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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.

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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.

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This is a problem that has also an inequality constraint, alongside the budget constraint that is here given as an equality. For simplicity, substitute the equality constraint into the utility function and form a Lagrangian

$$\Lambda = u(x_1,x_2(x_1)) + \lambda(T-x_1),\;\; \lambda \geq 0$$

where $\lambda$ is a Karush-Kuhn-Tucker multiplier. Then maximize with respect to $x_1$.

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