Price Elasticity (Midpoint Form)

Elasticity is one measurement, and the sign of the answer tells you which curve you are standing on. A negative result means the quantity fell as the price rose: that is a demand curve. A positive result means the quantity rose with the price: that is a supply curve. Same subtraction, same division, same number of arguments — nothing in the arithmetic knows or needs to know which side of the market the two observations came from. Presenting demand elasticity and supply elasticity as two different formulas hides the one fact that connects them, so this page presents them as what they are: a single ratio, read by its sign.

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 move, 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. Below −1 the demand is elastic, between −1 and 0 inelastic, −1 exactly unit elastic. On the other side of zero, above +1 the supply is elastic, below +1 inelastic, and 0 is a stock that does not move at any price. Half the sources you will meet quote demand elasticity 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. This page accepts either sign and says what each one means, because both are ordinary answers here. The pages downstream of it are strict: enter a positive elasticity on the marginal-revenue page and it 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, and it applies to both curves. 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.

READING A NEGATIVE ANSWER: the demand side. Three things move a demand elasticity 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. If demand is all you came for, the last two paragraphs of this page — on what the two observations assume — still apply to you; the three cases in between are about supply.

READING A POSITIVE ANSWER: the supply side. What a supply elasticity measures is how easily output can be expanded, and that is overwhelmingly a question about TIME and about spare capacity. Three cases are worth having in mind, because almost everything sits between them.

Perfectly inelastic, $E = 0$. The quantity is fixed and no price moves it. A fixed stock of land, the seats in a stadium tonight, a harvest already in the barn, an original painting by a dead artist. Everything interesting about taxation, rent and windfall gains hangs on this case, because when supply cannot move, a tax changes who gets the money and changes nothing else — which is the entire argument behind a land value tax, and it holds exactly as far as the inelasticity does.

Perfectly elastic, $E \to \infty$. Any quantity at all is available at one price and none at a penny less. A small buyer in a world commodity market faces this. In practice it describes an industry with easily replicable capacity and no scarce input.

In between, which is nearly everything. A factory with idle machines and a labour market to hire from has an elastic supply. The same factory at full capacity has an inelastic one — until it can build another factory, at which point it becomes elastic again. This is why supply elasticity is really a statement about a time horizon, and why the same industry can be quoted at 0.2 and at 3.0 without either figure being wrong.

The practical consequence needs both signs at once, which is the best argument for one page rather than two. It is about who bears a tax, and it is the one result from this subject that people actually use. The side of the market that can move away from the tax pays less of it. A tax on a good with inelastic supply and elastic demand falls almost entirely on the producers, because the buyers can go elsewhere and the sellers cannot stop producing. Reverse the elasticities and the buyers pay almost all of it. Nothing about who physically writes the cheque to the government affects this, which is why "the employer pays half the payroll tax" is a statement about paperwork rather than about incidence.

Finally, the assumption underneath every number this page returns: the two observations must lie on ONE 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, or because a cost shock or a strike moved the whole supply curve, then something shifted between the two measurements and the ratio you compute is not an elasticity at all — it is a measurement of the shock. That is the identification problem, and it is the reason serious elasticity estimation is an econometrics exercise rather than a subtraction.

Which leaves the case where the sign surprises you. A positive result on something you were sure was demand, or a negative one on something you were sure was supply, is usually two observations that do not belong to one curve, or two columns entered the wrong way round — check those first. But not always. Backward-bending labour supply is real: past a high enough wage, people buy leisure instead of selling hours, and the curve genuinely turns back on itself. Giffen and Veblen goods, where quantity demanded rises with price, are rare but not mythical. The formula will not tell you which of these you have. It will only tell you, honestly, which way the quantity moved.