2.3 - Price, Income & Cross Elasticities of Demand - notes

The concept of price elasticity of demand (PED)

Price elasticity of demand (PED) measures how the quantity demanded of a product responds to a change in its price.

Formula for calculating PED

$\text{PED}=\frac{\text{percentage change in quantity demanded}}{\text{percentage change in price}}$

Where:

• Percentage change in quantity demanded = $\frac{\text{change in quantity demanded}}{\text{original quantity demanded}}\times 100$
• Percentage change in price = $\frac{\text{change in price}}{\text{original price}}\times 100$

PED is typically negative for most goods, as demand usually decreases when price rises.

Worked example - Calculating PED

When the price of a soft drink increased from $1.50 to $2.00, the demand fell from 50 bottles to 35 bottles. Calculate the PED.

Step 1: Identify the values

• Original price = $1.50
• New price = $2.00
• Original quantity demanded = 50 bottles
• New quantity demanded = 35 bottles

Step 2: Calculate percentage change in quantity demanded

$\text{Percentage change in quantity demanded}=\frac{35-50}{50}\times 100=\frac{-15}{50}\times 100=-30\%$

Step 3: Calculate percentage change in price

$\text{Percentage change in price}=\frac{2.00-1.50}{1.50}\times 100=\frac{0.50}{1.50}\times 100\approx 33.33\%$

Step 4: Calculate PED

$\text{PED}=\frac{-30\%}{33.33\%}\approx -0.9$

Types of PED: elastic, inelastic, and unit elastic

PED values indicate how sensitive demand is to price changes, categorised as elastic, inelastic, or unit elastic.

Elastic demand (PED > 1)

PED (ignoring the negative sign) is greater than 1, meaning a percentage change in price causes a larger percentage change in quantity demanded. The diagram below illustrates this with a relatively flat demand curve D: a fall in price from P1 to P2 leads to a proportionally larger rise in quantity demanded, from Q1 to Q2.

Perfectly elastic demand has a PED of infinity, where any price increase leads to zero demand. As the diagram below shows, the demand curve D is horizontal at the price P: consumers will buy all they can obtain at P, but nothing at a higher price.

Inelastic demand (0 < PED < 1)

PED (ignoring the negative sign) is between 0 and 1, meaning a percentage change in price causes a smaller percentage change in quantity demanded. In the diagram below, the demand curve D is steep: even a large fall in price from P1 to P2 increases quantity demanded only slightly, from Q1 to Q2.

Perfectly inelastic demand has a PED of 0, where quantity demanded remains unchanged regardless of price. The demand curve D in the diagram below is vertical: at any price, such as P1 or P2, the quantity demanded stays the same.

Unit elastic demand (PED = 1)

PED (ignoring the negative sign) equals 1, meaning the percentage change in price equals the percentage change in quantity demanded. The diagram below shows this as a demand curve D along which a fall in price from P1 to P2 causes an equal percentage rise in quantity demanded, from Q1 to Q2 (for example, a 20% fall in price leads to a 20% rise in quantity demanded).

Worked example - Identifying type of PED

The price of a video game decreases from $45 to $30, increasing demand from 100 units to 160 units. Calculate the PED and identify its type.

Step 1: Identify the values

• Original price = $45
• New price = $30
• Original quantity demanded = 100 units
• New quantity demanded = 160 units

Step 2: Calculate percentage change in quantity demanded

$\text{Percentage change in quantity demanded}=\frac{160-100}{100}\times 100=\frac{60}{100}\times 100=60\%$

Step 3: Calculate percentage change in price

$\text{Percentage change in price}=\frac{30-45}{45}\times 100=\frac{-15}{45}\times 100\approx -33.33\%$

Step 4: Calculate PED and identify type

$\text{PED}=\frac{60\%}{-33.33\%}\approx -1.8$ Since the value (ignoring sign) is greater than 1, demand is elastic.

Income elasticity of demand (YED) and its types

Income elasticity of demand (YED) measures how the quantity demanded of a product changes with variations in real income.

Formula for calculating YED

$\text{YED}=\frac{\text{percentage change in quantity demanded}}{\text{percentage change in real income}}$

Where:

• Percentage change in quantity demanded = $\frac{\text{change in quantity demanded}}{\text{original quantity demanded}}\times 100$
• Percentage change in real income = $\frac{\text{change in real income}}{\text{original real income}}\times 100$

YED can be positive or negative, depending on the type of good.

Types of YED

• Income elastic (YED > 1) - Demand changes more than proportionally with income.
• Income inelastic (0 < YED < 1) - Demand changes less than proportionally with income.
• Perfectly inelastic (YED = 0) - Demand remains constant regardless of income changes.
• Negative YED - Indicates inferior goods, where demand falls as income rises.

The diagrams below show the first three cases, with real income on the vertical axis. When demand is income elastic, the demand curve D is relatively flat: a rise in real income from Y1 to Y2 causes a proportionally larger rise in quantity demanded, from Q1 to Q2.

When demand is income inelastic, the demand curve D is steep: the same rise in real income, from Y1 to Y2, causes a proportionally smaller rise in quantity demanded, from Q1 to Q2.

When demand is perfectly inelastic, the demand curve D is vertical: the quantity demanded stays the same however much real income changes.

Worked example - Calculating YED

If real incomes rise by 5% and the demand for organic food increases by 9%, calculate the YED.

Step 1: Identify the values

• Percentage change in real income = 5%
• Percentage change in quantity demanded = 9%

Step 2: Apply the YED formula

$\text{YED}=\frac{9\%}{5\%}=1.8$

Step 3: Interpretation

Since YED is greater than 1, demand is income elastic.

Cross elasticity of demand (XED) for substitutes and complements

Cross elasticity of demand (XED) measures how the quantity demanded of one product responds to a price change in another product.

Formula for calculating XED

$\text{XED}=\frac{\text{percentage change in quantity demanded of good A}}{\text{percentage change in price of good B}}$

Where:

• Percentage change in quantity demanded of good A = $\frac{\text{change in quantity demanded of good A}}{\text{original quantity demanded of good A}}\times 100$
• Percentage change in price of good B = $\frac{\text{change in price of good B}}{\text{original price of good B}}\times 100$

XED for substitutes and complements

• Substitutes (positive XED) - Goods that can replace each other; price rise in one increases demand for the other.
• Complements (negative XED) - Goods used together; price rise in one decreases demand for the other.
• Unrelated goods (XED = 0) - No demand relationship.

Worked example - Calculating XED

Smartwatches and traditional watches are substitutes. If the price of smartwatches rises by 30%, demand for traditional watches increases by 15%. Calculate the XED.

Step 1: Identify the values

• Percentage change in price of smartwatches = 30%
• Percentage change in quantity demanded of traditional watches = 15%

Step 2: Apply the XED formula

$\text{XED}=\frac{15\%}{30\%}=0.5$

Step 3: Interpretation

The positive XED confirms they are substitutes.