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2.2.2 formulae for and calculation of price elasticity, income elasticity and cross elasticity of demand

2.2.2 formulae for and calculation of price elasticity, income elasticity and cross elasticity of demand

Calculating Demand Elasticities

Definition

Percentage change: the change in a variable expressed as a proportion of its original value and multiplied by 100, so that responses in differently sized markets can be compared on the same scale.

% ΔX=ΔXX0×100 \%\,\Delta X = \dfrac{\Delta X}{X_0} \times 100 %ΔX=X0​ΔX​×100
  1. Each demand elasticity is a ratio of two percentage changes, so the units cancel and the coefficient is a pure number.
  2. The top line is always the percentage change in quantity demanded of the good in question.
  3. The bottom line is the percentage change in the cause: the good's own price, real income, or another good's price.
Key Idea
  • PED divides the percentage change in quantity demanded by the percentage change in the good's own price.
  • YED uses the percentage change in real income, and XED uses the percentage change in the price of a related good.

The Three Formulae

Price elasticity of demand (PED)

PED=% ΔQd% ΔP PED = \dfrac{\%\,\Delta Q_d}{\%\,\Delta P} PED=%ΔP%ΔQd​​

Income elasticity of demand (YED)

YED=% ΔQd% ΔY YED = \dfrac{\%\,\Delta Q_d}{\%\,\Delta Y} YED=%ΔY%ΔQd​​

Cross elasticity of demand (XED)

XED=% ΔQd,A% ΔPB XED = \dfrac{\%\,\Delta Q_{d,A}}{\%\,\Delta P_B} XED=%ΔPB​%ΔQd,A​​

Worked Calculations

Example
  • The price of a good rises from £20 to £24 (a 20% rise) while the quantity demanded falls from 100 to 90 (a 10% fall).
PED=−10%+20%=−0.5 PED = \dfrac{-10\%}{+20\%} = -0.5 PED=+20%−10%​=−0.5
  • The negative value reflects the inverse law of demand, and because it lies between 0 and 1 demand is price inelastic, so raising the price increases total revenue.
  • Real income rises 5% and the quantity of restaurant meals demanded rises 10%.
YED=+10%+5%=+2 YED = \dfrac{+10\%}{+5\%} = +2 YED=+5%+10%​=+2
  • The positive value makes meals a normal good, and because it is greater than 1 the good is income-elastic, behaving as a luxury whose demand grows faster than income.
  • The price of Pepsi rises 10% and the quantity of Coca-Cola demanded rises 4%.
XED=+4%+10%=+0.4 XED = \dfrac{+4\%}{+10\%} = +0.4 XED=+10%+4%​=+0.4
  • The positive value marks Pepsi and Coca-Cola as substitutes, and at only 0.4 they are weak substitutes, so the price change pulls only a little extra demand across.

Reading the Size

  1. Elastic: the coefficient is greater than 1 in size, so quantity demanded responds more than proportionately to the change.
  2. Inelastic: the coefficient is between 0 and 1 in size, so quantity demanded responds less than proportionately to the change.
  3. Unit elastic: the coefficient equals 1, so quantity demanded changes in exact proportion to the change.
Exam technique
  • Convert each raw change into a percentage of its original value before dividing.
  • Keep the sign of the answer and state it, because the sign carries economic meaning.
  • Interpret the coefficient in words, naming the good and whether demand is elastic or inelastic.
Common Mistake
  • Do not divide the raw changes instead of the percentage changes, because elasticity always compares proportionate changes.
  • Do not drop the minus sign on PED, because it signals the inverse link between price and quantity demanded.
Self review
  • State the formula for PED.
  • If price rises 20% and quantity demanded falls 10%, calculate PED and say whether demand is elastic or inelastic.
  • What does a positive YED tell you about a good, and what does a value above 1 add?
  • What does a positive XED tell you about the relationship between two goods?
  • Why must each change be converted to a percentage first?
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Demand elasticity measures how responsive quantity demanded is to a change in something that influences demand. It compares percentage changes, which lets economists compare markets of very different sizes.

To calculate a percentage change, divide the change by the original value and multiply by 100. This uses the original value, X0X_0X0​, as the base for comparison.

% ΔX=ΔXX0×100 \%\,\Delta X = \frac{\Delta X}{X_0} \times 100 %ΔX=X0​ΔX​×100

All demand elasticities are ratios of percentage changes, so the percentage units cancel. The final elasticity coefficient is therefore a pure number.

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2.2.2 formulae for and calculation of price elasticity, income elasticity and cross elasticity of demand Revision Guide

  1. Intl A Level
  2. /Economics
  3. /2.2.2 formulae for and calculation of price elasticity, income elasticity and cross elasticity of demand

Revision notes for CIE Intl A Level Economics 2.2.2 formulae for and calculation of price elasticity, income elasticity and cross elasticity of demand: explanations and worked examples.

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