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The stability and duality of dynamic Cournot and Bertrand duopoly model with comprehensive preference

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  • Yu, Yu
  • Yu, Weisheng

Abstract

In this paper, a Cournot model and its dual Bertrand model where firms have comprehensive preferences are developed. The preference was proposed by Bowles (2004) based on the results of Rabin (1993) and Lebine (1998). Comparable static method is used to illustrate the impact of parameters on equilibrium. Three scenarios are classified by preference parameter: completely cooperative, hostile to each other and general Cournot or Bertrand model. In each situation, Cournot model and its dual model present different chaos phenomenon which have various and abundant strange attractors, while the shapes of the stability regions are similar in the same situation. In the static setting, the local stability analysis of equilibrium gives the bounded region of quantity or price. The one-dimensional Logistic mapping is applied to studying the dynamics of the models on invariant axes. The critical curves classify different regions according to the number of preimages (q1, q2) or (p1, p2). Besides, simulations give more intuitive results: the cycle attractor, chaotic attractor and the basin of attraction with “holes” are presented. The article also provides new findings that under the assumption of comprehensive preference, the Cournot model with substitutes (complements) and the Bertrand model with complements (substitutes) are still duality models in the dynamic settings.

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  • Yu, Yu & Yu, Weisheng, 2021. "The stability and duality of dynamic Cournot and Bertrand duopoly model with comprehensive preference," Applied Mathematics and Computation, Elsevier, vol. 395(C).
    • Handle: RePEc:eee:apmaco:v:395:y:2021:i:c:s0096300320308055
    DOI: 10.1016/j.amc.2020.125852
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    1. Bischi, Gian Italo & Kopel, Michael, 2001. "Equilibrium selection in a nonlinear duopoly game with adaptive expectations," Journal of Economic Behavior & Organization, Elsevier, vol. 46(1), pages 73-100, September.
    2. Elsadany, A.A., 2017. "Dynamics of a Cournot duopoly game with bounded rationality based on relative profit maximization," Applied Mathematics and Computation, Elsevier, vol. 294(C), pages 253-263.
    3. S. S. Askar & A. Al-khedhairi, 2019. "Investigations of Nonlinear Triopoly Models with Different Mechanisms," Complexity, Hindawi, vol. 2019, pages 1-15, December.
    4. Luca Gori & Luca Guerrini & Mauro Sodini, 2017. "A characterisation of duopoly dynamics with frictions in production adjustments," Journal of Evolutionary Economics, Springer, vol. 27(5), pages 963-988, November.
    5. Junhai Ma & Hongwu Wang, 2013. "Complex Dynamics Analysis for a Cournot-Bertrand Mixed Game Model with Delayed Bounded Rationality," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-11, October.
    6. Joaquín Andaluz & Gloria Jarne, 2016. "Stability of vertically differentiated Cournot and Bertrand-type models when firms are boundedly rational," Annals of Operations Research, Springer, vol. 238(1), pages 1-25, March.
    7. Wei Chen & Wentao Wang, 2016. "Global Exponential Stability of Cournot Duopolies with Delays," Computational Economics, Springer;Society for Computational Economics, vol. 48(1), pages 147-154, June.
    8. Agliari, A. & Naimzada, A.K. & Pecora, N., 2016. "Nonlinear dynamics of a Cournot duopoly game with differentiated products," Applied Mathematics and Computation, Elsevier, vol. 281(C), pages 1-15.
    9. Yi Wang & Hui Wang & Shubing Guo, 2019. "Research on Dynamic Game Model and Application of China’s Imported Soybean Price in the Context of China-US Economic and Trade Friction," Complexity, Hindawi, vol. 2019, pages 1-13, November.
    10. 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.
    11. Wei Zhou & Xiao-Xue Wang, 2019. "On the Stability and Multistability in a Duopoly Game with R&D Spillover and Price Competition," Discrete Dynamics in Nature and Society, Hindawi, vol. 2019, pages 1-20, August.
    12. Naimzada, Ahmad & Ricchiuti, Giorgio, 2011. "Monopoly with local knowledge of demand function," Economic Modelling, Elsevier, vol. 28(1), pages 299-307.
    13. S. S. Askar & A. Al-khedhairi, 2017. "Analysis of Nonlinear Duopoly Games with Product Differentiation: Stability, Global Dynamics, and Control," Discrete Dynamics in Nature and Society, Hindawi, vol. 2017, pages 1-13, June.
    14. TAKASHI Sato, 1996. "On Cournot-Bertrand Mixed Duopolies," The Japanese Economic Review, Japanese Economic Association, vol. 47(4), pages 412-420, December.
    15. Arya, Anil & Mittendorf, Brian & Sappington, David E.M., 2008. "Outsourcing, vertical integration, and price vs. quantity competition," International Journal of Industrial Organization, Elsevier, vol. 26(1), pages 1-16, January.
    16. Rabin, Matthew, 1993. "Incorporating Fairness into Game Theory and Economics," American Economic Review, American Economic Association, vol. 83(5), pages 1281-1302, December.
    17. Naimzada, Ahmad Kabir & Raimondo, Roberto, 2018. "Chaotic congestion games," Applied Mathematics and Computation, Elsevier, vol. 321(C), pages 333-348.
    18. Villena, Marcelo J. & Araneda, Axel A., 2017. "Dynamics and stability in retail competition," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 134(C), pages 37-53.
    19. Cerboni Baiardi, Lorenzo & Naimzada, Ahmad K., 2018. "An oligopoly model with best response and imitation rules," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 193-205.
    20. Cerboni Baiardi, Lorenzo & Naimzada, Ahmad K., 2019. "An oligopoly model with rational and imitation rules," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 156(C), pages 254-278.
    21. Puu, Tonu & Tramontana, Fabio, 2019. "Can Bertrand and Cournot oligopolies be combined?," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 97-107.
    22. 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.
    23. Victor J. Tremblay & Carol Horton Tremblay & Kosin Isariyawongse, 2013. "Endogenous Timing And Strategic Choice: The Cournot-Bertrand Model," Bulletin of Economic Research, Wiley Blackwell, vol. 65(4), pages 332-342, October.
    24. 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.
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