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

  • Luciano Fanti
  • Luca Gori

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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Paper provided by Dipartimento di Economia e Management (DEM), University of Pisa, Pisa, Italy in its series Discussion Papers with number 2011/120.

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Date of creation: 09 Jan 2011
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Handle: RePEc:pie:dsedps:2011/120
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  1. Tramontana, Fabio, 2010. "Heterogeneous duopoly with isoelastic demand function," Economic Modelling, Elsevier, vol. 27(1), pages 350-357, January.
  2. Vives, Xavier, 1985. "On the efficiency of Bertrand and Cournot equilibria with product differentation," Journal of Economic Theory, Elsevier, vol. 36(1), pages 166-175, June.
  3. 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, vol. 320(C), pages 512-524.
  4. 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, vol. 27(1), pages 107-22, February.
  5. Ghosh, Arghya & Mitra, Manipushpak, 2010. "Comparing Bertrand and Cournot in mixed markets," Economics Letters, Elsevier, vol. 109(2), pages 72-74, November.
  6. Zhang, Jixiang & Da, Qingli & Wang, Yanhua, 2007. "Analysis of nonlinear duopoly game with heterogeneous players," Economic Modelling, Elsevier, vol. 24(1), pages 138-148, January.
  7. Nirvikar Singh & Xavier Vives, 1984. "Price and Quantity Competition in a Differentiated Duopoly," RAND Journal of Economics, The RAND Corporation, vol. 15(4), pages 546-554, Winter.
  8. Luciano Fanti & Nicola Meccheri, 2010. "The Cournot-Bertrand profit differential in a differentiated duopoly with unions and labour decreasing returns," Discussion Papers 2010/107, Dipartimento di Economia e Management (DEM), University of Pisa, Pisa, Italy.
  9. 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, vol. 48(3), pages 681-696, June.
  10. Bischi, Gian-Italo & Stefanini, Luciano & Gardini, Laura, 1998. "Synchronization, intermittency and critical curves in a duopoly game," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 44(6), pages 559-585.
  11. Häckner, Jonas, 1999. "A Note on Price and Quantity Competition in Differentiated Oligopolies," Research Papers in Economics 1999:9, Stockholm University, Department of Economics.
  12. Medio,Alfredo & Gallo,Giampaolo, 1995. "Chaotic Dynamics," Cambridge Books, Cambridge University Press, number 9780521484619.
  13. Qiu, Larry D., 1997. "On the Dynamic Efficiency of Bertrand and Cournot Equilibria," Journal of Economic Theory, Elsevier, vol. 75(1), pages 213-229, July.
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