There are three agents: a manager, an auditor, and a shareholder.

The shareholder (owns 100 percent of the firm) takes no action but receives the value of the company at the end. The total value of the firm at the start is $10.

The manager has the option to abuse their position to earn extra money from the company (at a cost to the shareholder). The manager chooses the extent of this abuse (denote A). A must be between zero and one.*A also equals the benefit to the manager. The cost of A to the shareholder is 10 x A.

The manager will choose A = 1 & receive a payoff of $1. The shareholder will receive a payoff of $1010 x $1 = $0.

The auditor is to figure out if the manager is abusing their power. To find out, the auditor exerts effort (E), which costs her E2. Restrict E to between zero and one. For a given amount of abuse and effort, there is a probability (p) that the auditor will detect the abuse, with p = A2E.

The probability of detection = zero if there is no abuse or if the auditor doesn't work at all.

*The probability of detection is never more than 1.

If auditor discovers nothing (probability 1 - p), then nothing will happen. If the auditor discovers abuse (probability p), then the manager suffers some loss. This loss equals four. If the auditor discovers the abuse, the auditor receives a benefit equal to two.

1. What is the payout to each of the three agents for A and E?

2. Find an equilibrium for A and an E such that the manager's choice of A is optimal given E, and the auditor's choice of E is optimal given

2a. What is the shareholder's loss in this equilibrium?

3. Assume the Regulators proposes additional disclosure (which will make the auditor's job easier). It will cost the company (and thus the shareholder) money to comply with (call this cost C). Everything is the same except the auditor's costs are decreased from E2 to (E2)/8. Find the new equilibrium. For what values of C would the shareholder want to implement the new rule, and for what values of C would the shareholder prefer not to implement the new rule?

4.) Redo question 3, with the assumption that the new rule reduces the auditor's costs from E2 to E2 - 0.1.

5.) Why would the shareholder want to pay to help the auditor?