Revenue Response to a Price Change

This is the single result that makes elasticity worth measuring, and it can be stated in one line: whether a price rise raises or lowers revenue depends entirely on whether the elasticity is above or below −1. Everything else on this page is detail.

The mechanism is a tug of war. Revenue is price times quantity. Raise the price and you earn more on every unit you still sell, which pushes revenue up. But you sell fewer units, which pushes it down. Which effect wins is exactly the question of whether quantity moves more or less than proportionately — which is exactly what elasticity measures. The bracket $(1 + E_d)$ is the score.

Work the three cases with a 10% price rise. Inelastic, $E_d = -0.4$: revenue changes by $0.10 \times 0.60 = +6\%$. Revenue rises, and the firm sells less while collecting more. Elastic, $E_d = -2.5$: revenue changes by $0.10 \times (-1.50) = -15\%$. The same price rise destroys revenue. Unit elastic, $E_d = -1$: the bracket is zero and revenue does not move at all. This is the top of the revenue curve.

Two consequences follow that are worth carrying around. First, on a straight-line demand curve the unit-elastic point sits exactly at the midpoint of the price range, so revenue is maximized at half the choke price — and a firm anywhere below that price is leaving revenue on the table by not raising it. Second, and less often noticed, revenue is not profit. The inelastic price rise raised revenue by 6% while shipping fewer units, which also cut the cost of goods; the profit improvement is larger than the revenue improvement. And in the elastic case a price rise can still be worth making if the margin gained per unit outweighs the volume lost. This page answers a revenue question and nothing more.

The approximation, and when it breaks. The exact statement is $R = PQ$, so proportionally $\%\Delta R = \%\Delta P + \%\Delta Q + (\%\Delta P)(\%\Delta Q)$ — and this page drops the last term, the product of two small numbers. For a 5% price change that term is a rounding error. For a 40% change it is not, and the answer here will be visibly off. Over a large price move, compute the two quantities and multiply the revenues directly; that is exact and no harder.

Two further limits. The elasticity used has to be the elasticity over the range you are moving through, not one measured somewhere else on the curve — and on a linear demand curve it changes as you move, so a big move passes through several elasticities. And the whole thing assumes nothing else changed: no competitor response, no shift in income, no seasonal effect. In a market with a handful of large sellers, a price rise that a competitor matches gives a completely different quantity response than one they undercut, and no own-price elasticity contains that information.