Calculate hicksian demand with utility function (with restriction)

Question

$U(x_1, x_2) = 1/2 * x_1 $

I am trying to calculate the Hicksian demand when when $U(x_1, x_2) = 2$ and the value of the minimum expenditure when $p_1 = 9$ and $p_2 = 16$

For the hicksian demand I tried using Lagrange but it did not work out as the partial derivatives turned out to be only constants, not equations. I'm not completely sure if I should substitute with $p_1$ and $p_2$ at any point.

Am I missing something ? Is there another way to solve this kind of problems without using Lagrange and in a more straightforward manner ? Any help is appreciated.

Answer 1

The fact that you get strange results using the Lagrangian is because you have corner solutions. For this, you should use the Karush-Kuhn-Tucker conditions.

However, for your case, you can solve it much easier. Consider the expenditure minimisation problem: $$ \min_{x_1, x_2} p_1 x_1 + p_2 x_2 \text{ s.t. } U(x_1, x_2) = 2. $$ Then using your functional form we get: $$ \min_{x_1, x_2} p_1 x_1 + p_2 x_2 \text{ s.t. } \frac{x_1}{2} = 2. $$ Notice that the constraint already allows you to solve for $x_1$. As such, you can simply substitute this solution into your objective and then solve for $x_2$. No need to set up the Lagrangian.