Elasticity (economics)

In economics, an elasticity is a measure that indicates the relative change of a dependent variable to a relative change of one of its independent variables. Not quite correct (see "Mathematical Representation"), but illustrative is the following question: By how much percent does a variable change in response to a one percent change in the other variable ? This relative change is called the elasticity of with respect to or the -elasticity of .

For example, if we look at the relative change in demand given a relative change in price, this is the elasticity of demand with respect to price, or the price elasticity of demand, also known as price elasticity for short.

In theoretical studies, point elasticity is usually assumed (continuous changes), whereas in practice or empirical studies, only arc elasticity - also called distance elasticity - with discrete changes is often used (distinction, see Mathematical representation).

Motivation

The motivation for using elasticities stems from the fact that the absolute change in the dependent variable provides insufficient information about the structure of a response.

For example, consider a product whose price is increased by €1, whereupon sales fall by 10,000 units. The absolute values reveal little about the scope of the change in demand. The benchmark is missing: Was the price at the starting point €10 or €100? Did sales drop from 50,000 to 40,000 or from 1,000,000 to 990,000? In contrast, a useful measure of the impact of an instrument is elasticity, which assumes relative changes. Since elasticity does not contain a dimension (such as "€" or "units"), it enables the comparability of similar values.

Mathematical representation

An independent variable

To grasp this verbal definition mathematically, consider a function y=f(x) .

Analogous to the concept of the difference quotient as a lead-in to the differential quotient, the so-called arc elasticity (also called distance elasticity) is first assumed. Consider a finitely small change Δ in $\Delta x$ the variable and Δ $\Delta y$ the variable , resulting in the relative changes Δ ${\tfrac {\Delta x}{x}}$ and Δ ${\tfrac {\Delta y}{y}}$ . The average relative change of respect to a relative change of gives the arc elasticity

$\varepsilon _{y,x}:={\frac {\frac {\Delta y}{y}}{\frac {\Delta x}{x}}}$

an. Letting Δ $\Delta x\rightarrow 0$ go, one obtains as an infinitesimal notion the elasticity function of with respect to all , for which is differentiable and not a $x|f(x)=y$ zero,

$\varepsilon _{y,x}:={\frac {\frac {\mathrm {d} y}{y}}{\frac {\mathrm {d} x}{x}}}$

which also

$\varepsilon _{y,x}:={\frac {\mathrm {d} y}{\mathrm {d} x}}\cdot {\frac {x}{y}}=y'\cdot {\frac {x}{y}}$

can be written. This elasticity is also called point elasticity.

It can also be shown that elasticity can also be represented as

$\varepsilon_{y,x} = \frac{\mathrm d \ln y}{\mathrm d \ln x}$ .

Multiple independent variables

Consider a function $y=f(x_{1},x_{2},\dotsc ,x_{n})$ , which $x_1, x_2, \dotsc, x_n$ depends on one or more influence variables An elasticity ε $\varepsilon_i$ specifies the relative amount Δ $\Delta y/y$ by which, ceteris paribus, the function value changes when an influence quantity changes by the relative amount Δ . $\Delta x_{i}/x_{i}$ This results in the following for the arc elasticity

$\varepsilon_{y,x_i} = \frac{\Delta y/y}{\Delta x_i / x_i}$

and with infinitesimal consideration

$\varepsilon _{y,x_{i}}=\lim _{\Delta x_{i}\rightarrow 0}{\frac {\Delta y/y}{\Delta x_{i}/x_{i}}}={\frac {\partial y/y}{\partial x_{i}/x_{i}}}={\frac {x_{i}}{y}}{\frac {\partial y}{\partial x_{i}}}$ ,

where ∂ $\partial$ denotes a partial derivative. Following this, this case with multiple independent variables is also called partial elasticity.