Please answer the questions below regarding the manager's choice.

1. Suppose that the utility that the manager obtains from the income of w can be expressed as a natural logarithmic function, u(w) = ln(w). Currently, the manager has a performance-based pay contract. If the company succeeds in the project, he gets a payoff of R, and if it fails, he gets a payoff of r(

2. Now let's say that the probability of success q depends on the level of managerial effort, which is invisible to the shareholders. Specifically, the manager can choose a probability q, which must bear the cost of personal effort (ie, reduced utility) of 0.5q2. Show that if this manager has a fixed salary (ie, R = r), then he chooses the project success probability q = 0. (Hint: Please consider the cost of individual effort in addition to the expected utility function obtained above.)

3. Given a performance pay (R,r), find the success probability q that the manager chooses. Also, discuss how management's choice of probability of success is related to the difference (R-r) between project success and failure remuneration. (For the second question, please explain with economic logic even if you cannot solve the success probability q selected by the manager.)