Annals of Operations Research

, Volume 274, Issue 1–2, pp 211–240 | Cite as

Dynamics and chaos control of a duopolistic Bertrand competitions under environmental taxes

  • A. A. ElsadanyEmail author
  • A. M. Awad
Original Research


This paper investigates the difference between price and quantity competition in a mixed duopoly game. We describe the behavior of a duopolistic Bertrand competition market with environmental taxes. There are two cases. In the first, the public firm is privatized and in the second, it is not privatized. In case I, private duopoly (postprivatization) where players use different production methods and choose their prices with (bounded rationality and naive). In case II, mixed duopoly (preprivatization) in this case there are two levels for the market including standard objective of the private firm is to maximize profits and including another objective function of the public firm namely “private welfare maximization”. We study numerically the dynamical behaviors of the models. The Nash equilibrium loses stability through a period-doubling bifurcation and the market in the end gets to be disordered. The disordered behavior of the market has been controlled by using feedback control method.


Price competition Quantity competition Environmental taxes Delay feedback control 


  1. Ahmed, E., Elsadany, A. A., & Puu, T. (2015). On Bertrand duopoly game with differentiated goods. Applied Mathematics and Computations, 251, 169–179.CrossRefGoogle Scholar
  2. Ahmed, E., Hegazi, A. S., & Abd El-Hafez, A. T. (2003). On multiobjective oligopoly. Nonlinear Dynamics, Psychology, and Life Sciences, 7(2), 205–219.CrossRefGoogle Scholar
  3. Beladi, H., & Chao, C. C. (2006). Does privatization improve the environment? Economics Letters, 93, 343–347.CrossRefGoogle Scholar
  4. Bertrand, J. (1883). Thorie mathmatique de la richesse sociale. Journal des Savants, 48, 499–508.Google Scholar
  5. Bhattacharjee, T., & Pal, R. (2013). Price vs. quantity in duopoly with strategic delegation: Role of network externalitie. Mumbai: Indira Gandhi Institute of Development Research.
  6. Brcena-Ruiz, J. C., & Sedano, M. (2011). Endogenous timing in a mixed duopoly: Weighted welfare and price competition. The Japanese Economic Review, 62, 485–503.CrossRefGoogle Scholar
  7. Carraro, C., & Soubeyaran, A. (1996). Environmental taxation, market share, and profits in oligopoly. In C. Carraro, Y. Katsoulacos, A. Xepapadeas (Eds.), Environmental policy and market structure (pp. 23–44).Google Scholar
  8. Chen, F., Ma, J. H., & Chen, X. Q. (2009). The study of dynamic process of the triopoly games in chinese 3G telecommunication market. Chaos, Solitons and Fractals, 42, 1542–1551.CrossRefGoogle Scholar
  9. Cournot, A. (1838). Rcherches sur les principes mathmatiques de la thorie des richesses. Paris: Dunod.Google Scholar
  10. Dixit, A. K. (1979). A model of duopoly suggesting a theory of entry barriers. Bell Journal Economics, 10, 20–32.CrossRefGoogle Scholar
  11. D-Teleszynski, T. (2010). Complex dynamics in a Bertrand duopoly game with heterogeneous players. Central European Journal of Economic Modelling and Econometrics, 2, 95–116.Google Scholar
  12. Elabbasy, E. M., Agiza, H. N., EL-Metwally, H., & Elsadany, A. A. (2007). Bifurcation analysis, chaos and control in the Burgers mapping. International Journal of Nonlinear Science, 4, 171–185.Google Scholar
  13. Elabbasy, E. M., Agiza, H. N., Elsadany, A. A., & EL-Metwally, H. (2007). The dynamics of triopoly game with heterogeneous players. International Journal of Nonlinear Science, 3, 83–90.Google Scholar
  14. Elsadany, A. A. (2012). Competition analysis of a triopoly game with bounded rationality. Chaos, Solitons & Fractals, 45, 1343–1348.CrossRefGoogle Scholar
  15. Elsadany, A. A. (2015). A dynamic Cournot duopoly model with different strategies. Journal of the Egyptian Mathematical Society, 23, 56–61.CrossRefGoogle Scholar
  16. Elsadany, A. A., Agiza, H. N., & Elabbasy, E. M. (2013). Complex dynamics and chaos control of heterogeneous quadropoly game. Applied Mathematics and Computation, 219, 11110–11118.CrossRefGoogle Scholar
  17. Fanti, L., Gori, L., Mammana, C., & Michetti, E. (2013). The dynamics of a Bertrand duopoly with differentiated products: Synchronization, intermittency and global dynamics. Chaos, Solitons & Fractals, 52, 73–86.CrossRefGoogle Scholar
  18. Ghosh, A., & Mitra, M. (2010). Comparing Bertrand and Cournot in mixed markets. Economic Letters, 109, 72–74.CrossRefGoogle Scholar
  19. Hu, R., & Xia, H-s. (2011). Chaotic dynamics in differentiated Bertrand model with heterogeneous players. In Grey systems and intelligent services (GSIS), IEEE international conference (pp. 675–678).Google Scholar
  20. Jury, E. I., & Blanchard, J. (1961). A stability test for linear discrete systems in table form. Proceedings of the Institute of Radio Engineers, 49, 1947–1948.Google Scholar
  21. Kangsik, C. (2012). Cournot and Bertrand competition with asymmetric costs in a mixed duopoly revisited. MPRA Paper No. 37704, posted 28. March 2012 12:29 UTC.
  22. Kopel, M. (2014). Price and quantity contracts in a mixed duopoly with a socially concerned firm. Managerial and Decision Economics.
  23. Liu, J., Liu, G., Li, N. & Xu, H. (2014). Dynamics analysis of game and chaotic control in the Chinese fixed broadband telecom market. Discrete Dynamics in Nature and Society, vol. 2014, Article ID 275123, 8 pages.Google Scholar
  24. Ma, J., Zhang, F., & He, Y. (2014). Complexity analysis of a master-slave oligopoly model and chaos control. Abstract and Applied Analysis, vol. 2014, Article ID 970205, 13 pages.Google Scholar
  25. Nakamura, Y. (2014). Social welfare under quantity competition and price competition in a mixed duopoly with network effects: An analysis. Theoretical Economics Letters, 4, 133–138.CrossRefGoogle Scholar
  26. Ohori, S. (2013). Price and quantity competition in a mixed duopoly with emission tax. Theoretical Economics Letters, 3, 211–215.CrossRefGoogle Scholar
  27. Pu, X., & Ma, J. (2013). Complex dynamics and chaos control in nonlinear four-oligopolist game with different expectations. International Journal of Bifurcation and Chaos, 23, 1350053. (15 pages) .CrossRefGoogle Scholar
  28. Pyrages, K. (1992). Continuous control of chaos by self-controlling feedback. Physics Letters A, 170, 421–4281.CrossRefGoogle Scholar
  29. Singh, N., & Vives, X. (1984). Price and quantity competition in a differentiated duopoly. The Rand Journal Economics, 15, 546–554.CrossRefGoogle Scholar
  30. Sun, L., & Ma, J. (2015). Study and simulation on discrete dynamics of Bertrand triopoly team-game. Mathematical Problems in Engineering, vol. 2015, Article ID 960380, 12 pages.Google Scholar
  31. Wang, H., & Ma, J. (2013). Complexity analysis of a Cournot–Bertrand duopoly game model with limited information. Discrete Dynamics in Nature and Society, vol. 2013, Article ID 287371, 6 pages.Google Scholar
  32. Wang, L. F. S., & Wang, J. (2009). Environmental taxes in a differentiated mixed duopoly. Economic Systems, 33, 389–396.CrossRefGoogle Scholar
  33. Wang, T., Wang, X., & Wang, M. (2011). Chaotic control of Hénon map with feedback and nonfeedback methods. Communications in Nonlinear Science and Numerical Simulation, 16, 3367–3374.CrossRefGoogle Scholar
  34. Wu, F., & Ma, J. (2014). The stability, bifurcation and chaos of a duopoly game in the market of complementary products with mixed bundling pricing. Wseas Transactions on Mathematics, 13, 374–384.Google Scholar
  35. Wu, F., & Ma, J. (2015). The complex dynamics of a multi-product mixed duopoly model with partial privatization and cross-ownerships. Nonlinear Dynamics, 80, 1391–1401.CrossRefGoogle Scholar
  36. Xin, B., & Li, Y. (2013). Bifurcation and chaos in a price game of irrigation water in a coastal irrigation district. Discrete Dynamics in Nature and Society, vol. 2013, Article ID 408904, 10 pages.Google Scholar
  37. Yali, L. (2015). Analysis of duopoly output game with different decision-making rules. Management Science and Engineering, 9, 19–24.Google Scholar
  38. Zhang, J., Da, Q., & Wang, Y. (2007). Analysis of nonlinear duopoly game with heterogeneous players. Economic Modelling, 24, 138–148.CrossRefGoogle Scholar
  39. Zhang, J., & Wang, Y. (2013). Complex dynamics of Bertrand duopoly games with bounded rationality. International Journal of Social, Education, Economics and Management Engineering, 7, 795–799.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Science and Humanities Studies Al-Kharj, Mathematics DepartmentPrince Sattam Bin Abdulaziz UniversityAl-KharjSaudi Arabia
  2. 2.Department of Basic Science, Faculty of Computers and InformaticsSuez Canal UniversityIsmailiaEgypt

Personalised recommendations