Tricia owns a small manufacturing firm. In the past, when she worked hard (high effort) during a year, her income before manager compensation was $1,600 (60 percent of the time) or $400 (40 percent of the time). She anticipates this income model to continue going forward. However, Tricia has been ill. When ill (low effort), her income before manager compensation is expected to $1,600 (30 percent of the time) or $400 (70 percent of the time). Due to illness, Tricia needs to hire a replacement manager for one year. She is risk neutral, with utility equal to the amount of net income for the year, minus managerial compensation. Tricia is negotiating with Jack to be hired as her replacement manager. She ascertains that Jack is risk averse, with a utility function equal to the square root of the dollar compensation received. Jack is willing to work for Tricia as long as he receives expected utility equal to six. Jack tells Tricia he is effort averse (aren’t we all?), with disutility of effort equal to two if he works hard (high effort) and one if he does not work hard (low effort). 

a) Tricia suggests a salary of $64 which Jack says he would accept immediately without further negotiation. Why is Jack willing to accept without further negotiation? 

b) Before formally extending the offer, Tricia speaks with you. You advise against formalizing the offer and suggest instead offering a proportion of net income before manager compensation. Why do you advise against the fixed salary? What proportion of net income do you recommend (showing your relevant calculations)? 

c) Show calculations verifying that Tricia’s expected utility is higher if she takes your advice instead of paying a salary. Why is her expected utility higher (showing your relevant calculations)? 

d) Assuming that Jack accepts the proportional pay offer, verify that he will work mathematically. Show all relevant calculations.