Coordinating a supply chain with green innovation in a dynamic setting

Abstract

This paper addresses the channel coordination problem in a green supply chain consisting of a manufacturer and a retailer, in which the manufacturer controls green innovation and wholes price, while the retailer controls sales price. Pricing and green innovation strategies in integrated and decentralized channels are computed and compared, and a two-part tariff contract is designed to coordinate the decentralized supply chain. A Nash bargaining model is further developed to distribute the extra-profit between channel members. A numerical example is conducted to explore the impacts of green effectiveness and operational inefficiency effect on optimal/equilibrium solutions and coordination. The main results show that the green innovation investment, energy efficiency level and channel profit of integrated channel are larger than those of decentralized one, but the relationship of sales prices under two channel structures depends on system parameters. Green effectiveness exerts a positive effect on optimal/equilibrium solutions. The coordinator’s coordination capability is improved by green effectiveness, but weakened by operational inefficiency effect.

Appendix

For simplification, the time argument is omitted in the Appendices.

1.1 Proof of Proposition 1

Let $V^I$ denote the value function of the integrated channel. Taking the energy efficiency level evolution into account, the Hamilton-Jacobi-Bellman (HJB) equation for the whole supply chain is

$$\begin{aligned} \rho V^I(G)=\max \limits _{p, u}\{(p-c_0-c_1x)(\alpha -\beta p+\gamma x)-\frac{h}{2}u^2+\frac{\partial V^I}{\partial x}(\theta u-\delta x)\}. \end{aligned}$$

Maximization of the right-hand side of the HJB equation with respect to p and u yields

$$\begin{aligned} p= & {} \frac{\alpha +\beta c_0+(\gamma +\beta c_1) x}{2\beta },\end{aligned}$$

(30)

$$\begin{aligned} u= & {} \frac{\theta }{h}\frac{\partial V^I}{\partial x}. \end{aligned}$$

(31)

Inserting (30) and (31) on the right-hand side of the HJB equation provides

$$\begin{aligned} \rho V^I=\frac{(\alpha -\beta c_0+(\gamma -\beta c_1) x)^2}{4\beta }+\frac{\theta ^2}{2h}\left( \frac{\partial V^I}{\partial x}\right) ^2-\frac{\partial V^I}{\partial x}\delta x. \end{aligned}$$

(32)

Guided by the model’s linear-quadratic structure, we conjecture that the integrated channel value function is quadratic and given by

$$\begin{aligned} V^{I}(x)=\frac{I_{1}}{2}x^{2}+I_{2}x+I_{3}, \end{aligned}$$

(33)

and follows from (33) that

$$\begin{aligned} \frac{\partial V^I}{\partial x}=I_1x+I_2. \end{aligned}$$

(34)

Substituting (33) and (34) into (32) yields

$$\begin{aligned} \frac{\rho }{2}I_1x^2+\rho I_2x+\rho I_3= & {} \left( \frac{(\gamma -\beta c_1)^2}{4\beta }+\frac{\theta ^2}{2h}I_1^2-\rho I_1\right) x^2+\left( \frac{(\alpha -\beta c_0)(\gamma -\beta c_1)}{2\beta }\right. \nonumber \\&\quad \left. +\frac{\theta ^2}{h}I_1I_2-\delta I_2\right) x+\frac{(\alpha -\beta c_0)^2}{4\beta }+\frac{\theta ^2}{2h}I_2^2. \end{aligned}$$

(35)

Equating the coefficients of $x^2, x$ on both sides of (35), we get the expressions of $I_1, I_2$ shown in Proposition 1. Similarly, $I_3$ is obtained as

$$\begin{aligned} I_{3}=\frac{(\alpha -\beta c_0)^2\left( (\rho \beta h+\eta _1)^2+2\beta h\theta ^2(\gamma -\beta c_1)^2\right) }{4\beta \rho (\rho \beta h+\eta _1)^2}. \end{aligned}$$

(36)

Accordingly, the channel profit is given by

$$\begin{aligned} J^{I}=\frac{I_{1}}{2}x_0^{2}+I_{2}x_0+I_{3}. \end{aligned}$$

(37)

Inserting (35) into (1) produces a differential equation for the optimal energy efficiency level. Integrating the differential equation one obtains the trajectory in (10), where the steady-state energy efficiency level $x^{I}_{\infty }$ is globally stable if and only if $R_1>0$, i.e.,

$$\begin{aligned} 2\beta \delta h(\delta +\rho )-\theta ^2(\gamma -\beta c_1)^2>0. \end{aligned}$$

(38)

1.2 Proof of Corollary 2

Note from Proposition 2 that the monotonicity of sales price depends on the relationship of $x_0$ and $x^I_\infty$, of which $R_1>0$. Specifically, when $x_0>x^I_\infty$, the sales price is monotonically decreasing with time, namely, skimming pricing; when $x_0<x^I_\infty$, the sales price monotonically increases with time, namely, penetration pricing.

Since $0<c_1<\frac{\gamma }{\beta }$ which is assumed to reflect operational inefficiency effect and to ensure a positive $x^I_\infty$, we have $0<x^I_\infty <\frac{\gamma \theta ^2(\alpha -\beta c_0)}{2\beta h\delta (\delta +\rho )-\theta ^2\gamma ^2}$.

As such, if $x_0>\frac{\gamma \theta ^2(\alpha -\beta c_0)}{2\beta h\delta (\delta +\rho )-\theta ^2\gamma ^2}$, meaning that $x_0>x^I_\infty$, a skimming pricing strategy is adopted.

However, if $x_0<\frac{\gamma \theta ^2(\alpha -\beta c_0)}{2\beta h\delta (\delta +\rho )-\theta ^2\gamma ^2}$, there exists $\widetilde{c}_1$ which satisfies that $x_0=\frac{\theta ^2(\gamma -\beta \widetilde{c}_1)(\alpha -\beta c_0)}{2\beta h\delta (\delta +\rho )-\theta ^2(\gamma -\beta \widetilde{c}_1)^2}$. When $0<c_1<\widetilde{c}_1$, one has $x_0>x^I_\infty$, which implies a skimming pricing strategy; when $\widetilde{c}_1<c_1<\frac{\gamma }{\beta }$, it is found that $x_0<x^I_\infty$, implying a penetration pricing.

1.3 Proof of Proposition 3

To obtain a Stackelberg equilibrium, we first determine the retailer’s pricing strategy p as a function of the manufacturer’s decisions w and u. Let $V_R^D, V_M^D$ denote the value functions for the retailer and the manufacturer. The retailer’s HJB equation can be specified as

$$\begin{aligned} \rho V_R^D=\max \limits _{p}\{(p-w)(\alpha -\beta p+\gamma x)+\frac{\partial V_R^D}{\partial x}(\theta u-\delta x)\}. \end{aligned}$$

(39)

The maximization with respect to p yields the retailer’s reaction function:

$$\begin{aligned} p=\frac{\alpha +\beta w+\gamma x}{2b}. \end{aligned}$$

(40)

Anticipating the retailer’s response in (40), the manufacturer’s HJB equation is given by

$$\begin{aligned} \rho V_M^D=\max \limits _{w, u}\{(w-c_0-c_1x)(\alpha -\beta p+\gamma x)-\frac{h}{2}u^2+\frac{\partial V_M^D}{\partial x}(\theta u-\delta x)\}. \end{aligned}$$

(41)

Performing the maximization on the right-hand side with respect to w and u yields

$$\begin{aligned} w=\frac{\alpha +\beta c_0+(\gamma +\beta c_1) x}{2\beta }, u=\frac{\theta }{h}\frac{\partial V_M^D}{\partial x}. \end{aligned}$$

(42)

Substituting the expression of w above into (40) produces

$$\begin{aligned} p=\frac{3\alpha +\beta c_0+(3\gamma +\beta c_1) x}{4\beta }. \end{aligned}$$

(43)

Substitute (42) and (43) into HJB equations (39) and (41), and assume the following quadratic value functions:

$$\begin{aligned} V^{D}_{M}= & {} \frac{A_{1}}{2}x^{2}+A_{2}x+A_{3},\nonumber \\ V^{D}_{R}= & {} \frac{B_{1}}{2}x^{2}+B_{2}x+B_{3}. \end{aligned}$$

(44)

Following the same procedure as that in the proof of Proposition 1, the six Riccati equations that characterize the coefficients of the value functions $A_i, B_i, i=1, 2, 3$ are determined by identification. The coefficients $A_1, A_2$ are given in Proposition 3, and other coefficients are presented as follows.

$$\begin{aligned} A_{3}= & {} \frac{(\alpha -\beta c_0)^2\left( (\rho \beta h+\eta _2)^2+\beta h\theta ^2(\gamma -\beta c_1)^2\right) }{8\beta \rho (\rho \beta h+\eta _2)^2},\nonumber \\ B_{1}= & {} \frac{h(\gamma -\beta c_1)^2}{8\eta _2},\nonumber \\ B_{2}= & {} \frac{h(\alpha -\beta c_0)(\gamma -\beta c_1)(\theta ^ 2\beta h(\gamma -\beta c_1)^2 +2\eta _2(\rho \beta h+\eta _2))}{8\eta _2(\rho \beta h+\eta _2)^2},\nonumber \\ B_{3}= & {} \frac{(\alpha -\beta c_0)^2(\eta _2(\rho \beta h+\eta _2)((\rho \beta h+\eta )^2+2\beta h\theta ^2(\gamma -\beta c_1)^2)+h^2\theta ^4\beta ^2(\gamma -\beta c_1)^4)}{16\rho \beta \eta _2(\rho \beta h+\eta _2)^3}.\nonumber \\ \end{aligned}$$

(45)

Then, the equilibrium profits of the manufacturer, the retailer and the whole channel are

$$\begin{aligned}&J^{D}_{M}=\frac{A_{1}}{2}x_0^{2}+A_{2}x_0+A_{3},\nonumber \\&J^{D}_{R}=\frac{B_{1}}{2}x_0^{2}+B_{2}x_0+B_{3},\nonumber \\&J^{D}=\frac{1}{2}(A_1+B_1)x_0^{2}+(A_{2}+B_{2})x_0+A_{3}+B_{3}. \end{aligned}$$

(46)

Similarly, the optimal time path of the energy efficiency level can be written as in (17), where the steady state $x^{D}_{\infty }$ is globally stable if and only if $R_2>0$, i.e.,

$$\begin{aligned} 4\beta \delta h(\delta +\rho )-\theta ^2(\gamma -\beta c_1)^2>0. \end{aligned}$$

(47)

1.4 Proof of Proposition 5

Let $V_R^C, V^C$ denote the value functions for the retailer and the manufacturer, respectively. The retailer’s HJB equation can be specified as

$$\begin{aligned} \rho V_R^C=\max \limits _{p}\{(p-c_0-c_1x)(\alpha -\beta p+\gamma x)-k+\frac{\partial V_R^C}{\partial x}(\theta u-\delta x)\}. \end{aligned}$$

(48)

The equilibrium sales price is given below by maximizing the right-hand side of (48) with respect to p, i.e.,

$$\begin{aligned} p^C=\frac{\alpha +\beta c_0+(\gamma +\beta c_1)x}{2\beta }. \end{aligned}$$

(49)

The manufacturer’s optimization problem is given by

$$\begin{aligned} \max \limits _{u}\int _0^\infty e^{-\rho t}\left( (p-c_0-c_1x)(\alpha -\beta p+\gamma x)-\frac{h}{2}u^{2}\right) \mathrm{d}t, \end{aligned}$$

(50)

subject to the dynamic evolution of energy efficiency level (1).

The corresponding HJB equation is

$$\begin{aligned} \rho V^C=\max \limits _{u}\{(p-c_0-c_1x)(\alpha -\beta p+\gamma x)-\frac{h}{2}u^2+\frac{\partial V^C}{\partial x}(\theta u-\delta x)\}. \end{aligned}$$

(51)

Substituting (49) into (51), then maximizing the right-hand side with respect to u yields

$$\begin{aligned} u^C=\frac{\theta }{h}\frac{\partial V^C_M}{\partial x}. \end{aligned}$$

(52)

Conjecture the following value functions:

$$\begin{aligned} V^C_R(x)= & {} \frac{N_1}{2}x^{2}+N_{2}x+N_{3},\end{aligned}$$

(53)

$$\begin{aligned} V^C(x)= & {} \frac{C_{1}}{2}x^{2}+C_{2}x+C_{3}. \end{aligned}$$

(54)

By means of the procedure of the proof for Proposition 1, we have

$$\begin{aligned} C_1= & {} I_1, C_2=I_2, C_3=I_3,\nonumber \\ N_1= & {} \frac{h(\gamma -\beta c_1)^2}{2\eta _1}, \end{aligned}$$

(55)

$$\begin{aligned} N_2= & {} \frac{h(\alpha -\beta c_0)(\gamma -\beta c_1)(\theta ^2\beta h(\gamma -\beta c_1)^2 +\eta _1(\rho \beta h+\eta _1))}{\eta _1(\rho \beta h+\eta _1)^2}, \end{aligned}$$

(56)

$$\begin{aligned} N_3(k)= & {} \frac{\theta ^2(\alpha -\beta c_0)(\gamma -\beta c_1)}{\rho (\rho \beta h+\eta _1)}N_2+\frac{(\alpha -\beta c_0)^{2}}{4\beta \rho }-\frac{k}{\rho }. \end{aligned}$$

(57)

Consequently, $u^C=u^I, x^C=x^I, p^C=p^I$. These equilibrium strategies are identical with the integrated solutions.

Similarly, the profits of the manufacturer and retailer, $J^C_M$, $J^C_R$, are given by the value functions in (24) and (25), where

$$\begin{aligned} M_1= & {} -\frac{(\beta h(2\delta +\rho )-\eta _1)^2}{4\theta ^2\beta \eta _1}, \end{aligned}$$

(58)

$$\begin{aligned} M_2= & {} \frac{h(\eta _1^2-\beta ^2h^2(2\delta +\rho )^2)(\alpha -\beta c_0)(\gamma -\beta c_1)}{2\eta _1(\rho \beta h+\eta _1)^2}, \end{aligned}$$

(59)

$$\begin{aligned} M_3(k)= & {} \frac{k}{\rho }+\frac{\theta ^2(\alpha -\beta c_0)(\gamma -\beta c_1)}{\rho (\rho \beta h+\eta _1)}M_2-\frac{h\theta ^2(\alpha -\beta c_0)^2(\gamma -\beta c_1)^2}{2\rho (\rho \beta h+\eta _1)^2}. \end{aligned}$$

(60)

1.5 Proof of Proposition 6

According to (26), we have

$$\begin{aligned} \frac{M_1}{2}x_0^2+M_2x_0+M_3(k)-\frac{A_1}{2}x_0^2-A_2x_0-A_3>0, \end{aligned}$$

i.e.,

$$\begin{aligned} \frac{k}{\rho }&>-\left( \frac{M_1}{2}x_0^2+M_2x_0+\frac{\theta ^2(\alpha -\beta c_0)(\gamma -\beta c_1)}{\rho (\rho \beta h+\eta _1)}M_2-\frac{h\theta ^2(\alpha -\beta c_0)^2(\gamma -\beta c_1)^2}{2\rho (\rho \beta h+\eta _1)^2}\right) \nonumber \\&\quad +\frac{A_1}{2}x_0^2+A_2x_0+A_3=J_M^{D}-J_{M}^C(0).\nonumber \end{aligned}$$

Thus, k satisfies $k>\rho (J_M^{D}-J_{M}^C(0))$.

Similarly, according to (27), we have $k<\rho (J_R^C(0)-J_R^{D})$.

Let $k_1=\rho (J_M^{D}-J_{M}^C(0)), k_2=\rho (J_R^C(0)-J_R^{D})$. It’s easy to verify that $J_{M}^C(0)<0, J_M^{D}>0$, so $k_1>0$. Also,

$$\begin{aligned} k_2-k_1&=J_{M}^C(0)+J_R^C(0)-J_M^{D}-J_R^D\nonumber \\&=J^{I}-J^{D}\nonumber \\&>0.\nonumber \end{aligned}$$

Consequently, k should satisfy $\rho (J_M^{D}-J_{M}^C(0))<k<\rho (J_R^C(0)-J_R^{D})$.