Feng is the owner of a small business. When Feng has worked hard (a1) during the year, net income before manager compensation has been $ 1,600 60% of the time and $ 400 40% of the time. More recently, Feng has been ill and has had to shirk (a2). Net income has been $ 1,600 only 30% of the time and $ 400 70% of the time. Feng realizes that he must hire a manager for one year while he devotes full time to his recovery. Feng is risk neutral, with utility equal to the amount of net income for the year after manager compensation. Feng is negotiating with Yuan for the manager job. He ascertains that Yuan is risk averse, with utility equal to the square root of the dollar compensation received. Yuan is willing to work for Feng providing she receives expected utility of at least 6. Yuan advises Feng that she is effort averse, with disutility of effort of 2 if she works hard, and 1 if she does not work hard.
a. Feng suggests a salary of $ 64, Yuan immediately says that she would accept a salary of $ 64. Which act would she take?
b. However, surprised by her quick acceptance, Feng consults you. You immediately advise against such an offer, suggesting instead a proportion of net income before manager compensation. Why do you advise against? What proportion of net income do you recommend? Show calculations.
c. Show calculations to verify that Feng’s expected utility is higher if he takes your advice instead of paying Yuan a salary. Why is his expected utility higher?
d. Assuming that Yuan accepts Feng’s new offer, verify that she will in fact work hard.