Income Elasticity of Demand vs Cross-Price Elasticity of Demand
Income Elasticity of Demand and Cross-Price Elasticity of Demand are two Elasticity concepts in AP Economics that students often mix up. Income elasticity of demand measures how responsive the quantity demanded is to a change in consumers' income. Cross-price elasticity of demand measures how responsive the quantity demanded of one good is to a change in the price of another good. Here is how they compare side by side.
It is calculated as the percentage change in quantity demanded divided by the percentage change in income. Demand is considered a normal good if the ratio is positive, meaning demand increases as income increases. Demand is considered an inferior good if the ratio is negative.
It is calculated as the percentage change in quantity demanded of Good A divided by the percentage change in price of Good B. If the ratio is positive, the goods are considered substitutes. If the ratio is negative, the goods are considered complements.
Income Elasticity vs Cross-Price Elasticity: Same Numerator, Different Denominator
| Income Elasticity of Demand | Cross-Price Elasticity of Demand | |
|---|---|---|
| What sits in the denominator | The percentage change in consumer income | The percentage change in the price of a second good |
| How many goods are involved | One | Two |
| Formula | Percentage change in quantity demanded divided by percentage change in income | Percentage change in quantity of good A divided by percentage change in the price of good B |
| What a positive value means | The good is normal | The two goods are substitutes |
| What a negative value means | The good is inferior | The two goods are complements |
| What a value near zero means | Demand barely responds to income, as with salt | The two goods are unrelated, as with shoelaces and milk |
| What size adds beyond sign | Above 1 marks a luxury, between 0 and 1 a necessity | A larger absolute value means a stronger link between the pair |
Run both through the midpoint method and only the denominator changes
Use the midpoint method for each so the answer does not depend on the direction of travel. Income elasticity first. Suppose weekly household income rises from an illustrative $400 to $500, and restaurant meals per month go from 6 to 9. The income change is 100 over the average of 450, which is 22.22 percent. The quantity change is 3 over the average of 7.5, which is 40 percent. Divide 40 by 22.22 and you get 1.8, a positive number above 1, so restaurant meals are a normal good and a luxury for this household. Cross-price elasticity next, with the same machinery. Coffee rises from $4 to $5, and tea sales climb from 90 to 110 packets. The price change is 1 over the average of 4.50, again 22.22 percent. The tea quantity change is 20 over the average of 100, which is 20 percent. Divide 20 by 22.22 and you get 0.9, positive, so tea and coffee are substitutes for these buyers. Identical arithmetic, identical top line, and the only real difference is what you put underneath. Check your own figures at /calculate/income-elasticity-of-demand and /calculate/cross-price-elasticity.
The sign carries the answer, which is why dropping it costs marks
Price elasticity of demand is often quoted as an absolute value, because everyone already knows the quantity moves against the price. Neither of these two elasticities may be treated that way. Their signs are the entire classification. Return to the household whose weekly income went from $400 to $500, a rise of 22.22 percent on the midpoint. Suppose its bus trips fall from 30 to 20 a month, a change of 10 over the average of 25, which is a 40 percent drop. Income elasticity is negative 40 divided by 22.22, or negative 1.8, so bus travel is inferior for that household even though the magnitude matches the restaurant figure exactly. Write 1.8 without the minus sign and you have reported the opposite conclusion. The same trap sits in cross-price work: negative 0.9 for two goods means they are complements, and a marker reading only the digits cannot award the point. Two further cautions. These labels describe one household over one income stretch, not the good in general. And a value near zero is a real answer, not a failed calculation.
Frequently asked questions
What is the difference between income elasticity and cross-price elasticity of demand?
Income elasticity measures how the quantity demanded of one good responds to a change in consumer income, while cross-price elasticity measures how the quantity demanded of one good responds to a change in the price of a different good. Both put a percentage change in quantity on top, and they differ in what goes underneath. Income elasticity classifies a good as normal or inferior; cross-price elasticity classifies a pair as substitutes or complements.
Does a negative elasticity always mean the good is inferior?
No, a negative value means inferior only for income elasticity, because a negative cross-price elasticity instead means the two goods are complements. Always check which denominator produced the number before naming the good. A negative price elasticity of demand carries no classification at all, since almost every demand curve slopes downward.
How do you tell a luxury from a necessity using elasticity?
A luxury has an income elasticity greater than 1, meaning quantity demanded grows faster in percentage terms than income, and a necessity has a value between 0 and 1. Both are normal goods because both values are positive. A negative value takes the good out of that ranking entirely and marks it as inferior.
Live Elasticity graph. Drag the curves, or open the full version.
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