Fanti, Luciano and Gori, Luca (2011): The dynamics of a Bertrand duopoly with differentiated products and bounded rational firms revisited.

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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 BertrandNash equilibrium, in the case of both linear and nonlinear 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 twodimensional system through a perioddoubling bifurcation. Moreover, by using numerical simulations (i.e., phase portraits, sensitive dependence on initial conditions and Lyapunov exponents), we find that a “quasiperiodic” route to chaos and a large gamma of strange attractors for the cases of both substitutability and complementarity can occur.
Item Type:  MPRA Paper 

Original Title:  The dynamics of a Bertrand duopoly with differentiated products and bounded rational firms revisited 
English Title:  The dynamics of a Bertrand duopoly with differentiated products and bounded rational firms revisited 
Language:  English 
Keywords:  Bifurcation; Chaos; Differentiated products; Duopoly; Price competition 
Subjects:  L  Industrial Organization > L1  Market Structure, Firm Strategy, and Market Performance > L13  Oligopoly and Other Imperfect Markets D  Microeconomics > D4  Market Structure, Pricing, and Design > D43  Oligopoly and Other Forms of Market Imperfection C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C62  Existence and Stability Conditions of Equilibrium 
Item ID:  33268 
Depositing User:  Luca Gori 
Date Deposited:  09. Sep 2011 14:51 
Last Modified:  31. Mar 2015 06:41 
References:  Agiza, H.N., Elsadany, A.A., 2003. Nonlinear dynamics in the Cournot duopoly game with heterogeneous players. Physica A 320, 512–524. Agiza, H.N., Elsadany, A.A., 2004. Chaotic dynamics in nonlinear duopoly game with heterogeneous players. Applied Mathematics and Computation 149, 843–860. Andronov, A.A., Leontovich, E.A., Gordon, I.E., Maier, A.G., 1971. The Theory of Bifurcations of Dynamical Systems on a Plane. New York (NY): Wiley. Bertrand, J., 1883. Théorie mathématique de la richesse sociale. Journal des Savants 48, 499–508. Bergé, P., Pomeau, Y., Vidal, C., 1986. Order Within Chaos. Towards a Deterministic Approach to Turbulence. New York (NY): WileyInterscience. Bischi, G.I., Gardini, L., 2000. Global properties of symmetric competition models with riddling and blowout phenomena. Discrete Dynamics in Nature and Society 5, 149–160. Bischi, G.I., Naimzada, A., 1999. Global analysis of a dynamic duopoly game with bounded rationality. Advanced in Dynamics Games and Application, vol. 5. Birkhauser, Basel. Bischi, G.I., Gallegati, M., Naimzada, A., 1999. Symmetrybreaking bifurcations and representative firm in dynamic duopoly games. Annals of Operations Research 89, 253–272. 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. Chamberlin, E., 1933. The Theory of Monopolistic Competition. Cambridge (MA): Harvard University Press. CorreaLópez, M., Naylor, R.A., 2004. The CournotBertrand profit differential: a reversal result in a differentiated duopoly with wage bargaining. European Economic Review 48, 681–696. Curry, J., Yorke, J., 1977. A transition from Hopf bifurcation to chaos: computer experiments with maps in R2. In: The Structure of Attractors in Dynamical Systems, Springer Notes in Mathematics, vol. 668, 48–56. Dixit, A.K., 1979. A model of duopoly suggesting a theory of entry barriers. Bell Journal of Economics 10, 20–32. Dixit, A.K., 1986. Comparative statics for oligopoly. International Economic Review 27, 107–122. Elhadj Z., Sprott, J.C., 2008. A minimal 2D quadratic map with quasiperiodic route to chaos. International Journal of Bifurcations & Chaos, 1567–1577. Elhadj Z., Sprott, J.C., 2010. Quadratic maps of the plane: tutorial and review. In: 2D Quadratic Maps and 3D ODE Systems: A Rigorous Approach. World Scientific Series on Nonlinear Science Series A. Fanti, L., Meccheri, N., 2011. The CournotBertrand profit differential in a differentiated duopoly with unions and labour decreasing returns. Economics Bulletin 31, 233–244. Feigenbaum, M.J., Kadanoff, L.P., Shenker, S.J., 1982. Quasiperiodicity in dissipative systems: a renormalization group analysis. Physica D, 370–386. Gandolfo, G., 2010. Economic Dynamics. Forth Edition. Heidelberg: Springer. Gosh, A., Mitra, M., 2010. Comparing Bertrand and Cournot in mixed markets. Economics Letters 109, 72–74. Häckner, J., 2000. A note on price and quantity competition in differentiated oligopolies. Journal of Economic Theory 93, 233–239. Hotelling, H., 1929. Stability in competition. Economic Journal 39, 41–57. Qiu, L.D., 1997. On the dynamic efficiency of Bertrand and Cournot equilibria. Journal of Economic Theory 75, 213–229. Medio, A., 1992. Chaotic Dynamics. Theory and Applications to Economics. Cambridge (UK): Cambridge University Press. Moon, H.T., 1997. Twofrequency motion to chaos with fractal dimension . Physical Review Letters 79, 403–406. Ruelle, D., Takens, F., 1971. On the nature of turbulence. Communications in Mathematical Physics 20, 167–192. Shilnikov, L., Shilnikov, A., Turaev, D., Chua, L., 2001. Methods of Qualitative Theory in Nonlinear Dynamics. Parts II. Singapore: World Scientific. Singh, N., Vives, X., 1984. Price and quantity competition in a differentiated duopoly. RAND Journal of Economics 15, 546–554. Vives, X., 1985. On the efficiency of Bertrand and Cournot equilibria with product differentiation. Journal of Economic Theory 36, 166–175. Tramontana, F., 2009. Heterogeneous duopoly with isoelastic demand function. Economic Modelling 27, 350–357. Zhang, J., Da, Q., Wang, Y., 2007. Analysis of nonlinear duopoly game with heterogeneous players. Economic Modelling 24, 138–148. Zhang, J., Da, Q., Wang, Y., 2009. The dynamics of Bertrand model with bounded rationality. Chaos, Solitons and Fractals 39, 2048–2055. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/33268 