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Stability of vertically differentiated Cournot and Bertrand-type models when firms are boundedly rational

Abstract

This paper compares the dynamic of Cournot and Bertrand duopolies with vertical product differentiation and under bounded rationality. We find that an increase in the product differentiation degree destabilizes the Nash equilibrium under quantity competition, while the Bertrand–Nash equilibrium becomes more stable. From a global dynamic analysis, we show that an increase in the firms’ adjustment speed constitutes a source of complexity in both models. There is a cascade of flip bifurcations leading to increasingly complex attractors, and there are global bifurcations generating complex basins of attraction.

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Notes

  1. 1.

    Otherwise, the Cournot and Bertrand models can be interpreted as conjectural variation models (Bowley 1924; Frisch 1933). In Cournot’s original model, each firm’s conjecture is that the other firms are satisfied to continue selling their current quantity of output. However, from a purely game-theory perspective, the conjectural variations approach is theoretically unsatisfactory (see Tirole 1988).

  2. 2.

    Horizontal product differentiation has been developed through the non-address approach, dating back to Bowley (1924) and Chamberlin (1933), and the address models, dating back to Hotelling (1929).

  3. 3.

    The original version of the model was introduced by Musa and Rosen (1978).

  4. 4.

    Vertical product differentiation studies traditionally distinguish between two alternative cost formulations: (1) they assume that improvements in quality imply a higher fixed cost, while marginal costs need not differ much across firms (Shaked and Sutton 1982; Häckner 1994); and (2) they assume that such improvements lead to an increase in the variable production cost (Musa and Rosen 1978; Champsaur and Rochet 1989).

  5. 5.

    We deduce that \(E_0\), \(E_1\) and \(E_2\) are unstable equilibria. \(E_0\) is a repelling node, and \(E_1\) and \(E_2\) are either repelling nodes or saddle points, depending on the parameter values. If \(\alpha \overline{{v}}s<2 (\alpha \overline{{v}}s>2)\) then \(E_1\) is a saddle point (a repelling node) and if \(\alpha \overline{{v}}<2 (\alpha \overline{{v}}>2)\) then \(E_2\) is a saddle point (a repelling node).

  6. 6.

    This bifurcation corresponds to the bifurcation of the map \(T_{C}\) described in footnote 5 that transforms the saddle point \(E_{1}\) into a repelling node.

  7. 7.

    We deduce that \(E_0\) and \(E_1\) are unstable equilibria, with \(E_0\) being a saddle point. \(E_1\) is a repelling node if \(\alpha \overline{{v}}>2\) and a saddle point if \(\alpha \overline{{v}}<2.\)

  8. 8.

    Note that in a vertical product differentiation context this axis has no economic meaning. However, it must be considered to analyze the global dynamic of \(T_{B}\).

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Acknowledgments

The authors wish to thank the Spanish Ministry of Economics and Competitiveness (ECO2012-34828 and ECO2013-41353-P) and the Government of Aragon and FEDER (consolidated groups S10 and S13) for their financial support. This paper benefited from comments made by two anonymous referees of this journal.

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Correspondence to Joaquín Andaluz.

Appendix

Appendix

From (8) and (9) the trace and the determinant of the matrix \(JT_C (E_C^{*} )\) can be given by:

$$\begin{aligned} \left\{ {\begin{array}{l} Tr=2-2\alpha (sq_1^{*} +q_2^{*} )=2-4\frac{\alpha s^{2}\overline{{v}}}{4s-1} \\ Det=Tr-1+M,\hbox { with }M=\alpha ^{2}(4s-1)q_1^{*} q_2^{*} =\alpha ^{2}\frac{(2s-1)s\overline{{v}}^{2}}{4s-1}>0 \\ \end{array}} \right. \end{aligned}$$
(32)

Thus, the eigenvalues of the matrix \(JT_C (E_C^{*} )\) in terms of Tr and M are:

$$\begin{aligned} \lambda _1 =\frac{Tr}{2}-\sqrt{\left( {\frac{Tr-2}{2}} \right) ^{2}-M}\le \lambda _2 =\frac{Tr}{2}+\sqrt{\left( {\frac{Tr-2}{2}} \right) ^{2}-M} \end{aligned}$$

Analyzing \(\lambda _1 \) and \(\lambda _2 \), we can characterize the equilibrium \(E_C^{*} \) and subsequently the evolution of the solution trajectories of the dynamic system in a neighborhood of this equilibrium. The following proposition is established.

Proposition 4

Under Cournot competition:

(i) If \(\alpha \overline{{v}}<(\alpha \overline{{v}})_3^C =\frac{2s}{2s-1}-\frac{\sqrt{s(4s^{3}-8s^{2}+6s-1}}{s(2s-1)}\), then \(0<\lambda _1 <\lambda _2 <1\) (region A in Fig. 1).

(ii) If \((\alpha \overline{{v}})_3^C <\alpha \overline{{v}}<(\alpha \overline{{v}})_1^C \), then \(-1<\lambda _1 <0<\lambda _2 <1\) (region B in Fig. 1).

(iii) If \(\alpha \overline{{v}}=(\alpha \overline{{v}})_1^C \), then \(-1=\lambda _1 <\lambda _2 <1\), and there is a Flip bifurcation (curve \((\alpha \overline{{v}})_F^C =(\alpha \overline{{v}})_1^C \) in Fig. 1).

(iv) If \((\alpha \overline{{v}})_1^C <\alpha \overline{{v}}<(\alpha \overline{{v}})_2^C \), then \(\lambda _1 <-1<\lambda _2 <1\) (region C in Fig. 1).

(v) If \((\alpha \overline{{v}})_2^C <\alpha \overline{{v}}\), then \(\lambda _1 <\lambda _2 <-1\) (region D in Fig. 1).

(vi) There is neither a transcritical bifurcation nor a Neimark–Sacker bifurcation.

Proof

We deduce the following:

  1. (a)

    \(\left( {\frac{Tr-2}{2}} \right) ^{2}-M\ge 0\) for all \(s > 1\), \(\overline{{v}}>0\) and \(\alpha >0\) given that:

    $$\begin{aligned} (Tr-2)^{2}-4M\ge & {} 0\Leftrightarrow s^{2}q_1^{*2}+q_2^{*2}-(2s-1)q_1^{*} q_2^{*} \\= & {} \left( {s^{2}-\frac{(2s-1)^{2}}{4}} \right) q_1^{*2}+\left( {q_2^{*} -\frac{2s-1}{s}q_1^{*} } \right) ^{2}\ge 0 \end{aligned}$$

    Therefore, \(\lambda _1 \hbox { and }\lambda _2 \) cannot be complex values. Consequently, there is no Neimark–Sacker bifurcation.

  2. (b)

    As \(M > 0\), \(\forall s>1,\alpha >0,\overline{{v}}>0\), then \(\lambda _2 <1\), and there is no transcritical bifurcation.

  3. (c)

    If \(Tr<\frac{-M}{2}\), then \(\lambda _1 <-1<\lambda _2 <1\), and the Nash equilibrium is a saddle point.

  • Substituting (32) into the inequality \(Tr<\frac{-M}{2}\) we deduce \((\alpha \overline{{v}})_1^C <\alpha \overline{{v}}<(\alpha \overline{{v}})_2^C \), defining region C in Fig. 1, being:

    $$\begin{aligned} (\alpha \overline{{v}})_1^C= & {} \frac{4s}{2s-1}-\frac{2\sqrt{s(4s^{3}-8s^{2}+6s-1)}}{s(2s-1)}\hbox { and }\\ (\alpha \overline{{v}})_2^C= & {} \frac{4s}{2s-1}+\frac{2\sqrt{s(4s^{3}-8s^{2}+6s-1)}}{s(2s-1)} \end{aligned}$$
  1. (d)

    If \(\frac{-M}{2}<Tr<2-2\sqrt{M}\) and \(M>4\), then \(\lambda _1 <\lambda _2 <-1\), and for any initial condition near to equilibrium, the trajectory moves away in a fluctuating manner.

  • Substituting (32) into the above inequalities, we deduce \((\alpha \overline{{v}})_2^C <\alpha \overline{{v}}\), leading to region D in Fig. 1.

  1. (e)

    If \(Tr=\frac{-M}{2}\) and \(M<4\), then \(\lambda _1 =-1<\lambda _2 <1\). In this case, there is a flip bifurcation.

  • Substituting (32) into the above expressions, we deduce \(\alpha \overline{{v}}=(\alpha \overline{{v}})_1^C \), obtaining the curve \((\alpha \overline{{v}})_F^C =(\alpha \overline{{v}})_1^C \) in Fig. 1.

  1. (f)

    In the stability area (\(\alpha \overline{{v}}<(\alpha \overline{{v}})_1^C\)), according to the sign of the eigenvalues, we can distinguish the following three subregions:

    • If \(1-M<Tr<2-2\sqrt{M}\) and \(0<M<1\), then \(0<\lambda _1 <\lambda _2 <1\) and the Nash equilibrium is a stable node. Substituting (32) into these expressions, it follows that \(\alpha \overline{{v}}<(\alpha \overline{{v}})_3^C =\frac{2s}{2s-1}-\frac{\sqrt{s(4s^{3}-8s^{2}+6s-1)}}{s(2s-1)}\), giving rise to region A in Fig. 1.

    • If \(-\frac{M}{2}<Tr<1-M\), then \(-1<\lambda _1 <0<\lambda _2 <1\) and the Nash equilibrium is asymptotically stable; however, there is no stable node. As a negative eigenvalue exists (with modulus lower than one), there may appear to be a fluctuating convergence. Introducing (32) into the above expressions holds that \((\alpha \overline{{v}})_3^C <\alpha \overline{{v}}<(\alpha \overline{{v}})_1^C \), leading to region B in Fig. 1.

    • If \(1-M<Tr<2-2\sqrt{M}\) and \(1<M<2\) or \(\frac{-M}{2}<Tr<2-2\sqrt{M}\) and \(2<M<4\), then \(-1<\lambda _1 <\lambda _2 <0\) and for any initial condition near to equilibrium, the trajectory converges to it in a fluctuating manner.

      Substituting (32) into these expressions, we conclude that the set of parameters \((s, \alpha , \overline{{v}})\), such that the above inequalities hold, is empty. \(\square \)

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Andaluz, J., Jarne, G. Stability of vertically differentiated Cournot and Bertrand-type models when firms are boundedly rational. Ann Oper Res 238, 1–25 (2016). https://doi.org/10.1007/s10479-015-2057-4

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Keywords

  • Vertical product differentiation
  • Bertrand competition
  • Cournot competition
  • Bounded rationality
  • Dynamic stability
  • Bifurcation

JEL Classification

  • C62
  • D43
  • L13