# Markup rule

A markup rule refers to the pricing practice of a producer with market power, where a firm charges a fixed mark up over its marginal cost.[1][2]

## Derivation of the markup rule

Mathematically, the markup rule can be derived for a firm with price-setting power by maximizing the following equation for "Economic 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 (Economic) 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. Profit of a firm is given by total revenue (price times quantity sold) minus total cost:

${\displaystyle P'(Q)\cdot Q+P-C'(Q)=0}$
where
Q = quantity sold,
P'(Q) = the partial derivative of the inverse demand function.
C'(Q) = Marginal Cost, or the partial 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 quantity at which the demand 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 believes ${\displaystyle \eta =0}$ which means that it sets price equal to marginal cost.

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.

## References

1. ^ Roger LeRoy Miller, Intermediate Microeconomics Theory Issues Applications, Third Edition, New York: McGraw-Hill, Inc, 1982.
2. ^ Tirole, Jean, "The Theory of Industrial Organization", Cambridge, Massachusetts: The MIT Press, 1988.