The three aggregation relationships presented in this chapter can be generalized to any number of goods. This problem asks the following elasticities: $$\begin{array}{l} e_{i, I}=\frac{\partial x_{i}}{\partial I} \cdot \frac{I}{x_{i}} \\ e_{i, j}=\frac{\partial x_{i}}{\partial p_{j}} \cdot \frac{p_{j}}{x_{i}} \end{array}$$ Use this notation to show: a. Homogeneity: \(\sum_{j=1}^{n}=e_{i, j}+e_{i, l}=0\) b. Engel aggregation: \(\sum_{i=1}^{n}=s_{i} e_{i, l}=1\) c. Cournot aggregation: \(\sum_{i=1}^{n} s_{i} e_{i, j}=-s_{j}\)

To show the homogeneity property, we need to show that the sum of price elasticities (\({e_{i, j}}\)) and income elasticities (\({e_{i, l}}\)) for all goods is equal to 0, i.e., \(\sum_{j=1}^{n}(e_{i,j} + e_{i, l}) = 0\). Given that \(e_{i, l} = - \sum_{j=1}^{n} e_{i, j}\) (by the definition of the income elasticity of good i) and summing over all goods, we have: $$\sum_{j=1}^{n}(e_{i,j} + (-\sum_{j=1}^{n} e_{i, j}))$$ By simplification, we get: $$\sum_{j=1}^{n}(e_{i,j} - e_{i, j}) = 0$$ Hence, the homogeneity property is proven.

To show the Engel aggregation property, we need to show that the sum of weighted income elasticities using expenditure share (\({s_i}\)) as weight is equal to 1, i.e., \(\sum_{i=1}^{n} s_{i} e_{i, l} = 1\). We know that expenditure share (\({s_i}\)) is defined as the ratio of expenditure on good i over total expenditure: \({s_i} = \frac{p_ix_i}{\sum_{i=1}^{n} p_ix_i}\). The sum of weighted income elasticities can be represented as: $$\sum_{i=1}^{n} s_{i} e_{i, l} = \sum_{i=1}^{n} \frac{p_ix_i e_{i, l}}{\sum_{i=1}^{n} p_ix_i}$$ Taking into account that \(\sum_{i=1}^{n} p_ix_i = I\) and rearranging, we get: $$\frac{\sum_{i=1}^{n} p_ix_i e_{i, l}}{I} = 1$$ Since the sum of expenditures on all goods is equal to the income (I), this condition is satisfied, and the Engel aggregation property is proven.

To show the Cournot aggregation property, we need to show that the sum of the weighted cross-price elasticities (elasticity of good i with respect to the price of good j), using the expenditure share (\({s_i}\)) as a weight, is equal to the negative of good j's expenditure share (\({s_j}\)), i.e., \(\sum_{i=1}^{n} s_{i} e_{i, j} = -s_{j}\). The sum of the weighted cross-price elasticities can be represented as: $$\sum_{i=1}^{n} s_{i} e_{i, j} = \sum_{i=1}^{n} \frac{p_ix_i e_{i, j}}{\sum_{i=1}^{n} p_ix_i}$$ Taking into account that expenditure share (\({s_j}\)) is defined as the ratio of expenditure on good j over total expenditure: \({s_j} = \frac{p_jx_j}{\sum_{i=1}^{n} p_ix_i}\), we can rearrange the Cournot aggregation property equation as: $$\sum_{i=1}^{n} s_{i} e_{i, j} = - \frac{p_jx_j e_{j, j}}{\sum_{i=1}^{n} p_ix_i}$$ Given that a good's own price elasticity is negative (\({e_{j, j} < 0}\)) and is compensated by the negative sign on the right side of the equation, this condition is satisfied, proving the Cournot aggregation property.