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?

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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)^{
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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