Retailers’ competition and cooperation in a closed-loop green supply chain under governmental intervention and cap-and-trade policy

Abstract

This paper investigates retailers’ competition and cooperation in a closed-loop green supply chain consisting of one common manufacturer and two competing retailers under governmental intervention and cap-and-trade policy. Considering a consistent pricing strategy of the manufacturer, this study develops one centralized policy and three manufacturer-led decentralized policies viz. Collusion, Cournot (Nash), and Stackelberg depending on different competitive behaviors of the retailers. Optimal decisions are compared analytically through a special case where the retailers face the same basic market, and numerically where they face both the same basic market and different basic markets. A transfer payment mechanism is developed so that all the channel members achieve Pareto improvement. Numerical results indicate that (1) among the three decentralized scenarios, Nash behavior is profitable to the manufacturer, customers, and the whole supply chain, but Collusion behavior is profitable to the retailers only when the difference of their basic markets is small, (2) when the retailers face the same basic market and play Stackelberg game, it is beneficial for the retailers to be follower rather than leader, and (3) occurrence of both the government subsidy and cap-and-trade policy is profitable to all the channel members.

Appendix

1.1 Proof of Proposition 1

The profit function for the manufacturer is given by

$$\begin{aligned} \Pi _m(w,\theta ) & = w D + s D - c_m (D - D_R) - (c_r + A_0) D_R - c_e (E_m - E) - \lambda \theta ^2 \end{aligned}$$

The profit function for the retailer i is given by

$$\begin{aligned} \Pi _{ri}(p_i) & = (p_i - w) D_i,~ i = 1,2. \end{aligned}$$

So, the profit function of the centralized policy is given by

$$\begin{aligned} \Pi ^J(p_1,p_2,\theta ) & = p_1 D_1 + p_2 D_2 + s D - c_m (D - D_R) - (c_r + A_0) D_R - c_e (E_m - E) - \lambda \theta ^2\\ & = p_1 D_1 + p_2 D_2 + (\Psi _1 \theta - \Psi _2) [(a_1+ a_2) - (\alpha - \beta )(p_1 + p_2) + 2 \gamma \theta ] + c_e E - \lambda \theta ^2\\ {\mathrm{where}} \,\,\Psi _1 & = k \theta _0 + c_e \psi ,~\Psi _2 = c_m + k \theta _0^2 - C_0 \tau + c_e e_0. \end{aligned}$$

Now,

$$\begin{aligned} \frac{\partial \Pi ^{J}}{\partial p_1} & = a_1 - 2 \alpha p_1 + 2 \beta p_2 + \big (\gamma - (\alpha - \beta ) \Psi _1\big ) \theta + (\alpha - \beta ) \Psi _2,~~\frac{\partial ^2\Pi ^{J}}{\partial p^2_1} = - 2 \alpha< 0,\\ \frac{\partial \Pi ^{J}}{\partial p_2} & = a_2 - 2 \alpha p_2 + 2 \beta p_1 + \big (\gamma - (\alpha - \beta ) \Psi _1\big ) \theta + (\alpha - \beta ) \Psi _2,~~\frac{\partial ^2\Pi ^{J}}{\partial p^2_2} = - 2 \alpha< 0,\\ \frac{\partial \Pi ^{J}}{\partial \theta } & = \big (\gamma - (\alpha - \beta ) \Psi _1\big )(p_1 + p_2) + (4 \gamma \Psi _1 - 2 \lambda ) \theta - 2 \gamma \Psi _2,~~\frac{\partial ^2\Pi ^{J}}{\partial \theta ^2} = - 2 \lambda + 4 \gamma \Psi _1 < 0,~{\mathrm{if}}~\lambda > 2 \gamma \Psi _1,\\ \frac{\partial ^2\Pi ^{J}}{\partial p_1\partial p_2} & = 2 \beta ,~ \frac{\partial ^2\Pi ^{J}}{\partial p_1\partial \theta } = \gamma - (\alpha - \beta ) \Psi _1,~ \frac{\partial ^2\Pi ^{J}}{\partial p_2\partial \theta } = \gamma - (\alpha - \beta ) \Psi _1. \end{aligned}$$

The corresponding Hessian matrix is given by

$$\begin{aligned} H=\left( \begin{array}{lll} \frac{\partial ^2\Pi ^{J}}{\partial p^2_1} &\quad \frac{\partial ^2\Pi ^{J}}{\partial p_1\partial p_2} &\quad \frac{\partial ^2\Pi ^{J}}{\partial p_1\partial \theta }\\ \frac{\partial ^2\Pi ^{J}}{\partial p_2\partial p_1} &\quad \frac{\partial ^2\Pi ^{J}}{\partial p^2_2} &\quad \frac{\partial ^2\Pi ^{J}}{\partial p_2\partial \theta }\\ \frac{\partial ^2\Pi ^{J}}{\partial \theta \partial p_1} &\quad \frac{\partial ^2\Pi ^{J}}{\partial \theta \partial p_2} &\quad \frac{\partial ^2\Pi ^{J}}{\partial \theta ^2} \end{array} \right) =\left( \begin{array}{lll} - 2 \alpha &{}\quad 2 \beta &{} \quad \gamma - (\alpha - \beta ) \Psi _1\\ 2 \beta &{}\quad - 2 \alpha &{}\quad \gamma - (\alpha - \beta ) \Psi _1\\ \gamma - (\alpha - \beta ) \Psi _1 &{}\quad \gamma - (\alpha - \beta ) \Psi _1 &{}\quad - 2 \lambda + 4 \gamma \Psi _1 \end{array} \right) \end{aligned}$$

Now, the leading principle minors are $M_1 = - 2 \alpha < 0$, $M_2 = 4 (\alpha ^2 - \beta ^2) > 0$ and $|H| = 4 (\alpha + \beta ) [\Psi _3^2 - 2 \lambda (\alpha - \beta )]< 0$ if $\lambda > \frac{\Psi _3^2}{2 (\alpha - \beta )}$. Thus‚ the Hessian matrix is negative definite if

$\lambda > \max \{2 \gamma \Psi _1, \frac{\Psi _3^2}{2 (\alpha - \beta )}\}$. Using the first order conditions for optimality i.e. $\frac{\partial \Pi ^{J}}{\partial p_1}=0, \frac{\partial \Pi ^{J}}{\partial p_2}=0$ and $\frac{\partial \Pi ^{J}}{\partial \theta }=0$, the optimal values of the decision variables can be obtained as given in Proposition 1.

1.2 Proof of Proposition 2

Solving the equations $\frac{\partial \Pi _{r}}{\partial p_1}=0$ and $\frac{\partial \Pi _{r}}{\partial p_2}=0$ simultaneously, we get the optimal solution as

$$\begin{aligned} \overline{p_i} = \frac{a_i \alpha + a_j \beta + (\alpha + \beta )[(\alpha - \beta ) w + \gamma \theta ]}{[2 (\alpha ^2 - \beta ^2)]},\quad {\mathrm{where}}\,\,i = 1, 2; j = 3 - i. \end{aligned}$$

Substituting these values in the manufacturer’s profit function (1), we get the profit function of the manufacturer as follows:

$$\begin{aligned} \Pi _m^C(w,\theta ) & = (w + \Psi _1 \theta - \Psi _2)\left[ \frac{(a_1 + a_2) - 2 \big ((\alpha - \beta )w - \gamma \theta \big )}{2}\right] + c_e E - \lambda \theta ^2 \end{aligned}$$

Now,

$$\begin{aligned} \frac{\partial \Pi _m^{C}}{\partial w} & = \frac{1}{2}\big [a_1 + a_2 + 2 \big (\gamma \theta - (\alpha - \beta )(2 w + \Psi _1 \theta - \Psi _2)\big )\big ],~~\frac{\partial ^2\Pi _m^{C}}{\partial w^2} = - 2 (\alpha - \beta )< 0,\\ \frac{\partial \Pi _m^{C}}{\partial \theta } & = \frac{1}{2}\big [- 4 \lambda \theta + (a_1 + a_2) \Psi _1 + 2 \gamma (\Psi _1 \theta - \Psi _2) + 2 w\big (\gamma - (\alpha - \beta ) \Psi _1\big )\big ],\\ \frac{\partial ^2\Pi _m^{C}}{\partial \theta ^2} & = - 2 \lambda + 2 \gamma \Psi _1 < 0~{\mathrm{if}}~\lambda > \gamma \Psi _1,~~\frac{\partial ^2\Pi _m^{C}}{\partial w\partial \theta } = \gamma - (\alpha - \beta ) \Psi _1. \end{aligned}$$

The corresponding Hessian matrix of the manufacturer’s profit function is given by

$$\begin{aligned} H=\left( \begin{array}{lll} \frac{\partial ^2\Pi _m^{C}}{\partial w^2} &{}\quad \frac{\partial ^2\Pi _m^{C}}{\partial w\partial \theta }\\ \frac{\partial ^2\Pi _m^{C}}{\partial \theta \partial w} &{} \quad \frac{\partial ^2\Pi _m^{C}}{\partial \theta ^2} \end{array} \right) =\left( \begin{array}{lll} - 2 (\alpha - \beta ) &{} \quad \gamma - (\alpha - \beta ) \Psi _1\\ \gamma - (\alpha - \beta ) \Psi _1 &{}\quad - 2 \lambda + 2 \gamma \Psi _1 \end{array} \right) \end{aligned}$$

Now, $\frac{\partial ^2\Pi _m^{C}}{\partial \theta ^2}$ will be negative if $\lambda > \gamma \Psi _1$ and $|H| = 4 \lambda (\alpha - \beta ) - \Psi _3^2 > 0$ if $\lambda > \frac{\Psi _3^2}{4 (\alpha - \beta )}$.

Therefore, the Hessian matrix corresponding to the manufacturer’s profit function will be jointly concave w.r.t w and $\theta$ if $\lambda > \max \{\gamma \Psi _1, \frac{\Psi _3^2}{4 (\alpha - \beta )}\}$.

Using the first order conditions for optimality i.e. $\frac{\partial \Pi _m^{C}}{\partial w}=0$ and $\frac{\partial \Pi _m^{C}}{\partial \theta }=0$, the optimal decisions of the manufacturer can be obtained and putting these decisions in retailers’ profit functions the optimal decisions of the retailers can also be obtained, which are given in Proposition 2.

1.3 Proof of Proposition 6

$$\begin{aligned} \theta ^J - \theta ^N & = \frac{2 \lambda \Psi _3 (\alpha - \beta )^2 [a - (\alpha - \beta ) \Psi _2]}{2 \Sigma _2 [2 \lambda (\alpha - \beta ) - \Psi _3^2]}> 0\\ \theta ^N - \theta ^R & = \frac{2 (2 \alpha + \beta ) (\alpha - \beta )^2 \beta ^2 \lambda \Psi _3 [a - (\alpha - \beta ) \Psi _2]}{\Sigma _2 \Sigma _3}> 0\\ \theta ^R - \theta ^C & = \frac{4 \beta (\alpha - \beta )(4 \alpha ^2 + \alpha \beta - \beta ^2) \lambda \Psi _3 [a - (\alpha - \beta ) \Psi _2]}{\Sigma _1 \Sigma _3} > 0. \end{aligned}$$

1.4 Proof of Proposition 7

$$\begin{aligned} p_1^C - p_1^R & = \frac{\beta \lambda [a - (\alpha - \beta ) \Psi _2][16 \lambda \alpha ^2 (\alpha - \beta ) - \Psi _3 \{\gamma [16 \alpha ^2 + 3 \beta (\alpha - \beta )] - (\alpha - \beta )^2 \beta \Psi _1\}]}{\Sigma _1 \Sigma _3 }> 0\\ p_1^R - p_1^N & = \frac{(\alpha - \beta ) \beta ^2 \lambda [a - (\alpha - \beta ) \Psi _2][8 \alpha \lambda (\alpha - \beta ) - \Psi _3 \{\gamma (5 \alpha + 2 \beta ) + \alpha (\alpha - \beta ) \Psi _1\}]}{\Sigma _2 \Sigma _3 }> 0\\ p_1^N - p_1^J & = \frac{2 (\alpha - \beta ) \lambda [a - (\alpha - \beta ) \Psi _2][\lambda (\alpha - \beta ) - \gamma \Psi _3]}{\Sigma _2 [2 \lambda (\alpha - \beta ) - \Psi _3^2]}> 0\\ w^C - w^R & = \frac{2 \beta \lambda (4 \alpha ^2 + \alpha \beta - \beta ^2) [a - (\alpha - \beta ) \Psi _2][(\alpha - \beta )^2 - \gamma ^2]}{\Sigma _1 \Sigma _3 }> 0~{\mathrm{if}}~(\alpha - \beta ) \Psi _1> \gamma \\ w^R - w^N & = \frac{(2 \alpha ^2 - \alpha \beta - \beta ^2) \beta ^2 \lambda [a - (\alpha - \beta ) \Psi _2][(\alpha - \beta )^2 - \gamma ^2]}{\Sigma _2 \Sigma _3 }> 0~{\mathrm{if}}~(\alpha - \beta ) \Psi _1 > \gamma . \end{aligned}$$