Clare manages a piano store. Her utility function is given by
Utility = w – 100
Where w is the total of all monetary payments to her and 100 represents the cost to her of the effort of running the store. Clare's next best alternative to managing the store provides her with zero utility. The store's gross profit depends on random factors. There is a 50% chance it earns $1,000 (where by earnings we mean gross profits, not including payments to the manager) and a 50% chance it earns only $400.
a. If shareholders offered to share half of the store's gross profit, what would her expected utility be? Would she accept such a contract? What if she were only given a quarter share? What would be the lowest share she would accept to manage the firm?
b. What is the most Clare would pay to buy out the store if shareholders decided to sell it to her?
c. Suppose instead that shareholders decided to offer her a $100 bonus if the store earns $1,000. What fixed salary would Clare need to be paid in addition to get her to accept the contract?