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.

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


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

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


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

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