121A Industrial Organization - Professor Giacomo Bonanno
THE DOUBLE MARGINALIZATION PROBLEM
Upstream firm: Goodyear. Goodyear is a monopolist in the market for tires
Downstream firm: Ford. Ford is a monopolist in the market for cars
The demand for cars is Q = 1 − P
Ford's total costs (EXCLUDING the cost of obtaining a set of 5 tires) are C = c Q
where
0 < c < 1 is a constant. Thus to produce 1 car it costs $c PLUS the cost of obtaining a set of 5 tires.
Goodyear has zero costs.
Let us calculate the demand function for tires. It will be a derived demand, that is, derived
from the demand for cars (assume for simplicity that tires are only bought by Ford, that is, consumers
don't buy tires).
Let w denote the price (set by Goodyear) of a set of 5 tires. Given w, Ford will decide how
many sets of 5 tires to buy from Goodyear. This decision will be a by-product of the decision of how
many cars to sell. Ford's profit function is:
π(P) = Revenue − Costs = P. ( 1
P)
(c
w) . ( 1
P ) simplifies to P
P
2
c
c. P
w
w. P
To find the profit-maximizing price we take the derivative of the profit function, set it equal to
zero and solve for P
d
P
dP
P
2
the solution to 1
c
c. P
2. P
c
w. P
1
w 0 is
P=
w
2. P
c
w
1+c+w
. This is the profit-maximizing price for
2
Ford. The corresponding number of cars sold is obtained from the demand function:
Q=1 − P=1 −
1 + c + w 1 −c −w
=
2
2
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121A Industrial Organization - Professor Giacomo Bonanno
Thus the number of cars sold is
1 −c −w
. Since each car requires a set of 5 tires, Ford
2
1 −c −w
sets of 5 tires from Goodyear (at the price of w per set). Thus the demand
2
1 −c −w
function of Goodyear is:
2
will buy
Goodyear's profit function is therefore given by (recall the assumption of zero costs)
1
w.
c
w
2
1
d
w.
dw
Goodyear will choose that w which maximizes its profits:
The solution to
1
1.
2
2
c
w
2
1
1.
2
2
c
w
1 −c
w 0 is w = 2
. Thus Goodyear will charge
c
(1 - c) / 2 for a set of 5 tires.
Given this price Ford will charge
cars will be
1
c
1− c
2 = 3 + c . Thus the price of
2
4
1+ c +
w
=
2
3
3+c
1
4 and the number of cars sold will be
c
4
=
1−c
4 . Hence this is also the
number of sets of 5 tires that Ford buys from Goodyear.
Ford's profits:
3
c . 1
4
Goodyear's profits:
c
c
4
1
c . 1
2
1
c
4
c . 1
2
=
Sum of Goodyear's and Ford's profits:
c
4
(1
c)
(1
=
c)
2
16
2
8
(1
c)
2
8
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(1
c)
16
2
=
3. ( 1
16
c)
2
121A Industrial Organization - Professor Giacomo Bonanno
Now consider a VERTICAL MERGER
Suppose that Goodyear and Ford merge into a single firm, called Speedy. Thus the new
firm produces its own tires internally (at zero cost, just like Goodyear used to).
Speedy's profit function is P. ( 1
d
P
dP
P
2
c
Speedy would charge
c. P
1
P)
2. P
c. ( 1
P) = P
P
2
c . The solution to 1
c
2. P
c. P
c 0 is
1+c
2 . Thus
1+c
1−c
1+c
for a car and sell 1 − 2 = 2 cars.
2
Thus P =
Speedy’s profits are
1
c . 1
2
c
2
1+c
1−c
and
Q
=
2
2
1
c.
c
2
=
(1
c)
2
4
Now let us compare the two cases:
Thus after the merger consumers are better off (they pay a lower price for cars) and the
shareholders of the two firms make higher profits. Thus the initial situation was inefficient. The
vertical merger increased social welfare.
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121A Industrial Organization - Professor Giacomo Bonanno
RESALE PRICE MAINTENANCE
1+c
to Ford. Then Ford would sell 1 −
2
Suppose Goodyear imposed a price of cars of
(1+c)/2 = (1−c)/2 cars and its profict would be (given w = (1−c)/2)
1
c . 1
2
c
2
c
1
c . 1
2
Goodyear's profits, on the other hand, would be
c
= 0
2
1
c
2
2
=
(1
c)
2
4
Then Goodyear could offer the following deal to Ford: you charge a price of (1+c)/2 per
car and pay me (1−c)/2 for each set of tires and I will give you the following sum of money:
(1
c)
2
(1
16
c)
32
2
=
3. ( 1
c)
2
32
which is greater than
(1
c)
2
16
Then Ford will be better off than without the contract and so will Goodyear be, since its
profits will then be
(1
c)
4
2
3. ( 1
32
c)
2
=
5. ( 1
32
c)
2
which is greater than
(1
c)
8
2
.
Thus they will both be better off and so will be the consumers. Hence this is an example
where Resale Price Maintenance increases social welfare (yet it is illegal in the U.S.).
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