A manufacturer wishes to contract a manager for its retail outlet. The profits of the business depend on the state of the world $(,$ 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

$(e,)=Ae-e^2$

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

a$.$ If the manufacturer could observe the state $()$ in advance, what would the optimal contract be $($assume that the manufacturer must hire a manager$)?[5$ marks$]$

b$.$ Now suppose that the manager does not observe the state $$ until after the contract is signed, but does learn the state before choosing $e.$ Also assume that the manufacturer cannot observe anything in advance, and only observes profit afterwards, and therefore offers a profit$-$sharing contract that pays the manager a proportion $$ of profit. Set up the firm's optimisation problem and find the optimal profit$-$sharing level and expected payoffs to the firm and the manager. $[15$ marks$]$

c$.$ Describe in words what how to set up the problem if the manager observed the state before signing the contract. What you would expect to happen to the payoffs of the two parties? $[15$ marks$]$

d$.$ Now suppose that $($as in part a$)$ the manager does not observe the state $$ until after the contract is signed, but faces no personal cost