Optimal control model of an enterprise for single and inheriting periods of carbon emission reduction

Abstract

In this paper, based on the framework of the single-period model, we establish an optimal control model for the inheriting period which allows inter-phase banking and borrowing of allowances under the cap-and-trade system. By considering the abatement control policy and the initial auction amount of allowances, we optimize the problem in two steps. The two models can then be expressed using the Hamilton–Jacobi–Bellman (HJB) equations. In the framework of viscosity solution, we prove that the value functions in two models are the unique viscosity solutions of the corresponding HJB equations. Finally, we analyze the properties of optimal policy by referring to numerical results and a comparison of the two models is presented. Our results show that under the same circumstances, allowing inter-phase banking and borrowing of allowances indeed reduces the company’s abatement costs and has an incentive-based effect on the company’s emission reductions.

Appendix

Proof of Proposition 1

(i) First we prove that $V_1(x,t)$ is continuous with respect to x, uniformly in t.

Fix $t\in [0,T]$, $\forall x_2\ge x_1$, $\exists u_1^*.$, $u_2^*.\in {\mathscr {U}}$, such that

$$\begin{aligned}J_1(x_i,t;u_i^*.)-\inf _{u_i.\in {\mathscr {U}}}J_1(x_i,t;u_i.)\le \frac{1}{2}|x_2-x_1|,\quad i=1,2.\end{aligned}$$

According to Proposition 2, we have

$$\begin{aligned}&J_1(x_2,t;u.)\ge J_1(x_1,t;u.),\quad \forall u.\in {\mathscr {U}}.\\&V_1(x_2,t)\ge V_1(x_1,t). \end{aligned}$$

Then

$$\begin{aligned}&|V_1(x_2,t)-V_1(x_1,t)|=|\inf _{u_2.\in {\mathscr {U}}}J_1(x_2,t;u_2.)-\inf _{u_1.\in {\mathscr {U}}}J_1(x_1,t;u_1.)|\\&\quad =\inf _{u_2.\in {\mathscr {U}}}J_1(x_2,t;u_2.)-\inf _{u_1.\in {\mathscr {U}}}J_1(x_1,t;u_1.)\\&\quad \le J_1(x_2,t;u_1^*.)-\inf _{u_1.\in {\mathscr {U}}}J_1(x_1,t;u_1.)\\&\quad \le (J_1(x_2,t;u_1^*.)-\inf _{u_1.\in {\mathscr {U}}}J_1(x_1,t;u_1.))+|J_1(x_1,t;u_2^*.)-J_1(x_2,t;u_2^*.)+\frac{1}{2}|x_2-x_1||\\&\quad \le (J_1(x_2,t;u_1^*.)-J_1(x_1,t;u_1^*.)+\frac{1}{2}|x_2-x_1|)+|J_1(x_1,t;u_2^*.)-J_1(x_2,t;u_2^*.)\\&\qquad +\frac{1}{2}|x_2-x_1||\\&\quad \le |J_1(x_2,t;u_1^*.)-J_1(x_1,t;u_1^*.)+\frac{1}{2}|x_2-x_1||+|J_1(x_1,t;u_2^*.)-J_1(x_2,t;u_2^*.)\\&\qquad +\frac{1}{2}|x_2-x_1||\\&\quad \le |J_1(x_2,t;u_1^*.)-J_1(x_1,t;u_1^*.)|+|J_1(x_1,t;u_2^*.)-J_1(x_2,t;u_2^*.)|+|x_2-x_1|. \end{aligned}$$

$\forall u^*.\in {\mathscr {U}}$, we have

$$\begin{aligned}&|J_1(x_2,t;u^*.)-J_1(x_1,t;u^*.)|\\&\quad \le E\Big [e^{-r(T-t)}|P((X_{2,T}-N_1-N_2)^+-(X_{1,T}-N_1-N_2)^+)|\Big |X_{1,t}=x_1,X_{2,t}=x_2\Big ]\\&\quad \le E\Big [e^{-r(T-t)}P|X_{2,T}-X_{1,T}|\Big |X_{1,t}=x_1,X_{2,t}=x_2\Big ]\\&\quad =E\Big [e^{-r(T-t)}P|x_2-x_1|\Big ]. \end{aligned}$$

where $E[X_{1,T}|X_{1,t}=x_1]$ denotes the expectation of X in (3) at T with initial value $X_t=x_1$ and $E[X_{2,T}|X_{2,t}=x_2]$ denotes the expectation of X in (3) at T with initial value $X_t=x_2$.

That is, $\exists M_1>0$, such that

$$\begin{aligned}|J_1(x_2,t;u^*.)-J_1(x_1,t;u^*.)|<\frac{PM_1}{2}|x_2-x_1|.\end{aligned}$$

Then,

$$\begin{aligned}|V_1(x_2,t)-V_1(x_1,t)|<(PM_1+1)|x_2-x_1|.\end{aligned}$$

Below we prove the continuity of $V_1(x,t)$ with respect to t.

Fix $x\in R$, $\forall 0\le t_1\le t_2\le T$, $\exists u_1^*.$, $u_2^*.\in {\mathscr {U}}$, such that

$$\begin{aligned}0<J_1(x,t_i;u_i^*.)-\inf _{u_i.\in {\mathscr {U}}}J_1(x,t_i;u_i.)\le \frac{1}{2}|t_1-t_2|,\quad i=1,2.\end{aligned}$$

Similar to before, we have

$$\begin{aligned}|V_1(x,t_2)-V_1(x,t_1)|\le & {} |J_1(x,t_2;u_1^*.)-J_1(x,t_1;u_1^*.)|+|J_1(x,t_2;u_2^*.)\\&-J_1(x,t_1;u_2^*.)|+|t_1-t_2|.\end{aligned}$$

$\forall u^*.\in {\mathscr {U}}$, we have

$$\begin{aligned}&|J_1(x,t_2;u^*.)-J_1(x,t_1;u^*.)|\\&\quad \le E\Big [|\int _{t_2}^Te^{-r(s-t_2)}C(u^*)ds-\int _{t_1}^Te^{-r(s-t_1)}C(u^*)ds+e^{-r(T-t_2)}P(X_{2,T}-N_1-N_2)^+\\&\qquad -e^{-r(T-t_1)}P(X_{1,T}-N_1-N_2)^+|\Big |X_{2,t_2}=x,X_{1,t_1}=x\Big ]\\&\quad \le \frac{m{\bar{u}}^2}{2r}\Big [1-e^{-r(t_2-t_1)}\Big ]\\&\qquad +e^{-r(T-t_2)}E\Big [|P(X_{2,T}-N_1-N_2)^+-P(X_{1,T}-N_1-N_2)^+|\Big |X_{2,t_2}=x,X_{1,t_1}=x\Big ]\\&\qquad +|e^{-r(T-t_2)}-e^{-r(T-t_1)}|E\Big [|P(X_{1,T}-N_1-N_2)^+|\Big |X_{2,t_2}=x,X_{1,t_1}=x\Big ]. \end{aligned}$$

where $E[X_{1,T}|X_{1,t_1}=x]$ denotes the expectation of X in (3) at T with initial value $X_{t_1}=x$ and $E[X_{2,T}|X_{2,t_2}=x]$ denotes the expectation of X in (3) at T with initial value $X_{t_2}=x$.

Since

$$\begin{aligned}&E\Big [|P(X_{2,T}-N_1-N_2)^+-P(X_{1,T}-N_1-N_2)^+|\Big |X_{2,t_2}=x,X_{1,t_1}=x\Big ]\\&\quad \le E\Big [P|X_{2,T}-X_{1,T}|\Big |X_{2,t_2}=x,X_{1,t_1}=x\Big ]\\&\quad \le PC_1(1+|x|)\sqrt{t_2-t_1}, \end{aligned}$$

and

$$\begin{aligned}E\Big [|P(X_{1,T}-N_1-N_2)^+|\Big |X_{2,t_2}=x,X_{1,t_1}=x\Big ]\le PC_2(1+|x|),\end{aligned}$$

then, $\exists M_2>0$,$M_3>0$ such that

$$\begin{aligned}|J_1(x,t_2;u^*.)-J_1(x,t_1;u^*.)|<\frac{M_2}{2}(1+|x|)\sqrt{t_2-t_2}+\frac{M_3}{2}|t_2-t_1|.\end{aligned}$$

Therefore,

$$\begin{aligned}|V_1(x,t_2)-V_1(x,t_1)|<M_2(1+|x|)\sqrt{t_2-t_2}+(M_3+1)|t_2-t_1|.\end{aligned}$$

For (ii), similar to the technique in (i), we can prove $V_2(x,s,t)$ is continuous with respect to x, uniformly in t and continuous with respect to s, uniformly in t.

Below we prove the continuity of $V_2(x,s,t)$ with respect to t. Fix $x,s\in R\times R^+$, $\forall 0\le t_1\le t_2\le T$, $\exists u_1^*.$, $u_2^*.\in {\mathscr {U}}$, such that

$$\begin{aligned}0<J_2(x,s,t_i;u_i^*.)-\inf _{u_i.\in {\mathscr {U}}}J_2(x,s,t_i;u_i.)\le \frac{1}{2}|t_2-t_1|,\quad i=1,2.\end{aligned}$$

Similar to previous, we have

$$\begin{aligned}&|V_2(x,s,t_2)-V_2(x,s,t_1)|\\&\quad \le |J_2(x,s,t_2;u_1^*.)-J_2(x,s,t_1;u_1^*.)|+|J_2(x,s,t_2;u_2^*.)-J_2(x,s,t_1;u_2^*.)|+|t_2-t_1|. \end{aligned}$$

$\forall u^*.\in {\mathscr {U}}$, let

$$\begin{aligned}Q_i:=\min \{P,S_{i,T}\}(X_{i,T}-N_1-N_2)^+-S_{i,T}(X_{i,T}-N_1-N_2)^-,\quad i=1,2.\end{aligned}$$

Then, we have

$$\begin{aligned}&|J_2(x,s,t_2;u^*.)-J_2(x,s,t_1;u^*.)|\\&\quad \le \frac{m{\bar{u}}}{2r}|1-e^{-r(t_2-t_1)}|+e^{-r(T-t_2)}E\Big [|Q_2-Q_1|\Big |X_{2,t_2}\!=\!x,S_{2,t_2}\!=\!s,X_{1,t_1}\!=\!x,S_{1,t_1}\!=\!s\Big ]\\&\qquad +|e^{-r(T-t_2)}-e^{-r(T-t_1)}|E\Big [|Q_1|\Big |X_{2,t_2}=x,S_{2,t_2}=s,X_{1,t_1}=x,S_{1,t_1}=s\Big ]. \end{aligned}$$

In the above equation, according to the existence and uniqueness theorem of strong solution [26], we have

$$\begin{aligned}&E\Big [|Q_2-Q_1|\Big |X_{2,t_2}=x,S_{2,t_2}=s,X_{1,t_1}=x,S_{1,t_1}=s\Big ]\\&\quad =E\Big [|\min \{P,S_{2,T}\}(X_{2,T}-N_1-N_2)^+-S_{2,T}(X_{2,T}-N_1-N_2)^-\\&-\min \{P,S_{1,T}\}(X_{1,T}-N_1-N_2)^+\\&\quad \quad +S_{1,T}(X_{1,T}-N_1-N_2)^-|\Big |X_{2,t_2}=x,S_{2,t_2}=s,X_{1,t_1}=x,S_{1,t_1}=s\Big ]\\&\quad \le E\Big [|\min \{P,S_{2,T}\}((X_{2,T}-N_1-N_2)^+-(X_{1,T}-N_1-N_2)^+)|\\&\qquad +|(\min \{P,S_{2,T}\}-\min \{P,S_{1,T}\})(X_{1,T}-N_1-N_2)^+|\\&\quad \quad +|S_{2,T}((X_{2,T}-N_1-N_2)^--(X_{1,T}-N_1-N_2)^-)|\\&\quad \quad +|(S_{2,T}-S_{1,T})(X_{1,T}-N_1-N_2)^-|\Big |X_{2,t_2}=x,\\&S_{2,t_2}=s,X_{1,t_1}=x,S_{1,t_1}=s\Big ]\\&\quad \le E\Big [2S_{2,T}|X_{2,T}-X_{1,T}|+2|S_{2,T}-S_{1,T}||X_{1,T}-N_1-N_2|\Big |X_{2,t_2}=x,\\&S_{2,t_2}=s,X_{1,t_1}=x,S_{1,t_1}=s\Big ]\\&\quad \le M_3s|x|\sqrt{t_2-t_1}+M_4s\sqrt{t_2-t_1}, \end{aligned}$$

where $M_3>0$, $M_4>0$ are constants.

And

$$\begin{aligned}&E\Big [|Q_1|\Big |X_{2,t_2}=x,S_{2,t_2}=s,X_{1,t_1}=x,S_{1,t_1}=s\Big ]\\&\quad =E\Big [|\min \{P,S_{1,T}\}(X_{1,T}-N_1-N_2)^+-S_{1,T}(X_{1,T}-N_1-N_2)^-|\Big |X_{2,t_2}=x,\\&\quad S_{2,t_2}=s,X_{1,t_1}=x,S_{1,t_1}=s\Big ]\\&\quad \le E\Big [2|S_{1,T}||X_{1,T}-N_1-N_2||X_{2,t_2}=x,\\&\quad S_{2,t_2}=s,X_{1,t_1}=x,S_{1,t_1}=s\Big ]\\&\quad \le M_5s|x|+M_6s, \end{aligned}$$

where $M_5>0$, $M_6>0$ are constants.

That is, $\exists M_7>0$, $M_8>0$, such that

$$\begin{aligned}|J_2(x,s,t_2;u^*.)-J_2(x,s,t_1;u^*.)|\le \frac{M_7}{2}(x+1)s\sqrt{t_2-t_1}+\frac{M_8}{2}|t_2-t_1|.\end{aligned}$$

Then,

$$\begin{aligned}|V_2(x,s,t_2)-V_2(x,s,t_1)|<M_7(x+1)s\sqrt{t_2-t_2}+(M_8+1)|t_2-t_1|.\end{aligned}$$

Therefore, $V_2(x,s,t)$ is continuous with respect to t. $\square$