Price elasticity of demand explained with worked examples

Price elasticity of demand answers one question: when the price of something changes, how much does the quantity people buy change in response? Not in units, but in percentages. It is the first real calculation in a principles of microeconomics course, and it comes back later in pricing, tax incidence and policy questions, so it is worth getting the setup right once.

The definition and the three categories

Price elasticity of demand = percentage change in quantity demanded / percentage change in price.

Because price and quantity demanded move in opposite directions, the raw number is negative. Economists report its absolute value. Then:

Elasticity has no units, since it is a ratio of two percentages. A cafe that raises latte prices by 20% and sees sales fall by 5% has an elasticity of 5 / 20 = 0.25: inelastic. A store that cuts a price by 10% and sees quantity rise by 25% has 25 / 10 = 2.5: elastic.

Why the midpoint formula exists

Ordinary percentage change uses the starting value as the base. That causes a problem: the answer depends on which direction you travel.

Take a smartphone whose price falls from $70 to $60 while quantity demanded rises from 2,800 to 3,000.

• Going down: price changes by -10 / 70 = -14.29%, quantity by 200 / 2,800 = 7.14%. Elasticity = 7.14 / 14.29 = 0.50.
• Going up between the same two points: price changes by 10 / 60 = 16.67%, quantity by -200 / 3,000 = -6.67%. Elasticity = 6.67 / 16.67 = 0.40.

Same two points, two different answers. The midpoint method fixes this by using the average of the two values as the base:

Midpoint % change = (new - old) / ((new + old) / 2) x 100

Worked example 1: an inelastic price cut

Use the same smartphone, $70 to $60, quantity 2,800 to 3,000.

1. Quantity: (3,000 - 2,800) / ((3,000 + 2,800) / 2) = 200 / 2,900 = 6.90%
2. Price: (60 - 70) / ((60 + 70) / 2) = -10 / 65 = -15.38%
3. Elasticity: 6.90 / 15.38 = 0.45 (absolute value)

Now run it in the other direction, $60 to $70: the bases are still 2,900 and 65, so the answer is still 0.45. That symmetry is the whole point. Compute with unrounded percentages and round only the final elasticity to 2 decimal places, which is the rounding rule the course cards use.

Demand here is inelastic: a 15% price cut brought only about a 7% rise in quantity.

Worked example 2: an elastic price rise

The price rises from $120 to $130, and quantity demanded falls from 1,800 to 1,600.

1. Quantity: (1,600 - 1,800) / 1,700 = -11.76%
2. Price: (130 - 120) / 125 = 8.00%
3. Elasticity: 11.76 / 8.00 = 1.47

Greater than 1, so demand is elastic in this price range.

The total revenue rule

Total revenue (TR) is price times quantity sold. Elasticity tells you which way TR moves when price changes, without doing the multiplication:

• Elastic demand: raise the price and TR falls; cut it and TR rises.
• Inelastic demand: raise the price and TR rises; cut it and TR falls.
• Unitary: TR stays the same.

Check it against the two examples:

• Example 1 (inelastic, price cut): $70 x 2,800 = $196,000 before, $60 x 3,000 = $180,000 after. Revenue fell by $16,000, as the rule predicts.
• Example 2 (elastic, price rise): $120 x 1,800 = $216,000 before, $130 x 1,600 = $208,000 after. Revenue fell by $8,000, again as predicted.

One more from the course: a firm raises its price from $4 to $5 and sales fall from 1,000 to 900. TR goes from $4,000 to $4,500, up $500. The midpoint elasticity is (100 / 950) / (1 / 4.5) = 0.47, inelastic, so a price rise raising revenue is exactly what the rule says.

This is also why a bumper harvest can leave farmers with less revenue. Food demand is inelastic, so the extra supply only sells at a much lower price, and the price falls by a larger percentage than quantity rises.

Elasticity is not slope

A straight-line demand curve has the same slope everywhere, but its elasticity changes along the line. Demand is more elastic at high prices and low quantities, and less elastic at low prices and high quantities. The reason is the base: at low quantities, a one-unit change is a big percentage; at high prices, a dollar change is a small one.

Take the demand curve P = 48 - 3Q as quantity rises from 5 to 6. The price goes from 48 - 15 = 33 to 48 - 18 = 30.

1. Quantity: 1 / 5.5 = 18.18%
2. Price: -3 / 31.5 = -9.52%
3. Elasticity: 18.18 / 9.52 = 1.91

Elastic, at the high-price end of this line. Revenue confirms it: $33 x 5 = $165 rises to $30 x 6 = $180 after the price cut. Further down the same line, with the same slope of -3, demand becomes inelastic.

The polar cases are the exception where the shape tells you everything. A horizontal curve is perfectly elastic. A vertical curve has zero elasticity: a concert hall with exactly 2,000 seats has a perfectly inelastic supply of tickets.

What makes demand elastic or inelastic

The course cards point to three drivers:

• Close substitutes. When Netflix raised the price of its combined plan by about 60% in 2011, it lost more subscribers than it expected, because people had other options.
• Necessity. Insulin for a diabetic patient has highly inelastic demand. Designer handbags and cruises do not.
• Time. Elasticities are larger in the long run. Gasoline demand is inelastic in the short run and more elastic once people can change cars, commutes or homes.

Elasticity also predicts quantity directly: predicted % change in quantity = elasticity x % change in price. With an elasticity of 0.45, a 10% price rise cuts quantity demanded by 4.5%. And it decides who bears a tax: the more inelastic side of the market pays more of it, whoever legally writes the check.

Other elasticities use the same formula

Income elasticity and cross-price elasticity use the same midpoint arithmetic, but you keep the sign, because the sign carries the answer.

Income rises from $25,000 to $38,000 and bread purchases fall from 30 to 22 loaves. Quantity changes by -8 / 26 = -30.77%, income by 13,000 / 31,500 = 41.27%, so income elasticity = -30.77 / 41.27 = -0.75. Negative, so bread is an inferior good for this household. A positive cross-price elasticity means substitutes; a negative one means complements, like printers and ink.

Practicing it

Elasticity is a setup problem more than a math problem. The usual errors are using the old value as the base, forgetting to take the absolute value, rounding the percentages too early, and confusing elastic with steep. Work fresh problems from a blank page rather than rereading solved ones; the generation effect is why producing the answer yourself sticks better. Then keep the formula on a spaced schedule so it is still there at the final; what spaced repetition actually does explains the timing.

Elasticity is unit 5 of the course, and what's in Principles of Microeconomics shows where it sits. How to study for microeconomics covers the cost tables and market structures that follow. Pricing decisions in the real world also need costs, which is where the break-even point calculator helps.

Unit 5 is shared with macroeconomics, where demand works differently at the scale of a whole economy: aggregate demand slopes down for reasons that have nothing to do with substitutes, as aggregate demand and aggregate supply explained shows. How to study for macroeconomics covers the rest of that course. Both are listed in free college gen-ed course flashcards.

Every elasticity calculation in the course is a typed card with a script-checked answer, free in Encodr's microeconomics course.