Price elasticity of demand

In economics, 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.

Introduction

Price elasticity of demand is measured as the percentage change in quantity demanded that occurs in response to a percentage change in price. For example, if, in response to a 10 % fall in the price of a good, the quantity demanded increases by 20 %, the price elasticity of demand would be 20 %/(− 10 %) = −2. (Case & Fair, 1999: 109).

In general, a fall in the price of a good is expected to increase the quantity demanded, so the price elasticity of demand is negative as above. Note that in economics literature the minus sign is often omitted and the elasticity is given as an absolute value. (Case & Fair, 1999: 110). Because both the denominator and numerator of the fraction are percent changes, price elasticities of demand are dimensionless numbers and can be compared even if the original calculations were performed using different currencies or goods.

An example of a good with a highly inelastic demand curve is salt: people need salt, so for even relatively large changes in the price of salt, the amount demanded will not be significantly altered. Similarly, a product with a highly elastic demand curve is red cars: if the price of red cars went up even a small amount, demand is likely to go down since substitutes are readily available for purchase (cars of other colors).

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.

Various research methods are used to calculate price elasticity:

• Test markets
• Analysis of historical sales data
• Conjoint analysis

Mathematical definition

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

$E_{d}=\left|{\frac {\%\ {\rm {{change}\ {\rm {{in}\ {\rm {{quantity}\ {\rm {{demanded}\ {\rm {{of}\ {\rm {{product}\ X}}}}}}}}}}}}}{\%\ {\rm {{change}\ {\rm {{in}\ {\rm {{price}\ {\rm {{of}\ {\rm {{product}\ X}}}}}}}}}}}}\right|={\frac {\Delta Q_{d}/Q_{d}}{\Delta P_{d}/P_{d}}}$

Using all the differential calculus:

$E_{d}={P \over Q}{{\partial Q} \over {\partial P}}$

where:

$P$ = price (Average of earlier & new price)

$Q$ = quantity (Average of earlier & new quantity)

Elasticity and revenue

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 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 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 = ∞), 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 straght 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.

Types of Elasticity of Demand

Point Elasticity Cross or substitute Elasticity Arc Elasticity Price Elasticity Income Elasticity

Point-Price Elasticity

Point Elasticity = (% change in Q)(% change in X)

Point Elasticity = (∆Q/Q)/(∆P/P)

Point Elasticity = (∆Q/Q)(P/∆P)

Point Elasticity = (∆Q/∆P)(P/Q) Note: "(∆Q/∆P)" is merely the first derivative of the demand function when Q is written as a function of 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