# Multivariable optimization

I've been sitting here all afternoon trying to show that if we have a function:

$u(x) = x^{\alpha}y^{\beta}$ and I maximize it subject to:

i) $x \ge 0$
ii) $y \ge 0$
iii) $p_1x + p_2y = w$

Then I get: $x = \frac{\alpha w}{p_1}$ and $y = \frac{\beta w}{p_2}$

I've been using the lagrangian to optimize this and have got to this point:

$x = \frac{ \alpha(p_1x-p_2+w)}{p_1+p_1x}$

I don't know if I'm being stupid or what but I can't seem to find this relationship.

If anyone can give me steps on this I would be grateful.

• is this $u(x)$ or $u(x,y)$? – Dr. Sonnhard Graubner Oct 24 '15 at 16:00
• I assume you mean that $\alpha + \beta =1$? – Chris Kerridge Oct 24 '15 at 16:11
• Well it's not given in the question that beta = 1 - alpha but it has to right?? And yes this is u(x,y) – gloveman998 Oct 24 '15 at 16:19

If $p_1 x + p_2 y = w$ then solve for $y$ and substitute to turn it into a problem that can be solved with basic calculus finding solutions to $\frac{du}{dx} = 0$.