Suppose that all you do in a day is work, play and sleep. Let $x_1$ be the number of hours per day you spend playing, $x_2$ is the number of hours you spend sleeping, and $x_3$ is the number of hours you spend working. Suppose that sleeping is free, but playing costs you $15 an hour. Furthermore, you spend all the money you earn working on playing.

The utility you get from sleeping and playing is given by a Cobb-Douglas utility function:
$U = x_1^{a_1}x_2^{a_2}$, where $a_1 + a_2 = 1$

NOTE: By construction, $x_2$ can represent the hours you spend consuming anything that is free, and $x_1$ can be the number of hours consuming goods you have to pay for. This does not change the question, it is just interesting that this set-up can be applied to a more general setting.

Let $a_1 = \frac{3}{4}$
If your hourly wage is $w$. Find the number of hours you should work a day $(x_3^*)$ in order to maximize your utility as a function of $w$.
$x_3^* =$

If $w = 30$:
$x_3^* =$

The easiest way to solve this question is using Lagrange multipliers.
In general: suppose you want to maximize $f(x_1,x_2,x_3)$, subject to $h(x_1,x_2,x_3)=0$, and $g(x_1,x_2,x_3)=0$.
We define the Lagrange function to be:
$\displaystyle \Lambda(x_1,x_2,x_3,\lambda_1,\lambda_2) = f(x_1,x_2,x_3) -\lambda_1 g(x_1,x_2,x_3) - \lambda_2 h(x_1,x_2,x_3)$

$f(x_1,x_2,x_3)$ is maximized when all of the partial derivatives of the Lagrange function are equal to $0$.

(you will lose 50% of your points if you do)