$13\mathrm{.}4$ Reserve Prices $($Harder$)$: Consider a seller who must sell a single private value good. There are two potential buyers, each with a valuation that is drawn independently and uniformly from the interval $\backslash ([0,1]\backslash ).$ The seller will offer the good using a second$-$price sealed$-$bid auction, but he can set a "reserve price" of $\backslash ($ r $\backslash$geq $0\backslash )$ that modifies the rules of the auction as follows. If both bids are below $\backslash ($ r $\backslash )$ then neither bidder obtains the good and it is destroyed. If both bids are at or above $\backslash ($ r $\backslash )$ then the regular auction rules prevail. If only one bid is at or above $\backslash ($ r $\backslash )$ then that bidder obtains the good and pays $\backslash ($ r $\backslash )$ to the seller.

a$.$ Show that choosing $\backslash ($ r$=0\backslash )$ is not optimal for the seller. What is the intuition for this fact?

b$.$ What is the optimal reserve price for the seller?