Price elasticity of demand

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In economics and business studies, the price elasticity of demand (PED) is an elasticity that measures the nature and degree of the relationship between changes in quantity demanded of a good and changes in its price.

1 Introduction
2 Mathematical definition
3 Elasticity and revenue
4 Point-price elasticity
5 See also
6 References

[edit] Introduction

Price elasticity of demand describes the relationship between changes in quantity demanded of a good (or service) and changes in the price for that good (or service.) When prices fall, the quantity consumers demand normally rises--if it costs less, consumers buy more. If price drops 10% but the quantity demanded increases by 20%, the good (or service) is relatively elastic. More specifically, the price elasticity of demand would be 20 %/10 % = 2. Any number above 1 indicates relative elasticity. (In determining elasticity, absolute values govern--negative and positive signs may be omitted.) Numbers between 0 and 1 indicate relatively inelastic demand.

In general (for normal goods), a price drop results in an increase in the quantity demanded by consumers. The demand for a good is relatively inelastic when the quantity demanded does not change much with the price change. Goods and services for which no substitutes exist are generally inelastic. Demand for an antibiotic, for example, becomes highly inelastic when it alone can kill an infection resistant to all other anti-biotics. Rather than die of an infection, a patient will happily pay whatever s/he can/must to acquire enough of the anti-biotic to kill the infection.

Price elasticity of demand is rarely constant throughout the ranges of quantity demanded and price. A good (or service) can have relatively inelastic demand up to a certain price, above which demand becomes elastic. Even if automobiles, for example, were extremely inexpensive, parking or other related ownership issues would presumably keep most people from owning more than some "maximum" number of automobiles. For these and other reasons, elasticity of demand remains valid only over a specific (and small) range of price. Demand for cars (as well as other goods and services) is not elastic or inelastic for all prices. Elasticity of demand can change dramatically across a range of prices.

One popular misinterpretation is that demand for "necessities" is highly inelastic, but this explanation masks real causes and yields false conclusions. Demand for salt, for instance, at its modern levels of supply is highly inelastic not because it is a necessity but because it is such a small part of the household budget. (Technology has increased the supply of salt modernly and reduced its historically high price.) Demand for water, another necessity, is highly inelastic for similar supply side reasons. Demand for other goods, like cocaine, which is not a necessity, can be highly inelastic.

Substitution serves as a much more reliable predictor of elasticity of demand than "necessity." Few substitutes for oil and gasoline exist. Predictably, demand for these goods is relatively inelastic. However, products with a high elasticity usually have many substitutes. Lots of substitutes exist for potato chips, from apples and carrot sticks to corn chips. No substitutes exist for cigarettes other than (other) cigarettes. Predictably, demand for chips is relatively elastic but demand for chocolate and cigarettes becomes increasingly more inelastic.

It may be possible that quantity demanded for a good rises as its price rises, even under conventional economic assumptions of consumer rationality. Two such classes of goods are known as Giffen goods or Veblen goods. Another case is the price inflation during an economic bubble. Consumer perception plays an important role in explaining the demand for products in these categories. A starving musician who offers lessons at a bargain basement rate of $5.00 per hour will continue to starve, but if s/he raises the price to $35.00 per hour, s/he will discover increased demand. Those who need lessons want someone who is worth $35.00 an hour not someone who can only charge $5.00.

Various research methods are used to calculate price elasticity:

Test markets
Analysis of historical sales data
Conjoint analysis

[edit] Mathematical definition

The formula used to calculate the coefficient of price elasticity of demand is

$E_d = \left|\frac{\%\ \rm{change}\ \rm{in}\ \rm{quantity}\ \rm{demanded}\ \rm{of}\ \rm{product}\ X}{\%\ \rm{change}\ \rm{in}\ \rm{price}\ \rm{of}\ \rm{product}\ X}\right| = \frac{\Delta Q_d/Q_d}{\Delta P_d/P_d}$

Or, using the differential calculus:

$E_d={P \over Q }\times{{\partial Q}\over{\partial P}}$

where:

P

= price

Q

= quantity

Q d

= original quantity

P d

= original price

Δ Q d

Q d n e w

Q d o l d

Δ P d

P d n e w

P d o l d

One caveat in using this (valid) formula: Will you use the original or the new quantity demanded in the denominator? Will you use the original or new price in the denominator? Be consistent at least and use one set or the other. A more elegant and reliable calculation uses a midpoint analysis--instead of the original or the new, try using the point half way between the two for your calculations.

Interestingly, repeated use of the chain rule reveals that:

$E_d={{\partial \ln (Q)}\over{\partial \ln (P)}}$

From the following process:

Note:

${{\partial \ln (Q)}\over{\partial \ln (P)}}={{\partial \ln (Q)}\over{\partial Q}}\times{{\partial Q}\over{\partial \ln (P)}}={{1}\over{Q}}\times{{\partial Q}\over{\partial \ln (P)}}$

Further Note:

${{\partial Q}\over{\partial P}}={{\partial \ln (P)}\over{\partial P}}\times{{\partial Q}\over{\partial \ln (P)}}={{1}\over{P}}\times{{\partial Q}\over{\partial \ln (P)}}$

Implying:

${{\partial Q}\over{\partial \ln (P)}}={P}\times{{\partial Q}\over{\partial P}}$

Substitution reveals:

${{\partial \ln (Q)}\over{\partial \ln (P)}}={{1}\over{Q}}\times{{\partial Q}\over{\partial P}}\times{P}=E_d$

[edit] Elasticity and revenue

See also: Total revenue test

A set of graphs shows the relationship between demand and total revenue. As elasticity decreases in the elastic range, revenue increases, but in the inelastic range, revenue decreases.

When the price elasticity of demand for a good is inelastic (|Ed| < 1), the percentage change in quantity is smaller than that in price. Hence, when the price is raised, the total revenue of producers rises, and vice versa.

When the price elasticity of demand for a good is elastic (|Ed| > 1), the percentage change in quantity is greater than that in price. Hence, when the price is raised, the total revenue of producers falls, and vice versa.

When the price elasticity of demand for a good is unit elastic (or unitary elastic) (|Ed| = 1), the percentage change in quantity is equal to that in price. Hence, when the price is raised, the total revenue remains unchanged. The demand curve is a rectangular hyperbola.

When the price elasticity of demand for a good is perfectly elastic (Ed is undefined), 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 of producers falls to zero. The demand curve is a horizontal straight line. A ten-dollar banknote is an example of a perfectly elastic good; nobody would pay $10.01, yet everyone will pay $9.99 for it.

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. The demand curve is a vertical straight line; this violates the law of demand. An example of a perfectly inelastic good is a human heart for someone who needs a transplant; nobody would buy more than the exact amount of hearts demanded, no matter how low the price is.

[edit] Point-price elasticity

Point Elasticity = (% change in Quantity)/(% change in Price)
Point Elasticity = (∆Q/Q)/(∆P/P)
Point Elasticity = (P ∆Q)/(Q ∆P)
Point Elasticity = (P/Q)(∆Q/∆P) Note: In the limit (or "at the margin"), "(∆Q/∆P)" is the derivative of the demand function with respect to P. "Q" means 'Quantity' and "P" means 'Price'.
Example
demand curve: Q = 1,000 - .6P
a.) Given this demand curve determine the point price elasticity of demand at P = 80 and P = 40 as follows.
i.) obtain the derivative of the demand function when it's expressed Q as a function of P.
${{\partial Q}\over{\partial P}} = -.6$
ii.) next apply the above equation to the sought ordered pairs: (40, 976), (80, 952)
$E_p={{\partial Q}\over{\partial P}}{P \over Q }$
e = -.6(40/976) = -.02
e = -.6(80/952) = -.05