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The dynamics of a Bertrand duopoly with differentiated products: Synchronization, intermittency and global dynamics

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  • Fanti, Luciano
  • Gori, Luca
  • Mammana, Cristiana
  • Michetti, Elisabetta

Abstract

We study the dynamics of a duopoly game à la Bertrand with horizontal product differentiation as proposed by Zhang et al. (2009) [35] by introducing opportune microeconomic foundations. The final model is described by a two-dimensional non-invertible discrete time dynamic system T. We show that synchronized dynamics occurs along the invariant diagonal being T symmetric; furthermore, we show that when considering the transverse stability, intermittency phenomena are exhibited. In addition, we discuss the transition from simple dynamics to complex dynamics and describe the structure of the attractor by using the critical lines technique. We also explain the global bifurcations causing a fractalization in the basin of attraction. Our results aim at demonstrating that an increase in either the degree of substitutability or complementarity between products of different varieties is a source of complexity in a duopoly with price competition.

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  • Fanti, Luciano & Gori, Luca & Mammana, Cristiana & Michetti, Elisabetta, 2013. "The dynamics of a Bertrand duopoly with differentiated products: Synchronization, intermittency and global dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 52(C), pages 73-86.
    • Handle: RePEc:eee:chsofr:v:52:y:2013:i:c:p:73-86
    DOI: 10.1016/j.chaos.2013.04.002
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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, vol. 320(C), pages 512-524.
    2. Avinash Dixit, 1979. "A Model of Duopoly Suggesting a Theory of Entry Barriers," Bell Journal of Economics, The RAND Corporation, vol. 10(1), pages 20-32, Spring.
    3. Brianzoni, Serena & Mammana, Cristiana & Michetti, Elisabetta, 2009. "Nonlinear dynamics in a business-cycle model with logistic population growth," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 717-730.
    4. 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.
    5. Motta, Massimo, 1993. "Endogenous Quality Choice: Price vs. Quantity Competition," Journal of Industrial Economics, Wiley Blackwell, vol. 41(2), pages 113-131, June.
    6. Avner Shaked & John Sutton, 1982. "Relaxing Price Competition Through Product Differentiation," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 49(1), pages 3-13.
    7. Bignami Fernando & Agliari Anna, 2010. "Synchronization and On-Off Intermittency Phenomena in a Market Model with Complementary Goods and Adaptive Expectations," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 14(2), pages 1-31, March.
    8. 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.
    9. Zhang, Jixiang & Da, Qingli & Wang, Yanhua, 2009. "The dynamics of Bertrand model with bounded rationality," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2048-2055.
    10. Jaskold Gabszewicz, J. & Thisse, J. -F., 1979. "Price competition, quality and income disparities," Journal of Economic Theory, Elsevier, vol. 20(3), pages 340-359, June.
    11. Giant-italo Bischi & Laura Gardini, 2000. "Global properties of symmetric competition models with riddling and blowout phenomena," Discrete Dynamics in Nature and Society, Hindawi, vol. 5, pages 1-12, January.
    12. 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.
    13. Dana, Rose-Anne & Montrucchio, Luigi, 1986. "Dynamic complexity in duopoly games," Journal of Economic Theory, Elsevier, vol. 40(1), pages 40-56, October.
    14. 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.
    15. Gian Italo Bischi & Carl Chiarella & Michael Kopel & Ferenc Szidarovszky, 2010. "Nonlinear Oligopolies," Springer Books, Springer, number 978-3-642-02106-0, December.
    16. 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 107-122, February.
    17. Agliari, Anna & Chiarella, Carl & Gardini, Laura, 2006. "A re-evaluation of adaptive expectations in light of global nonlinear dynamic analysis," Journal of Economic Behavior & Organization, Elsevier, vol. 60(4), pages 526-552, August.
    18. Tramontana, Fabio, 2010. "Heterogeneous duopoly with isoelastic demand function," Economic Modelling, Elsevier, vol. 27(1), pages 350-357, January.
    19. 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.
    20. Shaked, Avner & Sutton, John, 1983. "Natural Oligopolies," Econometrica, Econometric Society, vol. 51(5), pages 1469-1483, September.
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    2. Fanti, Luciano & Gori, Luca & Mammana, Cristiana & Michetti, Elisabetta, 2014. "Local and global dynamics in a duopoly with price competition and market share delegation," Chaos, Solitons & Fractals, Elsevier, vol. 69(C), pages 253-270.
    3. Luca Gori & Nicolò Pecora & Mauro Sodini, 2017. "Market share delegation in a nonlinear duopoly with quantity competition: the role of dynamic entry barriers," Journal of Evolutionary Economics, Springer, vol. 27(5), pages 905-931, November.
    4. Zhou, Jie & Zhou, Wei & Chu, Tong & Chang, Ying-xiang & Huang, Meng-jia, 2019. "Bifurcation, intermittent chaos and multi-stability in a two-stage Cournot game with R&D spillover and product differentiation," Applied Mathematics and Computation, Elsevier, vol. 341(C), pages 358-378.
    5. Li, Hui & Zhou, Wei & Elsadany, A. A & Chu, Tong, 2021. "Stability, multi-stability and instability in Cournot duopoly game with knowledge spillover effects and relative profit maximization," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    6. Ahmed, E. & Elsadany, A.A. & Puu, Tonu, 2015. "On Bertrand duopoly game with differentiated goods," Applied Mathematics and Computation, Elsevier, vol. 251(C), pages 169-179.
    7. Puu, Tonu & Tramontana, Fabio, 2019. "Can Bertrand and Cournot oligopolies be combined?," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 97-107.
    8. Gori, Luca & Sodini, Mauro, 2017. "Price competition in a nonlinear differentiated duopoly," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 557-567.
    9. Liu, Yuxia & Zhou, Wei & Wang, Qian, 2022. "Global dynamics of an oligopoly competition model with isoelastic demand and strategic delegation," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    10. Liu, Cui & He, Rui-chun & Zhou, Wei & Li, Hui, 2021. "Dynamic analysis of airline bidding game based on nonlinear cost," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 565(C).
    11. Ciprian Rusescu & Mihai Daniel - Roman, 2021. "Dynamic Behaviour In A Bertrand Model With Bounded Rational Players," Annals - Economy Series, Constantin Brancusi University, Faculty of Economics, vol. 3, pages 45-55, June.
    12. S. S. Askar & Mona F. EL-Wakeel & M. A. Alrodaini, 2018. "Exploration of Complex Dynamics for Cournot Oligopoly Game with Differentiated Products," Complexity, Hindawi, vol. 2018, pages 1-13, February.
    13. Brianzoni, Serena & Gori, Luca & Michetti, Elisabetta, 2015. "Dynamics of a Bertrand duopoly with differentiated products and nonlinear costs: Analysis, comparisons and new evidences," Chaos, Solitons & Fractals, Elsevier, vol. 79(C), pages 191-203.
    14. Li, Hui & He, Ruichun & Zhou, Wei, 2022. "Dynamic behaviors in a Cournot duopoly model with knowledge spillover effect based on constant conjectural variation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 201(C), pages 305-323.
    15. Askar, S.S. & Alshamrani, Ahmad M. & Alnowibet, K., 2015. "Dynamic Cournot duopoly games with nonlinear demand function," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 427-437.
    16. Joaquín Andaluz & Gloria Jarne, 2019. "On price stability and the nature of product differentiation," Journal of Evolutionary Economics, Springer, vol. 29(2), pages 741-762, April.
    17. Brianzoni, Serena & Campisi, Giovanni & Colasante, Annarita, 2022. "Nonlinear banking duopoly model with capital regulation: The case of Italy," Chaos, Solitons & Fractals, Elsevier, vol. 160(C).
    18. Xiaoliang Li & Bo Li, 2023. "A Bertrand duopoly game with differentiated products reconsidered," Papers 2301.01007, arXiv.org.
    19. A. A. Elsadany & A. M. Awad, 2016. "Dynamical analysis and chaos control in a heterogeneous Kopel duopoly game," Indian Journal of Pure and Applied Mathematics, Springer, vol. 47(4), pages 617-639, December.
    20. Yi, Qi Guo & Zeng, Xiang Jin, 2015. "Complex dynamics and chaos control of duopoly Bertrand model in Chinese air-conditioning market," Chaos, Solitons & Fractals, Elsevier, vol. 76(C), pages 231-237.
    21. Mikhail Anufriev & Davide Radi & Fabio Tramontana, 2018. "Some reflections on past and future of nonlinear dynamics in economics and finance," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 41(2), pages 91-118, November.
    22. Zhang, Xinli & Sun, Deshan & Ma, Sijia & Zhang, Shuning, 2020. "The dynamics of a quantum Bertrand duopoly with differentiated products and heterogeneous expectations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 557(C).
    23. Jianbo Zhu & Qianqian Shi & Peng Wu & Zhaohan Sheng & Xiangyu Wang, 2018. "Complexity Analysis of Prefabrication Contractors’ Dynamic Price Competition in Mega Projects with Different Competition Strategies," Complexity, Hindawi, vol. 2018, pages 1-9, September.

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