Green channel coordination under asymmetric information

Abstract

The increased environmental awareness of consumers has led supply chains (SC) to green their operations. To extract a higher portion from the expanded demand due to greening activities, SC parties may hide key information regarding their green activities. This paper investigates the channel coordination problem in a green SC consisting of a manufacturer who sells a green product through a retailer. Both parties may involve in greening operations to expand an environmental-aware market; however, the retailer is privy to the information about his green sales effort. The analysis of the first-best outcome characterizes the conditions for (i) hold-up problem under which the retailer benefits from free ride on the manufacturer's greening operations effort, (ii) commitment strategy from the retailer to cover for the market expansion due to the manufacturer’s underinvestment in greening operations, and (iii) synergy in greening efforts. We then solve for the optimal incentive contracts under asymmetric information. Our analysis suggests that the manufacturer can include her greening effort in the contract to work as an incentive-fee; the higher level of greening effort by the manufacturer incentivizes the retailer to increase his green sales effort. We also show that the wholesale price term works as a screening tool to avoid the low efficient retailer from mimicking the high efficient one. Finally, we show that information asymmetry reduces the social welfare in a green market; it leads to a higher market price and a lower greening effort level.

Appendix

1.1 Proof of Proposition 1

The total channel profit under centralized setting would be:

$$ \mathop {\max }\limits_{{\tau_{m} ,\tau_{r} ,p}} \;\;\Pi_{C} = \Pi_{r} + \Pi_{m} = \left( {p - c} \right)\left( {\gamma - kp + \alpha_{m} \tau_{m} + \alpha_{r} \tau_{r} } \right) - \beta_{m} \tau_{m}^{2} - \beta_{r} \tau_{r}^{2} $$

(A.1)

Due to the concavity of the objective function (see Swami & Shah, 2013), the first order conditions result in the following optimal solutions:

$$ \tau_{m}^{C} = \frac{{\beta_{r} \alpha_{m} }}{{\left( {4\beta_{r} \beta_{m} k - \beta_{m} \alpha_{r}^{2} - \beta_{r} \alpha_{m}^{2} } \right)}} \times \left( {\gamma - ck} \right) $$

(A.2)

$$ \tau_{r}^{C} = \frac{{\beta_{m} \alpha_{r} }}{{\left( {4\beta_{r} \beta_{m} k - \beta_{m} \alpha_{r}^{2} - \beta_{r} \alpha_{m}^{2} } \right)}} \times \left( {\gamma - ck} \right) $$

(A.3)

$$ p^{C} = c + \frac{{2\beta_{r} \beta_{m} \left( {\gamma - ck} \right)}}{{\left( {4\beta_{r} \beta_{m} k - \beta_{m} \alpha_{r}^{2} - \beta_{r} \alpha_{m}^{2} } \right)}} $$

(A.4)

1.2 Proof of Lemma 1

The profit of the retailer is:

$$ \Pi_{r} \left( {p,\tau_{r} |w} \right) = \left( {p - w} \right) \times \left( {\gamma - kp + \alpha_{m} \tau_{m} + \alpha_{r} \tau_{r} } \right) - \beta_{r} \tau_{r}^{2} $$

(A.5)

Due to the concavity of the above objective function, the first best conditions for $\tau_{r}^{{\text{d}}}$ and $p^{{\text{d}}}$ yield:

$$ \tau_{r}^{{\text{d}}} = \frac{{\left( {\gamma - kw + \alpha_{m} \tau_{m} } \right)\alpha_{r} }}{{2\beta_{r} - \alpha_{r}^{2} }} $$

(A.6)

$$ p^{{\text{d}}} = \frac{1}{2k}\left( {kw + \gamma + \alpha_{m} \tau_{m} + \alpha_{r} \tau_{r}^{{\text{d}}} } \right) $$

(A.7)

Proof of Proposition 2. Applying backward induction, we first solve the retailer’s profit considering the manufacturer’s decisions are known. i.e.:

$$ \frac{{\partial \Pi_{r} }}{\partial p} = 0 $$

(A.8)

$$ \frac{{\partial \Pi_{r} }}{{\partial \tau_{r} }} = 0 $$

(A.9)

So we have:

$$ p = \frac{{ - w\alpha_{r}^{2} + 2\beta_{r} \gamma + 2\alpha_{m} \beta_{r} \tau_{m} + 2\beta_{r} kw}}{{\left( {4\beta_{r} k - \alpha_{r}^{2} } \right)}} $$

(A.10)

$$ \tau_{r} = \frac{{\alpha_{r} \left( {\gamma + \alpha_{m} \tau_{m} - kw} \right)}}{{\left( {4\beta_{r} k - \alpha_{r}^{2} } \right)}} $$

(A.11)

Now we solve the manufacturer’s optimal response function. Plugging the found solutions into $\Pi_{m}$, we have:

$$ \Pi_{m} \!=\! \frac{{\left( {\beta_{m} \alpha_{r}^{2} \tau_{m}^{2} \!-\! 2\beta_{r} k^{2} w^{2} \!+\! 2\beta_{r} ck^{2} w \!-\! 4\beta_{m} \beta_{r} k\tau_{m}^{2} \!+\! 2\alpha_{m} \beta_{r} k\tau_{m} w \!-\! 2\alpha_{m} \beta_{r} ck\tau_{m} \!+\! 2\beta_{r} \gamma kw \!-\! 2\beta_{r} c\gamma k} \right)}}{{\left( {4\beta_{r} k \!-\! \alpha_{r}^{2} } \right)}} $$

(A.12)

Applying the first-best conditions for $\Pi_{m}$, we obtain the optimal values of $w$ and $\tau_{m}$:

$$ w^{d} = ck + \frac{{\left( {4\beta_{m} \beta_{r} k - \beta_{m} \alpha_{r}^{2} } \right)}}{{\left( {8\beta_{m} \beta_{r} k - 2\beta_{m} \alpha_{r}^{2} - \beta_{r} \alpha_{m}^{2} } \right)}} \times \left( {\gamma - ck} \right) $$

(A.13)

$$ \tau_{m}^{d} = \frac{{\beta_{r} \alpha_{m} }}{{\left( {8\beta_{m} \beta_{r} k - 2\beta_{m} \alpha_{r}^{2} - \beta_{r} \alpha_{m}^{2} } \right)}} \times \left( {\gamma - ck} \right) $$

(A.14)

Inserting the extracted equilibriums into equations (A.10) and (A.11) we get:

$$ p^{d} = ck + \frac{{\left( {6\beta_{m} \beta_{r} k - \beta_{m} \alpha_{r}^{2} } \right)}}{{\left( {8\beta_{m} \beta_{r} k - 2\beta_{m} \alpha_{r}^{2} - \beta_{r} \alpha_{m}^{2} } \right)}} \times \left( {\gamma - ck} \right) $$

(A.15)

$$ \tau_{r}^{d} = \frac{{\beta_{m} \alpha_{r} }}{{\left( {8\beta_{m} \beta_{r} k - 2\beta_{m} \alpha_{r}^{2} - \beta_{r} \alpha_{m}^{2} } \right)}} \times \left( {\gamma - ck} \right) $$

(A.16)

Proof of Corollary 1. To study the impact of $\beta_{r}$ and $\beta_{m}$ on the manufacturer’s greening effort, we need to find $\partial \tau_{m\theta } /\partial \beta_{r}$ and $\partial \tau_{m\theta } /\partial \beta_{m}$, respectively;

$$ \frac{{\partial \tau_{m\theta } }}{{\partial \beta_{r} }} = - \frac{{2\left( {\gamma - c} \right)\alpha_{m} \beta_{m} \alpha_{r}^{2} }}{{\left( {\left( {2\alpha_{r}^{2} - 8\beta_{r} } \right)\beta_{m} + \beta_{r} \alpha_{m}^{2} } \right)^{2} }} < 0 $$

(A.17)

$$ \frac{{\partial \tau_{m\theta } }}{{\partial \beta_{m} }} = \frac{{2\beta_{r} \alpha_{m} \left( {\gamma - c} \right)\left( {\alpha_{r}^{2} - 4\beta_{r} } \right)}}{{\left( {\left( {\alpha_{m}^{2} - 8\beta_{m} } \right)\beta_{r} + 2\beta_{m} \alpha_{r}^{2} } \right)^{2} }} $$

(A.18)

Note that, from Eq. (A.17), we have $\partial \tau_{m\theta } /\partial \beta_{r} < 0$. However, from Eq. (A.18), one can verify that $\partial \tau_{m\theta } /\partial \beta_{m} \ge 0$ if $\alpha_{r}^{2} \ge 4\beta_{r}$, or $\alpha_{r} \ge 2\sqrt {\beta_{r} }$.

Similarly, we need to find $\partial \tau_{r\theta } /\partial \beta_{m}$ and $\partial \tau_{r\theta } /\partial \beta_{r}$, respectively;

$$ \frac{{\partial \tau_{r\theta } }}{{\partial \beta_{m} }} = - \frac{{\left( {\gamma - c} \right)\alpha_{r} \beta_{r} \alpha_{m}^{2} }}{{\left( {\left( {\alpha_{m}^{2} - 8\beta_{m} } \right)\beta_{r} + 2\beta_{m} \alpha_{r}^{2} } \right)^{2} }} < 0 $$

(A.19)

$$ \frac{{\partial \tau_{r\theta } }}{{\partial \beta_{r} }} = \frac{{\beta_{m} \alpha_{r} \left( {\gamma - c} \right)\left( {\alpha_{m}^{2} - 8\beta_{m} } \right)}}{{\left( {\left( {2\alpha_{r}^{2} - 8\beta_{r} } \right)\beta_{m} + \beta_{r} \alpha_{m}^{2} } \right)^{2} }} $$

(A.20)

Note that, therefore, from Eq. (A.19), we have $\partial \tau_{m\theta } /\partial \beta_{r} < 0$. However, from Eq. (A.20), one can verify that $\partial \tau_{r\theta } /\partial \beta_{r} \ge 0$ if $\alpha_{m}^{2} \ge 8\beta_{m}$, or $\alpha_{m} \ge 2\sqrt {2\beta_{m} }$.

Proof of Proposition 3. The problem can be written as follows:

$$ \max E\left( {\Pi_{m} } \right) = v\left( {\begin{array}{*{20}c} {\left( {w_{H} - c} \right) \times \left( {\begin{array}{*{20}c} {\gamma - kp_{HH} + } \\ {\alpha_{m} \tau_{mH} + \alpha_{r} \tau_{rH} } \\ \end{array} } \right)} \\ { - \beta_{m} \tau_{mH}^{2} } \\ \end{array} } \right) + \left( {1 - v} \right)\left( {\begin{array}{*{20}c} {\left( {w_{L} - c} \right) \times \left( {\begin{array}{*{20}c} {\gamma - kp_{LL} + } \\ {\alpha_{m} \tau_{mL} + \alpha_{r} \tau_{rL} } \\ \end{array} } \right)} \\ { - \beta_{m} \tau_{mL}^{2} } \\ \end{array} } \right) $$

(A.21)

s.t.

$$ \left( {p_{HH} - w_{H} } \right) \times \left( {\gamma - kp_{HH} + \alpha_{m} \tau_{mH} + \alpha_{r} \tau_{rH} } \right) - \beta_{r}^{H} \tau_{rH}^{2} \ge \left( {p_{HL} - w_{L} } \right) \times \left( {\gamma - kp_{HL} + \alpha_{m} \tau_{mL} + \alpha_{r} \tau_{rH} } \right) - \beta_{r}^{H} \tau_{rH}^{2} $$

(A.22)

$$ \left( {p_{LL} - w_{L} } \right) \times \left( {\gamma - kp_{LL} + \alpha_{m} \tau_{mL} + \alpha_{r} \tau_{rL} } \right) - \beta_{r}^{L} \tau_{rL}^{2} \ge \left( {p_{LH} - w_{H} } \right) \times \left( {\gamma - kp_{LH} + \alpha_{m} \tau_{mH} + \alpha_{r} \tau_{rL} } \right) - \beta_{r}^{L} \tau_{rL}^{2} $$

(A.23)

$$ \left( {p_{HH} - w_{H} } \right) \times \left( {\gamma - kp_{HH} + \alpha_{m} \tau_{mH} + \alpha_{r} \tau_{rH} } \right) - \beta_{r}^{H} \tau_{rH}^{2} \ge 0 $$

(A.24)

$$ \left( {p_{LL} - w_{L} } \right) \times \left( {\gamma - kp_{LL} + \alpha_{m} \tau_{mL} + \alpha_{r} \tau_{rL} } \right) - \beta_{r}^{L} \tau_{rL}^{2} \ge 0 $$

(A.25)

$$ p_{HH} ,p_{LL} ,\tau_{rH} ,\tau_{rL} ,\tau_{mH} ,\tau_{mL} ,w_{H} ,w_{L} \in {\mathbb{R}}^{ + } $$

For the sake of simplicity and based on $\left( {p_{HH}^{*} } \right) = argmax\Pi_{rHH}$ and $\left( {p_{LL}^{*} } \right) = argmax\Pi_{rLL}$, we have:

$$ \left\{ {\begin{array}{*{20}c} {\frac{{\partial \Pi_{rHH} }}{{\partial p_{HH} }} = 0} \\ {\frac{{\partial \Pi_{rLL} }}{{\partial p_{LL} }} = 0} \\ \end{array} } \right., $$

(A.26)

Thus, computing the first order derivative over market price and setting it equal to zero, we derive:

$$ p_{HH} = \frac{1}{2k}\left( {\gamma + kw_{H} + \alpha_{m} \tau_{mH} + \alpha_{r} \tau_{rH} } \right) $$

(A.27)

$$ p_{LL} = \frac{1}{2k}\left( {\gamma + kw_{L} + \alpha_{m} \tau_{mL} + \alpha_{r} \tau_{rL} } \right) $$

(A.28)

Similarly, we define:

$$ \left\{ {\begin{array}{*{20}c} {\frac{{\partial \Pi_{rHL} }}{{\partial p_{HL} }} = 0} \\ {\frac{{\partial \Pi_{rLH} }}{{\partial p_{LH} }} = 0} \\ \end{array} } \right., $$

(A.29)

So, we have:

$$ p_{HL} = \frac{1}{2k}\left( {\gamma + kw_{L} + \alpha_{m} \tau_{mL} + \alpha_{r} \tau_{rH} } \right) $$

(A.30)

$$ p_{LH} = \frac{1}{2k}\left( {\gamma + kw_{H} + \alpha_{m} \tau_{mH} + \alpha_{r} \tau_{rL} } \right) $$

(A.31)

Plugging $p_{HH}$, $p_{HL}$, $p_{LL}$, $p_{LH}$ into $\Pi_{rHH}$, $\Pi_{rHL}$, $\Pi_{rLL}$, $\Pi_{rLH}$, $\Pi_{mH}$ and $\Pi_{mL}$, we derive the retailer and the manufacturer’s profit functions:

$$ \Pi_{rHH} = \left( {\frac{1}{2k}\left( {\gamma + kw_{H} + \alpha_{m} \tau_{mH} + \alpha_{r} \tau_{rH} } \right) - {\text{w}}_{H} } \right) \times \left( {\gamma - \frac{1}{2k}\left( {\gamma + kw_{H} + \alpha_{m} \tau_{mH} + \alpha_{r} \tau_{rH} } \right) + {\upalpha }_{m} \tau_{mH} + {\upalpha }_{r} \tau_{rH} } \right) - \beta_{r} \tau_{rH}^{2} $$

(A.32)

$$ \Pi_{rHL} = \left( {\frac{1}{2k}\left( {\gamma + kw_{L} + \alpha_{m} \tau_{mL} + \alpha_{r} \tau_{rH} } \right) - w_{L} } \right) \times \left( {\gamma - \frac{1}{2k}\left( {\gamma + kw_{L} + \alpha_{m} \tau_{mL} + \alpha_{r} \tau_{rH} } \right) + \alpha_{m} \tau_{mL} + \alpha_{r} \tau_{rH} } \right) - \beta_{r} \tau_{rH}^{2} $$

(A.33)

$$ \Pi_{rLL} = \left( {\frac{1}{2k}\left( {\gamma + kw_{L} + \alpha_{m} \tau_{mL} + \alpha_{r} \tau_{rL} } \right) - w_{L} } \right) \times \left( {\gamma - \frac{1}{2k}\left( {\gamma + kw_{L} + \alpha_{m} \tau_{mL} + \alpha_{r} \tau_{rL} } \right) + \alpha_{m} \tau_{mL} + \alpha_{r} \tau_{rL} } \right) - \beta_{r} \tau_{rL}^{2} $$

(A.34)

$$ \Pi_{rLH} = \left( {\frac{1}{2k}\left( {\gamma + kw_{H} + \alpha_{m} \tau_{mH} + \alpha_{r} \tau_{rL} } \right) - w_{H} } \right) \times \left( {\gamma - \frac{1}{2k}\left( {\gamma + kw_{H} + \alpha_{m} \tau_{mH} + \alpha_{r} \tau_{rL} } \right) + \alpha_{m} \tau_{mH} + \alpha_{r} \tau_{rL} } \right) - \beta_{r} \tau_{rL}^{2} $$

(A.35)

$$ \Pi_{mH} = \left( {w_{H} - c} \right) \times \left( {\gamma - \frac{1}{2k}\left( {\gamma + kw_{H} + \alpha_{m} \tau_{mH} + \alpha_{r} \tau_{rH} } \right) + \alpha_{m} \tau_{mH} + \alpha_{r} \tau_{rH} } \right) - \beta_{m} \tau_{mH}^{2} $$

(A.36)

$$ \Pi_{mL} = \left( {w_{L} - c} \right) \times \left( {\gamma - \frac{1}{2k}\left( {\gamma + kw_{L} + \alpha_{m} \tau_{mL} + \alpha_{r} \tau_{rL} } \right) + \alpha_{m} \tau_{mL} + \alpha_{r} \tau_{rL} } \right) - \beta_{m} \tau_{mL}^{2} $$

(A.37)

Let $\lambda 1$, $\lambda 2$, $\lambda 3$, $\lambda 4$ denote the Lagrange multiplier of the participation constraints and the incentive compatibility constraints and $l$ represent the Lagrange function, we have:

$$ l = v\pi_{mH} + \left( {1 - v} \right)\pi_{mL} + M_{f} \lambda_{f} $$

(A.38)

We first set all the Lagrange multipliers to zero and derive the optimal solutions from first-order partial derivatives of the first fourth equations of the KKT condition (obtain optimal solutions). Then we check the other KKT conditions to see whether the obtained solution is feasible. Thus, we have:

$$ l = v\pi_{mH} + \left( {1 - v} \right)\pi_{mL} $$

(A.39)

The KKT condition will be:

$$ \left\{ {\begin{array}{*{20}c} {\frac{\partial l}{{\partial w_{H} }} = 0 } \\ {\frac{\partial l}{{\partial w_{L} }} = 0} \\ {\frac{\partial l}{{\partial \tau_{mH} }} = 0} \\ {\frac{\partial l}{{\partial \tau_{mL} }} = 0} \\ {M_{f} \lambda_{f} = 0} \\ {\lambda_{f} \ge 0} \\ {M_{f} \ge 0} \\ {f = 1,2,3,4} \\ \end{array} } \right. $$

(A.40)

As mentioned earlier, because of the complexity of the model, closed-forms are derived where we fix some parameters to specific values. Hence, the parameters that we listed in the numerical illustration section are fixed firstly and then, we checked the KKT condition. In doing so, the values in Table

Table 8 Values obtained from KKT conditions

8 are obtained:

Since we set all the Lagrange multipliers to zero, all the constraints must be positive or zero ($M_{f} \ge 0$). However, the first constraint is negative, so the results are infeasible. For the second round, we add the first constraint to the Lagrange function. Thus we have:

$$ l = v\pi_{mH} + \left( {1 - v} \right)\pi_{mL} + M_{1} \lambda_{1} $$

Applying KKT conditions we can obtain values illustrated in Table

Table 9 Values obtained from KKT conditions

9.

The obtained results apply in KKT conditions; thus, they are both optimal and feasible. In fact, with this method, since the complexity of the proposed model makes it hard or impossible to check all the KKT conditions, we search for the feasible solution in the optimal area.

Finally, from the KKT conditions with given Lagrange multipliers ($\lambda_{1} = 0.017,\lambda_{2} = \lambda_{3} = \lambda_{4} = 0$), we can obtain the closed-form expressions:

$$ \frac{\partial l}{{\partial w_{H} }} = v\left( {\frac{kc}{2} + \frac{\gamma }{2} - kw_{H} + \frac{{\alpha_{m} \tau_{mH} }}{2} + \frac{{\alpha_{r} \tau_{rH} }}{2}} \right) - \lambda 1\left( {\frac{\gamma }{2k} - \frac{{w_{H} }}{2} + \frac{{\alpha_{m} \tau_{mH} }}{2k} + \frac{{\alpha_{r} \tau_{rH} }}{2k}} \right) = 0 $$

(A.41)

$$ \frac{\partial l}{{\partial w_{L} }} = - \left( {v - 1} \right)\left( {\frac{kc}{2} + \frac{\gamma }{2} - kw_{L} + \frac{{\alpha_{m} \tau_{mL} }}{2} + \frac{{\alpha_{r} \tau_{rL} }}{2}} \right) + \lambda 1\left( {\frac{\gamma }{2k} - \frac{{w_{L} }}{2} + \frac{{\alpha_{m} \tau_{mL} }}{2k} + \frac{{\alpha_{r} \tau_{rH} }}{2k}} \right) = 0 $$

(A.42)

$$ \frac{\partial l}{{\partial \tau_{mH} }} = - v\left( {2\beta_{m} \tau_{mH} + \alpha_{m} k\left( {c - w_{H} } \right)} \right) + \alpha_{m} \lambda 1\left( {\frac{\gamma }{2k} - \frac{{w_{H} }}{2} + \frac{{\alpha_{m} \tau_{mH} }}{2k} + \frac{{\alpha_{r} \tau_{rH} }}{2k}} \right) = 0 $$

(A.43)

$$ \frac{\partial l}{{\partial \tau_{mL} }} = \left( {v - 1} \right)\left( {2\beta_{m} \tau_{mL} + \frac{{\alpha_{m} k\left( {c - w_{L} } \right)}}{2}} \right) - \alpha_{m} \lambda 1\left( {\frac{\gamma }{2k} - \frac{{w_{L} }}{2} + \frac{{\alpha_{m} \tau_{mL} }}{2k} + \frac{{\alpha_{r} \tau_{rH} }}{2k}} \right) = 0 $$

(A.44)

By setting the first order derivatives of Lagrange function $l$ and using the abbreviation Table 3, the optimal solutions can be achieved as:

$$ w_{H}^{*} = \frac{{ - \left( {\left( {16A_{1} k^{3} - 4A_{3} k^{2} } \right)\left( {1 - 2v} \right) - 48A_{2} k^{2} \left( {1 - \frac{2}{3}v} \right) + (A_{5} - A_{4} )\left( {1 - v} \right) + \left( {8A_{6} + 16A_{8} k + 16A_{9} - 8A_{10} } \right)k\left( {1 - v} \right) - 64A_{7} k^{2} \left( {1 - \frac{3}{4}v} \right)} \right)}}{{\left( {4\beta_{m} k^{2} \left( { - \alpha_{m}^{2} + 8\beta_{m} k} \right)} \right)}} $$

(A.45)

$$ w_{L}^{*} = \frac{{\left( {\left( {16A_{1} k^{3} - 4A_{3} k^{2} } \right)\left( {1 - 2v} \right) + 16A_{2} k^{2} \left( {1 + 2v} \right) + (A_{4} - A_{5} )v - \left( {8A_{6} - 16A_{9} + 8A_{10} } \right)kv + 48A_{7} k^{2} v + 16A_{8} k^{2} \left( {1 - v} \right)} \right)}}{{\left( {4\beta_{m} k^{2} \left( { - \alpha_{m}^{2} + 8\beta_{m} k} \right)} \right)}} $$

(A.46)

$$ \tau_{mH}^{*} = \frac{{\left( {4B_{1} k\left( {ck - \gamma - 2cvk + 2\gamma v} \right) + \left( {B_{2} - B_{3} + 4B_{5} k} \right)\left( {1 - v} \right) - 8B_{4} k\left( {1 - \frac{3}{2}v} \right)} \right)}}{{\left( {4\beta_{m} k\left( { - \alpha_{m}^{2} + 8\beta_{m} k} \right)} \right)}} $$

(A.47)

$$ \tau_{mL}^{*} = \frac{{\left( {4B_{1} k\left( {\gamma - ck - 2\gamma v + 2ckv} \right) + \left( {B_{2} - B_{3} - 12B_{4} k} \right)v + 4B_{5} k\left( {1 + v} \right)} \right)}}{{\left( {4\beta_{m} k\left( { - \alpha_{m}^{2} + 8\beta_{m} k} \right)} \right)}} $$

(A.48)

Finally, by plugging the found solutions into Eq. (A.27) and Eq. (A.28), we can obtain

$$ p_{HH}^{*} = \frac{{ - \left( {\left( {8A_{1} k^{3} - 4A_{3} k^{2} } \right)\left( {1 - 2v} \right) - 40A_{2} k^{2} \left( {1 - \frac{2}{5}v} \right) + \left( {A_{5} - A_{4} + 8A_{6} k + 8A_{8} k^{2} + 14A_{9} k - 6A_{10} k} \right)\left( {1 - v} \right) - 48A_{7} k^{2} \left( {1 - \frac{1}{2}v} \right)} \right)}}{{\left( {4\beta_{m} k^{2} \left( { - \alpha_{m}^{2} + 8\beta_{m} k} \right)} \right)}} $$

(A.49)

$$ p_{LL}^{*} = \frac{{\left( {8A_{1} k^{3} \!-\! 4A_{3} k^{2} } \right)\left( {1 \!-\! 2v} \right) \!+\! 24A_{2} k^{2} \left( {1 \!+\! \frac{2}{3}v} \right) \!+\! (A_{4} \!-\! A_{5} )v \!+\! \left( { - 8A_{6} \!+\! 24A_{7} \!-\! 14A_{9} \!+\! 6A_{10} } \right)kv \!+\! 24A_{8} k^{2} \left( {1 \!-\! \frac{1}{3}v} \right)}}{{\left( {4\beta_{m} k^{2} \left( { - \alpha_{m}^{2} \!+\! 8\beta_{m} k} \right)} \right)}} $$

(A.50)

Proof of Lemma 2. Given $\left( {w_{\theta } ,\tau_{m\theta } } \right)$, the profit of the $H$-type retailer would be:

$$ \Pi_{rH} = \left( {p_{H} - w_{\theta } } \right) \times \left( {\gamma - p_{H} + \alpha_{m} \tau_{m\theta } + \alpha_{r} H} \right) - H^{2} \beta_{r} $$

(A.51)

$$ \frac{{\partial \Pi_{rH} }}{{\partial p_{H} }} = 0 \to w_{\theta } - 2p_{H} + \gamma + \alpha_{r} H + \alpha_{m} \tau_{m\theta } = 0 $$

(A.52)

$$ \to p_{H} = \frac{{w_{\theta } }}{2} + \frac{\gamma }{2} + \frac{{\left( {\alpha_{r} H} \right)}}{2} + \frac{{\left( {\alpha_{m} \tau_{m\theta } } \right)}}{2} $$

(A.53)

$$ \to \Pi_{rH} = \frac{\gamma }{2} - \frac{{w_{\theta } }}{2} + \frac{{\left( {\alpha_{r} H} \right)}}{2} + \frac{{\left( {\alpha_{m} \tau_{m\theta } } \right)}}{2}^{2} - H^{2} \beta_{r} $$

(A.54)

In a similar way, given $\left( {w_{\theta } ,\tau_{m\theta } } \right)$, the profit of the $L$-type retailer would be:

$$ \Pi_{rL} = \left( {p_{L} - w_{\theta } } \right) \times \left( {\gamma - p_{L} + \alpha_{m} \tau_{m\theta } + \alpha_{r} L} \right) - L^{2} \beta_{r} $$

(A.55)

$$ \frac{{\partial \Pi_{rL} }}{{\partial p_{L} }} = 0 \to w_{\theta } - 2p_{L} + \gamma + \alpha_{r} L + \alpha_{m} \tau_{m\theta } = 0 $$

(A.56)

$$ \to p_{L} = \frac{{w_{\theta } }}{2} + \frac{\gamma }{2} + \frac{{\left( {\alpha_{r} L} \right)}}{2} + \frac{{\left( {\alpha_{m} \tau_{m\theta } } \right)}}{2} $$

(A.57)

$$ \to \Pi_{rL} = \frac{\gamma }{2} - \frac{{w_{\theta } }}{2} + \frac{{\left( {\alpha_{r} L} \right)}}{2} + \frac{{\left( {\alpha_{m} \tau_{m\theta } } \right)}}{2}^{2} - L^{2} \beta_{r} $$

(A.58)

The retailer is rational, so he exerts high green sales effort if it leads to more profit. By comparing his profit under both $H$ and $L$ conditions, we obtain:

$$ \Pi_{rH} > \Pi_{rL} \to \frac{\gamma }{2} - \frac{{w_{\theta } }}{2} + \frac{{\left( {\alpha_{r} H} \right)}}{2} + \frac{{\left( {\alpha_{m} \tau_{m\theta } } \right)}}{2}^{2} - H^{2} \beta_{r} > \frac{\gamma }{2} - \frac{{w_{\theta } }}{2} + \frac{{\left( {\alpha_{r} L} \right)}}{2} + \frac{{\left( {\alpha_{m} \tau_{m\theta } } \right)}}{2}^{2} - L^{2} \beta_{r} $$

(A.59)

$$ \to \gamma > w_{\theta } - \alpha_{m} \tau_{m\theta } $$

(A.60)