Marginal Revenue from Elasticity (Amoroso–Robinson)

Joan Robinson set this relation out in The Economics of Imperfect Competition in 1933, and it is usually called the Amoroso–Robinson relation after her and Luigi Amoroso. It answers a question that only exists once a seller has some pricing power: if selling one more unit means cutting the price on ALL of them, what does that extra unit actually bring in?

Less than the price, always, whenever demand slopes down. The new unit earns $P$, and the price cut needed to move it costs something on every unit that was already selling. The term $P/E_d$ is that cost, and because $E_d$ is negative the term is a subtraction. A firm in perfect competition is the limiting case: it faces a perfectly elastic demand curve, $E_d \to -\infty$, the correction term vanishes, and marginal revenue equals price. That is what "price taker" means expressed as arithmetic.

The result worth remembering is that marginal revenue goes NEGATIVE once demand turns inelastic. At $E_d = -0.5$, a $20 price gives $\mathit{MR} = 20(1 - 2) = -\$20$: the extra unit loses twenty dollars. It follows that no firm with pricing power ever operates on the inelastic part of its demand curve — it would raise the price, sell less, spend less on production, and collect more money, all at once. Any firm you find sitting there is either constrained by something outside this model (a regulator, a long-term contract, a strategic reason to hold share) or has not measured its elasticity.

The sign convention is load bearing here, more than anywhere else on the site. Enter $E_d = +2$ instead of −2 and the formula returns $1.5P$ — a marginal revenue half again above the price, which is impossible for anybody who has to cut the price to sell more. The number looks perfectly ordinary and the error is invisible unless you happen to notice that MR came out above P. This page rejects a positive demand elasticity rather than computing with it.

Set $\mathit{MR}$ equal to marginal cost — which is the profit-maximizing condition, since a firm should expand until the last unit stops paying for itself — and rearranging gives $(P - \mathit{MC})/P = -1/E_d$. That is the Lerner index, and it is the most useful thing this relation produces: the profit-maximizing markup is set entirely by the elasticity of demand. Face a demand elasticity of −4 and the optimal markup is 25% of price; face −1.5 and it is 67%. That, and not a cost-plus rule, is what pricing theory actually says.

Two limits. The relation is a point statement — it uses the elasticity at the price you are at, and that elasticity changes as you move — so it tells you the direction to move, not how far. And it assumes a single price for all buyers. A seller who can charge different buyers different prices is playing a different game, and the whole apparatus of price discrimination begins where this relation's assumption ends.