<p>Let <span class="math-container">$c(p, u)$</span> be the expenditure function. The Hicksian demand for good <span class="math-container">$j$</span> is the derivative of <span class="math-container">$c$</span> with respect to <span class="math-container">$p_j$</span>.
<span class="math-container">$$
\frac{\partial c(p,u)}{\partial p_j} = h_j(p,u).
$$</span>
From this, it follows (by Young's theorem) that:
<span class="math-container">$$
\frac{\partial h_j(p,u)}{\partial p_i} = \frac{\partial^2 c(p,u)}{\partial p_j \partial p_i} = \frac{\partial^2 c(p,u)}{\partial p_i \partial p_j} = \frac{\partial h_i(p,u)}{\partial p_j},
$$</span>
So the Hicksian cross price effects are symmetric. Using the Slutsky equation, we get:
<span class="math-container">$$
\begin{align*}
&amp;\frac{\partial x_i(p,m)}{\partial p_j} + \frac{\partial x_i(p,m)}{\partial m} x_i(p,m),\\
&amp;= \frac{\partial h_i(p,u)}{\partial p_j},\\
&amp;= \frac{\partial h_j(p,u)}{\partial p_i},\\
&amp;= \frac{\partial x_j(p,m)}{\partial p_i} + \frac{\partial x_j(p,m)}{\partial m} x_j(p,m).
\end{align*}
$$</span></p>

CC BY-SA 4.0
2021-10-20T13:26:29.143Z
2021-10-20T13:26:29.143Z
tdm
<p>I don't understand how to prove slutsky matrix is symmetric for L=2
<span class="math-container">$$\frac{\partial x_1}{\partial p_2}+\frac{\partial x_1}{\partial w}\cdot x_2= \frac{\partial x_2}{\partial p_1}+\frac{\partial x_2}{\partial w}\cdot x_1$$</span></p>

<microeconomics><slutsky-equation>
How to prove Slutsky matrix's symmetry for L=2
Danjar27
# How to prove Slutsky matrix's symmetry for L=2

(Asked by: username_0 on 2021-10-18)

I don't understand how to prove slutsky matrix is symmetric for L=2
$$\frac{\partial x\_1}{\partial p\_2}+\frac{\partial x\_1}{\partial w}\cdot x\_2= \frac{\partial x\_2}{\partial p\_1}+\frac{\partial x\_2}{\partial w}\cdot x\_1$$

**Tags:** microeconomics, slutsky-equation

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## 1 answers

### Answer by username_1 on 2021-10-20. Score: 1

Let $c(p, u)$ be the expenditure function. The Hicksian demand for good $j$ is the derivative of $c$ with respect to $p\_j$.
$$
\frac{\partial c(p,u)}{\partial p\_j} = h\_j(p,u).
$$
From this, it follows (by Young's theorem) that:
$$
\frac{\partial h\_j(p,u)}{\partial p\_i} = \frac{\partial^2 c(p,u)}{\partial p\_j \partial p\_i} = \frac{\partial^2 c(p,u)}{\partial p\_i \partial p\_j} = \frac{\partial h\_i(p,u)}{\partial p\_j},
$$
So the Hicksian cross price effects are symmetric. Using the Slutsky equation, we get:
$$
\begin{align\*}
&\frac{\partial x\_i(p,m)}{\partial p\_j} + \frac{\partial x\_i(p,m)}{\partial m} x\_i(p,m),\\
&= \frac{\partial h\_i(p,u)}{\partial p\_j},\\
&= \frac{\partial h\_j(p,u)}{\partial p\_i},\\
&= \frac{\partial x\_j(p,m)}{\partial p\_i} + \frac{\partial x\_j(p,m)}{\partial m} x\_j(p,m).
\end{align\*}
$$


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