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.
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.