Give solution for b and c as well Suppose the manufacturer produces at a cost of $\backslash (\backslash$$ $20/\backslash )$ unit. The distributor sells to end customers for $\backslash$$$50/$unit during season, unsold units are sold for $\backslash$$$10/$unit after season.

a$.$ What is the system optimal production quantity and expected profit under global optimization? $\backslash (50\backslash$quad $143\backslash )$

b$.$ Suppose the manufacturer is make$-$to$-$order; that is$,$ the timing of events is as follows:

$-$ The distributor orders before it receives demand from end customers.

$-$ The manufacturer produces the amount ordered by the distributor.

$-$ Customer demand is observed.

i$.$ Suppose the manufacturer sells to the distributor at $\backslash (\backslash$$ $40/\backslash )$ unit, how much will the distributor order? What is the expected profit for the manufacturer and distributor?

ii$.$ Find an option contract such that both the manufacturer and distributor enjoy a higher expected profit than $($b$)($i$).$ What is the expected profit for the manufacturer and the distributor?

c$.$ Suppose the manufacturer is make$-$to$-$stock; that is$,$ the timing of events is as follows:

$-$ The manufacturer produces a certain amount.

$-$ The distributor observes demand.

$-$ The distributor orders from the manufacturer.

i$.$ Using the same wholesale price contract as part $($b$)($i$),$ calculate the production$/$ inventory level of the manufacturer. What is the expected profit for the manufacturer and distributor? Compare your results with part $($b$)($i$).$

ii$.$ Find a cost$-$sharing contract such that both the manufacturer and distributor enjoy a higher expected profit that that in $($c$)($i$)$ and calculate their expected profits.