Elasticity is a general idea
In Topics 2.3 and 2.4 you measured how much quantity responds to a change in a good’s own price. But price is only one influence. In Topic 2.1 you met the other determinants of demand: income, the prices of related goods, tastes, expectations and the number of buyers. Elasticity can measure the response to any of them.
Every elasticity has the same shape:
elasticity = (percentage change in quantity) ÷ (percentage change in the cause)
Change the “cause” and you get a new elasticity. This topic covers the two that the course asks you to calculate and interpret:
| Elasticity | Cause | Formula |
|---|---|---|
| Income elasticity of demand (YED) | a change in buyers’ income | % change in quantity demanded ÷ % change in income |
| Cross-price elasticity of demand (XED) | a change in the price of another good | % change in quantity demanded of good X ÷ % change in the price of good Y |
The same idea works on the supply side too. For example, you could measure how much the quantity supplied of furniture responds to a change in the price of timber. You will not be asked to name every possible elasticity, but you should recognise the pattern when you meet a new one.
An example with another determinant
A fictional bakery, Pellow’s, raises its monthly advertising spending from $10,000 to $14,000. At the same prices, the number of cakes it sells rises from 5,000 to 5,500 per month. Using the midpoint method:
- % change in quantity = 500 ÷ 5,250 × 100 = 9.52%.
- % change in advertising = 4,000 ÷ 12,000 × 100 = 33.33%.
- Advertising elasticity of demand = 9.52 ÷ 33.33 = 0.29.
Demand responds to advertising, but by a smaller percentage than the change in spending. The method is exactly the one you already know; only the cause has changed.
The midpoint method again
When you are given two values of quantity and two values of the cause, use the midpoint method from Topic 2.3:
- % change in quantity = (Q₂ − Q₁) ÷ [(Q₁ + Q₂) ÷ 2] × 100
- % change in the cause = (X₂ − X₁) ÷ [(X₁ + X₂) ÷ 2] × 100
Here X is income for YED, or the other good’s price for XED. If a question gives you the percentage changes directly (“income rises by 10% and quantity demanded rises by 15%”), just divide: YED = 15 ÷ 10 = 1.5.
One rule is different from Topic 2.3. For price elasticity of demand, the answer is always negative, so we often drop the sign. For YED and XED, keep the sign. It is the most important part of the answer.
Income elasticity of demand
Income elasticity of demand (YED) measures how responsive the quantity demanded of a good is to a change in buyers’ income, with the good’s price held constant.
YED = (% change in quantity demanded) ÷ (% change in income)
Economists use YED to decide whether a good is normal or inferior:
| Value of YED | What happens when income rises | Type of good |
|---|---|---|
| YED > 0 (positive) | quantity demanded rises | normal good |
| YED < 0 (negative) | quantity demanded falls | inferior good |
| YED = 0 | quantity demanded does not change | demand is unrelated to income |
The size of YED tells you how strongly demand responds. If incomes rise 10% and the quantity demanded of a good rises 15%, YED = 1.5. If incomes rise 10% and the quantity demanded falls 4%, YED = −0.4: an inferior good whose demand falls by a smaller percentage than income rises.
Background (useful, but not needed for the course). Many textbooks split normal goods further. When YED is greater than 1, demand is income elastic: spending on the good rises faster than income, as with many luxuries. When YED is between 0 and 1, demand is income inelastic: the quantity rises, but more slowly than income, as with many necessities. Use these labels only alongside “normal”, never instead of it.
What YED looks like on a graph
A change in income is not a change in the good’s own price. So it does not move buyers along the demand curve. It shifts the whole curve. When incomes rise:
- for a normal good, demand increases: the curve shifts right;
- for an inferior good, demand decreases: the curve shifts left.
When incomes fall, the shifts go the other way.
YED tells you how far the curve moves, measured as the percentage change in quantity demanded at each price.
Cross-price elasticity of demand
Cross-price elasticity of demand (XED) measures how responsive the quantity demanded of one good (X) is to a change in the price of a different good (Y), with X’s own price held constant.
XED = (% change in quantity demanded of X) ÷ (% change in the price of Y)
Economists use XED to decide how two goods are related:
| Value of XED | What happens when the price of Y rises | Relationship |
|---|---|---|
| XED > 0 (positive) | buyers switch to X: quantity demanded of X rises | substitutes |
| XED < 0 (negative) | buyers use less Y, so they need less X: quantity demanded of X falls | complements |
| XED = 0 (or very close to it) | no change in X | not related |
The size matters too. A large positive XED means the goods are close substitutes: a small rise in Y’s price sends many buyers to X. A large negative XED means strong complements, goods that are almost always used together. A value near zero means the link is weak.
Always say which good is X and which is Y. The XED of X with respect to the price of Y does not have to equal the XED of Y with respect to the price of X. A rise in the price of cars cuts demand for car tyres a lot, but a rise in the price of tyres hardly changes demand for cars, because tyres are a small part of the cost of owning a car.
What XED looks like on a graph
A change in Y’s price shifts the demand curve for X:
- if X and Y are substitutes, a rise in Y’s price increases demand for X (shift right);
- if X and Y are complements, a rise in Y’s price decreases demand for X (shift left).
Worked example 1: income elasticity in Corvale
Question. Corvale is a fictional town. Over two years, average household income rises from $800 to $1,000 per week. Prices of the goods below stay the same. A survey records the typical household’s monthly purchases:
| Good | Quantity before (per month) | Quantity after (per month) |
|---|---|---|
| Theatre tickets | 4 | 6 |
| Tinned soup (cans) | 30 | 24 |
| Milk (litres) | 20 | 21 |
| Table salt (kg) | 1 | 1 |
(a) Calculate the YED of each good using the midpoint method. (b) Classify each good. (c) Next year, incomes are expected to rise by a further 5%. Predict the percentage change in the quantity demanded of theatre tickets and of tinned soup at current prices.
(a) The income change is the same for every good:
- Change in income = 1,000 − 800 = 200. Average income = (800 + 1,000) ÷ 2 = 900.
- % change in income = 200 ÷ 900 × 100 = 22.22%.
Now each good:
| Good | Change in Q | Average Q | % change in Q | YED |
|---|---|---|---|---|
| Theatre tickets | +2 | 5 | +40.00% | 40 ÷ 22.22 = +1.80 |
| Tinned soup | −6 | 27 | −22.22% | −22.22 ÷ 22.22 = −1.00 |
| Milk | +1 | 20.5 | +4.88% | 4.88 ÷ 22.22 = +0.22 |
| Table salt | 0 | 1 | 0% | 0 |
(b)
- Theatre tickets: normal (YED positive). It is also large: quantity rises by a bigger percentage than income.
- Tinned soup: inferior (YED negative). As households get richer, they swap tinned soup for fresh food they prefer.
- Milk: normal (YED positive), but the response is small: households already buy most of the milk they want.
- Table salt: YED = 0, so demand is unrelated to income over this range.
(c) Rearrange the formula: % change in quantity = YED × % change in income.
- Theatre tickets: 1.8 × 5 = +9%.
- Tinned soup: −1.0 × 5 = −5%.
Check. If you divide by starting values instead of averages, tinned soup gives (−6 ÷ 30) ÷ (200 ÷ 800) = −0.20 ÷ 0.25 = −0.80. The sign is the same, so the classification is the same, but the number is not the midpoint answer. Use the midpoint method whenever you have two values of each.
Interpretation. The prediction in (c) is “at current prices”. Higher incomes shift the demand curves; they do not tell you the new market price. That needs supply as well, which comes in Topics 2.6 and 2.7.
Worked example 2: cross-price elasticity for games consoles
Question. Nova and Lumen are two fictional brands of games console. Nova games run only on Nova consoles. The price of a Nova console rises from $360 to $440. Over the same months, with their own prices unchanged:
- monthly sales of Nova games fall from 12,000 to 9,000;
- monthly sales of Lumen consoles rise from 5,000 to 6,000;
- monthly sales of board games stay at 2,000.
(a) Calculate the XED of each good with respect to the price of a Nova console. (b) Classify each relationship and explain it. (c) The maker of Nova consoles estimates that the XED of Nova consoles with respect to the price of Lumen consoles is +0.6. Lumen cuts its price by 10%. Predict the effect on the quantity of Nova consoles demanded at the current Nova price.
(a) First the price change, which is the same for all three:
- Change in price = 440 − 360 = 80. Average price = (360 + 440) ÷ 2 = 400.
- % change in the price of Nova consoles = 80 ÷ 400 × 100 = 20%.
Then each good:
| Good | Change in Q | Average Q | % change in Q | XED |
|---|---|---|---|---|
| Nova games | −3,000 | 10,500 | −28.57% | −28.57 ÷ 20 = −1.43 |
| Lumen consoles | +1,000 | 5,500 | +18.18% | 18.18 ÷ 20 = +0.91 |
| Board games | 0 | 2,000 | 0% | 0 |
(b)
- Nova games and Nova consoles are complements (XED negative). Games are useless without the console. When the console costs more, fewer people buy one, so fewer people need games. The size, 1.43, shows a strong link.
- Lumen consoles and Nova consoles are substitutes (XED positive). When Nova’s price rises, some buyers choose Lumen instead.
- Board games are not related to Nova consoles (XED = 0) over this range.
(c) % change in quantity of Nova consoles = XED × % change in Lumen’s price = 0.6 × (−10) = −6%. A cheaper substitute draws buyers away, so demand for Nova consoles falls: its curve shifts left by 6% at each price.
Check. The signs make sense. For complements, the price of one and the quantity of the other move in opposite directions (price up, games down), so XED is negative. For substitutes they move in the same direction, so XED is positive.
Interpretation. Firms use XED to plan. The maker of Nova consoles knows that a price rise on consoles will also cut its games sales. That is why some firms keep the price of a “base” product low and earn more on the complement.
Common misconceptions
- “Inferior goods are low-quality goods.” “Inferior” is about the sign of YED, not quality. A good is inferior if buyers want less of it when their income rises.
- “I can drop the minus sign, as with price elasticity of demand.” Not here. A YED of −1.0 and +1.0 describe opposite kinds of goods.
- “A rise in income moves buyers along the demand curve.” Income is not the good’s own price. It shifts the demand curve.
- “Complements have positive XED because the goods go together.” Going together means the price of one and the quantity of the other move in opposite directions, so XED is negative.
- “XED of A on B is always the same as XED of B on A.” It need not be. Always name which good’s quantity and which good’s price you are using.
- “XED = change in quantity of X ÷ change in price of Y.” Convert both changes to percentages first, as with every elasticity.
- “Income or cross-price elasticity tells me what happens to total revenue through the total revenue test.” That test is for a good’s own price only.
Where this leads
Elasticities measure how strongly demand and supply respond. Next, Topic 2.6 brings demand and supply together to find the market equilibrium and consumer and producer surplus, and Topic 2.7 shows how shifts like those in Figures 1 and 2 change the equilibrium. Test yourself with the practice questions, then use the revision notes and the checklist to consolidate. For own-price elasticity, look back at price elasticity of supply.
Quick questions
Why do I keep the minus sign here when I could drop it for price elasticity of demand?
For price elasticity of demand the sign is always negative, so it carries no news. For income and cross-price elasticity the sign can be positive or negative, and it is the sign that tells you whether a good is normal or inferior, or whether two goods are substitutes or complements.
Can I use the total revenue test with income or cross-price elasticity?
No. The total revenue test links a change in a good's own price to spending on that good. Income and cross-price elasticities describe shifts of the whole demand curve, so they need a different kind of reasoning.