Dynamics of a Bertrand duopoly with differentiated products and nonlinear costs: Analysis, comparisons and new evidences

Abstract

a r t i c l e i n f o Article history: Available online xxx a b s t r a c tThis paper studies mathematical properties and dynamics of a duopoly with price competition and horizontal product differentiation by introducing quadratic production costs (decreasing returns to scale), thus extending the model with linear costs (constant returns to scale) of Fanti et al. [11]. The economy is described by a two-dimensional non-invertible discrete time dynamic system. The paper first determines fixed points and other invariant sets, showing that synchronized dynamics can occur. Then, stability properties are compared in the cases of quadratic costs and linear costs by considering the degree of product differentiation and the speed of adjustment of prices as key parameters. It is also shown that synchronization takes place if products tend to be relatively complements and stressed similarities and differences between models with quadratic and linear costs. Finally, the paper focuses on the phenomenon of multistability thus underlying new evidences in comparison with the model with linear costs.

Complex Dynamics of a Continuous Bertrand Duopoly Game Model with Two-Stage Delay

“…In order to show more clearly the effect of chaos control, we only let 1 Figure 9, and Equation (21) is chaotic. Without chaos control, the dynamic properties of the system (21) are shown in Figure 16.…”

“…Some researchers have studied the Bertrand duopoly with differentiated products. The results show that the degree of product differentiation has a great influence on the price and quantity of sale [1][2][3]. In the Bertrand game model, the amount of information that the duopoly has will affect the stability of the system, mainly reflected in the change of the basin of attraction [4].…”

On the Stability and Multistability in a Duopoly Game with R&D Spillover and Price Competition

“…And they found that the Nash equilibrium point may lose its stability through a flip bifurcation or a Neimark-Sacker bifurcation. In addition, a lot of researchers have discussed the complex dynamical behaviors of nonlinear oligopolies from different aspects, such as differentiated goods [21][22][23][24][25], heterogeneous firms [26][27][28][29][30][31], and delayed decisions [32,33].…”

Complexity Dynamic Character Analysis of Retailers Based on the Share of Stochastic Demand and Service

“…Kawabata and Takarada [14] probed into the effect of free trade agreements on the welfare in Bertrand and Cournot competition model under differentiated products. Brianzoni et al [15] studied the mathematical properties and dynamics of a Bertrand duopoly with horizontal differentiated products and nonlinear costs, and the new evidence has been offered. Liu and Wang [16] investigated the quality control game that discussed the different risk attitudes affecting quality control game of supply chain especially in logistics service supply chain, instead of considering one member's risk attitude and ignoring the combination of two members' risk attitudes.…”