For

*Q* units of the option purchased by the retailer at an exercise price

*e*, the value of the

*Q*th of the option is given by

$$\displaystyle{ \begin{array}{lll} MV _{opt.}(Q_{th}) = z(e)[1 - F(Q)]. \end{array} }$$

(6.5.9)

Obviously, the retailer achieves the maximum expected profit only at the option quantity that makes the value of the last unit of the option just equal to the marginal cost of purchasing the option. Therefore, the optimal option purchase quantity for the retailer is given by the following equation

$$\displaystyle{ \begin{array}{lll} z(e)[1 - F(Q)] =\lambda z(e)[\frac{Q-\int _{0}^{Q}F(x)dx} {Q} ],\end{array} }$$

(6.5.10)

which leads to

$$\displaystyle{ \begin{array}{lll} L(Q) = [1 - F(Q)] -\lambda [Q -\int _{0}^{Q}F(x)dx]/Q = 0. \end{array} }$$

(6.5.11)

Thus Theorem

6.3.4(i) follows. To show Theorem

6.3.4(ii), we first derive the first-best production quantity in the channel. To this end, we take the supply chain as a centralized entity. Since the spot price

*w* of the product is bounded in the range (

*c*,

*p*), the centralized entity will satisfy the demand by using in-house production prior to purchasing from the spot market; otherwise, the centralized entity never achieves the optimal performance. Hence, the overall profit of the centralized supply chain is given by

$$\displaystyle{ \begin{array}{lll} E\pi _{s}(Q) = E_{W}E_{X}[pX - W\max \{X - Q,0\} - cQ], \end{array} }$$

(6.5.12)

where

*Q* denotes the production quantity of the channel. The first term of (

6.5.12) is the sales revenue, the second term is the spot procurement cost, and the last term is the in-house production cost. Furthermore, by (

6.5.12) we obtain

$$\displaystyle{ \begin{array}{lll} E\pi _{s}(Q)& =&\int _{c}^{p}[\int _{0}^{+\infty }pxdF(x) - w\int _{Q}^{+\infty }(x - Q)dF(x) - cQ]dG(w) \\ & =&(\overline{w} - c)Q + (p -\overline{w})\overline{x} -\overline{w}\int _{0}^{Q}F(x)dx,\end{array} }$$

(6.5.13)

where

\(\overline{w} =\int _{ c}^{p}wdG(w)\) represents the expected spot price of the underlying product and

\(\overline{x} =\int _{ 0}^{+\infty }xdF(x)\) represents the expected demand. Since

\(\frac{d^{2}E\pi _{ s}(Q)} {dQ^{2}} = -\overline{w}f(Q) <0\),

*E π*_{ s }(

*Q*) is strictly concave in

*Q*. Thus the first-order optimality condition works. Letting

\(\frac{dE\pi _{s}(Q)} {dQ} = \overline{w} - c -\overline{w}F(Q) = 0\), we obtain the first-best production quantity of the channel as

\(\hat{Q} = F^{-1}(\frac{\overline{w}-c} {\overline{w}} )\).

Obviously, an option pricing scheme that can provide the retailer with an incentive to purchase as many as

\(\hat{Q}\) units of the option can be used to coordinate the channel. The option pricing scheme making

\(\hat{Q}\) the unique solution to (

6.5.10) just provides the retailer with such an incentive. Hence, letting

\(Q =\hat{ Q}\) in (

6.5.10) leads to

$$\displaystyle{ \begin{array}{lll} z(e)[1 - F(\hat{Q})] =\lambda z(e)[\frac{\hat{Q}-\int _{0}^{\hat{Q}}F(x)dx} {\hat{Q}} ], \end{array} }$$

(6.5.14)

from which

\(\hat{\lambda }= \frac{\hat{Q}[1-F(\hat{Q})]} {\hat{Q}-\int _{0}^{\hat{Q}}F(x)dx}\). To summarize, the proof of Theorem

6.3.4 is completed.

\(\quad \square\)