elasticity of demand (PED or Ed) is a measure used in economics to show the responsiveness, or elasticity, of the

quantity demanded of a good or service to a change in its price. More precisely, it gives the percentage change in

quantity demanded in response to a one percent change in price (ceteris paribus, i.e. holding constant all the other

determinants of demand, such as income). It was devised by Alfred Marshall.

Price elasticities are almost always negative, although analysts tend to ignore the sign even though this can lead to

ambiguity. Only goods which do not conform to the law of demand, such as Veblen and Giffen goods, have a positive

PED. In general, the demand for a good is said to be inelastic (or relatively inelastic) when the PED is less than one

(in absolute value): that is, changes in price have a relatively small effect on the quantity of the good demanded. The

demand for a good is said to be elastic (or relatively elastic) when its PED is greater than one (in absolute value): that

is, changes in price have a relatively large effect on the quantity of a good demanded.

Revenue is maximized when price is set so that the PED is exactly one. The PED of a good can also be used to

predict the incidence (or "burden") of a tax on that good. Various research methods are used to determine price

elasticity, including test markets, analysis of historical sales data and conjoint analysis.

1 Definition

o 1.1 Point-price elasticity

o 1.2 Arc elasticity

2 History

3 Determinants

4 Interpreting values of price elasticity coefficients

5 Effect on total revenue

6 Effect on tax incidence

7 Optimal pricing

o 7.1 Constant elasticity and optimal pricing

o 7.2 Non-constant elasticity and optimal pricing

o 7.3 Limitations of revenue-maximizing and profit-maximizing pricing strategies

8 Selected price elasticities

9 See also

10 Notes

11 References

12 External links

Definition[edit]

It is a measure of responsiveness of the quantity of a raw good or service demanded to changes in its price.[1] The

formula for the coefficient of price elasticity of demand for a good is:[2][3][4]

The above formula usually yields a negative value, due to the inverse nature of the relationship between price and

quantity demanded, as described by the "law of demand".[3] For example, if the price increases by 5% and

quantity demanded decreases by 5%, then the elasticity at the initial price and quantity = −5%/5% = −1. The only

classes of goods which have a PED of greater than 0 are Veblen and Giffen goods.[5]Because the PED is negative

for the vast majority of goods and services, however, economists often refer to price elasticity of demand as a

positive value (i.e., in absolute value terms).[4]

This measure of elasticity is sometimes referred to as the own-price elasticity of demand for a good, i.e., the

elasticity of demand with respect to the good's own price, in order to distinguish it from the elasticity of demand for

that good with respect to the change in the price of some other good, i.e., a complementary or substitute good.[1] The latter type of elasticity measure is called a cross -price elasticity of demand .[6][7]

As the difference between the two prices or quantities increases, the accuracy of the PED given by the formula

above decreases for a combination of two reasons. First, the PED for a good is not necessarily constant; as

explained below, PED can vary at different points along the demand curve, due to its percentage nature.[8]

[9] Elasticity is not the same thing as the slope of the demand curve, which is dependent on the units used for both

price and quantity.[10][11] Second, percentage changes are not symmetric; instead, the percentage change between

any two values depends on which one is chosen as the starting value and which as the ending value. For

example, if quantity demanded increases from 10 units to 15 units, the percentage change is 50%, i.e., (15 − 10)

÷ 10 (converted to a percentage). But if quantity demanded decreases from 15 units to 10 units, the percentage

change is −33.3%, i.e., (10 − 15) ÷ 15.[12][13]

Two alternative elasticity measures avoid or minimise these shortcomings of the basic elasticity formula: point-

price elasticity and arc elasticity.

Point-price elasticity[edit]

One way to avoid the accuracy problem described above is to minimise the difference between the starting and

ending prices and quantities. This is the approach taken in the definition of point-price elasticity, which

uses differential calculus to calculate the elasticity for an infinitesimal change in price and quantity at any given

point on the demand curve: [14]

In other words, it is equal to the absolute value of the first derivative of quantity with respect to price (dQd/dP)

multiplied by the point's price (P) divided by its quantity (Qd).[15]

In terms of partial-differential calculus, point-price elasticity of demand can be defined as follows:

[16] let   be the demand of goods   as a function of parameters price and wealth,

and let  be the demand for good  . The elasticity of demand for good   with respect to

price   is

However, the point-price elasticity can be computed only if the formula for the demand

function,  , is known so its derivative with respect to price,  , can be determined.

Arc elasticity[edit]

A second solution to the asymmetry problem of having a PED dependent on which of the two given

points on a demand curve is chosen as the "original" point and which as the "new" one is to compute the

percentage change in P and Q relative to the average of the two prices and the average of the two

quantities, rather than just the change relative to one point or the other. Loosely speaking, this gives an

"average" elasticity for the section of the actual demand curve—i.e., the arc of the curve—between the

two points. As a result, this measure is known as the arc elasticity, in this case with respect to the price of

the good. The arc elasticity is defined mathematically as:[13][17][18]

This method for computing the price elasticity is also known as the "midpoints formula", because the average price

and average quantity are the coordinates of the midpoint of the straight line between the two given points.[12]

[18] However, because this formula implicitly assumes the section of the demand curve between those points is linear,

the greater the curvature of the actual demand curve is over that range, the worse this approximation of its elasticity

will be.[17][19]

History[edit]

The illustration that accompanied Marshall's original definition of PED, the ratio of PT to Pt

Together with the concept of an economic "elasticity" coefficient, Alfred Marshall is credited with defining PED

("elasticity of demand") in his book Principles of Economics, published in 1890.[20] He described it thus: "And we may

say generally:— the elasticity (or responsiveness) of demand in a market is great or small according as the amount

demanded increases much or little for a given fall in price, and diminishes much or little for a given rise in price".[21] He

reasons this since "the only universal law as to a person's desire for a commodity is that it diminishes... but this

diminution may be slow or rapid. If it is slow... a small fall in price will cause a comparatively large increase in his

purchases. But if it is rapid, a small fall in price will cause only a very small increase in his purchases. In the former

case... the elasticity of his wants, we may say, is great. In the latter case... the elasticity of his demand is

small."[22] Mathematically, the Marshallian PED was based on a point-price definition, using differential calculus to

calculate elasticities.[23]

Determinants[edit]

The overriding factor in determining PED is the willingness and ability of consumers after a price change to postpone

immediate consumption decisions concerning the good and to search for substitutes ("wait and look").[24] A number of

factors can thus affect the elasticity of demand for a good:[25]

Availability of substitute goods: the more and closer the substitutes available, the higher the elasticity is likely to be,

as people can easily switch from one good to another if an even minor price change is made;[25][26][27] There is a strong

substitution effect.[28] If no close substitutes are available, the substitution effect will be small and the demand inelastic.[28]

Breadth of definition of a good: the broader the definition of a good (or service), the lower the elasticity. For

example, Company X's fish and chips would tend to have a relatively high elasticity of demand if a significant number

of substitutes are available, whereas food in general would have an extremely low elasticity of demand because no

substitutes exist.[29]

Percentage of income: the higher the percentage of the consumer's income that the product's price represents, the

higher the elasticity tends to be, as people will pay more attention when purchasing the good because of its cost;[25]

[26] The income effect is substantial.[30] When the goods represent only a negligible portion of the budget the income

effect will be insignificant and demand inelastic,[30]

Necessity: the more necessary a good is, the lower the elasticity, as people will attempt to buy it no matter the price,

such as the case of insulin for those that need it.[10][26]

Duration: for most goods, the longer a price change holds, the higher the elasticity is likely to be, as more and more

consumers find they have the time and inclination to search for substitutes.[25][27] When fuel prices increase suddenly,

for instance, consumers may still fill up their empty tanks in the short run, but when prices remain high over several

years, more consumers will reduce their demand for fuel by switching to carpooling or public transportation, investing

in vehicles with greater fuel economy or taking other measures.[26] This does not hold for consumer durables such as

the cars themselves, however; eventually, it may become necessary for consumers to replace their present cars, so

one would expect demand to be less elastic.[26]

Brand loyalty : an attachment to a certain brand—either out of tradition or because of proprietary barriers—can

override sensitivity to price changes, resulting in more inelastic demand.[29][31]

Who pays: where the purchaser does not directly pay for the good they consume, such as with corporate expense

accounts, demand is likely to be more inelastic.[31]

Interpreting values of price elasticity coefficients[edit]

Perfectly inelastic demand[10]

Perfectly elastic demand[10]

Elasticities of demand are interpreted as follows:[10]

Value Descriptive Terms

Perfectly inelastic demand

Inelastic or relatively inelastic demand

Unit elastic, unit elasticity, unitary elasticity, or unitarily elastic demand

Elastic or relatively elastic demand

Perfectly elastic demand

A decrease in the price of a good normally results in an increase in the quantity demanded by consumers because of

the law of demand, and conversely, quantity demanded decreases when price rises. As summarized in the table

above, the PED for a good or service is referred to by different descriptive terms depending on whether the elasticity

coefficient is greater than, equal to, or less than −1. That is, the demand for a good is called:

relatively inelastic when the percentage change in quantity demanded is less than the percentage change in price (so

that Ed > - 1);

unit elastic, unit elasticity, unitary elasticity, or unitarily elastic demand when the percentage change in quantity

demanded is equal to the percentage change in price (so that Ed = - 1); and

relatively elastic when the percentage change in quantity demanded is greater than the percentage change in price

(so that Ed < - 1).[10]

As the two accompanying diagrams show, perfectly elastic demand is represented graphically as a horizontal line,

and perfectly inelastic demand as a vertical line. These are the only cases in which the PED and the slope of the

demand curve (∆P/∆Q) are both constant, as well as the only cases in which the PED is determined solely by the

slope of the demand curve (or more precisely, by the inverse of that slope).[10]

Effect on total revenue[edit]

See also: Total revenue test

A set of graphs shows the relationship between demand and total revenue (TR) for a linear demand curve. As price decreases in the

elastic range, TR increases, but in the inelastic range, TR decreases. TR is maximised at the quantity where PED = 1.

A firm considering a price change must know what effect the change in price will have on total revenue. Revenue is

simply the product of unit price times quantity:

Generally any change in price will have two effects:[32]

the price effect : For inelastic goods, an increase in unit price will tend to increase revenue, while a decrease in price

will tend to decrease revenue. (The effect is reversed for elastic goods.)

the quantity effect : an increase in unit price will tend to lead to fewer units sold, while a decrease in unit price will tend

to lead to more units sold.

For inelastic goods, because of the inverse nature of the relationship between price and quantity demanded (i.e., the

law of demand), the two effects affect total revenue in opposite directions. But in determining whether to increase or

decrease prices, a firm needs to know what the net effect will be. Elasticity provides the answer: The percentage

change in total revenue is approximately equal to the percentage change in quantity demanded plus the percentage

change in price. (One change will be positive, the other negative.)[33] The percentage change in quantity is related to

the percentage change in price by elasticity: hence the percentage change in revenue can be calculated by knowing

the elasticity and the percentage change in price alone.

As a result, the relationship between PED and total revenue can be described for any good:[34][35]

When the price elasticity of demand for a good is perfectly inelastic (Ed = 0), changes in the price do not affect the

quantity demanded for the good; raising prices will always cause total revenue to increase. Goods necessary to

survival can be classified here; a rational person will be willing to pay anything for a good if the alternative is death.

For example, a person in the desert weak and dying of thirst would easily give all the money in his wallet, no matter

how much, for a bottle of water if he would otherwise die. His demand is not contingent on the price.

When the price elasticity of demand for a good is relatively inelastic (-1 < Ed < 0), the percentage change in quantity

demanded is smaller than that in price. Hence, when the price is raised, the total revenue rises, and vice versa.

When the price elasticity of demand for a good is unit (or unitary) elastic (Ed = 1), the percentage change in quantity is

equal to that in price, so a change in price will not affect total revenue.

When the price elasticity of demand for a good is relatively elastic ( -∞ < Ed < -1), the percentage change in quantity

demanded is greater than that in price. Hence, when the price is raised, the total revenue falls, and vice versa.

When the price elasticity of demand for a good is perfectly elastic (Ed is − ∞), any increase in the price, no matter how

small, will cause demand for the good to drop to zero. Hence, when the price is raised, the total revenue falls to zero.

This situation is typical for goods that have their value defined by law (such as fiat currency); if a 5 dollar bill were sold

for anything more than 5 dollars, nobody would buy it, so demand is zero.

Hence, as the accompanying diagram shows, total revenue is maximized at the combination of price and quantity

demanded where the elasticity of demand is unitary.[35]

It is important to realize that price-elasticity of demand is not necessarily constant over all price ranges. The linear

demand curve in the accompanying diagram illustrates that changes in price also change the elasticity: the price

elasticity is different at every point on the curve.

Effect on tax incidence[edit]

When demand is more inelastic than supply, consumers will bear a greater proportion of the tax burden than producers will.

Main article: tax incidence

PEDs, in combination with price elasticity of supply (PES), can be used to assess where the incidence (or "burden") of

a per-unit tax is falling or to predict where it will fall if the tax is imposed. For example, when demand is perfectly

inelastic, by definition consumers have no alternative to purchasing the good or service if the price increases, so the

quantity demanded would remain constant. Hence, suppliers can increase the price by the full amount of the tax, and

the consumer would end up paying the entirety. In the opposite case, when demand is perfectly elastic, by definition

consumers have an infinite ability to switch to alternatives if the price increases, so they would stop buying the good or

service in question completely—quantity demanded would fall to zero. As a result, firms cannot pass on any part of

the tax by raising prices, so they would be forced to pay all of it themselves.[36]

In practice, demand is likely to be only relatively elastic or relatively inelastic, that is, somewhere between the extreme

cases of perfect elasticity or inelasticity. More generally, then, the higher the elasticity of demand compared to PES,

the heavier the burden on producers; conversely, the more inelastic the demand compared to PES, the heavier the

burden on consumers. The general principle is that the party (i.e., consumers or producers) that

has fewer opportunities to avoid the tax by switching to alternatives will bear the greater proportion of the tax burden.[36] In the end the whole tax burden is carried by individual households since they are the ultimate owners of the means

of production that the firm utilises (see Circular flow of income).

Optimal pricing[edit]

Among the most common applications of price elasticity is to determine prices that maximize revenue or profit.

Constant elasticity and optimal pricing[edit]

If one point elasticity is used to model demand changes over a finite range of prices, elasticity is implicitly assumed

constant with respect to price over the finite price range. The equation defining price elasticity for one product can be

rewritten (omitting secondary variables) as a linear equation.

where

 is the elasticity, and   is a constant.

Similarly, the equations for cross elasticity for n products can be written as a set of n simultaneous linear equations.

where

 and  , and   are constants; and appearance of a letter index

as both an upper index and a lower index in the same term implies summation over that index.

This form of the equations shows that point elasticities assumed constant over a price range cannot determine what

prices generate maximum values of  ; similarly they cannot predict prices that generate maximum   or

maximum revenue.

Constant elasticities can predict optimal pricing only by computing point elasticities at several points, to determine the

price at which point elasticity equals -1 (or, for multiple products, the set of prices at which the point elasticity matrix is

the negative identity matrix).

Non-constant elasticity and optimal pricing[edit]

If the definition of price elasticity is extended to yield a quadratic relationship between demand units ( ) and price,

then it is possible to compute prices that maximize  ,  , and revenue. The fundamental equation for one

product becomes

and the corresponding equation for several products becomes

Excel models are available that compute constant elasticity, and use non-constant elasticity to estimate prices that

optimize revenue or profit for one product[37] or several products.[38]

Limitations of revenue-maximizing and profit-maximizing pricing strategies[edit]

In most situations, revenue-maximizing prices are not profit-maximizing prices. For example, if variable costs per unit

are nonzero (which they almost always are), then a more complex computation of a similar kind yields prices that

generate optimal profits.

In some situations, profit-maximizing prices are not an optimal strategy. For example, where scale economies are

large (as they often are), capturing market share may be the key to long-term dominance of a market, so maximizing

revenue or profit may not be the optimal strategy.

Selected price elasticities[edit]

Various research methods are used to calculate price elasticities in real life, including analysis of historic sales data,

both public and private, and use of present-day surveys of customers' preferences to build uptest markets capable of

modelling such changes. Alternatively, conjoint analysis (a ranking of users' preferences which can then be

statistically analysed) may be used.[39]

Though PEDs for most demand schedules vary depending on price, they can be modeled assuming constant

elasticity.[40] Using this method, the PEDs for various goods—intended to act as examples of the theory described

above—are as follows. For suggestions on why these goods and services may have the PED shown, see the above

section on determinants of price elasticity.

Cigarettes (US)[41]

−0.3 to −0.6 (General)

−0.6 to −0.7 (Youth)

Alcoholic beverages (US)[42]

−0.3 or −0.7 to −0.9 as of 1972 (Beer)

−1.0 (Wine)

−1.5 (Spirits)

Airline travel (US)[43]

−0.3 (First Class)

−0.9 (Discount)

−1.5 (for Pleasure Travelers)

Livestock

−0.5 to −0.6 (Broiler Chickens)[44]

Oil (World)

−0.4

Car fuel[45]

−0.09 (Short run)

−0.31 (Long run)

Medicine (US)

−0.31 (Medical insurance)[46]

−.03 to −.06 (Pediatric Visits)[47]