# 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":

$\pi = P(Q)\cdot Q - C(Q)$
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.
$\pi$ = Economic Profit

Profit maximization means that the derivative of $\pi$ 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:

$P'(Q)Q+P-C'(Q)=0$
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:

$P'(Q)*Q + P = C'(Q)$

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.
$P(P'(Q/P)+1)=MC$

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

$P(1+1/{\epsilon})=MC$

This gives the markup rule:

$P=\frac {\epsilon} {\epsilon+1} MC$

or, letting $\eta$ be the reciprocal of the price elasticity of demand,

$P=\frac {1} {1+\eta} MC$

Thus a firm with market power chooses the quantity at which the demand price satisfies this rule. Since for a price setting firm $\eta<0$ 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 $\eta=0$ 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.