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# Markup rule

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 Title: Markup rule Author: World Heritage Encyclopedia Language: English Subject: Collection: Microeconomics Publisher: World Heritage Encyclopedia Publication Date:

### Markup rule

A markup rule refers to the pricing practice of a producer with 

## 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.