Markup rule

Mathematically, the markup rule can be derived for a firm with price-setting power by maximizing the following expression for profit:

${\displaystyle \pi =P(Q)\cdot Q-C(Q)}$

where

Q = quantity sold,

P(Q) = inverse demand function, and thereby the price at which Q can be sold given the existing demand

C(Q) = total cost of producing Q.

${\displaystyle \pi }$ = economic profit

Profit maximization means that the derivative of ${\displaystyle \pi }$ with respect to Q is set equal to 0:

${\displaystyle P'(Q)\cdot Q+P-C'(Q)=0}$

where

P'(Q) = the derivative of the inverse demand function.

C'(Q) = marginal cost–the derivative of total cost with respect to output.

This yields:

${\displaystyle P'(Q)\cdot Q+P=C'(Q)}$

or "marginal revenue" = "marginal cost".

A firm with market power will set a price and production quantity such that marginal cost equals marginal revenue. A competitive firm's marginal revenue is the price it gets for its product, and so it will equate marginal cost to price.

${\displaystyle P\cdot (P'(Q)\cdot Q/P+1)=MC}$

By definition ${\displaystyle P'(Q)\cdot Q/P}$ is the reciprocal of the price elasticity of demand (or ${\displaystyle 1/\epsilon }$). Hence

${\displaystyle P\cdot (1+1/{\epsilon })=P\cdot \left({\frac {1+\epsilon }{\epsilon }}\right)=MC}$

Letting ${\displaystyle \eta }$ be the reciprocal of the price elasticity of demand,

${\displaystyle P=\left({\frac {1}{1+\eta }}\right)\cdot MC}$

Thus a firm with market power chooses the output quantity at which the corresponding price satisfies this rule. Since for a price-setting firm ${\displaystyle \eta <0}$ this means that a firm with market power will charge a price above marginal cost and thus earn a monopoly rent. On the other hand, a competitive firm by definition faces a perfectly elastic demand; hence it has ${\displaystyle \eta =0}$ which means that it sets the quantity such that marginal cost equals the price.

The rule also implies that, absent menu costs, a firm with market power will never choose a point on the inelastic portion of its demand curve (where ${\displaystyle \epsilon \geq -1}$ and ${\displaystyle \eta \leq -1}$). Intuitively, this is because starting from such a point, a reduction in quantity and the associated increase in price along the demand curve would yield both an increase in revenues (because demand is inelastic at the starting point) and a decrease in costs (because output has decreased); thus the original point was not profit-maximizing.