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Elasticity Arithmetic & the Total Revenue Test

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Elasticity is a ratio of percentage changes

Price elasticity of demand is the percentage change in quantity demanded divided by the percentage change in price. Because demand slopes downward the raw value is negative, so it is conventionally reported as an absolute value. Greater than 1 is elastic — quantity responds proportionally more than price. Less than 1 is inelastic. Exactly 1 is unit elastic.

The midpoint (arc) method
E = [ΔQ / ((Q₁+Q₂)/2)] ÷ [ΔP / ((P₁+P₂)/2)]
Dividing by the average rather than the starting value makes the answer the same whichever direction you compute it. Using the starting value gives two different answers for the same pair of points.

The total revenue test

Total revenue is P × Q, and a price change moves those two in opposite directions — so which wins depends on elasticity. If demand is elastic, quantity dominates: raising the price lowers revenue. If inelastic, price dominates: raising the price raises revenue. At unit elastic, revenue is at its maximum and unchanged by a small price move. This is the practical payoff of the whole topic, and it is how the exam asks about elasticity without using the word.

What makes demand elastic

Four things. Availability of substitutes — more substitutes means more elastic. Share of budget — a good taking a large share of income is more elastic, because a price change matters more. Necessity versus luxury — necessities are inelastic. Time horizon — demand is more elastic over longer periods, since consumers can find alternatives. Insulin is the standard inelastic example; a particular brand of soft drink the standard elastic one.

Worked example

When price rises from $8 to $12, quantity demanded falls from 60 to 40. Compute the price elasticity of demand using the midpoint method, classify it, and state what happens to total revenue.

  1. 1.ΔQ = 40 − 60 = −20. Average Q = (60 + 40)/2 = 50. So %ΔQ = −20/50 = −0.40.
  2. 2.ΔP = 12 − 8 = 4. Average P = (8 + 12)/2 = 10. So %ΔP = 4/10 = 0.40.
  3. 3.E = −0.40 / 0.40 = −1.0, so |E| = 1.0.
  4. 4.Check revenue: before, 8 × 60 = $480; after, 12 × 40 = $480. Unchanged.
Answer: |E| = 1.0, unit elastic. Total revenue is unchanged at $480, which is exactly what unit elasticity predicts — the percentage gain in price is exactly offset by the percentage loss in quantity.
Watch out

A straight-line demand curve does not have constant elasticity. Elasticity varies along it — elastic at high prices and low quantities, inelastic at low prices and high quantities, unit elastic at the midpoint. Slope and elasticity are not the same thing.

Checkpoint

A firm raises its price and finds total revenue falls. Demand for its product must be:

Checkpoint

Price falls from $20 to $16 and quantity rises from 90 to 110. Using the midpoint method, |E| is approximately:

Checkpoint

Which good would you expect to have the most inelastic demand?

On the exam

Use the midpoint method unless the question explicitly says otherwise, and show the two averages. Rubrics award the setup separately from the value, so a visible ΔQ/average-Q calculation earns credit even if the division slips.

Answer the 3 checkpoints as you read.

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