Midpoint formula for elasticity

where Xnew is the NEW value of X and Xold is the OLD value of X.

For example, if price increased from 10 dollars to 12 dollars, the percentage change, as usually defined, would be:

(12 - 10) / 10 = 2/10 = 20 percent.

If at the same time the quantity demanded fell from 30 to 20 items, the percentage change in quantity demanded -- again using the usual definition of percentage -- would be

(20 - 30) / 30 = - 10/30 or - 33 percent.

If we used the usual percentage change formula in our calculation of elasticity, we would arrive at a coefficient of elasticity of:

That we define the midpoint percentage change as

Where Xaverage is the sum of the old and new values divided by 2.

Implementing this is straightforward. In our previous example:

The percentage change in price, caluclated by the midpoint formula would be

since the average price is 11.

The percentage change in quantity, calculated bye the midpoint formula would be

since the average quantity is 25.

And the coefficient of elasticity, calculated by the midpoint formula is

We are coming up with a different and higher value for elasticity than we did with the midpoint formula. Why have economists adopted this formula? Not to make beginning students suffer, but for two compelling reasons:

1. the usual formula is dependent on which value is taken as the new value and which is taken as the old value. The midpoint formula is not.

   To illustrate the point, consider a company which, having fallen on bad times, asks its employees to take a pay cut of 50 percent this year, and promises it will make it up to them with a pay raise of 60 percent next year. Sound good? It shouldn't, if the percentage calculations are to be carried out in the usual way.

   Suppose you are making $3000 a month. The pay cut will be a cut of $1500, leaving you with $1500 a month.

   The pay increase of 60 percent will add .6 x $ 1500 = $ 900 to your pay next year, for a total of $1500 + $900 or $2400. You did not get back to where you started from, because the base for the percentage calculation changed in the meantime.

   If our company had used the midpoint percentage change formula, a cut of 50 percent would have been a cut to $1800; a subsequent raise of 50 percent would have taken you back to exactly $3000, and a raise of 60 percent up to $3343.

2. the usual formula makes the percentage change algebra very inexact unless the changes involved are quite small. Much better approximations are obtained using the midpoint formula.

   To illustrate, consider the example above, where price was increased from 10 to 12 dollars and quantity fell from 30 to 20.

   The usual formula treats this as a 20 percent increase in price and a 33 percent fall in quantity. Applying percentage change algebra to the revenue formula R = PQ, we have:

   Pct. change in R = Pct. Change in P + Pct. change in Q

   or

   -13 = 20 - 33

   But if we compute the percentage change in revenue directly (using the usual formula) we find that -- since the old revenue was $10 x 30 = $300 and the new revenue is $12 x 20 = $240

   Percent change in revenue = (240 - 300)/300 = -60/300 = - 20 percent

   This is very far off the 13 percent decline predicted by the percentage change algebra.

   Now consider the percentage changes in price and quantity computed by the midpoint formula. We found above that the percentage change in price was 18.2 percent, and the percentage fall in quantity was 40 percent. Applying the percentage change algebra which tells us that

   %delta R = %delta P + %delta Q

   or

   -21.8 = 18.2 - 40

   Applying the midpoint formula to the change in revenue, we find that the

   Percent change in Revenue = (240 - 300)/270 = - 22 percent.

   The error is less than 1 percent using the midpoint formula; it was 7 percent using the usual formula.