<p>If you neither observe the utility level nor the expenditure (or income) level, it seems not possible to identify separately Hicksian and Marshallian demands. So it is not advisable to impose symmetry.<br />
Related questions have been answered here:</p>
<ul>
<li><a href="https://economics.stackexchange.com/questions/50316/what-is-a-simple-demand-function-that-allows-for-different-price-and-income-elas?noredirect=1&amp;lq=1">What is a simple demand function that allows for different price and income elasticities than 1 and -1?</a></li>
<li><a href="https://economics.stackexchange.com/questions/50748/relative-prices-in-demand-system-estimation">Relative prices in demand system estimation</a></li>
</ul>

CC BY-SA 4.0
2022-10-17T12:20:18.413Z
Do you think there is an option to estimate the income as the sum of product of quantities and prices? Let's say the consumer has an income $\hat{M}$ which is equal to the total expenditures he or she left in a shop per single shopping?
Athaeneus
CC BY-SA 4.0
2022-10-17T12:43:01.703Z
This is often done this way. Empirically a careful treatment of endogeneity is required in this case, as the random term in the demand function is positively correlated with the expenditures (which becomes an endogenous explanatory variable).
Bertrand
CC BY-SA 4.0
2022-10-17T13:15:04.187Z
Thank you! Do you have any idea how to treat this endogeneity in this scenario? I cannot use simultaneous equations because I cannot see anything else than bills. I cannot use panel data as well since I do not see the consumer per se.
Athaeneus
CC BY-SA 4.0
2022-10-17T13:37:45.793Z
If preferences are homothetic, the income explanatory variable, which occurs in the demand functions, vanishes from the *relative* demands. This is basically what you have proposed in the last equation of your question. You could maybe add a second step to identify the income effects using predicted expenditures from the first step. But I need more place and time to find out whether this idea is meaningful.
Bertrand
CC BY-SA 4.0
2022-10-17T08:52:54.687Z
2022-10-17T08:52:54.687Z
Bertrand
<h2>Estimation of complements/substitutes by Slutsky matrix from observable data</h2>
<p>Hello everyone,</p>
<p>I was curious about the following problem: I can observe price <span class="math-container">$P_i$</span> of <span class="math-container">$n$</span> goods and the amount of goods <strong><span class="math-container">$x$</span></strong> bought by each consumer. Sadly I cannot see their income <span class="math-container">$M$</span> (only average). Now I would like to estimate whether each consumer considers good <span class="math-container">$x_i$</span> to be a complement/substitute to <span class="math-container">$x_j$</span>.</p>
<p>I can go with regression equation: <span class="math-container">$ln(x_i) = ln(P_j) + \epsilon $</span>, but this option would lack the symmetry, since now <span class="math-container">$x_i$</span> might be a complement to <span class="math-container">$x_j$</span> but the same might not hold <em>vice versa</em>.</p>
<p>That is why I would like to use Slutsky matrix defined over Hicksian compensated demand <strong><span class="math-container">$x^H$</span></strong> since it should provide symmetric results:</p>
<p><span class="math-container">\begin{bmatrix}
    \partial x_{1}^H/\partial P_1 &amp; \partial x_{1}^H/\partial P_2 &amp; \dots  &amp; \partial x_{1}^H/\partial P_n \\
    \partial x_{2}^H/\partial P_1 &amp; \partial x_{2}^H/\partial P_2 &amp; \dots  &amp; \partial x_{2}^H/\partial P_n \\
    \vdots &amp; \vdots &amp; \ddots &amp; \vdots \\
    \partial x_{n}^H/\partial P_1 &amp; \partial x_{n}^H/\partial P_2 &amp; \dots  &amp; \partial x_{n}^H/\partial P_n
\end{bmatrix}</span></p>
<p>However, here is when I suddenly had to stop since I do not know how to estimate <span class="math-container">$\partial x_{i}^H/P_j$</span> from my data. Slutsky equation requires to use the expenditure function <span class="math-container">$E^F$</span>, which needs to be derived with the use of income <span class="math-container">$M$</span> and reservational utility <span class="math-container">$U^{(0)}$</span> which I do not have.</p>
<p>I thought of using the equation for the elasticity of substitution <strong><span class="math-container">$\sigma_{ij}$</span></strong>:</p>
<p><span class="math-container">$$\sigma_{ij} = \frac{\frac{\partial (x_j/x_i)}{x_j/x_i}}{\frac{\partial MRS_{ij}}{MRS_{ij}}}$$</span></p>
<p>because I know the First order condition and can plug it for <span class="math-container">$MRS_{ij}$</span> such that:</p>
<p><span class="math-container">$$
\hat{\sigma_{ij}} = \frac{\frac{\partial (x_j/x_i)}{x_j/x_i}}{\frac{\partial P_i/P_j}{P_i/P_j}}
$$</span></p>
<p>which would result in a regression equation:</p>
<p><span class="math-container">$$
ln \left( \frac{x_j}{x_i} \right) = \hat{\sigma_{ij}}*ln \left( \frac{P_i}{P_j} \right) + \epsilon
$$</span></p>
<p>But I still consider it to be suboptimal decision since it requires many other assumptions and it does not provide symmetrical solution for non-infinitesimal changes. Thus I would like to use Slutsky matrix instead. Is there any way?</p>
<p><strong>EDIT:</strong> I have one idea how to get the income... Do you think it would be possible to determine the income <span class="math-container">$\hat{M}$</span> as the total expenditures a consumer has taken per shopping? Because I know the prices and I know what the consumer has bought, so I could determine the <span class="math-container">$\hat{M} = \sum P_i*\hat{x_i}$</span>. Is it correct assumption?</p>
<p>Thank you very much!</p>

<microeconomics><demand><elasticity-of-substitution><micro-data><slutsky-equation>
Complements/substitutes estimation from data (Slutsky matrix)
Athaeneus
# Complements/substitutes estimation from data (Slutsky matrix)

(Asked by: username_0 on 2022-10-16)

## Estimation of complements/substitutes by Slutsky matrix from observable data

Hello everyone,

I was curious about the following problem: I can observe price $P\_i$ of $n$ goods and the amount of goods **$x$** bought by each consumer. Sadly I cannot see their income $M$ (only average). Now I would like to estimate whether each consumer considers good $x\_i$ to be a complement/substitute to $x\_j$.

I can go with regression equation: $ln(x\_i) = ln(P\_j) + \epsilon $, but this option would lack the symmetry, since now $x\_i$ might be a complement to $x\_j$ but the same might not hold *vice versa*.

That is why I would like to use Slutsky matrix defined over Hicksian compensated demand **$x^H$** since it should provide symmetric results:

\begin{bmatrix}
\partial x\_{1}^H/\partial P\_1 & \partial x\_{1}^H/\partial P\_2 & \dots & \partial x\_{1}^H/\partial P\_n \\
\partial x\_{2}^H/\partial P\_1 & \partial x\_{2}^H/\partial P\_2 & \dots & \partial x\_{2}^H/\partial P\_n \\
\vdots & \vdots & \ddots & \vdots \\
\partial x\_{n}^H/\partial P\_1 & \partial x\_{n}^H/\partial P\_2 & \dots & \partial x\_{n}^H/\partial P\_n
\end{bmatrix}

However, here is when I suddenly had to stop since I do not know how to estimate $\partial x\_{i}^H/P\_j$ from my data. Slutsky equation requires to use the expenditure function $E^F$, which needs to be derived with the use of income $M$ and reservational utility $U^{(0)}$ which I do not have.

I thought of using the equation for the elasticity of substitution **$\sigma\_{ij}$**:

$$\sigma\_{ij} = \frac{\frac{\partial (x\_j/x\_i)}{x\_j/x\_i}}{\frac{\partial MRS\_{ij}}{MRS\_{ij}}}$$

because I know the First order condition and can plug it for $MRS\_{ij}$ such that:

$$
\hat{\sigma\_{ij}} = \frac{\frac{\partial (x\_j/x\_i)}{x\_j/x\_i}}{\frac{\partial P\_i/P\_j}{P\_i/P\_j}}
$$

which would result in a regression equation:

$$
ln \left( \frac{x\_j}{x\_i} \right) = \hat{\sigma\_{ij}}\*ln \left( \frac{P\_i}{P\_j} \right) + \epsilon
$$

But I still consider it to be suboptimal decision since it requires many other assumptions and it does not provide symmetrical solution for non-infinitesimal changes. Thus I would like to use Slutsky matrix instead. Is there any way?

**EDIT:** I have one idea how to get the income... Do you think it would be possible to determine the income $\hat{M}$ as the total expenditures a consumer has taken per shopping? Because I know the prices and I know what the consumer has bought, so I could determine the $\hat{M} = \sum P\_i\*\hat{x\_i}$. Is it correct assumption?

Thank you very much!

**Tags:** microeconomics, demand, elasticity-of-substitution, micro-data, slutsky-equation

---

## 1 answers

### Answer by username_1 on 2022-10-17. Score: 1

If you neither observe the utility level nor the expenditure (or income) level, it seems not possible to identify separately Hicksian and Marshallian demands. So it is not advisable to impose symmetry.  
Related questions have been answered here:

* [What is a simple demand function that allows for different price and income elasticities than 1 and -1?](https://economics.stackexchange.com/questions/50316/what-is-a-simple-demand-function-that-allows-for-different-price-and-income-elas?noredirect=1&lq=1)
* [Relative prices in demand system estimation](https://economics.stackexchange.com/questions/50748/relative-prices-in-demand-system-estimation)

> **Comments:**
> **username_0**: Do you think there is an option to estimate the income as the sum of product of quantities and prices? Let's say the consumer has an income $\hat{M}$ which is equal to the total expenditures he or she left in a shop per single shopping?  
> **username_1**: This is often done this way. Empirically a careful treatment of endogeneity is required in this case, as the random term in the demand function is positively correlated with the expenditures (which becomes an endogenous explanatory variable).  
> **username_0**: Thank you! Do you have any idea how to treat this endogeneity in this scenario? I cannot use simultaneous equations because I cannot see anything else than bills. I cannot use panel data as well since I do not see the consumer per se.  
> **username_1**: If preferences are homothetic, the income explanatory variable, which occurs in the demand functions, vanishes from the *relative* demands. This is basically what you have proposed in the last equation of your question. You could maybe add a second step to identify the income effects using predicted expenditures from the first step. But I need more place and time to find out whether this idea is meaningful.  

---

