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

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Abstract

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.

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Notes

  1. We mention here, the second order condition for the optimal environmental tax in the stage of p–p game \(\frac{\partial \tau ^{ pp }}{\partial d}=\frac{ 8(1+d)^{2}\left( A-1\right) (9d^{2}+29d+23)}{\left( 18d^{2}+47d+31\right) ^{2}}\ge 0\), since \(d^{2}\ge \frac{29d+23}{-\,9}.\)

  2. We mention here, the second order condition for the optimal environmental tax in the stage of q–q game \(\frac{\partial \tau ^{ qq }}{\partial d}=\frac{ 3\left( A-1\right) (d^{2}-14d-11)}{\left( d^{2}+6d-31\right) ^{2}}\ge 0,\) since \(d^{2}\ge 14d+11.\)

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Authors and Affiliations

  1. College of Science and Humanities Studies Al-Kharj, Mathematics Department, Prince Sattam Bin Abdulaziz University, Al-Kharj, Saudi Arabia

    A. A. Elsadany

  2. Department of Basic Science, Faculty of Computers and Informatics, Suez Canal University, Ismailia, 41522, Egypt

    A. A. Elsadany & A. M. Awad

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Elsadany, A.A., Awad, A.M. Dynamics and chaos control of a duopolistic Bertrand competitions under environmental taxes. Ann Oper Res 274, 211–240 (2019). https://doi.org/10.1007/s10479-018-2837-8

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