solution

(Dynamic Pricing in Markdown) A department store has 700 pairs of purple Capri stretch pants that it must sell in the next three weeks. The store manager knows that demand by week for the next three weeks will be linear each week, with the following price-response functions:

Week 1: d1(p1) = (1000 – 100p1)+

Week 2: d2(p2) = (800 – 100p2)+

Week 3: d3(p3) = (700 – 100p3)+

Assume that the demands in the different weeks are independent, that is, that customers who do not buy in a given week do not come back in subsequent weeks.

(a) The retailer wants to find the optimal charge per pair given that she can only set one price for all three weeks. To do so, draw the aggregate price response function as a function of the price.

(b) Find the optimal price. What is her corresponding revenue?

(c) Assume she can charge a different price each week. What are the optimum prices by week she should charge? What is her corresponding revenue?

(d) Consider the optimal value of of the first week price you computed in part (c). Denote it by p*1. The retailer sets this as the first period price. For this price, she was expecting to see a demand of d1(p*1 ) = (1000 – 100p * 1 )+, but instead she sees a demand that is 100 less than expected due to some unforeseen event. She decides to accordingly adjust the prices in weeks 2 and 3. Assuming that the future demand (in weeks 2 and 3) will be as expected, what would be the new optimal prices for weeks 2 and 3?