<p>The necessary and sufficient conditions on a demand system are homogeneity of degree 0, Slutsky symmetry and Slutsky negativity.</p>
<p>From a theoretical point of view, symmetry of the Slutsky matrix is due to the fact that the cross partial derivatives of the expenditure function are equal (Young's theorem). If <span class="math-container">$e(p,u)$</span> is the expenditure function, then the Hicksian demand for good <span class="math-container">$i$</span> is the derivative of <span class="math-container">$e(p,u)$</span> with respect to the price of good <span class="math-container">$i$</span>, say <span class="math-container">$p_i$</span>. So:
<span class="math-container">$$
\frac{\partial h_i(p,u)}{\partial p_j} = \frac{\partial^2 e(p,u)}{\partial p_j \partial p_i} = \frac{\partial^2 e(p,u)}{\partial p_i \partial p_j} = \frac{\partial h_j(p,u)}{\partial p_i}.
$$</span>
So symmetry gives the basic integrability condition.</p>
<p>Negativity of the Slutsky matrix, is equivalent to the requirement that the expenditure function is concave. As such we get the intuitive connections:</p>
<ul>
<li>symmetry <span class="math-container">$\sim$</span> integrability condition</li>
<li>negativity <span class="math-container">$\sim$</span> concavity condition</li>
</ul>
<p>Here I use the symbol <span class="math-container">$\sim$</span> to say that they are related (although) not necessarily one to one from a pure mathematical perspective.</p>
<p>There also seems also te be a strong connnection between the various revealed preference axioms and the Slutsky conditions. In particular, WARP (the weak axiom of revealed preference) appears to be tightly connected to the negativity of the Slutsky matrix (see <a href="https://doi.org/10.2307/2548836" rel="nofollow noreferrer">(Samuelson,1938)</a> for an exposition).</p>
<p>For example let <span class="math-container">$\widetilde{x}$</span> be the optimal bundle at price <span class="math-container">$p$</span>, so <span class="math-container">$\widetilde{x} = x(p, p \cdot \widetilde{x})$</span> where <span class="math-container">$x(p,m)$</span> is the Marshallian demand at prices <span class="math-container">$p$</span> and income <span class="math-container">$m$</span>. Consider a compensated price change to <span class="math-container">$\widehat{p}$</span> giving rise to a new demand <span class="math-container">$\widehat{x} = x(\widehat{p}, \widehat{p} \cdot \widetilde{x})$</span> (so the new budget can still afford the bundle <span class="math-container">$\widetilde{x}$</span>. Then,
<span class="math-container">$$
\widehat{p} \cdot \widehat{x} = \widehat{p} \cdot \widetilde{x}
$$</span>
and WARP requires that:
<span class="math-container">$$
p \cdot \widetilde{x} &lt; p \cdot \widehat{x}.
$$</span>
Then adding up and rewriting gives
<span class="math-container">$$
(\widehat{p} - p) \cdot (\widehat{x} - \widetilde{x}) &lt; 0.
$$</span>
Now if we specify <span class="math-container">$\widehat{p} = p + t v$</span> for some vector <span class="math-container">$v$</span>, we get:
<span class="math-container">$$
t v \cdot (x(p + v t, (p+vt)\cdot \widetilde{x}) - x(p, p \cdot \widetilde{x}) &lt; 0.
$$</span>
dividing by <span class="math-container">$t^2$</span> and taking the limit for <span class="math-container">$t \to 0$</span> gives,
<span class="math-container">$$
v \cdot S \cdot v &lt; 0.
$$</span>
where <span class="math-container">$S$</span> is the Slutsky matrix. As <span class="math-container">$v$</span> can be any vector, this means that <span class="math-container">$S$</span> needs to be negative definite.</p>
<p>Now, given that WARP does not exhaust all testable implications on a demand function, the remaining part should be related to the symmetry of the Slutsky matrix. In particular, Slutsky symmetry seems to be tightly connected to transitivity of the (revealed) preference relation. I refer to <a href="https://doi.org/10.2307/1910408" rel="nofollow noreferrer">Hurwicz &amp; Richter (1979)</a> who link symmetry to the so called Ville axiom as the argument is not that straightforward.</p>
<p>Anyway, if we summarize it seems that:</p>
<ul>
<li>WARP <span class="math-container">$\sim$</span> negativity of Slutsky matrix</li>
<li>'transitivity´ of the (revealed) preferences <span class="math-container">$\sim$</span> symmetry of the Slutsky matrix</li>
</ul>
<p>Of course this is not a theoretical one to one mapping, but it should give some behavioural intuition behind the two Slutsky requirements.</p>
<p>Together WARP and transitivity give rise to the Strong Axiom of Revealed Preference (SARP) (see <a href="https://doi.org/10.2307/2549382" rel="nofollow noreferrer">Houthakker,1950</a>). It is known that this exhausts all testable implications on a (homogeneous) demand functions (to be consistent with utility maximisation).</p>
<p>Interesting note: it is known that if there are only two goods then WARP is equivalent to SARP (see <a href="https://doi.org/10.2307/2296210" rel="nofollow noreferrer">Rose,1958</a>). It is also known that in the two goods case, Slutsky symmetry is always satisfied (follows from Engel aggregation and homogeneity of degree 0). So WARP (or  negativity of the Slutsky matrix) is sufficient in the 2 goods case.</p>

CC BY-SA 4.0
2023-09-06T07:16:58.670Z
2023-09-06T07:16:58.670Z
tdm
<p>In demand system estimation, theoretically we require this matrix to be symmetric. This unfortunately is not the case most of the time.</p>
<p>What are some reasons for why symmetry of the slutsky matrix may fail empirically?</p>

CC BY-SA 4.0
2023-09-05T17:36:20.320Z
Might be relevant: https://economics.brown.edu/sites/default/files/papers/Bravo%20Working%20Paper%202022_004.pdf
Michael Greinecker
CC BY-SA 4.0
2023-09-06T04:41:00.647Z
@MichaelGreinecker thank you for the resource! I am a bit iffy on the examples provided but the idea of price paths is something new to me. Very interesting.
EconJohn
<microeconomics><consumer-theory><demand-system-estimation>
Reasons for why slutsky matrix may be non symmetric
EconJohn
# Reasons for why slutsky matrix may be non symmetric

(Asked by: username_0 on 2023-09-05)

In demand system estimation, theoretically we require this matrix to be symmetric. This unfortunately is not the case most of the time.

What are some reasons for why symmetry of the slutsky matrix may fail empirically?

**Tags:** microeconomics, consumer-theory, demand-system-estimation
> **Comments:**
> **username_1**: Might be relevant: https://economics.brown.edu/sites/default/files/papers/Bravo%20Working%20Paper%202022_004.pdf  
> **username_0**: @MichaelGreinecker thank you for the resource! I am a bit iffy on the examples provided but the idea of price paths is something new to me. Very interesting.  

---

## 1 answers

### Answer by username_2 on 2023-09-06. Score: 3

The necessary and sufficient conditions on a demand system are homogeneity of degree 0, Slutsky symmetry and Slutsky negativity.

From a theoretical point of view, symmetry of the Slutsky matrix is due to the fact that the cross partial derivatives of the expenditure function are equal (Young's theorem). If $e(p,u)$ is the expenditure function, then the Hicksian demand for good $i$ is the derivative of $e(p,u)$ with respect to the price of good $i$, say $p\_i$. So:
$$
\frac{\partial h\_i(p,u)}{\partial p\_j} = \frac{\partial^2 e(p,u)}{\partial p\_j \partial p\_i} = \frac{\partial^2 e(p,u)}{\partial p\_i \partial p\_j} = \frac{\partial h\_j(p,u)}{\partial p\_i}.
$$
So symmetry gives the basic integrability condition.

Negativity of the Slutsky matrix, is equivalent to the requirement that the expenditure function is concave. As such we get the intuitive connections:

* symmetry $\sim$ integrability condition
* negativity $\sim$ concavity condition

Here I use the symbol $\sim$ to say that they are related (although) not necessarily one to one from a pure mathematical perspective.

There also seems also te be a strong connnection between the various revealed preference axioms and the Slutsky conditions. In particular, WARP (the weak axiom of revealed preference) appears to be tightly connected to the negativity of the Slutsky matrix (see [(Samuelson,1938)](https://doi.org/10.2307/2548836) for an exposition).

For example let $\widetilde{x}$ be the optimal bundle at price $p$, so $\widetilde{x} = x(p, p \cdot \widetilde{x})$ where $x(p,m)$ is the Marshallian demand at prices $p$ and income $m$. Consider a compensated price change to $\widehat{p}$ giving rise to a new demand $\widehat{x} = x(\widehat{p}, \widehat{p} \cdot \widetilde{x})$ (so the new budget can still afford the bundle $\widetilde{x}$. Then,
$$
\widehat{p} \cdot \widehat{x} = \widehat{p} \cdot \widetilde{x}
$$
and WARP requires that:
$$
p \cdot \widetilde{x} < p \cdot \widehat{x}.
$$
Then adding up and rewriting gives
$$
(\widehat{p} - p) \cdot (\widehat{x} - \widetilde{x}) < 0.
$$
Now if we specify $\widehat{p} = p + t v$ for some vector $v$, we get:
$$
t v \cdot (x(p + v t, (p+vt)\cdot \widetilde{x}) - x(p, p \cdot \widetilde{x}) < 0.
$$
dividing by $t^2$ and taking the limit for $t \to 0$ gives,
$$
v \cdot S \cdot v < 0.
$$
where $S$ is the Slutsky matrix. As $v$ can be any vector, this means that $S$ needs to be negative definite.

Now, given that WARP does not exhaust all testable implications on a demand function, the remaining part should be related to the symmetry of the Slutsky matrix. In particular, Slutsky symmetry seems to be tightly connected to transitivity of the (revealed) preference relation. I refer to [Hurwicz & Richter (1979)](https://doi.org/10.2307/1910408) who link symmetry to the so called Ville axiom as the argument is not that straightforward.

Anyway, if we summarize it seems that:

* WARP $\sim$ negativity of Slutsky matrix
* 'transitivity´ of the (revealed) preferences $\sim$ symmetry of the Slutsky matrix

Of course this is not a theoretical one to one mapping, but it should give some behavioural intuition behind the two Slutsky requirements.

Together WARP and transitivity give rise to the Strong Axiom of Revealed Preference (SARP) (see [Houthakker,1950](https://doi.org/10.2307/2549382)). It is known that this exhausts all testable implications on a (homogeneous) demand functions (to be consistent with utility maximisation).

Interesting note: it is known that if there are only two goods then WARP is equivalent to SARP (see [Rose,1958](https://doi.org/10.2307/2296210)). It is also known that in the two goods case, Slutsky symmetry is always satisfied (follows from Engel aggregation and homogeneity of degree 0). So WARP (or negativity of the Slutsky matrix) is sufficient in the 2 goods case.


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