What it is: Elasticity arc (arc elasticity) is a measure of elasticity based on the midpoint of the two points. If you measure the price elasticity of a product, it is the percentage change in quantity demanded divided by the percentage change in price between two points.
The arc elasticity produces the same elasticity value whether the price moves up or down to a certain level. We use the mean as the denominator to calculate changes in changes in quantity demanded and in price. Thus, the difference between the starting and ending points does not affect the calculation results.
Why is arc elasticity important
You can measure the responsiveness of quantity demanded to changes in price even if you do not have information about the entire demand curve . To calculate elasticity, you can use two observations of price and quantity demanded.
This method produces a consistent value of elasticity, regardless of whether the price is rising or falling. That’s because we use the two-point average as the denominator, both for the price and for the quantity demanded.
Calculating arc elasticity
You must have two records for the price and quantity demanded. To calculate the percentage change, you subtract the difference between the two data points and divide by the respective midpoint. 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 price elasticity of demand for the product as follows:
Elasticity = [(20 – 18)/((20+18)/2)]/ [(6-7)/((6+7)/2)] = 0.68
The difference between arc elasticity and point elasticity
We can use two ways to calculate the elasticity of demand, the point elasticity and the 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 a bias. Therefore, you have to find it through statistical inference from actual observations.
On the other hand, you can measure arc elasticity directly and don’t need such a mathematical function. To do so you need two points of view for 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 are likely to come up with different numbers.
Let’s take an example to explain it.
Say, because the price of a product decreases from $10 to $8, the quantity demanded increases from 40 units to 60 units. Using the above point elasticity formula, we get:
Elasticity = ((60 – 40)/40)/((8 – 10)/10) = -2.5
Now, let’s use the same data in reverse but with a different starting point. Assume that the price increases from CU8 to CU10 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, they should be the same because we use the same demand function and demand curve, namely:
y = −10x + 140
You can use manual solution to get above equation or instantaneously. 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 can 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 which point is the starting point and the end point. Since we use the midpoint (average) as the denominator, we get the same elasticity value whether the price goes up or down.
On the other hand, below the point elasticity, an increase or decrease in price affects the denominator we use. That ends up producing two different elasticity figures.
Thus, arc elasticity is useful when there is a large enough change in price. However, if the changes in price and quantity demanded are very small, the two tend to produce close values.