Optimality Conditions: Mixed Constraints (KKT)

Question

Solve the following optimization problem:

$\max \left\{ x_1^{\alpha_1}x_2^{\alpha_2}: p_1x_1+p_2x_2 = w, x_1>0, x_2>0 \right\}$

where: $p_1, p_2>0$ and $\alpha_1+\alpha_2=1$.

I need an idea to solve the system

Answer 1

Using $\alpha_{2} = 1 -\alpha_{1}$ and $ x_2 = \frac{w-p_1x_1}{p_2}$ you get :

$$f(x_1) = x_1^{\alpha_1} \cdot (\frac{w-p_1x_1}{p_2})^{1 -\alpha_{1}}$$

From here you can use the 0 derivative condition to find the value of $x_1$

EDIT : this works only when $\alpha_1,\alpha_2 > 0$