Price Elasticity of Demand

E_d = \frac{\%\Delta Q}{\%\Delta P} = \frac{\Delta Q / \bar{Q}}{\Delta P / \bar{P}}

$Q_0, Q_1$: quantity demanded before and after the price change
$P_0, P_1$: the price before and after
$\bar{Q}, \bar{P}$: the midpoints: averages of the two quantities and the two prices, used as bases
$E_d$: elasticity: % quantity response per 1% price change (negative for normal demand)
$|E_d| > 1$: elastic — buyers flee faster than price rises, so a price rise LOWERS revenue

Why it must be true

Raise the price and customers buy less — but how much less is the entire game. Elasticity is the exchange rate between the two percentage changes: the % change in quantity per 1% change in price. Beyond −1 ("elastic"), buyers flee faster than the price rises and revenue FALLS when you raise prices; between 0 and −1 ("inelastic"), they mostly stay and revenue rises. Gasoline is inelastic; one brand of gasoline among many is highly elastic.

The midpoint (arc) convention divides each change by the average of the endpoints rather than the starting value — so the answer is the same whether the price went from $8 to $10 or $10 to $8. It removes the embarrassing dependence on which direction you tell the story.

The derivation

Percentage changes need a base, and using the starting point makes A→B differ from B→A. The midpoint convention uses the average as the base for both changes:

\%\Delta Q = \frac{Q_1 - Q_0}{(Q_0 + Q_1)/2}, \qquad \%\Delta P = \frac{P_1 - P_0}{(P_0 + P_1)/2}

Their ratio is the arc elasticity:

E_d = \frac{(Q_1 - Q_0)/\bar{Q}}{(P_1 - P_0)/\bar{P}}

For demand it comes out negative (price up, quantity down) — the sign is the law of demand showing through; magnitude is what gets classified as elastic or inelastic.

When to reach for it

Measuring demand's sensitivity to a price change between two observed points — and classifying it as elastic or inelastic for revenue reasoning.

Listen for

“price elasticity of demand”“midpoint / arc method”“quantity demanded falls from … to … when price rises from … to …”“is demand elastic or inelastic”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

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Eyeball the two percentage moves against their midpoints before dividing: quantity swings ~20% on a ~10% price move → elasticity ≈ −2. The division is the easy part.
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Revenue check beats memory: if the price rose and total revenue (P×Q) fell, demand was elastic — |E| > 1. Compute revenue at both points as a cross-check.
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Sign discipline: normal demand gives a NEGATIVE elasticity. Choices offering the same magnitude with both signs? Price up, quantity down → minus.

Traps in applying it

✗Using the starting values as bases instead of the midpoints — the answer then depends on the direction of the story.
✗Inverting the ratio — price change over quantity change measures the wrong sensitivity.
✗Dropping the sign AND the interpretation: elastic vs inelastic classification uses the magnitude, revenue logic uses the sign.

Limits & criticisms

An arc elasticity is an average over the segment measured — demand curves have different elasticities at every point (a linear curve runs from elastic to inelastic along its length), so the number moves with the interval chosen. It also assumes nothing else changed between the two observations — income, rivals' prices, seasons — an assumption real sales data almost never grants; that is why econometricians exist.

Where it came from

Alfred Marshall gave elasticity its name and notation in his Principles of Economics (1890) — the book that trained generations of economists — formalizing a question tax authorities and monopolists had fumbled for centuries: who bears a tax, and what price maximizes revenue? Today elasticities run airline yield-management systems, utility rate cases, sin-tax policy design and every pricing team's revenue model; "our demand is inelastic" is a claim worth billions when it's true.

One identity, 1 questions

The exam can hide any variable. Each face below is the same equation solved for a different unknown — drill them separately.

Demand's sensitivity dial

E_d = \frac{\Delta Q / \bar{Q}}{\Delta P / \bar{P}}

The pricing-power face: |E| > 1 means price rises destroy revenue; |E| < 1 means the seller can squeeze.

Drill this face →

On the BA II Plus

Worked example: A price increase from $9.00 to $13.00 cuts weekly sales from 360 to 290 units. Compute the arc (midpoint) price elasticity of demand.

1.290 [−] 360 [=] [÷] [(] 360 [+] 290 [)] [×] 2 [=]%ΔQ on the midpoint base
2.[STO] 1park it
3.13 [−] 9 [=] [÷] [(] 9 [+] 13 [)] [×] 2 [=]%ΔP on the midpoint base
4.[STO] 2 [RCL] 1 [÷] [RCL] 2 [=]quantity response per price change

→ -0.5923

Drill this formula Back to the library