Feng is the owner of a small business. When Feng has worked hard (an) during the year, net income before manager compensation has been $1,600 for 60 percent of the time and $400 for 40 percent of the time. More recently, Feng has been ill and has had to shirk (az). Net income has been $1,600 only 30 percent of the time and $400 for 70 percent 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 Shui for the manager job. He ascertains that Shui is risk-averse, with utility equal to the square root of the dollar compensation received. Shui is willing to work for Feng providing she receives expected utility of at least 6. Shui advises Feng that she is effort-averse, with disutility of effort of 3 if she works hard, and 2 if she does not work hard. Required
`Question:
Feng is the owner of a small business. When Feng has worked hard (an) during the year, net income before manager compensation has been $1,600 for 60 percent of the time and $400 for 40 percent of the time.
More recently, Feng has been ill and has had to shirk (az). Net income has been $1,600 only 30 percent of the time and $400 for 70 percent 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 Shui for the manager job. He ascertains that Shui is risk-averse, with utility equal to the square root of the dollar compensation received. Shui is willing to work for Feng providing she receives expected utility of at least 6. Shui advises Feng that she is effort-averse, with disutility of effort of 3 if she works hard, and 2 if she does not work hard.
Required
a. Feng suggests a salary of $81, Shui immediately says that she would accept a salary of $81. 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 Shui a salary. Why is his expected utility higher?
d. Assuming that Shui accepts Feng's new offer, verify that she will in fact work hard.
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