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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!

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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.

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