The same percent-change ratio (Price Elasticity of Demand) measures how demand responds to income and to the prices of related goods (Substitutes and Complements). This time the sign is the point.
Key idea
Income elasticity $= \dfrac{\%\Delta Q_d}{\%\Delta \text{income}}$. Positive: a normal good (above 1, a luxury; between 0 and 1, a necessity). Negative: an inferior good. Cross-price elasticity of good X with respect to the price of Y $= \dfrac{\%\Delta Q_{d,X}}{\%\Delta P_Y}$. Positive: substitutes. Negative: complements. Near zero: unrelated. Keep the signs.
Why it works
For price elasticity of demand the sign is always negative, so dropping it loses nothing. Here the sign is not known in advance; it carries the answer. If quantity rises when income rises, both changes are positive and so is the ratio: normal. If tea sales rise when coffee's price rises, both changes have the same sign: substitutes. Opposite directions give a negative ratio: inferior goods or complements.
Worked example: Income elasticity
When average income in a town rises from \$40,000 to \$60,000, sales of a meal kit rise from 10,000 to 14,000 a month. Find the income elasticity (midpoint method) and classify the good.
- $\%\Delta Q = \dfrac{4{,}000}{12{,}000} \approx 33.3\%$.
- $\%\Delta \text{income} = \dfrac{20{,}000}{50{,}000} = 40\%$.
- Income elasticity $\approx \dfrac{33.3\%}{40\%} \approx +0.83$: positive and below 1, a normal good and a necessity for these buyers.
Answer: About +0.83: a normal good (a necessity).
Worked example: Cross-price elasticity
The price of printers rises by 10%, and the quantity of ink cartridges demanded falls by 15%. Find the cross-price elasticity and say how the goods are related.
- $$\dfrac{\%\Delta Q_{\text{ink}}}{\%\Delta P_{\text{printers}}} = \dfrac{-15\%}{+10\%} = -1.5$$
- Negative: complements, as you'd expect for goods used together.
Answer: −1.5: complements.
Common mistake
Dropping the sign as you would for price elasticity of demand. An income elasticity of −0.4 and one of +0.4 describe opposite kinds of goods. Check: before dividing, write the sign of each percent change. Same signs give a positive ratio; opposite signs give a negative one.