# Elasticity (economics)

In economics, elasticity is the ratio of the proportional change in one variable with respect to proportional change in another variable. Price elasticity, for example, is the sensitivity of quantity demanded or supplied to changes in prices. Elasticity is usually expressed as a negative number but shown as a positive percentage value.

## Applications

One typical application of the concept of elasticity is to consider what happens to consumer demand for a good (for example, apples) when prices increase. As the price of a good rises, consumers will usually demand a lower quantity of that good, perhaps by consuming less, substituting other goods, and so on. The greater the extent to which demand falls as price rises, the greater the price elasticity of demand. However, there may be some goods that consumers require, cannot consume less of, and cannot find substitutes for even if prices rise (for example, certain prescription drugs). Another example is oil and its derivatives such as gasoline. For such goods, the price elasticity of demand might be considered inelastic.

Further, elasticity will normally be different in the short term and the long term. For example, for many goods the supply can be increased over time by locating alternative sources, investing in an expansion of production capacity, or developing competitive products which can substitute. One might therefore expect that the price elasticity of supply will be greater in the long term than the short term for such a good, that is, that supply can adjust to price changes to a greater degree over a longer time.

This applies to the demand side as well. For example, if the price of petrol rises, consumers will find ways to conserve their use of the resource. However, some of these ways, like finding a more fuel-efficient car, take time. So consumers as well may be less able to adapt to price shocks in the short term than in the long term.

The concept of elasticity has an extraordinarily wide range of applications in economics. In particular, an understanding of elasticity is useful to understand the dynamic response of supply and demand in a market, in order to achieve an intended result or avoid unintended results. For example, a business considering a price increase might find that doing so lowers profits if demand is highly elastic, as sales would fall sharply. Similarly, a business considering a price cut might find that it does not increase sales, if demand for the product is price inelastic.

An example of how elasticity can be useful in business situations can be shown by the equation MR = P * (1+E)/E, where MR is marginal revenue, P is price of the good, and E is the own price elasticity of demand for the good. Notice that when E is less than negative one, demand is elastic. When E is between negative one and zero, demand is inelastic. And at E=-1, demand is unit elastic (or unitary elastic), and thus MC=MB and MNB=0.

## Mathematical definition

In economics, the definition of elasticity is based on the mathematical notion of point elasticity. For example, it applies to price elasticity of demand and price elasticity of supply, in which case the functions of the interest are $Q^{d}(P)$ and $Q^{s}(P)$ . When working with graphs, it is common to put Quantity on x-axis and Price on y-axis, thus the function of the interest is x(y) rather than commonly used in mathematics y(x).

In general, the "y-elasticity of x" is:

$E_{x,y}=\left|{\frac {\partial \ln x}{\partial \ln y}}\right|=\left|{\frac {\partial x}{\partial y}}\cdot {\frac {y}{x}}\right|$ .

or, in terms of percentage change

$E_{x,y}\simeq \left|{\frac {{\rm {percent\ change\ in}}\ x}{{\rm {percent\ change\ in}}\ y}}\right|,$ The "y-elasticity of x" is also called "the elasticity of x with respect to y".

It is typical to represent elasticity as 'E', 'e' or lowercase epsilon, 'ε'.

## Examples

File:Unit elasticity.svg
Unit elasticity for a supply line passing through the origin.

A common mistake for students and teachers of economics is to confuse elasticity with slope. (Case & Fair, 1999: 108, 109). Elasticity is the slope of a curve on a loglog graph only, not on a regular graph (taking into account whether the independent variable is on the horizontal or the vertical axis). Consider the information in the figure. This is a special case which illustrates that slope and elasticity are different. In the above example the slope of S1 is clearly different from the slope of S2, but since the rate of change of P relative to Q is always proportionate, both S1 and S2 are unit elastic (i.e. E = 1).

(Keeping in mind the example of price elasticity of demand, these figures show x = Q horizontal and y = P vertical).

Illustrations of perfect elasticity and perfect inelasticity.

 File:Elasticity-elastic.png The demand curve (D1) is perfectly ("infinitely") elastic. File:Elasticity-inelastic.png The demand curve (D2) is perfectly inelastic.

## Importance

Elasticity is an important concept in understanding the incidence of indirect taxation, marginal concepts as they relate to the theory of the firm, distribution of wealth and different types of goods as they relate to the theory of consumer choice and the Lagrange multiplier. Elasticity is also crucially important in any discussion of welfare distribution, in particular consumer surplus, producer surplus, or government surplus. The concept of elasticity was also an important component of the Singer-Prebisch thesis which is a central argument in dependency theory as it relates to development economics.

An elasticity, defined as a ratio of proportional or percent changes, is necessarily dimensionless -- meaning that it is independent of units of measurement. For example, the value of the price elasticity of demand for gasoline would be the same whether prices were measured in dollars or francs, or quantities in tonnes or gallons. This unit-independence is the main reason why elasticity is so popular a measure of the responsiveness of economic behaviour. 