<p>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</p>

<p>$$ x\left(p_1,p_2,m\right) = \left( \frac{2}{3} \frac{m}{p_1} , \frac{1}{3} \frac{m}{p_2} \right) $$</p>

<p>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.</p>

CC BY-SA 3.0
2016-12-22T06:40:45.543Z
2016-12-22T06:40:45.543Z
Theoretical Economist
<p>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</p>

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

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

CC BY-SA 3.0
2016-12-31T15:02:37.357Z
2016-12-31T15:02:37.357Z
Alecos Papadopoulos
<p>I am given the following utility function:</p>

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

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

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

<p>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.</p>

<microeconomics><utility>
Budget constraint with leisure and consumption
Omrane
# Budget constraint with leisure and consumption

(Asked by: username_0 on 2016-12-21)

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.

**Tags:** microeconomics, utility

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## 2 answers

### Answer by username_1 on 2016-12-22. Score: 1

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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### Answer by username_2 on 2016-12-31. Score: 1

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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