<p>Let <span class="math-container">$f(U)=U^{6/5}$</span>. This is a positive monotone transformation of <span class="math-container">$U$</span> on <span class="math-container">$\mathbb{R}_0^+$</span>. So the preferences represented by <span class="math-container">$U$</span> are also represented by <span class="math-container">$V(x_1,x_2):=f(U(x_1,x_2))=x_1^{3/5}x_2^{2/5}$</span>. The utility function <span class="math-container">$V$</span> has Cobb-Douglas form and you can use the formula for the Hicksian demand for Cobb-Douglas utilities: <span class="math-container">$$x_1^*=\left(\frac{3p_2}{2p_1}\right)^{2/5}V,\quad x_2^*=\left(\frac{2p_1}{3p_2}\right)^{3/5}V.$$</span> Substituting for <span class="math-container">$V$</span> you get <span class="math-container">$$x_1^*=\left(\frac{3p_2}{2p_1}\right)^{2/5}U^{6/5},\quad x_2^*=\left(\frac{2p_1}{3p_2}\right)^{3/5}U^{6/5},$$</span> or <span class="math-container">$$x_1^*=\left[\left(\frac{3p_2}{2p_1}\right)U^3\right]^{2/5},\quad x_2^*=\left[\left(\frac{2p_1}{3p_2}\right)U^2\right]^{3/5}.$$</span></p>

CC BY-SA 4.0
2020-06-10T15:12:00.837Z
2020-06-10T15:12:00.837Z
VARulle
<p>Consider the utility function <span class="math-container">$U(x_1,x_2) = x_1^{1/2}x_2^{1/3}$</span> and the budget line <span class="math-container">$p_1x_1+p_2x_2 = m$</span>. Then I have to find the Hicks demand function.</p>

<p>I know that to do this I have to solve the following expenditure minimization problem
<span class="math-container">$$
\min_{x_1,x_2} p_1x_1+p_2x_2
$$</span>
subject to
<span class="math-container">$$
x_1^{1/2}x_2^{1/3} = U
$$</span>
for some utility level <span class="math-container">$U$</span>. I have already found the Marshallian demand functions to be <span class="math-container">$x_1^* = \frac{3m}{5p_1}$</span> and <span class="math-container">$x^* = \frac{2m}{5p_2}$</span>.</p>

<p>I thought maybe we could use the well-known Cobb-Douglas demand functions but I don't think we can as <span class="math-container">$1/2 + 1/3 = 5/6 \neq 1$</span>.</p>

<p>I then tried to use Lagrange and got that</p>

<p>By Lagrange we have that 
<span class="math-container">$$ 
L = p_1x_1+p_2x_2 - \lambda \left( x_1^{1/2}x_2^{1/3}-U \right)
$$</span>
Thus the three FOC will be
<span class="math-container">\begin{align}
\frac{\partial L}{\partial x_1}: p_1- \lambda \left( \frac{1}{2} x_1^{-1/2}x_2^{1/3} \right)
\\
\frac{\partial L}{\partial x_2}: p_2 - \lambda \left( \frac{1}{3} x_1^{1/2}x_2^{-2/3} \right)
\\
\frac{\partial L}{\partial \lambda}: x_1^{1/2}x_2^{1/3} = U
\end{align}</span> 
Dividng the first equation with the second equation one gets
<span class="math-container">$$
\frac{3x_2}{2x_1} = \frac{p_1}{p_2}
$$</span>
Thus
<span class="math-container">$$
x_1 = \frac{3p_2x_2}{2p_1}
$$</span>
Inserting this into <span class="math-container">$U(x_1,x_2)$</span> we have that
<span class="math-container">$$
\left(\frac{3p_2x_2}{2p_1} \right)^{1/2} x_2^{1/3}
$$</span>
which gives me after some few calculations that 
<span class="math-container">$$
x_1^* = \frac{3p_2}{2p_1} \left( \frac{U\sqrt{2}\sqrt{p_1}}{ \sqrt{3} \sqrt{p_2}} \right)^{6/5}
$$</span>
<span class="math-container">$$
x_2^* = \left( \frac{ U \sqrt{2} \sqrt{p_1}}{ \sqrt{3} \sqrt{p_2} } \right)^{1/(5/6)}
$$</span>
but this doesn't seem right. Is there an easier solution to this kind of problem?</p>

<microeconomics>
Derive the Hicks demand function for $U(x_1,x_2) = x_1^{1/2}x_2^{1/3}$
Mathias
# Derive the Hicks demand function for $U(x_1,x_2) = x_1^{1/2}x_2^{1/3}$

(Asked by: username_0 on 2020-06-10)

Consider the utility function $U(x\_1,x\_2) = x\_1^{1/2}x\_2^{1/3}$ and the budget line $p\_1x\_1+p\_2x\_2 = m$. Then I have to find the Hicks demand function.

I know that to do this I have to solve the following expenditure minimization problem
$$
\min\_{x\_1,x\_2} p\_1x\_1+p\_2x\_2
$$
subject to
$$
x\_1^{1/2}x\_2^{1/3} = U
$$
for some utility level $U$. I have already found the Marshallian demand functions to be $x\_1^\* = \frac{3m}{5p\_1}$ and $x^\* = \frac{2m}{5p\_2}$.

I thought maybe we could use the well-known Cobb-Douglas demand functions but I don't think we can as $1/2 + 1/3 = 5/6 \neq 1$.

I then tried to use Lagrange and got that

By Lagrange we have that
$$
L = p\_1x\_1+p\_2x\_2 - \lambda \left( x\_1^{1/2}x\_2^{1/3}-U \right)
$$
Thus the three FOC will be
\begin{align}
\frac{\partial L}{\partial x\_1}: p\_1- \lambda \left( \frac{1}{2} x\_1^{-1/2}x\_2^{1/3} \right)
\\
\frac{\partial L}{\partial x\_2}: p\_2 - \lambda \left( \frac{1}{3} x\_1^{1/2}x\_2^{-2/3} \right)
\\
\frac{\partial L}{\partial \lambda}: x\_1^{1/2}x\_2^{1/3} = U
\end{align}
Dividng the first equation with the second equation one gets
$$
\frac{3x\_2}{2x\_1} = \frac{p\_1}{p\_2}
$$
Thus
$$
x\_1 = \frac{3p\_2x\_2}{2p\_1}
$$
Inserting this into $U(x\_1,x\_2)$ we have that
$$
\left(\frac{3p\_2x\_2}{2p\_1} \right)^{1/2} x\_2^{1/3}
$$
which gives me after some few calculations that
$$
x\_1^\* = \frac{3p\_2}{2p\_1} \left( \frac{U\sqrt{2}\sqrt{p\_1}}{ \sqrt{3} \sqrt{p\_2}} \right)^{6/5}
$$
$$
x\_2^\* = \left( \frac{ U \sqrt{2} \sqrt{p\_1}}{ \sqrt{3} \sqrt{p\_2} } \right)^{1/(5/6)}
$$
but this doesn't seem right. Is there an easier solution to this kind of problem?

**Tags:** microeconomics

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## 1 answers

### Answer by username_1 on 2020-06-10. Score: 3

Let $f(U)=U^{6/5}$. This is a positive monotone transformation of $U$ on $\mathbb{R}\_0^+$. So the preferences represented by $U$ are also represented by $V(x\_1,x\_2):=f(U(x\_1,x\_2))=x\_1^{3/5}x\_2^{2/5}$. The utility function $V$ has Cobb-Douglas form and you can use the formula for the Hicksian demand for Cobb-Douglas utilities: $$x\_1^\*=\left(\frac{3p\_2}{2p\_1}\right)^{2/5}V,\quad x\_2^\*=\left(\frac{2p\_1}{3p\_2}\right)^{3/5}V.$$ Substituting for $V$ you get $$x\_1^\*=\left(\frac{3p\_2}{2p\_1}\right)^{2/5}U^{6/5},\quad x\_2^\*=\left(\frac{2p\_1}{3p\_2}\right)^{3/5}U^{6/5},$$ or $$x\_1^\*=\left[\left(\frac{3p\_2}{2p\_1}\right)U^3\right]^{2/5},\quad x\_2^\*=\left[\left(\frac{2p\_1}{3p\_2}\right)U^2\right]^{3/5}.$$


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