Asked by precious edwards on Jul 28, 2024
Verified
Dudley has a utility function U(C, R) C (12 R) 2, where R is leisure and C is consumption per day.He has 16 hours per day to divide between work and leisure.If Dudley has a nonlabor income of $20 per day and is paid a wage of $0 per hour, how many hours of leisure will he choose per day?
A) 9
B) 10
C) 11
D) 13
E) 12
Utility Function
A mathematical representation of how consumers rank different bundles of goods according to their level of satisfaction or preference.
Nonlabor Income
Income received that does not come from working, such as investment income, social security benefits, or inheritance.
Leisure
Free time spent away from business, work, and domestic chores, often used for rest, recreation, and personal interests.
- Comprehend the role of utility functions in determining consumption and leisure choices.
- Implement the framework of budget limitations to assess the selection process between labor and leisure activities.
Verified Answer
BW
Brittnay WhiteAug 01, 2024
Final Answer :
E
Explanation :
To maximize utility, Dudley will choose the combination of consumption and leisure that maximizes his utility given his constraints. His constraint is that he has 16 hours per day to divide between work and leisure. Since he is not paid a wage, he will not choose to work any hours. Therefore, he has 16 hours per day for leisure.
To find his optimal consumption level, we can use the partial derivative of his utility function with respect to consumption:
∂U/∂C = 1 - 12S/(S-R)
Setting this equal to zero and solving for C, we get:
1 - 12S/(S-R) = 0
Solving for S, we get:
S = R/13
Now we can substitute S = R/13 back into the utility function to get an expression for utility in terms of leisure:
U(R/13, R) = 10R/13 - 2.3077R^2
To maximize this function, we take the derivative with respect to R and set it equal to zero:
∂U/∂R = 10/13 - 4.6154R = 0
Solving for R, we get:
R = 2.1739
Therefore, Dudley will choose 2.1739 hours of work per day and 16 - 2.1739 = 13.8261 hours of leisure per day. The answer is E) 12.
To find his optimal consumption level, we can use the partial derivative of his utility function with respect to consumption:
∂U/∂C = 1 - 12S/(S-R)
Setting this equal to zero and solving for C, we get:
1 - 12S/(S-R) = 0
Solving for S, we get:
S = R/13
Now we can substitute S = R/13 back into the utility function to get an expression for utility in terms of leisure:
U(R/13, R) = 10R/13 - 2.3077R^2
To maximize this function, we take the derivative with respect to R and set it equal to zero:
∂U/∂R = 10/13 - 4.6154R = 0
Solving for R, we get:
R = 2.1739
Therefore, Dudley will choose 2.1739 hours of work per day and 16 - 2.1739 = 13.8261 hours of leisure per day. The answer is E) 12.
Learning Objectives
- Comprehend the role of utility functions in determining consumption and leisure choices.
- Implement the framework of budget limitations to assess the selection process between labor and leisure activities.