Yumi is a lawyer specialized in M&A. When she takes a case, she receives $1000 as a basic payment. If she manages to successfully defend her client and wins the lawsuit, she receives an extra $1500 as a bonus. But if she loses the case, she only receives the basic payment. If Yumi works very hard and prepares her closing argument in detail, her probability of winning the lawsuit is 50%. When Yumi works hard, she also needs an intern to help her, to whom she pays $450. On the other hand, if Yumi show up in the court without any preparation, her probability of winning becomes 20%. In the latter case, Yumi does not hire an intern. Yumi's utility function is U(W) = W0.5.
(a) What is Yumi's expected revenue when she works hard? What is Yumi's expected revenue when she does not work hard?
(b) Will Yumi work hard or not?
Yumi's boss asks her to consider working with an economist, Zhao, who has a deep understanding of M&A structure. Yumi would need to pay Zhao $800 for him to work with her. When they work together as a team, the probability of winning the lawsuit is always 80% whether Yumi works hard or not.
(c) Should Yumi work with Zhao as a team?
Yumi's boss is considering an alternative wage contract, where Yumi gets paid a fixed amount M, regardless of the lawsuit outcome. If Yumi does not sign this new wage contract, she continues to receive the highest possible wage in (b)-(c).
(d) What is the minimal wage that must be offered to Yumi for her to sign this new wage contract?