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Elasticity
Calculating the Price Elasticity of Demand

There are two methods for calculating elasticity: using percentage changes and using the midpoint formula.

We will first examine the formula using percentage changes. Consider the following demand curve:

fg04_01200.gif

To calculate the elasticity, we must find the percentage change in price and the percentage change in quantity demanded moving from point A to point B. To find the percentage change in price we use the formula for a percentage change:

% change = (Q2 - Q1) / ((Q1 + Q2)/2) x 100%

So the change in price moving from A to B is -1 and the denominator is 19/2 so the percentage change in price is -10.5%.

Similarly, the change in quantity demanded moving from A to B is 2, the denominator is 2/((4+2)/2), and the percentage change in quantity demanded is 2/3 x 100% = 66.7%. Therefore, the price elasticity of demand is 66.7%/(-10.5%) = -6.4. And since we are ignoring the negative sign, we say the price elasticity of demand between point A and point B is 6.4, and (using the definitions above) that demand is elastic between those two points.

Let us look at two other points on this demand curve.

fg04_01200.gif

Let's find the price elasticity of demand when we move from point C to point D. The change in price is -1 and the denominator is((3+2)/2), therefore the percentage change in price is (-1/(5/2)) = .40 x 100%. The change in quantity is 2 while the denominator is ((16+18)/2) = 17, so the percentage change in quantity is (2/17) x 100% = 11.76%. Thus the price elasticity of demand between point C and D is 11.76%/(-40%) = -.29 and again (using the definitions above), demand is inelastic between those two points.

It is important to notice that the price elasticity of demand changed when we looked at a different point on the demand curve. It was 6.4 between A and B, but it was .29 between C and D.

As you move to the right along a demand curve, the price elasticity of demand will always fall. Demand curves are more elastic at higher prices and less elastic at lower prices.

In the illustration above, the slope of the demand curve is -1. But the slope is -1 everywhere on the demand curve, while the elasticity is not the same everywhere on the demand curve. There is a significant difference between slope and elasticity!

There is another problem: when we want to find the price elasticity of demand over the same range, but using different starting points, we get a different answer. For example, when we started with C, the elasticity was 2/7, but when we start with D the elasticity is 1/8.

The solution to this problem is to make the two points on the demand curve infinitely close together. We get an approximation of the true price elasticity of demand by using the midpoint formula. The midpoint formula for the price elasticity of demand uses a slightly different method for finding the percentage change. Instead of computing a percentage change as:

(the change) / (starting point)

the midpoint formula computes percentage change as:

(the change) / (the average of the starting point and the ending point)

To find the price elasticity of demand between point A and B using the midpoint formula, first, find the percentage change in price. The change in price is -1 and the average of the starting and ending prices is (1/2)(6 + 5) = 5.5. So the percentage change in price is -1/(5.5) = -0.18.

The change in quantity demanded is 1, and the average of the starting and ending quantities is (1/2)(2 + 3) = 2.5. Thus, the percentage change in quantity demanded is 1/(2.5) = 0.4. Then dividing 0.4 by 0.18 (and ignoring the sign) results in a price elasticity of demand = 2.22. So when the price elasticity of demand is 2.22 for a 1% change in price, we can expect a 2.22% change in quantity demanded.

This is a slightly different number than that from using the other formula, but notice it does not matter which point is the starting point when using the midpoint formula.

To better understand how elasticity changes when price changes, try the Active Graph exercise below:

active_mini.jpgActive Graph Level 1: Elasticity Changes Along a Straight-Line Demand Curve

For a further look into the concept of elasticity of demand, check out the current events story "A Little Energy Conservation Would Go A Long Way" (Business Week, July 17, 2000).

To see the effect of taxes and elasticity, see the current events story "Glendening's Tobacco Tax Permeates Budget Discussions" (The Washington Post, February 18, 1999).



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