By (

7.7.7), we know that under the option contract (

*o*,

*e*), the retailer’s expected profit function is given by

$$\displaystyle{ \begin{array}{lll} E\Pi _{or}(Q_{or}) = (p - o - e)Q_{or} - (p - e)\int _{0}^{Q_{or}}F(x)dx,\end{array} }$$

(7.7.15)

where

*Q*_{ or } denotes the retailer’s reservation quantity with the option contract. By (

7.5.1), the supply chain system’s profit function is given by

$$\displaystyle{ \begin{array}{lll} E\Pi _{s}(Q_{s}) = (p - c)Q_{s} - (p - v)\int _{0}^{Q_{s}}F(x)dx.\end{array} }$$

(7.7.16)

where

*Q*_{ s } denotes the production quantity of the supply chain system. Let

$$\displaystyle{ \left \{\!\!\begin{array}{llll} p - o - e =\lambda (p - c),\\ p - e =\lambda (p - v). \\ \end{array} \right. }$$

(7.7.17)

By the assumption that

*e* >

*v*, we require

\(\lambda \in [0,1)\). Substituting (

7.7.17) into (

7.7.15), we obtain

$$\displaystyle{ \begin{array}{lll} E\Pi _{or}(Q_{or}) =\lambda [(p - c)Q_{or} - (p - v)\int _{0}^{Q_{or}}F(x)dx].\end{array} }$$

(7.7.18)

Comparing (

7.7.18) with (

7.7.16), we know that any option contract (

*o*,

*e*) satisfying (

7.7.17) will push the retailer to reserve as much as

\(\overline{Q}_{s}\), i.e.,

\(\overline{Q}_{or} = \overline{Q}_{s},\) where

\(\overline{Q}_{s} = F^{-1}(\frac{p-c} {p-v})\) corresponds to the system-wide optimal production quantity for the supply chain. Again, since the manufacture adopts the “make-to-order” production policy, any option contract (

*o*,

*e*) satisfying (

7.7.17) will make the supply chain system achieve the maximum expected profit for the channel. Taking

*o* and

*e* as variables and solving (

7.7.17), we obtain the following option contract set, denoted as

*M*,

$$\displaystyle{ M =\{ (o,e): o =\lambda (c - v),e = (1-\lambda )p +\lambda v,\lambda \in [0,1)\}. }$$

(7.7.19)

Substituting

\(o =\lambda (c - v)\) and

\(e = (1-\lambda )p +\lambda v\) into

\(\overline{Q}_{om} = F^{-1}(\frac{e+o-c} {e-v} )\) leads to

\(\overline{Q}_{om} = F^{-1}(\frac{p-c} {p-v}) = \overline{Q}_{s}\). Thus, with any option contract (

*o*,

*e*) in

*M*, we have

\(\overline{Q}_{or} = \overline{Q}_{om} = \overline{Q}_{s}\). Again, substituting

\(o =\lambda (c - v),e = (1-\lambda )p +\lambda v\) into (

7.7.11), together with (

7.7.18), we see that under the option contract (

*o*,

*e*) in the set

*M* associated with the parameter

\(\lambda\), the retailer’s expected profit, denoted as

\(\pi _{or}(\lambda )\), is given by

$$\displaystyle{ \begin{array}{lll} \pi _{or}(\lambda )& =&E\Pi _{or}(\overline{Q}_{or}) =\lambda [(p - c)\overline{Q}_{or} - (p - v)\int _{0}^{\overline{Q}_{or}}F(x)dx] \\ & =&\lambda [(p - c)\overline{Q}_{s} - (p - v)\int _{0}^{\overline{Q}_{s}}F(x)dx] \\ & =&\lambda E\Pi _{s}(\overline{Q}_{s}) =\lambda \pi _{c}, \end{array} }$$

(7.7.20)

where

\(\pi _{c} = E\Pi _{s}(\overline{Q}_{s})\) denotes the system-wide optimal profit, and the manufacturer’s expected profit, denoted as

\(\pi _{om}(\lambda )\), is given by

$$\displaystyle{ \begin{array}{lll} \pi _{om}(\lambda )& =&E\Pi _{om}(\overline{Q}_{om}) = (e - c + o)\overline{Q}_{om} - (e - v)\int _{0}^{\overline{Q}_{om}}F(x)dx \\ & =&(1-\lambda )[(p - c)\overline{Q}_{s} - (p - v)\int _{0}^{\overline{Q}_{s}}F(x)dx] \\ & =&(1-\lambda )E\Pi _{s}(\overline{Q}_{s}) = (1-\lambda )\pi _{c}. \end{array} }$$

(7.7.21)

By Proposition

7.4.1, we obtain that with the wholesale price mechanism, the entire supply chain’s total profit is

$$\displaystyle{ \begin{array}{lll} \pi _{wr} +\pi _{wm} = (p - c)\overline{Q}_{wr} - (p - v)\int _{0}^{\overline{Q}_{wr}}F(x)dx.\end{array} }$$

(7.7.22)

Hence, with any option contract (

*o*,

*e*) in

*M*, the increased profit of the supply chain system is

$$\displaystyle{ \begin{array}{lll} \Delta \pi & =&\pi _{c} - (\pi _{wr} +\pi _{wm}) \\ & =&(p - c)(\overline{Q}_{s} -\overline{Q}_{wr}) - (p - v)\int _{\overline{Q}_{ wr}}^{\overline{Q}_{s}}F(x)dx. \end{array} }$$

(7.7.23)

To summarize, Theorem

7.5.1 follows. □