8.Consider the charity auction. In many charity auctions, altruistic celebrities auction objects with special value for their fans to raise funds for charity. Madonna, for example, held an auction to sell clothing worn during her career and raised about 3.2 million dollars. In the charity auction the winner of the lot is the highest bidder. The difference with the standard auction is that all bidders are required to pay an amount equal to what they bid.

Suppose there are two bidders and assume bidders have valuations randomly drawn from the interval [2, 4] according to the uniform distribution.

Derive the equilibrium bidding function. Hint: Follow the procedure in class. In particular, after getting the differential equation given by the FOC, propose a non-linear bidding function b(v) = + v2 as solution. Your task is to find and .

9. (8 points) Derive the revenue of the seller in the charity auction