The image provided contains a question on the Constant Relative Risk Aversion (CRRA) Utility Function that is used in macroeconomics to model consumer behavior. The question asks about a specific utility function in a one-period model involving consumption and leisure choices, as well as providing some tasks. Let's break it down:

CRRA Utility Function:

The general form of the CRRA utility function is given by: $u(x) = \frac{x^{1-\sigma} - 1}{1 - \sigma}$ where $\sigma$ is the curvature parameter (typically $\sigma > 0$).

One-period Model:

For the household, utility over consumption $c$ and leisure $l$ is defined as: $u(c) = \frac{c^{1-\sigma} - 1}{1 - \sigma} + \frac{l^{1-\sigma} - 1}{1 - \sigma}$ The budget constraint is given by: $P_C = Wn$ where $P_C$ is the price of consumption goods, $W$ is the wage rate, and $n$ is labor, with the condition $n + l = 1$ (i.e., time is divided between labor and leisure).

Questions:

1. Indifference Curve:

   • You are asked to plot the indifference curve for a given utility level $\tilde{u} = \frac{2(0.5^{1-\sigma} - 1)}{1-\sigma}$ for $\sigma = 0.5$ and $\sigma = 2$. You should set the domain of leisure $l \in [0.4, 0.6]$ and carefully label the graph.

2. Optimality Condition for Utility Maximization:

   • This involves deriving the first-order condition (FOC) to solve for the optimal balance of labor $n$ and leisure $l$ by differentiating the utility function and solving for $W/P_C$ in terms of $c$ and $l$.
   • Investigate how the substitution and income effects vary with different values of $\sigma$. Specifically, for which values of $\sigma$ the substitution effect dominates the income effect and vice versa.

3. Behavior of the Utility Function as $\sigma \to 1$:

   • The utility function becomes undefined when $\sigma = 1$, but you are asked to derive the limiting behavior as $\sigma \to 1$. Using L'Hopital's Rule, this will lead to a logarithmic utility function.
   • The final part asks how labor supply behaves in this limiting case, providing insights into how labor supply reacts when utility is logarithmic.

Would you like help with specific parts of the question, such as plotting the indifference curve or deriving the optimality conditions? Let me know if you'd like a detailed breakdown of any step.

Here are five related questions:

1. How does the value of $\sigma$ affect risk aversion in the CRRA utility function?
2. What are the substitution and income effects, and how do they influence labor supply decisions?
3. How do you apply L'Hopital's Rule to find the limit of the utility function as $\sigma \to 1$?
4. What happens to the labor-leisure tradeoff when $\sigma = 1$, and what does that imply for wage elasticity?
5. Can you provide a general interpretation of the CRRA utility function in terms of household decision-making?

Tip: When using L'Hopital's Rule to resolve indeterminate forms, remember that you need to differentiate both the numerator and the denominator before evaluating the limit.