What’s it: Arc elasticity is a measure of elasticity based on two given points. Suppose you measure the own-price elasticity of demand. In that case, it is the percentage change in quantity demanded divided by the percentage change in price between two points.
Arc elasticity yields the same elasticity value, whether the price moves up or down to a certain level. We use the midpoint (mean) as the denominator to calculate the change in quantity demanded and price. Thus, the difference between the start and end points does not affect the calculation results.
Why the arc elasticity matters
You can measure the responsiveness of the quantity demanded to changes in price even though you don’t have information on the demand curve. To calculate elasticity, you can use two observations of price and quantity demanded.
This method produces a consistent elasticity value, regardless of whether the price is rising or falling. That’s because we are using the average as the denominator for both the price and the quantity demanded.
Calculating the arc elasticity
You must have two data for price and quantity demanded. To calculate the percentage change, you subtract the two data sets and divide them by the respective midpoints. Mathematically, the arc elasticity formula is as follows:
Take a simple example. The price of a product decreases from $7 to $6. As a result, the quantity demanded increases from 18 to 20 units.
From this case, we can calculate the demand price elasticity for the product as follows:
Elasticity = [(20 – 18)/((20 + 18)/2)]/[(6-7)/((6 + 7)/2)] = 0.68
Difference between arc elasticity and point elasticity
We can use two methods to calculate the elasticity of demand, point elasticity, and arc elasticity. Under point elasticity, you need a mathematical function (demand curve) to define the relationship between price and quantity demanded. You cannot calculate the point elastic directly because it produces bias. Therefore, you have to find it through statistical inferences from actual observations.
On the other hand, you can measure the arc elasticity directly and do not need such a mathematical function. To do this, you need two observation points for the price and quantity demanded.
Furthermore, arc elasticity overcomes the weakness in point elasticity. When you calculate the elasticity at two different points using the point elasticity, you will likely result in different numbers.
Let’s take an example to explain it.
Say, because a product’s price decreases from $10 to $8, the quantity demanded increases from 40 units to 60 units. Using the point elasticity formula above, we get:
Elasticity = ((60 – 40)/40)/((8 – 10)/10) = -2.5
Now, let’s use the same data but with a different starting point. Assume that the price increases from $8 to $10, and the quantity demanded decreases from 60 to 40. Then the point elasticity of this case is:
Elasticity = ((40 – 60)/60)/((10 – 8)/8) = -0.33/0.25 = -1.32
See, the result is different from the previous calculation (-2.5).
In fact, both should be the same because we use the same demand function and demand curve, namely:
y = −10x + 140
You can use the manual solution to get the equation above or instantly. For a manual solution, you can use the following formula to apply both cases:
(y – y1)/(y2 – y1) = (x – x1)/(x2 – x1)
- (x1, y1) = (10,40)
- (x2, y2) = (8,60)
(y – 40)/(60 – 40) = (x – 10)/(8 – 10)
(y – 40)/20 = (x – 10)/- 2
(y – 40) * -2 = (x – 10) * 20
-2y + 80 = 20x – 200
-2y = 20 x – 280
y = -10x + 140
- (x1, y1) = (8,60)
- (x2, y2) = (10,40)
(y – 60)/(40 – 60) = (x – 8)/(10 – 8)
(y – 40)/- 20 = (x – 10)/2
(y – 40) * 2 = (x – 10) * -20
2y – 80 = -20x + 200
2y = -20 x + 280
y = -10x + 140
To solve this problem, we use the arc elasticity formula. Here are the calculations for both cases:
- First case = [(60 – 40)/((60 + 40)]/[(8 – 10)/((8 + 10)/2)] = 0.4/-0.22 = -1.82
- Second case = [(40 – 60)/((40 + 60)/2)]/[(10 – 8)/((8 + 10)/2)] = -1.82
In conclusion, if we use arc elasticity, we don’t have to worry about the starting point and the endpoint. Using the midpoint (average) as the denominator, we get the same elasticity of whether prices go up or down.
Conversely, under the point elasticity, rising or falling prices affect the denominators we use. That ultimately yields two different numbers of elasticity.
Thus, arc elasticity is useful when there is a significant price change. However, if the change in price and quantity demanded is very small, the two methods tend to produce a close value.