Please answer all the questions step by step, with explaining all the symbols, and how to approach this type of question rather than answers only. Thanks!

2. Consider an auction for Elvis memorabilia. There are two bidders, i and j, with private values who compete in a first-price auction. Each bidder knows his private value and believes the other bidder's private value is drawn from a uniform distribution on the interval [0,6000] and this is common knowledge. (a) If bidder j is bidding half her valuation and bidder i bids half of wi, then show that bidder i 's expected surplus is (vi21wi)6000wi Use this to show that it is an equilibrium to use the bid function b(v)=21v Now assume the bidders are of different types. Assume that Bidder 2 believes that bidder 1's valuation is drawn from the distribution F1(v) which is uniform on the interval [0,6000] as before. However, bidder 1 knows that bidder 2 is an Elvis fanatic and his valuation is drawn from the distribution F2(v) where F2(v)F1(v) for all v. In the lecture next week we will see that in equilibrium they will now use different bid functions B1(v) and B2(v). (b) Explain why the outcome of this auction is not necessarily efficient. Intuitively, do you think bidder 1 will bid more or less than he did in part a ) now that he knows that he is up against an Elvis fanatic? 2. Consider an auction for Elvis memorabilia. There are two bidders, i and j, with private values who compete in a first-price auction. Each bidder knows his private value and believes the other bidder's private value is drawn from a uniform distribution on the interval [0,6000] and this is common knowledge. (a) If bidder j is bidding half her valuation and bidder i bids half of wi, then show that bidder i 's expected surplus is (vi21wi)6000wi Use this to show that it is an equilibrium to use the bid function b(v)=21v Now assume the bidders are of different types. Assume that Bidder 2 believes that bidder 1's valuation is drawn from the distribution F1(v) which is uniform on the interval [0,6000] as before. However, bidder 1 knows that bidder 2 is an Elvis fanatic and his valuation is drawn from the distribution F2(v) where F2(v)F1(v) for all v. In the lecture next week we will see that in equilibrium they will now use different bid functions B1(v) and B2(v). (b) Explain why the outcome of this auction is not necessarily efficient. Intuitively, do you think bidder 1 will bid more or less than he did in part a ) now that he knows that he is up against an Elvis fanatic