Moral Hazard) The board of directors of a major corporation is trying to determine how to structure the salary of the new CEO. One option is for the board to offer the new CEO a flat salary of$1million per year. A second option is to offer a profit-sharing plan with a base salary of$200,000plus10%of the firm's profit. If the CEO puts a lot of effort into the job, she will generate a$10million profit for the firm. If the CEO exerts modest effort, the corporation will earn$7million in profit. Expending a lot of effort costs the CEO$500,000; expending modest effort costs her$300,000. 1) Draw the game tree for the game played between the board and the CEO. Assume that the board moves first, choosing the type of salary offer. Assume that the CEO moves second, choosing her effort level. Be sure to enumerate payoffs to the board (and the shareholders they represent) and the CEO. 2) What is the equilibrium outcome for this game? What kind of contract should the board offer? What level of effort should the CEO choose? For remaining parts, assume that, as before, expending modest effort costs the CEO$300,000. But, assume now that expending high effort costs the CEO$750,000. 4) Draw the game tree for this game and find the equilibrium. Does the change in the cost of high effort alter the outcome? Explain. 5) Demonstrate that the directors cannot improve their outcome by changing the base salary in the profit-sharing plan. 6) What is the minimum share of the firm's profit that will induce the CEO to expend high effort?

1. To draw the game tree for the game played between the board and the CEO, we need to consider the different choices and payoffs for each player.

First, the board chooses between offering a flat salary of $1 million per year or a profit-sharing plan with a base salary of$200,000 plus 10% of the firm's profit. Let's label these choices as "Flat Salary" and "Profit-Sharing" respectively.

Next, the CEO chooses her effort level, which can be either "High Effort" or "Modest Effort".

Now, let's consider the payoffs for each player. If the board chooses the Flat Salary option and the CEO exerts high effort, the firm will generate a $10 million profit. In this case, the CEO's payoff will be$10 million profit minus the cost of high effort ($500,000), resulting in a payoff of$9.5 million. The board's payoff will be the same as the CEO's, as they represent the shareholders.

If the board chooses the Flat Salary option and the CEO exerts modest effort, the firm will generate a $7 million profit. The CEO's payoff will be$7 million profit minus the cost of modest effort ($300,000), resulting in a payoff of$6.7 million. Again, the board's payoff will be the same as the CEO's.

If the board chooses the Profit-Sharing option and the CEO exerts high effort, the firm will generate a $10 million profit. The CEO's base salary will be$200,000 plus 10% of the profit, which amounts to $1 million. After deducting the cost of high effort ($500,000), the CEO's payoff will be $1 million profit minus the cost of high effort, resulting in a payoff of$500,000. The board's payoff will be the remaining profit of $9 million.

If the board chooses the Profit-Sharing option and the CEO exerts modest effort, the firm will generate a $7 million profit. The CEO's base salary will be$200,000 plus 10% of the profit, which amounts to $900,000. After deducting the cost of modest effort ($300,000), the CEO's payoff will be $900,000 profit minus the cost of modest effort, resulting in a payoff of$600,000. The board's payoff will be the remaining profit of $6.4 million.

Based on these payoffs, we can now draw the game tree for the game played between the board and the CEO.

2. To find the equilibrium outcome for this game, we need to consider the best strategies for each player.

In this case, the board's best strategy is to offer the Profit-Sharing option, as it leads to higher payoffs for both the board and the CEO in both effort levels.

For the CEO, the best strategy is to exert high effort, as it leads to higher payoffs regardless of the board's choice.

Therefore, the equilibrium outcome for this game is for the board to offer the Profit-Sharing option and for the CEO to choose high effort.

4. Now, let's consider the case where the cost of high effort increases to $750,000. We need to draw a new game tree and find the equilibrium.

In this case, the payoffs for the CEO will be lower in both effort levels, as the cost of high effort has increased. However, the board's payoffs remain the same.

Despite the change in the cost of high effort, the equilibrium outcome remains the same. The board should still offer the Profit-Sharing option, and the CEO should still choose high effort. The change in the cost of high effort does not alter the outcome because it affects both effort levels equally.

5. To demonstrate that the directors cannot improve their outcome by changing the base salary in the profit-sharing plan, we need to consider different scenarios.

Let's assume the base salary in the profit-sharing plan is changed to $300,000. In this case, the CEO's payoff will be$300,000 plus 10% of the profit, resulting in a higher payoff compared to the previous scenario. However, the board's payoff remains the same.

Despite the increase in the CEO's payoff, the equilibrium outcome remains the same. The board should still offer the Profit-Sharing option, and the CEO should still choose high effort. Changing the base salary does not alter the outcome because the CEO's effort level has a greater impact on the payoffs.

6. To determine the minimum share of the firm's profit that will induce the CEO to expend high effort, we need to consider the CEO's payoff in both effort levels.

Let's assume the CEO's payoff for high effort is denoted as P(high) and the CEO's payoff for modest effort is denoted as P(modest). The CEO will choose high effort if P(high) is greater than or equal to P(modest).

In the Profit-Sharing option, the CEO's payoff for high effort is $1 million profit minus the cost of high effort. Let's denote the share of the firm's profit as x. Therefore, P(high) = x -$750,000.

The CEO's payoff for modest effort is $300,000 profit minus the cost of modest effort. Therefore, P(modest) =$300,000.

To induce the CEO to expend high effort, we need P(high) to be greater than or equal to P(modest). Therefore, x - $750,000 >=$300,000.

Simplifying the inequality, we find that x >= $1,050,000.

Therefore, the minimum share of the firm's profit that will induce the CEO to expend high effort is $1,050,000.