Risk hedging via option contracts in a random yield supply chain

Abstract

This article investigates the role of option contracts in a random yield supply chain in the presence of a spot market. Considering a single-period supplier-manufacturer system where the supplier with random yield produces key components for the manufacturer and the manufacturer assembles/processes the components into end products to meet a deterministic market demand, we develop game models to derive the manufacturer’s optimal ordering policy and the supplier’s optimal production policy under two contract mechanisms (with and without option contracts). Our results suggest that option contracts can coordinate the manufacturer’s order quantity as well as the supplier’s production quantity, and eventually achieve optimal supply chain performance, i.e. the random yield supply chain can be completely coordinated with option contracts in our setting. However, our study also reveals that the supplier and the manufacturer are not always better off with option contracts than without. Therefore, the conditions on which Pareto-improvement is achieved are provided in this paper. Finally, by adopting numerical examples, we draw additional managerial insights into managing random yield supply chains in the presence of spot market.

Appendix

Proof of Lemma 1

Equation (2) shows that $\frac{d\Pi ^{c}\left( {Q^{c}} \right) }{dQ^{c}}=\left( {\alpha {\mathop {p}\limits ^{=}}_{s}\mu -c} \right) +\left( {1-\alpha } \right) {\mathop {p}\limits ^{=}}_{s}\mathop {\int }\nolimits _0^{D/Q^{c}} t\varphi \left( t \right) dt$ and $\frac{d^{2}\Pi ^{c}\left( {Q^{c}} \right) }{d\left( {Q^{c}} \right) ^{2}}={-}\left( {1-\alpha } \right) {\mathop {p}\limits ^{=}}_{s}\frac{D}{\left( {Q^{c}} \right) ^{3}}\varphi \left( {\frac{D}{Q^{c}}} \right) <0$. Therefore, $\Pi ^{c}\left( {Q^{c}} \right)$ is concave in$Q^{c}$, and a unique production quantity $Q^{c*}$ which maximizes $\Pi ^{c}\left( {Q^{c}} \right)$ exists. Let $d\Pi ^{c}\left( {Q^{c}} \right) /dQ^{c}=0$, we can derive Eq. (3). $\square$

Proof of Lemma 2

Equation (4) shows that $\frac{d\Pi _{0}^{s} \left( {Q_0 } \right) }{dQ_0 }=\left( {\alpha {\mathop {p}\limits ^{=}}_{s}\mu -c} \right) +\left( {w-\alpha {\mathop {p}\limits ^{=}}_{s}} \right) \mathop {\int }\nolimits _0^{q_0 /Q_0 } t\varphi \left( t \right) dt$ and $\frac{d^{2}\Pi _{0}^{s} \left( {Q_0 } \right) }{d\left( {Q_0 } \right) ^{2}}=-\left( {w-\alpha {\mathop {p}\limits ^{=}}_{s}} \right) \frac{\left( {q_0 } \right) ^{2}}{\left( {Q_0 } \right) ^{3}}\varphi \left( {\frac{q_0 }{Q_0 }} \right) <0$. Therefore, $\Pi _{0}^{s} \left( {Q_0}\right)$ is concave in $Q_0$, and a unique production quantity $Q_0^{*} \left( {q_0} \right)$ which maximizes $\Pi _{0}^{s} \left( {Q_0 } \right)$ exists. Let $d\Pi _{0}^{s} \left( {Q_0 } \right) /dQ_0 =0$, we can derive Eq. (5). $\square$

Proof of Lemma 3

From Eq. (8), we can get $\frac{d\Pi _{01}^m \left( {q_0} \right) }{dq_0 }=\mathop {\int }\nolimits _0^{1/\vartheta } \left( {{\mathop {p}\limits ^{=}}_{s}-w} \right) \left( {t\vartheta -1} \right) \varphi \left( t \right) dt+{\mathop {p}\limits ^{=}}_{s}-w=\left( {{\mathop {p}\limits ^{=}}_{s}-w} \right) \left[ {\vartheta \times \frac{c-s\mu }{w-s}+1-\varPhi \left( {\frac{1}{\vartheta }} \right) } \right] >0$, and $\frac{d^{2}\Pi _{01}^m \left( {q_0 } \right) }{d\left( {q_0 } \right) ^{2}}=0$. The two equations imply that $\Pi _{01}^m \left( {q_0 } \right)$ linearly increases in $q_0$, so the optimal order quantity of the manufacturer is $q_0^*=D$ when $q_0 \le D$. From Eq. (9), we can get $\frac{d\Pi _{02}^m \left( {q_0 } \right) }{dq_0 }=\mathop {\int }\nolimits _0^{D/\vartheta q_0 } {\mathop {p}\limits ^{=}}_{s}t\vartheta \varphi \left( t \right) dt-\mathop {\int }\nolimits _0^{1/\vartheta } w\left( {t\vartheta -1} \right) \varphi \left( t \right) dt-w$, and $\frac{d^{2}\Pi _{02}^m \left( {q_0 } \right) }{d\left( {q_0 } \right) ^{2}}=-\frac{D^{2}}{\vartheta \left( {q_0 } \right) ^{3}}\varphi \left( {\frac{D}{\vartheta q_0 }} \right) <0$. Therefore, $\Pi _{02}^m \left( {q_0 } \right)$ is concave in $q_0$, and a unique production quantity $q_{0}^*$ which maximizes $\Pi _{01}^m \left( {q_0} \right)$ exists when $q_0 >D$. Let $d\Pi _{01}^m \left( {q_0 } \right) /dq_0 =0$, we can derive Eq. (10). In addition, Eq. (10) shows that $\frac{d\Pi _{02}^m \left( {q_0 } \right) }{dq_0 }|_{q_0 =D} =\mathop {\int }\nolimits _0^{1/\vartheta } {\mathop {p}\limits ^{=}}_{s}t\vartheta \varphi \left( t \right) dt-\mathop {\int }\nolimits _0^{1/\vartheta } w\left( {t\vartheta -1} \right) \varphi \left( t \right) dt-w$, from the analyses above, we can see that if only $d\Pi _{02}^m \left( {q_0 } \right) /dq_0 |_{q_0 =D} >0$, that is, $w<w_0 (w_0$ is determined by $\mathop {\int }\nolimits _0^{1/\vartheta } {\mathop {p}\limits ^{=}}_{s}t\vartheta \varphi \left( t \right) dt-\mathop {\int }\nolimits _0^{1/\vartheta } w_0 \left( {t\vartheta -1} \right) \varphi \left( t \right) dt-w_0 =0)$, the optimal order quantity of the manufacturer $q_{0}^*$ should be calculated by the Eq. (10), otherwise, $q_0^*=D$. $\square$

Proof of Proposition 1

(1) When $w\ge w_0$, the manufacturer’s optimal order quantity is $q_0^*=D$, thus the supplier’s optimal production quantity $Q_0^{*}$ can be solved by $\mathop {\int }\nolimits _0^{D/Q_0^{*}} t\varphi \left( t \right) dt=\frac{c-\alpha {\mathop {p}\limits ^{=}}_{s}\mu }{w-\alpha {\mathop {p}\limits ^{=}}_{s}}$. Because $w-\alpha {\mathop {p}\limits ^{=}}_{s}<{\mathop {p}\limits ^{=}}_{s}-\alpha {\mathop {p}\limits ^{=}}_{s}$, so $\mathop {\int }\nolimits _0^{D/Q_0^{*} } t\varphi \left( t \right) dt=\frac{c-\alpha {\mathop {p}\limits ^{=}}_{s}\mu }{w-\alpha {\mathop {p}\limits ^{=}}_{s}}>\frac{c-\alpha {\mathop {p}\limits ^{=}}_{s}\mu }{\left( {1-\alpha } \right) {\mathop {p}\limits ^{=}}_{s}}=\mathop {\int }\nolimits _0^{D/Q^{c*}} t\varphi \left( t \right) dt$. Since $\mathop {\int }\nolimits _0^x t\varphi \left( t \right) dt$ is increase in the x, so we can get $Q_0^{*} <Q^{c{*}}$ from $\mathop {\int }\nolimits _0^{D/Q_0^{*} } t\varphi \left( t \right) dt>\mathop {\int }\nolimits _0^{D/Q^{c*}} t\varphi \left( t \right) dt$. (2) When$w<w_0$, from Eqs. (10) and (3), we have $\mathop {\int }\nolimits _0^{D/Q_0^{*} } t\varphi \left( t \right) dt=\frac{w+\mathop {\int }\nolimits _0^{1/\vartheta } w\left( {t\vartheta -1} \right) \varphi \left( t \right) dt}{{\mathop {p}\limits ^{=}}_{s}\vartheta }=\frac{w\left( {c-\alpha {\mathop {p}\limits ^{=}}_{s}\mu } \right) }{{\mathop {p}\limits ^{=}}_{s}\left( {w-\alpha {\mathop {p}\limits ^{=}}_{s}} \right) }+\frac{w\left( {1-\varPhi \left( {\frac{1}{\vartheta }} \right) } \right) }{\vartheta {\mathop {p}\limits ^{=}}_{s}}>\frac{w\left( {c-\alpha {\mathop {p}\limits ^{=}}_{s}\mu } \right) }{{\mathop {p}\limits ^{=}}_{s}\left( {w-\alpha {\mathop {p}\limits ^{=}}_{s}} \right) }>\frac{c-\alpha {\mathop {p}\limits ^{=}}_{s}\mu }{\left( {1-\alpha } \right) {\mathop {p}\limits ^{=}}_{s}}=\mathop {\int }\nolimits _0^{D/Q^{c*}} t\varphi \left( t \right) dt$. It also implies that $Q_0^*<Q^{c*}$. In summarize, we have $Q_0^{*} <Q^{c{*}}$. $\square$

Proof of Lemma 4

Equation (12) shows that $\frac{d\Pi _1^s \left( {Q_1 } \right) }{dQ_1 }=\left( {\alpha {\mathop {p}\limits ^{=}}_{s}\mu -c} \right) +\left( {1-\alpha } \right) {\mathop {p}\limits ^{=}}_{s}\quad \mathop {\int }\nolimits _0^{D/Q_1 } t\varphi \left( t \right) dt$, and $\frac{d\Pi _1^s \left( {Q_1 } \right) }{d\left( {Q_1 } \right) ^{2}}=-\left( {1-\alpha } \right) {\mathop {p}\limits ^{=}}_{s}\frac{D}{\left( {Q_1 } \right) ^{3}}\varphi \left( {\frac{D}{Q_1 }} \right) <0$. Therefore, $\Pi _1^s \left( {Q_1 } \right)$ is concave in $Q_1$, and a unique production quantity $Q_1^*$ which maximizes $\Pi _1^s \left( {Q_1 } \right)$ exists. Let $d\Pi _1^s \left( {Q_1 } \right) /dQ_1 =0$, we can derive Eq. (13). $\square$

Proof of Proposition 2

Lemma 3 reveals that the optimal total order quantity of the manufacturer is no less than the demand D in the case of without option contracts, but with option contracts, the optimal total order quantity is D, so we can see that the optimal total order quantity of the manufacturer is less with option contracts than without. Comparing Eq. (13) with the Eq. (3), we have $Q_1^*=Q^{c{*}}$, and according to $Q_0^{*} <Q^{c{*}}$, it follows that $Q_1^*>Q_0^*$.

$\square$

Proof of Proposition 3

From Eqs. (13) and (3), we have $Q_1^*=Q^{c{*}}$. $\square$

Proof of Proposition 4

According to the Lemma 3 and Eq. (14), we firstly compare the expected profit of the manufacturer in the cases of with option contracts and without, which is discussed in four different cases below.

Case 1: When $w_{0}>w\ge \bar{V} (e)e+o+\underline{p}_{se}$, we can get

$$\begin{aligned} \Pi _{0}^{m}\left( q_{0}^{*} \right) =\mathop {\int }\nolimits _{0}^{D/ {\vartheta q_{0}^{*}}} {{\mathop {p}\limits ^{=}}_{s}\left( t\vartheta q_{0}^{*}-D \right) } \varphi \left( t \right) dt-\mathop {\int }\nolimits _0^{1/ \vartheta } w \left( t\vartheta q_{0}^{*}-q_{0}^{*} \right) \varphi \left( t \right) dt+rD-wq_{0}^{*}, \end{aligned}$$

and $\Pi _{1}^{m}\left( q_{1} \right) =\left( r-e\bar{V} (e)-o-\underline{p}_{se} \right) D+(-w+e\bar{V} (e)+o+\underline{p}_{se})q_{1}$.

From $\Pi _{1}^{m}\left( q_{1}^{m1} \right) {=\Pi }_{0}^{m}\left( q_{0}^{*} \right)$, we have $q_{1}^{m1}=\frac{-{\mathop {p}\limits ^{=}}_{s}\varPhi \left( \frac{D}{\vartheta q_{0}^{*}} \right) +e\bar{V} (e)+o+\underline{p}_{se}}{-w+e\bar{V} (e)+o+\underline{p}_{se}}D$.

• (1) If $\left| -w+e\bar{V} (e)+o+\underline{p}_{se} \right| \le \left| -{\mathop {p}\limits ^{=}}_{s}\varPhi \left( \frac{D}{\vartheta q_{0}^{*}} \right) +e\bar{V} (e)+o+\underline{p}_{se} \right|$, that is, $e\bar{V} (e)+o+\underline{p}_{se}\le w\le {\mathop {p}\limits ^{=}}_{s}\varPhi \left( \frac{D}{\vartheta q_{0}^{*}} \right)$, we can see $q_{1}^{m1}\ge D$. Because $\Pi _{1}^{m}\left( q_{1} \right)$ decreases in $q_{1}$, therefore, if $w\ge \bar{V} (e)e+o+\underline{p}_{se}$ and $w\le \min \left( w_{0},{\mathop {p}\limits ^{=}}_{s}\varPhi \left( \frac{D}{\vartheta q_{0}^{*}} \right) \right)$, then $\Pi _{1}^{m}\left( q_{1} \right) \ge \Pi _{0}^{m}\left( q_{0}^{*} \right)$.

• (2) If $\left| -w+e\bar{V} (e)+o+\underline{p}_{se} \right| >\left| -{\mathop {p}\limits ^{=}}_{s}\varPhi \left( \frac{D}{\vartheta q_{0}^{*}} \right) +e\bar{V} (e)+o+\underline{p}_{se} \right|$, that is, $\bar{V} (e)+o+\underline{p}_{se}\le {\mathop {p}\limits ^{=}}_{s}\varPhi \left( \frac{D}{\vartheta q_{0}^{*}} \right) \le w$, we can see $q_{1}^{m1}<D$. Let $\frac{-{\mathop {p}\limits ^{=}}_{s}\varPhi \left( \frac{D}{\vartheta q_{0}^{*}} \right) +e\bar{V} (e)+o+\underline{p}_{se}}{-w+e\bar{V} (e)+o+\underline{p}_{se}}=\propto _{1}$, we get $q_{1}^{m1}=\propto _{1}D$. Because $\Pi _{1}^{m}\left( q_{1} \right)$ decreases in $q_{1}$, therefore, if $w\ge \bar{V} (e)e+o+\underline{p}_{se}$ and ${\mathop {p}\limits ^{=}}_{s}\varPhi \left( \frac{D}{\vartheta q_{0}^{*}} \right) \le w<w_{0}$, when $q_{1}\le \propto _{1}D,$ then $\Pi _{1}^{m}\left( q_{1} \right) \ge \Pi _{0}^{m}\left( q_{0}^{*} \right)$,    otherwise $\Pi _{1}^{m}\left( q_{1} \right) {<\Pi }_{0}^{m}\left( q_{0}^{*} \right)$.

Case 2: When $e+o>w\ge \bar{V} (e)e+o+\underline{p}_{se}$ and $w_{0}\le w$, we can get

$$\begin{aligned} \Pi _{0}^{m}\left( q_{0}^{*} \right) =\left[ \mathop {\int }\nolimits _0^{1/\vartheta } {\left( {\mathop {p}\limits ^{=}}_{s}-w \right) \left( t\vartheta -1 \right) } \varphi \left( t \right) dt+r-w \right] D, \end{aligned}$$

and $\Pi _{1}^{m}\left( q_{1} \right) =\left( r-e\bar{V} (e)-o-\underline{p}_{se} \right) D+(-w+e\bar{V} (e)+o+\underline{p}_{se})q_{1}$.

From ${\Pi _{1}^{m}\left( q_{1}^{m2} \right) =\Pi }_{0}^{m}\left( q_{0}^{*} \right)$, we have ${ q}_{1}^{m2}=D\left[ 1-\frac{\mathop {\int }\nolimits _0^{1/\vartheta } {\left( {\mathop {p}\limits ^{=}}_{s}-w \right) \left( t\vartheta -1 \right) } \varphi \left( t \right) dt}{w-e\bar{V} (e)-o-\underline{p}_{se}} \right]$.

Due to ${\mathop {p}\limits ^{=}}_{s}-w>0$ and $\mathop {\int }\nolimits _0^{1/\vartheta } \left( t\vartheta -1 \right) \varphi \left( t \right) dt\le 0$, we can see ${q}_{1}^{m2}\ge D$. Because $\Pi _{1}^{m}\left( q_{1} \right)$ still decreases in $q_{1}$, therefore, if $w\ge \bar{V} (e)e+o+\underline{p}_{se}$ and $w_{0}\le w$,    then $\Pi _{1}^{m}\left( q_{1} \right) \ge \Pi _{0}^{m}\left( q_{0}^{*} \right)$.

Case 3: When $w<\bar{V} (e)e+o+\underline{p}_{se}$ and $w_{0}>w$, we can get

$$\begin{aligned} \Pi _{0}^{m}\left( q_{0}^{*} \right) =\mathop {\int }\nolimits _0^{D/{\vartheta q_{0}^{*}}} {{\mathop {p}\limits ^{=}}_{s}\left( t\vartheta q_{0}^{*}-D \right) } \varphi \left( t \right) dt-\mathop {\int }\nolimits _0^{1/\vartheta } w \left( t\vartheta q_{0}^{*}-q_{0}^{*} \right) \varphi \left( t \right) dt+rD-wq_{0}^{*}, \end{aligned}$$

and $\Pi _{1}^{m}\left( q_{1} \right) =\left( r-e\bar{V} (e)-o-\underline{p}_{se} \right) D+(-w+e\bar{V} (e)+o+\underline{p}_{se})q_{1}$ .

From $\Pi _{1}^{m}\left( { q}_{1}^{m3} \right) =\Pi _{0}^{m}\left( q_{0}^{*} \right)$, we have ${ q}_{1}^{m3}=\frac{-{\mathop {p}\limits ^{=}}_{s}\varPhi \left( \frac{D}{\vartheta q_{0}^{*}} \right) +e\bar{V} (e)+o+\underline{p}_{se}}{-w+e\bar{V} (e)+o+\underline{p}_{se}}D=\propto _{1}D$

• (1) If $0\le -{\mathop {p}\limits ^{=}}_{s}\varPhi \left( \frac{D}{\vartheta q_{0}^{*}} \right) +e\bar{V} (e)+o+\underline{p}_{se}\le -w+e\bar{V} (e)+o+\underline{p}_{se}$, that is, $w\le {\mathop {p}\limits ^{=}}_{s}\varPhi \left( \frac{D}{\vartheta q_{0}^{*}} \right)$, we can see ${q}_{1}^{m3}\le D$. Because $\Pi _{1}^{m}\left( q_{1} \right)$ increases in $q_{1}$, therefore, if $w<\bar{V} (e)e+o+\underline{p}_{se}$ and $w\le \min \left( w_{0},{\mathop {p}\limits ^{=}}_{s}\varPhi \left( \frac{D}{\vartheta q_{0}^{*}} \right) \right)$, when $q_{1}\ge \propto _{1}D$, then $\Pi _{1}^{m}\left( q_{1} \right) \ge \Pi _{0}^{m}\left( q_{0}^{*} \right)$,    otherwise $\Pi _{1}^{m}\left( q_{1} \right) < \Pi _{0}^{m}\left( q_{0}^{*} \right)$.

• (2) If $-{\mathop {p}\limits ^{=}}_{s}\varPhi \left( \frac{D}{\vartheta q_{0}^{*}} \right) +e\bar{V} (e)+o+\underline{p}_{se}\ge -w+e\bar{V} (e)+o+\underline{p}_{se}\ge 0$, that is, ${\mathop {p}\limits ^{=}}_{s}\varPhi \left( \frac{D}{\vartheta q_{0}^{*}} \right) \le w<\bar{V} (e)e+o+\underline{p}_{se}$, we can see $q_{1}^{m3}>D$. Because$\Pi _{1}^{m}\left( q_{1} \right)$increases in $q_{1}$, therefore, if $w<\bar{V} (e)e+o+\underline{p}_{se}$ and ${\mathop {p}\limits ^{=}}_{s}\varPhi \left( \frac{D}{\vartheta q_{0}^{*}} \right) \le w<w_{0}$, then $\Pi _{1}^{m}\left( q_{1} \right) < \Pi _{0}^{m}\left( q_{0}^{*} \right)$.

Case 4: When $w_{0}\le w<\bar{V} (e)e+o+\underline{p}_{se}$, we can get

$$\begin{aligned} \Pi _{0}^{m}\left( q_{0}^{*} \right) =\left[ \mathop {\int }\nolimits _0^{1/\vartheta } {\left( {\mathop {p}\limits ^{=}}_{s}-w \right) \left( t\vartheta -1 \right) } \varphi \left( t \right) dt+r-w \right] D, \end{aligned}$$

and $\Pi _{1}^{m}\left( q_{1} \right) =\left( r-e\bar{V} (e)-o-\underline{p}_{se} \right) D+(-w+e\bar{V} (e)+o+\underline{p}_{se})q_{1}$.

From $\Pi _{1}^{m}\left( { q}_{1}^{m4} \right) =\Pi _{0}^{m}\left( q_{0}^{*} \right)$, we have ${ q}_{1}^{m4}=D\left[ 1-\frac{\mathop {\int }\nolimits _0^{1/\vartheta } {\left( {\mathop {p}\limits ^{=}}_{s}-w \right) \left( t\vartheta -1 \right) } \varphi \left( t \right) dt}{w-e\bar{V} (e)-o-\underline{p}_{se}} \right]$.

Due to ${\mathop {p}\limits ^{=}}_{s}-w>0$ and $\mathop {\int }\nolimits _0^{1/\vartheta } \left( t\vartheta -1 \right) \varphi \left( t \right) dt\le 0$, we can see ${q}_{1}^{m4}\le D$. Let$\left[ 1-\frac{\mathop {\int }\nolimits _0^{1/\vartheta } {\left( {\mathop {p}\limits ^{=}}_{s}-w \right) \left( t\vartheta -1 \right) } \varphi \left( t \right) dt}{w-e\bar{V} (e)-o-\underline{p}_{se}} \right] =\propto _{2}$, we get $q_{1}^{m4}=\propto _{2}D$. Because $\Pi _{1}^{m}\left( q_{1} \right)$ increases in $q_{1}$, therefore, if $w_{0}\le w<\bar{V} (e)e+o+\underline{p}_{se}$, when $q_{1}\ge \propto _{2}D,$ then $\Pi _{1}^{m}\left( q_{1} \right) \ge \Pi _{0}^{m}\left( q_{0}^{*} \right)$,    otherwise $\Pi _{1}^{m}\left( q_{1} \right) < \Pi _{0}^{m}\left( q_{0}^{*}\right)$. $\square$

The above provides necessary and sufficient conditions on which the manufacturer’s expected profit is equal to, less or more with option contracts than without. For the supplier, we conduct a similar analysis as follows

Case 1: When $w_{0}>w\ge \bar{V} (e)e+o+\underline{p}_{se}$, we can get

$$\begin{aligned} \Pi _{0}^{s}\left( Q_{0}^{*} \right)= & {} \mathop {\int }\nolimits _0^{1/\vartheta } \left[ \left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) \left( t\vartheta q_{0}^{*}-q_{0}^{*} \right) \right] \varphi \left( t \right) dt+\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) q_{0}^{*}\\&+\left( \alpha {\mathop {p}\limits ^{=}}_{s}\mu -c \right) \vartheta q_{0}^{*},\\ \hbox {and}\, \Pi _{1}^{s}\left( Q_{1}^{*} \right)= & {} \mathop {\int }\nolimits _0^{D/Q_{1}^{*}} {\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\left( tQ_{1}^{*}-D \right) \varphi \left( t \right) dt+\left( w-o-e\bar{V} (e)-\underline{p}_{se} \right) q_{1}}\\&+\left( o+e\bar{V} (e)+\underline{p}_{se}-\alpha {\mathop {p}\limits ^{=}}_{s}\right) D-\left( c-\alpha {\mathop {p}\limits ^{=}}_{s}\mu \right) Q_{1}^{*}, \end{aligned}$$

which can be simplified as: $\Pi _{0}^{s}\left( Q_{0}^{*} \right) =\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) q_{0}^{*}\left[ 1-\varPhi (1/\vartheta ) \right]$ and $\Pi _{1}^{s}\left( Q_{1}^{*} \right) =\left( w-o-e\overline{E} -\underline{p}_{se} \right) q_{1}-\mathop {\int }\nolimits _0^{D/Q_{1}^{*}} {D\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varphi \left( t \right) dt} +\left( o+e\overline{E} +\underline{p}_{se}-\alpha {\mathop {p}\limits ^{=}}_{s}\right) D$.

From ${\Pi _{1}^{s}\left( Q_{1}^{*} \right) =\Pi }_{0}^{s}\left( Q_{0}^{*} \right)$, we have

$$\begin{aligned} {q}_{1}^{s1}=\frac{\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) \frac{q_{0}^{*}}{D}\left[ 1-\varPhi (1/\vartheta ) \right] +\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D/Q_{1}^{*} \right) -\left( o+e\bar{V} (e)+\underline{p}_{se}-\alpha {\mathop {p}\limits ^{=}}_{s}\right) }{w-o-e\bar{V} (e)-\underline{p}_{se}}D. \end{aligned}$$

In addition, because $\Pi _{1}^{s}\left( Q_{1}^{*} \right)$ increases in $q_{1}$, so from ${\Pi _{1}^{s}\left( Q_{1}^{*}\vert q_{1}=D \right) \ge \Pi }_{0}^{s}\left( Q_{0}^{*} \right)$, that is, $\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) D-\mathop {\int }\nolimits _0^{D / Q_{1}^{*}} {\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}D\varphi \left( t \right) dt} -\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) q_{0}^{*}\left[ 1-\varPhi (1/\vartheta ) \right] \ge 0$, we have ${ q}_{0}^{*}\le \frac{\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) -\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D/Q_{1}^{*} \right) }{\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) \left( 1-\varPhi (1/\vartheta ) \right) }D$.

• (1) If $\frac{\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) -\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D/Q_{1}^{*} \right) }{\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) \left( 1-\varPhi \left( 1 / \vartheta \right) \right) }<1$, that is, $w<\frac{\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D/Q_{1}^{*} \right) }{\varPhi \left( 1/\vartheta \right) }+\alpha {\mathop {p}\limits ^{=}}_{s}, \Pi _{1}^{s}\left( Q_{1}^{*}\vert q_{1}=D \right) \ge \Pi _{0}^{s}\left( Q_{0}^{*} \right)$ is true on condition that $q_{0}^{*}<D$ but which is not in line with Lemma 3, thus, ${\Pi _{1}^{s}\left( Q_{1}^{*}\vert q_{1}=D \right) \ge \Pi }_{0}^{s}\left( Q_{0}^{*} \right)$ is false. Let $\frac{{\mathop {p}\limits ^{=}}_{s}\varPhi \left( D / Q_{1}^{*} \right) }{\varPhi \left( 1/\vartheta \right) }+\alpha {\mathop {p}\limits ^{=}}_{s}=n$, therefore, if $w\ge \bar{V} (e)e+o+\underline{p}_{se }$ and $w<\min (w_{0},n)$, we know that $\Pi _{1}^{s}\left( Q_{1}^{*} \right) {< \varPi }_{0}^{s}\left( Q_{0}^{*} \right)$.

• (2) If $\frac{\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) -\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D/Q_{1}^{*} \right) }{\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) \left( 1-\varPhi \left( 1 / \vartheta \right) \right) }\ge 1$, that is, $w{\ge }\frac{\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D/Q_{1}^{*} \right) }{\varPhi \left( 1/\vartheta \right) }+\alpha {\mathop {p}\limits ^{=}}_{s}$, $\Pi _{1}^{s}\left( Q_{1}^{*}\vert q_{1}=D \right) \ge \Pi _{0}^{s}\left( Q_{0}^{*} \right)$ is true on condition that ${ D<q}_{0}^{*}\le \frac{\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) -\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D/Q_{1}^{*} \right) }{\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) \left( 1-\varPhi \left( 1 / \vartheta \right) \right) }D$. Let $\frac{\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) \frac{q_{0}^{*}}{D}\left[ 1-\varPhi \left( 1/\vartheta \right) \right] +\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D/Q_{1}^{*} \right) -\left( o+e\bar{V} (e)+\underline{p}_{se}-\alpha {\mathop {p}\limits ^{=}}_{s}\right) }{w-o-e\bar{V} (e)-\underline{p}_{se}}=\gamma _{1}$, and $\frac{\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) -\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D / Q_{1}^{*} \right) }{\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) \left( 1-\varPhi \left( 1 / \vartheta \right) \right) }=\beta$, so $q_{1}^{s1}=\gamma _{1}D$ and ${D<q}_{0}^{*}\le \beta D$. Therefore, if $w\ge \bar{V} (e)e+o+\underline{p}_{se}$ and $n\le w<w_{0}$, and $D<q_{0}^{*}\le \beta D$, when $q_{1}\ge \gamma _{1}D$, then $\Pi _{1}^{s}\left( Q_{1}^{*} \right) \ge \Pi _{0}^{s}\left( Q_{0}^{*} \right)$, otherwise $\Pi _{1}^{s}\left( Q_{1}^{*} \right) {< \Pi }_{0}^{s}\left( Q_{0}^{*} \right)$; if $q_{0}^{*}>\beta D$, then $\Pi _{1}^{s}\left( Q_{1}^{*} \right) {< \varPi }_{0}^{s}\left( Q_{0}^{*} \right)$.

Case 2: When $w\ge \bar{V} (e)e+o+\underline{p}_{se}$ and $w_{0}\le w$, we can get

$$\begin{aligned} \Pi _{0}^{s}\left( Q_{0}^{*} \right)= & {} \left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) D\left[ 1-\varPhi \left( 1/\vartheta \right) \right] , \hbox {and}\\ \Pi _{1}^{s}\left( Q_{1}^{*} \right)= & {} \left( w-o-e\bar{V} (e)-\underline{p}_{se} \right) q_{1}+\left( o+e\bar{V} (e)+\underline{p}_{se}-\alpha {\mathop {p}\limits ^{=}}_{s}\right) D\\&-\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}D\varPhi \left( D/Q_{1}^{*} \right) . \end{aligned}$$

From $\Pi _{1}^{s}\left( Q_{1}^{*} \right) =\Pi _{0}^{s}\left( Q_{0}^{*} \right)$, we have ${ q}_{1}^{s2}=\left[ 1-\frac{\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) \varPhi \left( 1/\vartheta \right) -\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D/Q_{1}^{*} \right) }{w-o-e\bar{V} (e)-\underline{p}_{se}} \right] D$.

• (1) If $\left( w-s \right) \varPhi \left( 1/\vartheta \right) -\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D/Q_{1}^{*} \right) \ge 0$, that is, $w\ge \frac{\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D/Q_{1}^{*} \right) }{\varPhi \left( 1/\vartheta \right) }+s$, we can see ${ q}_{1}^{s2}<D$. Let $1-\frac{\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) \varPhi \left( 1/\vartheta \right) -\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D / Q_{1}^{*} \right) }{w-o-e\bar{V} (e)-\underline{p}_{se}}=\gamma _{2}$, it follows that ${ q}_{1}^{s2}=\gamma _{2}D$. Because $\Pi _{1}^{s}\left( Q_{1}^{*} \right)$ increases in $q_{1}$, therefore, if $w\ge \bar{V} (e)e+o+\underline{p}_{se}$ and $w\ge \max (w_{0},n)$, when ${q}_{1}\ge \gamma _{2}D$, then $\Pi _{1}^{s}\left( Q_{1}^{*} \right) \ge \Pi _{0}^{s}\left( Q_{0}^{*} \right)$, otherwise $\Pi _{1}^{s}\left( Q_{1}^{*} \right) {< \varPi }_{0}^{s}\left( Q_{0}^{*} \right)$.

• (2) If $\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) \varPhi \left( 1/\vartheta \right) -\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D/Q_{1}^{*} \right) <0$, that is, $w<\frac{\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D/Q_{1}^{*} \right) }{\varPhi \left( 1/\vartheta \right) }+s$, we can see ${ q}_{1}^{s2}>D$. Therefore, if $w\ge \bar{V} (e)e+o+\underline{p}_{se}$ and $w_{0}\le w<n$, then $\Pi _{1}^{s}\left( Q_{1}^{*} \right) {< \varPi }_{0}^{s}\left( Q_{0}^{*} \right)$.

Case 3: When $w<\bar{V} (e)e+o+\underline{p}_{se}$ and $w_{0}>w$, similar to the analysis of Case 1, because $\Pi _{1}^{s}\left( Q_{1}^{*} \right)$ decreases in $q_{1}$, so from ${\Pi _{1}^{s}\left( Q_{1}^{*}\vert q_{1}=D \right) \le \Pi }_{0}^{s}\left( Q_{0}^{*} \right)$, that is,

$$\begin{aligned} \left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) D-\mathop {\int }\nolimits _0^{D/Q_{1}^{*}} {\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}D\varphi \left( t \right) dt} -\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) q_{0}^{*}\left[ 1-\varPhi \left( 1/\vartheta \right) \right] \le 0, \end{aligned}$$

we have ${ q}_{0}^{*}\ge \frac{\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) -\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D/Q_{1}^{*} \right) }{\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) \left( 1-\varPhi (1/\vartheta ) \right) }D$.

• (1) If $\frac{\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) -\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D/Q_{1}^{*} \right) }{\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) \left( 1-\varPhi (1/\vartheta ) \right) }<1$, that is, $w<\frac{\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D/Q_{1}^{*} \right) }{\varPhi (1/\vartheta )}+\alpha {\mathop {p}\limits ^{=}}_{s}$, $\Pi _{1}^{s}\left( Q_{1}^{*}\vert q_{1}=D \right) \le \Pi _{0}^{s}\left( Q_{0}^{*} \right)$ is true on condition that ${ q}_{0}^{*}>D\ge \beta D$. Therefore, if $w<\bar{V} (e)e+o+\underline{p}_{se}$ and $w<\min (w_{0},n)$, when $q_{1}\le \gamma _{1}D$, then $\Pi _{1}^{s}\left( Q_{1}^{*} \right) \ge \Pi _{0}^{s}\left( Q_{0}^{*} \right)$, otherwise $\Pi _{1}^{s}\left( Q_{1}^{*} \right) {< \varPi }_{0}^{s}\left( Q_{0}^{*} \right)$.

• (2) If $\frac{\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) -\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D/Q_{1}^{*} \right) }{\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) \left( 1-\varPhi (1/\vartheta ) \right) }\ge 1$, that is, $w\ge \frac{\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D/Q_{1}^{*} \right) }{\varPhi (1/\vartheta )}+\alpha {\mathop {p}\limits ^{=}}_{s}$, $\Pi _{1}^{s}\left( Q_{1}^{*}\vert q_{1}=D \right) \le \Pi _{0}^{s}\left( Q_{0}^{*} \right)$ on condition that ${ q}_{0}^{*}\ge \beta D>D$. Therefore, if $w<\bar{V} (e)e+o+\underline{p}_{se}$ and $n\le w<w_{0}$, and ${q}_{0}^{*}\ge \beta D$, when $q_{1}\le \gamma _{1}D$, then $\Pi _{1}^{s}\left( Q_{1}^{*} \right) \ge \Pi _{0}^{s}\left( Q_{0}^{*} \right)$, otherwise $\Pi _{1}^{s}\left( Q_{1}^{*} \right) {< \Pi }_{0}^{s}\left( Q_{0}^{*} \right)$; if $D<q_{0}^{*}<\beta D$, then $\Pi _{1}^{s}\left( Q_{1}^{*} \right) >\varPi _{0}^{s}\left( Q_{0}^{*} \right)$.

Case 4: When $w_{0}\le w<\bar{V} (e)e+o+\underline{p}_{se}$, similar to the analysis of Case 2, we have

• (1) If $\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) \varPhi (1/\vartheta )-\left( 1-\alpha \right) \varPhi \left( D/Q_{1}^{*} \right) \ge 0$, that is, $w\ge \frac{\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D / Q_{1}^{*} \right) }{\varPhi (1/\vartheta )}+\alpha {\mathop {p}\limits ^{=}}_{s}$, we can see ${ q}_{1}^{s4}\ge D$. Because $\Pi _{1}^{s}\left( Q_{1}^{*} \right)$ decreases in $q_{1}$, therefore, if $w<\bar{V} (e)e+o+\underline{p}_{se}$ and $w\ge \max (w_{0},n)$, then $\Pi _{1}^{s}\left( Q_{1}^{*} \right) \ge \Pi _{0}^{s}\left( Q_{0}^{*} \right)$.

• (2) If $\left( w-\alpha {\mathop {p}\limits ^{=}}_{s}\right) \varPhi (1/\vartheta )-\left( 1-\alpha \right) \varPhi \left( D/Q_{1}^{*} \right) <0$, that is, $w<\frac{\left( 1-\alpha \right) {\mathop {p}\limits ^{=}}_{s}\varPhi \left( D/Q_{1}^{*} \right) }{\varPhi (1/\vartheta )}+\alpha {\mathop {p}\limits ^{=}}_{s}$, we can see ${ q}_{1}^{s4}<D$. Therefore, if $w<\bar{V} (e)e+o+\underline{p}_{se}$ and $w_{0}\le w<n$, when ${q}_{1}\le \gamma _{2}D$, then $\Pi _{1}^{s}\left( Q_{1}^{*} \right) \ge \Pi _{0}^{s}\left( Q_{0}^{*} \right)$, otherwise $\Pi _{1}^{s}\left( Q_{1}^{*} \right) {< \varPi }_{0}^{s}\left( Q_{0}^{*} \right)$.

Under above discussion, Proposition 4 can be safely drawn. $\square$