Risk Pooling

The UGA Bookstore stocks two types of cashmere sweaters. The two sweaters are identical in every way except on the first sweater is stitched UGA FOOTBALL (FF) while on the second is stitched DAWGS FOOTBALL (RF) (we'll refer to these two types as UGA sweaters and DAWGS sweaters). Both sweaters retail for $100 apiece and cost the Bookstore $40 to procure. Because the procurement lead time is long relative to the length of the football season, the Bookstore places a single order to cover anticipated sales for the entire season. Any sweaters left over at the end of the season are shipped to a reseller for $20 apiece. The demand for UGA sweaters is normally distributed with a mean of 1000 and a standard deviation of 400. The demand for DAWGS sweaters is normally distributed with a mean of 800 and a standard deviation of 300. It's been noted that in previous years when the demand for one type of sweater is high, the demand for the other type of sweater is low, leading the Bookstore to estimate the correlation between the two sweaters at -0.40.

a. How many UGA sweaters should the Bookstore order for the season to maximize expected profit? What is the expected profit?

Co = Overage Cost = Cost - Salvage = 40-20 = 20

Cu = Underage cost = Price - Cost = 100-40 = 60

So Critical Ratio F(Q) = Cu/(Co+Cu) = 60/(60+20) = 0.75

Corresponding z-stat is NORM.S.INV(0.75) = 0.6745

Expected Pooled demand = Mean(FF) + Mean(RF) = 1000+800 = 1800

Std Dev = Sqrt ((400^2 + 300^2)*(1+ Corrreation))

= Sqrt((400^2 + 300^2)*(1-0.40))

= 387.30

So Optimal Order Qty Q = Mean + Std Dev*z = 1800 + 387.30*0.6745 = 2062

Number of UGA Sweaters to Order (units) 2062

If All sweaters are Sold, Expected Profit = Cu*Q = 60*2062 = $123,720

b. How many DAWGS sweaters should the Bookstore order for the season to maximize expected profit? What is the expected profit?