# Computing the price elasticity of demand

The price elasticity of demand in general terms, let’s be more precise about how it is measured. Economists compute the price elasticity of demand as the percentage change in the quantity demanded divided by the percentage change in the price. That is:

Price elasticity of demand = Percentage change in quantity demanded / Percentage change in price.

For example suppose that a 10 percent increase in the price of an ice cream cone causes the amount of ice cream you buy to fall by 20 percent. We calculate your elasticity of demand is:

Price elasticity of demand = 20 % / 10% = 2

In this example, the elasticity is 2, reflecting that the change in the quantity demanded is proportionately twice as large as the change in the price.

Because the quantity demanded of a good is negatively related to its price the percentage change in quantity will always have the opposite sign as the percentage change price. In this example, the percentage change in price is a positive 10 percent (reflecting an increase) and the percentage change in quantity demanded is a negative 20 percent (reflecting a decrease). For this reason price elasticity of demand are sometimes reported as negative numbers. The common practice of dropping the minus sign and reporting all price. With is convention larger price elasticity implies a greater responsiveness of quantity demanded to price.

The Mid point Method: A better way to calculate percentage changes and elasticities

If you try calculating the price elasticity of demand between two points on a demand curve you will quickly notice an annoying problem: The elasticity from point A to point B seems different from the elasticity from point B to point A. For example consider these numbers:

Point A: Price = \$4 Quantity = 120

Point B: Price = \$6 Quantity = 80

Going from point A to point B, the price rises by 50% and the quantity falls by 33%, indicating that the price elasticity of demand is 33/50 of 0.66. By contrast going from point B to point A, the price falls by 33% and the quantity rises by 50 %, indicating that the price elasticity of demand is 50/33or 1.5. The reason this differences arises is that the percentage changes are calculated from a different base.

One way to avoid this problem is to use the midpoint method for calculating elasticities. The standard way to compute a percentage change is to divide the change by the initial level. By contrast the midpoint method computes a percentage change by dividing the change by the midpoint (or average) of the initial and final levels. For instance \$5 is the midpoint of \$4 and \$6. Therefore according to the midpoint method, a change from \$4 to \$6 is considered a 40 percent rise because (6 – 4) / 5 x 100 = 40. Similarly a change from \$6 to \$4 is considered a 40 percent fall.

Because the midpoint method gives the same answer regardless of the direction of change, it’s often used when calculating the price elasticity of demand between two points. In our example the midpoint between point A and point B is:

Midpoint: Price = \$5 Quantity = 100

According to the midpoint method, when going from point A to point B the price rises by 40 percent and the quantity falls by 40 percent. Similarly, when going from point B to point A, the price falls by 40 percent, and the quantity rises by 40 percent. In both directions the price elasticity of demand equals 1.

We can express the midpoint method with the following formula for the price elasticity of demand between two points, denoted (Q1, P1) and (Q2, P2).

Price elasticity of demand =

= (Q2 – Q1 ) / [( Q2 + Q1 ) / 2] / (P2 – P1) / [ (P2 + P1) /2]

The numerator is the percentage change in quantity computed using the midpoint method and the denominator is the percentage change in price computed using the midpoint method. If you ever need to calculate elasticities you should use this formula.

For most of our purposes, what elasticity represents – the responsiveness of quantity demanded to price – is more important than how it is calculated.