Price change to maximize profit. A business sells n products,and is considering changing the price of one of the products toincrease its total profits. A business analyst develops aregression model that (reasonably accurately) predicts the totalprofit when the product prices are changed, given by Pˆ = ?T x + P, where the n-vector x denotes the fractional change in the productprices, xi = (pnew ? pi)/pi. Here P is the profit with thecurrentiprices, Pˆ is the predicted profit with the changed prices,pi is the current (positive) price of product i, and pnew is thenew price of product i.
(a) What does it mean if ?3 < 0? (And yes, this canoccur.)
(b) Suppose that you are given permission to change the price ofone product, by up to 1%, to increase total profit. Which productwould you choose, and would you increase or decrease the price? Byhow much?
(c) Repeat part (b) assuming you are allowed to change the priceof two products, each by up to 1%.
