Effects of carbon tax on environment under duopoly

Abstract

Environment protection and green development is of great importance. Therefore, this paper attempts to propose an environmentally friendly and sustainable development model that depends on environmental economics and game theory. This paper employs an analytical approach to show that the effects of carbon tax on environment regulation in case of the duopoly market structure. Some interesting conclusions are obtained. Firstly, factors affecting carbon tax, such as the importance of environment to the government, are captured. Secondly, this paper shows that low efficiency firm has more stimulation to reduce product substitutability and more motivation to practice product efficiency improving innovations. Thirdly, optimal carbon tax is compared between tax on energy input and unit carbon tax. The results show that unit tax is lower than energy input tax under optimal condition. More importantly, the numeric results of the paper clearly demonstrate that those countries committed to environment prefer input tax since it is more efficient in reducing energy consumption, while a profit maximization approach leads to the unit carbon tax. The findings of this paper can be helpful to those who are responsible for the environment and sustainable progress.

Appendix

1.1 Proof of Proposition 1

Equation (16) is denoted as

$$\begin{gathered} f = \frac{2\kappa }{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)\left[ {\frac{A}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right) - \frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\frac{4}{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right)} \right. \hfill \\ \;\;\;\left. { + \frac{4}{(2 + \gamma )}\frac{{\alpha + \tau_{1} }}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right)^{2} - \frac{6}{(2 + \gamma )}\left( {\frac{1}{{\theta_{2} }} + \frac{1}{{\theta_{1} }}} \right)\frac{A}{(2 + \gamma )} - \frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)} \right] = 0. \hfill \\ \end{gathered}$$

(A1)

By the implicit function theorem, we have

$$\frac{{\partial \tau_{1}^{*} }}{\partial \kappa } = - \frac{{\frac{\partial f}{{\partial \kappa }}}}{{\frac{\partial f}{{\partial \tau_{1}^{*} }}}} > 0.$$

The above inequality holds because $\frac{\partial f}{{\partial \kappa }} > 0$ and concavity of the GSW function implies $\frac{\partial f}{{\partial \tau_{1}^{*} }} < 0$. Thus, optimal carbon tax increases with the degree to attach importance of the government to the environment. By Eqs. (10, 12,13), we declare that the more importance attached to the environment is, the lower energy is consumed and fewer profits for firm are.

Here we address the effects of market power on optimal carbon tax ratio. Given $\theta_{{1}}$, by function (A1) and $\theta_{{2}} { = }\theta_{{1}} + \eta$, we achieve

$$\begin{gathered} \frac{\partial f}{{\partial \eta }} = \frac{\partial f}{{\partial \theta_{2} }} = \frac{2}{{\theta_{2}^{2} }}\left[ { - \frac{4}{(2 + \gamma )}\frac{{\alpha + \tau_{1} }}{(2 + \gamma )}(\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}) + \frac{3}{(2 + \gamma )}\frac{A}{(2 + \gamma )}} \right. \hfill \\ \;\;\;\;\;\; + \frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\frac{4}{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right) - \frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\frac{2\gamma }{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right) \hfill \\ \;\;\;\;\;\; - \frac{4\kappa }{{(4 - \gamma^{2} )}}\frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)\left( {\frac{\gamma }{{\theta_{1} }} - \frac{2}{{\theta_{2}^{{}} }}} \right) \hfill \\ \;\;\;\;\;\;\left. { - \frac{\kappa }{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)\frac{A}{(2 + \gamma )} + \frac{2\kappa }{{(4 - \gamma^{2} )}}\frac{A}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right)\left( {\frac{\gamma }{{\theta_{1} }} - \frac{2}{{\theta_{2}^{{}} }}} \right)} \right]. \hfill \\ \end{gathered}$$

(A2)

Moreover, $1 \le \theta_{1} \le \theta_{2} \le 2$ implies $\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }} > \frac{1}{{\theta_{1}^{2} }} + \frac{1}{{\theta_{2}^{2} }} - \frac{\gamma }{{\theta_{1} \theta_{2} }}$. Function (A1) and $A > > \alpha$ jointly indicate.

$\begin{gathered} \frac{4}{(2 + \gamma )}\frac{{\alpha + \tau_{1} }}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right)^{2} + \frac{2\kappa }{{(4 - \gamma^{2} )}}\frac{A}{(2 + \gamma )}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right) - \frac{6}{(2 + \gamma )}\left( {\frac{1}{{\theta_{2} }} + \frac{1}{{\theta_{1} }}} \right)\frac{A}{(2 + \gamma )} \hfill \\ = \frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\frac{4}{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right) + \frac{8\kappa }{{(4 - \gamma^{2} )}}\frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\left( {\frac{1}{{\theta_{1}^{2} }} + \frac{1}{{\theta_{2}^{2} }} - \frac{\gamma }{{\theta_{1} \theta_{2} }}} \right)^{2} , \hfill \\ \end{gathered}$and

$$\begin{gathered} \left[ { - \frac{2}{(2 + \gamma )}\frac{{\alpha + \tau_{1} }}{(2 + \gamma )}(\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}) + \frac{3}{(2 + \gamma )}\frac{A}{(2 + \gamma )} - \frac{\kappa }{{(4 - \gamma^{2} )}}(\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }})\frac{A}{(2 + \gamma )}} \right]\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right) \hfill \\ = - \frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\frac{2}{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right) - \frac{4\kappa }{{(4 - \gamma^{2} )}}\frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\left( {\frac{1}{{\theta_{1}^{2} }} + \frac{1}{{\theta_{2}^{2} }} - \frac{\gamma }{{\theta_{1} \theta_{2} }}} \right)^{2} . \hfill \\ \end{gathered}$$

Thus,

$$\begin{gathered} - \frac{2}{(2 + \gamma )}\frac{{\alpha + \tau_{1} }}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right) + \frac{3}{{(2{ + }\gamma )}}\frac{A}{(2 + \gamma )} - \frac{\kappa }{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)\frac{A}{(2 + \gamma )} \hfill \\ { = } - \frac{{2(\alpha + \tau_{1} )}}{{(4 - \gamma^{2} )^{2} (\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }})}}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right) - \frac{{4\kappa (\alpha + \tau_{1} )}}{{(4 - \gamma^{2} )^{2} (\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }})}}\left( {\frac{1}{{\theta_{1}^{2} }} + \frac{1}{{\theta_{2}^{2} }} - \frac{\gamma }{{\theta_{1} \theta_{2} }}} \right)^{2} . \hfill \\ \end{gathered}$$

Moreover,

$$\begin{gathered} \frac{\partial f}{{\partial \eta }} = \frac{2}{{\theta_{2}^{2} }}\left[ { - \frac{4}{(2 + \gamma )}\frac{{\alpha + \tau_{1} }}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right) + \frac{3}{{(2{ + }\gamma )}}\frac{A}{(2 + \gamma )}{ + }\frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\frac{4}{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right)} \right. \hfill \\ \;\;\;\;\; - \frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\frac{2\gamma }{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right) - \frac{4\kappa }{{(4 - \gamma^{2} )}}\frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)\left( {\frac{\gamma }{{\theta_{1} }} - \frac{2}{{\theta_{2}^{{}} }}} \right) \hfill \\ \left. {\;\;\;\;\; - \frac{\kappa }{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)\frac{A}{(2 + \gamma )} - \frac{2\kappa }{{(4 - \gamma^{2} )}}\frac{A}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right)\left( {\frac{2}{{\theta_{2}^{{}} }} - \frac{\gamma }{{\theta_{1} }}} \right)} \right] \hfill \\ \end{gathered}$$

$$\begin{gathered} = \frac{2}{{\theta_{2}^{2} }}\left[ { - \frac{2}{(2 + \gamma )}\frac{{\alpha + \tau_{1} }}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right) - \frac{{2(\alpha + \tau_{1} )}}{{(4 - \gamma^{2} )^{2} (\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }})}}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right) - \frac{{4\kappa (\alpha + \tau_{1} )}}{{(4 - \gamma^{2} )^{2} (\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }})}}\left( {\frac{1}{{\theta_{1}^{2} }} + \frac{1}{{\theta_{2}^{2} }} - \frac{\gamma }{{\theta_{1} \theta_{2} }}} \right)^{2} } \right. \hfill \\ { + }\frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\frac{4}{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right) - \frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\frac{2\gamma }{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right) + \frac{4\kappa }{{(4 - \gamma^{2} )}}\frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)\left( {\frac{2}{{\theta_{2}^{{}} }} - \frac{\gamma }{{\theta_{1} }}} \right) \hfill \\ \left. { - \frac{2\kappa }{{(4 - \gamma^{2} )}}\frac{A}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right)\left( {\frac{2}{{\theta_{2}^{{}} }} - \frac{\gamma }{{\theta_{1} }}} \right)} \right] \hfill \\ = \frac{2}{{\theta_{2}^{2} }}\left\{ { - \frac{2}{(2 + \gamma )}\frac{{\alpha + \tau_{1} }}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right) + \frac{{2(\alpha + \tau_{1} )}}{{(4 - \gamma^{2} )^{2} (\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }})}}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{2} }}} \right)} \right. \hfill \\ - \frac{{2(\alpha + \tau_{1} )}}{{(4 - \gamma^{2} )^{2} (\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }})}}\left[ {2\kappa \left( {\frac{1}{{\theta_{1}^{2} }} + \frac{1}{{\theta_{2}^{2} }} - \frac{\gamma }{{\theta_{1} \theta_{2} }}} \right)^{2} - \left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{2} }}} \right)} \right] - \frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\frac{2\gamma }{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right) \hfill \\ - \left. {\frac{2\kappa }{{(4 - \gamma^{2} )^{2} }}(\frac{2}{{\theta_{2}^{{}} }} - \frac{\gamma }{{\theta_{1} }})\left[ {A(2 - \gamma )(\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}) - 4(\alpha + \tau_{1} )(\frac{1}{{\theta_{1}^{2} }} + \frac{1}{{\theta_{2}^{2} }} - \frac{\gamma }{{\theta_{1} \theta_{2} }})} \right]} \right\} < 0. \hfill \\ \end{gathered}$$

The last inequality holds because $A\left( {2 - \gamma } \right)\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right) > 4\left( {\alpha + \tau_{1} } \right)\left( {\frac{1}{{\theta_{1}^{2} }} + \frac{1}{{\theta_{2}^{2} }} - \frac{\gamma }{{\theta_{1} \theta_{2} }}} \right)$ and $2\kappa \left( {\frac{1}{{\theta_{1}^{2} }} + \frac{1}{{\theta_{2}^{2} }} - \frac{\gamma }{{\theta_{1} \theta_{2} }}} \right)^{2} > \left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{2} }}} \right)$. Therefore, based on functions (A1) and equation (A2), we have

$$\frac{{\partial \tau_{1}^{*} }}{\partial \eta } = - \frac{{\frac{\partial f}{{\partial \eta }}}}{{\frac{\partial f}{{\partial \tau_{1}^{*} }}}} < 0.$$

Similarly, we have the relationship $\frac{{\partial \tau_{1}^{*} }}{{\partial \theta_{1} }} < 0.$ Therefore, product efficiency reduces the optimal carbon tax ratio.

By virtue of function (A1), we obtain

$$\begin{gathered} f_{1} = \frac{2\kappa }{{(2 - \gamma )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)\left[ {A\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right) - \frac{{\alpha + \tau_{1} }}{(2 - \gamma )}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)} \right] \hfill \\ \;\;\;\; + 4\left( {\alpha + \tau_{1} } \right)\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right)^{2} - 6A\left( {\frac{1}{{\theta_{2} }} + \frac{1}{{\theta_{1} }}} \right) - \frac{{\alpha + \tau_{1} }}{(2 - \gamma )}\frac{4}{(2 - \gamma )}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right) = 0. \hfill \\ \end{gathered}$$

(A3)

From function (A3), we have $\frac{{\partial f_{1} }}{{\partial \tau_{1}^{*} }} < 0$ and

$\begin{gathered} \frac{{\partial f_{1} }}{\partial \gamma } = \frac{1}{2 - \gamma }\left[ {\frac{2A\kappa }{{(2 - \gamma )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right) - \frac{{4\kappa (\alpha + \tau_{1} )}}{{(2 - \gamma )^{2} }}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)^{2} - \frac{{8(\alpha + \tau_{1} )}}{{(2 - \gamma )^{2} }}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right)} \right] \hfill \\ \;\;\;\;\;\; - \frac{4A\kappa }{{(2 - \gamma )\theta_{1} \theta_{2} }}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right) + \frac{{8\kappa (\alpha + \tau_{1} )}}{{(2 - \gamma )^{2} \theta_{1} \theta_{2} }}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right) + \frac{{4(\alpha + \tau_{1} )}}{{(2 - \gamma )^{2} }}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right) \hfill \\ \end{gathered}$$\begin{gathered} = \frac{1}{2 - \gamma }\left[ {2\left( {3 - \frac{2\kappa }{{\theta_{1} \theta_{2} }}} \right)A\left( {\frac{1}{{\theta_{2} }} + \frac{1}{{\theta_{1} }}} \right) - \frac{{4(\alpha + \tau_{1} )}}{{(2 - \gamma )^{2} }}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right)} \right] \hfill \\ + \frac{{2\kappa (\alpha + \tau_{1} )}}{{(2 - \gamma )^{3} }}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)\left( {\frac{8 - 2\gamma }{{\theta_{1} \theta_{2} }} - \frac{2}{{\theta_{1}^{2} }} - \frac{2}{{\theta_{2}^{2} }}} \right) - \frac{{4(\alpha + \tau_{1} )}}{{(2 - \gamma )^{2} }}\left( {\frac{4}{{\theta_{1} \theta_{2} }} - \frac{\gamma }{{\theta_{1}^{2} }} - \frac{\gamma }{{\theta_{2}^{2} }}} \right) \hfill \\ < \frac{1}{2 - \gamma }\left[ {2\left( {3 - \frac{2\kappa }{{\theta_{1} \theta_{2} }}} \right)A\left( {\frac{1}{{\theta_{2} }} + \frac{1}{{\theta_{1} }}} \right) - \frac{{4(\alpha + \tau_{1} )}}{{(2 - \gamma )^{2} }}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right)} \right] \hfill \\ + \frac{{2\kappa (\alpha + \tau_{1} )}}{{(2 - \gamma )^{2} }}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)\frac{2}{{\theta_{1} \theta_{2} }} - \frac{{4(\alpha + \tau_{1} )}}{{(2 - \gamma )^{2} }}\left( {\frac{4}{{\theta_{1} \theta_{2} }} - \frac{\gamma }{{\theta_{1}^{2} }} - \frac{\gamma }{{\theta_{2}^{2} }}} \right) \hfill \\ \end{gathered}$$\begin{gathered} = \frac{1}{2 - \gamma }\left[ {2\left( {3 - \frac{\kappa }{{\theta_{1} \theta_{2} }}} \right)A\left( {\frac{1}{{\theta_{2} }} + \frac{1}{{\theta_{1} }}} \right) - \frac{{4(\alpha + \tau_{1} )}}{{(2 - \gamma )^{2} }}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right)} \right] \hfill \\ - \frac{2\kappa }{{\theta_{1} \theta_{2} (2 - \gamma )^{2} }}\left[ {A\left( {\frac{1}{{\theta_{2} }} + \frac{1}{{\theta_{1} }}} \right)\left( {2 - \gamma } \right) - 2\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)(\alpha + \tau_{1} )} \right] - \frac{{4(\alpha + \tau_{1} )}}{{(2 - \gamma )^{2} }}\left( {\frac{4}{{\theta_{1} \theta_{2} }} - \frac{\gamma }{{\theta_{1}^{2} }} - \frac{\gamma }{{\theta_{2}^{2} }}} \right) < 0. \hfill \\ \end{gathered}$ The second equality above comes from function (A3) under $3 < \frac{\kappa }{{\theta_{1} \theta_{2} }}$ and $A\left( {\frac{1}{{\theta_{2} }} + \frac{1}{{\theta_{1} }}} \right)\left( {2 - \gamma } \right) > 2\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)\left( {\alpha + \tau_{1} } \right)$. Therefore, we get $\frac{{\partial \tau_{1}^{*} }}{\partial \gamma } = - \frac{{\frac{\partial f}{{\partial \gamma }}}}{{\frac{\partial f}{{\partial \tau_{1}^{*} }}}} < 0.$

Thus, product substitutability reduces the optimal carbon tax ratio. Conclusions are achieved and the proof is complete.

1.2 Proof of Proposition 2

By Eq. (19), we have

$$\begin{gathered} \frac{{\partial q_{1}^{{*,2}} }}{\partial \gamma }{ = }\frac{{\partial \left( {\frac{{A - \tau_{2} }}{2 + \gamma }} \right)}}{\partial \gamma } - \frac{2\gamma \alpha }{{(4 - \gamma^{2} )^{2} }}\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right) + \frac{\alpha }{{(4 - \gamma^{2} )}}\frac{1}{{\theta_{2} }}{ = }\frac{{\partial \left( {\frac{{A - \tau_{2} }}{2 + \gamma }} \right)}}{\partial \gamma } + \frac{\alpha }{{(4 - \gamma^{2} )^{2} }}\left( {\frac{{4 + \gamma^{2} }}{{\theta_{2} }} - \frac{4\gamma }{{\theta_{1} }}} \right), \hfill \\ \frac{{\partial q_{2}^{{*,2}} }}{\partial \gamma }{ = }\frac{{\partial \left( {\frac{{A - \tau_{2} }}{2 + \gamma }} \right)}}{\partial \gamma } - \frac{2\gamma \alpha }{{(4 - \gamma^{2} )^{2} }}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right) + \frac{\alpha }{{(4 - \gamma^{2} )}}\frac{1}{{\theta_{1} }}{ = }\frac{{\partial \left( {\frac{{A - \tau_{2} }}{2 + \gamma }} \right)}}{\partial \gamma } + \frac{\alpha }{{(4 - \gamma^{2} )^{2} }}\left( {\frac{{4 + \gamma^{2} }}{{\theta_{1} }} - \frac{4\gamma }{{\theta_{2} }}} \right). \hfill \\ \end{gathered}$$

Obviously, $\frac{{\partial q_{2}^{{*,2}} }}{\partial \gamma } - \frac{{\partial q_{1}^{{*,2}} }}{\partial \gamma } = \frac{{2(\theta_{2} - \theta_{1} )}}{{(2 - \gamma )^{2} }} > 0$ because $1 \le \theta_{1} \le \theta_{2} \le 2$. Therefore, firm with lower efficiency has more intention to reduce product substitutability (or reduce $\gamma$) than others.

Here we address effects of market power on firm’s innovation. By Eq. (19), we have

$$\frac{{\partial q_{1}^{{*,2}} }}{{\partial \theta_{1} }}{ = }\frac{ - 1}{{(2 + \gamma )}}\frac{{\partial \tau_{2} }}{{\partial \theta_{1} }} + \frac{\alpha }{{(4 - \gamma^{2} )}}\frac{2}{{\theta_{1}^{2} }},\frac{{\partial q_{2}^{{*,2}} }}{{\partial \theta_{2} }}{ = }\frac{ - 1}{{(2 + \gamma )}}\frac{{\partial \tau_{2} }}{{\partial \theta_{2} }} + \frac{\alpha }{{(4 - \gamma^{2} )}}\frac{2}{{\theta_{2}^{2} }}.$$

Combining the formulations of $\frac{{\partial \tau_{2} }}{{\partial \theta_{1} }}$ and $\frac{{\partial \tau_{2} }}{{\partial \theta_{2} }}$, we have $\frac{{\partial q_{1}^{{*,2}} }}{{\partial \theta_{1} }} - \frac{{\partial q_{2}^{{*,2}} }}{{\partial \theta_{2} }} = \frac{{2\alpha (\theta_{2}^{2} - \theta_{1}^{2} )}}{{(4 - \gamma^{2} )\theta_{1}^{2} \theta_{2}^{2} }} > 0.$ Conclusions are achieved and the proof is complete. □

1.3 Proof of Proposition 3

From Eq. (16), we have.

$\begin{gathered} \left. f \right|_{{\tau_{1} = \tau_{1}^{*} }} = \frac{4}{(2 + \gamma )}\frac{{\alpha + \tau_{1} }}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right)^{2} - \frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\frac{4}{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right) - \frac{6}{{(2{ + }\gamma )}}\left( {\frac{1}{{\theta_{2} }} + \frac{1}{{\theta_{1} }}} \right)\frac{A}{(2 + \gamma )} \hfill \\ \;\;\;\;\;\;\;\;\;\; + \frac{2\kappa }{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)\left[ {\frac{A}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right) - \frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)} \right] = 0. \hfill \\ \end{gathered}$Here we consider the sign of the following formulation.

$\begin{gathered} \left. f \right|_{{\tau_{1} = \theta_{1} \tau_{2}^{*} }} = \frac{4}{(2 + \gamma )}\frac{{\alpha + \tau_{1} }}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right)^{2} - \frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\frac{4}{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right) - \frac{6}{{(2{ + }\gamma )}}\left( {\frac{1}{{\theta_{2} }} + \frac{1}{{\theta_{1} }}} \right)\frac{A}{(2 + \gamma )} \hfill \\ \;\;\;\;\;\;\;\;\;\;\; + \frac{2\kappa }{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)\left[ {\frac{A}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right) - \frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)} \right] \hfill \\ \end{gathered}$Taking Eq. (21) into account, we achieve the following equation.

$\frac{6\alpha }{{(2 + \gamma )^{2} }}\left( {\frac{1}{{\theta_{2} }} + \frac{1}{{\theta_{1} }}} \right) - \frac{{12(A - \tau_{2}^{*} )}}{{(2 + \gamma )^{2} }} + \frac{\kappa }{(2 + \gamma )}\left[ {\frac{{A - \tau_{2}^{*} }}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right) - \frac{\alpha }{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)} \right] = 0.$Therefore, we immediately get the following formulation.

$\begin{gathered} \left. f \right|_{{\tau_{1} = \theta_{1} \tau_{2}^{*} }} = \frac{4}{(2 + \gamma )}\frac{{\alpha + \theta_{1} \tau_{2}^{*} }}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right)^{2} - \frac{{\alpha + \theta_{1} \tau_{2}^{*} }}{{(4 - \gamma^{2} )}}\frac{4}{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right) - \frac{6A}{{(2{ + }\gamma )^{2} }}\left( {\frac{1}{{\theta_{2} }} + \frac{1}{{\theta_{1} }}} \right) \hfill \\ \;\;\;\;\;\;\;\;\;\;\; + \frac{2\kappa }{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)\left[ {\frac{A}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right) - \frac{{\alpha + \theta_{1} \tau_{2}^{*} }}{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)} \right] \hfill \\ \end{gathered}$$\begin{gathered} \;\;\;\;\;\;\;\;\;\;\; = \frac{4}{(2 + \gamma )}\frac{{\alpha + \theta_{1} \tau_{2}^{*} }}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right)^{2} - \frac{{\alpha + \theta_{1} \tau_{2}^{*} }}{{(4 - \gamma^{2} )}}\frac{4}{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right) - \frac{2\kappa }{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)^{2} \frac{{\theta_{1} \tau_{2}^{*} }}{{(4 - \gamma^{2} )}} \hfill \\ \;\;\;\;\;\;\;\;\;\;\; + \frac{2\kappa }{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)\left[ {\frac{A}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right) - \frac{\alpha }{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)} \right] - \frac{6A}{{(2{ + }\gamma )^{2} }}\left( {\frac{1}{{\theta_{2} }} + \frac{1}{{\theta_{1} }}} \right) \hfill \\ \end{gathered}$$\begin{gathered} \;\;\;\;\;\;\;\;\;\;\; = \frac{4}{(2 + \gamma )}\frac{{\alpha + \theta_{1} \tau_{2}^{*} }}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right)^{2} - \frac{{\alpha + \theta_{1} \tau_{2}^{*} }}{{(4 - \gamma^{2} )}}\frac{4}{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right) - \frac{2\kappa }{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)^{2} \frac{{\theta_{1} \tau_{2}^{*} }}{{(4 - \gamma^{2} )}} \hfill \\ \;\;\;\;\;\;\;\;\;\;\; + \frac{2}{(2 - \gamma )}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)\left[ {\frac{{12(A - \tau_{2}^{*} )}}{{(2 + \gamma )^{2} }} - \frac{6\alpha }{{(2 + \gamma )^{2} }}\left( {\frac{1}{{\theta_{2} }} + \frac{1}{{\theta_{1} }}} \right) + \frac{{\kappa \theta_{1} \tau_{2}^{*} }}{{(2 + \gamma )^{2} }}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right)} \right] - \frac{6A}{{(2{ + }\gamma )^{2} }}\left( {\frac{1}{{\theta_{2} }} + \frac{1}{{\theta_{1} }}} \right) \hfill \\ \;\;\;\;\;\;\;\;\;\;\; = \frac{6A}{{(2{ + }\gamma )^{2} }}\left[ {\frac{4}{(2 - \gamma )}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right) - \left( {\frac{1}{{\theta_{2} }} + \frac{1}{{\theta_{1} }}} \right)} \right] + \frac{{2\kappa \theta_{1} \tau_{2}^{*} (\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }})}}{{(2 + \gamma )^{2} (2 - \gamma )^{2} }}\left[ {(2 - \gamma )\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right) - \left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)} \right] \hfill \\ \;\;\;\;\;\;\;\;\;\;\;{ + }\frac{4\alpha }{{(2 + \gamma )^{2} }}\left[ {\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right)^{2} - \frac{3}{(2 - \gamma )}\left( {\frac{1}{{\theta_{2} }} + \frac{1}{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right) - \frac{1}{{(2 - \gamma )^{2} }}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right)} \right] \hfill \\ \;\;\;\;\;\;\;\;\;\;\; + \frac{{4\theta_{1} \tau_{2}^{*} }}{{(2 + \gamma )^{2} }}\left[ {\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right)^{2} - \frac{1}{{(2 - \gamma )^{2} }}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right)} \right] - \frac{2}{(2 - \gamma )}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)\frac{{12\tau_{2}^{*} }}{{(2 + \gamma )^{2} }} \ge {0}{\text{.}} \hfill \\ \end{gathered}$The above inequality holds because

$(2 - \gamma )\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right) - \left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right) = \left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right)\left( {1 - \frac{1}{{\theta_{1} }}} \right) + \left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {1 - \frac{1}{{\theta_{2} }}} \right) > 0$,

$$\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right)^{2} - \frac{1}{{(2 - \gamma )^{2} }}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right) > 0,$$

$\begin{gathered} \frac{4}{{\left( {2 - \gamma } \right)}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} + \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right) - \left( {\frac{1}{{\theta_{2} }} + \frac{1}{{\theta_{1} }}} \right) = \frac{4}{{\left( {2 - \gamma } \right)}}\left( {\frac{2 - \gamma }{{\theta_{1}^{2} }} + \frac{2 - \gamma }{{\theta_{2}^{2} }} + \frac{\gamma }{{\theta_{1}^{2} }} + \frac{\gamma }{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}\frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right) - \left( {\frac{1}{{\theta_{2} }} + \frac{1}{{\theta_{1} }}} \right) \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; > \frac{{2}}{(2 - \gamma )}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right), \hfill \\ \end{gathered}$and $A - 2\tau_{2}^{*} - \alpha \left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right) > 0.$ By the concavity of social welfare, we obtain $\frac{{\tau_{1}^{*} }}{{\theta_{1} }} \ge \tau_{2}^{*}$.

Moreover, by the similar way, we have

$\begin{gathered} \left. f \right|_{{\tau_{1} = \theta_{2} \tau_{2}^{*} }} = \frac{4}{(2 + \gamma )}\frac{{\alpha + \tau_{1} }}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right)^{2} - \frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\frac{4}{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{2} }} - \frac{\gamma }{{\theta_{1} }}} \right)\left( {\frac{2}{{\theta_{1} }} - \frac{\gamma }{{\theta_{2} }}} \right) - \frac{6}{{(2{ + }\gamma )}}\left( {\frac{1}{{\theta_{2} }} + \frac{1}{{\theta_{1} }}} \right)\frac{A}{(2 + \gamma )} \hfill \\ \;\;\;\;\;\;\;\;\;\;\; + \frac{2\kappa }{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)\left[ {\frac{A}{(2 + \gamma )}\left( {\frac{1}{{\theta_{1} }} + \frac{1}{{\theta_{2} }}} \right) - \frac{{\alpha + \tau_{1} }}{{(4 - \gamma^{2} )}}\left( {\frac{2}{{\theta_{1}^{2} }} + \frac{2}{{\theta_{2}^{2} }} - \frac{2\gamma }{{\theta_{1} \theta_{2} }}} \right)} \right] \ge {0}{\text{.}} \hfill \\ \end{gathered}$Therefore, $\tau_{2}^{*} \le \frac{{\tau_{1}^{*} }}{{\theta_{2} }}$. In summary, we achieve $\frac{{\tau_{1}^{*} }}{{\theta_{1} }} \ge \frac{{\tau_{1}^{*} }}{{\theta_{2} }} \ge \tau_{2}^{*}$.

Conclusions are achieved and the proof is complete. □