Price Elasticity of Demand (Midpoint Form)

Alfred Marshall gave the idea its name in 1890, and the reason it needed a name is that "demand falls when price rises" is true of almost everything and useful for almost nothing. What a seller needs is a number: how much does it fall, and is that more or less than proportionately? Elasticity is that number, and because it is a ratio of two proportional changes it is unit free — the same figure whether the price is in dollars or yen and the quantity in cases or tonnes.

The sign is part of the answer, and this site keeps it. A normal demand curve slopes down, so quantity and price move opposite ways and $E_d$ comes out negative. Half the sources you will meet quote it as a positive number "for convenience", drop the minus sign, and then feed the result into $\mathit{MR} = P(1 + 1/E_d)$, where the sign decides whether marginal revenue is above or below the price. The convention here is the strict one: −1.8 is elastic, −0.4 is inelastic, −1 exactly is unit elastic, and a positive value entered on the demand pages is rejected rather than quietly repaired.

Which form is this? The midpoint, or arc, form — each change divided by the AVERAGE of the two observations rather than by the starting one. That choice has one specific virtue and it is worth being precise about it. Take a price moving from $10 to $12 with quantity falling from 100 to 80. Measured from the $10 end, the point form gives $(-20/100)/(2/10) = -1.00$. Measured from the $12 end, walking the same change backwards, it gives $(+20/80)/(-2/12) = -1.50$. Same two observations, two different elasticities, and no rule for which end is the right one. The midpoint form gives −1.222 from either direction, and that is the whole reason it exists.

Notice that −1.00 and −1.50 sit on opposite sides of the elastic/inelastic boundary. A published elasticity of "about −1.2" for some good almost never states which form produced it, and the difference is large enough to reverse the advice you would draw from it. When you fit your own, write down the form beside the number.

An elasticity belongs to a price RANGE, and does not travel outside it. This is the honesty problem of the whole subject. Along a straight-line demand curve the elasticity is different at every single point — steeply elastic near the top, unit elastic exactly at the midpoint, inelastic near the bottom — so a curve with one slope has an infinity of elasticities. The number measured between $10 and $12 says nothing dependable about what happens at $30, and applying it there is the most common way this arithmetic gets misused.

Three other things move the number, and none of them is in the formula. Time: nearly everything is more elastic in the long run than the short, because substitutes take time to find. The 1970s gasoline shocks moved consumption very little in the first year and a great deal over the following decade, and quoting a short-run elasticity as though it were permanent understates the eventual response badly. Substitutes: a specific brand of coffee is far more elastic than coffee in general, because the brand has close substitutes and the category has few — which is why a category elasticity is nearly useless for setting one product's price. Share of budget: a good that takes a small slice of income is usually inelastic simply because nobody bothers to shop around over it.

Finally, the arithmetic assumes the two observations lie on ONE demand curve, and in real data they usually do not. If the price went up because a competitor left the market, or because the season turned, then something moved the whole curve between the two measurements and the ratio you compute is not an elasticity at all. That is the identification problem, and it is the reason serious elasticity estimation is an econometrics exercise rather than a subtraction.