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.
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 $\frac{du}{dx} = 0$.