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The dynamics of a Bertrand duopoly with differentiated products and bounded rational firms revisited

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  • Fanti, Luciano
  • Gori, Luca

Abstract

We revisit the study of the dynamics of a duopoly game à la Bertrand with horizontal product differentiation and bounded rational firms analysed by Zhang et al. (2009), (Zhang, J., Da, Q., Wang, Y., 2009. The dynamics of Bertrand model with bounded rationality. Chaos, Solitons and Fractals 39, 2048–2055), by introducing sound microeconomic foundations. We study how an increase in the relative degree of product differentiation affects the stability of the unique positive Bertrand-Nash equilibrium, in the case of both linear and non-linear costs. We show that an increase in either the degree of substitutability or complementarity between goods of different variety may destabilise the equilibrium of the two-dimensional system through a period-doubling bifurcation. Moreover, by using numerical simulations (i.e., phase portraits, sensitive dependence on initial conditions and Lyapunov exponents), we find that a “quasi-periodic” route to chaos and a large gamma of strange attractors for the cases of both substitutability and complementarity can occur.

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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 33268.

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Date of creation: 09 Sep 2011
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Handle: RePEc:pra:mprapa:33268

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Keywords: Bifurcation; Chaos; Differentiated products; Duopoly; Price competition;

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  1. Agiza, H.N. & Elsadany, A.A., 2003. "Nonlinear dynamics in the Cournot duopoly game with heterogeneous players," Physica A: Statistical Mechanics and its Applications, Elsevier, Elsevier, vol. 320(C), pages 512-524.
  2. Luciano Fanti & Nicola Meccheri, 2011. "The Cournot-Bertrand profit differential in a differentiated duopoly with unions and labour decreasing returns," Economics Bulletin, AccessEcon, vol. 31(1), pages 233-244.
  3. Bischi, Gian-Italo & Stefanini, Luciano & Gardini, Laura, 1998. "Synchronization, intermittency and critical curves in a duopoly game," Mathematics and Computers in Simulation (MATCOM), Elsevier, Elsevier, vol. 44(6), pages 559-585.
  4. Zhang, Jixiang & Da, Qingli & Wang, Yanhua, 2007. "Analysis of nonlinear duopoly game with heterogeneous players," Economic Modelling, Elsevier, Elsevier, vol. 24(1), pages 138-148, January.
  5. Medio,Alfredo & Gallo,Giampaolo, 1995. "Chaotic Dynamics," Cambridge Books, Cambridge University Press, Cambridge University Press, number 9780521484619.
  6. Tramontana, Fabio, 2010. "Heterogeneous duopoly with isoelastic demand function," Economic Modelling, Elsevier, Elsevier, vol. 27(1), pages 350-357, January.
  7. Hackner, Jonas, 2000. "A Note on Price and Quantity Competition in Differentiated Oligopolies," Journal of Economic Theory, Elsevier, Elsevier, vol. 93(2), pages 233-239, August.
  8. Ghosh, Arghya & Mitra, Manipushpak, 2010. "Comparing Bertrand and Cournot in mixed markets," Economics Letters, Elsevier, Elsevier, vol. 109(2), pages 72-74, November.
  9. Dixit, Avinash K, 1986. "Comparative Statics for Oligopoly," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 27(1), pages 107-22, February.
  10. Qiu, Larry D., 1997. "On the Dynamic Efficiency of Bertrand and Cournot Equilibria," Journal of Economic Theory, Elsevier, Elsevier, vol. 75(1), pages 213-229, July.
  11. Lopez, Monica Correa & Naylor, Robin A., 2004. "The Cournot-Bertrand profit differential: A reversal result in a differentiated duopoly with wage bargaining," European Economic Review, Elsevier, Elsevier, vol. 48(3), pages 681-696, June.
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Cited by:
  1. Luciano Fanti & Luca Gori, 2011. "Stability analysis in a Bertrand duopoly with different product quality and heterogeneous expectations," Discussion Papers, Dipartimento di Economia e Management (DEM), University of Pisa, Pisa, Italy 2011/122, Dipartimento di Economia e Management (DEM), University of Pisa, Pisa, Italy.

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