Question

Ernesto is considering taking a part-time job while in college. He has a utility function of u(c, l) = 4c^1/4l ^3/4 . His hourly wage at the job would be $10, but gets $60 in additional income each day from his parents.

a) Calculate Ernesto’s optimal level of leisure and consumption.

b) Suppose that Ernesto can’t choose how many hours he works. He either works 4 hours a day or not. Should he take the job?

Answer 1

Ernesto is considering taking a part-time job while in college. He has a utility function of u(c, l) = 4c1/4l 3/4 . His hourly wage at the job would be $10, but gets $60 in additional income each day from his parents.

a) Calculate Ernesto’s optimal level of leisure and consumption.

From the utility function, MRS = -4*1/4*(l/c)^3/4 divided by 4*3/4*(c/l)^1/4 = -l/3c. Budget equation is

c = (24 - l)*10 + 60

c = 240 + 60 - 10l

c + 10l = 300

Slope of budget = -1/10. At the optimal choice, MRS = slope of budget equation

l/3c = 1/10

l = 0.3c

Use this in budget

c + 10*0.3c = 300

4c = 300

c = $75, This gives l = 0.3*75 = 22.5

Hence Ernesto enjoys a leisure of 22.5 hours. He works for 1.5 hours, earn $15 as labor income and $60 as non labor income.

b) Suppose that Ernesto can’t choose how many hours he works. He either works 4 hours a day or not. Should he take the job?

If l = 24 - 4 = 20, he earns a total income of 4*10 + 60 = 100. Hence c = 100. Now MRS = l/3c = 20/3*100 = -1/15 and slope of budget equation is -1/10. Since MRS < slope of the budget, he should take more leisure and less consumption. Hence he should not take the job. Moreover, compare the two utilities

U(with no job) = 4*(60^0.25)*(24^0.75) = 120.71..............(with no labor income consumption = non labor income, leisure = full 24 hours).

U(with job) = 4*(100^0.25)*(20^0.75) = 119.62

Since utility with job is less, Ernesto should not take the job.