Markup Rule

A markup rule is the pricing practice of a producer with market power, where a firm charges a fixed mark-up over its marginal cost.Roger LeRoy Miller, ''Intermediate Microeconomics Theory Issues Applications, Third Edition'', New York: McGraw-Hill, Inc, 1982.Tirole, Jean, "The Theory of Industrial Organization", Cambridge, Massachusetts: The MIT Press, 1988.

Derivation of the markup rule

Mathematically, the markup rule can be derived for a firm with price-setting power by maximizing the following expression for 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 cost of producing Q.
:$\pi$ = economic profit

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

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

or "marginal revenue" = "marginal cost".

:$P\cdot(P'(Q)\cdot Q/P+1)=MC$

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

:$P\cdot(1+1/)=P\cdot\left(\frac\right)=MC$

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

:$P=\left(\frac\right)\cdot MC$

Thus a firm with market power chooses the output quantity at which the corresponding price satisfies this rule. Since for a price-setting firm $\eta<0$ 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 $\eta=0$ 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 $\epsilon \ge -1$ and $\eta \le -1$). 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.

References

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