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$).$