Question

Suppose we are interested in bidding on a piece of land and we know one other bidder is interested. The seller announced that the highest bid in excess of $25,000 will be accepted. Assume that the competitor's bid x is a random variable that is uniformly distributed between $25,000 and $30,000.

(a)

Suppose you bid $27,000. What is the probability that your bid will be accepted?

(b)

Suppose you bid $29,000. What is the probability that your bid will be accepted?

(c)

What amount should you bid in dollars to maximize the probability that you get the property?

$

(d)

Suppose you know someone who is willing to pay you $31,000 for the property.

What is the expected profit in dollars if you bid the amount given in part (c)?

$

Find a bid in dollars which produces a greater expected profit than bidding the amount given in part (c). (If an answer does not exist, enter DNE.)

$

Would you consider bidding less than the amount in part (c)? Why or why not?

Yes. There is a bid which gives a greater expected profit than the bid given in part (c), and thus a higher expected profit is possible with a bid smaller than the amount in part (c).No. The bid which maximizes the expected profit is the amount given in part (c), thus it does not make sense to place a smaller bid.    

Answer 1

(a)

x ~ Uniform($25,000, $30,000)

Probability that your bid will be accepted = P(x < $27,000) = (27,000 - 25,000) / (30,000 - 25,000)

= 0.4

(b)

Probability that your bid will be accepted = P(x < $29,000) = (29,000 - 25,000) / (30,000 - 25,000)

= 0.8

(c)

In order  to maximize the probability that you get the property, you need to bid the highest value of x which is $30,000

(d)

Expected profit in dollars if you bid the amount $30,000 = $31,000 - $30,000 = $1,000

(e)

In order to get a greater expected profit, you need to bid the lowest value of x which is $25,000.

Expected profit in dollars if you bid the amount $30,000 = $31,000 - $25,000 = $6,000

Yes. There is a bid which gives a greater expected profit than the bid given in part (c), and thus a higher expected profit is possible with a bid smaller than the amount in part (c).