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11.4 MONOPOLY WITH MULTIPLE PLANTS AND MARKETS 465

LEARNING-BY-DOING EXERCISE 11.6
S
D
E
Determining the Optimal Output, Price, and Division of Production
for a Multiplant Monopolist
Suppose a monopolist faces a demand curve given by 1 1 1
2
T
1
P 120 3Q. The monopolist has two plants. The first Q Q 20 MC 12 5 MC T
2
has a marginal cost curve given by MC 1 10 20Q 1 ,
and the second plant’s marginal cost curve is given by 12.5 0.25MC T
MC 2 60 5Q 2 .
If we let Q Q 1 Q 2 denote the monopolist’s total
output, we can now solve this equation for MC T :
Problem Q 12.5 0.25MC T , or MC T 50 4Q.
Now we can equate marginal revenue to marginal
(a) Find the monopolist’s optimal total quantity and price.
cost in order to find the monopolist’s profit-maximizing
(b) Find the optimal division of the monopolist’s quan- quantity and price: MR MC T , or 120 6Q 50 4Q,
tity between its two plants. or Q 7. We can find the optimal price by substituting
this quantity back into the demand curve: P 120
Solution 3(7) 99.
(a) First, let’s construct the monopolist’s multiplant (b) To find the division of output across the monopolist’s
plants, we first determine the monopolist’s marginal cost
marginal cost curve MC T , the horizontal sum of MC 1
and MC 2 . To find the equation of MC T , you cannot just at the optimal quantity of Q 7: MC T 50 4(7) 78.
add MC 1 and MC 2 as follows: 10 20Q 60 5Q Now we can use the inverted marginal cost curves
70 25Q. This is incorrect because it gives us the vertical we derived above to find out how much we must produce
sum of the two curves. To get the horizontal sum, we first at each plant to attain a marginal cost of 78 at each plant:
must invert each marginal cost curve by expressing Q as
a function of MC: 1 1
Q (78) 3.4
1
2 20
1 1
Q MC 1 1
1
2 20 Q 12 (78) 3.6
2
5
1
Q 12 MC 2
2
5 Thus, of the total quantity of 7, plant 1 produces 3.4
units, while plant 2 produces 3.6 units.
Now we can add these two equations to get the horizon-
tal sum of MC 1 and MC 2 : Similar Problems: 11.21, 11.22
OUTPUT CHOICE WITH TWO MARKETS

Now consider a monopolist that serves two markets. In this section we will assume
that the monopolist must charge the same price in both markets. (In Chapter 12 we
will consider how the firm might behave if it can “price discriminate” by charging dif-
ferent prices in different markets.) The demand in market 1 is Q 1 (P), where Q 1 is the
quantity demanded in market 1 when the price is P. Similarly, the quantity demanded
in market 2 when the price is P is Q 2 (P). The firm’s total cost of production depends
on the total amount produced, Q, where Q Q 1 (P) Q 2 (P). The firm’s total cost of
production is C(Q), and its marginal cost is MC(Q). What price should the firm set if
it wishes to maximize profit in both markets together?
The firm’s profits in both markets will be the difference between the total revenues
in the two markets and the costs C(Q). To find the firm’s total revenues in both markets,
the firm will need to determine its aggregate demand Q Q 1 (P) Q (P). Graphically,
2