PROBLEM $(3)($Value of Information $-$ continuous states$)$ Suppose a farmer's payoff depends on the friendliness of weather $n[0,1]$ next year and the amount of fertilizer ain$[0,1]$ he uses, via the utility function $u(x,a)=-(x-a{)}^2,($assume the utility is in dollars$).$ He has a prior probability over possible next year weather $x,$ summarized as $U[0,1],$ that is$,$ next year's weather $x$ will be a draw from the uniform distribution on $[0,1].$

$($a$)$ What is the $($ex$-$ante$)$ optimal a for the farmer that maximizes expected utility?

$($b$)$ Now assume that an agency can tell him now the exact value of $x,$ so that he can tailor his action a for the particular value $x$ takes. How much is this perfect information worth for him?

$13\mathrm{.}4$ 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 $[0,1$

$13\mathrm{.}4$ 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 $[0,1$