A manufacturer wishes to contract a manager for its retail outlet. The profits of the business depend on the state of the world $($w$,$ distributed uniformly on $[0,1])$ and the effort $($e$)$ supplied by the manager. Effort is costly; there is a cost included in the profit function

P$($e$,$ w$)=$ Aew $$ e$^2$

In addition, the manager incurs a personal cost of Be and has an outside offer worth U $>=0.$ After the profits are earned, both the manager and the manufacturer learn the state and the manager$$s effort $($w and e$).$ Assume that A$=30,$ B $=0\mathrm{.}25$ and U $=50.$

a$.$ if the manufacturer could observe the state w in advance, what would the optimal contract be $($assume that the manufacturer must hire a manager$)?$