Option pricing and coordination in the fresh produce supply chain with portfolio contracts

Abstract

This paper studies a fresh produce supply chain that consists of a supplier and a retailer in a newsvendor framework. The supplier is the Stackelberg leader and the retailer is the follower. The retailer can obtain products from the supplier by wholesale price and call option portfolio contracts. The fresh produce incurs a circulation loss in quantity during its transportation. The retailer’s optimal ordering policy and the supplier’s optimal pricing policy are derived in the presence of portfolio contracts and circulation loss. It is demonstrated that, as the prices of option increase toward their optimal, the supplier’s expected profit increases whereas the retailer’s expected profit decreases, and the retailer is more sensitive to the price change. It is also found that the fresh produce supply chain can be coordinated by the portfolio contracts, and Pareto improvement for both chain members can also be achieved as compared with the non-coordinated contracts. However, when the supply chain is coordinated, the supplier cannot realize its optimal pricing strategy. Finally, it is shown that the supplier’s optimal option pricing policy is independent to the demand risk and wholesale price, and the circulation loss of fresh produce increases the management risks of the fresh produce supply chain.

Appendix

Proof of Proposition 1

From Eq. (1), $\frac{\partial E[\pi _r \left( {q_1 ,q} \right) ]}{\partial q_1 }=w_2 \left( {1-\beta } \right) {\bar{F}}\left[ {q_1 \left( {1-\beta } \right) } \right] +w_0 -w_1$, $\frac{\partial ^{2}E[\pi _r \left( {q_1 ,q} \right) ]}{\partial q_1^2 }=-w_2 \left( {1-\beta } \right) ^{2}f\left[ {q_1 \left( {1-\beta } \right) } \right] <0$, $\frac{\partial E[\pi _r \left( {q_1 ,q} \right) ]}{\partial q}=(p+g-w_2 )\left( {1-\beta } \right) {\bar{F}}\left[ {q\left( {1-\beta } \right) } \right] -w_0$, $\frac{\partial ^{2}E\left[ {\pi _r \left( {q_1 ,q} \right) } \right] }{\partial q^{2}}=-\left( {p+g-w_2 } \right) \left( {1-\beta } \right) ^{2}f\left[ {q\left( {1-\beta } \right) } \right]$, and $\frac{\partial ^{2}E\left[ {\pi _r \left( {q_1 ,q} \right) } \right] }{\partial q_1 \partial q}=\frac{\partial ^{2}E\left[ {\pi _r \left( {q_1 ,q} \right) } \right] }{\partial q\partial q_1 }=0$. Then $\left| {{\begin{array}{ll} {\frac{\partial ^{2}E[\pi _r \left( {q_1 ,q} \right) ]}{\partial q_1^2 }}&{} {\frac{\partial ^{2}E[\pi _r \left( {q_1 ,q} \right) ]}{\partial q_1 \partial q}} \\ {\frac{\partial ^{2}E[\pi _r \left( {q_1 ,q} \right) ]}{\partial q\partial q_1 }}&{} {\frac{\partial ^{2}E[\pi _r \left( {q_1 ,q} \right) ]}{\partial q^{2}}} \\ \end{array} }} \right| =(p+g-w_2 )w_2 \left( {1-\beta } \right) ^{4}f\left[ {q_1 \left( {1-\beta } \right) } \right] f\left[ {q\left( {1-\beta } \right) } \right] >0,$ it follows that the Hessian matrix of $E[\pi _r \left( {q_1 ,q}\right) ]$ is negative definite. Thus, $E[\pi _r \left( {q_1,q} \right) ]$ is jointly concave in $q_1$ and q. Then the fresh produce retailer’s optimal ordering policy exists and is unique. Let $\frac{\partial E[\pi _r \left( {q_1 ,q} \right) ]}{\partial q_1}=0$, we obtain the retailer’s optimal firm order quantity is $q_1^{*} =\frac{1}{1-\beta }F^{-1}\left[ {1-\frac{w_1 -w_0 }{w_2 \left( {1-\beta } \right) }} \right]$.

Let $\frac{\partial E[\pi _r \left( {q_1 ,q} \right) ]}{\partial q}=0$, we obtain the retailer’s optimal total order quantity is $q^{*}=\frac{1}{1-\beta }F^{-1}\left[ {1-\frac{w_0 }{\left( {p+g-w_2 } \right) \left( {1-\beta } \right) }} \right]$.

Since $q^{*}=q_1^*+q_2^*$, the retailer’s optimal option order quantity is $q_2^*=\frac{1}{1-\beta }$ $\left\{ {F^{-1}\left[ {1-\frac{w_0 }{\left( {p+g-w_2 } \right) \left( {1-\beta } \right) }} \right] -F^{-1}\left[ {1-\frac{w_1 -w_0 }{w_2 \left( {1-\beta } \right) }} \right] } \right\}$.

Proof of Lemma 1

Recall that the fresh produce retailer places a firm order and purchases call options simultaneously. We now consider a special case that the retailer’s optimal firm order quantity $q_1^*=0$. Then $E\left[ {\pi _s \left( {w_0 ,w_2 } \right) } \right]$ can be rewritten as $E\left[ {\pi _s \left( {w_0 ,w_2 } \right) } \right] =\left( {w_0 -c} \right) q^{*}+w_2 \int _0^{q^{*}\left( {1-\beta } \right) } {\bar{F}}\left( x \right) dx.$ Then

$$\begin{aligned} \frac{\partial E\left[ {\pi _s \left( {w_0 ,w_2 } \right) } \right] }{\partial w_0 }=q^{*}+\left\{ {w_0 +w_2 \left( {1-\beta } \right) {\bar{F}}\left[ {q^{*}\left( {1-\beta } \right) } \right] -c} \right\} \frac{\partial q^{*}}{\partial w_0 }. \end{aligned}$$

(4)

From Proposition 1,

$$\begin{aligned} \frac{\partial q^{*}}{\partial w_0 }=-\frac{1}{\left( {p+g-w_2 } \right) \left( {1-\beta } \right) ^{2}f\left[ {q^{*}\left( {1-\beta } \right) } \right] }. \end{aligned}$$

(5)

Since $q^{*}=\frac{1}{1-\beta }F^{-1}\left[ {1-\frac{w_0 }{\left( {p+g-w_2 } \right) \left( {1-\beta } \right) }} \right]$, $\left( {p+g-w_2 } \right) \left( {1-\beta } \right) =\frac{w_0 }{{\bar{F}}\left[ {q^{*}\left( {1-\beta } \right) } \right] }$. Then (5) can be rewritten as

$$\begin{aligned} \frac{\partial q^{*}}{\partial w_0 }=-\frac{{\bar{F}}\left[ {q^{*}\left( {1-\beta } \right) } \right] }{w_0 \left( {1-\beta } \right) f\left[ {q^{*}\left( {1-\beta } \right) } \right] }=-\frac{1}{w_0 \left( {1-\beta } \right) r\left[ {q^{*}\left( {1-\beta } \right) } \right] }. \end{aligned}$$

(6)

Taking (6) into (4), we obtain $\frac{\partial E\left[ {\pi _s \left( {w_0 ,w_2 } \right) } \right] }{\partial w_0 }=q^{*}-\frac{\left( {p+g} \right) w_0 -\left( {p+g-w_2 } \right) c}{w_0 \left( {1-\beta } \right) \left( {p+g-w_2 } \right) r\left[ {q^{*}\left( {1-\beta } \right) } \right] }$, $\frac{\partial ^{2}E\left[ \pi _s \left( {w_0 ,w_2 } \right) \right] }{\partial w_0^2 }=-\frac{w_0 +c}{w_0^2 \left( 1-\beta \right) r\left[ q^{*}\left( {1-\beta } \right) \right] }-\frac{\left[ \left( {p+g} \right) w_0 -\left( {p+g-w_2 } \right) c \right] r'[q*(1-\beta )]}{w_{0}^2(1-\beta )(\rho +g-w_{2})r^{3}[q*(1-\beta )]}$. Recall that, $\hbox {F}\left( \cdot \right)$ belongs to the IFR class, if $\left( {p+g} \right) w_0 -\left( {p+g-w_2 } \right) c>0$, then $\frac{\partial ^{2}E\left[ {\pi _s \left( {w_0 ,w_2 } \right) } \right] }{\partial w_0^2 }<0$.

Similarly, we obtain that $\frac{\partial E\left[ {\pi _s \left( {w_0 ,w_2 } \right) } \right] }{\partial w_2 }=\int _0^{q^{*}\left( {1-\beta } \right) } {\bar{F}}\left( x \right) dx-\frac{\left( {p+g} \right) w_0 -\left( {p+g-w_2 } \right) }{\left( {p+g-w_2 } \right) ^{2}\left( {1-\beta } \right) r\left[ {q^{*}\left( {1-\beta } \right) } \right] }$, $\frac{\partial ^{2}E\left[ \pi _s \left( {w_0 ,w_2 } \right) \right] }{\partial w_2^2 }=-\frac{w_0 +c}{\left( p+g-w_2 \right) ^{2}\left( 1-\beta \right) r\left[ q^{*}\left( {1-\beta } \right) \right] }-\frac{\left[ \left( {p+g} \right) w_0 -\left( {p+g-w_2 } \right) c \right] \left\{ 2r^{2}\left[ {q^{*}\left( {1-\beta } \right) } \right] +r'[q^{*}(1-\beta )]\right\} }{(p+g-w_2)^{3}(1-\beta )r^3[q*(1-\beta )]}$, $\frac{\partial ^{2}E\left[ \pi _s \left( {w_0 ,w_2 } \right) \right] }{\partial w_0 \partial w_2 }=\frac{\partial ^{2}E\left[ \pi _s \left( {w_0 ,w_2 } \right) \right] }{\partial w_2 \partial w_0 }=-\frac{2\left( p+g \right) -w_2 }{\left( p+g-w_2 \right) ^{2}\left( 1-\beta \right) r\left[ q^{*}\left( {1-\beta } \right) \right] }-\frac{\left[ \left( {p+g} \right) w_0 -\left( {p+g-w_2 } \right) c\right] r'[q^{*}(1-\beta )]}{(p+g-w_{2})^{2}w_{0}(1-\beta )r^{3}[q^{*}(1-\beta )]}$ Since $\frac{\partial ^{2}E\left[ {\pi _s \left( {w_0 ,w_2 } \right) } \right] }{\partial w_0^2 }<0\left( {if\left( {p+g} \right) w_0 -\left( {p+g-w_2 } \right) c>0} \right)$, $\left| {{\begin{array}{ll} {\frac{\partial ^{2}E\left[ {\pi _s \left( {w_0 ,w_2 } \right) } \right] }{\partial w_0^2 }}&{} {\frac{\partial ^{2}E\left[ {\pi _s \left( {w_0 ,w_2 } \right) } \right] }{\partial w_0 \partial w_2 }} \\ {\frac{\partial ^{2}E\left[ {\pi _s \left( {w_0 ,w_2 } \right) } \right] }{\partial w_2 \partial w_0 }}&{} {\frac{\partial ^{2}E\left[ {\pi _s \left( {w_0 ,w_2 } \right) } \right] }{\partial w_2^2 }} \\ \end{array} }}\right|$ $=-\frac{\left[ {\left( {p+g} \right) w_0 -\left( {p+g-w_2 } \right) c} \right] ^{2}}{\left( {p+g-w_2 } \right) ^{4}w_0^2 \left( {1-\beta } \right) ^{2}r^{2}\left[ {q^{*}\left( {1-\beta } \right) } \right] }<0,$ it follows that the Hessian matrix of $E\left[ {\pi _s \left( {w_0 ,w_2 } \right) } \right]$ is not negative definite. Thus, $E\left[ {\pi _s \left( {w_0 ,w_2 } \right) }\right]$ is not jointly concave in $w_0$ and $w_2$.

Proof of Proposition 2

From Eq. (2),

$$\begin{aligned} E\left[ {\pi _s \left( {w_0 } \right) } \right]= & {} \left( {w_0 -c} \right) q^{*}+w_2 \int _0^{q^{*}\left( {1-\beta } \right) } {\bar{F}}\left( x \right) dx+\left( {w_1 -w_0 } \right) q_1^*\nonumber \\&-\,w_2\int _0^{q_1^*\left( {1-\beta } \right) } {\bar{F}}\left( x\right) dx. \end{aligned}$$

(7)

$$\begin{aligned} \frac{dE\left[ {\pi _s \left( {w_0 } \right) } \right] }{dw_0}= & {} q^{*}+\left\{ {w_0 -c+w_2 \left( {1-\beta } \right) {\bar{F}}\left[ {q^{*}\left( {1-\beta } \right) } \right] -c} \right\} \frac{\partial q^{*}}{\partial w_0 }-q_1^*\nonumber \\&+\left\{ {w_1 -w_0 -w_2 \left( {1-\beta } \right) {\bar{F}}\left[ {q_1^*\left( {1-\beta } \right) } \right] } \right\} \frac{\partial q_1^*}{\partial w_0 }. \end{aligned}$$

(8)

From Proposition 1, $\frac{\partial q^{*}}{\partial w_0 }=-\frac{1}{w_0 \left( {1-\beta } \right) r\left[ {q^{*}\left( {1-\beta } \right) } \right] }$, $\frac{\partial q_1^*}{\partial w_0 }=\frac{1}{\left( {w_1 -w_0 } \right) \left( {1-\beta } \right) r\left[ {q_1^*\left( {1-\beta } \right) } \right] }$. Then, (8) can be rewritten as

$$\begin{aligned}&\frac{dE\left[ {\pi _s \left( {w_0 } \right) } \right] }{dw_0 }\nonumber \\&\quad =q^{*}-\left\{ {w_0 -c+w_2 \left( {1-\beta } \right) {\bar{F}}\left[ {q^{*}\left( {1-\beta } \right) } \right] -c} \right\} \frac{1}{w_0 \left( {1-\beta } \right) r\left[ {q^{*}\left( {1-\beta } \right) } \right] }\nonumber \\&\qquad -\,q_1^*+\left\{ {w_1 -w_0 -w_2 \left( {1-\beta } \right) {\bar{F}}\left[ {q_1^*\left( {1-\beta } \right) } \right] } \right\} \frac{1}{\left( {w_1 -w_0 } \right) \left( {1-\beta } \right) r\left[ {q_1^*\left( {1-\beta } \right) } \right] }.\nonumber \\ \end{aligned}$$

(9)

Since $\left( {p+g-w_2 } \right) \left( {1-\beta } \right) =\frac{w_0 }{{\bar{F}}\left[ {q^{*}\left( {1-\beta } \right) } \right] }$, $\left( {1-\beta } \right) {\bar{F}}\left[ {q^{*}\left( {1-\beta } \right) } \right] =\frac{w_0 }{p+g-w_2 }$. Since $w_2 \left( {1-\beta } \right) =\frac{w_1 -w_0 }{{\bar{F}}\left[ {q_1^*\left( {1-\beta } \right) } \right] }$, $\left( {1-\beta } \right) {\bar{F}}\left[ {q_1^*\left( {1-\beta } \right) } \right] =\frac{w_1 -w_0 }{w_2}$. Then, (9) can be rewritten as

$$\begin{aligned} \frac{dE\left[ {\pi _s \left( {w_0 } \right) } \right] }{dw_0 }=q^{*}-q_1^*-\frac{\left( {p+g} \right) w_0 -\left( {p+g-w_2 } \right) c}{w_0 \left( {1-\beta } \right) \left( {p+g-w_2 } \right) r\left[ {q^{*}\left( {1-\beta } \right) } \right] }. \end{aligned}$$

(10)

Further, $\frac{d^{2}E\left[ \pi _s \left( {w_0 } \right) \right] }{dw_0^2 }=-\frac{w_0 +c}{w_0^2 \left( 1-\beta \right) r\left[ q^{*}\left( {1-\beta } \right) \right] }-\frac{1}{(w_1 -w_0 )\left( 1-\beta \right) r\left[ q_1^*\left( {1-\beta } \right) \right] }--\frac{\left[ \left( {p+g} \right) w_0 -\left( {p+g-w_2 } \right) c \right] r'[q*(1-\beta )]}{w_0^{2}(1-\beta )(p+g-w_2)r^3[q*(1-\beta )]}.$

If $\left( {p+g} \right) w_0 -\left( {p+g-w_2 } \right) c>0$, i.e., $c<\frac{\left( {p+g} \right) w_0 }{p+g-w_2 }$, then $\frac{d^{2}E\left[ {\pi _s \left( {w_0}\right) }\right] }{dw_0^2}<0$. Thus, for a given $w_2$, suppose $c<\frac{\left( {p+g}\right) w_0}{p+g-w_2}$ holds, then the fresh produce supplier’s expected profit $E\left[ {\pi _s\left( {w_0}\right) }\right]$ is concave in $w_0$. Thereby, the fresh produce supplier’s optimal option price exists and is unique. Let $\frac{dE\left[ {\pi _s\left( {w_0}\right) }\right] }{dw_0}$, i.e., (10)$\,=0$, we obtain the fresh produce supplier’s optimal option price $\hbox {w}_0^{*}$ is $w_0^{*} =\frac{\left( {p+g-w_2 } \right) c}{p+g-\left( {p+g-w_2 } \right) \left( {1-\beta } \right) \left( {q^{*}-q_1^*}\right) r\left[ {q^{*}\left( {1-\beta } \right) } \right] }$.

Proof of Proposition 3

The proof of Proposition 2 shows that $E\left[ {\pi _s \left( {w_0 } \right) } \right]$ is concave in $\hbox {w}_{0}$. So when $\hbox {w}_{0} \le \hbox {w}_{0}^*$, the fresh produce supplier’s expected profit $E\left[ {\pi _s\left( {w_0}\right) }\right]$ is increasing in $w_0$. From Eq. (1), we obtain equality

$$\begin{aligned} \frac{-\partial E[\pi _r \left( {q_1 ,\hbox {}q} \right) ]}{\partial w_0}= & {} q-q_1 -\left[ {\left( {p+g-w_2 } \right) \left( {1-\beta } \right) {\bar{F}}\left[ {q\left( {1-\beta } \right) } \right] -w_0 } \right] \frac{\partial q}{\partial w_0}\nonumber \\&-\,\left[ {w_2 \left( {1-\beta } \right) {\bar{F}}\left[ {q_1 \left( {1-\beta } \right) } \right] +w_0 -w_1 } \right] \frac{\partial q_1 }{\partial w_0 }=q-q_1 \end{aligned}$$

(11)

Since $\frac{-\partial E[\pi _r \left( {q_1 ,q} \right) ]}{\partial w_0 }=q-q_1 >0$, the fresh produce retailer’s expected profit $E[\pi _r \left( {q_1 ,q} \right) ]$ is decreasing in $w_0$. From (10)

$$\begin{aligned} \frac{dE\left[ {\pi _s \left( {w_0 } \right) } \right] }{dw_0 }=q^{*}-q_1^*-\frac{\left( {p+g} \right) w_0 -\left( {p+g-w_2 } \right) c}{w_0 \left( {1-\beta } \right) \left( {p+g-w_2 } \right) r\left[ {q^{*}\left( {1-\beta } \right) } \right] } \end{aligned}$$

(12)

From Proposition 1, $q^{*}$ decreases with $w_0$, and $q_1^*$ increases with $w_0$. Thus, when $w_0 \le w_0^*$, we obtain $q>q^{*}$, and $q_1 <q_1^*$. Comparing (11) and (12), we can conclude that $-\partial E[\pi _r \left( {q_1 ,q} \right) ]/\partial w_0 >dE\left[ {\pi _s \left( {w_0 } \right) } \right] /dw_0$.

Proof of Proposition 4

From Eq. (2),

$$\begin{aligned} E\left[ {\pi _s \left( {w_2 } \right) } \right]= & {} \left( {w_0 -c} \right) q^{*}+w_2 \int _0^{q^{*}\left( {1-\beta } \right) } {\bar{F}}\left( x \right) dx+\left( {w_1 -w_0 } \right) q_1^*\nonumber \\&-\,w_2\int _0^{q_1^*\left( {1-\beta } \right) } {\bar{F}}\left( x\right) dx. \end{aligned}$$

(13)

$$\begin{aligned} \frac{dE\left[ {\pi _s \left( {w_2 } \right) } \right] }{dw_2 }= & {} \left( {w_0 -c} \right) \frac{\partial q^{*}}{\partial w_2 }+\int _0^{q^{*}\left( {1-\beta } \right) } {\bar{F}}\left( x \right) dx+w_2 \left( {1-\beta } \right) {\bar{F}}\left[ {q^{*}\left( {1-\beta } \right) } \right] \frac{\partial q^{*}}{\partial w_2 }\nonumber \\&+\,\left( {w_1 -w_0 } \right) \frac{\partial q_1^*}{\partial w_2 }-\int _0^{q_1^*\left( {1-\beta } \right) } {\bar{F}}\left( x \right) dx-w_2 \left( {1-\beta } \right) {\bar{F}}\left[ {q_1^*\left( {1-\beta } \right) } \right] \frac{\partial q^{*}}{\partial w_2 }.\nonumber \\ \end{aligned}$$

(14)

From Proposition 1, $\frac{\partial q^{*}}{\partial w_2 }=-\frac{1}{\left( {p+g-w_2 } \right) \left( {1-\beta } \right) r\left[ {q^{*}\left( {1-\beta } \right) } \right] }$, $\frac{\partial q_1^*}{\partial w_2 }=\frac{1}{w_2 \left( {1-\beta } \right) r\left[ {q_1^*\left( {1-\beta } \right) } \right] }$. Then, (14) can be rewritten as

$$\begin{aligned} \frac{dE\left[ {\pi _s \left( {w_2 } \right) } \right] }{dw_2 }= & {} \int _0^{q^{*}\left( {1-\beta } \right) } {\bar{F}}\left( x \right) dx-\int _0^{q_1^*\left( {1-\beta } \right) } {\bar{F}}\left( x \right) dx\nonumber \\&-\,\frac{\left( {p+g} \right) w_0 -\left( {p+g-w_2 } \right) c}{\left( {p+g-w_2 } \right) \left( {1-\beta } \right) r\left[ {q^{*}\left( {1-\beta } \right) } \right] }. \end{aligned}$$

(15)

Further, $\frac{d^{2}E\left[ \pi _s \left( {w_2 } \right) \right] }{dw_2^2 }=-\frac{w_0 +c}{\left( p+g-w_2 \right) ^{2}\left( 1-\beta \right) r\left[ q^{*}\left( {1-\beta } \right) \right] }-\frac{w_1 -w_0 }{w_2^2 \left( 1-\beta \right) r\left[ q_1^*\left( {1-\beta } \right) \right] }-\frac{\left[ \left( {p+g} \right) w_0 -\left( {p+g-w_2 } \right) c \right] \{2r^{2}[q*(1-\beta )]+r'[q*(1-\beta )]\}}{(p+g-w_2)^3(1-\beta )r^{3}[q*(1-\beta )]}.$

If $\left( {p+g} \right) w_0 -\left( {p+g-w_2 } \right) c>0$, i.e., $c<\frac{\left( {p+g} \right) w_0 }{p+g-w_2 }$, then $\frac{d^{2}E\left[ {\pi _s \left( {w_2 } \right) } \right] }{dw_2^2 }<0$. Thus, for a given $w_0$, suppose $c<\frac{\left( {p+g} \right) w_0 }{p+g-w_2 }$ holds, then the fresh produce supplier’s expected profit $E\left[ {\pi _s \left( {w_2 } \right) } \right]$ is concave in $w_2$. Thereby, the fresh produce supplier’s optimal exercise price exists and is unique. Let $\frac{dE\left[ {\pi _s \left( {w_2 } \right) } \right] }{dw_2 }=0$, i.e., (15) $=$ 0, we obtain the fresh produce supplier’s optimal exercise price $w_2^*$ satisfies $\left( {p+g-w_2^{*}}\right) ^{2}\left( {1-\beta }\right) r\left[ {q^{*}\left( {1-\beta }\right) } \right] \Big [ \int _0^{q^{*}\left( {1-\beta }\right) } {\bar{F}}\left( x \right) dx-\int _0^{q_1^*\left( {1-\beta } \right) } {\bar{F}}\left( x \right) dx \Big ]=\left( {p+g} \right) w_0 -\left( {p+g-w_2^{*}}\right) c$.

Proof of Proposition 5

The proof of Proposition 4 shows that $E\left[ {\pi _s \left( {w_2 } \right) } \right]$ is concave in $w_2$. So when $w_2 \le w_2^*$, the fresh produce supplier’s expected profit $E\left[ {\pi _s \left( {w_2 } \right) } \right]$ is increasing in $w_2$. From Eq. (1), we obtain equality

$$\begin{aligned} \frac{-\partial E[\pi _r \left( {q_1 ,q} \right) ]}{\partial w_2}= & {} \int _0^{q\left( {1-\beta } \right) } {\bar{F}}\left( x \right) dx-\int _0^{q_1 \left( {1-\beta } \right) } {\bar{F}}\left( x \right) dx\nonumber \\&-\,\left\{ {\left( {p+g-w_2 } \right) \left( {1-\beta } \right) {\bar{F}}\left[ {q\left( {1-\beta } \right) } \right] -w_0 } \right\} \frac{\partial q}{\partial w_2 }\nonumber \\&-\,\left\{ {w_2 \left( {1-\beta } \right) {\bar{F}}\left[ {q_1 \left( {1-\beta } \right) } \right] +w_0 -w_1 } \right\} \frac{\partial q_1 }{\partial w_2}\nonumber \\= & {} \int _0^{q\left( {1-\beta } \right) } {\bar{F}}\left( x \right) dx-\int _0^{q_1 \left( {1-\beta } \right) } {\bar{F}}\left( x \right) dx \end{aligned}$$

(16)

Since $\frac{-\partial E[\pi _r \left( {q_1 ,q} \right) ]}{\partial w_2 }=\int _0^{q\left( {1-\beta } \right) } {\bar{F}}\left( x \right) dx-\int _0^{q_1 \left( {1-\beta } \right) } {\bar{F}}\left( x \right) dx$, the fresh produce retailer’s expected profit $E[\pi _r \left( {q_1 ,q} \right) ]$ is decreasing in $w_2$. From (15)

$$\begin{aligned} \frac{dE\left[ {\pi _s \left( {w_2 } \right) } \right] }{dw_2 }= & {} \int _0^{q^{*}\left( {1-\beta } \right) } {\bar{F}}\left( x \right) dx-\int _0^{q_1^*\left( {1-\beta } \right) } {\bar{F}}\left( x \right) dx\nonumber \\&-\,\frac{\left( {p+g} \right) w_0 -\left( {p+g-w_2 } \right) c}{\left( {p+g-w_2 } \right) \left( {1-\beta } \right) r\left[ {q^{*}\left( {1-\beta } \right) } \right] }. \end{aligned}$$

(17)

From Proposition 1, $q^{*}$ decreases with $w_2$, and $q_1^*$ increases with $w_2$. Thus, when $w_2 \le w_2^*$, we obtain $q>q^{*}$, and $q_1 <q_1^*$. Comparing (16) and (17), we can conclude that $-\partial E[\pi _r \left( {q_1 ,q} \right) ]/\partial w_2 >dE\left[ {\pi _s \left( {w_2 } \right) } \right] /dw_2$.

Proof of Proposition 6

From Eq. (3),

$$\begin{aligned} \frac{dE\left[ {\pi _I \left( {Q_I } \right) } \right] }{dQ_I}= & {} \left( {p+g} \right) \left( {1-\beta } \right) -c-\left( {p+g} \right) \left( {1-\beta } \right) F\left[ {Q_I \left( {1-\beta } \right) } \right] ,\nonumber \\ \frac{d^{2}E\left[ {\pi _I \left( {Q_I } \right) } \right] }{dQ_I^2}= & {} -\left( {p+g} \right) \left( {1-\beta } \right) ^{2}f\left[ {Q_I \left( {1-\beta } \right) } \right] <0. \end{aligned}$$

(18)

Thus, $E\left[ {\pi _I \left( {Q_I } \right) } \right]$ is concave in $Q_I$. Let $\frac{dE\left[ {\pi _I \left( {Q_I } \right) } \right] }{dQ_I }=0$, i.e., (18) = 0, we obtain that the integrated fresh produce supply chain’s optimal supply quantity is $Q_I^*=\frac{1}{1-\beta }F^{-1}\left[ {1-\frac{c}{\left( {p+g} \right) \left( {1-\beta } \right) }} \right]$.

Proof of Proposition 7

As supply chain coordination requires that the decisions of a decentralized chain and an integrated chain are consistent with each other. With Proposition 1 and Proposition 6, let $Q_I^*=q^{*}$, we obtain

$$\begin{aligned} q^{*}= & {} \frac{1}{1-\beta }F^{-1}\left[ {1-\frac{w_0 }{\left( {p+g-w_2 } \right) \left( {1-\beta } \right) }} \right] \\= & {} Q_I^*=\frac{1}{1-\beta }F^{-1}\left[ {1-\frac{c}{\left( {p+g} \right) \left( {1-\beta } \right) }} \right] . \end{aligned}$$

i.e., when $w_0 =\frac{\left( {p+g-w_2 } \right) c}{p+g}$ holds, the fresh produce supply chain can be coordinated.

Proof of Proposition 8

Case 1: When $w_2$ is given, from Proposition 2, the fresh produce supplier’s optimal option price $w_0^*$ is

$$\begin{aligned} w_0^*=\frac{\left( {p+g-w_2 } \right) c}{p+g-\left( {p+g-w_2 } \right) \left( {1-\beta } \right) \left( {q^{*}-q_1^*} \right) r\left[ {q^{*}\left( {1-\beta } \right) } \right] }. \end{aligned}$$

(19)

However, form Proposition 7, fresh produce supply chain coordination must satisfy

$$\begin{aligned} w_0 =\frac{\left( {p+g-w_2 } \right) c}{p+g}. \end{aligned}$$

(20)

Comparing (19) and (20), we can obtain $w_0^*>w_0$.

Case 2: When $w_0$ is given, from Proposition 4, the fresh produce supplier’s optimal exercise price $w_2^*$ satisfies

$$\begin{aligned}&\left( {p+g-w_2^*} \right) ^{2}\left( {1-\beta } \right) r\left[ {q^{*}\left( {1-\beta } \right) } \right] \left[ {\int _0^{q^{*}\left( {1-\beta } \right) } {\bar{F}}\left( x \right) dx-\int _0^{q_1^*\left( {1-\beta } \right) } {\bar{F}}\left( x \right) dx} \right] \nonumber \\&\quad =\left( {p+g} \right) w_0 -\left( {p+g-w_2^*} \right) c. \end{aligned}$$

(21)

However, form Proposition 7, fresh produce supply chain coordination must satisfy $\left( {p+g} \right) w_0 =\left( {p+g-w_2 } \right) c$, i.e.,

$$\begin{aligned} \left( {p+g} \right) w_0 -\left( {p+g-w_2 } \right) c=0. \end{aligned}$$

(22)

Comparing (21) and (22), we can obtain $w_2^*>w_2$.

Combining Cases 1 and 2, we can conclude that when the fresh produce supply chain is coordinated, the fresh produce supplier cannot realize its optimal pricing strategy.