Emission allowance allocation mechanism design: a low-carbon operations perspective

Abstract

Governments around the world are seeking an effective mechanism to cope with air pollution and climate change. The allocation of emission allowances, which is a key mechanism in the cap-and-trade system, is an important and intricate puzzle faced by environmental agencies. In this paper, we build a Stackelberg model to explore the emission allowance allocation mechanism design from an operations perspective. We demonstrate the feasibility and effectiveness of a linear emission allowance allocation mechanism. The results show that the emission allowance allocated by the government should always be insufficient to satisfy the ex-post emission demand at the industry level, even with low-carbon investment. To analyze the impacts on firms’ decision-makings, we explore a scenario in which two firms in the same industry sell a homogenous product to the market. The optimal low-carbon investment and production decisions are significantly affected by these market and carbon-related factors. Numerical examples are presented to further demonstrate the results that our paper has derived and investigate the optimal operational decisions of the two firms. Several meaningful management insights on allocation mechanism design and low-carbon operations of firms are obtained.

Appendices

Appendix A: The linear form of allocation mechanism

Proof

we firstly discuss the situation where the emission allowances per product is linear with carbon emission reduction rate.

1.1 Step 1: Solve the industry’s profit maximization problem

The profit function of the industry has been given in Eq. (6). The first-order conditions of $\pi$ with respect to q and $\theta$ respectively are:

$$\begin{aligned} \left\{ \begin{array}{l} \frac{{\partial \pi }}{{\partial q}} = p{\bar{F}}(q) + {p_\mathrm{{e}}}\left[ {{K_o} - {e_o} + \left( {{e_o} - a} \right) \rho \left( \theta \right) } \right] - c - \theta = 0; \\ \frac{{\partial \pi }}{{\partial \theta }} = \left( {{p_e}\left( {{e_o} - a} \right) \rho '\left( \theta \right) - 1} \right) q = 0. \end{array} \right. \end{aligned}$$

(A.1)

Then, we get the equations of stationary point

$$\begin{aligned}&p{\bar{F}}({q^*}\left( {{K_o},a} \right) ) = c+{\theta ^*}\left( {{K_o},a} \right) - {p_e}\left[ {{K_o} - {e_o} + \left( {{e_o} - a} \right) \rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right] ; \qquad \end{aligned}$$

(A.2)

$$\begin{aligned}&{p_e}\left( {{e_o} - a} \right) \rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) = 1. \end{aligned}$$

(A.3)

Accordingly, we verify that the determinant of the Hessian matrix at the stationary point is negative definite:

$$\begin{aligned} \begin{array}{l} H_1 = \frac{{{\partial ^2}\pi }}{{\partial {q^2}}} = p\left( {0 - f\left( q \right) } \right) = - pf\left( q \right)< 0;\\ H_2 = \frac{{{\partial ^2}\pi }}{{\partial q\partial \theta }} = {p_e}\left( {{e_o} - a} \right) \rho '\left( \theta \right) - 1 = 0;\\ H_3 = \frac{{{\partial ^2}\pi }}{{\partial {\theta ^2}}} = {p_e}\left( {{e_o} - a} \right) \rho ''\left( \theta \right) q < 0;\\ {H_1}{H_3} - {\left( {{H_2}} \right) ^2} = - pf\left( q \right) {p_e}\left( {{e_o} - a} \right) \rho ''\left( \theta \right) q > 0. \end{array} \end{aligned}$$

So the $\pi$ reaches the maximum at the stationary point.

1.2 Step 2: Solve the optimization problem faced by the government

The government anticipates the behaviors of the industry and makes own decisions. Now, the objective function of the government has been given in Eq. (8). To find the optimal results, we firstly take the derivative of Eq. (A.2) with respect to a and ${K_o}$ respectively. Then, we have

$$\begin{aligned} \left\{ \begin{array}{l} - pf \left( {{q^*} \left( {{K_o},a} \right) } \right) \frac{{\partial {q^*} \left( {{K_o},a} \right) }}{{\partial a}} = \frac{{\partial {\theta ^*} \left( {{K_o},a} \right) }}{{\partial a}} - {p_e} \left( {{e_o} - a} \right) {\rho ^\prime } \left( {{\theta ^*} \left( {{K_o},a} \right) } \right) \frac{{\partial {\theta ^*} \left( {{K_o},a} \right) }}{{\partial a}} + {p_e}\rho \left( {{\theta ^*} \left( {{K_o},a} \right) } \right) ;\\ - pf \left( {{q^*}\left( {{K_o},a} \right) } \right) \frac{{\partial {q^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}} = \frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}} - {p_e} - {p_e}\left( {{e_o} - a} \right) {\rho ^\prime }\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) \frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}}. \end{array} \right. \nonumber \\ \end{aligned}$$

(A.4)

Reducing the above equations, we get

$$\begin{aligned} \frac{{\partial {q^*}\left( {{K_o},a} \right) }}{{\partial a}} = - \frac{{{p_e}\rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) }}{{pf\left( {{q^*}\left( {{K_o},a} \right) } \right) }};\begin{array}{*{20}{c}} {}&{} \end{array}\frac{{\partial {q^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}} = \frac{{{p_e}}}{{pf\left( {{q^*}\left( {{K_o},a} \right) } \right) }}. \end{aligned}$$

(A.5)

By taking the derivative of Eq. (A.3) with respect to a and ${K_o}$ respectively, we have

$$\begin{aligned} \left\{ \begin{array}{l} {p_e}\left( {{e_o} - a} \right) \rho ''\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) \frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial a}} = {p_e}\rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) ;\\ {p_e}\left( {{e_o} - a} \right) \rho ''\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) \frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}} = 0. \end{array} \right. \end{aligned}$$

(A.6)

Reducing the above equations, we get

$$\begin{aligned} \frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial a}} = \frac{{\rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) }}{{({e_o} - a)\rho ''\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) }};\begin{array}{*{20}{c}} {}&{} \end{array}\frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}} = 0. \end{aligned}$$

(A.7)

Finally, the first-order conditions of W regarding a and ${K_o}$ can be derived as:

$$\begin{aligned} \left\{ \begin{array}{l} \frac{{\partial W}}{{\partial a}}=\frac{{\partial W}}{{\partial {q^*}\left( {{K_o},a} \right) }}\frac{{\partial {q^*}\left( {{K_o},a} \right) }}{{\partial a}} + \frac{{\partial W}}{{\partial {\theta ^*}\left( {{K_o},a} \right) }}\frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial a}}\\ \qquad = \left( {p{\bar{F}}\left( {{q^*}\left( {{K_o},a} \right) } \right) - (c + {\theta ^*}\left( {{K_o},a} \right) ) - \delta {e_o} \left( {1 - \rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right) } \right) \times \left( { - \frac{{{p_e}\rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) }}{{pf\left( {{q^*}\left( {{K_o},a} \right) } \right) }}} \right) \\ \quad \qquad +\left( {\delta {e_o}{q^*}\left( {{K_o},a} \right) \rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) - {q^*}\left( {{K_o},a} \right) } \right) \times \frac{{\rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) }}{{({e_o} - a)\rho ''\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) }} = 0;\\ \frac{{\partial W}}{{\partial {K_o}}} = \frac{{\partial W}}{{\partial {q^*}\left( {{K_o},a} \right) }}\frac{{\partial {q^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}} + \frac{{\partial W}}{{\partial {\theta ^*}\left( {{K_o},a} \right) }}\frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}}\\ \;\,\qquad = \left( {p{\bar{F}}\left( {{q^*}\left( {{K_o},a} \right) } \right) - \left( {c + {\theta ^*}\left( {{K_o},a} \right) } \right) - \delta {e_o} \left( {1 - \rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right) } \right) \times \frac{{{p_e}}}{{pf\left( {{q^*}\left( {{K_o},a} \right) } \right) }}\\ \;\,\quad \qquad + \left( {\delta {e_o}{q^*}\left( {{K_o},a} \right) \rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) - {q^*}\left( {{K_o},a} \right) } \right) \times 0 = 0. \end{array} \right. \nonumber \\ \end{aligned}$$

(A.8)

Reducing the above equations, we get

$$\begin{aligned} \left\{ \begin{array}{l} \delta {e_o}\rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) - 1 = 0;\\ p{\bar{F}}({q^*}\left( {{K_o},a} \right) ) - (c + {\theta ^*}\left( {{K_o},a} \right) ) - \delta {e_o} \left( {1 - \rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right) = 0. \end{array} \right. \end{aligned}$$

(A.9)

We verify that the determinant of the Hessian matrix is negative definite:

$$\begin{aligned} \left\{ \begin{array}{l} \frac{{{\partial ^2}W}}{{\partial {{\left( {{q^*}\left( {{K_o},a} \right) } \right) }^2}}} = - pf\left( {{q^*}\left( {{K_o},a} \right) } \right) ;\\ \frac{{{\partial ^2}W}}{{\partial {{\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) }^2}}} = \delta {q^*}\left( {{K_o},a} \right) {e_o}{\rho ^{\prime \prime }}\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) ;\\ \frac{{{\partial ^2}W}}{{\partial {q^*}\left( {{K_o},a} \right) \partial {\theta ^*}\left( {{K_o},a} \right) }} = \delta {e_o}{\rho ^\prime }\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) - 1 = 0; \end{array} \right. \end{aligned}$$

$$\begin{aligned} {H_4}= & {} \frac{{{\partial ^2}W}}{{\partial {a^2}}}= \left( {\frac{{\partial {q^*} \left( {{K_o},a} \right) }}{{\partial a}},\frac{{\partial {\theta ^*} \left( {{K_o},a} \right) }}{{\partial a}}} \right) \left( {\begin{array}{*{20}{c}} {\frac{{{\partial ^2}W}}{{\partial {{\left( {{q^*} \left( {{K_o},a} \right) } \right) }^2}}}}&{}{\frac{{{\partial ^2}W}}{{\partial {q^*} \left( {{K_o},a} \right) \partial {\theta ^*} \left( {{K_o},a} \right) }}}\\ {\frac{{{\partial ^2}W}}{{\partial {q^*} \left( {{K_o},a} \right) \partial {\theta ^*} \left( {{K_o},a} \right) }}}&{}{\frac{{{\partial ^2}W}}{{\partial {{\left( {{\theta ^*} \left( {{K_o},a} \right) } \right) }^2}}}} \end{array}} \right) \left( \begin{array}{l} \frac{{\partial {q^*} \left( {{K_o},a} \right) }}{{\partial a}}\\ \frac{{\partial {\theta ^*} \left( {{K_o},a} \right) }}{{\partial a}} \end{array} \right) \\&+ \left( {\frac{{\partial W}}{{\partial {q^*}\left( {{K_o},a} \right) }},\frac{{\partial W}}{{\partial {\theta ^*}\left( {{K_o},a} \right) }}} \right) \left( \begin{array}{l} \frac{{{\partial ^2}{q^*}\left( {{K_o},a} \right) }}{{\partial {a^2}}}\\ \frac{{{\partial ^2}{\theta ^*}\left( {{K_o},a} \right) }}{{\partial {a^2}}} \end{array} \right) \\= & {} - \frac{{{{\left( {{p_e}\rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right) }^2}}}{{pf({q^*}\left( {{K_o},a} \right) )}} + \frac{{\delta {q^*}\left( {{K_o},a} \right) {e_o}{{\left( {\rho ^{\prime }\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right) }^2}}}{{{{\left( {{e_o} - a} \right) }^2}\rho ^{\prime \prime }\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) }} < 0;\\ {H_5}= & {} \frac{{{\partial ^2}W}}{{\partial a\partial {K_o}}} = \left( {\frac{{\partial {q^*} \left( {{K_o},a} \right) }}{{\partial {K_o}}},\frac{{\partial {\theta ^*} \left( {{K_o},a} \right) }}{{\partial {K_o}}}} \right) \left( {\begin{array}{*{20}{c}} {\frac{{{\partial ^2}W}}{{\partial {{\left( {{q^*} \left( {{K_o},a} \right) } \right) }^2}}}}&{}{\frac{{{\partial ^2}W}}{{\partial {q^*} \left( {{K_o},a} \right) \partial {\theta ^*} \left( {{K_o},a} \right) }}}\\ {\frac{{{\partial ^2}W}}{{\partial {q^*} \left( {{K_o},a} \right) \partial {\theta ^*} \left( {{K_o},a} \right) }}}&{}{\frac{{{\partial ^2}W}}{{\partial {{\left( {{\theta ^*} \left( {{K_o},a} \right) } \right) }^2}}}} \end{array}} \right) \left( \begin{array}{l} \frac{{\partial {q^*} \left( {{K_o},a} \right) }}{{\partial a}}\\ \frac{{\partial {\theta ^*} \left( {{K_o},a} \right) }}{{\partial a}} \end{array} \right) \\&+ \left( {\frac{{\partial W}}{{\partial {q^*}\left( {{K_o},a} \right) }},\frac{{\partial W}}{{\partial {\theta ^*}\left( {{K_o},a} \right) }}} \right) \left( \begin{array}{l} \frac{{{\partial ^2}{q^*}\left( {{K_o},a} \right) }}{{\partial a\partial {K_o}}}\\ \frac{{{\partial ^2}{\theta ^*}\left( {{K_o},a} \right) }}{{\partial a\partial {K_o}}} \end{array} \right) \\= & {} \frac{{{p_e}^{{2}}\rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) }}{{pf\left( {{q^*}\left( {{K_o},a} \right) } \right) }};\\ {H_6}= & {} \frac{{{\partial ^2}W}}{{\partial {K_o}^2}} = \left( {\frac{{\partial {q^*} \left( {{K_o},a} \right) }}{{\partial {K_o}}},\frac{{\partial {\theta ^*} \left( {{K_o},a} \right) }}{{\partial {K_o}}}} \right) \left( {\begin{array}{*{20}{c}} {\frac{{{\partial ^2}W}}{{\partial {{\left( {{q^*} \left( {{K_o},a} \right) } \right) }^2}}}}&{}{\frac{{{\partial ^2}W}}{{\partial {q^*} \left( {{K_o},a} \right) \partial {\theta ^*} \left( {{K_o},a} \right) }}}\\ {\frac{{{\partial ^2}W}}{{\partial {q^*} \left( {{K_o},a} \right) \partial {\theta ^*} \left( {{K_o},a} \right) }}}&{}{\frac{{{\partial ^2}W}}{{\partial {{\left( {{\theta ^*} \left( {{K_o},a} \right) } \right) }^2}}}} \end{array}} \right) \left( \begin{array}{l} \frac{{\partial {q^*} \left( {{K_o},a} \right) }}{{\partial {K_o}}}\\ \frac{{\partial {\theta ^*} \left( {{K_o},a} \right) }}{{\partial {K_o}}}\end{array}\right) \\&+ \left( {\frac{{\partial W}}{{\partial {q^*}\left( {{K_o},a} \right) }},\frac{{\partial W}}{{\partial {\theta ^*}\left( {{K_o},a} \right) }}} \right) \left( \begin{array}{l} \frac{{{\partial ^2}{q^*}\left( {{K_o},a} \right) }}{{\partial {K_o}^2}}\\ \frac{{{\partial ^2}{\theta ^*}\left( {{K_o},a} \right) }}{{\partial {K_o}^2}} \end{array} \right) \\= & {} - \frac{{{p_e}^{{2}}}}{{pf({q^*}\left( {{K_o},a} \right) )}}; \\ {H_4}{H_6} - {H_5}^2= & {} \left( { - \frac{{{{\left( {{p_e}\rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right) }^2}}}{{pf\left( {{q^*}\left( {{K_o},a} \right) } \right) }} + \frac{{\delta {q^*}\left( {{K_o},a} \right) {e_o}{{\left( {\rho ^{\prime } \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right) }^2}}}{{{{\left( {{e_o} - a} \right) }^2}\rho ^{\prime \prime }\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) }}} \right) \times \left( { - \frac{{{p_e}^{{2}}}}{{pf({q^*}\left( {{K_o},a} \right) )}}} \right) \\&- {\left( {\frac{{{p_e}^{{2}}\rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) }}{{pf({q^*}\left( {{K_o},a} \right) )}}} \right) ^2}\\= & {} \frac{{{{\left( {{p_e}^2\rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right) }^2}}}{{{{\left( {pf({q^*}\left( {{K_o},a} \right) )} \right) }^2}}} - \frac{{\delta {q^*}\left( {{K_o},a} \right) {e_o}{{\left( {\rho ^{\prime }\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right) }^2}{p_e}^2}}{{pf\left( {{q^*}\left( {{K_o},a} \right) } \right) {{\left( {{e_o} - a} \right) }^2}\rho ^{\prime \prime }\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) }} - \frac{{{{\left( {{p_e}^2\rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right) }^2}}}{{{{\left( {pf({q^*}\left( {{K_o},a} \right) )} \right) }^2}}}\\= & {} - \frac{{{q^*}\left( {{K_o},a} \right) {p_e}^{{2}}\rho ^{\prime }\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) }}{{pf\left( {{q^*}\left( {{K_o},a} \right) } \right) {{\left( {{e_o} - a} \right) }^2}\rho ^{\prime \prime }\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) }} > 0. \end{aligned}$$

So the solution to the first order conditions gives the unique solution.

1.3 Step 3: To find the equilibrium solutions

Combining Eqs. (A.2), (A.3) and (A.9), we can derive the optimal solutions. The optimum decisions for the industry and government are shown as follows.

(A.10)

The form of the carbon emission allowances allocation mechanism is given as:

$$\begin{aligned} K\left( {\rho \left( \theta \right) } \right) = {K_o} - a\rho \left( \theta \right) = \left( {1 - \frac{\delta }{{{p_e}}}} \right) \left( {1 - \rho \left( \theta \right) } \right) {e_o}. \end{aligned}$$

(A.11)

$\square$

Appendix B: The quadratic form of allocation mechanism

In this paper, we attempt to explore the other form of emission allowance allocation mechanism preliminarily. Here, we suppose that the carbon emission allowances per product may be a concave or convex function with regard to the industry-level carbon emission reduction rate. Then we adopt the quadratic form to represent the allocation mechanism, this form is formulated as follows:

$$\begin{aligned} K\left( {\rho \left( \theta \right) } \right) = A{\rho ^2}\left( \theta \right) + B\rho \left( \theta \right) + C. \end{aligned}$$

(B.1)

Additionally, we suppose that this allocation mechanism is a quadratic function with mathematical characteristics including $K'\left( {\rho \left( \theta \right) } \right) < 0$ or $K'\left( {\rho \left( \theta \right) } \right) > 0$, $K\left( 0 \right) ={K_o}$, $\mathop {\lim }\nolimits _{\rho \left( \theta \right) \rightarrow 1} K\left( {\rho \left( \theta \right) } \right) = 0$. And the second-order condition of the carbon emission allowances per product regarding the carbon emission reduction rate is constant. It can be described as:

$$\begin{aligned} K''\left( {\rho \left( \theta \right) } \right) = 2A = 2a. \end{aligned}$$

(B.2)

Based on the above characteristics of this quadratic function, the quadratic form of allocation mechanism is equivalent to:

$$\begin{aligned} K\left( {\rho \left( \theta \right) } \right) = a{\rho ^2}\left( \theta \right) - \left( {{K_o} + a} \right) \rho \left( \theta \right) + {K_o}. \end{aligned}$$

(B.3)

In this section, we try to examine the rationality and feasibility of this form. First of all, we discuss how this mechanism affects the production and carbon reduction decisions of the industry. Substituting Eq. (B.3) into Eq. (1), we have:

$$\begin{aligned} \pi= & {} pS\left( q \right) + \left\{ {{p_e}\left[ {K\left( {\rho \left( \theta \right) } \right) - {e_o}\left( {1 - \rho \left( \theta \right) } \right) } \right] - c - \theta } \right\} q\nonumber \\= & {} pS\left( q \right) + \left\{ {{p_e}\left[ {a{\rho ^2}\left( \theta \right) - \left( {{K_o} + a} \right) \rho \left( \theta \right) \mathrm{{ + }}{K_o} - {e_o}\left( {1 - \rho \left( \theta \right) } \right) } \right] - c - \theta } \right\} q. \end{aligned}$$

(B.4)

When the mechanism is declared by the government, the industry is to make decisions to realize its own interests. We can get the optimal production quantity and investment of the industry by taking the derivative of the profit function. The two decision variables are determined by:

$$\begin{aligned} \left\{ \begin{array}{l} p{\bar{F}} \left( {{q^*}\left( {{K_o},a} \right) } \right) = c + {\theta ^*}\left( {{K_o},a} \right) - {p_e}\left[ \begin{array}{l} a{\rho ^2}\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) - \left( {{K_o} + a} \right) \rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) \\ +{K_o} - {e_o}\left( {1 - \rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right) \end{array} \right] ;\\ {p_e}\left[ {2a\rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) - \left( {{K_o} + a} \right) + {e_o}} \right] = \frac{1}{{\rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) }}, \end{array} \right. \end{aligned}$$

(B.5)

where ${q^*}\left( {{K_o},a} \right)$ and ${\theta ^*}\left( {{K_o},a} \right)$ are the optimal reaction functions of the industry given ${K_o}$ and a. Here, the objective function of the government can be written as:

$$\begin{aligned} W= & {} pS\left( q \right) - \left( {c + \theta } \right) q - \delta q {e_o}\left( {1 - \rho \left( \theta \right) } \right) \nonumber \\= & {} pS \left( {{q^*} \left( {{K_o},a} \right) } \right) - \left( {c + {\theta ^*} \left( {{K_o},a} \right) } \right) {q^*} \left( {{K_o},a} \right) - \delta {q^*} \left( {{K_o},a} \right) {e_o}\left( {1 - \rho \left( {{\theta ^*} \left( {{K_o},a} \right) } \right) } \right) .\nonumber \\ \end{aligned}$$

(B.6)

The first-order conditions of W regarding a and ${K_o}$ respectively are:

$$\begin{aligned} \frac{{\partial W}}{{\partial a}}= & {} \left[ {p{\bar{F}} \left( {{q^*} \left( {{K_o},a} \right) } \right) - \left( {c + {\theta ^*} \left( {{K_o},a} \right) } \right) - \delta {e_o}} \left( {1 - \rho \left( {{\theta ^*} \left( {{K_o},a} \right) } \right) } \right) \right] \nonumber \\&\times \frac{{{p_e}\left[ {{\rho ^2}\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) - \rho \left( {{\theta ^*} \left( {{K_o},a} \right) } \right) } \right] }}{{p f\left( {{q^*}\left( {{K_o},a} \right) } \right) }}\nonumber \\&+ \left[ {\delta {e_o}{q^*} \left( {{K_o},a} \right) \rho '\left( {{\theta ^*} \left( {{K_o},a} \right) } \right) - {q^*} \left( {{K_o},a} \right) } \right] \nonumber \\&\times \frac{{{p_e}\left[ {1 - 2\rho \left( {{\theta ^*} \left( {{K_o},a} \right) } \right) } \right] {{\left[ {\rho '\left( {{\theta ^*} \left( {{K_o},a} \right) } \right) } \right] }^2}}}{{2a{p_e}\rho '\left( {{\theta ^*} \left( {{K_o},a} \right) } \right) {{\left[ {\rho '\left( {{\theta ^*} \left( {{K_o},a} \right) } \right) } \right] }^2} + \rho ''\left( {{\theta ^*} \left( {{K_o},a} \right) } \right) }} = 0; \end{aligned}$$

(B.7)

$$\begin{aligned} \frac{{\partial W}}{{\partial {K_o}}}= & {} \left[ {p{\bar{F}} \left( {{q^*}\left( {{K_o},a} \right) } \right) - \left( {c + {\theta ^*}\left( {{K_o},a} \right) } \right) - \delta {e_o} \left( {1 - \rho \left( {{\theta ^*} \left( {{K_o},a} \right) } \right) } \right) } \right] \frac{{{p_e}\left[ {1 - \rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right] }}{{pf\left( {{q^*}\left( {{K_o},a} \right) } \right) }}\nonumber \\&+\left[ {\delta {e_o}{q^*} \left( {{K_o},a} \right) \rho '\left( {{\theta ^*} \left( {{K_o},a} \right) } \right) - {q^*} \left( {{K_o},a} \right) } \right] \nonumber \\&\times \frac{{{p_e}{{\left[ {\rho '\left( {{\theta ^*} \left( {{K_o},a} \right) } \right) } \right] }^2}}}{{2a{p_e}\rho '\left( {{\theta ^*} \left( {{K_o},a} \right) } \right) {{\left[ {\rho '\left( {{\theta ^*} \left( {{K_o},a} \right) } \right) } \right] }^2} + \rho ''\left( {{\theta ^*} \left( {{K_o},a} \right) } \right) }} = 0. \end{aligned}$$

(B.8)

By reducing the above equations, we get $\rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) = 1$. When the carbon emission reduction rate is 1, the government doesn’t need to allocate any allowance to the industry. However, the operations with no emission would not happen in real life and it is so hard for the industry to achieve 100% carbon emission reduction. Based on the actual practice, this paper assumes the carbon emission reduction rate is less than 1. So it doesn’t make sense to discuss the solution, i.e. $\rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) = 1$. This solution is infeasible and discarded since it doesn’t satisfy the assumption. Thus, there is no equilibrium solution under the strict quadratic form. Finally, the quadratic form will turn into the linear form.

$$\begin{aligned} K\left( {\rho \left( \theta \right) } \right)= & {} A\rho {\left( \theta \right) ^2}+B\rho \left( \theta \right) +C\mathrm{{ = }} a{\rho ^2}\left( \theta \right) - \left( {{K_o} + a} \right) \rho \left( \theta \right) + {K_o}\nonumber \\= & {} {K_o} - {K_o}\rho \left( \theta \right) = \left( {{e_o} - \frac{{\delta {e_o}}}{{{p_e}}}} \right) \left( {1 - \rho \left( \theta \right) } \right) , \end{aligned}$$

(B.9)

where $a=0$.

Proof

We explore whether the quadratic form of allocation mechanism is executable. In the same manner which has been adopted in “Appendix A”, we address this problem by using the backward induction.

1.1 Step 1: Solve the industry’s profit maximization problem

The quadratic form of allocation mechanism and the profit function of the industry have been shown in Eqs. (B.3) and (B.4). The first-order conditions of $\pi$ can be derived as:

$$\begin{aligned}&\frac{{\partial \pi }}{{\partial q}} = p{\bar{F}} \left( q \right) \nonumber \\&\quad + \left\{ {{p_e}\left[ {a{\rho ^2}\left( \theta \right) - \left( {{K_o} + a} \right) \rho \left( \theta \right) \mathrm{{ + }}{K_o} - {e_o}\left( {1 - \rho \left( \theta \right) } \right) } \right] - c - \theta } \right\} = 0;\quad \end{aligned}$$

(B.10)

$$\begin{aligned}&\frac{{\partial \pi }}{{\partial \theta }} = \left\{ {{p_e}\left[ {2a\rho \left( \theta \right) - \left( {{K_o} + a} \right) + {e_o}} \right] \rho '\left( \theta \right) - 1} \right\} q = 0. \end{aligned}$$

(B.11)

Then, the production quantity and low-carbon investment are given as follows:

$$\begin{aligned}&p{\bar{F}} \left( {{q^*} \left( {{K_o},a} \right) } \right) = c + {\theta ^*} \left( {{K_o},a} \right) \nonumber \\&\quad - {p_e} \left[ \begin{array}{l} a{\rho ^2}\left( {{\theta ^*} \left( {{K_o},a} \right) } \right) - \left( {{K_o} + a} \right) \rho \left( {{\theta ^*} \left( {{K_o},a} \right) } \right) \\ +{K_o} - {e_o}\left( {1 - \rho \left( {{\theta ^*} \left( {{K_o},a} \right) } \right) } \right) \end{array} \right] ; \end{aligned}$$

(B.12)

$$\begin{aligned}&{p_e}\left[ {2a\rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) - \left( {{K_o} + a} \right) + {e_o}} \right] = \frac{1}{{\rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) }}. \end{aligned}$$

(B.13)

1.2 Step 2: Solve the optimization problem faced by the government

Likewise, the government makes decisions according to anticipated behaviors of the industry. The objective function of the government has been shown in Eq. (B.6). To get the first-order conditions of W, we firstly take the derivative of Eq. (B.12) with respect to a, we obtain

$$\begin{aligned}&- pf\left( {{q^*}\left( {{K_o},a} \right) } \right) \frac{{\partial {q^*}\left( {{K_o},a} \right) }}{{\partial a}} =\frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial a}}\nonumber \\&\qquad - {p_e}\left[ \begin{array}{l} {\rho ^2}\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) + 2a\rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) \rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) \frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial a}}\\ - \left( {{K_o} + a} \right) \rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) \frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial a}}\\ - \rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) + {e_o}\rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) \frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial a}} \end{array} \right] \nonumber \\&\quad = - {p_e}\left[ {{\rho ^2}\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) - \rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right] \nonumber \\&\qquad + \frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial a}}\left[ {1 - {p_e}\left[ \begin{array}{l} 2a\rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) \rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) \\ - \left( {{K_o} + a} \right) \rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) + {e_o}\rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) \end{array} \right] } \right] \nonumber \\&\quad = - {p_e}\left[ {{\rho ^2}\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) - \rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right] . \end{aligned}$$

(B.14)

Reducing the above equation, we get

$$\begin{aligned} \frac{{\partial {q^*}\left( {{K_o},a} \right) }}{{\partial a}} = \frac{{{p_e}\left[ {{\rho ^2}\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) - \rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right] }}{{pf\left( {{q^*}\left( {{K_o},a} \right) } \right) }}. \end{aligned}$$

(B.15)

Then, we take the derivative of Eq. (B.12) with respect to ${K_o}$, we obtain

$$\begin{aligned}&- pf\left( {{q^*}\left( {{K_o},a} \right) } \right) \frac{{\partial {q^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}} = \frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}} \nonumber \\&\qquad - {p_e}\left[ \begin{array}{l} 2a\rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) \rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) \frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}} + 1\\ - \left( {{K_o} + a} \right) \rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) \frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}} - \rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) \\ + {e_o}\rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) \frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}} \end{array} \right] \nonumber \\&\quad = - {p_e} \left[ {1 - \rho \left( {{\theta ^*} \left( {{K_o},a} \right) } \right) } \right] \nonumber \\&\qquad + \frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}} \left[ {1 - {p_e} \left[ \begin{array}{l} 2a\rho \left( {{\theta ^*} \left( {{K_o},a} \right) } \right) \rho '\left( {{\theta ^*} \left( {{K_o},a} \right) } \right) \\ - \left( {{K_o} + a} \right) \rho ' \left( {{\theta ^*} \left( {{K_o},a} \right) } \right) + {e_o}\rho ' \left( {{\theta ^*} \left( {{K_o},a} \right) } \right) \end{array} \right] } \right] \nonumber \\&\quad = - {p_e}\left[ {1 - \rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right] . \end{aligned}$$

(B.16)

Reducing the above equation, we get

$$\begin{aligned} \frac{{\partial {q^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}} = \frac{{{p_e}\left[ {1 - \rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right] }}{{pf\left( {{q^*}\left( {{K_o},a} \right) } \right) }}. \end{aligned}$$

(B.17)

Next, we take the derivative of Eq. (B.13) with respect to a and ${K_o}$ respectively, we have

$$\begin{aligned}&{p_e} \left[ {2\rho \left( {{\theta ^*} \left( {{K_o},a} \right) } \right) + 2a\rho ' \left( {{\theta ^*} \left( {{K_o},a} \right) } \right) \frac{{\partial {\theta ^*} \left( {{K_o},a} \right) }}{{\partial a}} - 1} \right] = - \frac{{\rho '' \left( {{\theta ^*} \left( {{K_o},a} \right) } \right) }}{{{{\left[ {\rho ' \left( {{\theta ^*} \left( {{K_o},a} \right) } \right) } \right] }^2}}}\frac{{\partial {\theta ^*} \left( {{K_o},a} \right) }}{{\partial a}}; \end{aligned}$$

(B.18)

$$\begin{aligned}&{p_e}\left[ {2a\rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) \frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}} - 1} \right] = - \frac{{\rho ''\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) }}{{{{\left[ {\rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right] }^2}}}\frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}}. \end{aligned}$$

(B.19)

Reducing the above equations, we get

$$\begin{aligned}&\frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial a}} = \frac{{{p_e}\left[ {1 - 2\rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right] {{\left[ {\rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right] }^2}}}{{2a{p_e}\rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) {{\left[ {\rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right] }^2} + \rho ''\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) }}; \qquad \end{aligned}$$

(B.20)

$$\begin{aligned}&\frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}} = \frac{{{p_e}{{\left[ {\rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right] }^2}}}{{2a{p_e}\rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) {{\left[ {\rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right] }^2} + \rho ''\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) }}. \end{aligned}$$

(B.21)

At last, we have the first-order conditions of W as follows:

$$\begin{aligned} \frac{{\partial W}}{{\partial a}}= & {} p{\bar{F}} \left( {{q^*}\left( {{K_o},a} \right) } \right) \frac{{\partial {q^*}\left( {{K_o},a} \right) }}{{\partial a}} - \left( {c + {\theta ^*}\left( {{K_o},a} \right) } \right) \frac{{\partial {q^*}\left( {{K_o},a} \right) }}{{\partial a}} \nonumber \\&-\, {q^*}\left( {{K_o},a} \right) \frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial a}} - \delta {e_o}\left( {1 - \rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right) \frac{{\partial {q^*}\left( {{K_o},a} \right) }}{{\partial a}}\nonumber \\&+ \,\delta {e_o}{q^*}\left( {{K_o},a} \right) \rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) \frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial a}} = \left[ p{\bar{F}} \left( {{q^*} \left( {{K_o},a} \right) } \right) - \left( {c + {\theta ^*} \left( {{K_o},a} \right) } \right) \right. \nonumber \\&\left. -\, \delta {e_o}\left( {1 - \rho \left( {{\theta ^*} \left( {{K_o},a} \right) } \right) } \right) \right] \frac{{{p_e}\left[ {{\rho ^2}\left( {{\theta ^*} \left( {{K_o},a} \right) } \right) - \rho \left( {{\theta ^*} \left( {{K_o},a} \right) } \right) } \right] }}{{pf\left( {{q^*} \left( {{K_o},a} \right) } \right) }}\nonumber \\&+ \, \left[ {\delta {e_o}{q^*} \left( {{K_o},a} \right) \rho '\left( {{\theta ^*} \left( {{K_o},a} \right) } \right) - {q^*} \left( {{K_o},a} \right) } \right] \nonumber \\&\frac{{{p_e}\left[ {1 - 2\rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right] {{\left[ {\rho '\left( {{\theta ^*} \left( {{K_o},a} \right) } \right) } \right] }^2}}}{{2a{p_e}\rho '\left( {{\theta ^*} \left( {{K_o},a} \right) } \right) {{\left[ {\rho '\left( {{\theta ^*} \left( {{K_o},a} \right) } \right) } \right] }^2} + \rho ''\left( {{\theta ^*} \left( {{K_o},a} \right) } \right) }} = 0;\end{aligned}$$

(B.22)

$$\begin{aligned} \frac{{\partial W}}{{\partial {K_o}}}= & {} p{\bar{F}} \left( {{q^*}\left( {{K_o},a} \right) } \right) \frac{{\partial {q^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}} - \left( {c + {\theta ^*}\left( {{K_o},a} \right) } \right) \frac{{\partial {q^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}} - {q^*}\left( {{K_o},a} \right) \frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}}\nonumber \\&- \,\delta {e_o}\left( {1 - \rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right) \frac{{\partial {q^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}} + \delta {e_o}{q^*}\left( {{K_o},a} \right) \rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) \frac{{\partial {\theta ^*}\left( {{K_o},a} \right) }}{{\partial {K_o}}}\nonumber \\= & {} \left[ {p{\bar{F}} \left( {{q^*}\left( {{K_o},a} \right) } \right) - \left( {c + {\theta ^*}\left( {{K_o},a} \right) } \right) - \delta {e_o} \left( {1 - \rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right) } \right] \frac{{{p_e}\left[ {1 - \rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right] }}{{pf\left( {{q^*}\left( {{K_o},a} \right) } \right) }}\nonumber \\&+\,\left[ {\delta {e_o}{q^*}\left( {{K_o},a} \right) \rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) - {q^*}\left( {{K_o},a} \right) } \right] \nonumber \\&\quad \times \frac{{{p_e}{{\left[ {\rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right] }^2}}}{{2a{p_e}\rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) {{\left[ {\rho '\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) } \right] }^2} + \rho ''\left( {{\theta ^*}\left( {{K_o},a} \right) } \right) }} = 0. \end{aligned}$$

(B.23)

We get the final solution by combining and reducing Eqs. (B.12), (B.13), (B.22) and (B.23), which is $\rho \left( {{\theta ^*}\left( {{K_o},a} \right) } \right) = 1$.

As we all know, this situation in which there is no carbon emission is unrealistic and it is difficult for the industry to reach such a perfect level of carbon emission reduction. With the constraint $0 \le \rho \left( \theta \right) < 1$, this solution should be discarded. $\square$

Appendix C: Proof of Corollary 2

Proof

Under the carbon emission allowances allocation mechanism, the industry’s optimal production and low-carbon investment are:

$$\begin{aligned}&{\bar{F}}\left( {{q^\mathrm{{*}}}} \right) = \frac{{ {c + {\theta ^ * }} + \delta {e_o}\left( {1 - \rho \left( {{\theta ^ * }} \right) } \right) }}{p}; \end{aligned}$$

(C.1)

$$\begin{aligned}&\delta {e_o}\rho '\left( {{\theta ^ * }} \right) = 1. \end{aligned}$$

(C.2)

To examine how those exogenous variables affect the optimal decisions, we introduce an exponential form of emission reduction rate, i.e., $\rho \left( \theta \right) = 1 - {e^{ - \beta \theta }}$. Substituting it into Eq. (C.2), the optimal low-carbon investment of the industry is determined by

$$\begin{aligned} \rho '\left( {{\theta ^ * }} \right) = - {e^{ - \beta {\theta ^ * }}} \times \left( { - \beta } \right) = \beta {e^{ - \beta {\theta ^ * }}} = \frac{1}{{\delta {e_o}}}. \end{aligned}$$

(C.3)

By calculating, we have

$$\begin{aligned} {\theta ^ * } = \frac{{\ln \left( {\delta {e_o}\beta } \right) }}{\beta }. \end{aligned}$$

(C.4)

In the following, we take the derivatives of ${\theta ^*}$ with respect to the related parameters.

$$\begin{aligned}&\frac{{\partial {\theta ^ * }}}{{\partial {e_o}}} = \frac{1}{\beta } \times \frac{1}{{\delta {e_o}\beta }} \times \delta \beta = \frac{1}{{{e_o}\beta }} > 0; \end{aligned}$$

(C.5)

$$\begin{aligned}&\frac{{\partial {\theta ^ * }}}{{\partial \delta }} = \frac{1}{\beta } \times \frac{1}{{\delta {e_o}\beta }} \times {e_o}\beta = \frac{1}{{\beta \delta }} > 0. \end{aligned}$$

(C.6)

Substituting Eq. (C.4) into Eq. (C.1), the optimal production is determined by

$$\begin{aligned} {\bar{F}}\left( {{q^\mathrm{{*}}}} \right) = \frac{1}{p}\left( {c + \frac{{\ln \left( {\delta {e_o}\beta } \right) + 1}}{\beta }} \right) . \end{aligned}$$

(C.7)

Then, we take the derivatives of ${q ^*}$ with respect to the related parameters.

$$\begin{aligned}&\frac{{\partial {q^ * }}}{{\partial p}} = \frac{1}{{{p^2}f\left( {q^ * } \right) }}\left( {c + \frac{{\ln \left( {\delta {e_o}\beta } \right) + 1}}{\beta }} \right) > 0; \end{aligned}$$

(C.8)

$$\begin{aligned}&\frac{{\partial {q^ * }}}{{\partial c}} = \frac{1}{p} \times \left( { - \frac{1}{{f\left( {q^ * } \right) }}} \right) = - \frac{1}{{pf\left( {q^ * } \right) }} < 0; \end{aligned}$$

(C.9)

$$\begin{aligned}&\frac{{\partial {q^ * }}}{{\partial {e_o}}} = \frac{1}{p} \times \frac{1}{\beta } \times \frac{1}{{\delta {e_o}\beta }} \times \delta \beta \times \left( { - \frac{1}{{f\left( {q^ * } \right) }}} \right) = - \frac{1}{{p{e_o}\beta f\left( {q^ * } \right) }} < 0; \end{aligned}$$

(C.10)

$$\begin{aligned}&\frac{{\partial {q^ * }}}{{\partial \delta }} = \frac{1}{p} \times \frac{1}{\beta } \times \frac{1}{{\delta {e_o}\beta }} \times {e_o}\beta \times \left( { - \frac{1}{{f\left( {q^ * } \right) }}} \right) = - \frac{1}{{p\delta \beta f\left( {q^ * } \right) }} < 0. \end{aligned}$$

(C.11)

$\square$

Appendix D: Proof of Proposition 3

Proof

In this section, we simulate the entire industry with two firms. The carbon emission allowances per product depends on the average carbon reduction level of the industry, which has been shown in Eq. (14). Here, the expected sale of firm 1 is

$$\begin{aligned} S\left( {{q_1}} \right)= & {} E\left( {\lambda x \wedge {q_1}} \right) = \left\{ {\begin{array}{*{20}{c}} {{q_1},}&{}{\lambda x \ge {q_1}}\\ {\lambda x,}&{}{\lambda x < {q_1}} \end{array}} \right. = \int _0^{\frac{{{q_1}}}{\lambda }} {\lambda xf\left( x \right) dx} + \int _{\frac{{{q_1}}}{\lambda }}^{ + \infty } {{q_1}f\left( x \right) dx} \nonumber \\= & {} \lambda x \times F\left( x \right) \left| {\begin{array}{*{20}{c}} {\frac{{{q_1}}}{\lambda }}\\ 0 \end{array}} \right. - \int _0^{\frac{{{q_1}}}{\lambda }} {F\left( x \right) d\left( {\lambda x} \right) } + {q_1}F\left( x \right) \left| {\begin{array}{*{20}{c}} { + \infty }\\ {\frac{{{q_1}}}{\lambda }} \end{array}} \right. \nonumber \\= & {} {q_1}F\left( {\frac{{{q_1}}}{\lambda }} \right) - \int _0^{\frac{{{q_1}}}{\lambda }} {F\left( x \right) d\left( {\lambda x} \right) } + {q_1} - {q_1}F\left( {\frac{{{q_1}}}{\lambda }} \right) \nonumber \\= & {} {q_1} - \lambda \int _0^{\frac{{{q_1}}}{\lambda }} {F\left( x\right) dx}. \end{aligned}$$

(D.1)

The expected sale of firm 2 is

$$\begin{aligned} S\left( {{q_2}} \right)= & {} E\left[ {\left( {1 - \lambda } \right) x \wedge {q_2}} \right] = \left\{ {\begin{array}{*{20}{c}} {\left( {1 - \lambda } \right) x,}&{}{\left( {1 - \lambda } \right) x < {q_2}}\\ {{q_2},}&{}{\left( {1 - \lambda } \right) x \ge {q_2}} \end{array}} \right. \nonumber \\= & {} \int _0^{\frac{{{q_2}}}{{1 - \lambda }}} {\left( {1 - \lambda } \right) xf\left( x \right) dx} + \int _{\frac{{{q_2}}}{{1 - \lambda }}}^{ + \infty } {{q_2}f\left( x \right) dx} \nonumber \\= & {} \left( {1 - \lambda } \right) x \times F\left( x \right) \left| {\begin{array}{*{20}{c}} {\frac{{{q_2}}}{{1 - \lambda }}}\\ 0 \end{array}} \right. - \int _0^{\frac{{{q_2}}}{{1 - \lambda }}} {F\left( x \right) d\left( {\left( {1 - \lambda } \right) x} \right) } + {q_2}F\left( x \right) \left| {\begin{array}{*{20}{c}} { + \infty }\\ {\frac{{{q_2}}}{{1 - \lambda }}} \end{array}} \right. \nonumber \\= & {} \left( {1 - \lambda } \right) \frac{{{q_2}}}{{1 - \lambda }}F\left( {\frac{{{q_2}}}{{1 - \lambda }}} \right) - \left( {1 - \lambda } \right) \int _0^{\frac{{{q_2}}}{{1 - \lambda }}} {F\left( x \right) dx} + {q_2} - {q_2}F\left( {\frac{{{q_2}}}{{1 - \lambda }}} \right) \nonumber \\= & {} {q_2} - \left( {1 - \lambda } \right) \int _0^{\frac{{{q_2}}}{{1 - \lambda }}} {F\left( x \right) dx}. \end{aligned}$$

(D.2)

Then, we obtain the profit functions of two firms that have been presented in Eq. (16). The first-order conditions can be derived as:

$$\begin{aligned} \left\{ \begin{array}{l} \frac{{\partial {\pi _1}}}{{\partial {q_1}}} = p \left( {1 - F \left( {\frac{{{q_1}}}{\lambda }} \right) } \right) + \left[ {{p_e}{e_o} \left( {1 - \frac{\delta }{{{p_e}}}} \right) \left[ {\lambda {e^{ - {\beta _1}{\theta _1}}} + \left( {1 - \lambda } \right) {e^{ - {\beta _2}{\theta _2}}}} \right] - {p_e}{e_o}{e^{ - {\beta _1}{\theta _1}}} - c - {\theta _1}} \right] = 0;\\ \frac{{\partial {\pi _1}}}{{\partial {\theta _1}}} = \left[ { - {p_e}{e_o}\left( {1 - \frac{\delta }{{{p_e}}}} \right) \lambda {e^{ - {\beta _1}{\theta _1}}}{\beta _1} + {p_e}{e_o}{e^{ - {\beta _1}{\theta _1}}}{\beta _1} - 1} \right] {q_1} = 0;\\ \frac{{\partial {\pi _2}}}{{\partial {q_2}}} = p \left( {1 - F \left( {\frac{{{q_2}}}{{1 - \lambda }}} \right) } \right) + \left[ {{p_e}{e_o} \left( {1 - \frac{\delta }{{{p_e}}}} \right) \left[ {\lambda {e^{ - {\beta _1}{\theta _1}}} + \left( {1 - \lambda } \right) {e^{ - {\beta _2}{\theta _2}}}} \right] - {p_e}{e_o}{e^{ - {\beta _2}{\theta _2}}} - c - {\theta _2}} \right] = 0;\\ \frac{{\partial {\pi _2}}}{{\partial {\theta _2}}} = \left[ { - {p_e}{e_o}\left( {1 - \frac{\delta }{{{p_e}}}} \right) \left( {1 - \lambda } \right) {e^{ - {\beta _2}{\theta _2}}}{\beta _2} + {p_e}{e_o}{e^{ - {\beta _2}{\theta _2}}}{\beta _2} - 1} \right] {q_2} = 0. \end{array} \right. \nonumber \\ \end{aligned}$$

(D.3)

According to $\frac{{\partial {\pi _1}}}{{\partial {\theta _1}}}=0$ and $\frac{{\partial {\pi _2}}}{{\partial {\theta _2}}}=0$, we have

$$\begin{aligned} \left\{ \begin{array}{l} {p_e}{e_o}{\beta _1}{e^{ - {\beta _1}{\theta _1}}}\left( {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) = 1;\\ {p_e}{e_o}{\beta _2}{e^{ - {\beta _2}{\theta _2}}}\left( {1 - \left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) = 1. \end{array} \right. \end{aligned}$$

(D.4)

Reducing the above equations, the optimal low-carbon investment of each firm can be derived as:

$$\begin{aligned} \left\{ \begin{array}{l} {{\theta _1} ^*} = \frac{{\ln \left( {{p_e}{e_o}{\beta _1}\left( {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) } \right) }}{{{\beta _1}}};\\ {{\theta _2} ^*} = \frac{{\ln \left( {{p_e}e{}_o{\beta _2}\left( {1 - \left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) } \right) }}{{{\beta _2}}}. \end{array} \right. \end{aligned}$$

(D.5)

$\square$

Appendix E: Proofs of Corollaries in Section 5

1.1 E.1: Proof of Corollary 3

Proof

The carbon emission reduction rate of each firm has been given in Eq. (18). Based on the assumption in Sect. 3, we have $0 \le {\rho _i}\left( {{\theta _i}} \right) < 1$. Hence, we can derive the boundary conditions of carbon price. From Eq. (18), we have

$$\begin{aligned}&0 < \frac{1}{{{e_o}{\beta _1}\left( {{p_e} - \lambda \left( {{p_e} - \delta } \right) } \right) }} \le 1; \end{aligned}$$

(E.1)

$$\begin{aligned}&0 < \frac{1}{{{e_o}{\beta _2}\left( {{p_e} - \left( {1 - \lambda } \right) \left( {{p_e} - \delta } \right) } \right) }} \le 1. \end{aligned}$$

(E.2)

Since ${1 - \frac{\delta }{{{p_e}}}} > 0$ and $\mathrm{{\lambda }} \in \left[ {0,1} \right]$, the above inequations $0 < \frac{1}{{{e_o}{\beta _1}\left( {{p_e} - \lambda \left( {{p_e} - \delta } \right) } \right) }}$ and $0 < \frac{1}{{{e_o}{\beta _2}\left( {{p_e} - \left( {1 - \lambda } \right) \left( {{p_e} - \delta } \right) } \right) }}$ always hold.

From Eqs. (E.1) and (E.2), we observe that ${e_o}{\beta _1}\left( {{p_e} - \lambda \left( {{p_e} - \delta } \right) } \right) \ge 1$ and ${e_o}{\beta _2}\left( {{p_e} - \left( {1 - \lambda } \right) \left( {{p_e} - \delta } \right) } \right) \ge 1$. So the carbon price should satisfy the conditions presented as follows:

$$\begin{aligned} \begin{array}{l} {p_e} \ge \frac{1}{{1 - \lambda }}\left( {\frac{1}{{{e_o}{\beta _1}}} - \lambda \delta } \right) ;\\ {p_e} \ge \frac{1}{\lambda }\left( {\frac{1}{{{e_o}{\beta _2}}} - \left( {1 - \lambda } \right) \delta } \right) . \end{array} \end{aligned}$$

(E.3)

Thus we derive the Corollary 3. According to the above conditions, we also get

$$\begin{aligned} {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) }\ge & {} \frac{1}{{{p_e}{e_o}{\beta _1}}}> 0;\nonumber \\ {1 - \left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) }\ge & {} \frac{1}{{{p_e}e{}_o{\beta _2}}} > 0. \end{aligned}$$

(E.4)

$\square$

1.2 E.2: Proof of Corollary 4

Proof

By taking the derivatives of the firm 1’s optimal low-carbon investment with respect to the related parameters, we have

$$\begin{aligned} \frac{{\partial {{\theta _1} ^*}}}{{\partial \lambda }}= & {} \frac{1}{{{\beta _1}}} \times \frac{1}{{{p_e}{e_o}{\beta _1} \left[ {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right] }} \times \left[ { - {p_e}{e_o}{\beta _1} \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right] = - \frac{{1 - \frac{\delta }{{{p_e}}}}}{{{\beta _1} \left[ {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right] }} < 0; \end{aligned}$$

(E.5)

$$\begin{aligned} \frac{{\partial {{\theta _1} ^*}}}{{\partial \delta }}= & {} \frac{1}{{{\beta _1}}} \times \frac{1}{{{p_e}{e_o}{\beta _1}\left[ {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right] }} \times \frac{{\lambda {p_e}{e_o}{\beta _1}}}{{{p_e}}} = \frac{\lambda }{{{\beta _1}{p_e}\left[ {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right] }} > 0; \end{aligned}$$

(E.6)

$$\begin{aligned} \frac{{\partial {{\theta _1} ^*}}}{{\partial {p_e}}}= & {} \frac{1}{{{\beta _1}}} \times \frac{1}{{{p_e}{e_o}{\beta _1}\left[ {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right] }} \times \left( {1 - \lambda } \right) {e_o}{\beta _1} = \frac{{1 - \lambda }}{{{p_e}{\beta _1}\left[ {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right] }} > 0; \end{aligned}$$

(E.7)

$$\begin{aligned} \frac{{\partial {{\theta _1} ^*}}}{{\partial {e_o}}}= & {} \frac{1}{{{\beta _1}}} \times \frac{1}{{{p_e}{e_o}{\beta _1}\left[ {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right] }} \times {p_e}{\beta _1}\left[ {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right] = \frac{1}{{{\beta _1}{e_o}}} > 0; \end{aligned}$$

(E.8)

$$\begin{aligned} \frac{{\partial {{\theta _1} ^*}}}{{\partial {\beta _1}}}= & {} \frac{{\frac{1}{{{p_e}{e_o}{\beta _1}\left( {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) }} \times {p_e}{e_o}\left( {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) \times {\beta _1} - \ln \left( {{p_e}{e_o}{\beta _1}\left( {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) } \right) }}{{{\beta _1}^2}}\nonumber \\= & {} \frac{{1 - \ln \left( {{p_e}{e_o}{\beta _1}\left( {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) } \right) }}{{{\beta _1}^2}}; \end{aligned}$$

(E.9)

• (1) If $\frac{1}{{{p_e}{e_o}\left( {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) }} \le {\beta _1} < \frac{e}{{{p_e}{e_o}\left( {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) }}$, then $\frac{{\partial {{\theta _1} ^*}}}{{\partial {\beta _1}}} > 0$;

• (2) If ${\beta _1} \ge \frac{e}{{{p_e}{e_o}\left( {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) }}$, then $\frac{{\partial {{\theta _1} ^*}}}{{\partial {\beta _1}}} \le 0$.

Similarly, by taking the derivatives of the firm 2’s optimal low-carbon investment with respect to the related parameters, we have

$$\begin{aligned} \frac{{\partial {{\theta _2} ^*}}}{{\partial \lambda }}= & {} \frac{1}{{{\beta _2}}} \times \frac{1}{{{p_e}{e_o}{\beta _2}\left[ {1 - \left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right] }} \times {p_e}{e_o}{\beta _2}\left( {1 - \frac{\delta }{{{p_e}}}} \right) = \frac{{1 - \frac{\delta }{{{p_e}}}}}{{{\beta _2}\left[ {1 - \left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right] }} > 0; \end{aligned}$$

(B.10)

$$\begin{aligned} \frac{{\partial {{\theta _2} ^*}}}{{\partial \delta }}= & {} \frac{1}{{{\beta _2}}} \times \frac{1}{{{p_e}{e_o}{\beta _2} \left[ {1 - \left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right] }} \times \frac{{\left( {1 - \lambda } \right) {p_e}{e_o}{\beta _2}}}{{{p_e}}} = \frac{{1 - \lambda }}{{{\beta _2}{p_e} \left[ {1 - \left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right] }} > 0; \end{aligned}$$

(B.11)

$$\begin{aligned} \frac{{\partial {{\theta _2} ^*}}}{{\partial {p_e}}}= & {} \frac{1}{{{\beta _2}}} \times \frac{1}{{{p_e}{e_o}{\beta _2} \left[ {1 - \left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right] }} \times \left( {{e_o}{\beta _2} - \left( {1 - \lambda } \right) {e_o}{\beta _2}} \right) = \frac{\lambda }{{{\beta _2}{p_e} \left[ {1 - \left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right] }} > 0; \end{aligned}$$

(B.12)

$$\begin{aligned} \frac{{\partial {{\theta _2} ^*}}}{{\partial {e_o}}}= & {} \frac{1}{{{\beta _2}}} \times \frac{1}{{{p_e}{e_o}{\beta _2}\left[ {1 - \left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right] }} \times {p_e}{\beta _2}\left[ {1 - \left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right] = \frac{1}{{{\beta _2}{e_o}}} > 0; \end{aligned}$$

(B.13)

$$\begin{aligned} \frac{{\partial {{\theta _2} ^*}}}{{\partial {\beta _2}}}= & {} \frac{{\frac{1}{{{p_e}{e_o}{\beta _2}\left( {1 - \left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) }} \times {p_e}{e_o}\left( {1 - \left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) \times {\beta _2} - \ln \left( {{p_e}{e_o}{\beta _2}\left( {1 - \left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) } \right) }}{{{\beta _2}^2}}\nonumber \\= & {} \frac{{1 - \ln \left( {{p_e}{e_o}{\beta _2} \left( {1 - \left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) } \right) }}{{{\beta _2}^2}}; \end{aligned}$$

(B.14)

• (1) If $\frac{1}{{{p_e}{e_o}\left( {1 - \left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) }} \le {\beta _2} < \frac{e}{{{p_e}{e_o}\left( {1 - \left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) }}$, then $\frac{{\partial {{\theta _2} ^*}}}{{\partial {\beta _2}}} > 0$;

• (2) If ${\beta _2} \ge \frac{e}{{{p_e}{e_o}\left( {1 - \left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) }}$, then $\frac{{\partial {{\theta _2} ^*}}}{{\partial {\beta _2}}} \le 0$. $\square$

1.3 E.3: Proof of Corollary 5

Proof

By taking the derivatives of the critical point ${\hat{\beta }_1}$ with respect to the related parameters, we can get

$$\begin{aligned}&\frac{{\partial {{\hat{\beta }}_1}}}{{\partial {p_e}}} = \frac{{ - e{e_o}\left( {1 - \lambda } \right) }}{{{e_o}^2{{\left( {{p_e} - \lambda {p_e} + \lambda \delta } \right) }^2}}} = \frac{{ - e\left( {1 - \lambda } \right) }}{{{e_o}{{\left( {{p_e} - \lambda {p_e} + \lambda \delta } \right) }^2}}} < 0; \end{aligned}$$

(E.15)

$$\begin{aligned}&\frac{{\partial {{\hat{\beta }}_1}}}{{\partial \delta }} = \frac{{ - e{e_o}\lambda }}{{{e_o}^2{{\left( {{p_e} - \lambda {p_e} + \lambda \delta } \right) }^2}}} = \frac{{ - e\lambda }}{{{e_o}{{\left( {{p_e} - \lambda {p_e} + \lambda \delta } \right) }^2}}} < 0; \end{aligned}$$

(E.16)

$$\begin{aligned}&\frac{{\partial {{\hat{\beta }}_1}}}{{\partial {e_o}}} = \frac{{ - e\left( {{p_e} - \lambda {p_e} + \lambda \delta } \right) }}{{{e_o}^2{{\left( {{p_e} - \lambda {p_e} + \lambda \delta } \right) }^2}}} = \frac{{ - e}}{{{e_o}^2\left( {{p_e} - \lambda {p_e} + \lambda \delta } \right) }} < 0; \end{aligned}$$

(E.17)

$$\begin{aligned}&\frac{{\partial {{\hat{\beta }}_1}}}{{\partial \lambda }} = \frac{{ - e{e_o}\left( { - {p_e} + \delta } \right) }}{{{e_o}^2{{\left( {{p_e} - \lambda {p_e} + \lambda \delta } \right) }^2}}} = \frac{{ - e\left( {\delta - {p_e}} \right) }}{{{e_o}{{\left( {{p_e} - \lambda {p_e} + \lambda \delta } \right) }^2}}} > 0. \end{aligned}$$

(E.18)

Then, by taking the derivatives of the critical point ${\hat{\beta }_2}$ with respect to the related parameters, we can get

$$\begin{aligned}&\frac{{\partial {{\hat{\beta }}_2}}}{{\partial {p_e}}} = \frac{{ - e{e_o}\lambda }}{{{e_o}^2{{\left( {\lambda {p_e} + \delta - \lambda \delta } \right) }^2}}} = \frac{{ - e\lambda }}{{{e_o}{{\left( {\lambda {p_e} + \delta - \lambda \delta } \right) }^2}}} < 0; \end{aligned}$$

(E.19)

$$\begin{aligned}&\frac{{\partial {{\hat{\beta }}_2}}}{{\partial \delta }} = \frac{{ - e{e_o}\left( {1 - \lambda } \right) }}{{{e_o}^2{{\left( {\lambda {p_e} + \delta - \lambda \delta } \right) }^2}}} = \frac{{ - e\left( {1 - \lambda } \right) }}{{{e_o}{{\left( {\lambda {p_e} + \delta - \lambda \delta } \right) }^2}}} < 0; \end{aligned}$$

(E.20)

$$\begin{aligned}&\frac{{\partial {{\hat{\beta }}_2}}}{{\partial {e_o}}} = \frac{{ - e\left( {\lambda {p_e} + \delta - \lambda \delta } \right) }}{{{e_o}^2{{\left( {\lambda {p_e} + \delta - \lambda \delta } \right) }^2}}} = \frac{{ - e}}{{{e_o}^2\left( {\lambda {p_e} + \delta - \lambda \delta } \right) }} < 0; \end{aligned}$$

(E.21)

$$\begin{aligned}&\frac{{\partial {{\hat{\beta }}_2}}}{{\partial \lambda }} = \frac{{ - e{e_o}\left( {{p_e} - \delta } \right) }}{{{e_o}^2{{\left( {\lambda {p_e} + \delta - \lambda \delta } \right) }^2}}} = \frac{{ - e\left( {{p_e} - \delta } \right) }}{{{e_o}{{\left( {\lambda {p_e} + \delta - \lambda \delta } \right) }^2}}} < 0. \end{aligned}$$

(E.22)

$\square$

1.4 E.4: Proof of Corollary 6

Proof

By taking the derivatives of the firm 1’s emission gap with respect to the related parameters, we have

$$\begin{aligned} \frac{{\partial \Delta {e_1}}}{{\partial {\beta _1}}}= & {} \frac{{ - {p_e}}}{{{p_e}^2{\beta _1}^2}} = - \frac{1}{{{p_e}{\beta _1}^2}} < 0; \end{aligned}$$

(E.23)

$$\begin{aligned} \frac{{\partial \Delta {e_1}}}{{\partial {\beta _2}}}= & {} - \frac{{\left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) }}{{{p_e}\left( {1 - \left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) }}\left( { - \frac{1}{{{\beta _2}^2}}} \right) = \frac{{\left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) }}{{{\beta _2}^2\left( {\lambda {p_e} + \left( {1 - \lambda } \right) \delta } \right) }} > 0;\qquad \qquad \end{aligned}$$

(E.24)

$$\begin{aligned} \frac{{\partial \Delta {e_1}}}{{\partial \delta }}= & {} \frac{{\left( {1 - \lambda } \right) {\beta _2} \left( {1 - \left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) + {{\left( {1 - \lambda } \right) }^2} \left( {1 - \frac{\delta }{{{p_e}}}} \right) {\beta _2}}}{{{p_e}^2{\beta _2}^2{{\left( {1 - \left( {1 - \lambda } \right) \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) }^2}}} \nonumber \\= & {} \frac{{1 - \lambda }}{{{{\left( {\lambda {p_e} + \left( {1 - \lambda } \right) \delta } \right) }^2} {\beta _2}}} > 0; \end{aligned}$$

(E.25)

$$\begin{aligned} \frac{{\partial \Delta {e_1}}}{{\partial \lambda }}= & {} \frac{{\left( {1 - \frac{\delta }{{{p_e}}}} \right) {p_e} \left( {\lambda + \left( {1 - \lambda } \right) \frac{\delta }{{{p_e}}}} \right) + {{\left( {1 - \frac{\delta }{{{p_e}}}} \right) }^2} \left( {1 - \lambda } \right) {p_e}}}{{{p_e}^2{\beta _2}{{\left( {\lambda + \left( {1 - \lambda } \right) \frac{\delta }{{{p_e}}}} \right) }^2}}} \nonumber \\= & {} \frac{{{p_e} - \delta }}{{{{\left( {\lambda {p_e} + \left( {1 - \lambda } \right) \delta } \right) }^2}{\beta _2}}} > 0. \end{aligned}$$

(E.26)

Then, by taking the derivatives of the firm 2’s emission gap with respect to the related parameters, we have

$$\begin{aligned} \frac{{\partial \Delta {e_2}}}{{\partial {\beta _2}}}= & {} \frac{{ - {p_e}}}{{{p_e}^2{\beta _2}^2}} = - \frac{1}{{{p_e}{\beta _2}^2}} < 0; \end{aligned}$$

(E.27)

$$\begin{aligned} \frac{{\partial \Delta {e_2}}}{{\partial {\beta _1}}}= & {} - \frac{{\lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) }}{{{p_e}\left( {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) }}\left( { - \frac{1}{{{\beta _1}^2}}} \right) = \frac{{\lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) }}{{{\beta _1}^2\left( {{p_e} - \lambda {p_e} + \lambda \delta } \right) }} > 0; \end{aligned}$$

(E.28)

$$\begin{aligned} \frac{{\partial \Delta {e_2}}}{{\partial \delta }}= & {} \frac{{\lambda \left( {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) + {\lambda ^2}\left( {1 - \frac{\delta }{{{p_e}}}} \right) }}{{{p_e}^2{\beta _1}{{\left( {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) }^2}}} = \frac{\lambda }{{{\beta _1}{{\left( {{p_e} - \lambda {p_e} + \lambda \delta } \right) }^2}}} > 0;\qquad \end{aligned}$$

(E.29)

$$\begin{aligned} \frac{{\partial \Delta {e_2}}}{{\partial \lambda }}= & {} \frac{{ - \left( {1 - \frac{\delta }{{{p_e}}}} \right) {p_e} \left( {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) - \lambda {p_e}{{\left( {1 - \frac{\delta }{{{p_e}}}} \right) }^2}}}{{{p_e}^2{\beta _1}{{\left( {1 - \lambda \left( {1 - \frac{\delta }{{{p_e}}}} \right) } \right) }^2}}} \nonumber \\= & {} \frac{{ - \left( {{p_e} - \delta } \right) }}{{{\beta _1}{{\left( {{p_e} - {p_e}\lambda + \lambda \delta } \right) }^2}}} < 0. \end{aligned}$$

(E.30)

$\square$