Elasticity and revenue
Learning goals
After this page, you can:
- State the revenue rule and apply it to a proposed price change.
- Explain why revenue is highest where |ε| = 1.
- Estimate the percent change in revenue from two observations or from an elasticity, and keep the revenue question separate from the profit question.
The idea
A streaming service raises its price. It loses subscribers. The same quarter, it reports higher revenue. The headline looks like a contradiction. It is not.
Revenue is a tug of war. Revenue is price times quantity: R = P × Q. When you raise the price, two things happen at once. Each unit brings in more (good for revenue). You sell fewer units (bad for revenue). Which side wins is exactly what elasticity measures. A useful shortcut: %ΔR ≈ %ΔP + %ΔQ. Raise price 10% and lose 4% of sales: revenue rises about 6%. Raise price 10% and lose 15% of sales: revenue falls about 5%.
The revenue rule.
- Inelastic demand (|ε| < 1): the quantity loss is smaller than the price gain. Raising price raises revenue. Cutting price lowers it.
- Elastic demand (|ε| > 1): the quantity loss is larger than the price gain. Raising price lowers revenue. Cutting price raises it.
- Unit elastic (|ε| = 1): the two cancel. This is where revenue is at its maximum.
The manager’s version: if demand is inelastic at today’s price, you are leaving money on the table. If it is elastic, you are past the peak, and a cut would bring in more. The streaming service, with |ε| ≈ 0.2, was deep in the inelastic zone. A price rise was almost sure to lift revenue, and it did.
Why the peak is at |ε| = 1. On a straight demand line, elasticity is high at the top (high price, few units) and low at the bottom (low price, many units). Start at the top and cut price: you are in the elastic zone, so revenue rises. Keep cutting: elasticity falls toward 1 and the gains shrink. Pass |ε| = 1 and further cuts lose revenue. The revenue curve is a hill. Its top is at the unit-elastic point — halfway down a straight demand line.
Two ways to estimate the revenue effect. From two observations: revenue before = P₁Q₁, revenue after = P₂Q₂. No elasticity needed. From an elasticity: a price change of x% changes revenue by about x% × (1 − |ε|). A 10% rise with |ε| = 0.6 gives about +4%. With |ε| = 1.5, about −5%. Both are approximations. They get rougher for big changes — one reason to test price changes in small steps and in one market first.
Revenue is not profit. Selling fewer units also saves the cost of making them. So when a price cut raises revenue, profit may still fall if the extra units cost more than they bring in. Keep two questions apart: what does the price change do to revenue? (this page) and what does it do to profit? (the monopoly pricing pages add costs).
R = P × Q · %ΔR ≈ %ΔP + %ΔQ ≈ %ΔP × (1 − |ε|)
Inelastic → raise price, revenue up. Elastic → raise price, revenue down. Peak at |ε| = 1.
Try it
Part 1: slide along the line. The grey rectangle under the demand line is revenue (P × Q). The lower chart shows revenue at every quantity.
- Find the price where revenue stops rising. What is |ε| there?
- Start at $4.50 and cut the price in 50-cent steps. When does a cut start to hurt revenue?
- Set the price to $1.00. Would you advise raising it? Why?
Part 2: your own two price points. Type a price and quantity before and after a change — your company’s, or the coffee cart’s. The calculator gives the arc elasticity, revenue before and after, and whether the revenue rule predicted the result.
Worked example
Problem. A coffee cart sells 450 cups a week at $3.00. It raises the price to $3.50 and sells 340. Was that a good idea for revenue?
Step 1. Arc elasticity. %ΔQ = −110 ÷ 395 = −27.8%. %ΔP = 0.50 ÷ 3.25 = 15.4%. |ε| = 27.8 ÷ 15.4 = 1.81 — elastic.
Step 2. Revenue. Before: 3.00 × 450 = $1,350. After: 3.50 × 340 = $1,190. Down 11.9%.
Step 3. Check the rule. Elastic demand, price up, revenue down. The rule predicted it.
Step 4. The profit twist. Fewer cups also means less milk and fewer paper cups. If each cup costs $1.20 to make, contribution falls from 450 × $1.80 = $810 to 340 × $2.30 = $782. Still down, but by less than revenue. The direction is the same here. It is not always the same — when variable cost is high, a price rise can lower revenue and still raise profit.
Check yourself
At its current price a toll bridge has |ε| = 0.4. If the operator raises the toll 10%, revenue will:
%ΔR ≈ %ΔP × (1 − |ε|).
A café cuts its latte price from $5 to $4 and daily sales rise from 200 to 300. What happened to revenue, and what does that say about elasticity?
Price down + revenue up → elastic.
On a straight-line demand curve, total revenue is highest where:
Revenue is a hill with its peak at unit elasticity.
A software firm’s demand is elastic (|ε| = 2) and its variable cost per copy is nearly zero. It is considering a 10% price cut. Which is the best summary?
Revenue rule + cost structure together answer the profit question.
Which statement correctly separates revenue from profit?
Keep the two questions apart: revenue (this page) and profit (Week 5).
Video
Video coming soon.