Melanie’s preferences over annual income (in thousands of dollars) I and average
ID: 1201401 • Letter: M
Question
Melanie’s preferences over annual income (in thousands of dollars) I and average daily hours of daily leisure time on the job (“shirking”) S are expressed by the Utility function U(S,I) = S1/5I4/5. For this function Melanie is on the job, either working or shirking, 8 hours each workday. Each hour that she works rather than shirks on an average workday through the year adds $100,000 to the firm’s annual profit. If Melanie shirks all day the firm’s annual profit equals zero.
The firm is considering two different compensation packages for Melanie:
PACKAGE A: She receives a base salary of $100,000 + 5 percent of the firm’s profits
PACKAGE B: She receives 20 percent of the firm’s profits
1. If Melanie is compensated under Package A, she maximizes her satisfaction earning ________ thousand dollars per year.
a) 112 b) 116 c) 128 d) 140 e) None of the above
2. If Melanie is compensated under Package B, she maximizes her satisfaction shirking ________ hours on each average work day.
a) 0.8 b) 1.6 c) 3.2 d) 5.6 e) None of the above
Explanation / Answer
Answers:
Utility function U(S, I) = S1/5I4/5
The firm is considering two different compensation packages for Melanie:
PACKAGE A: She receives a base salary of $100,000 + 5 percent of the firm’s profits
PACKAGE B: She receives 20 percent of the firm’s profits
If we derivate the utility function with respect to ‘S’ and ‘I’, then we get:
dU(S, I)/dS = (1/5)S-4/5 I4/5, and
dU(S, I)/dI = (4/5)S1/5 I-1/5
MRS = MUS/MUI
= [(1/5)S-4/5 I4/5]/[( 4/5)S1/5 I-1/5]
= (1/4).(I/S)
For I = 160 and S = 2
Then, MRS = (1/4).(160/2)
MRS = 20 thousand dollars per hour shirked.
1. If Melanie is compensated under Package A, she maximizes her satisfaction earning 112 thousand dollars per year.
Package A: I = 140 – 5 (5.6) = 112 thousand dollars
2. If Melanie is compensated under Package B, she maximizes her satisfaction shirking 1.6 hours on each average work day.
Package B: S = 160 / 100 = 1.6 hours.