Price elasticity of demand is defined as $\frac{\delta q}{q}\frac{p}{\delta p}$. If aggregate demand $q = q_1 + \ldots + q_n$, then $\delta q = \delta q_1 + \ldots + \delta q_n$, and the price elasticity should essentially become $\frac{\sum\delta q_i}{\sum q_i}\frac{p}{\delta p}$, and go no further right? Can this be simplified?
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Let $Q = \sum_{i} q_i$ be aggregate demand where $q_i$ is individual demand.
Then: $$ \frac{\partial Q}{\partial p} = \sum_{i} \frac{\partial q_i}{\partial p}, $$ So: $$ \frac{\partial Q}{\partial p} \frac{p}{Q} = \sum_i \frac{\partial q_i}{\partial p} \frac{p}{q_i} \frac{q_i}{Q}. $$ Now let $w_i = \frac{q_i}{Q}$ be the share of $i$'s demand in total demand then we can write: $$ \varepsilon^Q_p = \sum_i w_i \varepsilon^{q_i}_p. $$ where $\varepsilon$ is the price elasticity. This shows that the price elasticity of the aggregate demand, $\varepsilon^Q_p$, is a weighted average of the individual demand elasticities, $\varepsilon^{q_i}_p$, where the weights, $w_i$, are given by the share of the individual demands in the aggregate demand.
