Allocation policy considering firm’s time-varying emission reduction in a cap-and-trade system

Abstract

The global warming problem has attracted worldwide attention. Cap-and-trade has been increasingly used in many countries to reduce carbon emissions. However, some firms are concerned about the additional costs required for carbon reduction, and another important concern comes from grandfathering in permit allocation. This paper incorporates these costs and cap-and-trade concerns into a multi-period carbon reduction problem in a Stackelberg game. The findings show that neither cap-and-trade nor the firm’s carbon reduction choice will always benefit the environment. From the government’s perspective, we identify the optimal grandfathering scheme to maximize social welfare that incorporates economic and environmental concerns. We demonstrate that the socially optimal emissions level depends on the level of low-carbon technology and the environmental recovery cost.

Appendices

Appendix A: Proof of firm’s optimal solutions

Proof

We first give the optimal solutions if the firm chooses to reduce product carbon emissions at the carbon reduction rate ${\varphi _t}$, where the profit function is:

$$\begin{aligned} {\pi _t}\mathrm{{(}}{p_t},{\varphi _t}\mathrm{{) = }}\left( {{p_t} - c - k{\varphi _t} - {x_t}(1 - {\varphi _t})} \right) {q_t} + \delta {x_{t + 1}}{A_{t+1}} + \frac{{\ln (1 - {\varphi _t})}}{\lambda } \end{aligned}$$

Then, the first-order conditions can be derived as:

$$\begin{aligned} \begin{aligned} \frac{{\partial {\pi _t}}}{{\partial {p_t}}}&= a + c - 2{p_t} - \left( {{x_t} - \delta {\varepsilon _{t + 1}}{x_{t + 1}}} \right) \left( { - 1 + {\varphi _t}} \right) + k{\varphi _t} = 0\\ \frac{{\partial {\pi _t}}}{{\partial {\varphi _t}}}&= \left( {a - {p_t}} \right) \left( { - k + {x_t}} \right) - \delta {\varepsilon _{t + 1}}\left( {a - {p_t}} \right) {x_{t + 1}} - \frac{1}{{\lambda \left( {1 - {\varphi _t}} \right) }} = 0 \end{aligned} \end{aligned}$$

Accordingly, we have the second-order derivatives as follows.

$$\begin{aligned} \begin{aligned} A&= \frac{{{\partial ^2}{\pi _t}}}{{\partial p_t^2}} = - 2 < 0\\ B&= \frac{{{\partial ^2}{\pi _t}}}{{\partial {p_t}{\varphi _t}}}=k - {x_t} + {\varepsilon _{t + 1}}\delta {x_{t + 1}} \\ C&= \frac{{{\partial ^2}{\pi _t}}}{{\partial \varphi _t^2}} = - \frac{1}{{\lambda {{\left( { - 1 + {\varphi _t}} \right) }^2}}} \end{aligned} \end{aligned}$$

When $\lambda > \frac{8}{{{{(a - c - k)}^2}}}$, the determinant of Hessian is:

$$\begin{aligned} \left| H \right| = {\left( {k -\left( { {x_t} - \varepsilon _{t+1} \delta {x_{t+1}}} \right) } \right) ^2}\left( \frac{8}{{{{\left( { - a + c + k + \sqrt{{{(a - c - k)}^2} - 8/\lambda } } \right) }^2}}} - 1\right) > 0. \end{aligned}$$

Thus, the solution to the first order conditions gives the unique solution. From the first order conditions, we get the stationary point as:

$$\begin{aligned} \varphi _t^ *= & {} \frac{{\sqrt{{{(a - c - k)}^2} - 8/\lambda } + 2 \left( {{x_t} - \delta {\varepsilon _{t + 1}}{x_{t + 1}}} \right) - a + c - k}}{{2\left( {\left( {{x_t} - \delta {\varepsilon _{t + 1}}{x_{t + 1}}} \right) - k} \right) }}\\ p_t^ *= & {} \frac{1}{4}\left( {3a + c + k - \sqrt{{{(a - c - k)}^2} - 8/\lambda } } \right) \\ \pi _t^ *= & {} \frac{1}{{8 }}( - 4/\lambda + {(a - c - k)^2} + (a - c - k)\sqrt{{{(a - c - k)}^2} - 8/\lambda } ) \\&+ \,\frac{1}{\lambda }\ln \left[ {\frac{{(a - c - k) - \sqrt{{{(a - c - k)}^2} - 8/\lambda } }}{{2\left( {\left( {{x_t} - \delta {\varepsilon _{t + 1}}{x_{t + 1}}} \right) - k} \right) }}} \right] \end{aligned}$$

$\square$

Appendix B: Proof of Proposition 1

Proof

Note that $\lambda > \frac{8}{{{{(a - c - k)}^2}}}$ is required to guarantee the quantity under the square root sign is positive. We assume it is always satisfied for all the analyse. To guarantee the positivity of demands, the market size should satisfy the condition: $a>c+k$. We now compare the profits with and without carbon reduction to determine firm’s optimal selection. We get the carbon price threshold ${{\bar{x}}_t}$ by solving $\pi _t^ *={{\tilde{\pi }} }_t^ *$. The form of ${{\bar{x}}_t}$ is given as:

$$\begin{aligned} {{\bar{x}}_t}=\frac{1}{2}\left( {a - c + k + 2\delta {\varepsilon _{t + 1}}{x_{t + 1}}} \right) - \frac{1}{2}\sqrt{{{(a - c - k)}^2} - 8/\lambda }. \end{aligned}$$

It is easy to estimate that $\frac{1}{2}\left( {a - c + k} \right) > \frac{1}{2}\sqrt{{{(a - c - k)}^2} - 8/\lambda }$. Thus, the value of ${{\bar{x}}_t}$ above is positive. When $x_t>{{\bar{x}}_t}$, the firm choose to reduce product carbon emissions because $\pi _t^ *>{{\tilde{\pi }} }_t^ *$. Otherwise, $\pi _t^ *\le {{\tilde{\pi }} }_t^ *$ that the firm chooses not change the product carbon attribute but reduce the production quantity as the reaction to cap-and-trade regulation.

Within the constraint $a>c+k$ (which we is assumed to be satisfied to guarantee the positivity of the demand, i.e., $q_t^ *>0$), and $x_t>{{\bar{x}}_t}$, the inequity $0\le \varphi _t^ *<1$ holds. Note that the have assumed that condition $\lambda > \frac{8}{{{{(a - c - k)}^2}}}$ holds for all this paper. $\square$

Appendix C: Proof of Corollary 1

Proof

When $x_t>{{\bar{x}}_t}$, we give the deviations of the optimal carbon reduction rate with respect to the carbon policy relating parameters.

$$\begin{aligned} \frac{{\partial \varphi _t^ * }}{{\partial {x_t}}}= & {} \frac{{a - c - k - \sqrt{ - 8/\lambda + {{(a - c - k)}^2}} }}{{2{{\left( {k - {x_t} + {\varepsilon _{t + 1}}\delta {x_{t + 1}}} \right) }^2}}}>0 \\ \frac{{\partial \varphi _t^ * }}{{\partial {x_{t + 1}}}}= & {} - \delta {\varepsilon _{t + 1}}\frac{{a - c - k - \sqrt{ - 8/\lambda + {{(a - c - k)}^2}} }}{{2{{\left( {k - {x_t} + {\varepsilon _{t + 1}}\delta {x_{t + 1}}} \right) }^2}}}< 0 \\ \frac{{\partial \varphi _t^ * }}{{\partial {\varepsilon _{t + 1}}}}= & {} - \delta {x _{t + 1}}\frac{{a - c - k - \sqrt{ - 8/\lambda + {{(a - c - k)}^2}} }}{{2{{\left( {k - {x_t} + {\varepsilon _{t + 1}}\delta {x_{t + 1}}} \right) }^2}}} < 0 \end{aligned}$$

It is trivial to show that the optimal price $p_t^*$ is unaffected by the carbon policy because $\frac{{\partial p_t^ * }}{{\partial {x_t}}} = \frac{{\partial p_t^ * }}{{\partial {x_{t + 1}}}} = \frac{{\partial p_t^ * }}{{\partial {\varepsilon _{t + 1}}}} = 0$.

When $x_t \le {{\bar{x}}_t}$, the optimal price is ${\tilde{p}}_t^ * = \frac{{a + c + \left( {{x_t} - \delta {\varepsilon _{t + 1}}{x_{t + 1}}} \right) }}{2}$, and $\frac{{\partial {\tilde{p}}_t^ * }}{{\partial {x_t}}} > 0$, $\frac{{\partial {\tilde{p}}_t^ * }}{{\partial {x_{t + 1}}}} < 0$ and $\frac{{\partial {\tilde{p}}_t^ * }}{{\partial {\varepsilon _{t + 1}}}} < 0$. $\square$

Appendix D: Proof of Corollary 2

Proof

We give the deviations of the optimal profits with respect to the carbon policy relating parameters. In the product carbon reduction scenario, From the condition $x_t>{{\bar{x}}_t}$, we have $k<{x_t}- \delta \varepsilon _{t + 1} {x_{t+1}}$ as the necessary condition. In another word, the inequity $x_t>{{\bar{x}}_t}$ never holds if $k>{x_t}- \delta \varepsilon _{t + 1} {x_{t+1}}$.

$$\begin{aligned} \begin{aligned} \frac{{\partial \pi _t^ * }}{{\partial {x_t}}}&= \frac{1}{{\lambda \left( {k - {x_t} + \delta {\varepsilon _{t + 1}}{x_{t + 1}}} \right) }} < 0 \\ \frac{{\partial \pi _t^ * }}{{\partial {x_{t + 1}}}}&= \frac{{\delta {\varepsilon _{t + 1}}}}{{\lambda \left( {{x_t} - \delta {\varepsilon _{t + 1}}{x_{t + 1}} - k} \right) }}> 0 \\ \frac{{\partial \pi _t^ * }}{{\partial {\varepsilon _{t + 1}}}}&= \frac{{\delta {x_{t + 1}}}}{{\lambda \left( {{x_t} - \delta {\varepsilon _{t + 1}}{x_{t + 1}} - k} \right) }} > 0 \end{aligned} \end{aligned}$$

In the no product carbon reduction scenario:

$$\begin{aligned} \begin{aligned} \frac{{\partial \tilde{\pi }_t^ * }}{{\partial {x_t}}}&= - \frac{{a - c - {x_t} + \delta {\varepsilon _{t + 1}}{x_{t + 1}}}}{2} < 0 \\ \frac{{\partial \tilde{\pi }_t^ * }}{{\partial {x_{t + 1}}}}&= \delta {\varepsilon _{t + 1}}\frac{{a - c - {x_t} + \delta {\varepsilon _{t + 1}}{x_{t + 1}}}}{2}> 0 \\ \frac{{\partial \tilde{\pi }_t^ * }}{{\partial {\varepsilon _{t + 1}}}}&= \delta {x_{t + 1}}\frac{{a - c - {x_t} + \delta {\varepsilon _{t + 1}}{x_{t + 1}}}}{2} > 0 \end{aligned} \end{aligned}$$

$\square$

Appendix E: Proof of Corollary 3

Proof

Note that the current profits can increase in the next period’s carbon price but it does not implies that the overall profits in the carbon regulated periods can increase in the carbon price. Adding up the profits in period t and $t+1$ which is affected by ${x_{t+1}}$, note that all the derivations are zero except $\frac{{\partial \pi _{t - 1}^ * }}{{\partial {x_t}}}$ and $\frac{{\partial \pi _t^ * }}{{\partial {x_t}}}$. Then in the product carbon reduction scenario we have:

$$\begin{aligned} \begin{aligned} \frac{{\partial \sum _t {\pi _t^ * } }}{{\partial {x_t}}}&=\frac{{\partial \pi _{t - 1}^ * }}{{\partial {x_t}}} + \frac{{\partial \pi _t^ * }}{{\partial {x_t}}} = \frac{{\delta {\varepsilon _t}}}{{\lambda \left( {{x_{t - 1}} - \delta {\varepsilon _t}{x_t} - k} \right) }} - \frac{1}{{\lambda \left( {{x_t} - \delta {\varepsilon _{t + 1}}{x_{t\mathrm{{ + 1}}}} - k} \right) }}\\ \frac{{{\partial ^2}\sum _t {\pi _t^ * } }}{{\partial {x_t}^2} }&= \frac{{{\delta ^2}{\varepsilon _t}^2}}{{\lambda {{\left( {k - {x_{t - 1}} + \delta {\varepsilon _t}{x_t}} \right) }^2}}} + \frac{1}{{\lambda {{\left( {k - {x_t} + \delta {\varepsilon _{t + 1}}{x_{t + 1}}} \right) }^2}}}>0 \end{aligned} \end{aligned}$$

From the first order condition we have:

$$\begin{aligned} \frac{{\partial \sum _t {\pi _t^ * } }}{{\partial {x_t}}}\;\mathrm{{> }}0 \Leftrightarrow \delta {\varepsilon _t} > \frac{{{x_{t - 1}} - \delta {\varepsilon _t}{x_t} - k}}{{{x_t} - \delta {\varepsilon _{t + 1}}{x_{t\mathrm{{ + 1}}}} - k}}. \end{aligned}$$

Or the condition can be equivalently written as:

$$\begin{aligned} \frac{{\partial \sum _t {\pi _t^ * } }}{{\partial {x_t}}}\;\mathrm{{> }}0 \Leftrightarrow {\varepsilon _t} > \frac{{{x_{t - 1}} - k}}{{\delta (2{x_t} - {\varepsilon _{t + 1}}{x_{t\mathrm{{ + 1}}}} - k)}}; \end{aligned}$$

or

$$\begin{aligned} \frac{{\partial \sum _t {\pi _t^ * } }}{{\partial {x_t}}}\;\mathrm{{> }}0 \Leftrightarrow {x_t} > \frac{{{x_{t - 1}} - k + \delta {\varepsilon _t}({\varepsilon _{t + 1}}{x_{t\mathrm{{ + 1}}}} + k)}}{{2\delta {\varepsilon _t}}}. \end{aligned}$$

In the no product carbon reduction scenario we have:

$$\begin{aligned} \begin{aligned} \frac{{\partial \sum _t {{\tilde{\pi }} _t^ * } }}{{\partial {x_t}}}&= \frac{{\partial {\tilde{\pi }} _{t - 1}^ * }}{{\partial {x_t}}} + \frac{{\partial {\tilde{\pi }} _t^ * }}{{\partial {x_t}}} = \frac{1}{2}\delta {\varepsilon _t}\left( {a - c - {x_{t - 1}} + \delta {\varepsilon _t}{x_t}} \right) - \frac{1}{2}\left( {a - c - {x_t} + \delta {\varepsilon _{t + 1}}{x_{t + 1}}} \right) \\ \frac{{{\partial ^2}\sum _t {{\tilde{\pi }} _t^ * } }}{{\partial {x_t}^2}}&= \frac{{1 + {\delta ^2}{\varepsilon _t}^2}}{2}>0 \end{aligned} \end{aligned}$$

From the first order condition we have

$$\begin{aligned} \frac{{\partial \sum _t {{\tilde{\pi }} _t^ * } }}{{\partial {x_t}}}\;> 0 \Leftrightarrow \delta {\varepsilon _t} > \frac{{a - c - {x_t} + \delta {\varepsilon _{t + 1}}{x_{t + 1}}}}{{a - c - {x_{t - 1}} + \delta {\varepsilon _t}{x_t}}}. \end{aligned}$$

Or the condition can be equivalently written as:

$$\begin{aligned} \frac{{\partial \sum _t {{\tilde{\pi }} _t^ * } }}{{\partial {x_t}}}\;> 0 \Leftrightarrow {x_t} > \frac{{a - c + \delta {\varepsilon _{t + 1}}{x_{t + 1}} - \delta {\varepsilon _t}(a - c - {x_{t - 1}})}}{{1 + {\delta ^2}{\varepsilon _t}^2}}. \end{aligned}$$

$\square$

Appendix F: Proof of Proposition 2

Proof

We compare the single period’s carbon emissions in different scenarios to give the conclusions in Proposition 2.

Comparing the total carbon emissions with and without product carbon reduction with all other parameters fixed, the inequity $E_t^* <{{\tilde{E}}}_t^*$ holds when ${{\bar{x}}_t}<{x_t} < \frac{1}{2}\sqrt{{{(a - c - k)}^2} - 8/\lambda } + \frac{1}{2}\left( {a - c + k + 2\delta {\varepsilon _{t + 1}}{x_{t + 1}}} \right)$. where ${{\bar{x}}_t}$ is given in Eq. 5 as the carbon price threshold for firm’s product carbon reduction. For simplicity, we denote ${{{\bar{x}}'}_t}\mathrm{{ = }}\frac{1}{2}\sqrt{{{(a - c - k)}^2} - 8/\lambda } + \frac{1}{2}\left( {a - c + k + 2\delta {\varepsilon _{t + 1}}{x_{t + 1}}} \right)$. So, the inequity $E_t^* <{{\tilde{E}}}_t^*$ holds when ${{\bar{x}}_t}<{x_t}< {{{\bar{x}}'}_t}$. If the carbon price exceeds ${{{\bar{x}}'}_t}$, we have $E_t^* >{{\tilde{E}}}_t^*$ where the firm emits more emissions in the product carbon reduction choice. The emissions increase comes from the production quantities increment. By comparing the production quantities, we have showed that ${q_t}^ * > {{\tilde{q}}}_t^ *$ always holds. $\square$

Appendix G: Proof of Corollary 4

Proof

We analyse the multi-periods’ carbon emissions in two product carbon reduction choices to give the conclusions in Corollary 4.

We can analysis the impacts of carbon price $x_t$ all over the carbon regulated periods in the sequential two periods t and $t-1$. If $x_t \le {{\bar{x}}_t}$, the total emission is ${{\tilde{E}}}_t^*\mathrm{{ = }}\frac{{a - c - \left( {{x_t} - \delta {\varepsilon _{t + 1}}{x_{1 + t}}} \right) }}{2}$ and $\frac{{\partial \sum _t {{{\tilde{E}}}_t^ * } }}{{\partial {x_t}}} = \frac{{\partial {{\tilde{E}}}_{t - 1}^ * }}{{\partial {x_t}}} + \frac{{\partial {{\tilde{E}}}_t^ * }}{{\partial {x_t}}} = \frac{1}{2}(\delta {\varepsilon _t} - 1)<0$. The carbon price in period t curb the current carbon emissions and promote the former period’s emission, while it has a overall negative impact on carbon emissions. If $x_t >{{\bar{x}}_t}$,

$$\begin{aligned} \begin{aligned} \frac{{\partial \sum _t {E_t^ * } }}{{\partial {x_t}}}&= \frac{{\partial E_{t - 1}^ * }}{{\partial {x_t}}} + \frac{{\partial E_t^ * }}{{\partial {x_t}}} = \frac{{\delta {\varepsilon _t}}}{{\lambda {{\left( {k - {x_{t - 1}} + \delta {\varepsilon _t}{x_t}} \right) }^2}}} - \frac{1}{{\lambda {{\left( {k - {x_t} + \delta {\varepsilon _{t + 1}}{x_{t + 1}}} \right) }^2}}} \\ \frac{{{\partial ^2}\sum _t { E_t^ * } }}{{\partial {x_t}^2}}&= - \frac{{2{\delta ^2}\varepsilon _t^2}}{{\lambda {{\left( {k - {x_{t - 1}} + \delta {\varepsilon _t}{x_t}} \right) }^3}}} - \frac{2}{{\lambda {{\left( {k - {x_t} + \delta {\varepsilon _{t + 1}}{x_{t + 1}}} \right) }^3}}}<0 \end{aligned} \end{aligned}$$

From the second order condition we know that the overall carbon emissions in the product carbon reduction scenario are convex in period t’s carbon price. In particular, from the first order condition we have:

$$\begin{aligned} \frac{{\partial \sum _t {E_t^ * } }}{{\partial {x_t}}}> 0 \Leftrightarrow \delta {\varepsilon _t} > \frac{{{{\left( {k - {x_{t - 1}} + \delta {\varepsilon _t}{x_t}} \right) }^2}}}{{{{\left( {k - {x_t} + \delta {\varepsilon _{t + 1}}{x_{t + 1}}} \right) }^2}}} \end{aligned}$$

The grandfathering proportion affects the overall carbon emissions by affecting only his previous period’s emissions and we have:

$$\begin{aligned} \begin{aligned} \frac{{\partial \sum _t {{{\tilde{E}}}_t^ * } }}{{\partial {\varepsilon _t}}}&= \frac{{\partial {{\tilde{E}}}_{t - 1}^ * }}{{\partial {\varepsilon _t}}}\mathrm{{ = }}\frac{{\delta {x_t}}}{2}> 0 \\ \frac{{\partial \sum _t {E_t^ * } }}{{\partial {\varepsilon _t}}}&= \frac{{\partial E_{t - 1}^ * }}{{\partial {\varepsilon _t}}} = \frac{{\delta {x_t}}}{{\lambda {{\left( {k - {x_{t - 1}} + \delta {\varepsilon _t}{x_t}} \right) }^2}}} > 0 \end{aligned} \end{aligned}$$

(G.1)

$\square$

Appendix H: Proof of Proposition 3

Proof

The optimal $\varepsilon _1, \varepsilon _2,\ldots \varepsilon _T$ is determined by maximizing the social welfare overall periods of all the firms. In period t, $\varepsilon _{t+1}$ effects the welfare independently, we can maximize the overall periods’ welfare by maximizing period t’s welfare through determining $\varepsilon _{t+1}$. Here, we find that $\varepsilon _{1}$ can not be determined. In period 1, the permits allocation works as subsidy and we do not discuss it in the later analyze. The government’s problem is then given as: $\mathop {\max }\limits _{{\varepsilon _{t + 1}}} {SW}^\prime ({\varepsilon _{t + 1}}) =$

$$\begin{aligned} \mathop {\max }\limits _{{\varepsilon _{t + 1}}} \sum _{i = 1}^n {\left( \left( {p_t^i({\varepsilon _{t + 1}}) - {c^i} - {k^i}\varphi _t^i({\varepsilon _{t + 1}})} \right) q_t^i + \frac{{\ln (1 - \varphi _t^i({\varepsilon _{t + 1}}))}}{{{\lambda ^i}}} - {w_t}E_t^i({\varepsilon _{t + 1}})\right) } \end{aligned}$$

If there are m firms make the product carbon reduction decision, and the number of the firms do not make this choice is $n-m$, the first order condition is given as $\frac{{dS{W^\prime }({\varepsilon _{t + 1}})}}{{d{\varepsilon _{t + 1}}}} = \frac{{dS{W^\prime }(\left. {{\varepsilon _{t + 1}}} \right| \varphi _t^i > 0)}}{{d{\varepsilon _{t + 1}}}} + \frac{{dS{W^\prime }(\left. {{\varepsilon _{t + 1}}} \right| \varphi _t^i = 0)}}{{d{\varepsilon _{t + 1}}}}=0$

For the product carbon reduction firms $i=1,2,\ldots m$, the social welfare’s first and second derivatives with respect to ${\varepsilon _{t + 1}}$ is:

$$\begin{aligned} \begin{aligned} \frac{{dS{W^\prime }(\left. {{\varepsilon _{t + 1}}} \right| \varphi _t^i> 0)}}{{d{\varepsilon _{t + 1}}}}&= \left( {{w_t} - {x_t} + \delta {\varepsilon _{t + 1}}{x_{t + 1}}} \right) \sum _{i = 1}^m { - \frac{{\delta {x_{t + 1}}}}{{{\lambda ^i}{{\left( {{k^i} - {x_t} + \delta {\varepsilon _{t + 1}}{x_{t + 1}}} \right) }^2}}}} \\ \frac{{{d^2}S{W^\prime }(\left. {{\varepsilon _{t + 1}}} \right| \varphi _t^i > 0)}}{{d\varepsilon _{t + 1}^2}}&= \sum _{i = 1}^m {\frac{{{\delta ^2}x_{t + 1}^2\left( { - {k^i} + 2{w_t} - {x_t} + \delta {\varepsilon _{t + 1}}{x_{t + 1}}} \right) }}{{{\lambda ^i}{{\left( {{k^i} - {x_t} + \delta {\varepsilon _{t + 1}}{x_{t + 1}}} \right) }^3}}}} \end{aligned} \end{aligned}$$

For the other firms $i=m+1, \ldots n$ choose ${\varphi _t^i({\varepsilon _{t + 1}})}=0$, the social welfare’s first derivative with respect to ${\varepsilon _{t + 1}}$ is:

$$\begin{aligned} \begin{aligned} \frac{{d{SW^\prime }(\left. {{\varepsilon _{t + 1}}} \right| \varphi _t^i = 0)}}{{d{\varepsilon _{t + 1}}}}&= \frac{{d\sum _{i = m + 1}^n {(({\tilde{p}}_t^i({\varepsilon _{t + 1}}) - {c^i}){{\tilde{q}}}_t^i({\varepsilon _{t + 1}}) - {w_t}{{\tilde{E}}}_t^i({\varepsilon _{t + 1}}))} }}{{d{\varepsilon _{t + 1}}}} \\&= \frac{{(n - m)\delta {x_{t + 1}}}}{2}\left( {{w_t} - {x_t} + \delta {\varepsilon _{t + 1}}{x_{t + 1}}} \right) \\ \frac{{{d^2}S{W^\prime }(\left. {{\varepsilon _{t + 1}}} \right| \varphi _t^i = 0)}}{{d{\varepsilon _{t + 1}}^2}}&= - \frac{{(n - m)}}{2}{\delta ^2}x_{t + 1}^2 < 0 \end{aligned} \end{aligned}$$

Therefore, we have the first order condition as $\frac{{dS{W^\prime }({\varepsilon _{t + 1}})}}{{d{\varepsilon _{t + 1}}}} =$

$$\begin{aligned} - \left( {{w_t} - {x_t} + \delta {\varepsilon _{t + 1}}{x_{t + 1}}} \right) \sum _{i = 1}^m {(\frac{{\delta {x_{t + 1}}}}{{{\lambda ^i}{{\left( {{k^i} - {x_t} + \delta {\varepsilon _{t + 1}}{x_{t + 1}}} \right) }^2}}}} + \frac{{(n - m)\delta {x_{t + 1}}}}{2})=0 \end{aligned}$$

It gives the solution $\varepsilon _{t + 1} = \frac{{{x_t} - {w_t}}}{{\delta {x_{t + 1}}}}$, at this point, the second derivative of social welfare is

$$\begin{aligned} \frac{{{d^2}S{W^\prime }({\varepsilon _{t + 1}})}}{{d{\varepsilon _{t + 1}}^2}}\mathrm{{ = }} - \sum _{i = 1}^m {\frac{{{\delta ^2}x_{t + 1}^2}}{{{\lambda ^i}{{\left( {{k^i} - {w_t}} \right) }^2}}}} - \frac{{(n - m)}}{2}{\delta ^2}x_{t + 1}^2 < 0. \end{aligned}$$

So we get the optimal grandfathering proportion $\varepsilon _{t + 1}^ * = \frac{{{x_t} - {w_t}}}{{\delta {x_{t + 1}}}}$ to maximize social welfare.

At $\varepsilon _{t + 1}^ * = \frac{{{x_t} - {w_t}}}{{\delta {x_{t + 1}}}}$, the optimal social welfare is

$$\begin{aligned} \begin{aligned}&\sum _{i=1}^m \frac{1}{8}\left( ({a^i} - {c^i} - {k^i})({a^i} - {c^i} - {k^i} + \sqrt{{{({a^i} - {c^i} - {k^i})}^2} - 8/{\lambda ^i}} ) - \frac{{{\lambda ^i}}}{4}\right) \\&\quad + \frac{1}{{{\lambda ^i}}}\ln \left( \frac{{{a^i} - {c^i} - {k^i} - \sqrt{{{({a^i} - {c^i} - {k^i})}^2} - 8/{\lambda ^i}} }}{{2\left( {{w_t} - {k^i}} \right) }}+ \sum _{i = m+1}^n {\frac{{{{({a^i} - {c^i} - {w_t})}^2}}}{4}}\right) \end{aligned} \end{aligned}$$

The social optimal welfare is decreasing in the social weight on environment because we have the first order conditions that $\sum _{i =1}^m {\frac{1}{{{\lambda ^i}({k^i} - {w_t})}}} - \sum _{i = m+1}^n {\frac{{{a^i} - {c^i} - {w_t}}}{2}} < 0$. In the second expression, $a^i> c^i+ {w_t}$ because the inequity $a> c + {x_t} - \delta {\varepsilon _{t + 1}}{x_{t + 1}}=c+ {w_t}$ holds to guarantee a positive demand in the no product carbon reduction scenario. $\square$

Appendix I: Proof of Corollary 5

Proof

Replacing $\varepsilon _t$ with the social optimal grandfathering proportion $\varepsilon _t^ *$ into the optimal solutions in the product carbon reduction scenario we have firm i’s optimal solutions at $\varepsilon _t =\varepsilon _t^ *$:

$$\begin{aligned} {\varphi _t^i}= & {} \frac{{\sqrt{{{(a^i - c^i - k^i)}^2} - 8/\lambda ^i } + 2w_t - a^i + c^i - k^i}}{{2\left( {w_t - k^i} \right) }} \\ {p_t^i}= & {} \frac{1}{4}\left( {3a^i + c^i + k^i - \sqrt{{{(a^i - c^i - k^i)}^2} - 8/\lambda ^i } } \right) \\ {\pi _t^i }= & {} \frac{1}{{8 }}\left( - 4/\lambda ^i + {(a^i - c^i - k^i)^2} + (a^i - c^i - k^i)\sqrt{{{(a^i - c^i - k^i)}^2} - 8/\lambda ^i }\right) \\&+\frac{1}{\lambda }\ln \frac{{(a - c - k) - \sqrt{{{(a - c - k)}^2} - 8/\lambda } }}{{2\left( {{w_t} - k} \right) }} \\ {E_t^i}= & {} \frac{1}{{\lambda ^i \left( {{w_t} - k^i} \right) }} \end{aligned}$$

In the above functions, $k^i<w_t$ is always satisfied in firm’s product carbon reduction scenario. Because in Proposition 1 the condition for this choice is $x_t>{\bar{x}}_t$, the inequity $k^i<{x_t}- \delta \varepsilon _{t + 1} {x_{t+1}}=w_t$ always holds as a necessary condition. Thus at social optimal, ${\varphi _t^ *}^i$ and ${E_t^\mathrm{{*}}}^i$ are positive. From the first derivation of profits $\frac{{\partial {\pi _t^ * }^i}}{{\partial {w_t}}}\mathrm{{ = }}\frac{1}{{\lambda ^i (k^i - {w_t})}} < 0$, we prove the firm’s profit in period t decreases in the social weight on environment. And the other derivations are: $\frac{{\partial {\varphi _t^ * }^i}}{{\partial {w_t}}} > 0$, $\frac{{\partial {p_t^ *}^i }}{{\partial {w_t}}} = 0$ and $\frac{{\partial {E_t^ * }^i}}{{\partial {w_t}}} < 0$. It implies that at social optimal the more the weight social puts on the environment, the higher the product carbon reduction rate the firms choose. $\square$