The dynamics of a differentiated duopoly with quantity competition

Fanti, Luciano and Gori, Luca (2011): The dynamics of a differentiated duopoly with quantity competition.

Abstract

We analyse the dynamics of a Cournot duopoly game with heterogeneous players to investigate the effects of micro-founded differentiated products demand. The present analysis, which modifies and extends Zhang et al. (2007) (Zhang, J., Da, Q., Wang, Y., 2007. Analysis of nonlinear duopoly game with heterogeneous players. Economic Modelling 24, 138–148) and Tramontana, F., (2010) (Tramontana, F., 2010. Heterogeneous duopoly with isoelastic demand function. Economic Modelling 27, 350–357), reveals that a higher degree of product differentiation may destabilise the market equilibrium. Moreover, we show that a cascade of flip bifurcations may lead to periodic cycles and ultimately chaotic motions. Since a higher degree of product differentiation implies weaker competition, then a theoretical implication of our findings, that also constitute a policy warning for firms, is that a fiercer (weaker) competition tends to stabilise (destabilise) the unique positive Cournot-Nash equilibrium of the economy.

• Item Type: MPRA Paper
• Language: English
• Keywords: Bifurcation; Chaos; Cournot; Oligopoly; Product differentiation
• 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

References

Agiza, H.N., 1999. On the analysis of stability, bifurcation, chaos and chaos control of Kopel map. Chaos, Solitons & Fractals 10, 1909–1916.

Agiza, H.N., Hegazi, A.S., Elsadany, A.A., 2002. Complex dynamics and synchronization of duopoly game with bounded rationality. Mathematics and Computers in Simulation 58, 133–146.

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.

Agliari, A., Gardini, L., Puu, T., 2005. Some global bifurcations related to the appearance of closed invariant curves. Mathematics and Computers in Simulation 68, 201–219.

Agliari, A., Gardini, L., Puu, T., 2006. Global bifurcations in duopoly when the Cournot point is destabilized via a subcritical Neimark bifurcation. International Game Theory Review 8, 1–20.

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

Bischi, G.I., Chiarella, C., Kopel, M., Szidarovszky, F., 2010. Nonlinear Oligopolies. Stability and Bifurcations. Berlin: Springer-Verlag.

Chamberlin, E., 1933. The Theory of Monopolistic Competition. Cambridge (MA): Harvard University Press.

Correa-López, M., Naylor, R.A., 2004. The Cournot-Bertrand profit differential: a reversal result in a differentiated duopoly with wage bargaining. European Economic Review 48, 681–696.

Cournot, A., 1838. Recherches sur les Principes Mathématiques de la Théorie des Richessess. Paris: Hachette.

Den Haan, W.J., 2001. The importance of the number of different agents in a heterogeneous asset-pricing model. Journal of Economic Dynamics and Control 25, 721–746.

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.

Fanti, L., Manfredi, P., 2007. Chaotic business cycles and fiscal policy: An IS-LM model with distributed tax collection lags. Chaos, Solitons & Fractals 32, 736–744.

Fanti, L., Meccheri, N., 2011. The Cournot-Bertrand profit differential in a differentiated duopoly with unions and labour decreasing returns. Economics Bulletin 31, 233–244.

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.

Kaplan, J.L., Yorke, J.A., 1979. Chaotic behavior of multidimensional difference equations. In: Peitgen, H.O., Walther, H.O., Functional Differential Equations and Approximation of Fixed Points. New York (NY): Springer.

Kopel, M., 1996. Simple and complex adjustment dynamics in Cournot duopoly models. Chaos, Solitons & Fractals 7, 2031–2048.

Leonard, D., Nishimura, K., 1999. Nonlinear dynamics in the Cournot model without full information. Annals of Operations Research 89, 165–173.

Medio, A., 1992. Chaotic Dynamics. Theory and Applications to Economics. Cambridge (UK): Cambridge University Press.

Puu, T., 1991. Chaos in duopoly pricing. Chaos, Solitons & Fractals 1, 573–581.

Qiu, L.D., 1997. On the dynamic efficiency of Bertrand and Cournot equilibria. Journal of Economic Theory 75, 213–229.

Singh, N., Vives, X., 1984. Price and quantity competition in a differentiated duopoly. RAND Journal of Economics 15, 546–554.

Tramontana F., 2010. 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.