# Econometric estimation of Engel curves (or demand?)

The problem is the following: Data is a family survey of expenditures on commodity groups.

An exercise told I should estimate the next equation: $$w_{j}=ß_{j}+\gamma_{j}log(cons)+a_{j}z+u_{i}$$ Here, $w_{j}$ is the consuming share (expenditure share) of the j-th commodity group like food, fuel, alcohol etc. $log(cons)$ is the total consume(expenditure) in a family. $z$ is dummy, 0 if 1 child in the family and 1 if 2 child in the family. There are 6 commodity groups which cover the full range of products.

$\text{Question 1.}$ Is this an engel curve? I have no idea, because as I know demand function need to contain prices and the income.

$\text{Question 2.}$ Because of the dependent variable $w_{j}$ (which is a share $\frac{(p_{j}*x_{j})}{cons}$) the $a_{j},ß_{j},\gamma_{j}$ estimates contains the price effects. What's more the exercise mention these are somehow the function of the logarithm of Prices. Can anyone explain why?

$\text{Question 3.}$ What are the restrictions for the parameters, when the results consistent with the consumer theory? [so when: homogenous of degree is 0 in demand functions $(a_{j},ß_{j})$, from walras's law restrictions to $\gamma_{j}$, and also how to derive the Slutsky-matrix (i don't know how to get the elasticities from Engel curves).

This is my first post, so sorry for long description and thanks in advance!