Stability Analysis in a Bertrand Duopoly with Different Product Quality and Heterogeneous Expectations

Abstract

We study the local stability properties of a nonlinear Bertrand duopoly with vertical differentiation and heterogeneous players under the hypotheses of both covered and uncovered markets. In the former case, the unique pure strategy Nash equilibrium can undergo a flip bifurcation when the extent of consumer’s heterogeneity increases. In the latter, the quality differential determines the stability of prices over time. Numerical evidence is provided to show the occurrence of endogenous fluctuations.

Appendices

Appendix A. Endogenous fluctuations (covered market)

In this appendix we show with numerical experiments that endogenous fluctuations can occur when a reduces (covered market). The parameter values are: b = 1, α = 4, u _H  = 2 and u _L  = 1. Then, a _BH/NL  = 1.4, α ₁ = 6/5 and α ₂ = 8/5 (BH/NL), and a _NH/BL  = 0.2 and α ₃ = 12/5 (NH/BL). Under this parameter constellation, we find a _BH/NL > b/2 so that the Nash equilibrium $E_{{BH/NL}}^{{CM}} = E_{{NH/BL}}^{{CM}}$ in the BH/NL model is unstable and trajectories are non-convergent for any a ∊ [0, 1/2). With regards to the NH/BL model, 4 = α > α ₃ implying that $E_{{BH/NL}}^{{CM}} = E_{{NH/BL}}^{{CM}}$ is stable or unstable depending on the relative size of a. To this purpose, Fig. 3a and b show the bifurcation diagrams for a under NH/BL, and depict the limit point of p _L and p _H , respectively, when the initial conditions are p _H (0) = 0.1 and p _L (0) = 0.05. The figures reveal that the long-run values of prices increase and they are locally asymptotically stable when a reduces from 1/2 to 0.2. Then, a flip bifurcation occurs at a _NH/BL  = 0.2. Then, a two-period cycle (broken off in the range a ∊ (0.05776, 0.05727) by more complicated dynamic events) emerges. As long as a reduces, we observe four-period cycles, eight-period cycles, and cycles of higher periodicity when a becomes lower. Figure 4 also depicts the chaotic attractor (black-coloured) and the corresponding basin of attraction (red-coloured) for a = 0.03.

Fig. 3

Covered market. Bifurcation diagram for a under NH/BL expectations. Initial conditions: p _H (0) = 0.1 and p _L (0) = 0.05

Fig. 4

Covered market. Chaotic attractor and their basic of attraction for a = 0.03 under NH/BL expectations

Appendix B. Endogenous fluctuations (uncovered market)

We show here that endogenous fluctuations occur when u _L raises (uncovered market). We depict bifurcation diagrams and basins of attraction only under BH/NL expectations, since the case NH/BL shows similar dynamic events.

The parameter set is: b = 1 (so that α ₄ = 6/5 and α ₅ = 2) and a = 0.5. Then, we choose α = 1.9, u _H  = 2 and let u _L increase from 0 to 2. These parameter values generate $u_L^{{BH/NL}} = 0.{2}0{51}$. We note that since α ₄ < α < α ₅, then the Nash equilibrium $E_{{BH/NL}}^{{UM}} = E_{{NH/BL}}^{{UM}}$ is locally stable or unstable depending on the relative size of u _L . Figure 5a and b show the bifurcation diagrams for u _L under BH/NL, and depict the limit point of p _L and p _H , respectively, when the initial conditions are p _H (0) = 0.1 and p _L (0) = 0.05. The Nash equilibrium is locally asymptotically stable when u _L is small. A flip bifurcation occurs at $u_L^{{BH/NL}} = 0.{2}0{51}$. Then, a two-period cycle emerges and cycles of higher periodicity occur when u _L becomes larger. It is interesting to note that in the long run the trend of the high-quality price is monotonically decreasing, while the low-quality price increases (decreases) when the index u _L is small (large). Figure 6 depicts the chaotic attractor (black-coloured) and the corresponding basin of attraction (red-coloured) for u _L  = 1.85.

Fig. 5

Uncovered market. Bifurcation diagram for u _L under BH/NL expectations. Initial conditions: p _H (0) = 0.1 and p _L (0) = 0.05

Fig. 6

Uncovered market. Chaotic attractor and their basic of attraction for u _L  = 1.85 under BH/NL expectations