Dominick Salvatore Schaums Outline of Microeconomics, 4th edition Schaums Outline Series 2006

CHAP. 9] PRICE AND OUTPUT UNDER PURE MONOPOLY 237
PRICE AND OUTPUT UNDER PURE MONOPOLY
9.28
Derive, with the use of calculus, the formula relating marginal revenue to price and the price elasticity of
demand.
Let P equal the price and Q equal the quantity of a commodity. The total revenue (TR) of the seller of the
commodity is then
TR ¼ PQ
9.29
and the marginal revenue is
MR ¼ d(TR)
dQ
¼ P þ Q dP
dQ
Manipulating the above equation algebraically, we get
MR ¼ P 1 þ Q
dP
¼ P 1
P dQ
where e equals 21 times the price elasticity of demand.
Derive using calculus the condition for a monopolist to maximize profits in selling the commodity in
two different markets (third-degree price discrimination).
1
e
The total profits of the monopolist (p) are equal to the sum of the total revenue the monopolist receives from
selling the commodity in the two markets (that is, TR 1 þ TR 2 ) minus the total cost (TC) of producing the total
output. That is,
p ¼ TR 1 þ TR 2
Taking the first partial derivative of p with respect to Q 1 , (the quantity sold in the first market) and Q 2 (the
amount sold in the second market) and setting them equal to zero, we get
or
@p
@Q 1
¼ @TR 1
@Q 1
@TC
@Q 1
¼ 0
and
MR 1 ¼ MR 2 ¼ MC
TC
@p
@Q 2
¼ @TR 2
@Q 2
@TC
@Q 2
¼ 0
That is, in order to maximize the total profits p, the monopolist must distribute sales between the two markets in
such a way that the marginal revenue is the same in both markets and equals the common marginal cost.
The above represents the first-order condition for profit maximization. The second-order condition is that
@ 2 p
, 0 and
@Q 2 1
@ 2 p
, 0
@Q 2 2