Surplus Effects of Vertical Integration With
and Without Double Marginalization Examples1
What is Double Marginalization?
When firms have market power, they will set price above marginal cost,
which causes a welfare loss.2 This problem is accentuated when we have a
firm with market power that buys an input from another firm that also has
market power. The producer of the input will price above marginal cost when
it sells the input to the other firm, who will then price above marginal cost
again when they sell the final product that uses the input. This means the
input is being market up above marginal cost twice: once by the producer of
the input, and once again by the firm that uses the input to make its final
product - that’s double marginalization.
What is Vertical Integration?
Vertical integration is when “firms that previously formed a buyer-seller relationship merge”.
3
This means that the firm that makes inputs and the firm
that makes final products become one firm. Sometimes vertical integration
improves social welfare by eliminating the problem of double marginalization,
as is shown in example 1. This is the case if both the input and final good
producing firms have monopoly power. As shown in example 2, however, this
1
Disclaimer : This handout has not been reviewed by the professor. In the case of
any discrepancy between this handout and lecture material, the lecture material should
be considered the correct source. Despite all efforts, typos may find their way in - please
read with a wary eye.
2
Remember, total surplus is maximized when we are at marginal cost pricing. Any
price above marginal cost means that a less than optimal amount of the good is being
produced, which means a loss in total surplus.
3
Viscusi, W. Kip, Joseph Harrington, and John Vernon, Economics of Regulation and
Antitrust (2005, MIT Press, 4th Ed.), pg. 236.
Prepared by Nick Sanders, UC Davis Graduate Department of Economics 2008
isn’t always true.
Note that both examples in this handout assume a 1:1 ratio of inputs
to final products. In other words, every final product sold uses exactly one
input.
1
Vertical Integration with Double Marginalization - The Market for Bicycles and Bicycle Seats
The Setup: There are two producers: Bikes Aplenty and Seats ’R Us.
Bikes Aplenty makes bicycles, and they are the only bicycle makers in their
market region. They face an inverse demand curve for bikes of PB = 450 −
4QB where QB is the quantity of bicycles sold. They have no fixed costs.
Their marginal cost has two components. First, they have to buy bike seats
from Seats ’R Us at a cost of PS per seat. Then, they have to assemble
the bike with all the other parts, which costs $50 per unit. Seats ’R Us is
the only maker of bike seats around. They have no fixed costs either, and a
marginal cost of $16. There is one bike seat per bike, so the total quantity
of bike seats sold is the same as the total quantity of bikes sold.
The Problem, Part 1: Assuming the firms operate as independent monopolists in their respective industries, find prices and quantities in the market, the profits of each firm, and consumer surplus.
The Solution, Part 1: First, let’s lay out what we know. For Bikes
Aplenty, we know
PB = 450 − 4QB
M CB = 50 + PS
For Seats ’R Us, our information is a little more limited - we only know
marginal costs and that the number of bike seats sold equals the number of
2
bikes sold.
M CS = 16
QS = QB = Q
To solve this problem, we need the derived demand curve for bike seats.
Since the only bike maker in town is Bikes Aplenty, all of the demand for
seats must come from there. But how do we know how many seats they
will want? To find that, we solve the profit maximization problem for Bikes
Aplenty. Setting marginal revenue equal to marginal cost gives
M RB = M CB
450 − 8Q = 50 + PS → 8Q = 400 − PS
(1)
Equation (1) looks an awful lot like a demand function - it tells us how
many seats Bikes Aplenty will demand at any given price of seats PS . Since
Bikes Aplenty is the only consumer of bike seats, this is the market demand
function seen by Seats ’R Us. Rewriting (1) as an inverse demand function
gives PS = 400 − 8Q. Since it is a linear inverse demand curve, we can get
the marginal revenue curve by doubling the slope.
M RS = M CS
400 − 16Q = 16
Q = 24
(2)
(2) tells us the optimal quantity of bike seats produced. Since QB = QS = Q,
it also tells us the optimal number of bikes produced by Bikes Aplenty. We
now have the following information:
M CS = 16
PS = 400 − 8(Q) = 400 − 8(24) = 208
M CB = 50 + PS = 258
PB = 450 − 4(Q) = 450 − 4(24) = 354
We’ve found the prices and the quantities in the market, now we need profits
and consumer surplus.
Profits
ΠB = 354 ∗ 24 − 258 ∗ 24 = 2304
ΠS = 208 ∗ 24 − 16 ∗ 24 = 4608
3
Consumer Surplus
1
CS = (450 − 354)(24) = 1152
2
Total Surplus
ΠB + ΠS + CS = 8064
Note that to calculate consumer surplus, we use the price of the final
good (bicycles), not the price of the input (seats). The final product is all
the consumer sees, so the final good quantity and price are what we want to
use when calculating consumer surplus.
The Problem, Part 2: Bikes Aplenty decides to buy Seats ’R Us. Find
the new prices and quantities in the market, the new profits, and the new
consumer surplus.
The Solution, Part 2: Bikes Aplenty now owns Seats ’R Us, so their
marginal costs are different. There is no longer going to be a mark up on the
cost of seats, because the firm is now making seats themselves rather than
buying them from another firm. That means they get seats at the marginal
cost of production, which remains at $16. Solving the profit maximization
problem for Bikes Aplenty gives
M RB = M CB
450 − 8Q = 50 + 16
Q = 48
PB = 450 − 4(48) = 258
We don’t have to solve for PS anymore - it is fixed at $16.
Profits
ΠB = 258 ∗ 48 − 66 ∗ 48 = 9216
Consumer Surplus
1
CS = (450 − 258)(48) = 4608
2
4
Total Surplus
ΠB + CS = 13824
Looking at parts 1 and 2, it’s clear that after the integration both total
profits and consumer surplus went up (meaning total surplus has gone up).
Since total surplus is maximized at marginal cost pricing, and the two firms
becoming one firm got us one step closer to marginal cost pricing, this isn’t
surprising. So if vertical integration eliminates double marginalization, it can
improve overall surplus.
We can look at the before and after graphically as well. Figure 1 shows the
profits and consumer surplus before vertical integration. Consumer surplus,
shaded red (or lightest grey for black & white printers), is the area under
the demand curve and above the price of bikes. Profits for Bikes Aplenty are
shaded green (slightly darker grey), while profits for Seats ’R Us are shaded
purple (darkest grey). The sum of all three shaded areas is the total surplus.
In Figure 2, it is clear that consumer surplus (still shaded red) has gone up.
Profits for the single remaining firm, Bikes Aplenty, are again shaded green,
and are higher than the sum of the two individual firm profits before they
integrated.
2
Vertical Integration Without Double Marginalization - The Market for Computers and
Processors
The Setup: Processorium makes computer processors, and they’re the
only company that does so. They sell processors to many different companies, all of whom then put the processors into computers and sell them
under different generic names. There are enough producers in the computer
market that it is perfectly competitive. The demand for computers is given
by Q = 600 − PC , where PC is the price of a computer. The marginal cost of
producing a processor is $100. The marginal cost of producing a computer
is $50 + PP , where PP is the price of a processor.
5
450
CS
Profits for Bikes Aplenty
400
Profits for Seats 'R Us
350
300
D
em
an
d
100
50
0
20
40
fo
r
eats
for S
150
0
8Q
045
=
8Q
es
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40
for
=
ev.
ts
lR
Sea
na
or
f
rgi
nd
Ma
ma
6Q
0-1
= 40
200
De
Rev.
inal
Marg
Price
Marginal Cost of Bikes = 50 + Price of Seats
250
Bi
ke
s
=
45
0
-
4Q
Marginal Cost of Seats = 16
60
80
100
120
135
Quantity
Figure 1: Profits and consumer surplus under double marginalization.
450
CS
Profits for Bikes Aplenty
400
350
D
em
ina
250
an
d
lR
for
Bi
ke
s
Bik
200
fo
r
ev.
Price
rg
Ma
300
45
0
=
es
-
45
150
=
4Q
08Q
100
Marginal Cost of Bikes = 50 + 16
50
0
0
20
40
60
80
100
120
135
Quantity
Figure 2: Profits and consumer surplus after vertical integration.
6
The Problem, Part 1: Find the optimal quantities and prices, the profits
for each firm, and consumer surplus.
Solution, Part 1: We can solve this the same way we did things in example
1 . . . just remember that this time, the final product market is perfectly
competitive. That means that the computer producers will have price equal
to marginal cost.
600 − Q = 50 + PP → PP = 550 − Q
(3)
Now equation (3) is the derived inverse demand function for processors. Since
Prossesorium is still a monopolist, we set their marginal revenue equal to their
marginal cost
550 − 2Q = 100
Q = 225
and
PP = 325
Plugging Q into the inverse demand function for computers gives PC = 375.
We can now solve for profits and consumer surplus.
Profits
ΠC = 375 ∗ 225 − (50 + 325) ∗ 225 = 0
ΠP = 325 ∗ 225 − 100 ∗ 225 = 50625
Consumer Surplus
1
CS = (600 − 375)(225) = 25312.5
2
Total Surplus
ΠC + ΠP + CS = 75937.5
The Problem, Part 2: Say Processorium buys one of the computer companies they sell to, and starts making their own computers. As a result, they
stop selling processors to anyone else, which turns the formerly competitive
computer market into a monopoly market controlled by Processorium. Find
the change in total surplus.
7
The Solution, Part 2: As in example 1, once the firms vertically integrate
we only need to worry about the profit maximization problem of one firm Processorium. Since they are the only computer company left, they profit
maximize as a monopolist.
M RP = M CP
600 − 2Q = 100 + 50
Q = 225
P = 375
A couple of things to note here. First, note that there is no longer a markup
on processors - that makes sense, since Processorium is making everything,
so there’s no point in selling themselves chips at a markup. Second, note
that we have the exact same price and quantity result that we had before the
merger. Since quantities and prices are the same, consumer surplus hasn’t
changed. As for profits,
ΠP = 375 ∗ 225 − 150 ∗ 225 = 50625
it looks like overall profits haven’t changed either. Vertical integration in
this market doesn’t change total surplus at all, nor does it change surplus
distribution.
What’s the logic here? If the output market is perfectly competitive,
then it isn’t putting any markup on its product, so the profits for all firms
in the output market will be zero. The input firm does get markup above
marginal cost, so they get positive profits. When the two firms vertically
integrate, the former input firm now gets more revenue per unit (since before
they were selling at the price of the input, now they’re selling at the price of
the final good) but they incur additional costs per unit as well (since they
are now paying the additional marginal cost of turning their input into a
final product). Since the increase in price per unit is identical to the increase
in cost per unit, the two effects will cancel each other out.
This can be seen graphically as well. The marginal revenue curve for
Processorium after the integration has the same slope as the marginal revenue
curve before the integration. After integration, the intercept of the marginal
revenue curve has shifted up by exactly the cost of assembling a computer.
8
Their marginal cost of production has shifted up by exactly that cost as well.
Since the total profit for the firm is the area under the marginal revenue curve
minus the area under the marginal cost curve, and both have shifted by the
same amount, the area is the same and profits won’t change. The profit
triangle has just been shifted up by $50 - the cost of putting the chips into
computers. This is clear when looking at Figure 3.
650
Post-merger profits
Pre-merger profits
500
Co
Price
er
rg
me ger
r
stPo -me
MR Pre
ium MR
or
ss rium
ce
o
Pro cess
Pro
400
m
300
200
pu
te
rM
ar
ke
tD
em
an
d
Processorium MC Post-merger
Processorium MC Pre-merger
100
0
0
100
200
300
400
500
Quantity
Figure 3: Profits for Processorium pre- and post-merger.
9
650