Selling an object at the monopoly price Andrew is interested in selling a rare car (whose value in his eyes we will normalize to 0). Assume there are n buyers and that buyer i's private value of the car, Vi, is uniformly distributed over [0, 1].
The private values of the buyers are independent. Instead of conducting an auction, Andrew intends on setting a price for the car, publicizing this price, and selling the car only to buyers who are willing to pay this price; if no buyer is willing to pay the price, the car will not be sold, and if more than one buyer is willing to pay the price, the car will be sold to one of them based on a fair lottery that gives each of them equal probability of winning. Answer the following questions:
(a) Find Andrew's expected revenue as a function of the price x that he sets.
(b) Find the price x∗ that maximizes Andrew's expected revenue.
(c) What is the maximal expected revenue that Andrew can obtain, as a function of n?
(d) Compare Andrew's maximal revenue with the revenue he would gain if he sells the car by way of a sealed-bid first-price auction. For which values of n does a sealed-bid first-price auction yield a higher revenue?