Price elasticity of demand

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In economics and business studies, the price elasticity of demand (PED) is a measure of the sensitivity of quantity demanded to changes in price. It is measured as elasticity, that is it measures the relationship as the ratio of percentage changes between quantity demanded of a good and changes in its price. Water is a good example of a good that has inelastic characteristics in that people will pay anything for it (high or low prices with relatively equivalent quantity demanded), so it is not elastic. On the other hand, sugar is very elastic because as the price of sugar increases, there are many substitutions such as the pink, yellow, brown, blue, and other "stuff."

1 Mathematical definition
2 Interpretation of elasticity
3 Determinants
4 Elasticity and revenue
5 Point-price elasticity
6 See also
7 External links
8 References
- 8.1 Notes
- 8.2 General references

Mathematical definition

The formula used to calculate the coefficient of price elasticity of demand for a given product is

$E_d = - \frac{\%\ \mbox{change in quantity demanded}}{\%\ \mbox{change in price}} = - \frac{\Delta Q_d/Q_d}{\Delta P_d/P_d}$

Conventions differ regarding the minus sign, considering remarks like "price elasticity of demand is usually negative".

This simple formula has a problem, however. It yields different values for E_d depending on whether Q_d and P_d are the original or final values for quantity and price. This formula is usually valid either way as long as you are consistent and choose only original values or only final values.

A more elegant^{[citation needed]} and reliable calculation uses a midpoint calculation, which eliminates this ambiguity. Another benefit of using the following formula is that when E_d = -1, it means there will be no change in revenue when the price changes from P₁ (the original price) to P₂.

Q_av means the average of the original and final values of quantity demanded, and likewise for P_av.

$\begin{align} E_d &= - \frac{\Delta Q_d/Q_{av}}{\Delta P_d/P_{av}} \\ &= - \frac{(Q_2 - Q_1)\ /\ [(Q_1 + Q_2)/2]}{(P_2 - P_1)\ /\ [(P_1 + P_2)/2]} \\ &= - \frac{Q_2 - Q_1}{P_2 - P_1} \times \frac{P_1+P_2}{Q_1+Q_2} \end{align}$

Or, using the differential calculus form:

$E_d = - \frac{P}{Q}\frac{dQ}{dP}$

This can be rewritten in the form:

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

Interpretation of elasticity

A diagram is needed in this article

Value	Meaning
n = 0	Perfectly inelastic.
0 < n < 1	Relatively inelastic.
n = 1	Unitary elastic.
1 < n < ∞	Relatively elastic.
n = ∞	Perfectly elastic.

A price drop usually results in an increase in the quantity demanded by consumers (see Giffen good for an exception). 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 antibiotics. Rather than die of an infection, patients will generally be willing to pay whatever is necessary to acquire enough of the antibiotic to kill the infection.

Various research methods are used to calculate price elasticity:

Test markets
Analysis of historical sales data
Conjoint analysis

Determinants

A number of factors determine the elasticity:

Substitutes: The more substitutes, the higher the elasticity, as people can easily switch from one good to another if a minor price change is made
Percentage of income: The higher the percentage that the product's price is of the consumers income, the higher the elasticity, as people will be careful with purchasing the good because of its cost
Necessity: The more necessary a good is, the lower the elasticity, as people will buy it no matter the price, such as insulin
Time: The longer a price change holds, the higher the elasticity, as more and more people will stop demanding the good (i.e. if you go to the supermarket and find that blueberries have doubled in price, you'll buy it because you need it this time, but next time you won't, unless the price drops back down again)
Breadth of definition: The broader the definition, the lower the elasticity. For example, Company X's fried dumplings will have a relatively high elasticity, where as food in general will have an extremely low elasticity

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Elasticity and revenue

See also: Total revenue test

A set of graphs shows the relationship between demand and total revenue. As price 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 (|E_d| < 1), the percentage change in quantity demanded 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 (|E_d| > 1), the percentage change in quantity demanded 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) (|E_d| = 1), the percentage change in quantity is equal to that in price.

When the price elasticity of demand for a good is perfectly elastic (E_d 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 banknote is the classic example of a perfectly elastic good; nobody would pay £10.01 for a £10 note, yet everyone will pay £9.99 for it.

When the price elasticity of demand for a good is perfectly inelastic (E_d = 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; neither increases nor decreases in price affect the quantity demanded (no matter what the price, a person will pay for one heart but only one; nobody would buy more than the exact amount of hearts demanded, no matter how low the price is).

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 - 0.6P
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.
${{\partial Q}\over{\partial P}} = -0.6$
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 = -0.6(40/976) = -0.02
e = -0.6(80/952) = -0.05