Product rollover strategy and emission reduction with intertemporal carbon emission regulation versus consumer supervision

Abstract

As global warming has severely threatened the ecosystem and sustainable development of human beings, carbon trading scheme is introduced to mitigate global warming and consumer environmental awareness (CEA) is gradually enhanced. Government regulation and consumer supervision have required firms to seek efficient strategies of product rollover and emission abatement in order to sustain and increase market share. This paper constructs a two-period analytical model in the context of intertemporal carbon emission regulation to investigate how carbon emission regulations and CEA affect the optimal strategies of product rollover, emission abatement, and social welfare. The results reveal that without consumer supervision, the firm prefers to adopt dual product rollover strategy and the optimal product rollover strategy depends on costs and benefits when product recycling is considered. When CEA is high, welfare and emission abatement regulated by hybrid policy is lower than those regulated by carbon trading scheme. When CEA is low, emission abatement under hybrid policy is superior to those regulated by carbon trading scheme. These findings help provide implications for improving carbon emission management efficiency and prompting sustainable development.

Appendices

Appendix

Proof of relationship between emissions, product pricing and carbon price under dual product rollover strategy.

$4{c}_{1}{c}_{2}{\theta }_{0}\left({\theta }_{0}-1\right)+2{\gamma }^{2}\left({c}_{2}+{c}_{1}{\theta }_{0}\right)-{\gamma }^{4}$ is simplified as:

$$4{c}_{1}{c}_{2}{\theta }_{0}\left({\theta }_{0}-1\right)+2{\gamma }^{2}\left({c}_{2}+{c}_{1}{\theta }_{0}\right)-{\gamma }^{4}=4{c}_{1}{c}_{2}{\theta }_{0}^{2}+\left({\gamma }^{2}-2{c}_{1}{\theta }_{0}\right)\left(2{c}_{2}-{\gamma }^{2}\right)$$

If ${\theta }_{0}>1$ and $2{c}_{2}>{\gamma }^{2}$,then we have $4{c}_{1}{c}_{2}{\theta }_{0}\left({\theta }_{0}-1\right)+2{\gamma }^{2}\left({c}_{2}+{c}_{1}{\theta }_{0}\right)-{\gamma }^{4}>0$.

If ${\theta }_{0}<1$ and $2{c}_{1}{\theta }_{0}<{\gamma }^{2}<2{c}_{2}$, we have $4{c}_{1}{c}_{2}{\theta }_{0}\left({\theta }_{0}-1\right)+2{\gamma }^{2}\left({c}_{2}+{c}_{1}{\theta }_{0}\right)-{\gamma }^{4}>0$.

We have the following first order condition:

$$\begin{array}{c}\frac{\partial {e}_{2d}^{**}}{\partial {p}_{c}^{2}}=\frac{\partial {e}_{2d}^{***}}{\partial {\tau }_{3}}=\frac{\partial {e}_{2d}^{***}}{\partial {p}_{c}^{2}}=\frac{2{\gamma }^{2}\left(1+{\theta }_{0}\right)+4{c}_{1}{\theta }_{0}\left({\theta }_{0}-1\right)}{4{c}_{1}{c}_{2}{\theta }_{0}\left({\theta }_{0}-1\right)+2{\gamma }^{2}\left({c}_{2}+{c}_{1}{\theta }_{0}\right)-{\gamma }^{4}}\\ \frac{\partial {p}_{2d}^{**}}{\partial {p}_{c}^{2}}=\frac{\partial {p}_{2d}^{***}}{\partial {\tau }_{3}}=\frac{\partial {p}_{2d}^{***}}{\partial {p}_{c}^{2}}=\frac{{\gamma }^{3}\left(1+{\theta }_{0}\right)+2{c}_{1}{\theta }_{0}\gamma \left({\theta }_{0}-1\right)}{4{c}_{1}{c}_{2}{\theta }_{0}\left({\theta }_{0}-1\right)+2{\gamma }^{2}\left({c}_{2}+{c}_{1}{\theta }_{0}\right)-{\gamma }^{4}}\\ \begin{array}{c}\frac{\partial {e}_{1d}^{**}}{\partial {p}_{c}^{2}}=\frac{\partial {e}_{1d}^{***}}{\partial {p}_{c}^{2}}=\frac{\partial {e}_{1d}^{***}}{\partial {\tau }_{3}}=\frac{4{\theta }_{0}\left({c}_{2}\left({\theta }_{0}-1\right)+{\gamma }^{2}\right)}{4{c}_{1}{c}_{2}{\theta }_{0}\left({\theta }_{0}-1\right)+2{\gamma }^{2}\left({c}_{2}+{c}_{1}{\theta }_{0}\right)-{\gamma }^{4}}\\ \frac{\partial {p}_{1d}^{**}}{\partial {p}_{c}^{2}}=\frac{\partial {p}_{1d}^{***}}{\partial {p}_{c}^{2}}=\frac{{\theta }_{0}}{2}\left(\frac{4\gamma \left({c}_{2}\left({\theta }_{0}-1\right)+{\gamma }^{2}\right)}{4{c}_{1}{c}_{2}{\theta }_{0}\left({\theta }_{0}-1\right)+2{\gamma }^{2}\left({c}_{2}+{c}_{1}{\theta }_{0}\right)-{\gamma }^{4}}\right)\\ \frac{\partial {p}_{1d}^{***}}{\partial {\tau }_{3}}=\frac{2{\theta }_{0}\gamma \left({c}_{2}\left({\theta }_{0}-1\right)+{\gamma }^{2}\right)}{4{c}_{1}{c}_{2}{\theta }_{0}\left({\theta }_{0}-1\right)+2{\gamma }^{2}\left({c}_{2}+{c}_{1}{\theta }_{0}\right)-{\gamma }^{4}}\end{array}\end{array}$$

Therefore, the following conclusions can be drawn:

• (1) If ${\theta }_{0}>1$ and $2{c}_{2}>{\gamma }^{2}$, then we have $\frac{\partial {e}_{2d}^{**}}{\partial {p}_{c}^{2}}>0$, $\frac{\partial {e}_{1d}^{**}}{\partial {p}_{c}^{2}}>0$, $\frac{\partial {p}_{2d}^{**}}{\partial {p}_{c}^{2}}>0$, $\frac{\partial {p}_{1d}^{**}}{\partial {p}_{c}^{2}}>0$, $\frac{\partial {e}_{2d}^{***}}{\partial {p}_{c}^{2}}>0$, $\frac{\partial {e}_{1d}^{***}}{\partial {p}_{c}^{2}}>0$, $\frac{\partial {p}_{2d}^{***}}{\partial {p}_{c}^{2}}>0$, $\frac{\partial {p}_{1d}^{***}}{\partial {p}_{c}^{2}}>0$, $\frac{\partial {e}_{2d}^{***}}{\partial {\tau }_{3}}>0$, $\frac{\partial {e}_{1d}^{***}}{\partial {\tau }_{3}}>0$, $\frac{\partial {p}_{2d}^{***}}{\partial {\tau }_{3}}>0$ and $\frac{\partial {p}_{1d}^{***}}{\partial {\tau }_{3}}>0$.

• (2) When ${\theta }_{0}<1$ and $2{c}_{1}{\theta }_{0}<{\gamma }^{2}<2{c}_{2}$, if ${\gamma }^{2}\left(1+{\theta }_{0}\right)>2{c}_{1}{\theta }_{0}\left(1-{\theta }_{0}\right)$, we have $\frac{\partial {e}_{2d}^{**}}{\partial {p}_{c}^{2}}>0$,$\frac{\partial {p}_{2d}^{**}}{\partial {p}_{c}^{2}}>0$, $\frac{\partial {e}_{2d}^{***}}{\partial {p}_{c}^{2}}>0$, $\frac{\partial {p}_{2d}^{***}}{\partial {p}_{c}^{2}}>0$, $\frac{\partial {e}_{2d}^{***}}{\partial {\tau }_{3}}>0$, $\frac{\partial {p}_{2d}^{***}}{\partial {\tau }_{3}}>0$. If ${\gamma }^{2}\left(1+{\theta }_{0}\right)<2{c}_{1}{\theta }_{0}\left(1-{\theta }_{0}\right)$, then, we have $\frac{\partial {e}_{2d}^{**}}{\partial {p}_{c}^{2}}<0$, $\frac{\partial {p}_{2d}^{**}}{\partial {p}_{c}^{2}}<0$, $\frac{\partial {e}_{2d}^{***}}{\partial {p}_{c}^{2}}<0$, $\frac{\partial {p}_{2d}^{***}}{\partial {p}_{c}^{2}}<0$, $\frac{\partial {e}_{2d}^{***}}{\partial {\tau }_{3}}<0$, $\frac{\partial {p}_{2d}^{***}}{\partial {\tau }_{3}}<0$.

• (3) When ${\theta }_{0}<1$ and $2{c}_{1}{\theta }_{0}<{\gamma }^{2}<2{c}_{2}$, if ${\gamma }^{2}>{c}_{2}\left(1-{\theta }_{0}\right)$, then, we have $\frac{\partial {e}_{1d}^{**}}{\partial {p}_{c}^{2}}>0$,$\frac{\partial {p}_{1d}^{**}}{\partial {p}_{c}^{2}}>0$, $\frac{\partial {e}_{1d}^{***}}{\partial {p}_{c}^{2}}>0$, $\frac{\partial {p}_{1d}^{***}}{\partial {p}_{c}^{2}}>0$, $\frac{\partial {e}_{1d}^{***}}{\partial {\tau }_{3}}>0$, and $\frac{\partial {p}_{1d}^{***}}{\partial {\tau }_{3}}>0$. If ${\gamma }^{2}<{c}_{2}\left(1-{\theta }_{0}\right)$, we have $\frac{\partial {e}_{1d}^{**}}{\partial {p}_{c}^{2}}<0$, $\frac{\partial {p}_{1d}^{**}}{\partial {p}_{c}^{2}}<0$, $\frac{\partial {e}_{1d}^{***}}{\partial {p}_{c}^{2}}<0$, $\frac{\partial {p}_{1d}^{***}}{\partial {p}_{c}^{2}}<0$, $\frac{\partial {e}_{1d}^{***}}{\partial {\tau }_{3}}<0$, and $\frac{\partial {p}_{1d}^{***}}{\partial {\tau }_{3}}<0$.

Proof of proposition 1

Proof: With ETS regulation, ${\Delta }_{1}$ is the difference that total profits under DPRS minus total profits under SPRS in the whole phase.

$${\Delta }_{1}=\frac{\delta {\left({p}_{c}^{2}\right)}^{2}}{2{c}_{1}}+\delta \left({C}_{SE}^{R}-{C}_{DE}^{R}+{S}_{DE}^{R}-{S}_{SE}^{R}\right)$$

With hybrid regulation, ${\Delta }_{2}$ is the difference that total profits under DPRS minus total profits under SPRS in the whole phase.

$${\Delta }_{2}=\frac{\delta {\left({p}_{c}^{2}+{\tau }_{3}\right)}^{2}}{2{c}_{1}}+\delta \left({C}_{SH}^{R}-{C}_{DH}^{R}+{S}_{DH}^{R}-{S}_{SH}^{R}\right)$$

Therefore, we have proposition 1.

Proof of proposition 3

• (1) Single product rollover strategy

When CEA is equal to zero, ${\Delta }_{5}$ represents the difference between total profit regulated by hybrid policy and total profit regulated by ETS with single product rollover strategy.

$${\Delta }_{5}=\frac{{\tau }_{3}\delta \left({c}_{1}\left(2{p}_{c}^{2}+{\tau }_{3}\right)+\delta {c}_{2}\left(2{p}_{c}^{2}\left({r}_{2}-2\right)+{\tau }_{3}\right)\right)}{2{c}_{1}{c}_{2}}+\delta \left({C}_{SE}^{R}-{C}_{SH}^{R}-{S}_{SE}^{R}+{S}_{SH}^{R}\right)$$

When product recycling is not considered, then we have the following conclusions.

• (i) If ${c}_{1}\left(2{p}_{c}^{2}+{\tau }_{3}\right)>\delta {c}_{2}\left(2{p}_{c}^{2}\left(2-{r}_{2}\right)-{\tau }_{3}\right)$ and ${C}_{SE}^{R}+{S}_{SH}^{R}={C}_{SH}^{R}+{S}_{SE}^{R}$, we have ${\Delta }_{5}>0$.

• (ii) If ${c}_{1}\left(2{p}_{c}^{2}+{\tau }_{3}\right)<\delta {c}_{2}\left(2{p}_{c}^{2}\left(2-{r}_{2}\right)-{\tau }_{3}\right)$ and ${C}_{SE}^{R}+{S}_{SH}^{R}={C}_{SH}^{R}+{S}_{SE}^{R}$, we have ${\Delta }_{5}<0$.

When product recycling is considered, then we have the following conclusions.

• (i) If ${\tau }_{3}\delta {H}_{1}+{k}_{1}>0$, we have ${\Delta }_{5}>0$.

• (ii) If ${\tau }_{3}\delta {H}_{1}+{k}_{1}<0$, we have ${\Delta }_{5}<0$.

where ${H}_{1}={c}_{1}\left(2{p}_{c}^{2}+{\tau }_{3}\right)+\delta {c}_{2}\left(2{p}_{c}^{2}\left({r}_{2}-2\right)+{\tau }_{3}\right)$, ${k}_{1}=2{c}_{1}{c}_{2}\delta \left({C}_{SE}^{R}-{C}_{SH}^{R}-{S}_{SE}^{R}+{S}_{SH}^{R}\right)$.

• (2) Dual product rollover strategy

When CEA is equal to zero, ${\Delta }_{6}$ represents the difference between total profit regulated by hybrid policy and total profit regulated by ETS with dual product rollover strategy.

$${\Delta }_{6}=\frac{{\tau }_{3}\delta \left(\left({c}_{1}+{c}_{2}\right)\left(2{p}_{c}^{2}+{\tau }_{3}\right)+\delta {c}_{2}\left(2{p}_{c}^{2}\left({r}_{2}-2\right)+{\tau }_{3}\right)\right)}{2{c}_{1}{c}_{2}}+\delta \left({C}_{DE}^{R}-{C}_{DH}^{R}-{S}_{DE}^{R}+{S}_{DH}^{R}\right)$$

When product recycling is not considered, then we have the following conclusions.

• (i) If $\left({c}_{1}+{c}_{2}\right)\left(2{p}_{c}^{2}+{\tau }_{3}\right)>\delta {c}_{2}\left(2{p}_{c}^{2}\left(2-{r}_{2}\right)-{\tau }_{3}\right)$ and ${C}_{DE}^{R}+{S}_{DH}^{R}={C}_{DH}^{R}+{S}_{DE}^{R}$, we have ${\Delta }_{6}>0$.

• (ii) If $\left({c}_{1}+{c}_{2}\right)\left(2{p}_{c}^{2}+{\tau }_{3}\right)<\delta {c}_{2}\left(2{p}_{c}^{2}\left(2-{r}_{2}\right)-{\tau }_{3}\right)$ and ${C}_{DE}^{R}+{S}_{DH}^{R}={C}_{DH}^{R}+{S}_{DE}^{R}$, we have ${\Delta }_{6}<0$.

When product recycling is considered, then we have the following conclusions.

• (i) With considering product recycling and if ${\tau }_{3}\delta {H}_{2}+{k}_{2}>0$, we have ${\Delta }_{6}>0$.

• (ii) With considering product recycling and if ${\tau }_{3}\delta {H}_{2}+{k}_{2}<0$, we have ${\Delta }_{6}<0$.

where ${H}_{2}=\left({c}_{1}+{c}_{2}\right)\left(2{p}_{c}^{2}+{\tau }_{3}\right)+\delta {c}_{2}\left(2{p}_{c}^{2}\left({r}_{2}-2\right)+{\tau }_{3}\right),{k}_{2}=2{c}_{1}{c}_{2}\delta \left({C}_{DE}^{R}-{C}_{DH}^{R}-{S}_{DE}^{R}+{S}_{DH}^{R}\right)$.