Income & Cross-Price Elasticity
Measure how demand responds to buyer income (YED) or to a related good’s price (XED), using the midpoint method.
YED = 1.80 — normal good — luxury.
Income elasticity (YED) = %ΔQ ÷ %Δincome; cross-price elasticity (XED) = %ΔQ of A ÷ %Δprice of B, both by the midpoint method. If income rises from $40,000 to $50,000 and meals bought rise from 20 to 30, YED ≈ 1.8 — a normal luxury good. A positive XED means substitutes; a negative XED means complements.
Income vs. cross-price elasticity
Income elasticity of demand (YED) measures how much quantity demanded responds to a change in buyers’ income, holding prices fixed. It is the percentage change in quantity demanded divided by the percentage change in income. Its sign tells you the good type: a positive YED marks a normal good, whose demand rises with income, while a negative YED marks an inferior good, whose demand falls as people grow richer and trade up to something better.
Cross-price elasticity of demand (XED) measures how the quantity demanded of one good (A) responds to a change in the price of another good (B). Here the sign reveals the relationship between the two goods. A positive XED means they are substitutes — good B getting dearer pushes buyers toward good A. A negative XED means they are complements, bought together, so a higher price for B drags down demand for A. An XED near zero means the goods are unrelated.
Midpoint method: each %Δ = (new − old) ÷ ((old + new) ÷ 2). YED divides by the change in income; XED divides by the change in the price of the related good B.
Worked example
A household’s income rises from $40,000 to $50,000, and the restaurant meals it buys each month rise from 20 to 30.
- 1 Find the percentage change in quantity. %ΔQ = (Q₂ − Q₁) ÷ ((Q₁ + Q₂) ÷ 2) = (30 − 20) ÷ 25 = 0.40.
- 2 Find the percentage change in the driver. For YED use income: %Δincome = (50,000 − 40,000) ÷ 45,000 = 0.2222. For XED, use the price of good B instead.
- 3 Divide %ΔQ by the driver’s %Δ. YED = 0.40 ÷ 0.2222 = 1.8.
- 4 Read the sign. YED is positive, so meals are a normal good. A negative YED would flag an inferior good; for XED, positive means substitutes and negative means complements.
- 5 Read the size (income only). Because YED > 1, demand rises faster than income, so restaurant meals are a normal luxury. A YED between 0 and 1 would be a necessity.
Reading the elasticity value
Income elasticity (YED) is classified by both sign and size; cross-price elasticity (XED) by sign.
| Measure | Value | Good type | Example |
|---|---|---|---|
| YED | Less than 0 | Inferior good — demand falls as income rises | Bus travel, instant noodles, store-brand staples |
| YED | Between 0 and 1 | Normal good, necessity — demand rises slower than income | Food at home, electricity, toothpaste |
| YED | Greater than 1 | Normal good, luxury — demand rises faster than income | Restaurant meals, foreign holidays, jewellery |
| XED | Greater than 0 | Substitutes — buy one instead of the other | Tea and coffee, Coke and Pepsi |
| XED | Near 0 | Unrelated goods — little to no link | Textbooks and toothpaste |
| XED | Less than 0 | Complements — bought and used together | Printers and ink, cars and petrol |
Signs, sizes, and common pitfalls
Sign first, then size. For income elasticity, the sign separates inferior from normal goods, and the size (above or below 1) then splits normal goods into luxuries and necessities. For cross-price elasticity there is no size threshold — the sign alone tells you whether goods are substitutes or complements, and a larger magnitude just means a stronger relationship.
The midpoint method keeps results symmetric. As with own-price elasticity, dividing each change by the average of the old and new values (rather than the starting value) means you get the same elasticity whether income or price rose or fell between the two points. That is why intro courses pair YED and XED with the same arc formula.
Watch what changes. Income elasticity assumes prices are held constant while income moves; cross-price elasticity assumes income and the price of good A are held constant while only the price of good B moves. If more than one thing changes at once, the ratio no longer isolates a single effect, and the good-type label can mislead.