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


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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  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 \(T_{B}\).


  1. Agiza, H. N. (1998). Explicit stability zones for Cournot games with 3 and 4 competitors. Chaos, Solitons and Fractals, 9, 1955–1966.

    Article  Google Scholar 

  2. Agiza, H. N. (1999). On the stability, bifurcations, chaos and control of Kopel map. Chaos, Solitons and Fractals, 11, 1909–1916.

    Article  Google Scholar 

  3. Agiza, H. N., & Elsadany, A. A. (2004). Chaotic dynamics in nonlinear duopoly game with heterogenous players. Applied Mathematics and Computation, 149, 843–860.

    Article  Google Scholar 

  4. Askar, S. S. (2014). On Courmot–Bertrand competition with differentiated products. Annals of Operations Research. doi:10.1007/s10479-014-1612-8.

  5. Bertrand, J. (1883). Révue de la Théorie Mathématique de la Richesse Sociale et des Recherches sur les Principles Mathématiques de la Théorie des Richesses. Journal des Savants, 48, 499–508.

    Google Scholar 

  6. Bischi, G. I., Stefanini, L., & Gardini, L. (1998). Synchronization, intermittency and critical curves in a duopoly game. Mathematics and Computers in Simulation, 44, 559–585.

    Article  Google Scholar 

  7. Bischi, G. I., & Naimzada, A. (2000). Global analysis of a dynamic duopoly game with bounded rationality. In J. A. Filar, V. Gaitsgory, & K. Mizukami (Eds.), Advances in dynamic games and applications (Vol. 5). Basel: Birkhauser.

    Google Scholar 

  8. Bischi, G. I., & Kopel, M. (2001). Equilibrium selection in a nonlinear duopoly game with adaptive expectations. Journal of Economic Behavior and Organization, 46, 73–100.

    Article  Google Scholar 

  9. Bischi, G. I., Chiarella, C., Kopel, M., & Szidarovsky, F. (2010). Nonlinear oligopolies. Stability and bifurcations. Berlin: Springer.

    Book  Google Scholar 

  10. Bowley, A. L. (1924). The Mathematical Groundwork of Economics. Oxford: Oxford University Press.

    Google Scholar 

  11. Chamberlin, E. (1933). The theory of monopolistic competition. Cambridge, MA: Harvard University Press.

    Google Scholar 

  12. Champsaur, P., & Rochet, J. C. (1989). Multiproduct duopolists. Econometrica, 57, 533–557.

    Article  Google Scholar 

  13. Cournot, A. (1838). Recherches sur les principes mathématiques de la théorie des richesses. Paris: Hachette.

    Google Scholar 

  14. Devaney, R. L. (1989). An introduction to chaotic dynamical systems (2nd ed.). Boston: Addison Wesley.

    Google Scholar 

  15. Dixit, A. (1986). Comparative statics in oligopoly. International Economic Review, 27(1), 107–122.

    Article  Google Scholar 

  16. Fanti, L., & Gori, L. (2012). The dynamics of a differentiated duopoly with quantity competition. Economic Modelling, 29, 421–427.

    Article  Google Scholar 

  17. Fanti, L., & Gori, L. (2013). Stability analysis in a Bertrand Duopoly with different product quality and heterogeneous expectations. Journal of Industry, Competition and Trade, 13, 481–501.

    Article  Google Scholar 

  18. Fanti, L., Gori, L., & Sodini, M. (2012). Nonlinear dynamics in a Cournot duopoly with relative profit delegation. Chaos, Solitons and Fractals, 45, 1469–1478.

    Article  Google Scholar 

  19. Fershtman, C., & Judd, K. L. (1987). Equilibrium incentives in oligopoly. American Economic Review, 77, 927–940.

    Google Scholar 

  20. Friedman, J. W., & Mezzetti, C. (2002). Bounded rationality, dynamic oligopoly, and conjectural variations. Journal of Economic Behavior and Organization, 49, 287–306.

    Article  Google Scholar 

  21. Frisch, R. (1933). Monopole-polypole. La Notion de Force dans l’Economie. Til Harald Westergaard. Copenhagen: Gyldendalske.

    Google Scholar 

  22. Gandolfo, G. (2010). Economic dynamics (Forth ed.). Springer: Heidelberg.

    Google Scholar 

  23. Häckner, J. (1994). Collusive pricing in markets for vertically differentiated products. International Journal of Industrial Organization, 12, 155–177.

    Article  Google Scholar 

  24. Häckner, J. (2000). A note on price and quantity competition in differentiated oligopolies. Journal of Economic Theory, 93, 233–239.

    Article  Google Scholar 

  25. Hotelling, H. (1929). Stability in competition. Economic Journal, 39, 41–57.

    Article  Google Scholar 

  26. Klemperer, P., & Meyer, M. (1986). Price competition versus quantity competition: The role of the uncertainty. Rand Journal of Economics, 17, 618–638.

    Article  Google Scholar 

  27. Kopel, M. (1996). Simple and complex adjustment dynamics in Cournot duopoly models. Chaos, Solitons and Fractals, 12, 2031–2048.

    Article  Google Scholar 

  28. Motta, M. (1993). Endogenous quality choice: Price versus quantity competition. The Journal of Industrial Economics, 49, 113–131.

    Article  Google Scholar 

  29. Musa, M., & Rosen, S. (1978). Monopoly and product quality. Journal of Economic Theory, 18, 301–317.

    Article  Google Scholar 

  30. Padmanabhan, V., Tsetlin, I., & Van Zandt, T. (2010). Setting prices or quantity: Depends on what the seller is more uncertain about. Quantitative Marketing and Economics, 8, 35–60.

    Article  Google Scholar 

  31. Puu, T. (1991). Chaos in duopoly pricing. Chaos, Solitons and Fractals, 1, 573–581.

    Article  Google Scholar 

  32. Puu, T. (1998). The chaotic duopolists revisited. Journal of Economic Behavior and Organization, 33, 385–394.

    Article  Google Scholar 

  33. Shaked, A., & Sutton, J. (1982). Relaxing price competition through product differentiation. Review of Economic Studies, 49, 3–14.

    Article  Google Scholar 

  34. Singh, N., & Vives, X. (1984). Price and quantity competition in a differentiated duopoly. Rand Journal of Economics, 15, 546–554.

    Article  Google Scholar 

  35. Sklivas, S. D. (1987). The strategic choice of managerial incentives. Rand Journal of Economics, 18, 452–458.

    Article  Google Scholar 

  36. Spiegler, R. (2014). Bounded rationality in industrial organization. Oxford: Oxford University Press.

    Google Scholar 

  37. Symeonidis, G. (2003). Comparing Cournot and Bertrand equilibria in a differentiated duopoly with product R&D. International Journal of Industrial Organization, 21, 39–55.

    Article  Google Scholar 

  38. Tirole, J. (1988). The theory of industrial organization. Cambridge, MA: The MIT Press.

    Google Scholar 

  39. Tramontana, F. (2010). Heterogeneous duopoly with isoelastic demand function. Economic Modelling, 27, 350–357.

    Article  Google Scholar 

  40. Tremblay, C., & Tremblay, V. (2011). The Cournot–Bertrand model and the degree of product differentiation. Economic Letters, 11(3), 233–235.

    Article  Google Scholar 

  41. Tremblay, V., Tremblay, C., & Isayirawongse, K. (2013). Endogenous timing and strategic choice: The Cournot–Bertrand model. Bulletin of Economic Research, 65(4), 332–342.

    Article  Google Scholar 

  42. Zhang, J., Da, Q., & Wang, Y. (2007). Analysis of nonlinear duopoly game with heterogeneous players. Economic Modelling, 24, 138–148.

    Article  Google Scholar 

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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.



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

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

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.


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).

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

JEL Classification

  • C62
  • D43
  • L13