What arc elasticity measures
– Arc elasticity quantifies the percentage responsiveness of one variable to another between two specific points. In economics, it’s most often used to express how quantity demanded responds to a change in price over a finite interval on a demand curve.
– Because it uses the midpoint (average) of the two prices and the two quantities as the base for percentage changes, arc elasticity gives the same value whether the price moves up or down between those two points.
Brief definitions
– Elasticity: the percentage change in one variable divided by the percentage change in another variable.
– Price elasticity of demand (Ed): percentage change in quantity demanded divided by percentage change in price.
– Arc elasticity of demand: Ed calculated using the midpoint (average) of the two prices and the two quantities as the base for the percentage changes.
Why use arc elasticity
– The usual percent-change formula depends on whether you measure change from the starting point or the ending point, so results differ depending on direction. Arc elasticity removes that asymmetry by using midpoint bases. It is therefore preferred when price or quantity changes are relatively large.
Formulas
1) Basic (point-to-point) price elasticity of demand:
Ed = (% change in quantity demanded) / (% change in price)
where % change = (new − old) / old
2) Arc (midpoint) elasticity of demand:
Ed_arc = [(Q2 − Q1) / ((Q1 + Q2) / 2)] ÷ [(P2 − P1) / ((P1 + P2) / 2)]
Equivalently:
Ed_arc = (ΔQ / average Q) / (ΔP / average P)
Notes on sign and interpretation
– Demand typically falls when price rises, so Ed is negative. Analysts often report the absolute value.
– |Ed| > 1 → elastic (quantity responds proportionally more than price).
– |Ed| 1 the change is elastic (quantity changed proportionally more than price). The negative sign simply reflects the inverse relationship between price and quantity demanded (law of demand).
Summary of the worked example (numbers):
– Price: $10 → $8 (ΔP = −2; midpoint P = 9; %ΔP ≈ −22.22%)
– Quantity: 40 → 60 (ΔQ = +20; midpoint Q = 50; %ΔQ = +40%)
– Arc elasticity: Ed_arc ≈ −1.8 (magnitude 1.8 → elastic)
Practical notes and checklist
– Use arc (midpoint) elasticity for discrete changes over an interval to avoid asymmetry from choosing P1/Q1 vs P2/Q2.
– Compute ΔP and ΔQ, compute midpoints (average P and average Q), compute percentage changes as Δ / midpoint, then divide %ΔQ by %ΔP.
– Remember the sign: negative for normal demand curves. Compare the absolute value to 1 to classify elasticity (>|1| elastic; <|1| inelastic; =|1| unit elastic).
– Limitations: arc elasticity gives an average elasticity over the interval, not the instantaneous (point) elasticity. It assumes linearity across the interval; results can differ for large changes or non-linear demand curves.
When to prefer point elasticity
– Use point elasticity (ε = (dQ/dP) × (P/Q)) when you have a demand function and want the instantaneous responsiveness at a specific price–quantity point.
Quick numerical check
– If you instead used initial values as the base (%ΔP = −20%, %ΔQ = +50%), you would get elasticity = 50% / (−20%) = −2.5. The midpoint method avoids this asymmetry and yields −1.8 instead.
Sources
– Investopedia — Arc Elasticity (midpoint method): https://www.investopedia.com/terms/a/arc-elasticity.asp
– Khan Academy — Price elasticity of demand: https://www.khanacademy.org/economics-finance-domain/microeconomics/elasticity-tutorial
– Encyclopedia Britannica — Price elasticity: https://www.britannica.com/topic/price-elasticity
Educational disclaimer: This explanation is for educational purposes only and is not individualized investment advice.