Mr. Cog has 18 hours per day to divide between labor and leisure. His utility function is U(C, R) = CR, where C is dollars per year spent on consumption and R is hours of leisure. If he has 5 dollars of nonlabor income per day and gets a wage rate of 11 dollars per hour when he works, his budget equation, expressing combinations of consumption and leisure that he can afford to have, can be written as

Question

Quizplus · Accepted Answer

The budget equation can be obtained by considering that Mr. Cog can spend his time either working or enjoying leisure, and that his total income comes from both sources:

Total Income = Income from Labor + Nonlabor Income
Total Income = Wage x Hours of Labor + Nonlabor Income
Total Income = 11R + 5

This total income must be equal to his total expenditure on consumption and leisure:

Total Income = Expenditure on Consumption + Expenditure on Leisure
11R + 5 = C + R x Wage
11R + 5 = C + 11R
C = 11R - 5 + 11R
C = 22R - 5

Now we substitute the budget equation into the utility function:

U(C,R) = CR
U(22R-5, R) = (22R-5)R
U(R) = 22R^2 - 5R

To find the maximum, we take the first derivative of U(R) with respect to R, and set it equal to zero:

dU/dR = 44R - 5 = 0
R = 5/44

We plug this value of R back into the budget equation to find the corresponding level of consumption:

C = 22R - 5 = 220/44 - 5 = 5

Therefore, the optimal choice for Mr. Cog is to work 5/44 hours per day and to consume 5 dollars per day. The answer is D.

Mr Cog Has 18 Hours Per Day to Divide Between