit form for the Nash equilibrium point is obtained under certain conditions. The stability analysis of Nash point is considered. Furthermore, some numerical simulations are carried out to confirm the chaotic behavior of Nash equilibrium point. This analysis includes bifurcation, attractor, maximum Lyapunov exponent, and sensitivity to initial conditions.

show abstract

“…For instance, Bertrand duopoly models were investigated by Brianzoni et al [33], Zhao and Zhang [34], and Fanti et al [35] depending on differentiated products. They have discussed that the degree of products differentiation had a major impact on the sale quantity and price.…”

Section: Introductionmentioning

confidence: 99%

Exploration of Complex Dynamics for Cournot Oligopoly Game with Differentiated Products

2018

View full text Add to dashboard Cite

This paper proposes a Cournot game organized by three competing firms adopting bounded rationality. According to the marginal profit in the past time step, each firm tries to update its production using local knowledge. In this game, a firm's preference is represented by a utility function that is derived from a constant elasticity of substitution (CES) production function. The game is modeled by a 3-dimensional discrete dynamical system. The equilibria of the system are numerically studied to detect their complex characteristics due to difficulty to get an explicit form for those equilibria. For the proposed utility function, some cases with different value parameters are considered. Numerical simulations are used to provide an experimental evidence for the complex behavior of the evolution of the system. The obtained results show that the system loses its stability due to different types of bifurcations.

show abstract

The dynamics of a Bertrand duopoly with differentiated products: Synchronization, intermittency and global dynamics

Cited by 56 publications

References 31 publications

Dynamic Cournot duopoly games with nonlinear demand function

Dynamic Cournot duopoly games with nonlinear demand function

Analysis of Nonlinear Duopoly Games with Product Differentiation: Stability, Global Dynamics, and Control

Exploration of Complex Dynamics for Cournot Oligopoly Game with Differentiated Products

Contact Info

Product

Resources

About