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I have many doubts with this exercise of microeconomics. I do not know if anyone could help me please. Thanks in advance.

Let $x$ be the food consumption of a household, and be $y$ the consumption of clothes. The preferences of a household can be represented as $U(x,y)=3\ln x + 5\ln y$.

Additionally, this household faces the unit prices: $p_x=\$ \, 10 \,$ and $\,p_y=\$ \, 4.$

Determine the Marshallian demands of each good considering a budget of $\$ \, 100.$

And it also determines the level of the level of utility reached.

mathsalomon
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1 Answers1

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Here are the steps to determine the Marshallian demands:

$\textbf{1.}$ Maximizing the Lagrange function: $$\max\mathcal L=3\ln x + 5\ln y+\lambda\cdot (100-10x-4y)$$

$\textbf{2}$. Calculating the partial derivatives w.r.t $x,y$ and $\lambda$.

$\textbf{3}$. Setting the partial derivatives equal to $0$.

$$\frac{\partial \mathcal L}{\partial x}=\frac{3}{x}-10\lambda=0\Rightarrow \frac{3}{x}=10\lambda$$

$$\frac{\partial \mathcal L}{\partial y}=\frac{5}{y}-4\lambda=0\Rightarrow \frac{5}{y}=4\lambda$$

$$\frac{\partial \mathcal L}{\partial \lambda}=100-10x-4y=0$$

$\textbf{4}$. Divide the first equation by the second equation. $\lambda$ can be cancelled.

$\textbf{5}$. Solve the result of step 4 for $x$ and insert the corresponding expression into the third equation of step 3. Then solve the equation for $y$ to obtain the Marshallian demand of good $y$.

$\textbf{6}$. Solve the result of step 4 for $y$ and insert the corresponding expression into the third equation of step 3. Then solve the equation for $x$ to obtain the Marshallian demand of good $x$.

$\textbf{7}$. Finally use the results of step 6 and step 7 and the utility function to calculate the level of utility.

callculus42
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  • Thank you. And the utility function mentioned in step 7 is $U(x,y)=3\ln x+5\ln y$, right? – mathsalomon Apr 01 '19 at 17:48
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    Yes, it is the utility function $U(x,y)=...$. – callculus42 Apr 01 '19 at 17:53
  • "If the consumption of good $y$ a tax of $r =3$ is applied for each unit consumed. What are the optimal demands and the level of utility achieved with the application of the tax?". I would like to know at least how step 3 changes. – mathsalomon Apr 02 '19 at 07:16
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    @mathsalomon The price which has to be payed for good y increases from $4$ to $7$. So the new budget constraint is $100-10x-7y=0$. – callculus42 Apr 02 '19 at 10:30