On price stability and the nature of product differentiation


In a spatial competition model, we analyze the stability of the Nash-price equilibrium under horizontal and vertical product differentiation, considering both homogenous and heterogeneous expectations. Regardless of the nature of product differentiation, assuming that firms behave according to an adaptive expectations rule, it is found that the Nash-price equilibrium is asymptotically stable. If at least one firm follows the gradient rule based on marginal profit, an increase in the adjustment speed turns out to be a source of complexity. Moreover, the influence of the locations on price stability depends on the nature of product differentiation and on the expectations scheme.

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  1. 1.

    As exceptions, we can cite the works of Fanti and Gori (2013) and Andaluz and Jarne (2016), who analyze the price stability in a vertically-differentiated duopoly, and Yu and Yu (2014) who investigate the dynamic price competition in the Hotelling model.

  2. 2.

    The second order conditions that ensure the existence of the maximum are the following: \( \frac{\partial^2{\prod}_1}{\partial {p_1}^2}=\frac{-1}{1-b-a}<0\kern0.5em \mathrm{and}\kern0.5em \frac{\partial^2{\prod}_2}{\partial {p_2}^2}=\frac{-1}{1-b-a}<0 \), for a ≥ 0, b ≥ 0 and a + b < 1.

  3. 3.

    The second order conditions for maximum are the following: \( \frac{\partial^2{\prod}_1}{\partial {p_1}^2}=\frac{-1}{b-a}<0\ \mathrm{and}\ \frac{\partial^2{\prod}_2}{\partial {p_2}^2}=\frac{-1}{b-a}<0 \) for b > a.

  4. 4.

    If a + b > 4, the Nash price equilibrium is given by: \( {p}_{1V}^{\ast }=\left(b-a\right)\left(b+a-2\right),{p}_{2V}^{\ast }=0 \). See Gabszewicz and Thisse (1986).

  5. 5.

    This assumption is usually made in the literature in order to simplify the dynamic analysis and to obtain formal results. See, among others, Fanti and Gori (2012), Fanti et al. (2013) and Askar (2014).

  6. 6.

    \( \left(0,0\right),\left(\frac{\left(1-b-a\right)\left(1-b+a\right)}{2},0\right)\ \mathrm{and}\ \left(0,\frac{\left(1-b-a\right)\left(1-b+a\right)}{2}\right) \) are unstable boundary equilibria of (14).

  7. 7.

    The Nash equilibrium never loses its stability through either a transcritical or a Neimark-Sacker bifurcation.

  8. 8.

    The maximum Lyapunov exponent showed in Figure 6 and Figure 7 has been obtained from the method proposed by Sandri (1996).

  9. 9.

    \( \left(0,0\right)\ \mathrm{and}\ \left(\frac{\left(b-a\right)\left(a+b\right)}{2},0\right) \) are two unstable boundary fixed points of (20).

  10. 10.

    If firm 1 is a gradient player and firm 2 is an adaptive player, the dynamic system will be \( {T}_H^{GA}:\left\{\begin{array}{l}{p}_{1,t+1}={p}_{1,t}+\alpha {p}_{1,t}\frac{p_{2,t}-2{p}_{1,t}+\left(1-b-a\right)\left(1-b+a\right)}{2\left(1-b-a\right)}\\ {}{p}_{2,t+1}={\theta}_2{p}_{2,t}+\left(1-{\theta}_2\right)\frac{p_{2,t}-2{p}_{1,t}+\left(1-b-a\right)\left(1-b+a\right)}{2\left(1-b-a\right)}\end{array}\right. \) being the equilibrium points \( {E}_H^{\ast } \) and the unstable boundary equilibrium\( \left(0,\frac{\left(1-b-a\right)\left(1+b-a\right)}{2}\right) \).

  11. 11.

    \( \left(\frac{\left(1-b-a\right)\left(1-b+a\right)}{2},0\right) \) is an unstable boundary fixed point of (24).

  12. 12.

    If firm 1 is a gradient player and firm 2 is an adaptive player, the dynamic system will be \( {T}_V^{GA}:\left\{\begin{array}{l}{p}_{1,t+1}={p}_{1,t}+\alpha {p}_{1,t}\frac{p_{2,t}-2{p}_{1,t}+\left(b+a\right)\left(b-a\right)}{2\left(b-a\right)}\\ {}{p}_{2,t+1}={\theta}_2{p}_{2,t}+\left(1-{\theta}_2\right)\frac{p_{2,t}-2{p}_{1,t}+\left(b+a\right)\left(b-a\right)}{2\left(b-a\right)}\end{array}\right. \)being the Nash equilibrium \( {E}_V^{\ast } \) the only steady state.

  13. 13.

    \( \left(\frac{\left(b-a\right)\left(b+a\right)}{2},0\right) \) is an unstable boundary fixed point of (28).


  1. Agiza HN (1998) Explicit stability zones for Cournot games with 3 and 4 competitors. Chaos Soliton. Fract. 9:1955–1966

    Article  Google Scholar 

  2. Agiza HN (1999) On the analysis of stability, bifurcation, chaos and chaos control of Kopel map. Chaos Soliton. Fract. 10:1909–1916

    Article  Google Scholar 

  3. Agiza HN, Elsadany AA (2004) Chaotic dynamics in nonlinear duopoly game with heterogenous players. Appl Math Comput 149:843–860

    Google Scholar 

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

    Article  Google Scholar 

  5. Andaluz J, Elsadany AA, Jarne G (2017) Nonlinear Cournot and Bertrand-type dynamic triopoly with differentiated products and heterogeneous expectations. Math Comput Simulat 132:86–99

    Article  Google Scholar 

  6. Askar SS (2014) On Cournot-Bertrand competition with differentiated products. Ann Oper Res 223:81–93

    Article  Google Scholar 

  7. Bischi GI, Kopel M (2001) Equilibrium selection in a nonlinear duopoly game with adaptive expectations. J Econ Behav Organ 46:73–100

    Article  Google Scholar 

  8. Bischi GI, Naimzada A (2000) Global analysis of a dynamic duopoly game with bounded rationalit. In: Filar JA, Gaitsgory V, Mizukami K (eds) Advances in Dynamic Games and Applications, vol 5, Birkhauser

  9. Bischi GI, Chiarella C, Kopel M, Scidarovszky F (2010) Nonlinear oligopolies. Stability and bifurcations. Springer-Verlag, Berlin Heidelberg

    Google Scholar 

  10. Chamberlin E, 1933. The theory of monopolistic competition. Harvard University press. Cambridge (MA)

  11. D’Aspremont C, Gabszewicz JJ, Thisse J-F (1979) On Hotelling’s stability in competition. Econometrica 47:1145–1150

    Article  Google Scholar 

  12. Dixit AK (1986) Comparative statics for oligopoly. Int Econ Rev 27:107–122

    Article  Google Scholar 

  13. Elsadany AA, Agiza HN, Elabbasy EM (2013) Complex dynamics and chaos control of heterogeneous quadropoly game. Appl Math Comput 219:11110–11118.

  14. Fanti L, Gori L (2012) The dynamics of a differentiated duopoly with quantity competition. Econ Model 29:421–427

    Article  Google Scholar 

  15. Fanti L, Gori L (2013) Stability analysis in a Bertrand duopoly with different product quality and heterogeneous expectations. J Ind Compet Trade 13:481–501

    Article  Google Scholar 

  16. Fanti L, Gori L, Mammana C, Michetti E (2013) The dynamics of a Bertrand duopoly with differentiated products: synchronization, intermittency and global dynamics. J Chaos Soliton Fract 52:73–86

    Article  Google Scholar 

  17. Fudenberg D, Tirole J (1984) The fat-cat effect, the puppy-dog ploy, and the lean and hungry look. Am Econ Rev 74(2):361–366

    Google Scholar 

  18. Gabszewicz J, Thisse J-F (1986) On the nature of competition with differentiated products. Econ J 96:160–172

    Article  Google Scholar 

  19. Gandolfo G (2010) Economic dynamics. Forth. Springer, Heidelberg

    Google Scholar 

  20. Hotelling H (1929) Stability in competition. Econ J 39:41–57

    Article  Google Scholar 

  21. Kopel M (1996) Simple and complex adjustment dynamics in Cournot duopoly models. Chaos Soliton. Fract. (12):2031–2048

  22. Lewis M (2008) Price dispersion and competition with differentiated sellers. J Ind Econ 56(3):654–678

    Article  Google Scholar 

  23. Liu Q, Zhang D (2013) Dynamic pricing competition with strategic costumers under vertical product differentiation. Manag Sci 59(1):84–101

    Article  Google Scholar 

  24. Naimzada AK, Tramontana F (2012) Dynamic properties of a Cournot-Bertrand duopoly game with differentiated products. Econ Model 29(4):1436–1439

    Article  Google Scholar 

  25. Puu T (1991) Chaos in duopoly pricing. Chaos Soliton Fract 1:573–581

    Article  Google Scholar 

  26. Puu T (1998) The chaotic duopolists revisited. J Econ Behav Organ 33:385–394

    Article  Google Scholar 

  27. Sandri M (1996) Numerical calculation of Lyapunov exponents. Math J 6(3):78–84

    Google Scholar 

  28. Tramontana F (2010) Heterogeneous duopoly with isoelastic demand function. Econ Model 27:350–357

    Article  Google Scholar 

  29. Tremblay CH, Tremblay VJ (2011) The Cournot-Bertrand model and the degree of product differentiation. Econ Lett 111(3):233–235

    Article  Google Scholar 

  30. Yu W, Yu Y (2014) The complexion of dynamic duopoly game with horizontal differentiated products. Econ Model 41:289–297

    Article  Google Scholar 

  31. Zhang J, Da Q, Wang Y (2009) The dynamics of Bertrand model with bounded rationality. Chaos Soliton. Fract. 39:2048–2055

    Article  Google Scholar 

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This study was funded by the Spanish Ministry of Economics and Competitiveness (ECO2016–74940-P) and the Government of Aragon and FEDER (S10/2016 and S13/2016 Consolidated Groups) and S40-17R Reference Group.

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Correspondence to Joaquín Andaluz.

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This paper benefited from comments made by two anonymous referees of this journal.

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Andaluz, J., Jarne, G. On price stability and the nature of product differentiation. J Evol Econ 29, 741–762 (2019). https://doi.org/10.1007/s00191-018-0584-2

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  • Horizontal product differentiation
  • Vertical product differentiation
  • Bounded rationality
  • Dynamic stability

JEL classification

  • C62
  • D43
  • L13