Amoroso–Robinson relation

The Amoroso–Robinson relation, named after economists Luigi Amoroso and Joan Robinson, describes the relation between price, marginal revenue, and price elasticity of demand. It is a mathematical consequence of the definitions of the quantities. For example, it holds true both when perfect competition holds and when a monopoly is present.
The relation states that
where

Proof

The revenue accrued when amount of a good is sold at price is. Taking a derivative with respect to quantity sold gives us
The elasticity of demand is defined as the fractional change in the quantity demanded given a fractional change in price
Thus,
so that
Substituting into the marginal revenue equation gives us the desired relation

The relation is used to derive the Lerner Rule: a monopolist will choose its price and production such that
where is the marginal cost of production.
This condition is derived by substituting the Amoroso-Robinson relation into the condition that at maximum profit the marginal revenue equals the marginal cost.

In 1967, Ernst Lykke Jensen published two extensions, one deterministic, the other probabilistic, of Amoroso–Robinson's formula.