A local department store puts out products at an initial price, and every week t

Question

A local department store puts out products at an initial price, and every week the product goes unsold its price is discounted by 25% of the original price. If it is not sold after 4 weeks, it is sent back to the warehouse. There is a set of butcher knives that was just put out for the price of $200. Your willingness to pay for the knives (your dollar value) is $180, so if you buy them at a price P, your payoff isu=180-P, and if you don't buy the knives, your payoff is 0. If you don't buy the knives, the chances that they are sold to someone else conditional on not selling in the week before are given as follows:

Week 1: 0.2

Week 2: 0.4

Week 3: 0.6

Week 4: 0.8

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

2.Let W denote your willingness to pay, so that your payoff is u=W-P when you pay price P. Find W such that you are indifferent between buying in week 3 and waiting until week 4.

Explanation / Answer

Payoff, U = W - P

(a) Week 1

I'll not buy in week 1 as per question.

(b) Week 2

Discounted price = $200 x 0.75 = $150

Probability that knife is still available = 0.8

I'll not buy in week 2 as per question.

(c) Week 3

Discounted price = $150 x 0.75 = $112.5

Probability that knife is still available = 0.8 x 0.6 = 0.48

So, expected payoff = [0.48 x $(W - 112.5)] + [0.52 x 0]

= $(0.48W - 54)

(d) Week 4

Discounted price = $112.5 x 0.75 = $84.375

Probability that knife is still available = 0.8 x 0.6 x 0.52 = 0.2496

So, expected payoff = [0.2496 x $(W - 84.375)] + [0.7504 x 0]

= $(0.2496W - 21.06)

I shall be indifferent between purchasing in week 3 versus week 4 if expected payoffs are equal.

0.48W - 54 = 0.2496W - 21.06

0.2304W = 32.94

W = 32.94 / 0.2304 = $143 (rounded off)

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