[Efficient collusion in a SPA] In class we learned that when bidders have SIPV the optimal reserve price is invariant to the number of bidders. For example, when bidders valuations are distributed (iid) Uniform [0, 1], the optimal reserve price is r regardless of how many bidders the seller faces. = The previous result relied on the fact that bidders were not colluding. If all the bidders in a SPA engage in efficient collusion, the optimal reserve price not only will be higher than but it will be increasing in the number of bidders in the ring. To see this, consider a SPA with n bidders who have SIPV and whose valuations are distributed Uniform [0, 1]. Suppose that all the bidders participate in the ring and that the ring is able to engage in efficient collusion. (a) Obtain an expression for the ring's expected gains when the seller does not set a reserve price. This expression will depend on n. (b) Show that the ring's gains increase with n. (c) Suppose now that the seller is aware that the bidders are colluding and decides to set a reserve price to limit the ring's damages. Obtain an expression for the expected revenue as a function of the reserve price, r, and the number of bidders, n. (d) Solve for the reserve price that maximizes expected revenue (which will be a function of n). (e) Show that the optimal reserve increases with n