Question

Question 1 Alice has a painting that she wants to sell. Bertie values the painting at $200. Alice tells Bertie that she is willing to sell the painting if he makes her a sufficiently attractive offer. Bertie doesn't know how much the painting is worth to Alice, but he thinks the painting's value to her is equally likely to be any amount between zero and $100. That is, the probability that Alice's value of the painting is less than or equal to v is F(u) = 100 for any 0 <v < 100. (1) If Bertie wants to maximize his expected surplus, what should he bid for the painting. Bertie is making a "take it or leave it" offer, which Alice is assumed to accept if and only if Bertie's offer is greater than the value of the painting to her.) (2) What is Alice's expected revenue from the auction? (3) What are the potential gains from trade? What are the actual gains from trade using this auction? Question 2 Alice has a painting that she wants to sell at auction at which is the only bidder. Alice doesn't know what Bertie's value for the painting is but she thinks it is uniformly distributed on the interval (50, 200). That is, the probability that Bertie's value of the painting is less than or equal to v is F(u) = -50 if 50 SU <200. (1) If Alice wants to maximize her expected revenue from the auction, what reservation price should she set. (2) What is Alice's expected revenue from the auction? (3) What are the potential gains from trade (assuming Alice's value for the painting is $0? What are the actual gains from trade using this auction? Question 3 Jo and Jim are bidding in a second-price, independent, private value auction. Their values for the object are independently and uniformly distributed on the interval (1,2). (1) What is the expected value of the winning bidder? (2) What is the expected value of the price?

Answer 1

Question-1 Answer: 1. yp Bertie wants to maximize his expected surplus. The bid b will be accepted by alice only it andonly If b2V, thus the probability that Bertieis bidis accepted equal o when the painting is purchased at a price b by Bertie the susplus will be 200-b, this expected Busplus will equal Il (b) - 60 (200-6) = ab- 6² 100 For the maximisation , 11'(b4) - 0 (ov) 2 - 680 The unique bid which satisfies the First osder Condition is b*= 100 2. Alice accepts the bid of Bertieis bid when the values the painting at less than 100, thus alepts with the Probability one and the expected revenue equal 100 3. Because Bertie values the painting for the amount of $200 and Alice expects the value to be $50 ( = 100/2 =50) * The potential gains from trade will be $50(-200-50). • As the obfect is exchanged with probability one when the bid equals 100, thue actual gaine lising this auction will be the same as the potential games.