Consider the following model of a simple supply chain: Consider a single SKU, sold by a wholesaler to a retailer, and sold by the retailer to end consumers. Index the retailer by 1, and the wholesaler by 2. The wholesaler has a cost per unit of the SKU of c2 = 11 . The retailer has a cost per unit of the SKU of c1 = 11, excluding the money paid to the wholesaler. Before each selling season, the retailer chooses the amount Q of the SKU to order from the wholesaler. No inventory replenishment takes place during the selling season (the retailer faces a newsvendor problem). The retailer's selling price is r1 (the retailer's selling price is not a decision variable in this simple model, but we will consider how the results depend on the value of r1). The wholesaler sells at a price r2 per unit to the retailer, independent of Q (no quantity discounts, no returns). The end consumer demand X is a uniformly distributed random variable on (0,100). For each unit of unsatisfied demand the retailer incurs a penalty (loss of customer goodwill) of g1 = 2 and the wholesaler incurs a penalty of g2 = 2. If the retailer decides not to participate in the supply chain, then the retailer will not incur the goodwill cost of g1 for unsatisfied demand. If the wholesaler decides not to participate in the supply chain, then the wholesaler will not incur the goodwill cost of g2 for unsatisfied demand. The salvage value of unsold inventory at the end of the selling season is v = 6 per unit.

1. Analyze the supply chain if the wholesaler first chooses its price r2 per unit of the SKU, and then the retailer chooses the amount Q of the SKU to order from the wholesaler. Assume that the retailer knows the costs and the demand distribution, and chooses Q to maximize its expected profit. Also assume that the wholesaler is smart enough to figure out how the retailer's order quantity Q depends on r2, and that the wholesaler chooses r2 to maximize its expected profit.

(a) Calculate the wholesaler's price r2(r1), the retailer's order quantity Q(r1), the whole- saler's expected profit (or cost) E[G2(r2(r1),X)], and the retailer's expected profit (or cost) E[G1(Q(r1),X)], all as functions of r1. Neatly display your calculations. Note that these calculations are valid as long as r1 > 18. However, that does not mean that the wholesaler and retailer will both participate in the supply chain as long as r1 > 18. To make it profitable for the wholesaler and retailer to both participate in the supply chain under these conditions, a larger value of r1 will be needed, as determined later.

(b) For which values of r1 will it hold that r2(r1) > c2, that is, the wholesaler's price can cover its variable cost?

(c) For which values of r1 will it hold that r1 > c1 + r2(r1), that is, the retailer's price can cover its variable cost?

(d) What is the minimum price r1 (one decimal point precision is sufficient) to make it

profitable for the retailer to participate in the supply chain?

(e) What is the minimum price r1 (one decimal point precision is sufficient) to make it profitable for the wholesaler to participate in the supply chain?