# Intuition behind Engel Aggregation and Cournot Aggregation

Could anyone provide a possibly intuitive and friendly explanation to the Engel Aggregation $(\sum s_i \eta_i = 1)$ and Cournot Aggregation$(\sum s_i \epsilon_{ij} = -s_j)$? Here, $s_i = \frac{p_i,x_i}{I} \ge 0$ is the income share, $\epsilon_{ij}, \eta_i$ are price and income elasticities, respectively. I was not sure how this theory of aggregation of consumer demand tell us about the consumption behavior. Thank you in advance for your help!

These expressions can be easily derived formally by taking the derivatives of Walras' law w.r.t $I$ (Engel aggregation) and $p_j$ (Cournot aggregation). The Engel aggregation means that not all goods are luxuries ($\eta_i>1$), or necessities ($\eta_i<1$), or inferior goods ($\eta_i<0$). The Cournot aggregation says that the budget share $s_j$ of the good $j$ is small when the good $j$ has many substitutes ($\epsilon_{ij}>0$), and big when it has many complements ($\epsilon_{ij}<0$), or their respective budget shares are large.