Amoroso–Robinson relation

The Amoroso–Robinson relation, named after economists Luigi Amoroso and Joan Robinson,^[1] 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

${\displaystyle {\frac {\partial R}{\partial x}}=p\left(1+{\frac {1}{\epsilon _{x,p}}}\right)}$

1

where

• ${\displaystyle {\frac {\partial R}{\partial x}}}$ is the marginal revenue,
• ${\displaystyle x}$ is the quantity of a particular good,
• ${\displaystyle p}$ is the good's price,
• ${\displaystyle \epsilon _{x,p}}$ is the price elasticity of demand.

Proof

The revenue accrued when ${\displaystyle x}$ amount of a good is sold at price ${\displaystyle p}$ is ${\displaystyle R=px}$. Taking a derivative with respect to quantity sold gives us (using the product rule)

${\displaystyle {\frac {\partial R}{\partial x}}=p+{\frac {\partial p}{\partial x}}x}$

2

The elasticity of demand is defined as the fractional change in the quantity demanded given a fractional change in price (often expressed as a percentage)

${\displaystyle \epsilon _{x,p}={\frac {\partial x/x}{\partial p/p}}={\frac {\partial x}{\partial p}}{\frac {p}{x}}}$

Thus,

${\displaystyle {\frac {1}{\epsilon _{x,p}}}={\frac {\partial p}{\partial x}}{\frac {x}{p}}}$

so that

${\displaystyle {\frac {p}{\epsilon _{x,p}}}={\frac {\partial p}{\partial x}}x}$

Substituting into the marginal revenue equation (2) gives us the desired relation (1) ${\displaystyle {\frac {\partial R}{\partial x}}=p+{\frac {p}{\epsilon _{x,p}}}=p\left(1+{\frac {1}{\epsilon _{x,p}}}\right)}$

Application

The relation is used to derive the Lerner Rule: a monopolist (or any firm with enough market power) will choose its price and production such that

${\displaystyle {\frac {P-MC}{P}}=-{\frac {1}{\epsilon _{x,p}}}}$

where ${\displaystyle MC}$ 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 (so that the marginal profit is 0).

Extension and generalization

In 1967, Ernst Lykke Jensen published two extensions, one deterministic, the other probabilistic, of Amoroso–Robinson's formula.^[2]

See also

• Lerner index
• Ramsey problem