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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!

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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:

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  • $\begingroup$ 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? $\endgroup$
    – Athaeneus
    Commented Oct 17, 2022 at 12:20
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    $\begingroup$ 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). $\endgroup$
    – Bertrand
    Commented Oct 17, 2022 at 12:43
  • $\begingroup$ 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. $\endgroup$
    – Athaeneus
    Commented Oct 17, 2022 at 13:15
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    $\begingroup$ 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. $\endgroup$
    – Bertrand
    Commented Oct 17, 2022 at 13:37

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