Q$2$ Imagine a two$-$firm supply chain that consists of a supplier and a retailer. The supplier has a

marginal cost $c=$$$40$ and a wholesale price $w=$$$60.$ The retailer is looking to sell its product at $p$

$=$$$100$; at this price point, demand over the lifespan of the product is distributed normally with

mean $200$ and standard deviation of $10.$

a$.$ What is the expected profit for the supplier and the retailer assuming each of them

maximizes their own profit?

b$.$ Suppose the supply chain was integrated $($or alternatively, the supply chain is perfectly

coordinated$).$ What is the optimal order quantity of the retailer? What is the expected

total profit for the supply chain?

c$.$ Now assume the firms seek to form a revenue sharing contract. Let $f$ be the upfront fee

and $$ be the revenue share percentage. Suppose they decide on an upfront fee of $$30$ the

retailer per unit. What is the share $$ so that the chain is perfectly coordinated? What is

the expected profit for the retailer and the supplier under this contract?

d$.$ Assume now that the firms are looking to form a buyback contract. What would be the

buy$-$back price that perfectly coordinates the supply chain assuming the wholesale

remains $60$ per unit? Find a buy$-$back contract for which both the supplier and the retailer

are better off comparing to their profits in part a $($the original wholesale price$).$