Elasticity

Saying that demand fell by ten units when the price rose by three pounds is almost useless, because both halves of that sentence depend on arbitrary choices of unit.

Measure petrol in gallons rather than litres and the slope of the demand curve changes by a factor of 4.55. Quote the price in dollars rather than pounds and it changes again. Nothing about the consumers has altered, so the number cannot be the thing worth reporting. What is needed is a measure that survives every change of unit, and comparing percentages to percentages is the way to get one.

Elasticity is a ratio of proportional changes

The price elasticity of demand is the proportional change in quantity divided by the proportional change in price:

ε = \frac{Δ q / q}{Δ p / p}

Both numerator and denominator are pure numbers, so $ε$ is too, and it means the same thing in any currency and any measure of volume. Taking the limit gives the point elasticity,

ε = \frac{d q}{d p} \cdot \frac{p}{q} = \frac{d \ln q}{d \ln p}

which is the cleanest definition: elasticity is the derivative of demand on log-log axes. That is also why empirical demand studies almost always regress the log of quantity on the log of price, since the coefficient then is the elasticity with no further work.

Because demand curves slope downwards, $ε$ is normally negative, and the sign is a persistent nuisance in the literature. Many sources quote elasticities as positive numbers with the sign understood, so an "elasticity of 0.4" for cigarettes and an elasticity of $- 0.4$ are the same claim. This course keeps the sign and says "more elastic" to mean larger in magnitude.

The vocabulary follows the magnitude. Demand is elastic when $| ε | > 1$ , meaning quantity moves proportionally more than price; inelastic when $| ε | < 1$ ; and unit elastic at exactly 1. The dividing line is not arbitrary, as the revenue section shows.

Elasticity is not the slope

The most common error in the subject is treating a flat demand curve as an elastic one. Elasticity contains the slope, but multiplied by $p / q$ , and that factor varies from point to point along any curve. A single straight line therefore has every elasticity on it.

Example. Demand is $q = 120 - 4 p$ . Find the elasticity at $p = 10$ and at $p = 20$ , and find the price at which demand is unit elastic.

The slope $d q / d p = - 4$ everywhere. At $p = 10$ , $q = 80$ , so $ε = - 4 \times 10 / 80 = - 0.5$ : inelastic. At $p = 20$ , $q = 40$ , so $ε = - 4 \times 20 / 40 = - 2$ : elastic. The same line, the same slope, elasticities differing by a factor of four. Unit elasticity needs $4 p = q = 120 - 4 p$ , so $p = 15$ and $q = 60$ , which is the exact midpoint of the line between its intercepts at $p = 30$ and $q = 120$ . That midpoint result holds for every linear demand curve.

Now you. Demand is $q = 200 - 5 p$ . Find the elasticity at $p = 10$ and $p = 30$ , and the unit-elastic price.

Answer

At $p = 10$ , $q = 150$ and $ε = - 5 \times 10 / 150 = - 0.33$ . At $p = 30$ , $q = 50$ and $ε = - 5 \times 30 / 50 = - 3$ . Unit elasticity is at $p = 20$ , $q = 100$ , again the midpoint of a line running from $p = 40$ to $q = 200$ .

The two extremes are worth naming because they appear as assumptions later. Perfectly inelastic demand is a vertical line, $ε = 0$ : the quantity does not respond at all, which is roughly true of insulin for a diabetic in the short run. Perfectly elastic demand is horizontal, $ε$ unbounded: any price above the going rate sells nothing. That is the demand curve a single wheat farmer faces, and it is the formal content of the price taking assumption three lessons from now.

Elasticity and revenue

Revenue is $R = p q$ . Differentiate with respect to price using the product rule:

\frac{d R}{d p} = q + p \frac{d q}{d p} = q (1 + ε)

Since $q > 0$ , the sign of the revenue response is the sign of $1 + ε$ . Where demand is inelastic, $ε$ lies between $- 1$ and $0$ , so $1 + ε > 0$ and raising the price raises revenue. Where it is elastic, $ε < - 1$ , and raising the price loses revenue. Revenue is maximised exactly where $ε = - 1$ .

This single result explains a great deal of pricing behaviour that otherwise looks contradictory. It is why a tobacco tax raises money while cutting smoking only modestly, and why a rail operator can find that a fare cut raises takings on leisure routes and loses money on commuter ones. The two answers are not inconsistent; they are the same formula on two different parts of the demand curve.

On the linear example above, revenue at $p = 10$ is £800, at the unit-elastic $p = 15$ it is £900, and at $p = 20$ it is £800 again. The symmetric fall on either side of the peak is the visible signature of $d R / d p = q (1 + ε)$ .

Constant elasticity demand

A straight line has a constant slope and a varying elasticity. The function with a constant elasticity is a power law, $q = A p^{ε}$ , since $\ln q = \ln A + ε \ln p$ has constant log-log slope. This is the form fitted in almost all empirical work, so it is worth being able to use directly.

Example. Cigarette demand in high-income countries is repeatedly estimated at about $ε = - 0.4$ . Take $q = 1000 p^{- 0.4}$ packs per week at a price of £10 and suppose a tax raises the price to £11. What happens to sales and to spending?

At £10, $q = 1000 \times 10^{- 0.4} = 398$ packs and revenue is £3,981. At £11, $q = 1000 \times 11^{- 0.4} = 383$ packs and revenue is £4,215. Sales fall by 3.74 per cent and spending rises by 5.89 per cent. The point elasticity predicts a 4 per cent fall for a 10 per cent price rise, and the exact answer is 3.74 per cent, because the elasticity applies to proportional changes and a 10 per cent step is not infinitesimal. The gap is the same first-derivative caveat that appeared on the production frontier in the first lesson.

Now you. A restaurant meal has estimated demand $q = 500 p^{- 1.6}$ per week. The price rises from £4 to £4.40. What happens to covers and to revenue?

Answer

At £4, $q = 500 \times 4^{- 1.6} = 54.4$ covers and revenue is £217.6. At £4.40, $q = 46.7$ and revenue is £205.5. Covers fall 14.1 per cent and revenue falls 5.6 per cent, since demand is elastic and $1 + ε < 0$ .

What makes demand elastic

Four things, and they are worth knowing because they let an elasticity be estimated to within a factor of two without any data.

Substitutes. The more closely something can be replaced, the more elastic its demand. This is the dominant factor and it is really a statement about how the good is defined. Demand for salt is very inelastic; demand for Saxa salt is elastic, because Cerebos will do. Any market defined narrowly enough has elastic demand.

Budget share. A good taking a large share of spending has a large income effect attached to any price change, which the previous lesson showed adds to the substitution effect for a normal good. Housing is more elastic than shoelaces partly for this reason alone.

Necessity against luxury. Insulin, tap water and heating in January have few substitutes and no comfortable option to go without. Second holidays have both.

Time. This is the one most often left out and it is frequently the largest. In the short run a household faces a fixed car, boiler and commute, so a fuel price rise can only be absorbed. Over a decade they change all three. Molly Espey's 1998 meta-analysis of published petrol demand studies found a median short-run elasticity near $- 0.23$ and a long-run one near $- 0.43$ , roughly double. Any statement about elasticity that does not say over what horizon is incomplete.

Cross-price and income elasticity

The same construction applied to other variables classifies goods.

The cross-price elasticity is $ε_{x y} = (\partial q_{x} / \partial p_{y}) (p_{y} / q_{x})$ . Positive means the goods are substitutes, since a rise in one price sends buyers to the other. Negative means complements, bought together, so a rise in one price kills demand for both. Near zero means unrelated, which is the case for the overwhelming majority of pairs of goods and is the assumption implicitly made whenever a single market is analysed on its own.

The income elasticity is $η = (\partial q / \partial m) (m / q)$ . Positive is a normal good, negative an inferior one, and the further split at $η = 1$ separates necessities, whose budget share falls as income rises, from luxuries, whose share rises. Engel's law is the statement that food has $η$ below 1.

Example. When the price of coffee rises 8 per cent, tea sales rise 3.2 per cent. When the price of printers falls 20 per cent, cartridge sales rise 6 per cent. Compute both cross elasticities and classify the pairs.

Coffee and tea: $3.2 / 8 = + 0.4$ , positive, so substitutes, though a fairly weak pair. Printers and cartridges: $6 / (- 20) = - 0.3$ , negative, so complements. Note that the second calculation says nothing about causation running the other way; cross elasticities are not symmetric in general, and the printer market is a case where they are strikingly asymmetric.

Now you. A household's income rises 12 per cent and its spending on bus journeys falls 9 per cent, while its restaurant visits rise 24 per cent. Compute both income elasticities and classify the goods.

Answer

Bus journeys: $- 9 / 12 = - 0.75$ , so an inferior good. Restaurant visits: $24 / 12 = + 2.0$ , positive and greater than one, so a normal good and a luxury: the share of income spent on eating out rises.

Cross elasticity is not merely a classification exercise. Competition authorities use it to define the boundaries of a market, since two products belong in the same market when a price rise on one drives enough substitution to the other. The formal version is the SSNIP test, introduced in the United States merger guidelines in 1982: ask whether a hypothetical monopolist over a candidate set of products could profitably impose a small but significant non-transitory increase in price, conventionally five per cent for a year. If buyers escape to something outside the set, the set was drawn too narrowly.

What an elasticity is not

Two honest limits are worth stating, because published elasticities are used far more confidently than they deserve.

An elasticity is a local number. Fitting a constant-elasticity curve to data spanning a 20 per cent price range and then using it to predict a doubling is extrapolation of the crudest sort, and the curve $q = A p^{ε}$ has no upper price at which demand goes to zero, which is certainly false.

And an elasticity is not a constant of nature. Jonathan Hughes, Christopher Knittel and Daniel Sperling compared American petrol demand across two periods and found the short-run price elasticity had fallen sharply, from roughly $- 0.21$ to $- 0.34$ in the late 1970s to something close to zero in the early 2000s. Suburban sprawl, higher incomes and more fuel-efficient vehicles had all changed the underlying situation. An elasticity summarises a population's circumstances, and circumstances move.

Finally, estimating one at all is harder than it looks. Observed prices and quantities are where supply meets demand, so a scatter of historical points traces out neither curve: it traces the intersections. Elmer Working made this point in 1927, and it is the origin of the whole econometric machinery of instrumental variables. Getting a demand elasticity requires something that shifts supply without shifting demand, which is why weather shocks and tax changes are the workhorses of applied demand estimation.

Demand is now fully described: where it comes from, how it responds, and how big the response is. What is entirely missing is the other side. Something has to produce the goods, and the price at which it is willing to is the subject of the next three lessons.