. The following payoff table summarizes gross profits (before compensation), ari
ID: 1252293 • Letter: #
Question
. The following payoff table summarizes gross profits (before compensation), arising from two different levels of effort and factors outside of the manager’s control (stage of the business cycle and resulting sales).Gross profits
Recession Probability = 0.5 Expansion Probability = 0.5
Low effort $75,000 $200,000
High effort $150,000 $400,000
Assume that the disutility of “high” effort is 50 for the manager and that he or she is risk averse so that Utility = (Compensation)0.5.
A. What is the expected gross profit of the firm after compensation is the manager is paid a flat salary of $40,000? (4 POINTS)
B. What bonus (percentage of gross profit) should be paid to the manager in lieu of this flat salary in order to maximize gross profit to the corporation after compensation? (3 POINTS)
C. What is the dollar amount of expected gross profit after the appropriate compensation bonus is paid to the manager? (3 POINTS)
Explanation / Answer
I'm going to assume you mean that Utility = (Compensation)^0.5 because simply dividing compensation by two does not imply risk aversion. In order to be risk averse, the utility function must be concave. Taking the square root does that. If this assumption is incorrect, let me know. But I'm pretty sharp on these things. A. If the manager is paid a flat salary of 40,000, he will always use low effort because he has a dis-utility of effort. As a result, the expected gross profit is: 75000*0.5 + 200000*0.5 - 40000 = 97,500 B. Here, the manager's utilities can be calculated by (GP*B)^0.5 - 50. We want the manager to have a dominant strategy of engaging in high effort. This means: (1) (150000*B)^0.5 - 50 > (75000*B)^0.5 (2) (400000*B)^0.5 - 50 > (200000*B)^0.5 Solve for B using (1) (150000*B)^0.5 - 50 > (75000*B)^0.5 (150000*B)^0.5 - (75000*B)^0.5 > 50 (B^0.5)*(150000^0.5 - 75000^0.5) > 50 (B^0.5)*113.43 > 50 (B^0.5) > 50/113.43 (B^0.5) > 0.44 B > 0.44^2 B > 0.19 Solve for B using (2) (400000*B)^0.5 - 50 > (200000*B)^0.5 (400000*B)^0.5 - (200000*B)^0.5 > 50 (B^0.5)*(400000^0.5 - 200000^0.5) > 50 (B^0.5)*185.24 > 50 B^0.5 > 50/185.24 B > (50/185.24)^2 B > 0.07 The greater of these two solutions is 0.19. So, the bonus must be greater than about 19% of the gross profit in order to ensure that the manager has a dominant strategy to give high effort. To be safe, the firm should probably set it at 20% because of rounding. C. Now that the manager has a dominant strategy to use high effort, we get to use the higher gross profits. But the firm only gets to keep 80% of them because it pays 20% to the manager. 150000*.8*0.5 + 400000*.8*0.5 = 220,000 So, we see it is much better off paying the manager using the bonus than the flat fee even though the manager earns more money. This example can be used to explain why CEOs earn such high salaries.
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.