Problem 3: Constrained Optimization
Andrea maximizes the following utility function:
$u(x_1, x_2) = x_1^{\frac{2}{5}}x_2^{\frac{3}{5}}$
subject to the budget constraint
$p_1x_1 + p_2x_2 = I$
where $p_1, p_2, x_1, x_2, I > 0$
(a) Find the $(x_1, x_2)$ that maximizes $u(x_1, x_2)$. Hint: Use the Lagrangian method.
(b) Show that the maximizer $x_2^*(p_1, p_2, I)$ is decreasing in $p_2$ and increasing in $I$ (Hint:
Use the partial derivatives).
(c) Does $x_2^*(p_1, p_2, I)$ change if $p_1$ changes? (Hint: partial derivatives).
