Markup rule

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 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

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.