Problem $2($A Second Price Auction with an Optimally Chosen Reserve Price$).$ Two bid$-$

ders, indexed by i$=1,2,$ compete for an item, which the seller does not value. Each

bidder i$$s valuation vi is distributed on $[1,$ $)$ with the cumulative distribution function

F $($vi$)=11/$vi$.$ The valuations are distributed independently across the bidders.

$1$Market Design Final Examination Fall $2024$

$1.$ Compute the seller$$s expected revenue from posting a price and selling to any bid$-$

der willing to buy at this price. What posted price maximizes the seller$$s expected

revenue, and what is the associated revenue?

$2.$ Compute the seller$$s expected revenue from running the second price auction with

no reserve price.

$3.$ Compute the seller$$s expected revenue from running the second price auction with

a reserve price. What reserve price maximizes the seller$$s expected revenue, and

what is the associated expected revenue?

$4.$ What the hell is going on$?$