Let Mauricio's utility function be \(U(C, L) = \frac{(C^\alpha L^{1-\alpha})^{1-\sigma}}{1-\sigma}\), where \(\sigma = 2\) and \(\alpha = 0.5\). Suppose he has 110 hours to split b... Let Mauricio's utility function be \(U(C, L) = \frac{(C^\alpha L^{1-\alpha})^{1-\sigma}}{1-\sigma}\), where \(\sigma = 2\) and \(\alpha = 0.5\). Suppose he has 110 hours to split between work and leisure and he has no non-labor income. (a) Write down the optimization problem of Mauricio. (b) Derive Mauricio's optimal choice of consumption and leisure as a function of the wage. (c) What is Mauricio's reservation wage? Read more

Steps to Solve

1. Write down the optimization problem

Mauricio wants to maximize his utility $U(C, L)$ subject to his budget constraint and time constraint. His budget constraint says that consumption $C$ must be equal to his wage $w$ times the hours he works, where hours worked is his total available time (110 hours) minus his leisure time $L$. His time constraint simply states that leisure must be less than or equal to the total time available. The optimization problem can be written as:

$$ \max_{C, L} \frac{(C^\alpha L^{1-\alpha})^{1-\sigma}}{1-\sigma} $$ subject to $$ C = w(110 - L) $$ $$ 0 \le L \le 110 $$

2. Substitute the budget constraint into the utility function

Substitute $C = w(110 - L)$ into the utility function:

$$ U(L) = \frac{((w(110-L))^\alpha L^{1-\alpha})^{1-\sigma}}{1-\sigma} $$ Given that $\alpha = 0.5$ and $\sigma = 2$:

$$ U(L) = \frac{((w(110-L))^{0.5} L^{0.5})^{-1}}{-1} = -((w(110-L))^{0.5} L^{0.5})^{-1} $$ $$ U(L) = \frac{1}{\sqrt{w(110-L)L}} $$

3. Take the derivative of the utility function with respect to leisure and set it to zero

We want to find the optimal level of leisure by taking the derivative of $U(L)$ with respect to $L$ and setting it equal to 0.

$$ \frac{dU(L)}{dL} = -\frac{1}{2} (w(110-L)L)^{-3/2} (w(110-2L)) = 0 $$

4. Solve for optimal leisure $L^*$

Set the derivative equal to zero and solve for $L$:

$$ w(110 - 2L) = 0 $$ $$ 110 - 2L = 0 $$ $$ 2L = 110 $$ $$ L^* = 55 $$

5. Solve for optimal consumption $C^*$

Substitute $L^* = 55$ into the budget constraint $C = w(110 - L)$:

$$ C^* = w(110 - 55) $$ $$ C^* = 55w $$

6. Determine the reservation wage

The reservation wage is the wage at which Mauricio is indifferent between working and not working. Not working means $L=110$ and $C=0$. In this case, his utility is undefined. However, we can also find the wage for which he is indifferent between $L=110$ and $L=55$. The wage for $L=55$ is always positive. When $L=110$, then $C=0$, yielding $U(0,110)$ which is not defined given the utility function. To find the reservation wage directly, consider a slightly altered utility function: $U(C,L) = C^{\alpha}L^{1-\alpha}$. The reservation wage is found by setting utility at zero consumption equal to utility at optimal consumption and leisure. However, with the utility function $U(C, L) = \frac{(C^\alpha L^{1-\alpha})^{1-\sigma}}{1-\sigma}$ as given with $\sigma = 2$ and $\alpha = 0.5$, the utility becomes: $U(C,L) = \frac{(C^{0.5}L^{0.5})^{-1}}{-1} = \frac{1}{\sqrt{CL}}$ If Mauricio does not work, then $C = 0$, $L=110$. Then $U(0,110)$ is undefined. However, the lowest wage he could accept is when his utility is equal to zero, given that the utility function is always positive. When $L^* = 55$ and $C^* = 55w$, then $U(C,L) = \frac{1}{\sqrt{55w \cdot 55}} = \frac{1}{55\sqrt{w}}$ If $w=0$, then $C=0$, $L=110$ which is undefined. The reservation wage is some small positive value. Since the utility function approaches $\infty$ as $w$ approaches 0, the reservation wage is essentially 0.