Question

A student has a part-time job in a restaurant. For this she is paid $8 per hour. Her utility function for earning $I and spending S hours studying is U(I,S) = I^1/4 S^3/4 (The utility function is a measure of the `usefulness' or `worth' to the student of a certain combination of money and study time). The total amount of time she spends each week working in the restaurant and studying is 100 hours. How should she divide up her time in order to maximise her utility?

Answer 1

Solution:

U(I, S) = I^1/4S^3/4

To maximize utility, she should set: $\frac{MU_{I}}{P_{I}} = \frac{MU_{S}}{P_{S}}$

MU_I = 0.25I^-0.75S^0.75

MU_S = 0.75I^0.25S^-0.25

Let P_S = 3

$\frac{0.25I^{-0.75}S^{0.75} }{8} = \frac{0.75I^{0.25}S^{-0.25} }{3}$

S = 8I

Budget constraint: 100 = 8I + 3S

100 = 8I + 3(8I)

100 = 8I + 24I

100 = 32I

I = 3.125

S = 8I = 8 $\times$ 3.125 = 25