Sustainability investment under cap-and-trade regulation

Abstract

Carbon emission abatement is a hot topic in environmental sustainability and cap-and-trade regulation is regarded as an effective way to reduce the carbon emission. According to the real industrial practices, sustainable product implies that its production processes facilitate to reduce the carbon emission and has a positive response in market demand. In this paper, we study the sustainability investment on sustainable product with emission regulation consideration for decentralized and centralized supply chains. We first examine the order quantity of the retailer and sustainability investment of the manufacturer for the decentralized supply chain with one retailer and one manufacturer. After that, we extend our study to the centralized case where we determine the production quantity and sustainability investment for the whole supply chain. We derive the optimal order quantity (or production quantity) and sustainability investment, and find that the sustainability investment efficiency has a significant impact on the optimal solutions. Further, we conduct numerical studies and find surprisingly that the order quantity may be increasing in the wholesale price due to the effects of the sustainability and emission consideration. Moreover, we investigate the achievability of supply chain coordination by various contracts, and find that only revenue sharing contract can coordinate the supply chain whereas the buyback contract and two-part tariff contract cannot. Important insights and managerial implications are discussed.

Appendix

Proof of Proposition 1

Note that Eq. (1) is a newsvendor model, so we can obtain the results immediately. □

Proof of Proposition 2

By taking the first and second derivative of the profit function Π _m (s) over s, we have

$$\begin{aligned} \frac{d \varPi_m (s)}{d s} = & c_eb\biggl(F^{-1}\biggl( \frac{p-w}{p+c_h}\biggr)+d\biggr)+(w-c-c_ea)\beta+(2c_eb \beta-c_I)s; \\ \frac{d^2 \varPi_m (s)}{d s^2} = & 2c_eb\beta-c_I. \end{aligned}$$

Then Π _m (s) is concave in s, given that c _I ≥2c _e bβ. By solving the first-order condition, i.e., $\frac{d \varPi_{m} (s)}{d s}=0$, we obtain that

$$\begin{aligned} s^* = & \frac{c_eb(F^{-1}(\frac{p-w}{p+c_h})+d)+(w-c-c_ea)\beta }{c_I-2c_eb\beta}. \end{aligned}$$

 □

Proof of Proposition 3

The effects of b and c _I can be obtained by just looking at the formula of x ^∗ and s ^∗, i.e., Eqs. (2) and (4), respectively.

However, the effects of β and c _e are more complex and not monotones in general. For completeness, we show the value of $\frac{d s^{*}}{d \beta}$, $\frac{d x^{*}}{d \beta}$, $\frac{d s^{*}}{d c_{e}}$, and $\frac{d x^{*}}{d c_{e}}$ in this proof as follows:

Recalling that x(s) is determined by the first-order condition of the retailer’s profit function:

$$\begin{aligned} \frac{\partial\varPi_r (x)}{\partial x} = & (p+c_h)F(x-d-\beta s)-p+w=0, \end{aligned}$$

and s ^∗ is determined by the first-order condition of the manufacturer’s profit function:

$$\begin{aligned} \frac{d \varPi_m (s)}{d s} = & (w-c)\beta+c_ebx(s)-c_e \beta(a-bs)-c_Is=0. \end{aligned}$$

Let $G_{1}=\frac{\partial\varPi_{r} (x)}{\partial x} = (p+c_{h})F(x-d-\beta s)-p+w$ and $G_{2}=\frac{d \varPi_{m} (s)}{d s} = (w-c)\beta +c_{e}bx(s)-c_{e}\beta(a-bs)-c_{I}s$.

By taking the first derivatives of G ₁ and G ₂ with respect to β, we have

$$\begin{aligned} \frac{d G_1}{d \beta} = & \hat{p}\hat{f}\biggl(\frac{d x^*}{d \beta }-s^*-\beta \frac{d s^*}{d \beta}\biggr)=0; \\ \frac{d G_2}{d \beta} = & w-c+c_eb\frac{dx^*}{d\beta}-c_e \bigl(a-bs^*\bigr)+c_eb\beta\frac{d s^*}{d \beta}-c_I \frac{ds^*}{d\beta}=0, \end{aligned}$$

where $\hat{p}=p+c_{h}$ and $\hat{f}=f(x^{*}-d-\beta s^{*})$. Solving the above two equations obtains that $\frac{d s^{*}}{d \beta} = \frac {w-c-c_{e}(a-2bs^{*})}{2c_{e}b\beta-c_{I}}$, $\frac{d x^{*}}{d \beta} = s^{*}+\frac {d s^{*}}{d \beta}\beta$, but they may be positive or non-positive. By taking the above approach to consider the effects of β and c _e , we can obtain that , $\frac{d s^{*}}{d c_{e}} = \frac{-bx^{*}+\beta (a-bs^{*})}{2c_{e}b\beta-c_{I}}$, and $\frac{d x^{*}}{d c_{e}} = \frac{d s^{*}}{d c_{e}}\beta$, but they may be positive or non-positive too.

For the effects on the profits, by taking the first derivatives of $\varPi_{r}^{*}$, $\varPi_{m}^{*}$, and $\varPi_{d}^{*}$ with respect to b, we have

$$\begin{aligned} \frac{d \varPi_r^*}{d b} = & \frac{\partial\varPi_r}{\partial b}+\frac{\partial\varPi_r}{\partial x} \frac{d x^*}{d b}+\frac{\partial\varPi_r}{\partial s}\frac{d s^*}{d b} = \frac{\partial\varPi_r}{\partial b}\bigg|_{(x=x^*, s=s^*)}+ \frac{\partial\varPi_r}{\partial s}\frac{d s^*}{d b}\bigg|_{(x=x^*, s=s^*)} \\ = & \beta\bigl((p+c_h)F\bigl(x^*-d-\beta s^*\bigr)\bigr) \frac{c_ex^*+c_e\beta s^*}{c_I-2c_eb\beta}; \\ \frac{d \varPi_m^*}{d b} = & \frac{\partial\varPi_m}{\partial b}+\frac{\partial\varPi_m}{\partial x} \frac{d x^*}{d b}+\frac{\partial\varPi_m}{\partial s}\frac{d s^*}{d b} \\ = & \frac{\partial\varPi_m}{\partial b}\bigg|_{(x=x^*, s=s^*)}+\frac {\partial\varPi_m}{\partial x}\frac{d x^*}{d b}\bigg|_{(x=x^*, s=s^*)} \\ = & c_es^*x^*+\bigl(w-c-c_e\bigl(a-bs^*\bigr)\bigr) \frac{c_ex^*+c_e\beta s^*}{c_I-2c_eb\beta}\beta; \\ \frac{d \varPi_d^*}{d b} = & \frac{\partial\varPi_d}{\partial b}+\frac{\partial\varPi_d}{\partial x} \frac{d x^*}{d b}+\frac{\partial\varPi_d}{\partial s}\frac{d s^*}{d b} \\ = & c_e x^*+\bigl(p-c-c_e\bigl(a-bs^*\bigr) \beta-c_Is^*+c_ebx^*\bigr)\frac{c_ex^*+c_e\beta s^*}{c_I-2c_eb\beta}. \end{aligned}$$

In the first equation, the second equality holds because $\frac {\partial\varPi_{r}}{\partial x}=0$ when (x=x ^∗,s=s ^∗), and in the second equation, the second equality holds because $\frac{\partial \varPi_{m}}{\partial s}=0$ when (x=x ^∗,s=s ^∗). However, $\frac{d \varPi _{r}^{*}}{d b}$, $\frac{d \varPi_{m}^{*}}{d b}$, and $\frac{d \varPi_{d}^{*}}{d b}$ may be positive or non-positive. Similarly, we can obtain the values of the first derivative of $\varPi_{r}^{*}$, $\varPi_{m}^{*}$, and $\varPi_{d}^{*}$ with respect to β, c _e , and c _I . Unfortunately, they are complex and are not monotone in general. □

Proof of Proposition 4

By taking the first and second partial derivatives of the profit function Π _c (x,s) with respect to x, we have

$$\begin{aligned} \frac{\partial\varPi_c (x,s)}{\partial x} = & (p+c_h) \bigl(1-F(x-d-\beta s) \bigr)-c-c_h-c_e(a-bs); \\ \frac{\partial^2 \varPi_c (x,s)}{\partial x^2} = & - (p+c_h)f(x-d-\beta s)\le0. \end{aligned}$$

As the second partial derivative is non-positive, Π _c (x,s) is convex in x, and the optimal response of the production quantity is uniquely determined by the first order condition of the profit function, i.e., $\frac{\partial\varPi_{c} (x,s)}{\partial x}=0$. □

Proof of Corollary 2

By taking the derivative of $\frac{\partial\varPi_{c} (x,s)}{\partial x}$ with respect to s, we have

$$\begin{aligned} \frac{\partial^2 \varPi_c (x,s)}{\partial x \partial s} = & (p+c_h)f(x-d-\beta s)\beta +c_eb. \end{aligned}$$

Then, by the Implicit Function Theorem, i.e., $\frac{d x(s)}{d s}=-\frac {\frac{\partial^{2} \varPi_{c} (x,s)}{\partial x \partial s}}{\frac{\partial ^{2} \varPi_{c} (x,s)}{\partial x^{2}}}$, we have

$$\begin{aligned} \frac{d x(s)}{d s} = & -\frac{(p+c_h)f(x-d-\beta s)\beta +c_eb}{-(p+c_h)f(x-d-\beta s)} \\ = & \beta+\frac{c_eb}{(p+c_h)f(x-d-\beta s)}>0. \end{aligned}$$

 □

Proof of Proposition 5

Given that c _I ≥2c _e bβ, it is difficult to determine the sign of $\frac {d^{2} \varPi_{c} (x(s),s)}{d x^{2}}$ directly. So we take the third derivative of Π _c (x(s),s) over s, and we have

$$\begin{aligned} \frac{d^3 \varPi_c (x(s),s)}{d s^3} = & -\frac {(c_eb)^3f'(x(s)-d-\beta s)}{(p+c_h)^2(f(x(s)-d-\beta s))^3}. \end{aligned}$$

When f′(⋅)≥0, we have $\frac{d^{3} \varPi_{c} (x(s),s)}{d s^{3}}\le 0$, it implies that $\frac{d \varPi_{c} (x(s),s)}{d s}$ is concave in s. So $\frac{d \varPi_{c} (x(s),s)}{d s}=0$ has at most two roots and the larger of the two makes a change of sign for $\frac{d \varPi_{c} (x(s),s)}{d s}$ from positive to negative that corresponds to a local maximum of Π _c (x(s),s).

When f′(⋅)<0, we have $\frac{d^{3} \varPi_{c} (x(s),s)}{d s^{3}}>0$, it implies that $\frac{d \varPi_{c} (x(s),s)}{d s}$ is convex in s. So $\frac{d \varPi_{c} (x(s),s)}{d s}=0$ has at most two roots and the smaller of the two makes a change of sign for $\frac{d \varPi_{c} (x(s),s)}{d s}$ from positive to negative that corresponds to a local maximum of Π _c (x(s),s). □

Proof of Corollary 3

For the uniform distribution of the demand, ϵ∽U[A,B], then $f(z)=\frac{1}{B-A}$ and f′(z)=0. We have

$$\begin{aligned} \frac{d^2 \varPi_c (x(s),s)}{d s^2} = & 2c_eb\beta-c_I+ \frac{(c_eb)^2(B-A)}{(p+c_h)} \begin{cases} \ge0 &\mbox{if}\ c_I\le2c_eb\beta+\frac{(c_eb)^2(B-A)}{(p+c_h)};\\ <0 & \mbox{otherwise}, \end{cases} \end{aligned}$$

which means that Π _c (x(s),s) is a convex function if $c_{I}\le 2c_{e}b\beta+\frac{(c_{e}b)^{2}(B-A)}{(p+c_{h})}$, and concave otherwise. So there is at most one optimal point of s that satisfies dΠ _c (x(s),s)/ds=0 for the uniform distribution.

For the exponential distribution, ϵ∽Exp(1/θ), then $f(z)=\frac{1}{\theta}e^{-\frac{z}{\theta}}$ and $f'(z)=-\frac{1}{\theta }f(s)=-\frac{1}{\theta^{2}}e^{-\frac{z}{\theta}}$. We have

$$\begin{aligned} \frac{d^3 \varPi_c (x(s),s)}{d s^3} = & \frac {(c_eb)^3}{(p+c_h)^2(f(x(s)-d-\beta s))^2\theta}\ge0. \end{aligned}$$

So dΠ _c (x(s),s)/ds=0 has at most two roots, and the smaller of the two makes a change of sign for dΠ _c (x(s),s)/ds from positive to negative that corresponds to a local maximum of Π _c (x(s),s).

For the normal distribution, ϵ∽Normal(μ,σ), then $f(z)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(z-\mu)^{2}}{2\sigma^{2}}}$ and $f'(z)=-\frac{z-\mu}{\sigma^{2}}f(z)$. We have

$$\begin{aligned} \frac{d^3 \varPi_c (x(s),s)}{d s^3} = & \frac {(c_eb)^3}{(p+c_h)^2(f(x(s)-d-\beta s))^2}\frac{x(s)-d-\beta s-\mu }{\sigma^2} \\ &{} \begin{cases} \ge0 &\mbox{if}\ x(s)-d-\beta s\ge\mu;\\ <0 & \mbox{otherwise}. \end{cases} \end{aligned}$$

By Corollary 2, we obtain that

$$\begin{aligned} \frac{d (x(s)-d-\beta s)}{d s} = & \frac{c_eb}{(p+c_h)f(x-d-\beta s)}\ge0, \end{aligned}$$

which means that x(s)−d−βs increases in s. Let s _t be the solution of x(s)−d−βs=μ. Then we have that, if s<s _t , then dΠ _c (x(s),s)/ds is concave in s; if s≥s _t , then dΠ _c (x(s),s)/ds is convex in s. In other words, dΠ _c (x(s),s)/ds changes from a concave function to a convex function as s increases. Therefore, dΠ _c (x(s),s)/ds has at most three roots, and the one (and has at most one) makes a changes of sign for dΠ _c (x(s),s)/ds from positive to negative that corresponds to a local maximum of Π _c (x(s),s). □

Proof of Proposition 6

(a) Recalling that x(s) is determined by

$$\begin{aligned} \frac{\partial\varPi_c (x,s)}{\partial x} = & (p+c_h) \bigl(1-F(x-d-\beta s) \bigr)-c-c_h-c_e(a-bs)=0, \end{aligned}$$

and s ^∗ is determined by

$$\begin{aligned} \frac{d \varPi_c (x(s),s)}{d s} = & (p+c_h)F(x-d-\beta s)\beta -c_Is+c_ebx(s)=0. \end{aligned}$$

Let $G_{1}=-\frac{\partial\varPi_{c} (x,s)}{\partial x}=(p+c_{h})F(x-d-\beta s)-p+c+c_{e}(a-bs)$ and $G_{2}=\frac{d \varPi_{c} (x(s),s)}{d s} = (p+c_{h})F(x-d-\beta s)\beta-c_{I}s+c_{e}bx(s)$.

By taking the first derivatives of G ₁ and G ₂ with respect to b, we have

$$\begin{aligned} \frac{d G_1}{d b} = & -c_es^*+\hat{p}\hat{f}\frac{d x^*}{d b}-( \hat{p}\hat{f}\beta+c_eb)\frac{d s^*}{d b}=0; \\ \frac{d G_2}{d b} = & c_ex^*+(\hat{p}\hat{f}\beta+c_eb) \frac{d x^*}{d b}-\bigl(\hat{p}\hat{f}\beta^2+c_I\bigr) \frac{d s^*}{d b}=0, \end{aligned}$$

where $\hat{p}=p+c_{h}$ and $\hat{f}=f(x^{*}-d-\beta s^{*})$.

Solving the above two equations obtains that

$$\begin{aligned} \frac{d s^*}{d b} = & -\frac{c_ex^*+c_e\beta s^*+\frac {c_ebc_es^*}{\hat{p}\hat{f}}}{2c_eb\beta-c_I+\frac{(c_eb)^2}{\hat{p}\hat {f}}}\ge0; \\ \frac{d x^*}{d b} = & \frac{c_es^*+(\hat{p}\hat{f}\beta+c_eb)\frac{d s^*}{d b}}{\hat{p}\hat{f}}\ge0. \end{aligned}$$

The inequalities hold because that, when s is obtained at the optimal point, $2c_{e}b\beta-c_{I}+\frac{(c_{e}b)^{2}}{\hat{p}\hat{f}}=\frac{d^{2} \varPi _{c} (x(s),s)}{d s^{2}}\le0$.

Similarly, by taking the first derivatives of G ₁ and G ₂ with respect to c _I , we have

$$\begin{aligned} \frac{d G_1}{d c_I} = & \hat{p}\hat{f}\frac{d x^*}{d c_I}-(\hat{p}\hat{f} \beta+c_eb)\frac{d s^*}{d c_I}=0; \\ \frac{d G_2}{d c_I} = &-s^*+(\hat{p}\hat{f}\beta+c_eb) \frac{d x^*}{d c_I}-\bigl(\hat{p}\hat{f}\beta^2+c_I\bigr) \frac{d s^*}{d c_I}=0, \end{aligned}$$

Solving the above two equations obtains that

$$\begin{aligned} \frac{d s^*}{d c_I} = & \frac{s^*}{2c_eb\beta-c_I+\frac{(c_eb)^2}{\hat {p}\hat{f}}}\le0; \\ \frac{d x^*}{d c_I} = & \frac{(\hat{p}\hat{f}\beta+c_eb)\frac{d s^*}{d c_I}}{\hat{p}\hat{f}}\le0. \end{aligned}$$

The inequalities hold because we have that, here, $2c_{e}b\beta-c_{I}+\frac {(c_{e}b)^{2}}{\hat{p}\hat{f}}\le0$ when s=s ^∗.

By taking the above approach to consider the effects of β and c _e , we can obtain that, for β, $\frac{d s^{*}}{d \beta} = \frac {-c_{e}bs^{*}}{2c_{e}b\beta-c_{I}+\frac{(c_{e}b)^{2}}{\hat{p}\hat{f}}}\ge0$ and $\frac{d x^{*}}{d \beta} = s+\frac{\hat{p}\hat{f}\beta+c_{e}b}{\hat{p}\hat {f}}\frac{d s^{*}}{d \beta}\ge0$; for c _e , $\frac{d s^{*}}{d c_{e}} = \frac {-bx^{*}+\frac{\hat{p}\hat{f}\beta+c_{e}b}{\hat{p}\hat {f}}(a-bs^{*})}{2c_{e}b\beta-c_{I}+\frac{(c_{e}b)^{2}}{\hat{p}\hat{f}}}$ and $\frac{d x^{*}}{d c_{e}} = \frac{-(a-bs^{*})+(\hat{p}\hat{f}\beta+c_{e}b)\frac {d s^{*}}{d c_{e}}}{\hat{p}\hat{f}}$, but which may be positive or non-positive.

By taking the first derivative of $\varPi_{c}^{*}$ with respect to b, we have

$$\begin{aligned} \frac{d \varPi_c^*}{d b} = & \frac{\partial\varPi_c}{\partial b}+\frac{\partial\varPi_c}{\partial x} \frac{d x^*}{d b}+\frac{\partial\varPi_c}{\partial s}\frac{d s^*}{d b} \\ = & \frac{\partial\varPi_c}{\partial b}\bigg|_{(x=x^*, s=s^*)} \\ = & c_es^*x^*\ge0. \end{aligned}$$

The second equality holds because $\frac{\partial\varPi_{c}}{\partial x}=\frac{\partial\varPi_{c}}{\partial s}=0$ when (x=x ^∗,s=s ^∗). Similarly, we can obtain that $\frac{d \varPi_{c}^{*}}{d c_{I}}=-\frac {(s^{*})^{2}}{2}\le0$, $\frac{d \varPi_{c}^{*}}{d \beta}=s^{*}\hat {p}F(x^{*}-d-\beta s^{*})\ge0$, and $\frac{d \varPi_{c}^{*}}{d c_{e}}=K-(a-bs^{*})x^{*}$ but which may be positive or non-positive. □

Proof of Proposition 7

The retailer’s problem is a newsvendor problem, so we can easily obtain that

$$\begin{aligned} x = & F^{-1}\biggl(\frac{\phi p-w}{\phi p+c_h}\biggr)+d+\beta s. \end{aligned}$$

By substituting the x into the manufacturer’s profit function, and taking the derivatives with respect to s, we have

$$\begin{aligned} \frac{d \varPi_m}{d s} = &\bigl((1-\phi)p+w-c-c_e(a-bs)\bigr) \beta-c_Is+c_ebx(s); \\ \frac{d^2 \varPi_m}{d s^2} = & 2c_eb\beta-c_I\le0. \end{aligned}$$

The inequality holds because in this section we restrict our attention to the case in which the sustainability level are determined by the first-order condition of the profit function for the centralized supply chain, i.e., c _I ≥2c _e bβ. Thus, the optimal s is uniquely determined by $\frac{d \varPi_{m}}{d s}=0$. □

Proof of Proposition 8

The proof is similar to the proof for Proposition 7 and omitted. □

Proof of Proposition 9

The proof is similar to the proof for Proposition 7 and omitted. □