# Budget constraint with leisure and consumption

I am given the following utility function:

$u(x_{1}, x_{2})=x_{1}^2x_{2}$ where $x_{1}$ represents leisure and $x_{2}$ represents consumption.

I am also given the budget constraint with the price of $x_{2}$ normalized to 1:

$w(T-x_{1})=x_{2}$ where $w$ is the wage and $T$ is the number of hours available to the agent.

I am asked to derive Marshallian Demands. How am I supposed to interpret this problem? Should I consider $w$ the price of $x_{1}$ since it is the opportunity cost of leisure? If so, then the agent is constrained by both, time and wage. I'm not sure how to go about it.

The utility function is Cobb-Douglas, so we know that the Marshallian demand when the consumer faces the prices $\left(p_1,p_2\right)$ and has money $m$ is given by
$$x\left(p_1,p_2,m\right) = \left( \frac{2}{3} \frac{m}{p_1} , \frac{1}{3} \frac{m}{p_2} \right)$$
You should be able to derive the Marshallian demand for your problem by noticing what $p_1$, $p_2$, and $m$ are in your problem.
$$\Lambda = u(x_1,x_2(x_1)) + \lambda(T-x_1),\;\; \lambda \geq 0$$
where $\lambda$ is a Karush-Kuhn-Tucker multiplier. Then maximize with respect to $x_1$.