Elasticity (economy)

In economics , elasticity is a measure of the relative change in a dependent variable to a relative change in one of its independent variables. Not entirely correct (see “Mathematical Representation”), but the following question is clear: By what percentage does one variable change as a reaction to the one percent change in the other variable ? ${\displaystyle y}$ ${\displaystyle x}$This relative change is called the elasticity of with respect to or the elasticity of . ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle y}$

If one considers, for example, the relative change in demand with a relative change in price, this is the elasticity of demand with respect to the price or the elasticity of demand, also called price elasticity for short .

In theoretical investigations, point elasticity is usually assumed (constant changes), in practice or empirical evidence , however, often only the arch elasticity - also called stretch elasticity - is used with discrete changes (differentiation, see mathematical representation ).

motivation

The motivation for using elasticities arises from the fact that the absolute change in the dependent variable does not provide sufficient information about the structure of a reaction.

For example, consider a product whose price is increased by € 1, whereupon sales decrease by 10,000 pieces. On the basis of the absolute values, little can be seen about the scope of the change in demand. The standard of comparison is missing: Was the price at the starting point 10 or 100 €? Has sales dropped from 50,000 to 40,000 or from 1,000,000 to 990,000 pieces? On the other hand, a useful measure of the effectiveness of an instrument is its elasticity, which is based on relative changes . Since elasticity does not contain a dimension (such as “€” or “piece”), it enables similar values ​​to be compared.

Mathematical representation

An independent variable

In order to grasp this verb definition mathematically, consider a function . ${\displaystyle y=f(x)}$

Analogous to the concept of the difference quotient as an introduction to the differential quotient , the so-called arc elasticity (also called stretch elasticity ) is initially assumed. One considers a finitely small change in the variables and the variables , so that the relative changes and result. The average relative change in with respect to a relative change in gives the arch elasticity ${\displaystyle \Delta x}$ ${\displaystyle x}$ ${\displaystyle \Delta y}$ ${\displaystyle y}$ ${\displaystyle {\tfrac {\Delta x}{x}}}$ ${\displaystyle {\tfrac {\Delta y}{y}}}$ ${\displaystyle y}$ ${\displaystyle x}$

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

on. If one lets go, one obtains as an infinitesimal view the elasticity function of with respect to all , for which there is differentiable and no zero , ${\displaystyle \Delta x\rightarrow 0}$ ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle f}$ ${\displaystyle x|f(x)=y}$

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

,

which too

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

lets write. This elasticity is also referred to as point elasticity .

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

${\displaystyle \varepsilon _{y,x}={\frac {\mathrm {d} \ln y}{\mathrm {d} \ln x}}}$.

Multiple independent variables

You consider a function that depends on one or more influencing variables. An elasticity indicates the relative amount by which , ceteris paribus, the function value changes if an influencing variable changes by the relative amount . This results in the arch elasticity ${\displaystyle y=f(x_{1},x_{2},\dotsc ,x_{n})}$ ${\displaystyle x_{1},x_{2},\dotsc ,x_{n}}$ ${\displaystyle \varepsilon _{i}}$ ${\displaystyle \Delta y/y}$ ${\displaystyle y}$ ${\displaystyle \Delta x_{i}/x_{i}}$

${\displaystyle \varepsilon _{y,x_{i}}={\frac {\Delta y/y}{\Delta x_{i}/x_{i}}}}$

and with infinitesimal consideration

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

where denotes a partial derivative . Based on this, this case with several independent variables is also called partial elasticity . ${\displaystyle \partial }$

Mathematical properties of elasticity

The elasticity is dimensionless. Their range of values ​​is the set of real numbers.

• ${\displaystyle \varepsilon _{y,x}={\frac {1}{\varepsilon _{x,y}}}}$
• ${\displaystyle \varepsilon _{(y+z),x}={\frac {y\cdot \varepsilon _{y,x}+z\cdot \varepsilon _{z,x}}{y+z}}}$
• ${\displaystyle \varepsilon _{(y\cdot z),x}=\varepsilon _{y,x}+\varepsilon _{z,x}}$
• ${\displaystyle \varepsilon _{\left({\frac {y}{z}}\right),x}=\varepsilon _{y,x}-\varepsilon _{z,x}}$

Economic properties of elasticity

The elasticity is a measure of the extent to which a function is responsive to a change in the abscissa value . A negative elasticity means that the function falls within the relevant range.

The following findings can be derived with regard to elasticity:

value of ${\displaystyle \varepsilon _{y,x}}$	designation	impact
${\displaystyle \varepsilon =0}$	${\displaystyle y}$ is completely inelastic.	${\displaystyle y}$does not respond to a change in . ${\displaystyle x}$
${\displaystyle 0<|\varepsilon |<1}$	${\displaystyle y}$ is inelastic.	${\displaystyle y}$changes relatively less than . ${\displaystyle x}$
${\displaystyle |\varepsilon |=1}$	${\displaystyle y}$ is proportionally elastic.	The relative change in is equal to that in . ${\displaystyle y}$ ${\displaystyle x}$
${\displaystyle |\varepsilon |>1}$	${\displaystyle y}$ is elastic.	${\displaystyle y}$changes relatively more than . ${\displaystyle x}$
${\displaystyle |\varepsilon |\rightarrow \infty }$	${\displaystyle y}$ is completely elastic.	The relative change in is infinite, even with the smallest change in . ${\displaystyle y}$ ${\displaystyle x}$

Alternative notations

An elasticity with the value 1 is called proportionally elastic or flowing . In literature, such as B. in the widespread textbook by Varian "Grundzüge der Mikroökonomik" but also the term "unit elastic" for an elasticity with the absolute value 1. Values ​​below that are referred to as disproportionately elastic or inelastic , while values ​​above are disproportionately elastic or elastic are designated.

Peculiarities of elasticity

Special idealized cases are completely inelastic and completely elastic .

A linear function, as it is often used in economics, as a rule, like most functions, has a different elasticity at each point (exception: straight line through the origin ). Functions that have the same elasticity over their entire domain are called isoelastic functions .

Example of an isoelastic function

The elasticity function of is isoelastic because it is ${\displaystyle y={\frac {1}{x}}}$

${\displaystyle \varepsilon _{y,x}=y'\cdot {\frac {x}{y}}=-{\frac {1}{x^{2}}}\cdot {\frac {x}{1/x}}=-1}$.

${\displaystyle y={\frac {1}{x}}\,(x>0,y>0)}$could be interpreted as a model of a price-sales function . In this context one could say somewhat casually that in all areas of the price-sales function, demand falls by 1% when the price rises by 1%. Furthermore, in this case one can also say that the function is both isoelastic and unit elastic.

Another example of isoelasticity is a straight line through the origin with the elasticity . A sensible application would be a sales function in the polypolistic provider model. ${\displaystyle y=ax}$ ${\displaystyle \varepsilon =1}$

Selected elasticities

The following elasticities, among others, play a role in economics:

Elasticities related to the independent variable

• Price elasticities : What influence do price changes have on supply and demand ?
• Cross-price elasticities : What influence do price changes in one good have on supply and demand in other goods?
• Dynamic price elasticities : What influence does a current price change have on future sales?
• Income elasticities : How do changes in income affect the demand for a good?
• Sales value elasticities: What influence do marketing efforts have on the demand for a good?

In the case of price and cross-price elasticity, for example, a distinction is still made between supply and demand as dependent variables.

shortcut

	Supply as a dependent variable	Demand as a dependent variable
Price as an independent variable	(direct) price elasticity of supply : indicates how strongly the supply of a good reacts to changes in its own price.	(Direct) price elasticity of demand : indicates how strongly the demand for a good reacts to changes in its own price.
Cross price as an independent variable	Cross-price elasticity of supply : indicates how strongly the supply of a good reacts to changes in the price of a competing product.	Cross-price elasticity of demand : indicates how strongly the demand for a good reacts to changes in the price of another product.
Income as an independent variable		Income elasticity of demand : indicates how strongly the demand for a good reacts to changes in income.

The microeconomic concept of the price elasticity of demand and / or supply can be used economically not only where appropriate in-house data material is available, but can also be transferred to other independent variables than prices. Above all, trading companies with their own merchandise management system and scanner tills open up a wide range of options for analyzing success using elasticity indicators. For example, the change in demand or sales - even for a single variety - can be related as a dependent variable to independent variables such as use of advertising material, advertising intensity, change in price, change in placement, introduction of a double placement or other trade psychological measures . In principle, "the elasticity measurement is for commercial enterprises all the instruments of trade marketing and all market partners applicable service resilience, elasticity of retail space, front stretch elasticity or placement elasticity of suppliers, competitors and customers, etc. with appropriate cross elasticities."

Further economic elasticities

• Substitution elasticity indicates how “easily” one can replace one production factor (e.g. labor) with another (e.g. capital) for a given production function and output kept constant. (Compare for example the CES production function )
• Scale elasticity indicates how much the output can be increased if the input quantities are expanded.
• Tax amount elasticity measures the response of tax revenue to a change in the tax base .
• Interest elasticity indicates how an interest rate position reacts to a relative change in the interest rate.
• Production elasticity indicates approximately by how many percent the output (production) of a company or an economy changes if the use of a production factor is increased by one percent.

Examples

Example of a linear function

A straight line that does not start from the coordinate origin has a different elasticity at each point, as the following practical example shows.

The linear function is given . The elasticity at the point is to be examined, i.e. H. the percentage change in when increasing by one percent. ${\displaystyle y=f(x)=x+100}$ ${\displaystyle x=100}$ ${\displaystyle y}$ ${\displaystyle x}$

The function value belongs to . ${\displaystyle x=100}$ ${\displaystyle y=f(100)=100+100=200}$

${\displaystyle x}$is increased by 1% . So you get for . ${\displaystyle x+\Delta x=100+1}$ ${\displaystyle y=f(101)=101+100=201}$

After the 1% increase in the value increased from 200 to 201. It has increased by 1 in absolute terms, which corresponds to a percentage change of 0.5%. ${\displaystyle x}$ ${\displaystyle y}$

Using the elasticity function for a straight line , which can be specified as ${\displaystyle y=a+bx}$

${\displaystyle \varepsilon ={\frac {dy}{dx}}\cdot {\frac {x}{y}}=y'\cdot {\frac {x}{y}}=b\cdot {\frac {x}{a+bx}}}$,

would result for the example

${\displaystyle \varepsilon =b\cdot {\frac {x}{a+bx}}=1\cdot {\frac {100}{200}}=0{,}5}$,

it being noted that the elasticity function with positive slope of the line and positive absolute term with increasing increases. When it falls strictly from on of and strives with growing up. 1 ${\displaystyle a}$ ${\displaystyle x}$ ${\displaystyle a<0}$ ${\displaystyle x=-{\tfrac {a}{b}}}$ ${\displaystyle \infty }$ ${\displaystyle x}$

The elasticity is now calculated for the point that corresponds to the function value. is increased by 1%, i.e. absolutely by 2. It follows . The percentage change is there , i.e. 0.667%. ${\displaystyle x=200}$ ${\displaystyle y=f(x)=f(200)=200+100=300}$ ${\displaystyle x}$ ${\displaystyle y=f(x)=f(202)=202+100=302}$ ${\displaystyle 2/300=0{,}00667}$

The determination with the elasticity function results here

${\displaystyle \varepsilon =b\cdot {\frac {x}{a+bx}}=1\cdot {\frac {200}{300}}=0{,}667}$.

See also

• Amoroso-Robinson relation
• Sensitivity analysis

literature

• Karen Gedenk, Bernd Skiera: Marketing planning on the basis of reaction functions (I) - elasticities and sales reaction functions . 1993/94
• Hans-Otto Schenk: Psychology in Commerce . 2nd Edition. Munich / Vienna 2007, ISBN 978-3-486-58379-3 .

Individual evidence

1. ↑ Anton Frantzke: Fundamentals of Economics. Microeconomic theory and tasks of the state in a market economy . Schäffer-Poeschel, Stuttgart 1999, p. 80
2. ↑ Elasticity - definition in the Gabler Wirtschaftslexikon
3. ↑ Partial elasticities . Vienna University of Economics and Business.
4. ↑ Hans-Otto Schenk: Psychologie im Handel , 2nd edition, Munich-Vienna 2007, p. 270, ISBN 978-3-486-58379-3 .