A markup rule refers to the pricing practice of a producer with market power, where a firm charges a fixed mark up over its marginal cost.
Mathematically, the markup rule can be derived for a firm with price-setting power by maximizing the following equation for "Economic Profit":
π
=
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 (Economic) Cost of producing Q.
π
= Economic Profit
Profit maximization means that the derivative of
π
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".
P
⋅
(
P
′
(
Q
)
⋅
Q
/
P
+
1
)
=
M
C
By definition
P
′
(
Q
)
⋅
Q
/
P
is the reciprocal of the price elasticity of demand (or
1
/
ϵ
). Hence
P
⋅
(
1
+
1
/
ϵ
)
=
P
⋅
(
1
+
ϵ
ϵ
)
=
M
C
Letting
η
be the reciprocal of the price elasticity of demand,
P
=
(
1
1
+
η
)
⋅
M
C
Thus a firm with market power chooses the quantity at which the demand 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 believes
η
=
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.
