[Taken from Lecture Slides] A seller has a single indivisible object to sell. The seller values the object at zero, and seeks to maximize her expected revenue. There are 3 potential buyers: i = 1,2,3. Each buyer i's utility over possible allocations takes a quasilinear form: 0; -t; if he obtains the good and pays t;, and -t; if he does not obtain the good and pays t;. That is, i's type (private information) 0; indicates his value of the object. We assume that buyers' types are independently drawn from the following distributions:
01 and 02 are uniformly distributed over [0,2];
03 follows an exponential distribution 1 - e-1 with A = 2.
(a) Compute each buyer's virtual value function J;(0;) and show that it is nondecreasing in 0;.
(b) Characterize the allocation and transfer rules maximizing the seller's expected revenue.
(c) Discuss whether the revenue-maximizing mechanism achieves allocative efficiency.