Markup rule explained

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.^[1] ^[2]

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) ⋅ 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) ⋅ 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) ⋅ Q+P=C'(Q)

or "marginal revenue" = "marginal cost".

P ⋅ (P'(Q) ⋅ Q/P+1)=MC

By definition

P'(Q) ⋅ Q/P

is the reciprocal of the price elasticity of demand (or

1/\epsilon

). Hence

P ⋅ (1+1/{\epsilon})=P ⋅ \left(	1+\epsilon
	\epsilon

\right)=MC

Letting

be the reciprocal of the price elasticity of demand,

P=\left(	1
	1+η

\right) ⋅ 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

η<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

η=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

η\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.

Notes and References

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.