AP® Microeconomics review sheet from Aim for Five (aimforfive.com/micro/units/2/2-5)
Unit 2 · Topic 2.5
2.5 Other Elasticities
Two more elasticities use the same 'percent change over percent change' idea. Income elasticity tells you whether a good is normal or inferior, and cross-price elasticity tells you whether two goods are substitutes or complements. Here the sign matters, so never drop it.
Key terms
- income elasticity of demand
- normal good
- inferior good
- cross-price elasticity of demand
- substitutes
- complements
Income elasticity of demand
Income elasticity of demand = % change in quantity demanded ÷ % change in income.
It measures how demand responds when buyers' incomes change, with the good's price held constant.
| Sign | Type of good | Example |
|---|---|---|
| Positive (above 0) | Normal good: people buy more when income rises | restaurant meals, new cars |
| Negative (below 0) | Inferior good: people buy less when income rises | used clothing, generic cereal |
Cross-price elasticity of demand
Cross-price elasticity of demand = % change in quantity demanded of good A ÷ % change in the price of good B.
It shows how demand for one good responds when the price of a different good changes.
| Sign | Relationship | Why |
|---|---|---|
| Positive | Substitutes | B gets pricier, so people switch to A |
| Negative | Complements | B gets pricier, so people buy less B and also less A, which goes with it |
| Zero | Unrelated goods | B's price has no effect on demand for A |
How these connect to shifts
These elasticities measure the size and direction of demand shifts from topic 2.1. A positive income elasticity means rising income shifts demand right. A negative cross-price elasticity means a higher price of the other good shifts this good's demand left.
The size tells you how strong the relationship is. A cross-price elasticity of +3 means two goods are close substitutes, like two brands of bottled water. A value of +0.2 means they're only weak substitutes.
Notice the difference from topic 2.3. For price elasticity of demand you drop the minus sign because it's always negative. For income and cross-price elasticity, the sign is the whole point: it tells you what kind of good or relationship you have.
All three demand elasticities side by side
| Elasticity | Formula | Sign rule | What it tells you |
|---|---|---|---|
| Price elasticity of demand | % change in Qd ÷ % change in the good's own price | Always negative; drop the sign | Elastic, inelastic or unit elastic; the total revenue test |
| Income elasticity | % change in Qd ÷ % change in income | Keep the sign | Normal (positive) or inferior (negative) |
| Cross-price elasticity | % change in Qd of A ÷ % change in price of B | Keep the sign | Substitutes (positive), complements (negative) or unrelated (zero) |
Reading a question
Check which price is in the denominator. For cross-price elasticity it's the other good's price, not the good's own price. Also check the direction of each change. If income falls 5% and demand for bus rides rises 3%, the elasticity is +3% ÷ −5% = −0.6: negative, so bus rides are inferior. A quick check: the two changes went in opposite directions.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Income elasticity
When average income in a town rises by 10%, the quantity of bus rides demanded falls by 5%. Calculate the income elasticity of demand. Is a bus ride a normal or an inferior good there?
Show the solutionHide the solution
- Step 1: Income elasticity = % change in quantity demanded ÷ % change in income = −5% ÷ 10% = −0.5.
- Step 2: The value is negative, so demand falls as income rises.
- Step 3: That makes bus rides an inferior good in this town: as people earn more, they switch to driving.
Answer: −0.5; bus rides are an inferior good.
- Example 2Calculator allowed
Cross-price elasticity (classic trap)
The price of coffee rises by 20%, and the quantity of tea demanded rises by 8%. Calculate the cross-price elasticity of demand for tea with respect to coffee's price and identify the relationship. A classmate drops the sign and says 'the answer is 0.4, so they're complements.' What went wrong?
Show the solutionHide the solution
- Step 1: Cross-price elasticity = % change in quantity of tea ÷ % change in price of coffee = +8% ÷ +20% = +0.4.
- Step 2: Positive means that as coffee gets pricier, people buy more tea: they're substitutes.
- Step 3: The classmate's mistake: the sign isn't dropped for cross-price elasticity, and a positive sign means substitutes, not complements.
Answer: +0.4: coffee and tea are substitutes (weak ones, since 0.4 is small).
Common mistakes
- Dropping the minus sign. For income and cross-price elasticity, the sign tells you the type of good or relationship.
- Mixing up the signs: positive cross-price elasticity means substitutes; negative means complements.
- Using the good's own price change in a cross-price formula. The denominator is the percent change in the other good's price.
- Assuming a negative income elasticity means the good is bad or low quality. 'Inferior' only means demand falls as income rises.
On the exam
- Multiple-choice questions often give an elasticity value and ask you to classify the good or the pair of goods. Look at the sign first, then the size.
- Some questions run the logic backwards: they describe a reaction (incomes fell and demand rose) and ask what the elasticity's sign must be.
Connected topics
Videos
Check yourself
4 questions on 2.5 Other Elasticities. Pick an answer to see if you got it, and why.
| Good | Income elasticity of demand | Cross-price elasticity of demand with good Z |
|---|---|---|
| J | 1.2 | 0.8 |
| K | −0.6 | −1.5 |
| L | 0.3 | 0 |
Hypothetical elasticity estimates
Which of the goods in the table is an inferior good?
Which of the following describes the relationship between good Z and goods J, K and L?
If the price of good Z rises by 10 percent, the quantity demanded of good K will
If consumers' incomes rise by 10 percent, the quantity demanded of good J will
0 of 4 answered