Discount Prices: A local department store sells products at a given initial price, and every week a product goes unsold, its price is discounted by 25% of the original price. If it is not sold after four weeks, it is sent back to the regional warehouse. A set of kitchen knives was just put out for $200. Your willingness to pay for the knives (your dollar value) is $180, so if you buy them at a price P, your payoff is u = 180 - P. If you don't buy the knives, the chances that they will be sold to someone else, conditional on not having been sold the week before, are as follows:

For example, if you do not buy the knives during the first two weeks, the likelihood that they will be available at the beginning of the third week is the likelihood that they do not sell in either week 1 or week 2, which is 0.8 × 0.6 = 0.48.

a. Draw your decision tree for the four weeks after the knives are put out for sale.

b. At the beginning of which week, if any, should you run to buy the knives?

c. Find a willingness to pay for the knives that would make it optimal to buy at the beginning of the first week. d. Find a willingness to pay that would make it optimal to buy at the beginning of the fourth week.