# Calculate hicksian demand with utility function (with restriction)

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

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.

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.