mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the elasticity or point elasticity of a positive

differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...

''f'' of a positive variable (positive input, positive output) at point ''a'' is defined as :

Ef(a) = \fracf'(a)

=\lim_\frac\frac=\lim_\frac\frac=\lim_\frac\approx \frac

or equivalently :

Ef(x) = \frac.

It is thus the ratio of the relative (percentage) change in the function's output

f(x)

with respect to the relative change in its input

x

, for infinitesimal changes from a point

(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. The elasticity of a function is a constant

\alpha

if and only if the function has the form

f(x) = C x ^ \alpha

for a constant

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 Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ...

and

metabolic control analysis In biochemistry, metabolic control analysis (MCA) is a mathematical framework for describing Metabolic pathway, metabolic, Cell signaling#Signaling pathways, signaling, and genetic pathways. MCA quantifies how variables, such as fluxes and Chemi ...

(MCA); see

elasticity (economics) In economics, elasticity measures the responsiveness of one economic variable to a change in another. For example, if the price elasticity of the demand of a good is −2, then a 10% increase in price will cause the quantity demanded to fall by 2 ...

and

elasticity coefficient In chemistry, the Reaction rate, rate of a chemical reaction is influenced by many different factors, such as temperature, pH, reactant, the concentration of Product (chemistry), products, and other effectors. The degree to which these factors c ...

respectively for details.

Rules

Rules for finding the elasticity of products and quotients are simpler than those for derivatives. Let ''f, g'' be differentiable. Then :

E ( f(x) \cdot g(x) ) = E f(x) + E g(x)

E \frac = E f(x) - E g(x)

E ( f(x) + g(x) ) = \frac

E ( f(x) - g(x) ) = \frac

The derivative can be expressed in terms of elasticity as :

D f(x) = \frac

Let ''a'' and ''b'' be constants. Then :

E ( a ) = 0 \

E ( a \cdot f(x) ) = E f(x)

, :

E (b x^a) = a \

Estimating point elasticities

In economics, the

price elasticity of demand A good's price elasticity of demand (E_d, PED) is a measure of how sensitive the quantity demanded is to its price. When the price rises, quantity demanded falls for almost any good ( law of demand), but it falls more for some than for others. Th ...

refers to the elasticity of a

demand function In economics, an inverse demand function is the mathematical relationship that expresses price as a function of quantity demanded (it is therefore also known as a price function). Historically, the economists first expressed the price of a good a ...

''Q''(''P''), and can be expressed as (dQ/dP)/(Q(P)/P) or the ratio of the value of the marginal function (dQ/dP) to the value of the average function (Q(P)/P). This relationship provides an easy way of determining whether a demand curve is elastic or inelastic at a particular point. First, suppose one follows the usual convention in mathematics of plotting the independent variable (P) horizontally and the dependent variable (Q) vertically. Then the slope of a line tangent to the curve at that point is the value of the marginal function at that point. The slope of a ray drawn from the origin through the point is the value of the average function. If the absolute value of the slope of the tangent is greater than the slope of the ray then the function is elastic at the point; if the slope of the secant is greater than the absolute value of the slope of the tangent then the curve is inelastic at the point. If the tangent line is extended to the horizontal axis the problem is simply a matter of comparing angles created by the lines and the horizontal axis. If the marginal angle is greater than the average angle then the function is elastic at the point; if the marginal angle is less than the average angle then the function is inelastic at that point. If, however, one follows the convention adopted by economists and plots the independent variable ''P'' on the vertical axis and the dependent variable ''Q'' on the horizontal axis, then the opposite rules would apply. The same graphical procedure can also be applied to a supply function or other functions.

Semi-elasticity

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 :

Sf(x) = \fracf'(x) = \frac

The semi-elasticity will be constant for exponential functions of the form,

f(x) = C \alpha^x

since, :

\ln = \ln = \ln + x \ln \implies \frac = \ln.

An example of semi-elasticity is modified duration in bond trading. 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'' which would be :

\frac=\fracx

Rules

Estimating point elasticities

Semi-elasticity

See also

References

Further reading