Elasticity of a function

In mathematics, the elasticity or point elasticity of a positive differentiable function f of a positive variable (positive input, positive output)^[1] at point a is defined as^[2]

${\displaystyle Ef(a)={\frac {a}{f(a)}}f'(a)}$

${\displaystyle =\lim _{x\to a}{\frac {f(x)-f(a)}{x-a}}{\frac {a}{f(a)}}=\lim _{x\to a}{\frac {f(x)-f(a)}{f(a)}}{\frac {a}{x-a}}=\lim _{x\to a}{\frac {{\frac {f(x)}{f(a)}}-1}{{\frac {x}{a}}-1}}\approx {\frac {\%\Delta f(a)}{\%\Delta a}}}$

or equivalently

${\displaystyle Ef(x)={\frac {d\log f(x)}{d\log x}}.}$

It is thus the ratio of the relative (percentage) change in the function's output ${\displaystyle f(x)}$ with respect to the relative change in its input ${\displaystyle x}$, for infinitesimal changes from a point ${\displaystyle (a,f(a))}$. Equivalently, it is the ratio of the infinitesimal change of the logarithm of a function with respect to the infinitesimal change of the logarithm of the argument. Generalizations to multi-input–multi-output cases also exist in the literature.^[3][4]

The elasticity of a function is a constant ${\displaystyle \alpha }$ if and only if the function has the form ${\displaystyle f(x)=Cx^{\alpha }}$ for a constant ${\displaystyle C>0}$.

The elasticity at a point is the limit of the arc elasticity between two points as the separation between those two points approaches zero.

The concept of elasticity is widely used in economics and metabolic control analysis (MCA); see elasticity (economics) and elasticity coefficient respectively for details.