Other Elasticities — AP® Microeconomics Notes

When a city's factories hire again and paychecks rise, the steakhouse downtown fills up and the bus company sells fewer monthly passes. When a cafe raises the price of coffee, it sells more tea and less cream. Price elasticity of demand measures how buyers respond to a good's own price. Two more elasticities measure how they respond to their income and to the price of another good, and for both, the sign is the most important part of the answer.

Income elasticity of demand

$E_I = \frac{\%\Delta Q_d}{\%\Delta \text{income}}$

The income elasticity of demand is the percent change in quantity demanded divided by the percent change in buyers' income, with prices held the same. Keep the sign:

Positive: a normal good. More income, more demand. If the value is between 0 and 1, demand grows more slowly than income (bread, toothpaste). If it is above 1, demand grows faster than income (restaurant meals, travel).
Negative: an inferior good. More income, less demand, because buyers switch to something they prefer (bus passes when people can afford cars, store-brand cereal).

Worked example: three goods when incomes rise. In Wexmoor, an invented town, average income rises from $4,000 to $4,400 a month (invented data). Monthly purchases per household change as follows: restaurant meals 4 to 5, bus passes 20 to 19, loaves of bread 50 to 51.

Percent change in income. $\frac{4400 - 4000}{4000} \times 100$ $= 10\%$ .
Restaurant meals. $\frac{5 - 4}{4} \times 100 = 25\%$ , so $E_I = 25 \div 10 = 2.5$ : normal, and demand grows faster than income.
Bus passes. $\frac{19 - 20}{20} \times 100 = -5\%$ , so $E_I = -5 \div 10 = -0.5$ : inferior.
Bread. $\frac{51 - 50}{50} \times 100 = 2\%$ , so $E_I = 0.2$ : normal, growing slowly.

Dropping the minus sign on bus passes turns an inferior good into a normal one and loses the point.

Cross-price elasticity of demand

$E_{XY} = \frac{\%\Delta Q_d \text{ of X}}{\%\Delta P \text{ of Y}}$

The cross-price elasticity of demand is the percent change in quantity demanded of one good divided by the percent change in the price of another. Keep the sign:

Positive: substitutes. The price of coffee rises and buyers switch to tea, so tea sales rise: both changes go the same way.
Negative: complements. The price of coffee rises, people drink less coffee, so they buy less cream: the changes go opposite ways.
Near zero: the goods are unrelated (coffee and postcards).

Worked example: a cafe raises the price of coffee. The Lantern Cafe raises coffee from $2.50 to $3.00, a $\frac{0.50}{2.50} \times 100 = 20\%$ rise (invented data). Tea sales go from 100 to 110 cups (+10%); cream from 200 to 188 pitchers (−6%).

Tea. $10 \div 20 = 0.5$ . Positive: substitutes.
Cream. $-6 \div 20 = -0.3$ . Negative: complements.

Read the sign, then the size

Two number lines: a negative income elasticity means an inferior good and a positive one a normal good; a negative cross-price elasticity means complements, zero means unrelated goods, and a positive one means substitutes. — **Figure 1.** For income and cross-price elasticity, the sign is the answer: inferior or normal; complements, unrelated or substitutes. LevlPrep original diagram.

Price elasticity of demand is negative for nearly every good, so its sign tells you nothing new and you read its absolute value. Income and cross-price elasticity can be either sign, and the sign is the information. The size then tells you how strong the link is: a cross-price elasticity of 2 means close substitutes; 0.1 means weak ones.

Elasticity	Divide the percent change in quantity demanded by	Read
Price elasticity of demand	the percent change in the good's own price	absolute value against 1
Income elasticity	the percent change in income	sign: + normal, − inferior
Cross-price elasticity	the percent change in another good's price	sign: + substitutes, − complements

The same idea works for any determinant of demand: a percent change in quantity over a percent change in whatever caused it. You can also run it forward: with an income elasticity of 1.8, a 5% rise in income raises the quantity demanded by about $1.8 \times 5 = 9\%$ .

Linking it to a graph

These elasticities describe shifts of demand. When the price of coffee rises, the demand curve for tea (a substitute) shifts right. When incomes fall, demand for an inferior good such as used cars shifts right too. Name the elasticity's sign, then draw the shift it implies.