<p>For ease of notation, let's take the setting of two goods <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span>. The case is similar when there are more than two goods.</p>
<p>If utility is homogeneous, the demand functions are homogeneous of degree 1 in income, so for all <span class="math-container">$w$</span>, <span class="math-container">$x_1(p_1, p_2, w) = x_1(p_1, p_2,1) w$</span> and <span class="math-container">$x_2(p_1, p_2, w) = x_2(p_1, p_2, 1)w$</span>. Taking derivatives with respect to <span class="math-container">$w$</span> gives:
<span class="math-container">$$
\frac{\partial x_1}{\partial w}(p_1, p_2, w) = x_1(p_1, p_2,1),
$$</span>
and
<span class="math-container">$$
\frac{\partial x_2}{\partial w}(p_1, p_2, w) = x_2(p_1, p_2,1).
$$</span></p>
<p>Let <span class="math-container">$h_1(p_1,p_2, u)$</span> and <span class="math-container">$h_2(p_1, p_2,u)$</span> be the Hicksian demand functions. The Slutsky equation gives:
<span class="math-container">$$
\begin{align*}
\frac{\partial h_1}{\partial p_2}(p_1, p_2, u) &amp;= \frac{\partial x_1}{\partial p_2}(p_1, p_2, w) + \frac{\partial x_1}{\partial w}(p_1, p_2, w) x_2(p_1, p_2, w),\\
&amp;=\frac{\partial x_1}{\partial p_2}(p_1, p_2, w) + w\,\, x_1(p_1, p_2,1)x_2(p_1, p_2, 1).
\end{align*}
$$</span>
Similarly,
<span class="math-container">$$
\begin{align*}
\frac{\partial h_2}{\partial p_1}(p_1, p_2, u) &amp;= \frac{\partial x_2}{\partial p_1}(p_1, p_2, w) + \frac{\partial x_2}{\partial w}(p_1, p_2, w) x_1(p_1, p_2, w),\\
&amp;=\frac{\partial x_2}{\partial p_1}(p_1, p_2, w) + w\,\, x_1(p_1, p_2,1)x_2(p_1, p_2, 1).
\end{align*}
$$</span>
By Symmetry of the Slutsky matrix, we have <span class="math-container">$\frac{\partial h_1}{\partial p_2} = \frac{\partial h_2}{\partial p_1}$</span>. This gives:
<span class="math-container">$$
\begin{align*}
&amp;\frac{\partial x_1}{\partial p_2}(p_1, p_2, w) + w\,\, x_1(p_1, p_2,1)x_2(p_1, p_2, 1) = \frac{\partial x_2}{\partial p_1}(p_1, p_2, w) + w\,\, x_1(p_1, p_2,1)x_2(p_1, p_2, 1),\\
\leftrightarrow &amp;\frac{\partial x_1}{\partial p_2}(p_1, p_2, w) = \frac{\partial x_2}{\partial p_1}(p_1, p_2, w).
\end{align*}
$$</span></p>

CC BY-SA 4.0
2023-11-29T08:17:04.667Z
2023-11-29T08:17:04.667Z
tdm
<p>Prove that if <span class="math-container">$U(\alpha x)=\alpha U(x)$</span>, then <span class="math-container">$$\frac{\partial x_i(p,w)}{\partial p_j}=\frac{\partial x_j(p,w)}{\partial p_i}$$</span> for any <span class="math-container">$i$</span> and <span class="math-container">$j$</span>.</p>
<p>I've proved that <span class="math-container">$x(p,\alpha w)=\alpha x(p,w)$</span>, but I don't know how to handle partial derivatives.</p>

CC BY-SA 4.0
2023-11-29T07:05:35.950Z
@Giskard, it is a question from my lecturer. Can you provide a counterexample?
Paul R
CC BY-SA 4.0
2023-11-29T07:12:39.300Z
Oops, I am only now noticing that the equation is about the cross-derivatives (derivative of demand in price of the other good) not 'own-derivatives'. This is actually true.
Giskard
CC BY-SA 4.0
2023-11-29T07:15:06.240Z
What textbook do you use? You can prove this using the Envelope Theorem or Hotelling's Lemma, but unfortunately neither of these will be accessible at all if you don't know partial derivatives.
Giskard
CC BY-SA 4.0
2023-11-29T07:26:40.310Z
No actual textbook. Only lectures. Can you provide some proof?
Paul R
CC BY-SA 4.0
2023-11-29T07:55:48.777Z
Sorry, I don't want to type one in, but you could try looking it up in "Mas-Colell, Whinston Green: Microeconomic Theory*.
Giskard
CC BY-SA 4.0
2023-11-29T08:00:24.753Z
@Giskard, could you tell what section?
Paul R
<microeconomics><utility>
If utility function is homogenous of order 1, then partial derivatives of demand function are equal
Paul R
# If utility function is homogenous of order 1, then partial derivatives of demand function are equal

(Asked by: username_0 on 2023-11-28)

Prove that if $U(\alpha x)=\alpha U(x)$, then $$\frac{\partial x\_i(p,w)}{\partial p\_j}=\frac{\partial x\_j(p,w)}{\partial p\_i}$$ for any $i$ and $j$.

I've proved that $x(p,\alpha w)=\alpha x(p,w)$, but I don't know how to handle partial derivatives.

**Tags:** microeconomics, utility
> **Comments:**
> **username_0**: @Giskard, it is a question from my lecturer. Can you provide a counterexample?  
> **username_1**: Oops, I am only now noticing that the equation is about the cross-derivatives (derivative of demand in price of the other good) not 'own-derivatives'. This is actually true.  
> **username_1**: What textbook do you use? You can prove this using the Envelope Theorem or Hotelling's Lemma, but unfortunately neither of these will be accessible at all if you don't know partial derivatives.  
> **username_0**: No actual textbook. Only lectures. Can you provide some proof?  
> **username_1**: Sorry, I don't want to type one in, but you could try looking it up in "Mas-Colell, Whinston Green: Microeconomic Theory*.  
> **username_0**: @Giskard, could you tell what section?  

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## 1 answers

### Answer by username_2 on 2023-11-29. Score: 3 | Accepted answer

For ease of notation, let's take the setting of two goods $x\_1$ and $x\_2$. The case is similar when there are more than two goods.

If utility is homogeneous, the demand functions are homogeneous of degree 1 in income, so for all $w$, $x\_1(p\_1, p\_2, w) = x\_1(p\_1, p\_2,1) w$ and $x\_2(p\_1, p\_2, w) = x\_2(p\_1, p\_2, 1)w$. Taking derivatives with respect to $w$ gives:
$$
\frac{\partial x\_1}{\partial w}(p\_1, p\_2, w) = x\_1(p\_1, p\_2,1),
$$
and
$$
\frac{\partial x\_2}{\partial w}(p\_1, p\_2, w) = x\_2(p\_1, p\_2,1).
$$

Let $h\_1(p\_1,p\_2, u)$ and $h\_2(p\_1, p\_2,u)$ be the Hicksian demand functions. The Slutsky equation gives:
$$
\begin{align\*}
\frac{\partial h\_1}{\partial p\_2}(p\_1, p\_2, u) &= \frac{\partial x\_1}{\partial p\_2}(p\_1, p\_2, w) + \frac{\partial x\_1}{\partial w}(p\_1, p\_2, w) x\_2(p\_1, p\_2, w),\\
&=\frac{\partial x\_1}{\partial p\_2}(p\_1, p\_2, w) + w\,\, x\_1(p\_1, p\_2,1)x\_2(p\_1, p\_2, 1).
\end{align\*}
$$
Similarly,
$$
\begin{align\*}
\frac{\partial h\_2}{\partial p\_1}(p\_1, p\_2, u) &= \frac{\partial x\_2}{\partial p\_1}(p\_1, p\_2, w) + \frac{\partial x\_2}{\partial w}(p\_1, p\_2, w) x\_1(p\_1, p\_2, w),\\
&=\frac{\partial x\_2}{\partial p\_1}(p\_1, p\_2, w) + w\,\, x\_1(p\_1, p\_2,1)x\_2(p\_1, p\_2, 1).
\end{align\*}
$$
By Symmetry of the Slutsky matrix, we have $\frac{\partial h\_1}{\partial p\_2} = \frac{\partial h\_2}{\partial p\_1}$. This gives:
$$
\begin{align\*}
&\frac{\partial x\_1}{\partial p\_2}(p\_1, p\_2, w) + w\,\, x\_1(p\_1, p\_2,1)x\_2(p\_1, p\_2, 1) = \frac{\partial x\_2}{\partial p\_1}(p\_1, p\_2, w) + w\,\, x\_1(p\_1, p\_2,1)x\_2(p\_1, p\_2, 1),\\
\leftrightarrow &\frac{\partial x\_1}{\partial p\_2}(p\_1, p\_2, w) = \frac{\partial x\_2}{\partial p\_1}(p\_1, p\_2, w).
\end{align\*}
$$


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