Solving first order constraints; lagrangian function and utility maximisation

Question

I am supposed to find the demand curve if the following is given;

$U(x,y) = xy$

price of $x * x$ + price of $y * y = m$ (so a general case, and I will be adding certain prices and income levels later on).

How would I find the demand curves for the above?

What I did was that I found the lagrangian function then found the first order constraints (note: in the following, $L = \Lambda$ and $p_1 = $ price of $x$ and $p_2 =$ price of $y$, until someone perhaps edits it?);

1) $y - L p_1 = 0$

2) $x - L p_2 = 0$

3) $p_1 x + p_2 y - m = 0$

... so how do I continue from here on? I haven't use this method before, and no teacher to ask.

Answer 1

Your equations are correct.

1) $y - \lambda p_1 = 0$

2) $x - \lambda p_2 = 0$

3) $p_1 x + p_2 y - m = 0$

Just put $\lambda p_1$ and $\lambda p_1$ on the RHS.

$y = \lambda p_1 $

$x = \lambda p_2 $

Now divide one equation by another.

$\frac{x}{y}=\frac{\lambda p_1}{\lambda p_2}$

Cancelling out $\lambda$

$\frac{x}{y}=\frac{p_1}{ p_2}$

Solve for x

$x=\frac{p_1}{ p_2}\cdot y \ \ \quad (*)$

Now you can insert the expression for x in equation 3) and solve the equation 3) for y. Then you get $y^*$. Consider that $p_1,p_2$ and $m$ are constants.

After you have calculated $y^*$, you can calculate $x^*$ by using the equation $(*)$.