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 is u=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.

1.At the beginning of which week, if any, should you run to buy the knives? (Choose 0 if you should never buy the knives)

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.

3.Let W denote your willingness to pay, so that your payoff is u=W-P when you pay price P. Find x such that for W<x, it is never optimal to buy the knives.

Explanation / Answer

Payoff, U = 180 - P

(a) Week 1

If I buy, payoff = 180 - 200 = - 20

If I don't buy, payoff = 0

So I choose not to buy in week 1.

(b) Week 2

Discounted price = $200 x 0.75 = $150

Probability that knife is still available = 0.8

So, expected payoff = [0.8 x $(180 - 150)] + [0.2 x 0]

= $24

(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 $(150 - 112.5)] + [0.52 x 0]

= $18

(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 $(112.5 - 84.375)] + [0.7504 x 0]

= $7.02

So, purchase made in week 2 gives me highest payoff ($24), so I should purchase in week 2.

NOTE: Out of 3 questions, the first one has been answered.

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