Variable kernel density estimation

The Markup rule is used in economics to explain firm pricing decisions. It states that the price a firm with market power will charge is equal to a markup over the firm's marginal cost, equal to one over one minus the inverse of the price elasticity of demand.

A profit maximizing firm chooses the quantity of output to sell which equalizes its marginal revenue (the change in revenue from one extra unit sold) to its marginal cost (the change in total cost due to one extra unit produced). This results in the markup rule which captures the fact that the firm's ability to price its good over cost depends on the extent of its market power. This in turn depends on the price elasticity of demand faced by the firm.

Derivation of the markup rule

Profit of a firm is given by total revenue (price times quantity sold) minus total cost:

$\pi =P(Q)Q-C(Q)$

where P(Q) is the inverse demand function, Q is quantity and C(Q) is the total cost function. This implies that the firm chooses quantity so that

$P'Q+P-MC=0$

where P' is the partial derivative of the inverse demand function with respect to Q. Factoring out the price on the left hand side of the equation gives

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

By definition $P'(Q/P)=$inverse of price elasticity of demand$=1/\epsilon$. Hence

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

This gives the markup rule:

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

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

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

Since for a price setting firm $\eta >0$ this means that a firm with market power will charge a price above marginal cost. 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 monopolistic firm will never choose a point on the inelastic portion of its demand curve. Furthermore for an equilibrium to exist in a monopoly or an oligopoly market, the price elasticity of demand must be greater than one ($1/{\eta }<1$)(Mas-Colell).

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