Anika manages a piano store. Her utility function is given by

Utility $=$ w $?50$

where w is the total of all her monetary payments and $50$ represents the

cost of her e$$ort when running the store. Anika$$s next best alternative

to managing the store provides her with zero utility. The store$$s gross

pro...t depends on random factors. There is a $50\%$ chance it earns $$500$

$($where by earnings we mean gross pro...ts$,$ not including payments to the

manager$)$ and a $50\%$ chance it earns only $$50.$ Note that the $50$ in the

utility function above representing the disutility from the e$$ort does not

depend on random factors: it happens in all the scenarios $($both when

pro...ts are $$500$ and when they are $$50).$

$($a$)$ If shareholders o$$ered to share half of the store$$s gross pro...t$,$ what

would Anika$$s expected utility be$?$ Would she accept such a contract

when considering what her next best alternative is$?$ What if she

were only given a quarter share? What would be the lowest share

she would accept to manage the $...$rm$?($b$)$ What is the most Anika would pay to buy out the store if shareholders

decided to sell it to her? $[$hint: Anika$$s maximum willingness to

pay should result in an expected utility equal her utility in the best

alternative$]$

$($c$)$ Suppose instead that shareholders decided to o$$er her a $$75$ bonus

only if the store earns $$500.$ What $...$xed salary would Anika need

to be paid to get her to accept the contract $($note that a $...$xed salary

$-$call it f$-$is a payment that is not contingent on the $...$rm pro...ts: it

happens with certainty$)?$