# Complements/substitutes estimation from data (Slutsky matrix)

## 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!

• 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? Commented Oct 17, 2022 at 12:20