Sustainable Supply Chain Management

Abstract

Sustainability is the spirit of the time at the moment. As such, sustainable supply chain management is a very important and relevant topic and deserves to be in the concluding chapter. First, I premise that in addition to firms, there are two key players in the economy, the government and consumers, who should exert an influence on pollution reduction . I explore whether these two are substitutes or complements in improving environment. Then, I embark on analyzing the issue from the value chain perspective by looking into the role of consumer awareness in light of supply chain coordination strategy . I show why and how consumer awareness can make a huge impact on environmental sustainability.

The original version of this chapter was revised. The erratum to this chapter is available at 10.1007/978-981-10-3599-9_6

Appendices

Appendix 1

The Hamiltonian function is given by:

$$H = \left( {p - c} \right)\left( {\alpha - \beta p - \gamma y} \right) - c_{1} \left( {\alpha - \beta p - \gamma y - \overline{U} } \right)^{2} - ev^{2} - fy^{2} + \lambda \left[ {\overline{U} \left( {l - v} \right) - \rm{\delta }y} \right]$$

Assuming interior solutions, we obtain from the optimality conditions:

$$p = \frac{{\left( {1 + 2\beta c_{1} } \right)\left( {\alpha - \gamma y} \right) - 2\beta c_{1} \overline{U} + \beta c}}{{2\beta \left( {1 + \beta c_{1} } \right)}},$$

(21)

$$v = - \frac{{\lambda \overline{U} }}{2e}.$$

(22)

The solutions that satisfy the necessary conditions are optimal. The objective function is concave in $\left( {v, p} \right)$. All constraints are linear in $\left( {v, p} \right)$.

Costate equation, using (21), is as follows:

$$\begin{aligned} \dot{\lambda } = & \,\left( {r + \delta } \right)\lambda + \left[ {2f + 2\gamma^{2} c_{1} - \frac{{\gamma^{2} \left( {1 + 2\beta c_{1} } \right)^{2} }}{{2\beta \left( {1 + \beta c_{1} } \right)}}} \right]y \\ & + \,\left\{ {\frac{{\gamma \left( {1 + 2\beta c_{1} } \right)\left[ {\alpha \left( {1 + 2\beta c_{1} } \right) - 2\beta c_{1} \overline{U} + \beta c} \right]}}{{2\beta \left( {1 + \beta c_{1} } \right)}} - \left[ {\gamma c + 2\gamma c_{1} \left( {\alpha - \overline{U} } \right)} \right]} \right\} \\ \end{aligned}$$

(23)

From (22) and the state equation,

$$\lambda = \frac{2e}{{\overline{U}^{2} }}\left( {\dot{y} + \rm{\delta }y - \overline{\textit{U}} {\textit{l}}} \right).$$

(24)

Substituting (24) into (23) and solving the second-order differential equation of y yield:

$$y\left( t \right) = A_{1} e^{{m_{1} t}} + A_{2} e^{{m_{2} t}} + K_{2} ,$$

(25)

where

$$\begin{aligned} m_{1} & = \frac{{r + \sqrt {r^{2} + 4\delta \left( {r + \delta } \right) + \frac{{\overline{U}^{2} }}{e}\left[ {4f - \frac{{\gamma^{2} }}{{\beta \left( {1 + \beta c_{1} } \right)}}} \right]} }}{2} > r, \\ m_{2} & = \frac{{r - \sqrt {r^{2} + 4\delta \left( {r + \delta } \right) + \frac{{\overline{U}^{2} }}{e}\left[ {4f - \frac{{\gamma^{2} }}{{\beta \left( {1 + \beta c_{1} } \right)}}} \right]} }}{2}\, < \,0. \\ \end{aligned}$$

$$\begin{aligned} K_{1} & = \left\{ {\delta \left( {r + \delta } \right) + \frac{{\overline{U}^{2} }}{2e}\left[ {2f - \frac{{\gamma^{2} }}{{2\beta \left( {1 + \beta c_{1} } \right)}}} \right]} \right\} = \frac{{4e\beta \delta \left( {1 + \beta c_{1} } \right)\left( {r + \delta } \right) + 4f{\overline{U}}^{2} \beta \left( {1 + \beta c_{1} } \right) - \gamma^{2} {\overline{U}}^{2} }}{{4e\beta \left( {1 + \beta c_{1} } \right) }} \\ K_{2} & = \frac{{4\beta el\overline{U} \left( {1 + \beta c_{1} } \right)\left( {r + \delta } \right) - \gamma \overline{U}^{2} \left( {\alpha - \beta c + 2\beta c_{1} \overline{U} } \right)}}{{4\beta \left( {1 + \beta c_{1} } \right)\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right] - \gamma^{2} \overline{U}^{2} }} \\ \end{aligned}$$

Note that $K_{1} > 0$ would hold under the reasonable ranges of parameters, assuming a positive market demand.

From (24) and (25), $A_{1} = 0$ is obtained to guarantee that the limiting transversality condition $\mathop {\lim }\limits_{T \to \infty } e^{ - rT} \lambda \left( T \right) = 0$ holds under all parameters. Also, $A_{2} = y_{0} - K_{2}$.

Considering $m_{2} < 0$, the long-run equilibrium solutions in Table 2 are readily determined from (21), (22), and (25).

1.1 Proof of Theorem 1

$$\frac{{\partial y_{LR} }}{\partial f} = - \frac{{4\beta \left( {1 + \beta c_{1} } \right)\overline{U}^{2} K_{2} }}{{\left\{ {4\beta \left( {1 + \beta c_{1} } \right)\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right] - \gamma^{2} \overline{U}^{2} } \right\}}}.$$

Since $K_{1}$ is assumed to be positive and $K_{2}$ is nonnegative, ${\text{sgn}}\left( { \frac{{\partial y_{LR} }}{\partial f}} \right) < 0$.

Similarly, $\frac{{\partial v_{LR} }}{\partial f} = \frac{{4\beta \delta \left( {1 + \beta c_{1} } \right)\overline{U} }}{{\left\{ {4\beta \left( {1 + \beta c_{1} } \right)\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right] - \gamma^{2} \overline{U}^{2} } \right\}}} \cdot \,K_{2} > 0$.

Also it holds that $\frac{{\partial^{2} y_{LR} }}{{\partial f^{2} }} = \frac{{32 \beta^{2} \left( {1 + \beta c_{1} } \right)^{2} \overline{U}^{4} }}{{\left\{ {4\beta \left( {1 + \beta c_{1} } \right)\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right] - \gamma^{2} \overline{U}^{2} } \right\}^{2} }} \cdot \,K_{2} > 0$ and $\frac{{\partial^{2} v_{LR} }}{{\partial f^{2} }} = - \frac{\delta }{{\overline{U} }} \cdot \,\frac{{\partial^{2} y_{LR} }}{{\partial f^{2} }} < 0$

1.2 Proof of Theorem 2

$$\begin{aligned} \frac{{\partial^{2} y_{LR} }}{\partial f\partial \gamma } & = \frac{\partial }{\partial \gamma }\left( {\frac{{\partial y_{LR} }}{\partial f}} \right) \\ & = \frac{{4\beta \left( {1 + \beta c_{1} } \right)\overline{U}^{2} }}{{\left\{ {4\beta \left( {1 + \beta c_{1} } \right)\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right] - \gamma^{2} \overline{U}^{2} } \right\}^{3} }} \\ & \quad \times \,\left[ {3\overline{U}^{4} \left( {\alpha - \beta c + 2\beta c_{1} \overline{U} } \right)\gamma^{2} - 16\beta el\overline{U}^{3} \left( {1 + \beta c_{1} } \right)\left( {r + \delta } \right)\gamma } \right. \\ & \left. {\quad + \,4\beta \overline{U}^{4} \left( {1 + \beta c_{1} } \right)\left( {\alpha - \beta c + 2\beta c_{1} \overline{U} } \right)f + 4\beta e\delta \overline{U}^{2} \left( {r + \delta } \right)\left( {1 + \beta c_{1} } \right)\left( {\alpha - \beta c + 2\beta c_{1} \overline{U} } \right)} \right] \\ \end{aligned}$$

It is easily shown that $\frac{{\partial^{2} v_{LR} }}{\partial f\partial \gamma } = - \frac{\delta }{{\overline{U} }} \cdot \frac{{\partial^{2} y_{LR} }}{\partial f\partial \gamma }$.

Let $g\left( {\gamma ,f} \right) = A\gamma^{2} - B\gamma + Cf + D$, where $A = 3\overline{U}^{4} \left( {\alpha - \beta c + 2\beta c_{1} \overline{U} } \right) > 0$, $B = 16\beta el\overline{U}^{3} \left( {1 + \beta c_{1} } \right)\left( {r + \delta } \right) > 0$, $C = 4\beta \overline{U}^{4} \left( {1 + \beta c_{1} } \right)\left( {\alpha - \beta c + 2\beta c_{1} \overline{U} } \right) > 0$, $D = 4\beta e\delta \overline{U}^{2} \left( {r + \delta } \right)\left( {1 + \beta c_{1} } \right)\left( {\alpha - \beta c + 2\beta c_{1} \overline{U} } \right) > 0.$

Note that ${\text{sgn}}\left( {\frac{{\partial^{2} y_{LR} }}{\partial f\partial \gamma }} \right) = - {\text{sgn}}\left( { \frac{{\partial^{2} v_{LR} }}{\partial f\partial \gamma }} \right) = {\text{sgn}}\left[ {g\left( {\gamma ,f} \right)} \right]$ hold as $K_{1} > 0$.

Therefore,

• if $\left( {\gamma ,f} \right) \in \{ \left( {\gamma ,f} \right) | g\left( {\gamma ,f} \right) = A\gamma^{2} - B\gamma + Cf + D < 0\} ,\,\frac{{\partial^{2} y_{LR} }}{\partial f\partial \gamma } < 0$, and $\frac{{\partial^{2} v_{LR} }}{\partial f\partial \gamma } > 0$;

• if $\left( {\gamma ,f} \right) \in \{ \left( {\gamma ,f} \right) | g\left( {\gamma ,f} \right) = A\gamma^{2} - B\gamma + Cf + D \ge 0\} ,\frac{{\partial^{2} y_{LR} }}{\partial f\partial \gamma } \ge 0$, and $\frac{{\partial^{2} v_{LR} }}{\partial f\partial \gamma } \le 0$.

Let $h\left( \gamma \right) = - \frac{A}{C}\gamma^{2} + \frac{B}{C}\gamma - \frac{D}{C} = - \frac{A}{C}\left( {\gamma - \frac{B}{2A}} \right)^{2} + \frac{{B^{2} }}{4AC} - \frac{D}{C}$.

Let $G = B^{2} - 4AD = 16\beta e\overline{U}^{6} \left( {r + \delta } \right)\left( {1 + \beta c_{1} } \right)$ $\left[ {16\beta el^{2} \left( {1 + \beta c_{1} } \right)\left( {r + \delta } \right) - 3\delta \left( {\alpha - \beta c + 2\beta c_{1} \overline{U} } \right)^{2} } \right]$. G determines whether $h\left( \gamma \right) = 0$ has real root(s) and the sign of maximum value of $h\left( \gamma \right).$ Note that $g\left( {\gamma ,f} \right){ \gtreqless }0$ is equivalent to $f{ \gtreqless }h\left( \gamma \right)$ and $h\left( \gamma \right)$ has a concave form.

To summarize, if $G \le 0$, $g\left( {\gamma ,f} \right) \ge 0$ for all feasible f; if $G > 0$, $g\left( {\gamma ,f} \right) \ge 0$ for $f \ge h\left( \gamma \right)$; $g\left( {\gamma ,f} \right) < 0$ for $f < h\left( \gamma \right)$. Note that $\begin{aligned} {\text{sgn}}\left( G \right) & = {\text{sgn }}\left[ {16\beta el^{2} \left( {1 + \beta c_{1} } \right)\left( {r + \delta } \right) - 3\delta \left( {\alpha - \beta c + 2\beta c_{1} \overline{U} } \right)^{2} } \right] \\ & = - {\text{sgn}}\left[ {\alpha - \left( {\sqrt {\frac{{16\beta el^{2} \left( {1 + \beta c_{1} } \right)\left( {r + \delta } \right)}}{3\delta }} + \beta c - 2\beta c_{1} \overline{U} } \right)} \right]. \\ \end{aligned}$

Also, assuming $G > 0$, the real roots of $h\left( \gamma \right) = 0$ are strictly positive as $A,B,D > 0$.

Appendix 2

2.1 Analysis for Model 2

We present the solution procedure for Model 2 in detail and omit others, since they are similar to that for Model 2. Recall Model 2:

$$Maximize\,J^{r} = \mathop \int \limits_{0}^{\infty } e^{ - rt} \left[ {\left( {p_{2} - p_{1} } \right)\left( {\alpha - \beta p_{2} - \gamma y} \right) - c_{2} \left( {\alpha - \beta p_{2} - \gamma y} \right)^{2} } \right]{\text{d}}t$$

$$Maximize\,J^{m} = \mathop \int \limits_{0}^{\infty } e^{ - rt} \left[ {\left( {p_{1} - c} \right)\left( {\alpha - \beta p_{2} - \gamma y} \right) - c_{1} \left( {\alpha - \beta p_{2} - \gamma y - \overline{U} } \right)^{2} - ev^{2} - fy^{2} } \right]{\text{d}}t$$

$$\begin{aligned} {Subject}\,\,{to}\,\dot{y} & = \bar{U}\left( {l - v} \right) - \delta y \\ y\left( 0 \right) & = y_{0} > 0,{\text{where}}\; 0 \le v < l\;{\text{and}}\;p_{2} \ge 0. \\ \end{aligned}$$

The Hamiltonian for the manufacturer’s problem is

$$H^{m} = \left( {p_{1} - c} \right)\left( {\alpha - \beta p_{2} - \gamma y} \right) - c_{1} \left( {\alpha - \beta p_{2} - \gamma y - \overline{U} } \right)^{2} - ev^{2} - fy^{2} + \lambda_{1} \left[ {\overline{U} \left( {l - v} \right) - \delta y} \right]$$

(26)

Assuming interior solutions, necessary conditions for optimality lead to

$$v = - \frac{{\lambda_{1} \overline{U} }}{2e}$$

(27)

$$p_{1} = \left\{ {\begin{array}{*{20}c} 0 \\ {p_{1}^{s} } \\ M \\ \end{array} \in [0,M]} \right.\;{\text{if}}\;\alpha - \beta p_{2} - \gamma y\left\{ {\begin{array}{*{20}c} < \\ = \\ > \\ \end{array} } \right\} 0$$

(28)

Optimality condition for $p_{1}$ implies that $p_{1}^{*}$ is a bang-bang policy, with the possibility of a singular control whenever $\alpha - \beta p_{2} - \gamma y = 0$ holds. Note that $p_{1} = 0$ can be excluded when $\alpha - \beta p_{2} - \gamma y$, the market demand, is nonnegative. If the market demand is zero, we may assign $p_{1}$ an arbitrary value (e.g., M) because $p_{1}$ is indeterminate, and the choice of $p_{1}$ does not affect the Hamiltonian . Furthermore, the optimal $p_{1}$ is M for any positive market demand. Therefore, we can expect that the manufacturer always sets his maximum price at M (upper bound of $p_{1}$) as long as the market demand remains nonnegative. (see Jørgensen 1986)

The Hamiltonian for the retailer’s problem is

$$H^{r} = \left( {p_{2} - p_{1} } \right)\left( {\alpha - \beta p_{2} - \gamma y} \right) - c_{2} \left( {\alpha - \beta p_{2} - \gamma y} \right)^{2} + \lambda_{2} \left[ {\overline{U} \left( {l - v} \right) - \delta y} \right]$$

(29)

Assuming interior solutions, a necessary condition for optimality yields

$$p_{2} = \frac{{\left( {1 + 2\beta c_{2} } \right)\left( {\alpha - \gamma y} \right) + \beta p_{1} }}{{2\beta \left( {1 + \beta c_{2} } \right)}}.$$

(30)

The solutions that satisfy the necessary conditions are optimal. The objective function of the manufacturer is concave in ($v, p_{1} )$, and the objective function of retailer is concave in $p_{2}$. All constraints are linear in $\left( {v,p_{1} ,p_{2} } \right)$.

Costate equations are, using (30) as follows:

$$\dot{\lambda }_{1} = \left( {r + \delta } \right)\lambda_{1} + \left( {\frac{{\gamma^{2} c_{1} }}{{1 + \beta c_{2} }} + 2f} \right)y + \gamma \left( {p_{1} - c} \right) - 2\gamma c_{1} \left( {\alpha - \overline{U} } \right) + \frac{{\gamma c_{1} \left[ {\alpha \left( {1 + 2\beta c_{2} } \right) + \beta p_{1} } \right]}}{{\left( {1 + \beta c_{2} } \right)}},$$

(31)

$$\dot{\lambda }_{2} = \left( {r + \delta } \right)\lambda_{2} + \left[ {2\gamma^{2} c_{2} - \frac{{\gamma^{2} \left( {1 + 2\beta c_{2} } \right)^{2} }}{{2\beta \left( {1 + \beta c_{2} } \right)}}} \right]y + \frac{{\gamma \left( {1 + 2\beta c_{2} } \right)\left[ {\alpha \left( {1 + 2\beta c_{2} } \right) + \beta p_{1} } \right]}}{{2\beta \left( {1 + \beta c_{2} } \right)}} - \gamma \left( {p_{1} + 2\alpha c_{2} } \right).$$

(32)

Using (11) and (27) yields $\lambda_{1} = \frac{2e}{{\overline{U}^{2} }}\left( {\dot{y} + {\updelta }y - \overline{U} l} \right)$. Therefore, after rearranging, (31) becomes

$$\begin{aligned} & \ddot{y} - r\dot{y} - \left[ {\delta \left( {r + \delta } \right) + \frac{{\overline{U}^{2} }}{2e}\left( {\frac{{\gamma^{2} c_{1} }}{{1 + \beta c_{2} }} + 2f} \right)} \right]y \\ & \quad = \frac{{\overline{U}^{2} }}{2e}\left[ {\gamma \left( {p_{1} - c} \right) - 2\gamma c_{1} \left( {\alpha - \overline{U} } \right) + \frac{{\gamma c_{1} \left( {\alpha + 2\alpha \beta c_{2} + \beta p_{1} } \right)}}{{\left( {1 + \beta c_{2} } \right)}}} \right] - \left( {r + \delta } \right)\overline{U} l. \\ \end{aligned}$$

(33)

Corresponding homogenous equation is $\ddot{y} - r\dot{y} - \left[ {\delta \left( {r + \delta } \right) + \frac{{\overline{U}^{2} }}{2e}\left( {\frac{{\gamma^{2} c_{1} }}{{1 + \beta c_{2} }} + 2f} \right)} \right]y = 0$, and the auxiliary equation is $m_{2}^{2} - rm_{2} - \left[ {\delta \left( {r + \delta } \right) + \frac{{\overline{U}^{2} }}{2e}\left( {\frac{{\gamma^{2} c_{1} }}{{1 + \beta c_{2} }} + 2f} \right)} \right] = 0$, where the two roots are

$$\begin{aligned} m_{21} & = \frac{{r + \sqrt {r^{2} + 4\left[ {\delta \left( {r + \delta } \right) + \frac{{\overline{U}^{2} }}{2e}\left( {\frac{{\gamma^{2} c_{1} }}{{1 + \beta c_{2} }} + 2f} \right)} \right]} }}{2} > r, \\ m_{22} & = \frac{{r - \sqrt {r^{2} + 4\left[ {\delta \left( {r + \delta } \right) + \frac{{\overline{U}^{2} }}{2e}\left( {\frac{{\gamma^{2} c_{1} }}{{1 + \beta c_{2} }} + 2f} \right)} \right]} }}{2} < 0. \\ \end{aligned}$$

(34)

A particular solution of y from (33) is

$$K_{21} = \frac{{\overline{U}^{2} \left\{ {\gamma \left( {1 + \beta c_{2} } \right)\left[ { - \left( {p_{1} - c} \right) + 2c_{1} \left( {\alpha - \overline{U} } \right)} \right] - \gamma c_{1} \left( {\alpha + 2\alpha \beta c_{2} + \beta p_{1} } \right)} \right\} + 2el\overline{U} \left( {r + \delta } \right)\left( {1 + \beta c_{2} } \right)}}{{2\left( {1 + \beta c_{2} } \right)\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right] + \gamma^{2} \overline{U}^{2} c_{1} }}.$$

(35)

Therefore, a general solution of $y$ is $y\left( t \right) = A_{21} e^{{m_{21} t}} + A_{22} e^{{m_{22} t}} + K_{21}$ which yields

$$\lambda_{1} = \frac{2e}{{\overline{U}^{2} }}(m_{21} A_{21} e^{{m_{21} t}} + m_{22} A_{22} e^{{m_{22} t}} + \delta A_{21} e^{{m_{21} t}} + \delta A_{22} e^{{m_{22} t}} + \delta K_{21} - \overline{U} l).$$

(36)

Also, to guarantee that the limiting transversality condition $\mathop {\lim }\limits_{T \to \infty } e^{ - rT} \lambda_{1} \left( T \right) = 0$ holds for all parameters, $A_{21} = 0$ $\left( {\because\,m_{21} - r > 0} \right)$.

Therefore,

$$y\left( t \right) = A_{22} e^{{m_{22} t}} + K_{21}$$

(37)

$$\lambda_{1} \left( t \right) = \frac{2e}{{\overline{U}^{2} }}\left[ {\left( {m_{22} + \delta } \right)A_{22} e^{{m_{22} t}} + \delta K_{21} - \overline{U} l} \right]$$

(38)

$$v(t) = - \frac{{\lambda_{1} \overline{U} }}{2e} = - \frac{1}{{\overline{U} }}\left[ {\left( {m_{22} + \delta } \right)A_{22} e^{{m_{22} t}} + \delta K_{21} - \overline{U} l} \right]$$

(39)

$$p_{2} (t) = \frac{{\left( {1 + 2\beta c_{2} } \right)\left( {\alpha - \gamma y} \right) + \beta p_{1} }}{{2\beta \left( {1 + \beta c_{2} } \right)}} = \frac{{\alpha \left( {1 + 2\beta c_{2} } \right) + \beta p_{1} }}{{2\beta \left( {1 + \beta c_{2} } \right)}} - \frac{{\gamma \left( {1 + 2\beta c_{2} } \right)}}{{2\beta \left( {1 + \beta c_{2} } \right)}}y$$

(40)

Calculating the coefficient using the initial condition gives $A_{22} = y_{0} - K_{21}$, and the long-term equilibriums are as follows:

$$y_{LR}^{II} \to K_{21}$$

(41)

$$v_{LR}^{II} \to - \frac{1}{{\overline{U} }}\left( {\delta K_{21} - \overline{U} l} \right)$$

(42)

$$p_{2\,LR}^{II} \to \frac{{\alpha \left( {1 + 2\beta c_{2} } \right) + \beta p_{1} }}{{2\beta \left( {1 + \beta c_{2} } \right)}} - \frac{{\gamma \left( {1 + 2\beta c_{2} } \right)}}{{2\beta \left( {1 + \beta c_{2} } \right)}}K_{21} .$$

(43)

2.2 Proof of Theorem 3

From Table 6, it is obvious that $K_{11} = K_{41}$ holds, which leads to $v_{LR}^{I} = v_{LR}^{IV}$ and $y_{LR}^{I} = y_{LR}^{IV}$.

2.3 Proof of Theorem 4

Since $y_{LR}^{I} = y_{LR}^{IV}$ and $v_{LR}^{I} = v_{LR}^{IV}$ is proved in Theorem 3, we examine $y_{LR}^{IV} - y_{LR}^{II}$ and $v_{LR}^{II} - v_{LR}^{IV}$ only.

First,

$$y_{LR}^{IV} - y_{LR}^{II} = - \frac{1}{2f}K_{41} \left( {r + \delta } \right) - K_{21}$$

(44)

Plugging the values of $K_{41}$ and $K_{21}$ in Table 6 into (44) and rearranging the equation, $y_{LR}^{IV} - y_{LR}^{II} > 0$ is equivalent to

$$\begin{aligned} \left( {1 + \beta c_{1} + \beta c_{2} } \right)\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right]p_{1} > & \left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right]\left\{ {\left( {1 + \beta c_{2} } \right)\left[ {c + 2c_{1} \left( {\alpha - \overline{U} } \right)} \right]} \right. \\ & \left. { - \,\alpha c_{1} \left( {1 + 2\beta c_{2} } \right)} \right\} - el\left( {r + \delta } \right)\gamma \overline{U} c_{1} . \\ \end{aligned}$$

(45)

Since $\left( {1 + \beta c_{1} + \beta c_{2} } \right)\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right]$ is positive, (45) is equivalent to

$$p_{1} > \frac{{\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right]\left\{ {\left( {1 + \beta c_{2} } \right)\left[ {c + 2c_{1} \left( {\alpha - \overline{U} } \right)} \right] - \alpha c_{1} \left( {1 + 2\beta c_{2} } \right)} \right\} - el\gamma \overline{U} c_{1} \left( {r + \delta } \right)}}{{\left( {1 + \beta c_{1} + \beta c_{2} } \right)\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right]}} = \tilde{p}_{1} .$$

(46)

Therefore, it holds that $y_{LR}^{II} < y_{LR}^{I} = y_{LR}^{IV}$, when $p_{1} > \tilde{p}_{1}$.

Similarly,

$$v_{LR}^{II} - v_{LR}^{IV} = - \frac{1}{{\overline{U} }}\left( {\delta K_{21} - \overline{U} l} \right) + \frac{{\overline{U} }}{2e}K_{41} .$$

(47)

Plugging the values of $K_{21}$ and $K_{41}$ in Table 6 into (47) and rearranging the equation, $v_{LR}^{II} - v_{LR}^{IV} > 0$ is equivalent to $p_{1} > \tilde{p}_{1}$.

Therefore, it holds that $v_{LR}^{II} > v_{LR}^{I} = v_{LR}^{IV}$, when $p_{1} > \tilde{p}_{1}$.

2.4 Proof of Theorem 5

Let $Q_{1} = \left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right]$ and $Q_{2} = 4\beta \left( {1 + \beta c_{1} + \beta c_{2} } \right)\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right] - \gamma^{2} \overline{U}^{2}$.

Note that $Q_{2}$ is positive since $K_{31}$ in Table 6 is assumed to be positive.

Since $y_{LR}^{I} = y_{LR}^{IV}$ and $v_{LR}^{I} = v_{LR}^{IV}$ is proved in Theorem 3, we examine $y_{LR}^{IV} - y_{LR}^{III}$ and $v_{LR}^{III} - v_{LR}^{IV}$ only.

First,

$$y_{LR}^{IV} - y_{LR}^{III} = - \frac{1}{2f}K_{41} \left( {r + \delta } \right) - K_{32}$$

(48)

Plugging the values of $K_{41}$ and $K_{32}$ in Table 6 into (48) and utilizing $Q_{1}$ and $Q_{2}$, $y_{LR}^{IV} - y_{LR}^{III} > 0$ is equivalent to

$$\gamma \overline{U} Q_{1} \alpha > el\left( {r + \delta } \right)\gamma^{2} \overline{U}^{2} - \beta \gamma \overline{U} \left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right] \left( {2c_{1} \overline{U} - c} \right)$$

(49)

Since $\gamma \overline{U} Q_{1}$ is positive, (49) is equivalent to

$$\alpha > \frac{{el\left( {r + \delta } \right)\gamma \overline{U} }}{{\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right]}} - \beta \left( {2c_{1} \overline{U} - c} \right) = \tilde{\alpha }.$$

(50)

Therefore, it holds that $y_{LR}^{III} < y_{LR}^{I} = y_{LR}^{IV}$, when $\alpha > \tilde{\alpha }$.

Similarly,

$$v_{LR}^{III} - v_{LR}^{IV} = - \frac{1}{{\overline{U} }}\left( {{\updelta }K_{32} - \overline{U} l} \right) + \frac{{\overline{U}}}{2e}K_{41}$$

(51)

Plugging the values of $K_{32}$ and $K_{41}$ in Table 6 into (51) and rearranging the equation, $v_{LR}^{III} - v_{LR}^{IV} > 0$ is equivalent to $\alpha > \tilde{\alpha }$.

Therefore, it holds that $v_{LR}^{III} > v_{LR}^{I} = v_{LR}^{IV}$, when $\alpha > \tilde{\alpha }$.

Furthermore, since $D = \alpha - \beta p_{2} - \gamma y$ in Model 3, the long-term demand is

$$\begin{aligned} D_{LR}^{III} & = \alpha - \beta p_{2\,LR}^{III} - \gamma y_{LR}^{III} \\ & = \alpha - \beta \left[ {\frac{{\alpha \left( {1 + 2\beta c_{1} + 2\beta c_{2} } \right) - 2\beta c_{1} \overline{U} + \beta c}}{{2\beta \left( {1 + \beta c_{1} + \beta c_{2} } \right)}} - \frac{{\gamma \left( {1 + 2\beta c_{1} + 2\beta c_{2} } \right)}}{{2\beta \left( {1 + \beta c_{1} + \beta c_{2} } \right)}}K_{32} } \right] - \gamma K_{32} \\ & = \frac{1}{{2\left( {1 + \beta c_{1} + \beta c_{2} } \right)}}\alpha + \frac{{2\beta c_{1} \overline{U} - \beta c}}{{2\left( {1 + \beta c_{1} + \beta c_{2} } \right)}} - \frac{\gamma }{{2\left( {1 + \beta c_{1} + \beta c_{2} } \right)}} \\ & \quad \times \,\frac{{4\beta el\overline{U} \left( {1 + \beta c_{1} + \beta c_{2} } \right)\left( {r + \delta } \right) - \gamma \overline{U}^{2} \left( {\alpha - \beta c + 2\beta c_{1} \overline{U} } \right)}}{{4\beta \left( {1 + \beta c_{1} + \beta c_{2} } \right)\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right] - \gamma^{2} \overline{U}^{2} }}. \\ \end{aligned}$$

(52)

For the long-term demand to be positive, rearranging Eq. (52),

$\alpha + 2\beta c_{1} \overline{U} - \beta c - \frac{{4\beta \gamma el\overline{U} \left( {1 + \beta c_{1} + \beta c_{2} } \right)\left( {r + \delta } \right) - \gamma^{2} \overline{U}^{2} \left( {\alpha - \beta c + 2\beta c_{1} \overline{U} } \right)}}{{4\beta \left( {1 + \beta c_{1} + \beta c_{2} } \right)\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right] - \gamma^{2} \overline{U}^{2} }} > 0$ should hold which is equivalent to

$$\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right]\alpha + \left( {2\beta c_{1} \overline{U} - \beta c} \right)\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right] - \gamma el\overline{U} \left( {r + \delta } \right) > 0.$$

(53)

Therefore, after rearranging (53), it is obvious that $D_{LR}^{III} > 0$ holds if and only if $\alpha > \frac{{el\left( {r + \delta } \right)\gamma \overline{U} }}{{\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right]}} - \beta \left( {2c_{1} \overline{U} - c} \right) = \tilde{\alpha }$.

2.5 Proof of Theorem 6

Part 1

$$\frac{{\partial y_{LR}^{I} }}{\partial f} = - \frac{{\overline{U}^{2} }}{{\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right]^{2} }} \cdot e\overline{U} l\left( {r + \delta } \right) < 0$$

(54)

It is obvious that (54) holds because all parameters are positive.

Similarly,

$$\frac{{\partial v_{LR}^{I} }}{\partial f} = - \frac{{fl\overline{U}^{4} }}{{\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right]^{2} }} + \frac{{l\overline{U}^{2} }}{{f\overline{U}^{2} + e\delta \left( {r + \delta } \right)}} = \frac{{el\overline{U}^{2} \delta \left( {r + \delta } \right)}}{{\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right]^{2} }} > 0.$$

(55)

Since $y_{LR}^{I} = y_{LR}^{IV}$ and $v_{LR}^{I} = v_{LR}^{IV}$, it also holds that $\frac{{\partial y_{LR}^{IV} }}{\partial f} < 0$, $\frac{{\partial v_{LR}^{IV} }}{\partial f} > 0$.

Part 2

$$\begin{aligned} \frac{{\partial y_{LR}^{II} }}{\partial f} = & - \frac{{2\overline{U}^{2} \left( {1 + \beta c_{2} } \right)}}{{\left\{ {2\left( {1 + \beta c_{2} } \right)\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right] + \gamma^{2} \overline{U}^{2} c_{1} } \right\}^{2} }}\left\{ {\overline{U}^{2} \left\{ {\gamma \left( {1 + \beta c_{2} } \right)\left[ { - \left( {p_{1} - c} \right) + 2c_{1} \left( {\alpha - \overline{U} } \right)} \right]} \right.} \right. \\ & \left. { - \,\gamma c_{1} \left( {\alpha + 2\alpha \beta c_{2} + \beta p_{1} } \right)} \right\}\left. { + 2el\overline{U} \left( {r + \delta } \right)\left( {1 + \beta c_{2} } \right)} \right\} \\ \end{aligned}$$

(56)

Utilizing the value of $K_{21}$ in Table 6, $\frac{{\partial y_{LR}^{II} }}{\partial f} < 0$ is equivalent to:

$$- \frac{{2\overline{U}^{2} \left( {1 + \beta c_{2} } \right)}}{{\left\{ {2\left( {1 + \beta c_{2} } \right)\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right] + \gamma^{2} \overline{U}^{2} c_{1} } \right\}}} \cdot K_{21} < 0$$

(57)

Note that $K_{21}$ is assumed to be nonnegative for feasible controls. Therefore, (57) holds.

Similarly,

$$\frac{{\partial v_{LR}^{II} }}{\partial f} = \frac{{2\delta \overline{U} \left( {1 + \beta c_{2} } \right)}}{{\left\{ {2\left( {1 + \beta c_{2} } \right)\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right] + \gamma^{2} \overline{U}^{2} c_{1} } \right\}}} \cdot K_{21} > 0$$

(58)

Part 3

$$\begin{aligned} \frac{{\partial y_{LR}^{III} }}{\partial f} = & - \frac{{4\beta \left( {1 + \beta c_{1} + \beta c_{2} } \right)\overline{U}^{2} }}{{\left\{ {4\beta \left( {1 + \beta c_{1} + \beta c_{2} } \right)\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right] - \gamma^{2} \overline{U}^{2} } \right\}^{2} }} \\ & \times \,\left[ {4\beta el\overline{U} \left( {1 + \beta c_{1} + \beta c_{2} } \right)\left( {r + \delta } \right) - \gamma \overline{U}^{2} \left( {\alpha - \beta c + 2\beta c_{1} \overline{U} } \right)} \right] \\ \end{aligned}$$

(59)

Utilizing the value of $K_{32}$ in Table 6, $\frac{{\partial y_{LR}^{III} }}{\partial f} < 0$ is equivalent to

$$- \frac{{4\beta \left( {1 + \beta c_{1} + \beta c_{2} } \right)\overline{U}^{2} }}{{\left\{ {4\beta \left( {1 + \beta c_{1} + \beta c_{2} } \right)\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right] - \gamma^{2} \overline{U}^{2} } \right\}}} \cdot K_{32} < 0$$

(60)

Note that $\left\{ {4\beta \left( {1 + \beta c_{1} + \beta c_{2} } \right)\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right] - \gamma^{2} \overline{U}^{2} } \right\} > 0$ since $K_{31}$ is assumed to be positive. Also, $K_{32}$ is assumed to be nonnegative for feasible controls. Therefore, (60) holds.

Similarly,

$$\frac{{\partial v_{LR}^{III} }}{\partial f} = \frac{{4\beta \delta \left( {1 + \beta c_{1} + \beta c_{2} } \right)\overline{U} }}{{\left\{ {4\beta \left( {1 + \beta c_{1} + \beta c_{2} } \right)\left[ {f\overline{U}^{2} + e\delta \left( {r + \delta } \right)} \right] - \gamma^{2} \overline{U}^{2} } \right\}}} \cdot K_{32} > 0.$$

(61)

Appendix 3: Literature Review

In environmental economics, a number of studies examined how the regulators could effectively induce pollution reduction through diverse instruments and incentives (Benchekroun and van Long 1998; Chen and Sheu 2009; Jung et al. 1996; Krass et al. 2013; Li et al. 2014; Li 2013). Milliman and Prince (1989) investigated five regulatory regimes such as direct controls, emission subsidies, emission taxes, free marketable permits, and auctions marketable permits, and examined which policy would facilitate firms’ technological change in the pollution control most effectively. Jung et al. (1996) also evaluated the effectiveness of various regulatory instruments in terms of firms’ incentives to develop and adopt pollution abatement technology. Subramanian et al. (2007) studied how firms’ pollution reduction strategies would vary under a regulator’s decision on the permits for emissions.

More recently, in the literature, there have been emerging interests in operational and market factors beyond the regulatory policies in inducing firms’ environmental performance . In reviewing studies in environmentally and socially sustainable operations, Tang and Zhou (2012) emphasized that the role of environmentally conscious consumers and cooperation within a supply chain deserves further investigation. Despite the importance of supply chain coordination and consumer’s environmental awareness , however, how the two factors simultaneously affect firm’s environmental performance remains largely unexplored. One notable exception is a study of Zhang et al. (2015). They examined how consumer’s environmental awareness would influence the order quantity decision and profits in three supply chain scenarios, i.e., a centralized supply chain, a decentralized supply chain, and a decentralized supply chain with a return contract.

Regarding supply chain coordination , the commonly accepted view is that a cooperative supply chain leads to higher environmental performance and sustainability (Handfield et al. 1997; Hollos et al. 2012; Simpson 2010). Ni et al. (2010) found that socially responsible or environmental performance is the highest in the cooperative supply chain, where the supplier and the manufacturer jointly maximize the supply chain profit, mainly because the double marginalization problem is eliminated. Lou et al. (2015) also examined three supply chain configurations and found that the cooperative supply chain in which the manufacturer and the retailer act as a single firm and the supply chain coordinated by a revenue sharing contract invest in emission reduction more than the decentralized supply chain. Klassen and Vachon (2003) empirically showed that an increased collaboration in the supply chain helps the firms invest more in environmental programs.

Consumer’s increasing preference for environment-friendly products is another important mechanism to motivate firms to reconsider their environmental strategy. Lee (2010) described how consumer’s environmental awareness influenced Esquel, one of the leading suppliers of premium cotton, to improve its environmental sustainability . Several studies incorporated environmentally conscious consumers explicitly and analyzed its impact on firms’ decisions and environmental performance (Conrad 2005; Du et al. 2015; Ghosh and Shah 2012; Wang et al. 2014). Bagnoli and Watts (2003) investigated how firms’ competition for socially responsible or environment-friendly consumers influenced firm’s decisions. Yalabik and Fairchild (2011) showed that pressures from environment-conscious consumers and regulators both lead to lower emissions as long as the initial emissions are not severe. Also, they found that high environmental competition between firms not only induces lower emissions but also improves the effectiveness of environmental pressures from consumers or regulators. Liu et al. (2012) also examined the impact of consumer’s environmental awareness and firms’ competition in production or retail on the supply chain.

Another important issue in modeling the firm’s environmental effort is concerned with what actually generates pollution. For instance, is the pollution emission rate proportional to the production rate or production capacity? Examining firm’s effort to reduce pollution, Subramanian et al. (2007) put forth two types of pollution reduction , one independent of and the other dependent on the production volume. Similarly, Chung et al. (2013) specified two sources of manufacturer’s pollution emission, one due to the plant operations, e.g., the size of the capacity, independent of production rate and the other due to and proportional to the production rate. There is no shortage of empirical studies, which reported that the plant capacity is related with the firm’s pollution emission rate, e.g., a plant with a larger capacity emits more pollution and thus has lower environmental performance (Grant et al. 2002; Gray and Shadbegian 2004; Laplante and Rilstone 1996; Ludwig 2004; Vachon and Klassen 2006). We conjecture that as long as the firm utilizes its capacity sufficiently, the firm’s pollution emission rate is proportional to its plant capacity, which in turn is closely related to its production rate.