Arc Price Elasticity of Demand (Formula Guide)
Arc price elasticity of demand measures how much quantity demanded changes, in percentage terms, when price changes between two observations. It uses the average price and quantity as its reference points, so the result stays the same whichever endpoint you start from. The answer is usually negative for ordinary demand; its absolute value shows whether demand is elastic, inelastic, or unit elastic.
A price change can look small from one angle and large from another. If a product’s price rises from $10 to $12, that is a 20% increase when measured from $10, but a 16.7% decrease when measured back from $12. That difference can make a basic percentage method give different answers when you reverse the comparison.
The midpoint method avoids that problem by using the average of the two prices and the average of the two quantities. I use it when I want to compare two observed points on a demand curve without choosing one point as the only starting point.
Arc price elasticity of demand: definition, purpose, and formula
Arc price elasticity is a measure of how strongly quantity demanded responds to a price change across a specific interval. It compares percentage changes in quantity and price, with each percentage based on the average of the two observations. The result describes that interval, not every point on the demand curve.
The method is also called the midpoint method. “Arc” refers to the section of the demand curve between the two observations. It helps answer a practical question: between these two prices, how large was the quantity response compared with the price change?
The midpoint formula
The midpoint formula divides the percentage change in quantity by the percentage change in price. Each percentage uses the average of the starting and ending values as its denominator, which makes the calculation consistent when the endpoints are reversed.
[ E_d= \frac{(Q_2-Q_1)/[(Q_1+Q_2)/2]} {(P_2-P_1)/[(P_1+P_2)/2]} ]
Here:
- (P_1) is the initial price, and (P_2) is the final price.
- (Q_1) is the quantity demanded at the initial price, and (Q_2) is the quantity at the final price.
- (E_d) is the price elasticity of demand over the interval.
You can also use this equivalent form:
[ E_d=\frac{Q_2-Q_1}{P_2-P_1}\times\frac{P_1+P_2}{Q_1+Q_2} ]
Both versions give the same answer. The first makes the percentage changes clear. The second can be quicker once you understand the parts.
Midpoint calculations: variables, steps, and worked examples
A correct calculation starts with two comparable price-and-quantity observations. Keep the product, market, units, and time frame consistent, then label the earlier values with 1 and the later values with 2. This makes each step easier to check and helps prevent a misleading comparison.
Step-by-step calculation
Follow the same sequence each time. Keep the signs during the calculation, and round only after you have divided the two percentage changes. This reduces rounding errors and helps you explain whether the demand response moves in the expected direction.
- Record both observations. Write down (P_1, Q_1, P_2,) and (Q_2). For example, the first observation may be a price of $10 and sales of 100 units.
- Find the midpoint quantity change. Subtract (Q_1) from (Q_2), then divide by the average quantity: ((Q_2-Q_1)/[(Q_1+Q_2)/2]).
- Find the midpoint price change. Subtract (P_1) from (P_2), then divide by the average price: ((P_2-P_1)/[(P_1+P_2)/2]).
- Divide. Divide the quantity percentage change by the price percentage change. Report the signed result, and use its absolute value if you are classifying elasticity by size.
Worked example: price rises and quantity falls
Suppose a product’s price rises from $10 to $12. Over the same comparison, quantity demanded falls from 100 units to 80 units. Use these four values in the midpoint formula to estimate elasticity for that price-and-quantity interval.
First find the midpoint percentage change in quantity:
[ \frac{80-100}{(100+80)/2}=\frac{-20}{90}\approx -0.2222 ]
That is a decrease of about 22.22%. Next find the midpoint percentage change in price:
[ \frac{12-10}{(10+12)/2}=\frac{2}{11}\approx 0.1818 ]
That is an increase of about 18.18%. Divide the changes:
[ E_d=\frac{-0.2222}{0.1818}\approx -1.22 ]
The signed elasticity is approximately (-1.22). Its absolute value is about (1.22), so demand is elastic over this interval.
To check the midpoint method’s direction consistency, reverse both observations. Quantity then rises from 80 to 100, and price falls from $12 to $10. The numerator and denominator both change sign, so their ratio remains about (-1.22).
A comparison table
These examples show how to read different elasticity values. The first row uses the worked example above. The other rows are simple illustrations of the classification rule, not claims about a particular product or market.
| Signed elasticity | Absolute value | Classification | What the result says about the interval |
|---|---|---|---|
| (-1.22) | (1.22) | Elastic | Quantity changes by a larger percentage than price |
| (-0.60) | (0.60) | Inelastic | Quantity changes by a smaller percentage than price |
| (-1.00) | (1.00) | Unit elastic | Quantity and price change by equal percentages |
The minus sign and the classification answer different questions. The sign shows the direction of the relationship; the absolute value shows the size of the response. State which one you are reporting so a reader does not mistake a signed number for its magnitude.
Interpreting elasticity: sign, magnitude, and common errors
For an ordinary downward-sloping demand curve, price and quantity demanded move in opposite directions, so signed elasticity is negative. Many courses and textbooks classify demand using the absolute value instead. Check your class or report’s convention, and show the sign when clarity matters.
What the magnitude means
Elasticity compares percentage changes, not unit changes. A change of 20 units may be large for one product and small for another, while the ratio of percentage changes gives a scale-free measure for the interval being studied.
- If (|E_d|>1), demand is elastic: quantity changes by a larger percentage than price.
- If (|E_d|<1), demand is inelastic: quantity changes by a smaller percentage than price.
- If (|E_d|=1), demand is unit elastic: the percentage changes are equal in size.
These labels describe only the interval between the two observations. They do not prove that demand has the same elasticity at other prices or quantities.
Errors to avoid
Most mistakes come from mixing methods or comparing values that do not describe the same situation. I ask students to check the inputs and denominators before interpreting the result; a neatly written answer can still be misleading if the observations are not comparable.
- Mixing denominator methods: Do not use the initial value for one percentage and the midpoint for the other. The formula requires midpoint denominators for both changes.
- Rounding too early: Keep full precision through the division, then round the final result to a sensible number of decimal places.
- Changing units mid-calculation: If quantity is recorded in individual items at one point and cases at another, convert the values to the same unit first.
- Comparing unlike observations: The two points should refer to the same good, market, and time frame. If another factor also changed, the calculation alone cannot show what caused the quantity change.
- Treating the result as a point value: An arc estimate summarizes a stretch of the demand curve. It is not necessarily the elasticity at either endpoint.
- Using a zero price change: If (P_1=P_2), the percentage price change is zero, so this comparison cannot produce a finite arc elasticity.
For a normal demand response, a price rise pairs with a quantity fall, or a price drop pairs with a quantity rise. If both move in the same direction, check the data and context before interpreting the result. Other factors may have changed, or the observations may not be directly comparable.
Applications: revenue, pricing decisions, and limits
Arc elasticity can help describe how quantity responded between two observed prices. It can also support a revenue comparison when price and quantity are known at both points. It cannot, by itself, predict what will happen after a new price change or explain why customers changed their purchases.
Link to total revenue
Total revenue is price multiplied by quantity sold. In the worked example, revenue is $10 × 100 = $1,000 at the first observation and $12 × 80 = $960 at the second. Price rose, but total revenue fell across these two observations.
| Observation | Price | Quantity | Total revenue |
|---|---|---|---|
| Initial | $10 | 100 | $1,000 |
| Final | $12 | 80 | $960 |
This result fits the elastic classification for the measured interval: quantity fell by a larger percentage than price rose. Still, the example does not establish that elasticity alone caused the revenue change. It summarizes what happened across the same two observations.
Limits and careful use
An elasticity estimate is only as useful as its data. Make sure the price and quantity values describe the same product, market, and period. For example, comparing one region’s price with another region’s sales would not be a like-for-like estimate unless the difference in scope is clearly addressed.
The midpoint formula measures association across endpoints, not causation. If income, product availability, advertising, or other conditions also changed, the calculation does not separate their effects from the price response. It is a measure of the observed interval, not a controlled test or a guarantee about future demand.
When sharing a result, include the two endpoint observations, the signed elasticity, the absolute value if used, and the convention for classification. This gives readers enough information to check what the number means and what it does not show.
Conclusion
Arc elasticity gives a consistent way to compare price and quantity changes between two observations. Use midpoint denominators for both changes, keep the data comparable, and interpret the signed value and magnitude separately. Then report the interval so readers know exactly what the estimate describes.
Frequently asked questions
What is the arc elasticity formula?
Divide the midpoint percentage change in quantity by the midpoint percentage change in price. Each percentage uses the average of its starting and ending values.
Why use the midpoint formula?
It gives the same elasticity when you reverse the two endpoints. A basic percentage-change method can give different results depending on which observation you treat as the starting point.
Should price elasticity of demand be negative?
For an ordinary downward-sloping demand curve, yes. Price and quantity move in opposite directions, so the signed result is negative.
Why do some answers show elasticity as a positive number?
Many textbooks classify demand using the absolute value, which removes the negative sign. It is helpful to state whether your answer is signed or absolute.
What does an elasticity of (-1.22) mean?
Its magnitude is 1.22, which is greater than 1. Demand is elastic over the measured interval: quantity changed by a larger percentage than price.
Can I use units instead of percentages?
Not as a substitute for the formula’s percentage changes. Elasticity compares percentage responses, so it is not simply the number of units gained or lost per dollar.
Does arc elasticity show elasticity at one exact price?
No. It estimates the response across the interval between two observations. It does not necessarily equal elasticity at either endpoint.
Can I compare different products with this formula?
Only with care. The formula can calculate a ratio from entered values, but comparing different goods, markets, units, or time periods may not be meaningful.
What if price stays the same?
If both observed prices are equal, the price percentage change is zero. You cannot calculate a finite arc elasticity from that comparison.
Should I round each step?
No. Keep full precision while calculating both midpoint percentage changes and dividing them. Round only the final elasticity value.
(This article was written by one of our staff writers, Alan Westbrook. Visit our Meet the Team page.)
