An Explanation of the Midpoint Formula for Computing the Coefficient for Price Elasticity of Demand

The principle behind the midpoints formula is this: Because the elasticity of a demand curve is the percent change in quantity divided by the percent change in price. But computing percentage change can get messy. For example, the percent change from 1 to 2 is 100% while the percent change from 2 to 1 is -50%. To see how this works, consider the formula for percentage change:

Percent change in X = ((new X - old X)/old X) * 100

Based upon this formula, the percent change from 1 to 2 is -((2-1)/1)*100 = 100%

The percent change from 2 to 1 is: ((1-2)/2)*100 = -50%

As you can see, although in our example we moved 1 increment along the same scale, the direction plays an important role in determining percentage change. For another example, the percentage change from 8 to 10 is 25% while the percentage change from 10 to 8 is -20%. Let's compute the coefficient of price elasticity of demand for the following segment of a demand curve. Suppose that when the price of an item is 5, 20 units will be purchase. When the price falls to 4, 30 units will be purchased. What you will find is that if we use the unrefined formula to compute the coefficient of price elasticity of demand, we will get two answers, one answer if the price moves from 4 to 5 and another answer if the price moves from 5 to four.

Consider first the solution if the price moves from 4 to 5:

(((new Q - old Q)/old Q)*100)/(((new P - old P)/old P)*100) =

notice that the expressions "*100 will cancel out, thus

((new Q - old Q)/old Q)/((new P - old P)/old P) =

if you write this out as a fraction, it would be very easy to rearrange the terms as:

((new Q - old Q)/old Q) * (old P/(new P - old P))

Once again, you can rearrange this formula to be:

(new Q - old Q)/(new P - old P) * old P/old Q, or simply:

(change in Q / change in P) * old P/old Q

Now, when the price moves from 4 to 5, E(d) can be computed as:

10/1 * 4/30 = 40/30 = 1.333

But now, if we let the price change from 5 to 4, E(d) becomes

10/1 * 5/20 = 50/20 = 2.5

As you can see, using the unrefined version of the formula for computing the coefficient of price elasticity of demand, we wind up with two different answers depending upon the direction of the change in price. By examining the two equations above, we can immediately see the cause of this unacceptable problem -- the second expression in each equation is different and is determined by which value we designate as the old Q or the old P. In other words, we are computing the elasticity of the line segment at one of the extreme ends. But the way around this is to compute the  coefficient of the line segment at the middle -- at the average of the two extreme points. So, let's refine the formula step by step.

First, start with our original formula:

(((new Q - old Q)/old Q)*100)/(((new P - old P)/old P)*100) =

But from the above explanation, we can see that this equation can be rearranged as:

(change in Q / change in P) * old P/old Q

But from our example, we see that the value of the second part of this expression can vary depending upon which we designate as the old P or the old Q. To get around this, lets take an average of the two extreme points:

(change in Q / change in P) * ((new P + old P)/2)/((new Q + old Q)/2)

So now it doesn't matter which set of values we designate as new or old because we're taking an average. You can see why it's referred to as the midpoint formula. From here, it's easy to see that the expressions "/2" drop out. Thus our final formula for the coefficient of price elasticity of demand becomes:

(change in Q / change in P) * (new P + old P)/(new Q + old Q)

So, let's now compute the coefficient of price elasticity of demand for the problem above. First, let the price move from 4 to 5:

10/1 * 9/50 = 90/50 = 1.8

Now, let's let the price move from 5 to 4:

10/1 * 90/50 = 1.8

Conclusion, it doesn't matter which set of values you designate as new or old when you use the midpoint formula. In other words, directionality is no longer a problem.