What Is Arc Elasticity? Definition, Midpoint Formula, and Example

What Is Arc Elasticity?

Arc elasticity is the elasticity of one variable with respect to another between two given points. It is used when there is no general way to define the relationship between the two variables. Arc elasticity is also defined as the elasticity between two points on a curve.

The concept is used in both economics and mathematics. In economics, is it commonly used to measure the changes between the quantity of goods demanded and their prices.

Understanding Arc Elasticity

In economics, arc elasticity is commonly used in relation to the law of demand to measure percentage changes between the quantity of goods demanded and prices.

There are two possible ways of calculating elasticity—price (or point) elasticity of demand and arc elasticity of demand. Price elasticity of demand measures the responsiveness of quantity demanded to a price. It takes the elasticity of demand at a particular point on the demand curve, or between two points on the curve. Arc elasticity of demand uses a midpoint between the two points.

Formula for Price (Point) Elasticity of Demand


$PE_d = \dfrac{\text{\% Change in Qty}}{\text{\% Change in Price}}$PEd​=% Change in Price% Change in Qty​

How to Calculate the Price Elasticity of Demand

If the price of a product decreases from $10 to $8, leading to an increase in quantity demanded from 40 to 60 units, then the price elasticity of demand can be calculated as:

• % change in quantity demanded = (Qd₂ – Qd₁) / Qd₁ = (60 – 40) / 40 = 0.5
• % change in price = (P₂ – P₁) / P₁ = (8 – 10) / 10 = -0.2
• Thus, PE_d= 0.5 / -0.2 = 2.5

Since we’re concerned with the absolute values in price elasticity, the negative sign is ignored. You can conclude that the price elasticity of this good, when the price decreases from $10 to $8, is 2.5.

Arc Elasticity of Demand

One of the problems with the price elasticity of demand formula is that it gives different values depending on whether price rises or falls. If you were to use different start and end points in our example above—that is, if you assume the price increased from $8 to $10—and the quantity demanded decreased from 60 to 40, the Pe_d will be:

• % change in quantity demanded = (40 – 60) / 60 = -0.33
• % change in price = (10 – 8) / 8 = 0.25
• PE_d = -0.33 / 0.25 = 1.32, which is much different from 2.5

How to Calculate the Arc Elasticity of Demand

To eliminate this problem, the arc elasticity of demand can be used. Arc elasticity of demand measures elasticity at the midpoint between two selected points on the demand curve by using a midpoint between the two points. The arc elasticity of demand can be calculated as:

• Arc E_d = [(Qd₂ – Qd₁) / midpoint Qd] ÷ [(P₂ – P₁) / midpoint P]

Let’s calculate the arc elasticity following the example presented above:

• Midpoint Qd = (Qd₁ + Qd₂) / 2 = (40 + 60) / 2 = 50
• Midpoint Price = (P₁ + P₂) / 2 = (10 + 8) / 2 = 9
• % change in qty demanded = (60 – 40) / 50 = 0.4
• % change in price = (8 – 10) / 9 = -0.22
• Arc E_d = 0.4 / -0.22 = 1.82

When you use arc elasticity of demand you do not need to worry about which point is the starting point and which point is the ending point since the arc elasticity gives the same value for elasticity whether prices rise or fall.

The Bottom Line

Arc elasticity is commonly used in economics to determine the percentage of change between the demand for goods and their price. Elasticity can be calculated in two ways—price elasticity of demand and arc elasticity of demand. The latter is more useful when there is a significant change in price.