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]

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

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

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.

Rules

Rules for finding the elasticity of products and quotients are simpler than those for derivatives.^[5] Let f, g be differentiable. Then^[2]

${\displaystyle E(f(x)\cdot g(x))=Ef(x)+Eg(x)}$

${\displaystyle E{\frac {f(x)}{g(x)}}=Ef(x)-Eg(x)}$

${\displaystyle E(f(x)+g(x))={\frac {f(x)\cdot E(f(x))+g(x)\cdot E(g(x))}{f(x)+g(x)}}}$

${\displaystyle E(f(x)-g(x))={\frac {f(x)\cdot E(f(x))-g(x)\cdot E(g(x))}{f(x)-g(x)}}}$

The derivative can be expressed in terms of elasticity as

${\displaystyle Df(x)={\frac {Ef(x)\cdot f(x)}{x}}}$

Let a and b be constants. Then

${\displaystyle E(a)=0\ }$

${\displaystyle E(a\cdot f(x))=Ef(x)}$,

${\displaystyle E(bx^{a})=a\ }$.

Estimating point elasticities

In economics, the

A semi-elasticity (or semielasticity) gives the percentage change in f(x) in terms of a change (not percentage-wise) in x. Algebraically, the semi-elasticity S of a function f at point x is [7]^[8]

${\displaystyle Sf(x)={\frac {1}{f(x)}}f'(x)={\frac {d\ln f(x)}{dx}}}$

The semi-elasticity will be constant for exponential functions of the form, ${\displaystyle f(x)=C\alpha ^{x}}$ since,

${\displaystyle \ln {f}=\ln {C\alpha ^{x}}=\ln {C}+x\ln {\alpha }\implies {\frac {d\ln {f}}{dx}}=\ln {\alpha }.}$

An example of semi-elasticity is

The opposite definition is sometimes used in the literature. That is, the term "semi-elasticity" is also sometimes used for the change (not percentage-wise) in f(x) in terms of a percentage change in x^[9] which would be

${\displaystyle {\frac {df(x)}{d\ln(x)}}={\frac {df(x)}{dx}}x}$

See also

• Arc elasticity
• Elasticity (economics)
• Elasticity coefficient (biochemistry)
• Homogeneous function
• Logarithmic derivative

References