Price elasticity of demand

Summary

\(q\left( p \right)\) is a given demand function ( market demand \(X^l\) or individual demand \(x_i^l\) ) where \(p\) is the price of the good. \(q\) will depend on other variables as well .

\[ε= \frac{dq}{dp}⋅ \frac{p}{q\left( p \right)}\]

is called the price elasticity of demand or the elasticity of demand with respect to price.
Example ( \(c\) is an arbitrary constant)

\[q\left( p \right)=c/p \]

\[ \frac{dq}{dp}=- \frac{c}{p^2}\]

\[ε=- \frac{c}{p^2}⋅ \frac{p}{c/p}=-1\]

\[ε≈ \frac{Δq}{Δp}⋅ \frac{p}{q}= \frac{Δq/q}{Δp/p}\]

\(Δq/q⋅100\) is the percentage change in \(q\) .
\(ε\) is the approximate percentage increase in \(q\) when \(p\) increases by 1%