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

CC BY-SA 4.0
2021-09-12T08:48:25.113Z
2021-09-12T08:48:25.113Z
tdm
<p><span class="math-container">$U(x_1, x_2) =  1/2 * x_1 $</span></p>
<p>I am trying to calculate the Hicksian demand when when <span class="math-container">$U(x_1, x_2) = 2$</span> and the value of the minimum expenditure when <span class="math-container">$p_1 = 9$</span> and <span class="math-container">$p_2 = 16$</span></p>
<p>For the hicksian demand I tried using Lagrange but it did not work out as the partial derivatives turned out to be only constants, not equations. I'm not completely sure if I should substitute with <span class="math-container">$p_1$</span> and <span class="math-container">$p_2$</span> at any point.</p>
<p>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.</p>

<microeconomics><utility><demand><lagrangian>
Calculate hicksian demand with utility function (with restriction)
Coco Garazzo
# Calculate hicksian demand with utility function (with restriction)

(Asked by: username_0 on 2021-09-09)

$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.

**Tags:** microeconomics, utility, demand, lagrangian

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## 1 answers

### Answer by username_1 on 2021-09-12. Score: 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.


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