2. Suppose Jill derives utility from not only consuming goods, but also from enjoying leisure time. Let her utility function be defined as follows: U=C.25.R.75 where C is a consumption good that can be bought at a price of $1 and R is hours of leisure (relaxation) consumed per day. There are 24 hours in a day and leisure is defined as time spent not working. Jill has a job that pays $w per hour, a trust fund that pays her $M per day, and she can work any number of hours per day, L, she desires. C, consumption good; R, Leisure (relaxation); L, labor M, non-wage income; w, wage rate. a. Derive her labor supply function? b. Assume M = $100, at what wage is her quantity supplied of hours = 0?
2. Suppose Jill derives utility from not only consuming goods, but also from enjoying leisure time. Let her utility function be defined as follows: U=C.25.R.75 where C is a consumption good that can be bought at a price of $1 and R is hours of leisure (relaxation) consumed per day. There are 24 hours in a day and leisure is defined as time spent not working. Jill has a job that pays $w per hour, a trust fund that pays her $M per day, and she can work any number of hours per day, L, she desires. C, consumption good; R, Leisure (relaxation); L, labor M, non-wage income; w, wage rate. a. Derive her labor supply function? b. Assume M = $100, at what wage is her quantity supplied of hours = 0?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![### Utility Function and Labor Supply
**2.** Suppose Jill derives utility from not only consuming goods but also from enjoying leisure time. Let her utility function be defined as follows:
\[ U = C^{0.25} \cdot R^{0.75} \]
where:
- \( C \) is a consumption good that can be bought at a price of $1.
- \( R \) is hours of leisure (relaxation) consumed per day.
There are 24 hours in a day and leisure is defined as time spent not working. Jill has a job that pays \( \$w \) per hour, a trust fund that pays her \( \$M \) per day, and she can work any number of hours per day, \( L \), she desires.
Variables:
- \( C \) = consumption good
- \( R \) = leisure (relaxation)
- \( L \) = labor
- \( M \) = non-wage income
- \( w \) = wage rate
**Questions:**
a. Derive her labor supply function.
b. Assume \( M = \$100 \), at what wage is her quantity supplied of hours \( L \) = 0?
### Explanation for Graphs and Diagrams
*N/A (No graphs or diagrams are present in the given text)*
**Solution Approaches:**
a. **Deriving the Labor Supply Function:**
Jill's total time available per day \( T \) is 24 hours.
\( T = L + R \)
Therefore,
\( R = 24 - L \)
Jill's budget constraint combines her income from work \( w \cdot L \) and her non-wage income \( M \):
\( C = w \cdot L + M \)
Substituting \( C \) and \( R \) into her utility function:
\[ U = \big( w \cdot L + M \big)^{0.25} \cdot \big( 24 - L \big)^{0.75} \]
To maximize her utility function subject to her budget constraint, Jill needs to choose \( L \) appropriately. This involves setting up the Lagrangian as follows:
\[ \mathcal{L} = \big( w \cdot L + M \big)^{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fea0084d8-63e7-4f56-975c-add2c9289a98%2Fcd4df441-06d1-4c2c-b65e-fd7106ba96ad%2Fsi7v94_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Utility Function and Labor Supply
**2.** Suppose Jill derives utility from not only consuming goods but also from enjoying leisure time. Let her utility function be defined as follows:
\[ U = C^{0.25} \cdot R^{0.75} \]
where:
- \( C \) is a consumption good that can be bought at a price of $1.
- \( R \) is hours of leisure (relaxation) consumed per day.
There are 24 hours in a day and leisure is defined as time spent not working. Jill has a job that pays \( \$w \) per hour, a trust fund that pays her \( \$M \) per day, and she can work any number of hours per day, \( L \), she desires.
Variables:
- \( C \) = consumption good
- \( R \) = leisure (relaxation)
- \( L \) = labor
- \( M \) = non-wage income
- \( w \) = wage rate
**Questions:**
a. Derive her labor supply function.
b. Assume \( M = \$100 \), at what wage is her quantity supplied of hours \( L \) = 0?
### Explanation for Graphs and Diagrams
*N/A (No graphs or diagrams are present in the given text)*
**Solution Approaches:**
a. **Deriving the Labor Supply Function:**
Jill's total time available per day \( T \) is 24 hours.
\( T = L + R \)
Therefore,
\( R = 24 - L \)
Jill's budget constraint combines her income from work \( w \cdot L \) and her non-wage income \( M \):
\( C = w \cdot L + M \)
Substituting \( C \) and \( R \) into her utility function:
\[ U = \big( w \cdot L + M \big)^{0.25} \cdot \big( 24 - L \big)^{0.75} \]
To maximize her utility function subject to her budget constraint, Jill needs to choose \( L \) appropriately. This involves setting up the Lagrangian as follows:
\[ \mathcal{L} = \big( w \cdot L + M \big)^{
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