Appendix
1.1 Proof of Proposition 1
Proof
We claim that the shipper’s expected cost function (2) is convex in q. As the first and the second derivatives of \(\pi _\theta (B^d)\) with respect to q are given by
$$\begin{aligned} \frac{\partial {\pi _\theta (B^d)}}{\partial {q}}= & {} w^d +\frac{\eta \left( q-\overline{D}^{\theta }_{L}\right) }{2}-\frac{\left( \eta \mu _{H}-w^d\right) \left( \alpha \overline{D}_{H}-\alpha ^2 q\right) }{\overline{D}_{H}-\underline{D}_{H}}. \end{aligned}$$
(A.1)
And
$$\begin{aligned} \frac{\partial {\pi _\theta (B^d)}^{2}}{\partial ^{2}{q}}= & {} \frac{\eta }{2}+\frac{\left( \eta \mu _{H}-w^d\right) \alpha ^2}{\overline{D}_{H}-\underline{D}_{H}}>0. \end{aligned}$$
(A.2)
Obviously, the shipper’s optimal commitment quantity at the low season is determined by the first order condition. That is
$$\begin{aligned} q= & {} \frac{\left( \eta \mu ^{\theta }_{L}-w^d\right) \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \overline{D}_{H}\left( \eta \mu _{H}-w^d\right) }{\frac{\eta }{2}\left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha ^2\left( \eta \mu _{H}-w^d\right) }. \end{aligned}$$
(A.3)
Moreover, considering that the shipper’s commitment quantity should be non-negative and the proportion requirement parameter \(\alpha \) is no less than 1, hence we draw the following conclusion.
When the market state is \(\theta \), and the shipper selects the contract type d, the shipper’s optimal commitment quantity at the contract price in the low demand season, denoted by \(q^{d*}_{\theta }\), is uniquely determined by
$$\begin{aligned} q^{d*}_\theta =\left\{ \begin{array}{ll} \frac{\left( \eta \mu ^{\theta }_{L}-w^d\right) \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \overline{D}_{H}\left( \eta \mu _{H}-w^d\right) }{\frac{\eta }{2}\left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha ^2\left( \eta \mu _{H}-w^d\right) }, &{} \hbox {if } \alpha \ge \max \bigg \{1,\left( 1-\frac{\underline{D}_{H}}{\overline{D}_{H}}\right) \frac{w^d-\eta \mu ^{\theta }_L}{\eta \mu _H-w^d}\bigg \} ; \\ 0, &{} \hbox {if } \alpha <\max \bigg \{1, \left( 1-\frac{\underline{D}_{H}}{\overline{D}_{H}}\right) \frac{w^d-\eta \mu ^{\theta }_L}{\eta \mu _H-w^d}\bigg \}. \end{array} \right. \end{aligned}$$
\(\square \)
1.2 Proof of Proposition 2
Proof
\((\text {i})\) As \(\overline{D}_{H}-\underline{D}_{H}>0\), we may quickly get that \(\frac{\partial {q_\theta ^d}}{\partial {\mu ^{\theta }_{L}}}>0\).
\((\text {ii})\) From Proposition 1, we may easily derive the first derivative of \(q_\theta ^d\) with respect to \(\eta \),
$$\begin{aligned}&\frac{\partial {q_\theta ^d}}{\partial {\eta }} =\frac{\left[ \mu ^{\theta }_{L}\left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \mu _{H}\overline{D}_{H}\right] \left[ \frac{\eta }{2}\left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha ^2\left( \eta \mu _{H}-w^d\right) \right] }{\left[ \frac{\eta }{2}\left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha ^2\left( \eta \mu _{H}-w^d\right) \right] ^2}\nonumber \\&\quad -\frac{\left[ \frac{1}{2}\left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha ^2 \mu _H\right] \left[ \left( \eta \mu ^{\theta }_{L}-w^d\right) \left( \overline{D}_{H} -\underline{D}_{H}\right) +\alpha \overline{D}_{H} \left( \eta \mu _{H}-w^d\right) \right] }{\left[ \frac{\eta }{2} \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha ^2\left( \eta \mu _{H}-w^d\right) \right] ^2}\nonumber \\&\quad =\frac{\frac{w^d\left( \overline{D}_{H}-\underline{D}_{H}\right) ^2}{2}+\frac{\alpha w^d\overline{D}_{H}\left( \overline{D}_{H}-\underline{D}_{H}\right) }{2} +\alpha ^2\left( \mu ^{\theta }_{L}-\mu _{H}\right) w^d\left( \overline{D}_{H} -\underline{D}_{H}\right) }{\left[ \frac{\eta }{2}\left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha ^2\left( \eta \mu _{H}-w^d\right) \right] ^2}>0. \end{aligned}$$
(A.4)
\((\text {iii})\) From Proposition 1, the first derivative of \(q_\theta ^d\) with respect to \(\alpha \) is given by
$$\begin{aligned} \frac{\partial {q_\theta ^d}}{\partial {\alpha }}= & {} \frac{\overline{D}_{H}\left( \eta \mu _{H}-w^d\right) \left[ \frac{\eta }{2}\left( \overline{D}_{H} -\underline{D}_{H}\right) +\alpha ^2\left( \eta \mu _{H}-w^d\right) \right] }{\left[ \frac{\eta }{2}\left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha ^2\left( \eta \mu _{H}-w^d\right) \right] ^2}\\&-\frac{2\alpha \left( \eta \mu _{H}-w^d\right) \left[ \left( \eta \mu _{L}-w^d\right) \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \overline{D}_{H}\left( \eta \mu _{H}-w^d\right) \right] }{\left[ \frac{\eta }{2} \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha ^2\left( \eta \mu _{H}-w^d\right) \right] ^2}\\= & {} \frac{\left( \eta \mu _{H}-w^d\right) \left[ \frac{\eta }{2}\overline{D}_{H}\left( \overline{D}_{H} -\underline{D}_{H}\right) -\alpha ^2\overline{D}_{H}\left( \eta \mu _{H}-w^d\right) -2\alpha \left( \eta \mu _{L}-w^d\right) \left( \overline{D}_{H} -\underline{D}_{H}\right) \right] }{\left[ \frac{\eta }{2}\left( \overline{D}_{H} -\underline{D}_{H}\right) +\alpha ^2\left( \eta \mu _{H}-w^d\right) \right] ^2}. \end{aligned}$$
Clearly, if \((1-\frac{\underline{D}_{H}}{\overline{D}_{H}})\frac{w^d-\eta \mu ^{\theta }_L}{\eta \mu _H-w^d}<\alpha \le (1-\frac{\underline{D}_{H}}{\overline{D}_{H}})\frac{w^d-\eta \mu ^{\theta }_L}{\eta \mu _H-w^d}+\sqrt{(1-\frac{\underline{D}_{H}}{\overline{D}_{H}})^2(\frac{w^d-\eta \mu ^{\theta }_L}{\eta \mu _H-w^d})^2+\frac{\eta (\overline{D}_{H}-\underline{D}_{H})}{2(\eta \mu _H-w^d)}}\), we get \(\frac{\partial {q_\theta ^d}}{\partial {\alpha }}\ge 0\). Otherwise, if \(\alpha > (1-\frac{\underline{D}_{H}}{\overline{D}_{H}})\frac{w^d-\eta \mu ^{\theta }_L}{\eta \mu _H-w^d}+\sqrt{(1-\frac{\underline{D}_{H}}{\overline{D}_{H}})^2(\frac{w^d-\eta \mu ^{\theta }_L}{\eta \mu _H-w^d})^2+\frac{\eta (\overline{D}_{H}-\underline{D}_{H})}{2(\eta \mu _H-w^d)}}\), we derive \(\frac{\partial {q_\theta ^d}}{\partial {\alpha }}<0\).
As \(\alpha \ge 1\), we conclude that the optimal commitment quantity \(q_\theta ^d\) is increasing with \(\alpha \) if \( \max \left\{ 1, (1{-}\frac{\underline{D}_{H}}{\overline{D}_{H}})\frac{w^d-\eta \mu ^{\theta }_L}{\eta \mu _H-w^d}\right\} {\le }\alpha {\le } (1-\frac{\underline{D}_{H}}{\overline{D}_{H}})\frac{w^d-\eta \mu ^{\theta }_L}{\eta \mu _H-w^d}+\sqrt{(1{-}\frac{\underline{D}_{H}}{\overline{D}_{H}})^2(\frac{w^d-\eta \mu ^{\theta }_L}{\eta \mu _H-w^d})^2{+}\frac{\eta (\overline{D}_{H}-\underline{D}_{H})}{2(\eta \mu _H-w^d)}}\). While the optimal commitment quantity \(q_\theta ^d\) is decreasing with \(\alpha \) if \(\alpha \le (1-\frac{\underline{D}_{H}}{\overline{D}_{H}})\frac{w^d-\eta \mu ^{\theta }_L}{\eta \mu _H-w^d}+\sqrt{(1-\frac{\underline{D}_{H}}{\overline{D}_{H}})^2(\frac{w^d-\eta \mu ^{\theta }_L}{\eta \mu _H-w^d})^2+\frac{\eta (\overline{D}_{H}-\underline{D}_{H})}{2(\eta \mu _H-w^d)}}\). \(\square \)
1.3 Proof of Theorem 1
Proof
Note that when the state type \(\theta \), where \(\theta =B\) or G, is also know to the carrier, the contract design problem for the carrier is actually a special case of problem (4). Hence, the proof of Theorem 1 follows directly from Theorem 2 by setting \(B^B=B^G=B^\theta \), that is \(w^B=w^G=w^\theta \) and \(T^B=T^G=T^\theta \).
Next, we turn to justify the assumptions that the contract price is higher than the spot market price in the low season, while it is lower than the spot price in the high season.
$$\begin{aligned} \eta \mu _H-w^\theta _S= & {} \eta \mu _H-\frac{\eta \mu _{H} \alpha \left( \overline{D}_{H}-2\alpha \mu ^{\theta }_{L}\right) }{\overline{D}_{H}-\underline{D}_{H}+\alpha \left( \overline{D}_{H} -2\alpha \mu ^{\theta }_{L}\right) }\nonumber \\= & {} \frac{\eta \mu _H\left( \overline{D}_{H}-\underline{D}_{H}\right) }{\overline{D}_{H}-\underline{D}_{H}+\alpha \left( \overline{D}_{H} -2\alpha \mu ^{\theta }_{L}\right) }>0. \end{aligned}$$
(A.5)
$$\begin{aligned} \eta \mu ^{\theta }_L-w^\theta _S= & {} \eta \mu ^{\theta }_L-\frac{\eta \mu _{H} \alpha \left( \overline{D}_{H}-2\alpha \mu ^{\theta }_{L}\right) }{\overline{D}_{H} -\underline{D}_{H}+\alpha \left( \overline{D}_{H}-2\alpha \mu ^{\theta }_{L}\right) }\nonumber \\= & {} \frac{\eta \mu ^{\theta }_L\left( \overline{D}_{H}-\underline{D}_{H}\right) -\eta \alpha \left( \mu _H-\mu ^{\theta }_{L}\right) \left( \overline{D}_{H} -2\alpha \mu ^{\theta }_{L}\right) }{\overline{D}_{H}-\underline{D}_{H} +\alpha \left( \overline{D}_{H}-2\alpha \mu ^{\theta }_{L}\right) }. \end{aligned}$$
(A.6)
Obviously, only when \(\frac{\mu ^{\theta }_{L}}{\alpha (\mu _H-\mu ^{\theta }_{L})}\le \frac{\overline{D}_{H}-2\alpha \mu ^{\theta }_{L}}{\overline{D}_{H}-\underline{D}_{H}}\), we may get \(\eta \mu ^{\theta }_L-w^\theta _S<0\).
Finally, we claim that the reservation fee \(T^{\theta }_{S}\) is decreasing with the contract price \(w^{\theta }_{S}\),
$$\begin{aligned}&\frac{\partial {T^{\theta }_{S}}}{\partial {w^{\theta }_{S}}} =\frac{\left[ \eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2} \left( \eta \mu _{H}-w^{\theta }_{S}\right) \right] \{2\overline{D}_{H}(-\alpha -1) \left[ \alpha \left( \eta \mu _{H}-w^{\theta }_{S}\right) +\left( \eta \mu ^{\theta }_{L}-w^{\theta }_{S}\right) \right] }{\left[ \eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2} \left( \eta \mu _{H}-w^{\theta }_{S}\right) \right] ^2}\\&\quad \frac{-\underline{D}_{H}\left[ -\frac{\eta }{2}\underline{D}_{H} +2\alpha ^{2}\left( \eta \mu _{H}-w^{\theta }_{S}\right) -2\left( \eta \mu ^{\theta }_{L}-w^{\theta }_{S}\right) \right] \}}{\left[ \eta \left( \overline{D}_{H} -\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^{\theta }_{S}\right) \right] ^2}\\&\quad +\frac{2\alpha ^{2}\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^{\theta }_{S}\right) +\left( \eta \mu ^{\theta }_{L}-w^{\theta }_{S}\right) \right] ^{2}-2\alpha ^{2} \underline{D}_{H}\left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^{\theta }_{S}\right) -\alpha ^{2}\left( \eta \mu _{H}-w^{\theta }_{S}\right) ^{2} +\left( \eta \mu ^{\theta }_{L}-w^{\theta }_{S}\right) ^{2}\right] }{\left[ \eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H} -w^{\theta }_{S}\right) \right] ^2}\\&\quad =\frac{2\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^{\theta }_{S}\right) +\left( \eta \mu ^{\theta }_{L}-w^{\theta }_{S}\right) \right] \left[ -(1+\alpha )\eta \left( \overline{D}_{H} -\underline{D}_{H}\right) -\alpha {^3}\left( \eta \mu _{H}-w^{\theta }_{S}\right) -2\alpha {^2}\eta \mu _{H}+\alpha ^2\left( \eta \mu ^{\theta }_{L}+w^{\theta }_{S}\right) \right] }{\left[ \eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H} -w^{\theta }_{S}\right) \right] ^2}\\&\quad -\frac{2\alpha ^2\underline{D}_{H}\left[ \alpha ^2\left( \eta \mu _{H}-w^{\theta }_{S}\right) ^2 +\left( \eta \mu _{L}^{\theta }-w^{\theta }_{S}\right) ^2 -2\left( \eta \mu _{H}-w^{\theta }_{S}\right) \left( \eta \mu _{L}^{\theta }-w^{\theta }_{S}\right) \right] }{\left[ \eta \left( \overline{D}_{H} -\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^{\theta }_{S}\right) \right] ^2}\\&\quad -\frac{\underline{D}_{H}\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) \left[ -\frac{\eta }{2}\underline{D}_{H}+2\alpha ^2\left( \eta \mu _{H}-w^{\theta }_{S}\right) -2\left( \eta \mu _{L}^{\theta }-w^{\theta }_{S}\right) \right] }{\left[ \eta \left( \overline{D}_{H} -\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^{\theta }_{S}\right) \right] ^2}\\&\quad<\frac{2\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^{\theta }_{S}\right) +\left( \eta \mu ^{\theta }_{L}-w^{\theta }_{S}\right) \right] \left[ -(1+\alpha )\eta \left( \overline{D}_{H} -\underline{D}_{H}\right) -\alpha {^3}\left( \eta \mu _{H}-w^{\theta }_{S}\right) -2\alpha {^2}\eta \mu _{H}+\alpha ^2\left( \eta \mu _{L}^{\theta }+\eta \mu _H\right) \right] }{\left[ \eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H} -w^{\theta }_{S}\right) \right] ^2}\\&\quad -\frac{2\alpha ^2\underline{D}_{H}\left( \eta \mu _H-\eta \mu _L^\theta \right) ^2}{\left[ \eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H} -w^{\theta }_{S}\right) \right] ^2}-\frac{\underline{D}_{H}\eta \left( \overline{D}_{H} -\underline{D}_{H}\right) \left[ 2\alpha ^2\left( \eta \mu _{H}-w^{\theta }_{S}\right) -2\left( \eta \mu _{L}^{\theta } -w^{\theta }_{S}\right) -\frac{\eta }{2}\underline{D}_{H}\right] }{\left[ \eta \left( \overline{D}_{H} -\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^{\theta }_{S}\right) \right] ^2}\\&\quad<\frac{2\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^{\theta }_{S}\right) +\left( \eta \mu ^{\theta }_{L}-w^{\theta }_{S}\right) \right] \left[ -(1+\alpha )\eta \left( \overline{D}_{H} -\underline{D}_{H}\right) -\alpha {^3}\left( \eta \mu _{H}-w^{\theta }_{S}\right) \right] }{\left[ \eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H} -w^{\theta }_{S}\right) \right] ^2}\\&\quad -\frac{2\alpha ^2\underline{D}_{H}\left( \eta \mu _H-\eta \mu _L^\theta \right) ^2}{\left[ \eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^{\theta }_{S}\right) \right] ^2}-\frac{\underline{D}_{H}\eta \left( \overline{D}_{H} -\underline{D}_{H}\right) \left[ 2\alpha ^2\left( \eta \mu _{H}-w^{\theta }_{S}\right) -2\left( \eta \mu _{L}^{\theta }-w^{\theta }_{S}\right) -\frac{\eta }{2} \underline{D}_{H}\right] }{\left[ \eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^{\theta }_{S}\right) \right] ^2}\\&\quad <0. \end{aligned}$$
\(\square \)
1.4 Proof of Proposition 3
Proof
\((\text {i})\) From Theorem 1, we may quickly derive that the optimal contract price under symmetric information is increasing with \(\eta \).
\((\text {ii})\) From Theorem 1, we may also get
$$\begin{aligned} \frac{\partial {w^{\theta }_{S}}}{\partial \mu ^{\theta }_L}= & {} \frac{-2\eta \mu _H\alpha ^{2} \left[ \overline{D}_{H}-\underline{D}_{H}+\alpha \left( \overline{D}_{H} -2\alpha \mu ^{\theta }_L\right) \right] +2\alpha ^2\alpha \eta \mu _H\left( \overline{D}_{H} -2\alpha \mu ^{\theta }_L\right) }{\left[ \overline{D}_{H}-\underline{D}_{H} +\alpha \left( \overline{D}_{H}-2\alpha \mu ^{\theta }_L\right) \right] ^{2}}\\= & {} -\frac{\alpha ^{2}\eta \mu _H\left( \overline{D}_{H}-\underline{D}_{H}\right) }{\left[ \overline{D}_{H}-\underline{D}_{H}+\alpha \left( \overline{D}_{H} -2\alpha \mu ^{\theta }_L\right) \right] ^{2}}<0. \end{aligned}$$
\((\text {iii})\) By Theorem 1, we obtain
$$\begin{aligned} \frac{\partial {w^{\theta }_{S}}}{\partial \alpha }= & {} \frac{\eta \mu _H\left( \overline{D}_{H}-4\alpha \mu ^{\theta }_L\right) \left[ \overline{D}_{H}-\underline{D}_{H}+\alpha \left( \overline{D}_{H} -2\alpha \mu ^{\theta }_L\right) \right] -\alpha \eta \mu _H\left( \overline{D}_{H} -2\alpha \mu ^{\theta }_L\right) \left( \overline{D}_{H}-4\alpha \mu ^{\theta }_L\right) }{\left[ \overline{D}_{H}-\underline{D}_{H}+\alpha \left( \overline{D}_{H} -2\alpha \mu ^{\theta }_L\right) \right] ^{2}}\\= & {} \frac{\eta \mu _H\left( \overline{D}_{H}-4\alpha \mu ^{\theta }_L\right) \left( \overline{D}_{H}-\underline{D}_{H}\right) }{\left[ \overline{D}_{H} -\underline{D}_{H}+\alpha \left( \overline{D}_{H}-2\alpha \mu ^{\theta }_L\right) \right] ^{2}} \end{aligned}$$
Therefore, we conclude that if \(1<\alpha \le \frac{\overline{D}_{H}}{4\mu ^{\theta }_L}\), the optimal contract price is increasing with \(\alpha \); otherwise if \(\frac{\overline{D}_{H}}{4\mu ^{\theta }_L}<\alpha \le \frac{\overline{D}_{H}}{2\mu ^{\theta }_L}\), the optimal contract price is decreasing with \(\alpha \). \(\square \)
1.5 Proof of Theorem 2
Proof
After substituting the shipper’s expected cost function (3) into the carrier’s optimization problem (4), the carrier’s contract design problem can be simplified as follows:
$$\begin{aligned}&\Pi =\max _{B^{B}(\cdot ),B^{G}(\cdot )}\left\{ \rho \left[ T^B+w^{B}q^{*B}_{B} +(w^{B}-P_H)\min \left( \alpha q^{*B}_{B},D_H\right) +P_H K\right] \right. \nonumber \\&\quad \left. +(1-\rho )\left[ T^G+w^{G}q^{*G}_{G}+(w^{G}-P_H) \min (\alpha q^{*G}_{G},D_H)+P_H K\right] \right\} \nonumber \\&\quad s.t. \quad \left\{ \begin{array}{ll} \frac{\overline{D}_{H}\left[ \alpha (\eta \mu _{H}-w^B)+\left( \eta \mu ^{B}_{L}-w^B\right) \right] ^{2}-\underline{D}_{H}\left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^B\right) -\alpha ^{2}\left( \eta \mu _{H}-w^B\right) ^{2}+\left( \eta \mu ^{B}_{L}-w^B\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^B\right) }-T^B\ge 0, \\ \frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^G\right) +\left( \eta \mu ^{G}_{L}-w^G\right) \right] ^{2}-\underline{D}_{H} \left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^G\right) -\alpha ^{2}\left( \eta \mu _{H}-w^G\right) ^{2}+\left( \eta \mu ^{G}_{L}-w^G\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^G\right) }-T^G\ge 0, \\ \frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^B\right) +\left( \eta \mu ^{B}_{L}-w^B\right) \right] ^{2}-\underline{D}_{H} \left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^B\right) -\alpha ^{2}\left( \eta \mu _{H}-w^B\right) ^{2}+\left( \eta \mu ^{B}_{L}-w^B\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^B\right) }-T^B\ge \\ \frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^G\right) +\left( \eta \mu ^{B}_{L}-w^G\right) \right] ^{2}-\underline{D}_{H} \left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^G\right) -\alpha ^{2}\left( \eta \mu _{H}-w^G\right) ^{2}+\left( \eta \mu ^{B}_{L}-w^G\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^G\right) }-T^G,\\ \frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^G\right) +\left( \eta \mu ^{G}_{L}-w^G\right) \right] ^{2}-\underline{D}_{H} \left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^G\right) -\alpha ^{2}\left( \eta \mu _{H}-w^G\right) ^{2}+\left( \eta \mu ^{G}_{L}-w^G\right) ^{2}\right] }{\eta \left( \overline{D}_{H} -\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^G\right) }-T^G\ge \\ \frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^B\right) +\left( \eta \mu ^{G}_{L}-w^B\right) \right] ^{2}-\underline{D}_{H} \left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^B\right) -\alpha ^{2}\left( \eta \mu _{H}-w^B\right) ^{2}+\left( \eta \mu ^{G}_{L}-w^B\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^B\right) }-T^B. \end{array}\right. \nonumber \\ \end{aligned}$$
(A.7)
We claim that the first and last constraints are binding.
First of all, we claim that if the first and fourth constraints hold, the second constraint will hold automatically. To see this, notice that from the first constraint, we have
$$\begin{aligned} -T^{B}\ge -\frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^B\right) +\left( \eta \mu ^{B}_{L}-w^B\right) \right] ^{2}-\underline{D}_{H} \left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^B\right) -\alpha ^{2}\left( \eta \mu _{H}-w^B\right) ^{2}+\left( \eta \mu ^{B}_{L}-w^B\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^B\right) }.\nonumber \\ \end{aligned}$$
(A.8)
Since \(\mu ^{G}_{L}>\mu ^{B}_{L}\) and \(\alpha \overline{D}_{H}(\eta \mu _{H}-w^B)+(\overline{D}_{H} -\underline{D}_{H})(\eta \mu ^B_{L}-w^B)>0\), substituting (A.8) into the last constraint yields
$$\begin{aligned}&\frac{\overline{D}_{H}\left[ \alpha (\eta \mu _{H}-w^G)+\left( \eta \mu ^{G}_{L}-w^G\right) \right] ^{2} -\underline{D}_{H}\left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^G\right) -\alpha ^{2}\left( \eta \mu _{H}-w^G\right) ^{2}+\left( \eta \mu ^{G}_{L}-w^G\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^G\right) }-T^G\\&\quad \ge \frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^B\right) +\left( \eta \mu ^{G}_{L}-w^B\right) \right] ^{2} -\underline{D}_{H}\left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^B\right) -\alpha ^{2}\left( \eta \mu _{H}-w^B\right) ^{2}+\left( \eta \mu ^{G}_{L}-w^B\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^B\right) }-T^B\\&\quad \ge \frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^B\right) +\left( \eta \mu ^{G}_{L} -w^B\right) \right] ^{2}-\underline{D}_{H}\left[ \frac{\eta }{2}\underline{D}_{H} \left( \eta \mu _{H}-w^B\right) -\alpha ^{2}\left( \eta \mu _{H}-w^B\right) ^{2} +\left( \eta \mu ^{G}_{L}-w^B\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^B\right) }\\&\quad -\frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^B\right) +\left( \eta \mu ^{B}_{L} -w^B\right) \right] ^{2}-\underline{D}_{H}\left[ \frac{\eta }{2} \underline{D}_{H}\left( \eta \mu _{H}-w^B\right) -\alpha ^{2}\left( \eta \mu _{H}-w^B\right) ^{2}+\left( \eta \mu ^{B}_{L}-w^B\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^B\right) }\\&\quad =\frac{\left( \eta \mu ^{G}_{L}-\eta \mu ^{B}_{L}\right) \left[ 2\alpha \overline{D}_{H} \left( \eta \mu _{H}-w^B\right) +\left( \overline{D}_{H}-\underline{D}_{H}\right) \left( \eta \mu ^{G}_{L}+\eta \mu ^{B}_{L}-2w^{B}\right) \right] }{\eta \left( \overline{D}_{H} -\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^B\right) }\\&\quad \ge 0. \end{aligned}$$
Second, we show that in the optimal solution the first constraint must be binding. Recall that the second constraint is not binding and the shipper’s cost under the high market state is strictly negative. If the first constraint is not binding, we can increase \(T^{B}\) and \(T^{G}\) by the same amount \(\varepsilon \) such that all the constraints will still hold and the objective function will increase, which violates the optimality condition. Thus, the first constraint must be binding.
Third, we show that in the optimal solution the last constraint must be binding. As before, if in the optimal solution, the last constraint is not binding, we can increase \(T_{G}\) by \(\varepsilon \) such that the last constraint is binding. In this case, all other constraints will still hold and the objective function will increase, which violates the optimality condition. So the last constraint must be binding.
As the first and last constraints are binding, we obtain
$$\begin{aligned} \left\{ \begin{array}{ll} T^{B}=\frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^B\right) +\left( \eta \mu ^{B}_{L}-w^B\right) \right] ^{2}-\underline{D}_{H} \left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^B\right) -\alpha ^{2}\left( \eta \mu _{H}-w^B\right) ^{2}+\left( \eta \mu ^{B}_{L}-w^B\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^B\right) }\\ T^G=T^{B}+\frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^G\right) +\left( \eta \mu ^{G}_{L}-w^G\right) \right] ^{2}-\underline{D}_{H} \left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^G\right) -\alpha ^{2}\left( \eta \mu _{H}-w^G\right) ^{2}+\left( \eta \mu ^{G}_{L}-w^G\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^G\right) }\\ ~~~~~~~-\frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^B\right) +\left( \eta \mu ^{G}_{L}-w^B\right) \right] ^{2}-\underline{D}_{H} \left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^B\right) -\alpha ^{2}\left( \eta \mu _{H}-w^B\right) ^{2}+\left( \eta \mu ^{G}_{L}-w^B\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^B\right) }. \end{array}\right. \nonumber \\ \end{aligned}$$
(A.9)
After substituting \(T^B\) and \(T^G\) in (A.9) into the third constraint, we have
$$\begin{aligned}&\frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^B\right) +\left( \eta \mu ^{B}_{L}-w^B\right) \right] ^{2}-\underline{D}_{H} \left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^B\right) -\alpha ^{2}\left( \eta \mu _{H}-w^B\right) ^{2}+\left( \eta \mu ^{B}_{L}-w^B\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^B\right) }-T^B\\&\quad -\frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^G\right) +\left( \eta \mu ^{B}_{L}-w^G\right) \right] ^{2}-\underline{D}_{H}\left[ \frac{\eta }{2} \underline{D}_{H}\left( \eta \mu _{H}-w^G\right) -\alpha ^{2}\left( \eta \mu _{H} -w^G\right) ^{2}+\left( \eta \mu ^{B}_{L}-w^G\right) ^{2}\right] }{\eta \left( \overline{D}_{H} -\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^G\right) }+T^G\\&\qquad =\frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^B\right) +\left( \eta \mu ^{B}_{L}-w^B\right) \right] ^{2}-\underline{D}_{H}\left[ \frac{\eta }{2} \underline{D}_{H}\left( \eta \mu _{H}-w^B\right) -\alpha ^{2} \left( \eta \mu _{H}-w^B\right) ^{2}+\left( \eta \mu ^{B}_{L}-w^B\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^B\right) }\\&\qquad -\frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^G\right) +\left( \eta \mu ^{B}_{L}-w^G\right) \right] ^{2}-\underline{D}_{H} \left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^G\right) -\alpha ^{2}\left( \eta \mu _{H}-w^G\right) ^{2}+\left( \eta \mu ^{B}_{L}-w^G\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^G\right) }\\&\qquad +\frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^G\right) +\left( \eta \mu ^{G}_{L}-w^G\right) \right] ^{2}-\underline{D}_{H} \left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^G\right) -\alpha ^{2}(\eta \mu _{H}-w^G)^{2}+\left( \eta \mu ^{G}_{L} -w^G\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^G\right) }\\&\qquad -\frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^B\right) +\left( \eta \mu ^{G}_{L}-w^B\right) \right] ^{2}-\underline{D}_{H} \left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^B\right) -\alpha ^{2}\left( \eta \mu _{H}-w^B\right) ^{2} +\left( \eta \mu ^{G}_{L}-w^B\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^B\right) }\\&\qquad =\frac{\left( \eta \mu ^{G}_{L}-\eta \mu ^{B}_{L}\right) \left[ 2\alpha \overline{D}_{H}\left( \eta \mu _{H}-w^G\right) +\left( \overline{D}_{H} -\underline{D}_{H}\right) \left( \eta \mu ^{G}_{L}+\eta \mu ^{B}_{L}-2w^{G}\right) \right] }{\eta (\overline{D}_{H}-\underline{D}_{H})+2\alpha ^{2}\left( \eta \mu _{H}-w^G\right) }\\&\quad -\frac{\left( \eta \mu ^{G}_{L}-\eta \mu ^{B}_{L}\right) \left[ 2\alpha \overline{D}_{H}\left( \eta \mu _{H}-w^B\right) +\left( \overline{D}_{H}-\underline{D}_{H}\right) \left( \eta \mu ^{G}_{L} +\eta \mu ^{B}_{L}-2w^{B}\right) \right] }{\eta \left( \overline{D}_{H} -\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^B\right) } \end{aligned}$$
Define \(A=\frac{[2\alpha \overline{D}_{H}(\eta \mu _{H}-w)+(\overline{D}_{H}-\underline{D}_{H}) (\eta \mu ^{G}_{L}+\eta \mu ^{B}_{L}-2w)]}{\eta (\overline{D}_{H}-\underline{D}_{H})+2\alpha ^{2}(\eta \mu _{H}-w)}\), and we have
$$\begin{aligned} \frac{\partial {A}}{\partial {w}}= & {} \frac{\left[ \eta \left( \overline{D}_{H} -\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w\right) \right] \left[ -2\alpha \overline{D}_{H}-2\left( \overline{D}_{H} -\underline{D}_{H}\right) \right] }{\left[ \eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w\right) \right] ^2}\nonumber \\&+\frac{2\alpha ^2\left[ 2\alpha \overline{D}_{H}\left( \eta \mu _{H}-w\right) +\left( \overline{D}_{H}-\underline{D}_{H}\right) \left( \eta \mu ^{G}_{L} +\eta \mu ^{B}_{L}-2w\right) \right] }{\left[ \eta \left( \overline{D}_{H} -\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w\right) \right] ^2}\nonumber \\= & {} \frac{\left( \overline{D}_{H}-\underline{D}_{H}\right) \left[ -2\alpha \eta \overline{D}_{H}-2\eta \left( \overline{D}_{H} -\underline{D}_{H}\right) -4\alpha ^{2}\eta \mu _H+2\alpha ^2\eta \mu ^{G}_{L} +2\alpha ^2\eta \mu ^{B}_{L}\right] }{\left[ \eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w\right) \right] ^2}\nonumber \\< & {} 0. \end{aligned}$$
(A.10)
In this situation, we claim that as long as \(w^G<w^B\), the third constraint of Problem (A.7) holds automatically.
All of the above proves that the first and fourth constraints of Problem (A.7) are binding.
Let us now analyze the optimal contract price and the reservation fee. After relaxing the second and third constraints, and substituting \(T^B\) and \(T^G\) in (A.9) into the carrier’s objective function (A.7), we can then convert the original constrained optimization problem into an unconstrained one.
$$\begin{aligned} \Pi= & {} \rho \left[ T^B+\frac{\left( \eta \mu _{H}-w^{B}\right) \underline{D}^{2}_{H}}{2\left( \overline{D}_{H} -\underline{D}_{H}\right) }-\frac{\eta q^{B*2}_{B}}{4}-\frac{\left( \eta \mu _{H}-w^{B}\right) \alpha ^{2} q^{B*2}_{B}}{2\left( \overline{D}_{H}-\underline{D}_{H}\right) }-\frac{\eta q^{B*2}_{B}}{4}+\eta \mu ^{B}_{L}q^{B*}_{B}\right] \nonumber \\&+(1-\rho )\left[ T^G+\frac{\left( \eta \mu _{H}-w^{G}\right) \underline{D}^{2}_{H}}{2\left( \overline{D}_{H} -\underline{D}_{H}\right) }-\frac{\eta q^{G*2}_{G}}{4}-\frac{\left( \eta \mu _{H}-w^{G}\right) \alpha ^{2} q^{G*2}_{G}}{2\left( \overline{D}_{H}-\underline{D}_{H}\right) }-\frac{\eta q^{G*2}_{G}}{4}+\eta \mu ^{G}_{L}q^{G*}_{G}\right] \nonumber \\&=\rho \left( \eta \mu ^{B}_{L}q^{B*}_{B}-\frac{\eta q^{B*2}_{B}}{4}\right) +(1-\rho )\left( \eta \mu ^{G}_{L}q^{G*}_{G}-\frac{\eta q^{G*2}_{G}}{4}\right) \nonumber \\&-(1-\rho )\frac{\left( \eta \mu ^{G}_{L}-\eta \mu ^{B}_{L}\right) \left[ 2\alpha \overline{D}_{H}(\eta \mu _{H}-w^B)+\left( \overline{D}_{H}-\underline{D}_{H}\right) \left( \eta \mu ^{G}_{L} +\eta \mu ^{B}_{L}-2w^{B}\right) \right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^B\right) }. \end{aligned}$$
(A.11)
Note that the carrier’s objective function is separable in \(w^{B}\) and \(w^{G}\).
Then we get the first and the second derivatives of \(\Pi \) with respect to \(w^G\).
$$\begin{aligned} \frac{\partial {\Pi }}{\partial {w^{G}}}=\left( \eta \mu ^{G}_{L}-\frac{\eta q^{G*}_{G}}{2}\right) \frac{\partial {q^{G*}_{G}}}{\partial {w^{G}}} \end{aligned}$$
(A.12)
And
$$\begin{aligned} \frac{\partial ^{2}{\Pi }}{\partial {w^{G}}^{2}} =-\frac{\eta }{2}\left( \frac{\partial {q^{G*}_{G}}}{\partial {w^{G}}}\right) ^2+ \left( \eta \mu ^{G}_{L}-\frac{\eta q^{G*}_{G}}{2}\right) \frac{\partial ^2{q^{G*}_{G}}}{\partial {w^{G}}^{2}}<0. \end{aligned}$$
(A.13)
The optimal price in the high market state is uniquely determined by the first order condition:
$$\begin{aligned} w^{G*}\left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \eta \mu _{H}-w^{G*}\right) \left( 2\alpha \mu ^{G}_{L}-\overline{D}_{H}\right) =0. \end{aligned}$$
(A.14)
That is
$$\begin{aligned} w^{G*}=\frac{\eta \mu _{H}\alpha \left( \overline{D}_{H}-2\alpha \mu ^{G}_{L}\right) }{\overline{D}_{H}-\underline{D}_{H}+\alpha \left( \overline{D}_{H}-2\alpha \mu ^{G}_{L}\right) }. \end{aligned}$$
(A.15)
For the contract price in the low demand market, we have:
$$\begin{aligned}&\frac{\partial {\Pi }}{\partial {w^{B}}}=\rho \left( \eta \mu ^{B}_{L} -\frac{\eta q^{B*}_{B}}{2}\right) \frac{\partial {q^{B*}_{B}}}{\partial {w^{B}}}+(1-\rho )\left( \eta \mu ^{G}_{L}-\eta \mu ^{B}_{L}\right) \nonumber \\&\quad \cdot \frac{\left( \overline{D}_{H}-\underline{D}_{H}\right) \left[ 2\alpha \eta \overline{D}_{H}+2\eta \left( \overline{D}_{H} -\underline{D}_{H}\right) +4\alpha ^{2}\eta \mu _H-2\alpha ^2\eta \mu ^{B}_{L} -2\alpha ^2\eta \mu ^{G}_{L}\right] }{\left[ \eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^B\right) \right] ^2}\nonumber \\ \end{aligned}$$
(A.16)
Obviously, we may derive that \(\frac{\partial ^{2}{\Pi }}{\partial {w^{B}}^{2}}<0\), thus the optimal contract price in the low demand state is uniquely determined by the first order condition:
$$\begin{aligned} w^{B*}=\frac{\rho \alpha \eta \mu _H \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] +(1-\rho )\left( \mu ^G_L-\mu ^B_L\right) \left[ \eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^2\eta \mu _H\right] C}{\rho \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] +(1-\rho )2\alpha ^2(\mu ^G_L-\mu ^B_L)C}. \end{aligned}$$
where \(C=[2\alpha \overline{D}_{H}+2(\overline{D}_{H}-\underline{D}_{H})+4\alpha ^{2}\mu _H-2\alpha ^2\mu ^{G}_{L}-2\alpha ^2\mu ^{B}_{L}]\).
Finally, we need to verify \(w^{B*}>w^{G*}\). Since \(\mu ^G_L>\mu ^B_L\)
$$\begin{aligned} w^{B*}> & {} \frac{\rho \alpha \eta \mu _H\left( \overline{D}_{H} -2\alpha \mu ^B_L\right) C}{\rho \left[ \overline{D}_{H}-\underline{D}_{H} +\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \right] C}\nonumber \\= & {} \frac{\alpha \eta \mu _H\left( \overline{D}_{H}-2\alpha \mu ^B_L\right) }{\left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H} -2\alpha \mu ^B_L\right) }\nonumber \\\ge & {} \frac{\alpha \eta \mu _H\left( \overline{D}_{H}-2\alpha \mu ^G_L\right) }{\left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H}-2\alpha \mu ^G_L\right) }=w^{G*}. \end{aligned}$$
(A.17)
From all above, we conclude that with the payment structure \(B^{\theta }=\{(T^{\theta },w^{\theta })|\theta =B, G\}\) and under asymmetric information, the equilibrium contract parameters for the carrier are characterized as follows:
\((\text {i})\) In the bad demand state:
$$\begin{aligned} \left\{ \begin{array}{ll} w^{B*}=\frac{\rho \alpha \eta \mu _H\left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] +(1-\rho )\left( \mu ^G_L-\mu ^B_L\right) \left[ \eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^2\eta \mu _H\right] C}{\rho \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H} -2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] +(1-\rho )2 \alpha ^2\left( \mu ^G_L-\mu ^B_L\right) C},\\ T^{B*}=\frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^{B*}\right) +\left( \eta \mu ^{B}_{L}-w^{B*}\right) \right] ^{2}-\underline{D}_{H}\left[ \frac{\eta }{2} \underline{D}_{H}\left( \eta \mu _{H}-w^{B*}\right) -\alpha ^{2}\left( \eta \mu _{H}-w^{B*}\right) ^{2} +\left( \eta \mu ^{B}_{L}-w^{B*}\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^{B*}\right) }.\\ \end{array} \right. \end{aligned}$$
\((\text {ii})\) In the good demand state:
$$\begin{aligned} \left\{ \begin{array}{ll} w^{G*}=\frac{\eta \mu _{H}\alpha \left( \overline{D}_{H}-2\alpha \mu ^{G}_{L}\right) }{\overline{D}_{H}-\underline{D}_{H}+\alpha \left( \overline{D}_{H} -2\alpha \mu ^{G}_{L}\right) },\\ T^{G*}=T^{B*}+ \frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^{G*}\right) +\left( \eta \mu ^{G}_{L}-w^{G*}\right) \right] ^{2}-\underline{D}_{H} \left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^{G*}\right) -\alpha ^{2}\left( \eta \mu _{H}-w^{G*}\right) ^{2}+\left( \eta \mu ^{G}_{L} -w^{G*}\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2} \left( \eta \mu _{H}-w^{G*}\right) }-\\ ~~~~~~~\frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^{B*}\right) +\left( \eta \mu ^{G}_{L}-w^{B*}\right) \right] ^{2}-\underline{D}_{H} \left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^{B*}\right) -\alpha ^{2} \left( \eta \mu _{H}-w^{B*}\right) ^{2}+\left( \eta \mu ^{G}_{L}-w^{B*}\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^{B*}\right) }.\\ \end{array} \right. \end{aligned}$$
where \(C=[2\alpha \overline{D}_{H}+2(\overline{D}_{H}-\underline{D}_{H})+4\alpha ^{2}\mu _H-2\alpha ^2\mu ^{G}_{L}-2\alpha ^2\mu ^{B}_{L}]\). \(\square \)
1.6 Proof of Proposition 4
Proof
\((\text {i})\) In a good market state, from Theorem 1 and Theorem 2 we get that \(w^{G}_S=w^{G*}\). Under symmetric information, we have
$$\begin{aligned} T^{G}_{S}=\frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^{G}_{S}\right) +\left( \eta \mu ^{B}_{L}-w^{G}_{S}\right) \right] ^{2}-\underline{D}_{H} \left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^{G}_{S}\right) -\alpha ^{2}\left( \eta \mu _{H}-w^{G}_{S}\right) ^{2}+\left( \eta \mu ^{B}_{L}-w^{G}_{S}\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^{G}_{S}\right) } \end{aligned}$$
However, in the presence of asymmetric information, from the proof of Theorem 2 we get that the last constraint of Problem (A.7) is untight. That is
$$\begin{aligned}&\frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^{G*}\right) +\left( \eta \mu ^{G}_{L}-w^{G*}\right) \right] ^{2}-\underline{D}_{H} \left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^{G*}\right) -\alpha ^{2}\left( \eta \mu _{H}-w^{G*}\right) ^{2}+\left( \eta \mu ^{G}_{L}-w^{G*}\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^{G*}\right) }\\&\quad -T^{G*}>-T^{B*}\\&\quad +\frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^{B*}\right) +\left( \eta \mu ^{G}_{L}-w^{B*}\right) \right] ^{2} -\underline{D}_{H}\left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^{B*}\right) -\alpha ^{2}\left( \eta \mu _{H}-w^{B*}\right) ^{2}+\left( \eta \mu ^{G}_{L}-w^{B*}\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^{B*}\right) }. \end{aligned}$$
Since the right hand side of the above inequality is positive, we get
$$\begin{aligned}&\frac{\overline{D}_{H}\left[ \alpha \left( \eta \mu _{H}-w^{G*}\right) +\left( \eta \mu ^{G}_{L}-w^{G*}\right) \right] ^{2}-\underline{D}_{H} \left[ \frac{\eta }{2}\underline{D}_{H}\left( \eta \mu _{H}-w^{G*}\right) -\alpha ^{2}\left( \eta \mu _{H}-w^{G*}\right) ^{2}+\left( \eta \mu ^{G}_{L}-w^{G*}\right) ^{2}\right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^{G*}\right) }\\&>T^{G*}. \end{aligned}$$
Therefore, as \(w^{G}_{S}=w^{G*}\) we conclude that \(T^{G*}<T^{G}_{S}\).
\((\text {ii})\) We first claim that the contract price in the bad state \(w^{B*}\) is decreasing with \(\rho \), since
$$\begin{aligned}&\frac{\partial {w^{B*}}}{\partial {\rho }}=\frac{\rho \left[ C+2\alpha ^{2} \left( \mu ^G_L-\mu ^B_L\right) \right] ^2\left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \right] \alpha \eta \mu _H\left( \overline{D}_{H} -2\alpha \mu ^B_L\right) }{\left\{ \rho \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H} -2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] +(1-\rho )2 \alpha ^2\left( \mu ^G_L-\mu ^B_L\right) C\right\} ^2}\\&\quad -\frac{\rho C\left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] \left( \mu ^G_L-\mu ^B_L\right) \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H} -2\alpha \mu ^B_L\right) \right] \left[ \eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^2\eta \mu _H\right] }{\left\{ \rho \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] +(1-\rho )2\alpha ^2\left( \mu ^G_L-\mu ^B_L\right) C\right\} ^2}\\&\quad -\frac{\rho \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] ^2\left[ \left( \overline{D}_{H} -\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \right] \alpha \eta \mu _H\left( \overline{D}_{H}-2\alpha \mu ^B_L\right) }{\left\{ \rho \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2} \left( \mu ^G_L-\mu ^B_L\right) \right] +(1-\rho )2\alpha ^2\left( \mu ^G_L-\mu ^B_L\right) C\right\} ^2}\\&\quad +\frac{\rho C\left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] \left( \mu ^G_L-\mu ^B_L\right) 2\alpha ^2\alpha \eta \mu _H\left( \overline{D}_{H} -2\alpha \mu ^B_L\right) }{\left\{ \rho \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2} \left( \mu ^G_L-\mu ^B_L\right) \right] +(1-\rho )2\alpha ^2\left( \mu ^G_L-\mu ^B_L\right) C\right\} ^2}\\&\quad +\frac{(1-\rho )C\left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] 2\alpha ^2\left( \mu ^G_L-\mu ^B_L\right) \alpha \eta \mu _H\left( \overline{D}_{H} -2\alpha \mu ^B_L\right) }{\left\{ \rho \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2} \left( \mu ^G_L-\mu ^B_L\right) \right] +(1-\rho )2\alpha ^2\left( \mu ^G_L-\mu ^B_L\right) C\right\} ^2}\\&\quad -\frac{(1-\rho )C^22\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) ^2 \left[ \eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^2\eta \mu _H\right] }{\left\{ \rho \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] +(1-\rho )2\alpha ^2\left( \mu ^G_L-\mu ^B_L\right) C\right\} ^2}\\&\quad -\frac{(1-\rho )C\left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] \left( \mu ^G_L-\mu ^B_L\right) \left[ \eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^2\eta \mu _H\right] \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \right] }{\left\{ \rho \left[ \left( \overline{D}_{H} -\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] +(1-\rho )2\alpha ^2\left( \mu ^G_L-\mu ^B_L\right) C\right\} ^2}\\&\quad +\frac{(1-\rho )C^22\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) ^2 \left[ \eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^2\eta \mu _H\right] }{\left\{ \rho \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H} -2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] +(1-\rho )2\alpha ^2\left( \mu ^G_L-\mu ^B_L\right) C\right\} ^2}\\&\quad =-\frac{C\left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] \left( \mu ^G_L-\mu ^B_L\right) \left[ \eta \left( \overline{D}_{H}-\underline{D}_{H}\right) ^2+2\alpha ^2\eta \mu _H \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \eta \left( \overline{D}_{H} -\underline{D}_{H}\right) \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \right] }{\left\{ \rho \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H} -2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] +(1-\rho )2\alpha ^2 \left( \mu ^G_L-\mu ^B_L\right) C\right\} ^2}\\&\quad <0. \end{aligned}$$
Note that when \(\rho =1\), we have \(w^{B*}=w^{BS}\). Therefore, we conclude that \(w^{B*}\ge w^{B}_{S}\).
Moreover, from the proof of Theorem 1, we get \(\frac{\partial {T^{\theta }_{S}}}{\partial {w^{\theta }_{S}}}<0\). As \(w^{B*}\ge w^{B}_{S} \), we obtain that \(T^{B*}\ge T^{B}_{S} \) . \(\square \)
1.7 Proof of Proposition 5
Proof
\((\text {i})\) In the presence of symmetric information, where \(\rho =0\) or \(\rho =1\), from Theorem 1 we get that at optimality, the IR constraint of the carrier’s optimization problem is tight. From the perspective of the shipper, we have
$$\begin{aligned}&\pi ^{\theta }_S\left( w^{\theta }_S, T^{\theta }_S\right) - EP_L ED_L - EP_H ED_H=T^{\theta }_S\\&\quad +\frac{\left( \eta \mu _{H}-w^{\theta }_{S}\right) \underline{D}^{2}_{H}}{2\left( \overline{D}_{H}-\underline{D}_{H}\right) }-\frac{\eta q^{\theta *2}}{4}-\frac{\left( \eta \mu _{H}-w^{\theta }_{S}\right) \alpha ^2 q^{\theta *2}}{2\left( \overline{D}_{H}-\underline{D}_{H}\right) }-\eta \mu _{H}^{2}-\eta \mu ^{\theta 2}_{L}\\&\quad =0. \end{aligned}$$
From the perspective of the carrier, we get
$$\begin{aligned} \Pi ^{\theta }_S\left( w^{\theta }_S, T^{\theta }_S\right) -E P_H K= & {} T^{\theta }_S+\frac{\left( \eta \mu _{H}-w^{\theta }_{S}\right) \underline{D}^{2}_{H}}{2\left( \overline{D}_{H}-\underline{D}_{H}\right) }\\&-\frac{\eta q^{\theta *2}}{4}-\frac{\left( \eta \mu _{H}-w^{\theta }_{S}\right) \alpha ^2 q^{\theta *2}}{2\left( \overline{D}_{H}-\underline{D}_{H}\right) }-\frac{\eta q^{\theta *2}}{4}+\eta \mu _Lq^{\theta *}\\= & {} -\eta \mu ^{\theta 2}_{L}+2\eta \mu ^{\theta 2}_{L}\\= & {} \eta \mu ^{\theta 2}_{L}>0. \end{aligned}$$
Thus, in the presence of symmetric information, the optimal contract always creates a win–win situation for both the shipper and the carrier.
\((\text {ii})\) In the presence of asymmetric information, where \(\rho \in (0,1)\), from Theorem 2 we get that the first constraint of Problem (A.7) is tight. Thus, when the market state is bad, from the perspective of the shipper, we have
$$\begin{aligned} \pi ^{B*}-EP_L ED_L- EP_H ED_H=0. \end{aligned}$$
(A.18)
When the market state is good, we get
$$\begin{aligned}&\pi ^{G*}-EP_L ED_L- EP_H ED_H\nonumber \\&\quad =-\frac{\left( \eta \mu ^{G}_{L}-\eta \mu ^{B}_{L}\right) \left[ 2\alpha \overline{D}_{H}\left( \eta \mu _{H}-w^{B*}\right) +\left( \overline{D}_{H} -\underline{D}_{H}\right) \left( \eta \mu ^{G}_{L}+\eta \mu ^{B}_{L} -2w^{B*}\right) \right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^{B*}\right) }.\nonumber \\ \end{aligned}$$
(A.19)
To create the win–win situation for the shipper, we need the equation (A.19) to be non-positive. This is equivalent to testify
$$\begin{aligned} \frac{\left( \eta \mu ^{G}_{L}-\eta \mu ^{B}_{L}\right) \left[ 2\alpha \overline{D}_{H}\left( \eta \mu _{H}-w^{B*}\right) +\left( \overline{D}_{H}-\underline{D}_{H}\right) \left( \eta \mu ^{G}_{L} +\eta \mu ^{B}_{L}-2w^{B*}\right) \right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^{B*}\right) }\ge 0.\nonumber \\ \end{aligned}$$
(A.20)
Since
$$\begin{aligned}&\eta \mu _{H}-w^{B*}\nonumber \\&\quad =\frac{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) \left\{ \rho \mu _H\left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] -(1-\rho ) \left( \mu ^G_L-\mu ^B_L\right) C\right\} }{\rho \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] +(1-\rho )2\alpha ^2\left( \mu ^G_L-\mu ^B_L\right) C} \nonumber \\\end{aligned}$$
(A.21)
$$\begin{aligned}&\eta \mu ^{B}_{L}-w^{B*}\nonumber \\&\quad =\frac{\rho \left[ C+2\alpha ^{2} \left( \mu ^G_L-\mu ^B_L\right) \right] \left[ \eta \mu ^B_L\left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \left( \eta \mu ^B_L-\eta \mu _H\right) \right] }{\rho \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2} \left( \mu ^G_L-\mu ^B_L\right) \right] +(1-\rho )2\alpha ^2 \left( \mu ^G_L-\mu ^B_L\right) C}\nonumber \\&\qquad -\frac{(1-\rho )C\left( \mu ^G_L-\mu ^B_L\right) \left[ \eta \left( \overline{D}_{H} -\underline{D}_{H}\right) -2\alpha ^2\left( \eta \mu ^B_L-\eta \mu _H\right) \right] }{\rho \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H} -2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] +(1-\rho )2\alpha ^2\left( \mu ^G_L-\mu ^B_L\right) C}.\nonumber \\ \end{aligned}$$
(A.22)
Also
$$\begin{aligned}&\eta \mu ^{G}_{L}-w^{B*}\nonumber \\&\quad =\frac{\rho \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] \left[ \eta \mu ^G_L\left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H} -2\alpha \mu ^B_L\right) \left( \eta \mu ^G_L-\eta \mu _H\right) \right] }{\rho \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H} -2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2} \left( \mu ^G_L-\mu ^B_L\right) \right] +(1-\rho )2\alpha ^2 \left( \mu ^G_L-\mu ^B_L\right) C}\nonumber \\&\quad \quad -\frac{(1-\rho )C\left( \mu ^G_L-\mu ^B_L\right) \left[ \eta \left( \overline{D}_{H}-\underline{D}_{H}\right) -2\alpha ^2\left( \eta \mu ^G_L-\eta \mu _H\right) \right] }{\rho \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H} -2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2} \left( \mu ^G_L-\mu ^B_L\right) \right] +(1-\rho )2\alpha ^2\left( \mu ^G_L-\mu ^B_L\right) C}.\nonumber \\ \end{aligned}$$
(A.23)
Then
$$\begin{aligned}&\eta \mu ^{B}_{L}+\eta \mu ^{G}_{L}-2w^{B*}\nonumber \\&\quad =\frac{\rho \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] \left[ \eta \left( \mu ^G_L+\mu ^B_L\right) \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \left( \eta \mu ^B_L+\eta \mu ^G_L-2\eta \mu _H\right) \right] }{\rho \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H} -2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2} \left( \mu ^G_L-\mu ^B_L\right) \right] +\left( 1-\rho \right) 2\alpha ^2 \left( \mu ^G_L-\mu ^B_L\right) C}\nonumber \\&\qquad -\frac{\left( 1-\rho \right) C\left( \mu ^G_L-\mu ^B_L\right) \left[ 2\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) -2\alpha ^2\left( \eta \mu ^G_L+\eta \mu ^B_L-2\eta \mu _H\right) \right] }{\rho \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H} -2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2} \left( \mu ^G_L-\mu ^B_L\right) \right] +\left( 1-\rho \right) 2\alpha ^2\left( \mu ^G_L-\mu ^B_L\right) C}.\nonumber \\ \end{aligned}$$
(A.24)
After substituting (A.21) and (A.24) into the equation (A.20), we get that the optimal contract bundles create the win–win situation for the shipper if and only if the following condition holds
$$\begin{aligned}&\frac{2\alpha \overline{D}_{H}\left[ \rho \mu _H\left( C+2\alpha ^{2} \left( \mu ^G_L-\mu ^B_L\right) \right) -\left( 1-\rho \right) \left( \mu ^G_L-\mu ^B_L\right) C\right] }{\rho \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] + 2\alpha ^{2}\rho \mu _H\left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] }\\&\quad -\frac{\left( 1-\rho \right) C\left( \mu ^G_L-\mu ^B_L\right) \left[ 2\left( \overline{D}_{H} -\underline{D}_{H}\right) -2\alpha ^2\left( \mu ^G_L+\mu ^B_L-2\mu _H\right) \right] }{\rho \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] + 2\alpha ^{2}\rho \mu _H\left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] }\\&\quad +\frac{\rho \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] \left[ \left( \mu ^G_L+\mu ^B_L\right) \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \left( \mu ^G_L+\mu ^B_L-2\mu _H\right) \right] }{\rho \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H} -2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] + 2\alpha ^{2}\rho \mu _H\left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] }\ge 0. \end{aligned}$$
Obviously, the above inequality is equivalent to
$$\begin{aligned}&\rho \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] \left[ \left( \mu ^B_L+\mu ^G_L\right) \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha \overline{D}_{H}\mu _H\right. \nonumber \\&\quad \left. +\,\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \left( \mu ^B_L +\mu ^G_L-2\mu _H\right) \right] -\left( 1-\rho \right) \left( \mu ^G_L-\mu ^B_L\right) C^2\ge 0.\nonumber \\ \end{aligned}$$
(A.25)
From the perspective of the carrier, we need to verify that
$$\begin{aligned}&\Pi ^*-E P_H K=\rho \left( \eta \mu ^{B}_{L}q^{B*}_{B} -\frac{\eta q^{B*2}_{B}}{4}\right) +\left( 1-\rho \right) \{\eta \left( \mu ^{G}_{L}\right) ^2\nonumber \\&\quad -\frac{\left( \eta \mu ^{G}_{L}-\eta \mu ^{B}_{L}\right) \left[ 2\alpha \overline{D}_{H} \left( \eta \mu _{H}-w^{B*}\right) +\left( \overline{D}_{H}-\underline{D}_{H}\right) \left( \eta \mu ^{G}_{L}+\eta \mu ^{B}_{L}-2w^{B*}\right) \right] }{\eta \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha ^{2}\left( \eta \mu _{H}-w^{B*}\right) }\}\nonumber \\&\quad \ge 0. \end{aligned}$$
(A.26)
Since
$$\begin{aligned} q^{B*}_B= & {} 2\mu ^B_{L}-\frac{2\left( 1-\rho \right) \left( \mu ^{G}_{L}-\mu ^{B}_{L}\right) \left[ 2\alpha \overline{D}_{H}+2\left( \overline{D}_{H}-\underline{D}_{H}\right) +4\alpha ^{2}\mu _H-2\alpha ^2\mu ^{G}_{L}-2\alpha ^2\mu ^{B}_{L}\right] }{\rho \left[ 2\alpha \overline{D}_{H}+2\left( \overline{D}_{H}-\underline{D}_{H}\right) +4\alpha ^{2}\mu _H-4\alpha ^2\mu ^{B}_{L}\right] }\nonumber \\\le & {} 2\mu ^B_{L}. \end{aligned}$$
(A.27)
After substituting (A.21) and (A.24) into the equation (A.26), we get that the optimal contract bundles create the win–win situation for the carrier if the following condition holds
$$\begin{aligned}&\frac{\left( \mu ^{G}_{L}\right) ^2\left\{ \rho \left[ \left( \overline{D}_{H} -\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] + 2\alpha ^{2}\rho \mu _H\left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] \right\} }{\mu ^G_L-\mu ^B_L}\nonumber \\&\quad \ge \rho \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] \left[ \left( \mu ^B_L+\mu ^G_L\right) \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha \overline{D}_{H} \mu _H\right. \nonumber \\&\qquad \left. +\,\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \left( \mu ^B_L+\mu ^G_L-2\mu _H\right) \right] -\left( 1-\rho \right) \left( \mu ^G_L-\mu ^B_L\right) C^2. \end{aligned}$$
(A.28)
Combining (A.25) and (A.28), we get that the win–win situation is achieved under the optimal contract bundle for both the carrier and the shipper if the following condition holds
$$\begin{aligned}&\left( \mu ^{G}_{L}\right) ^2\left\{ \rho \left[ \left( \overline{D}_{H} -\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] \right. \nonumber \\&\left. \qquad + 2\alpha ^{2}\rho \mu _H\left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] \right\} \nonumber \\&\quad \ge \rho \left( \mu ^G_L-\mu ^B_L\right) \left[ C+2\alpha ^{2} \left( \mu ^G_L-\mu ^B_L\right) \right] \left[ \left( \mu ^B_L+\mu ^G_L\right) \left( \overline{D}_{H}-\underline{D}_{H}\right) \right. \nonumber \\&\qquad \left. +2\alpha \overline{D}_{H}\mu _H+\alpha \left( \overline{D}_{H}- 2\alpha \mu ^B_L\right) \left( \mu ^B_L+\mu ^G_L-2\mu _H\right) \right] \nonumber \\&\quad \ge \left( 1-\rho \right) \left( \mu ^G_L-\mu ^B_L\right) ^2C^2 \end{aligned}$$
(A.29)
Subsequently, we claim that the following inequality always holds.
$$\begin{aligned}&\left( \mu ^{G}_{L}\right) ^2\left\{ \rho \left[ \left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \right] \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] \right. \nonumber \\&\qquad \left. + 2\alpha ^{2}\rho \mu _H\left[ C+2\alpha ^{2}\left( \mu ^G_L -\mu ^B_L\right) \right] \right\} \\&\quad \ge \rho \left( \mu ^G_L-\mu ^B_L\right) \left[ C+2\alpha ^{2} \left( \mu ^G_L-\mu ^B_L\right) \right] \left[ \left( \mu ^B_L+\mu ^G_L\right) \left( \overline{D}_{H} -\underline{D}_{H}\right) \right. \\&\qquad \left. +2\alpha \overline{D}_{H}\mu _H+\alpha \left( \overline{D}_{H} -2\alpha \mu ^B_L\right) \left( \mu ^B_L+\mu ^G_L-2\mu _H\right) \right] . \end{aligned}$$
Since \(\rho [C+2\alpha ^{2}(\mu ^G_L-\mu ^B_L)]>0\), and \(\mu _H>\mu ^G_L>\mu ^B_L\), it is easy to show that
$$\begin{aligned}&\left[ \left( \mu ^{G}_{L}\right) ^2\left( \overline{D}_{H}-\underline{D}_{H}\right) +\alpha \left( \mu ^{G}_{L}\right) ^2\left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \right] + 2\alpha ^{2}\left( \mu ^{G}_{L}\right) ^2\mu _H\\&\quad >\left[ \left( \mu ^G_L-\mu ^B_L\right) \left( \mu ^B_L+\mu ^G_L\right) \left( \overline{D}_{H} -\underline{D}_{H}\right) +2\alpha \left( \mu ^G_L-\mu ^B_L\right) \overline{D}_{H}\mu _H\right. \\&\quad \left. +\alpha \left( \mu ^G_L-\mu ^B_L\right) \left( \overline{D}_{H}-2\alpha \mu ^B_L\right) \left( \mu ^B_L+\mu ^G_L-2\mu _H\right) \right] . \end{aligned}$$
In this scenario, we conclude that the inequality (A.29) is equivalent to
$$\begin{aligned}&\rho \left[ C+2\alpha ^{2}\left( \mu ^G_L-\mu ^B_L\right) \right] \left[ \left( \mu ^B_L+\mu ^G_L\right) \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha \overline{D}_{H}\mu _H\right. \nonumber \\&\qquad \left. +\alpha \left( \overline{D}_{H} -2\alpha \mu ^B_L\right) \left( \mu ^B_L+\mu ^G_L-2\mu _H\right) \right] \nonumber \\&\quad \ge \left( 1-\rho \right) \left( \mu ^G_L-\mu ^B_L\right) C^2. \end{aligned}$$
(A.30)
Therefore, we conclude that if the following condition holds
$$\begin{aligned} \frac{\rho }{1-\rho } \ge \frac{\left( \mu ^G_L-\mu ^B_L\right) \left[ 2\alpha \overline{D}_{H} +2\left( \overline{D}_{H}-\underline{D}_{H}\right) +4\alpha ^{2}\mu _H -2\alpha ^2\mu ^{G}_{L}-2\alpha ^2\mu ^{B}_{L}\right] ^2}{\left[ 2\alpha \overline{D}_{H}+2\left( \overline{D}_{H} -\underline{D}_{H}\right) +4\alpha ^{2}\left( \mu _H-\mu ^{B}_{L}\right) \right] \left[ \left( \mu ^B_L+\mu ^G_L\right) \left( \overline{D}_{H}-\underline{D}_{H}\right) +2\alpha \overline{D}_{H}\mu _H+\alpha \left( \overline{D}_{H} -2\alpha \mu ^B_L\right) \left( \mu ^B_L+\mu ^G_L-2\mu _H\right) \right] }. \end{aligned}$$
the contract creates a win–win situation for both the carrier and the shipper under asymmetric information. \(\square \)
