Arc Elasticity

Key takeaways

• Arc elasticity measures the responsiveness of one variable to another between two specific points using the midpoint method.
• It is commonly used to estimate how quantity demanded responds to a change in price when the change is large.
• The midpoint (arc) formula gives the same elasticity regardless of which point is treated as the start or end.
• Interpret elasticity by magnitude: |Ed| > 1 (elastic), |Ed| < 1 (inelastic), |Ed| = 1 (unitary).

What is arc elasticity?

Arc elasticity quantifies the percentage change in one variable (commonly quantity demanded) relative to the percentage change in another (commonly price) between two points on a curve. Unlike the simple point-percent method that references the initial value and can produce different results depending on direction of change, the arc (midpoint) method uses the average of the two values as the base. This makes the measure symmetric and more suitable for large changes.

Point elasticity vs. arc elasticity

• Point elasticity uses the initial value as the base for percentage changes. It is useful for very small (infinitesimal) changes or when one wants elasticity at a precise point on a curve.
• Arc elasticity (midpoint method) uses the average of the two values as the base, producing a single elasticity value that is independent of direction (price rise vs. fall). It is preferred for discrete or large changes between two observed points.

Formulas

Point elasticity (percent-change method):
Ed = (% change in Q) / (% change in P)
where % change in Q = (Q2 - Q1) / Q1 and % change in P = (P2 - P1) / P1.

Arc (midpoint) elasticity:
Arc Ed = [(Q2 - Q1) / ((Q1 + Q2)/2)] ÷ [(P2 - P1) / ((P1 + P2)/2)]

How to calculate (step-by-step)

1. Record the two prices P1 and P2 and the corresponding quantities Q1 and Q2.
2. Compute midpoint values:
3. Midpoint Q = (Q1 + Q2) / 2
4. Midpoint P = (P1 + P2) / 2
5. Compute percentage changes relative to midpoints:
6. %ΔQ = (Q2 – Q1) / Midpoint Q
7. %ΔP = (P2 – P1) / Midpoint P
8. Divide %ΔQ by %ΔP:
9. Arc Ed = %ΔQ / %ΔP
10. Interpret the result. Demand elasticities are typically negative (price up → quantity down), so consider absolute value for magnitude.

Example

Suppose price rises from $8 to $10 and quantity demanded falls from 60 to 40.

Point method (base = initial values):
* %ΔQ = (40 − 60) / 60 = −0.3333
%ΔP = (10 − 8) / 8 = 0.25
Ed = −0.3333 / 0.25 = −1.3333

If you reverse the start/end points (base = the other initial), you get a different value (e.g., −2.5). That asymmetry motivates the midpoint method.

Arc (midpoint) method:
* Midpoint Q = (60 + 40) / 2 = 50
Midpoint P = (8 + 10) / 2 = 9
%ΔQ = (40 − 60) / 50 = −0.4
%ΔP = (10 − 8) / 9 = 0.2222
Arc Ed = −0.4 / 0.2222 ≈ −1.8

Arc elasticity yields the same value (−1.8) regardless of which point is labeled 1 or 2.

Interpretation and use cases

• Magnitude: |Ed| > 1 → demand is elastic (quantity is relatively responsive to price changes). |Ed| < 1 → demand is inelastic. |Ed| = 1 → unitary elasticity.
• Sign: For demand, elasticity is typically negative; practitioners often quote absolute values.
• Use arc elasticity when comparing two observed points with a substantial change in price or quantity. Use point elasticity for marginal (infinitesimal) changes or when evaluating elasticity at a specific point on a smooth demand curve.

Conclusion

Arc elasticity (midpoint method) provides a consistent, symmetric measure of responsiveness between two points, making it the preferred approach for estimating elasticity when price or quantity changes are sizable. It avoids the ambiguity of the simple percent-change formula and yields a single interpretable elasticity value independent of direction.