The law of demand (The Law of Demand) says quantity demanded falls when price rises. Elasticity says by how much, in a way that doesn't depend on units. It is built from percent changes, which you know from Percent Increase and Decrease.
Key idea
The price elasticity of demand is $$E_d = \left|\frac{\%\Delta Q_d}{\%\Delta P}\right|.$$ In this course we report it as a positive number. Use the midpoint method: divide each change by the average of the two values. Demand is elastic if $E_d > 1$, inelastic if $E_d < 1$ and unit elastic if $E_d = 1$.
Why it works
Why percents? A slope depends on units: a price change of 1 dollar is 100 cents, and a quantity in pounds is 16 times bigger in ounces. Percent changes have no units, so elasticity can compare gasoline with movie tickets. Why the midpoint? A plain percent change depends on the direction: going from 4 to 6 is a 50% rise, but from 6 to 4 is a 33% fall. Dividing by the average, 5, gives 40% either way, so the elasticity between two points is the same in both directions. The ratio is negative for demand (price and quantity move opposite ways), so we drop the sign and remember it.
Worked example: An elastic stretch
When a café raises the price of a muffin from \$4 to \$6, the number sold falls from 100 to 60 a day. Find the price elasticity of demand.
- $\%\Delta Q = \dfrac{60 - 100}{(100 + 60)/2} = \dfrac{-40}{80} = -50\%$.
- $\%\Delta P = \dfrac{6 - 4}{(4 + 6)/2} = \dfrac{2}{5} = 40\%$.
- $E_d = \left|\dfrac{-50\%}{40\%}\right| = 1.25$. Greater than 1: elastic.
Answer: $E_d = 1.25$, elastic.
Worked example: An inelastic stretch
When the price of a commuter train ticket rises from \$2 to \$3, daily riders fall from 110 to 90 (in hundreds). Find the elasticity.
- $\%\Delta Q = \dfrac{90 - 110}{100} = -20\%$.
- $\%\Delta P = \dfrac{3 - 2}{2.5} = 40\%$.
- $E_d = \dfrac{20\%}{40\%} = 0.5$: inelastic. Riders have few alternatives for getting to work.
Answer: $E_d = 0.5$, inelastic.
Common mistake
Three slips are common: dividing the percent change in price by the percent change in quantity (upside down), using the slope $\Delta Q/\Delta P$ instead of percent changes, and dividing the two changes by different bases (one by the old value, the other by the average). Check: quantity goes on top ("how much does quantity respond?"), and every percent change uses the midpoint of its own two values.
A quick use: if $E_d = 2$, a 5% price rise cuts quantity demanded by about $2 \times 5\% = 10\%$.