Problem $2$

John, the owner of a family business, notices that when he works hard $($a$1)$

during the year the net income before manger compensation is $$1,764$ for

$70\%$ of the time, and $$441$ for $30\%$ of the time.

John recently had some issues and couldn$$t work hard. Because he shirked

$($a$2)$ the net income has been $$1,764$ for $40\%$ of the time and $$441$ for $60\%$ of

the time.

To remediate, John decides to hire Oliver as a manager. Oliver is both risk

and effort$-$adverse with utility equal to the square root of the amount of

money received. His disutility of effort is $3\mathrm{.}5,$ if he works hard, and $2\mathrm{.}1$ if he

shirks. His reservation utility is equal to $7.$

Required

$($Take calculations to two decimal places$).$

a$.$ John suggests a salary of $$85,$ Oliver accepts the offer. Which act

would he take a$1($Work Hard$)$ or a$2($Shirk$)?$

b$.$ John is not convinced that the contract offered to Oliver is the best

possible contract. He decides to hire a consultant who advises him to

offer a contract with a salary proportional to the net income $($before

manager compensation$).$ Why does the consultant think that such a

contract is better compared to the fixed salary offered in point a$)?$ What

proportion of net income should the consultant recommend? Show your

computations.

c$.$ John is risk neutral, with utility equal to the amount of profit received

after any manager$$s compensation. Is John expected utility higher $($or

lower$)$ under the contract offered in b$)$ compared to the contract offered

in point a$)?$ Show you computations.

d$.$ Which act would Oliver take if offered the contract in point b$)?$ Show

your computations.