How to Calculate Point Elasticity of Demand
Point elasticity of demand multiplies the slope dQ/dP by P/Q at one point on the curve: E = (dQ/dP) × (P/Q). For Q = a − bP, that is −b × P/Q.
The Point Elasticity formula
Calculator
For linear demand Q = a − bP, enter the intercept, the slope and a price to get the point elasticity there.
Quantity demanded when the price is zero.
Quantity lost for each $1 rise in price.
E = (dQ/dP) × (P/Q) = −bP ÷ (a − bP).
- Quantity at this price
- 60
- Type
- Inelastic
- Unit-elastic price
- $25
Compare the absolute value with 1.
Halfway down a linear demand curve, at P = a ÷ (2b).
How to calculate Point Elasticity, step by step
- 1Write demand with Q on the left. Put the curve in the form Q = a − bP. If you are given inverse demand P = c − mQ, solve for Q first so the slope you read is dQ/dP, not dP/dQ.
- 2Read the slope dQ/dP. For Q = a − bP the slope is −b everywhere on the line. For a curved demand function, differentiate and evaluate the derivative at the price you were given.
- 3Find Q at the chosen price. Plug the price into the demand equation to get the quantity at that exact point.
- 4Multiply the slope by P/Q. E = (dQ/dP) × (P/Q). The result is negative because demand slopes down.
- 5Classify with the absolute value. |E| > 1 is elastic, |E| < 1 is inelastic, and |E| = 1 is unit elastic. A price rise raises total revenue only where demand is inelastic.
Worked example: Point Elasticity
Demand is Q = 100 − 2P, so dQ/dP = −2. At P = $20, Q = 100 − 2 × 20 = 60, and E = −2 × (20 ÷ 60) = −40 ÷ 60 ≈ −0.67, which is inelastic. At P = $30, Q = 100 − 2 × 30 = 40, and E = −2 × (30 ÷ 40) = −60 ÷ 40 = −1.5, which is elastic. At P = $25, Q = 100 − 50 = 50, and E = −2 × (25 ÷ 50) = −1, unit elastic. Check with the shortcut E = P ÷ (P − 50), where $50 is the price at which quantity reaches zero: 20 ÷ (20 − 50) = 20 ÷ (−30) ≈ −0.67 and 30 ÷ (30 − 50) = 30 ÷ (−20) = −1.5, matching both answers.
Point elasticity versus the midpoint (arc) method
Point elasticity answers the question at one price. Arc elasticity, which the midpoint method computes, answers it across a stretch of the curve between two prices. They are different measurements, and on the same demand curve they usually give different numbers.
Take the example and move price from $20 to $30. The point elasticities at the two ends are −0.67 and −1.5. The midpoint method divides by averages instead: %ΔQ = (40 − 60) ÷ 50 = −40% and %ΔP = (30 − 20) ÷ 25 = 40%, so arc elasticity = −40% ÷ 40% = −1. That single number sits between the two endpoint values and describes the move as a whole.
On a straight-line demand curve the two methods meet at one place: the arc elasticity between two points equals the point elasticity at the midpoint price. Here the midpoint price is $25, and the point elasticity there is exactly −1. On a curved demand function the two drift apart, and the gap grows as the price change gets larger.
Use point elasticity when you have a demand equation and a single price, or when the change is very small. Use the midpoint method when a question hands you two observed price and quantity pairs, which is how most introductory exam questions are written.
Why elasticity falls as you move down a straight line
The slope of Q = 100 − 2P never changes, yet elasticity runs from very elastic near the top of the line to very inelastic near the bottom. The P/Q term does all the work. Near the price intercept of $50, quantity is small, so P/Q is large and |E| is large. Near the quantity intercept of 100, price is small, so P/Q is close to zero and so is |E|.
Halfway down, at P = a ÷ (2b) = 100 ÷ 4 = $25, the two effects balance and E = −1. That is also where total revenue peaks. Revenue is P × (100 − 2P) = 100P − 2P², which is largest at P = $25, giving 25 × 50 = $1,250. Above $25 demand is elastic, so a price cut raises revenue. Below $25 demand is inelastic, so a price rise raises revenue.
Point Elasticity questions
What is the difference between point elasticity and arc elasticity?
Point elasticity measures responsiveness at a single price using the slope there. Arc elasticity measures it between two prices using average values, which is the midpoint method. On a straight line, the arc value equals the point value at the midpoint price.
Can I use the slope of an inverse demand curve to find point elasticity?
Not directly. For P = 50 − 0.5Q, the slope is dP/dQ = −0.5, and the formula needs dQ/dP, its reciprocal: 1 ÷ (−0.5) = −2. Plugging in −0.5 makes every answer on this curve four times too small.
Does a straight-line demand curve have constant elasticity?
No. The slope is constant, but P/Q changes at every point, so elasticity shrinks in absolute value as you move down the line. Constant elasticity needs a curve of the form Q = A × P^(−e), where E = −e everywhere.
Why is point elasticity of demand negative?
Quantity demanded falls when price rises, so dQ/dP is negative while P and Q are positive. Most courses report the absolute value when labeling demand elastic or inelastic.
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