Elasticity

The elasticity of y with respect to x is the percentage change in y caused by a one percent change in x. Formally, this is

e = dy/dx · (x/y).

An intuitive version of this formula replaces the derivative with changes

e = ∆y/∆x · (x/y).

Reorganizing this yields

e = (∆y/y) / (∆x/x).

The form highlights the ratio of percentage changes.

A common functional form is

log(y) = a + b · log(x).

or y = exp(a+b·log(x)). Differentiating using the chain rule yields

dy/dx = (b/x)·exp(a+b·log(x)) = (b/x)·y = b·(y/x).

The elasticity of y with respect to x is then b, which is constant.

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