IS PRICE SQUEEZE THE OPTIMAL STRATEGY FORINTEGRATED FIRMS?

Chuan Yangand

Yasuo Kawashima

Faculty of Economics
Chuo University
October, 2011

Acknowledgement
Financial support from Chuo University (Research Grant, 0382) and the Japan Ministry of Education, Culture, Sports, Science and Technology (Grant-in-Aid for Scientiﬁc Research (C),20604001)
is highly appreciated. We would like to thank Professors Toichiro Asada, Akio Matsumoto,
Masahiro Yabuta of Chuo University and Moriki. Hosoe of Kumamoto Gakuin University for
their suggestions.
JEL classiﬁcation: L2, L4, L5
Keywords : vertically related market, integrated upstream monopolist, incentives tomonopolize, price squeeze, and upstream market structure.
Address: 742-1, Higashinakano, Hachioji-shi, Tokyo 192-0393,Japan
fax:+81 426 74 3425
e-mail: yasuokaw@tamacc.chuo-u.ac.jp

IS PRICE SQUEEZE THE OPTIMAL STRATEGY FORINTEGRATED FIRMS?
ABSTRACTThis paper seeks to analyze the eﬀects of a price squeeze by an upstream ﬁrm in vertically related
markets and to examine equilibrium in markets when one of the upstream ﬁrms integrates oneof the downstream ﬁrms. We show that an integrated monopolist has no incentives to employ
such a strategy, but that an integrated duopolist does have such incentives. Monopolization
does not give the maximum proﬁts to the monopolist with increased market prices. Such a
strategy employed by the integrated ﬁrms is anti-competitive in its nature, but it improves
market eﬃciency when employed by the integrated duopolist.

1

Introduction

The features of a vertically integrated ﬁrm (for example, a monopolist or a duopolist) are thatthe integrated ﬁrm supplies inputs to downstream rivals and at the same time competes tosupply outputs in a downstream market.1 As Armstrong (2008) pointed out, by using a pricing
strategy in the vertically related markets, the integrated ﬁrm can control not only input prices,
but also output prices. Then, Vickers (2008) points out that a price squeeze arises either when
price for inputs is unduly high or when price for outputs is unduly low.2 As the monopolization
is expected to give the monopolist the maximum proﬁts in a single market, the integrated ﬁrm
will have incentives to drive rival ﬁrms out of a market by strategies such as a price squeeze.3
In these settings, the Chicago School has long maintained that a price squeeze is not a rational
strategy of the integrated ﬁrm.4
In fact, Joskow (1985) observes through empirical analysis that the integrated ﬁrm has
incentives to employ a price squeeze to exclude rival ﬁrms from a market. King and Maddock(1999) show that the integrated ﬁrm has incentives to practice a price squeeze. Whinston (1990)
shows that a price squeeze is a proﬁtabler strategy for the integrated frim if the assumptionsof the Chicago School is relaxed. Chen (2001) shows that an integrated ﬁrm raises the costs
of independent ﬁrms by increasing switching costs. Thus, there is the fear that such a strategy
1Note that words “inputs” and “outputs” are defined from the viewpoint of downstream firms.
For an expository explanation of a price squeeze, see Tirole (1988), and see Joskow (1985) for its definition.
3
For a survey of price squeezes, see Crocioni and Veljanovski (2003).
4
See, for example, Bork (1978).
21

enables the monopolist to monopolize the downstream market.5
On the other hand, following Weisman (1995) and Sibley and Weisman (1998), Sappington(2006) concludes that the integrated ﬁrm does not have incentives to increase a rival’s costs.
Raising the rival’s costs will cause proﬁts of the rival ﬁrm to be squeezed to zero. Thus, it follows
from these observations that the integrated ﬁrm may not have incentives to exclude the rivalfrom a market by some strategies such as a price squeeze. These results are quite contrary toobservations by Joskow (1985), King and Maddock (1999), Whinston (1990), and Chen (2001).
Thus, the debate is not settled. From the viewpoint of regulation, it is very important to knowwhether the vertically integrated ﬁrm (a monopolist or a duopolist) has incentives to practice a
price squeeze.6
The present paper will show that the integrated upstream monopolists do not have incentives
to employ a price squeeze even if an downstream market is duopolistic, but that the integrated
duopolist has incentives to practice a price squeeze if both of an up- and downstream market
are duopoly. To be precise, if the upstream monopolist merges one of downstream ﬁrms, extra
proﬁts earned in the downstream market under a price squeeze is less than proﬁts lost in the
upstream market and market price goes up, which is equal to a monopoly price. On the other
hand, when the integrated duopolist employ such a strategy, additional proﬁts earned in adownstream market outweigh proﬁts lost in an upstream market. Thus, the integrated duopolist
has incentives to adopt a price squeeze. Moreover, it will be interesting to note that downstream
prices decrease under a price squeeze by the integrated duopolist. These are quite diﬀerent form
the results in the integrated monopolist. Depending upon the presence of competitive pressurein the upstream market, eﬀects of a price squeeze are quite diﬀerent. These are some of theimportant features of vertically related markets
The model presented in this paper has diﬀerent types of games. It is important to notethat all palyers are constrained to play their original roles once a game starts. This meansthat we do not focus on problems of palyers’ incentives, and that it is possible to compare theimportant characteristics of games. Our model is as follows. The ﬁrst stage corresponds to an
upstream market where an integrated upstream monopolist (or a duopolist) supplies inputs to anindependent downstream ﬁrm and the second stage corresponds to a downstream market where
the downstream division of the integrated ﬁrm and the independent downstream ﬁrm compete to5

Recently, Grout (2001), and Bouckaert and Verboven (2004) have investigated a price squeeze for the imple-

mentation of regulation policy.
6
This strategy is akin to predatory pricing (or limit pricing). See, for example, Kawashima (1983) for the
analysis of limit pricing.

2

supply a ﬁnal good. These markets are modeled as two-stage games with complete and perfect
information. In fact, given the linear demand for the ﬁnal good, the downstream ﬁrms compete
to supply them. Once equilibrium outputs of these ﬁrms are determined, demand for inputs of
the independent ﬁrm is derived from equilibrium outputs of the independent downstream ﬁrm.
The integrated ﬁrm maximizes proﬁts for the given demand for input. Thus, it is possible tosolve for upstream equilibrium price and input, from which downstream equilibrium prices andoutputs are determined.
The paper is organized as follows: Section 2 explains our basic settings and shows that the
backward induction enables us to determine not only equilibrium outputs and inputs, but alsoequilibrium prices in the up- and down-stream market. When we turn our attention to a gamein which the integrated ﬁrm has incentives to be a Stackelberg leader who can employ a pricesqueeze, such a strategy by the integrated monopolist is examined to see how proﬁts earned bythat strategy compare with those earned without that strategy. Then, we will show that the
integrated monopolist has no incentives to employ a price squeeze. In Section 3, it is shown thatthe integrated duopolist, who becomes a Stackelberg leader, has incentives to employ a pricesqueeze after the determination of equilibrium in one-sided integration is shown. We end up
with conclusion in Section 4.

3

2

Price Squeeze by Upstream Monopolist

Consider ﬁrst the integration in vertically related markets in which downstream ﬁrms D1 andD2 supply a consumer good in the downstream market, and a monopolist supplies inputs in the
upstream market. Without these inputs a consumer good can not be produced. The monopoly
ﬁrm is called U.
The demand for the consumer good is given by
P = a − x = a − (x1 + x2 ),(1)

where P represents the price of consumer good, and xi the quantity of output by Di, i=1,2.

To simplify our analysis, assume that αi units of inputs are translated into a unit of output(or a consumer good) by ﬁrm Di. It follows thatxi =

1yi , i = 1, 2,αi(2)where yi is the quantity of inputs by Di. This is the production function of D1 and D2. To
proceed with our analysis, it will be assumed that productivity of D2 is higher than that of D1,
and that the diﬀerence between them is not very large. Formally, this assumption is expressedas
1 < α = α1 /α2 ≤ 2.

(3)

Assume also that the monopolist (or U) produces inputs at a given constant marginal cost c.
Fixed costs are assumed away in what follows. Then, c is also the average cost of producing
inputs.
Together with these, it is also assumed thataci = αi p, i = 1, 2

where aci stands for the unit cost of the consumer good produced by Di and p for the price of
inputs. This means that Di using αi units of inputs produces one unit of the consumer good ata cost αi p.
To proceed with our analysis, assume further thata > 4α1 c.
4(4)This assumption is crucial to the analysis to follow although it is enough to assume that a > 3α1 c.
However, when games of multiple independent ﬁrms are considered, it is necessary to assume(4) to guarantee that each ﬁrm is viable in an upstream and a downstream market.
When an upstream monopolist U and D1 are integrated, D2 has to buy its essential inputs
from its competitor, the integrated ﬁrm I.7 Firm I is a bottleneck input provider and may
be called a partial monopolist, which is a monopolist in the upstream market, but it faces acompetitor in the downstream market. This is one of the features of vertically related markets.
Although U may have incentives to merge with an eﬃcient ﬁrm D2, merger with the eﬃcient ﬁrm
ends up with a pure monopoly and this game is not considered. This conversion of the market
into a pure monopoly will be analyzed later. When the merger does not lead the downstream
market to a pure monopoly, it is not known what a market will become after the integration.
This is why we take up the game in which U merges with less eﬃcient ﬁrm. This game is called
the partial monopoly game (henceforth, PM game).
In the PM game, assume that transfer price of throughputs is given by the marginal (oraverage) cost c. Thus, unit costs ĉ of producing ﬁnal goods by the integrated ﬁrm I are given byĉ = α1 c.
On the other hand, the independent ﬁrm D2 has to purchase inputs at a market price p,
which will be higher than the marginal cost c of its rival. Then, ﬁrm I can enjoy a cost advantage
over the independent D2. Thus, ﬁrm I, which merges with D1, may be able to exercise powerover price in the market and foreclose D2 from the market. Note also that focus here is mainlyon the eﬀects of vertical integration and that motivations for the integration are ignored.
Together with (1) and (2), and denoting by xI , x2 the outputs of the partial monopolist I
and D2, these proﬁts are given byπI= πd + πu = (P − ĉ)xI + (p − c)y2 = (P − α1 c)xI + α2 (p − c)x2
= (a − α1 c − (xI + x2 ))xI + α2 (p − c)x2 ,

π2 = (P − α2 p)x2 = (a − α2 p − (xI + x2 ))x2 ,where πd and πu stand for proﬁts in the downstream and upstream markets by ﬁrm I. The ﬁrst
order conditions for the maximization of proﬁts are

7

∂πI
∂xI

=

∂πd
= a − α1 c − 2xI − x2 = 0,∂xI

A game in which all firms are independent is considered by Vickers (1995). For further details, see also Yang

and Kawashima (2011).

5

∂π2
∂x2

= a − α2 p − xI − 2x2 = 0.

The Nash equilibrium outputs of these ﬁrms are expressed as
=

x̂I

x̂2 =

a − 2cα1 c + α2 p,
3a + α1 c − 2α2 p
.
3

(5)
(6)

D2 has to purchase inputs from I and its demand is derived from x̂2 . In view of (2), demand for
inputs by D2 is
α2 x̂2 = Y =α2 (a + α1 c − 2α2 p)
3Solving this equation for p, we have the inverse demand p for inputs:p=−3Y + α2 (a + α1 c).
2α22

The upstream division of ﬁrm I faces demand for its products derived above. Its proﬁt is
πI= πd + πu = (P − α1 c)x̂I + (p − c)Y = (P − α1 c)x̂I + (=−5Y 2 + 4Y (α1 − α2 )α2 c + α22 (a − α1 c)2.
4α22−3Y + α2 (a + α1 c)
− c)Y
2α22

It follows that the condition for optimality is
5Y
α1
dπI
= − 2 + (−1 +)c = 0.
dY2α2
α2
The partial monopolist supplies products to D2. It is given byY∗ =2(α1 − α2 )α2 c> 0,5where inequality is due to (3). Price charged for this input by the bottleneck provider is given
by substituting Y ∗ into the inverse demand function of inputs, and is given byp∗I =−3Y ∗ + α2 (a + α1 c)
5a − α1 c + 6α2 c
=
> 0,
2
10α2
2α2

(7)

where the sign comes from (4). It would be of some interest to check whether the equilibrium
price in the upstream market is higher than the provider’s marginal cost c. Subtracting c fromp∗I , we havep∗I − c =5a − α1 c + 6α2 c
5a − α1 c − 4α2 c
−c=
> 0,
10α2
10α2

6

which is positive under our assumptions (3) and (4). Thus, the integrated ﬁrm can make positive
proﬁts by supplying inputs to D2 and has cost advantage over its competitor because the ﬁrm
can get throughputs at the marginal costs. However, it is not certain that the independent ﬁrm
can reap positive proﬁts in the market because the ﬁrm has disadvantages in costs. It follows
from these that D2 may have to exit from the market on account of its cost disadvantage. Thismay be a prospect that D2 has to face.
Thus, we can now summarize our analysis above as:
Lemma 1. Equilibrium prices in the upstream and the downstream markets where there is apartial monopolist in vertically related markets are given by
p∗I =
PI

=

5a − α1 c + 6α2 c.
10α2
5a + c(3α1 + 2α2 ),10where PI is equilibrium downstream price and is higher than α2 p∗I . Moreover, outputs of D2 andI are given by
x∗2 =
x∗I =

2(α1 − α2 )c> 0,
52(α1 − α2 )c> 0.
5Proof. Substituting (7) into (5) and (6) yields,
x∗I

=
=

a − 2α1 c + α2
a − 2α1 c + α2 p∗I=33
5a − 7α1 c + 2α2 c> 0,105a−α1 c+6α2 c
10α2

and
1 c+6α2 c
a + α1 c − 2α2 5a−α10α2(α1 − α2 )ca + α1 c − 2α2 p∗I
∗
2
=
=
> 0,
x2 =

3

3

5

where the inequalities above come from (3) and (4). Then, it follows from (1), x∗I and x∗2 that
we have
PI = a − x∗I − x∗2 =5a + c(3α1 + 2α2 ).
10Using PI and p∗I derived above, we get
PI − α2 p∗I =2c(α1 − α2 )> 0,5where the sign comes from (3). Firm I has cost advantage over D2, but the independent ﬁrm
D2 can reap proﬁts to the extent that D2 has more eﬃcient production technology.
7

Lemma 1 shows the essential featrures of the partial monopoly game. Thus, a downstream
competitor can be viable as long as it has a superior technology: i.e. α2 < α1 ≤ 2α2 . If
this assumption does not hold, the vertical integration results in excluding a rival ﬁrm fromthe market and the downstream market changes into a pure monopoly. Thus, technologicalsuperiority enables an independent ﬁrm to cope eﬀectively with the partial monopolist even if
the monopolist has cost advantages. As mentioned before, this is the reason why it is assumed
that the upstream monopolist integrates less eﬃcient ﬁrm D1.
Even if the independent ﬁrm D2 can make proﬁts, the partial monopolist I has several strategies to drive a competitor out of a market. For example, the partial monopolist I may employ
a price squeeze strategy through which I sets the monopoly price in the market. The partialmonopolist I can set the price for the input high enough so that the downstream competitors can
not compete with it. After the competitor exits from the market, ﬁrm I will set higher downstream (or monopoly) prices to maximize its proﬁts. Thus, concern about vertical integrationby the upstream monopolist or duopolist is that it exploits leverage in the downstream market
to reap the maximum proﬁts and that this eventually aggravates market eﬃciency.
Then, turn our attention to a game in which the integrated ﬁrm changes its role and becomesa leader. For example, the leader practices a price-squeeze strategy and its inﬂuences on market
outcomes will be examined in what follows. Following Joskow (1985), this is deﬁned as a strategyof the monopolist which charges so a high price for its input to its downstream competitors thatthe competitors cannot make any proﬁts. Prices in the downstream and the upstream market
under a price squeeze are denoted by PSQ and psq respectively. Formally, assume that
PSQ = α2 psq .

(8)Note that individual equilibrium outputs of I and D2 are given by (5) and (6). Total output isthe sum of the individual outputs. Therefore, we havex̂I + x̂2 =2a − α1 c − α2 pa − 2α1 c + α2 p a + α1 c − 2α2 p
+
=
.
3
3
3

Taking this, (1) and (8) into account, the price PSQ set under the price-squeeze strategy isexpressed as
PSQ = a − x̂I − x̂2 = a −a + α1 c + α2 psq
a + α1 c + PSQ
2a − α1 c − α2 psq
=
=
.
3
3
3

This gives us
PSQ =a + α1 c
.
2
8

(9)

Together with (8), the price of input, which is derived from the price-squeeze strategy by ﬁrm
I, is reduced to
psq =a + α1 c
.
2α2

(10)

Note that if ﬁrm I sets input price equal to psq given above, output of D2 is equal to zero. In
fact, substituting (10) into (6) reveals that x̂2 indeed equals zero.
By deﬁnition, under a price squeeze, outputs of D2 are equal to zero. This in turn meansthat the integrated ﬁrm I can be a monopolist. Thus, price under this strategy is equal to amonopoly price PM . Then, we have
PM = PSQ =a + α1 c.
2(11)

Monopoly output xm is given by
xm = a −a − α1 c
a + α1 c
=
> 0,
2
2

where the inequality is due to (4).
These results are summarized in
Proposition 1. Price under a price-squeeze strategy in PM game is the same as monopoly
price. Moreover, the price under a price-squeeze strategy is higher than equilibrium price in thevertical integration game. Thus
PI < PSQ = PM .
Proof. Note that the price set under the price-squeeze strategy is given in (9), while the monopolist charges price equal to (11). These are the same. It is easy to see that
PSQ − PI =(α1 − α2 )ca + α1 c 5a + (3α1 + 2α2 )c
−
=
> 0.
2
10
5

where the inequality comes from (3).
It is interesting to note that monopoly price equals price set under a price squeeze. Proﬁts
earned under this strategy are shown to be equal to those under monopoly because the monopoly
is a market in which the monopolist has the maximum control over the market and hence it canreap the highest proﬁts. It follows from P roposition 1 that a price squeeze is anti-competitivein its nature and then market eﬃciency is shown to be aggravated. Moreover, our P roposition 1
suggests that a price squeeze by the partial monopolist I in these markets achieves the maximum
9

proﬁts. In fact, proﬁts earned in monopoly and under a price squeeze in the partial monopoly
games are equal.
However, when we consider vertically related markets, results may diﬀer because ﬁrm I
has two sources of proﬁts. Monopolization by some strategy enables the ﬁrm I to achieve themaximum proﬁts in the downstream market, but it causes a reduction in proﬁts in the upstream
market. Then, the monopolization has positive and negative eﬀects on proﬁts simultaneously.
These considerations suggest that a price squeeze may not be optimal for ﬁrm I. This possibility
is conjectured by Sappington (2006), who suggests that proﬁts lost in the upstream market,
which are due to exclusion of the downstream rival, will not be compensated by additional
proﬁts earned in the downstream market.
It will be interesting to examine if ﬁrm I has incentives to control integrated market structure.
If it has, the market structure will turn into a pure monopoly under some strategies and willthen aggravate economic eﬃciency. Deregulation, which permits the ﬁrm I to employ foreclosing
strategies, will then result in the formation of a pure monopoly and hence the partial monopoly
game will change into an ineﬃcient market structure. Then, deregulation should be banned.
Proposition 2. The integrated monopolist would prefer to keep the market structure unchangedin the partial monopoly game because it is not optimal for the monopolist to engage in a pricesqueeze in the partial monopoly game.
Proof. The integrated monopolist I can make proﬁts not only in the downstream market, butalso in the upstream market. If the monopolist I succeeds in driving competitors out of thedownstream market, the resulting increase in proﬁt earned in that market using perfect control
may be greater or smaller than the proﬁt lost in the upstream market. It is not clear which islarger. Therefore, we should make a precise comparison of proﬁts earned under a monopolistic
price squeeze with that earned by the partial monopolist.
If the monopolist I can perfectly control the downstream market, its proﬁt is

a + α1 ca − α1 c− α1 c)πm = (PM − α1 c)xm = (
2
2





=

a − α1 c
2

2.

On the other hand, the monopolist I in the integration game can make a proﬁt of πI , which
consists of proﬁts πd in the downstream market and πu in the upstream market. In view ofLemma 1 and its proof, the proﬁt in the downstream market isπd =(PI − α1 c)x∗I =(5a − 7α1 c + 2α2 c)25a + c(3α1 + 2α2 )− α1 c x∗I =
.
10
100

10

Similarly, the proﬁt earned in the upstream market isπu = (p∗I − c)α2 x∗2 =c(α1 − α2 )(5a − (α1 + 4α2 )c).
25

Total proﬁts of I are given byπI(5a − 7α1 c + 2α2 c)2 c(α1 − α2 )(5a − (α1 + 4α2 )c)+100
25
5a2 − 10aα1 c + (9α21 − 8α1 α2 + 4α22 )c2.
20= πd + πu ==Therefore, the diﬀerence between proﬁt πm earned by a pure monopoly and proﬁt πI earned
by the partial monopolist I isπm − πI =a − α1 c22−(α1 − α2 )2 c2
5a2 − 10aα1 c + (9α21 − 8α1 α2 + 4α22 )c2
=−
< 0.
20
5

As monopoly proﬁt is lower than the proﬁt earned by I, a price-squeeze strategy results inlower proﬁts for the partial monopolist. This means that ﬁrm I can make higher proﬁts if it
does not change this integrated market structure.
Although our results diﬀer from those obtained by Joskow (1985), they are quite consisitent
with Sappington (2006). In fact, it has been shown here that ﬁrm I has no incentives tomonopolize the market. The fear that the partial monopolist will try to monopolize a market andraise market price is a misconception under our assumptions. On the contrary, integrated market
structures that include a partial monopoly lead to larger welfare relative to a pure monopoly
and hence the formation of a partial monopoly should not be prevented by regulation.8 Thus,
this result shows that the presence of competitors is crucial for achieving market eﬃciency even
if a ﬁrm has partial controlling power over markets. These observations are important features
when we consider integrated market structures.

8

Yang and Kawashima (2011) has shown that equilibrium market prices are lower in the partial monopoly

game than in a pre-integration game.

11

3

Price Squeeze by Upstream Duopolist

Next, consider integration in a double duopoly game (or DD game), where there are two independent ﬁrms in both markets. Upstream ﬁrms are called U1 and U2 whose marginal (andaverage) costs are ﬁxed constants c1 and c2 . Without loss of generality, assume that c1 ≤ c2 .
To proceed with our analysis, assume thata > 4α1 c2 ,(12)which takes the place of the previous assumption (4).
There are concerns that the integration causes the aggravation of market eﬃciency. In
particular, when eﬃcient ﬁrms are integrated, serious concerns will emerge. Then, a game that
we take up is a game in which eﬃcient ﬁrm D2 integrates with upstream ﬁrm U1, and thedownstream division of the integrated ﬁrm F is superior in technologies to the downstream rival
D1. For simplicity, call this game a one-sided integration game (or OI game).
The market structure of this model is depicted in Figure 1. It follows that F can get accessto inputs at lower costs than D1. In other words, F is superior in costs and productivity. The
integrated ﬁrm has stronger market power than the other pair of ﬁrms and there will be concernsthat it will employ a strategy such as a price squeeze to reap more proﬁts.
In order to analyze eﬀects of the vertical integration, it is necessary to show the features of
the OI game. F may be able to make proﬁts by supplying outputs produced by using inputs
not only from the upstream division, but also from an upstream market. However, assume that
F does not purchase inputs from the upstream market because prices of inputs are higher thanthe marginal costs of inputs. Thus, proﬁts of F and D1 are expressed as
πF= (P − α2 c1 )xF + (p − c1 )y1 = (a − α2 c1 − xF − x1 )xF + (p − c1 )y1 ,π1 = (P − α1 p)x1 = (a − α1 p − xF − x1 )x1 ,where xF stands for output of F produced by through-puts and p for input price.
The ﬁrst order conditions for maximum proﬁts are
∂πF
∂xF
∂π1
∂x1

= a − α2 c1 − 2xF − x1 = 0,
= a − α1 p − xF − 2x1 = 0.

Solving these equations for xF and x1 , we have
x∗F

=

x∗1 =

a + α1 p − 2α2 c1,
3
a − 2α1 p + α2 c1
.
3
12

U1

U2

?

Upstream market

?

D2

?
Cournot competition

D1

Q

Q

Q

Q

Q

Q

s
Q
+

Consumers

Figure 1: One-Sided Integration Game

As noted above, derived demand d for inputs by D1 is given byd = α1 x∗1 = α1

a − 2α1 p + α2 c1.
3Solving this for p yields inverse demand for the inputs, and is given byp=(a + α2 c1 )α1 − 3d
.
2α21

(13)

Given this demand for the inputs, the upstream division of F and U2 supply the inputs tothe downstream ﬁrm, and their proﬁts areπF= (a − α2 c1 − x∗F − x∗1 )x∗F + (p − c1 )y1 = (a − α2 c1 − x∗F − x∗2 )x∗F + ((a + α2 c1 )α1 − 3d− c)y12α21(a + α2 c1 )α1 − 3(y1 + y2 )− c1 )y1 ,2α21(a + α2 c1 )α1 − 3d(a + α2 c1 )α1 − 3(y1 + y2 )− c2 )y2 = (− c2 )y2 ,πU 2 = (p − c2 )y2 = (
2
2α1
2α21
d = y1 + y2 .
= (a − α2 c1 − x∗F − x∗1 )x∗F + (Substituting x∗F , x∗2 , and p derived above into πF and πU 2 , the ﬁrst order condition for
maximum proﬁt yields
∂πF
∂y1

= −

5y1 + 2(y2 + α1 (α1 − α2 )c1 )≤ 0, f or y1 , y2 ≥ 0,2α2113

where the inequality is due to (3). It follows from the Kuhn-Tucker conditions that optimal
supply ŷ1 of inputs of F is equal to 0. Thus, the integrated ﬁrm produces throughputs and does
not supply inputs to the upstream market.
It follows from these arguments that U2 is the sole supplier of inputs, which are demanded
by D1. U2 maximizes its proﬁts given the derived demand for inputs. Then, we have
∂πU 2
∂y2

=

−6y2 + α1 (a − 2α1 c2 + α2 c1 )
= 0,
2α21

where (a + α2 c1 − 2α1 c2 ) is positive because of (12). Solving this equation for y2 , we haveŷ2 =α1 (a − 2α1 c2 + α2 c1 )> 0,
6where the inequality is again due to (12). Noting that d = y2 and substituting ŷ2 into (13), equilibrium price p̂ of inputs is determined and then substituting this p̂ into x∗F , x∗1 yields equilibrium
outputs x̂F , x̂1 of the two ﬁrms.
Now we can summarize our arguments above as:
Lemma 2. If the more productive ﬁrm D2 in the downstream market merges with the more
eﬃcient upstream ﬁrm U1, the one-sided vertically merged ﬁrm F can maximize proﬁts by notsupplying inputs to the upstream market. Equilibrium outputs, prices and inputs of ﬁrms in the
post-merger game are given byŷ1 = 0,α1 (a − 2α1 c2 + α2 c1 )
,
ŷ2 =
6
a + 2α1 c2 + α2 c1
,
p̂ =
4α1
5a + 2α1 c2 − 7α2 c1
,
x̂F =
12
a − 2α1 c2 + α2 c1
,
x̂1 =
6
5a + 2α1 c2 + 5α2 c1
.
P̂ =
12
Proof. In the post-merger game, U2 maximizes proﬁts by supplying products for the given
demand (13) for inputs. Note also that d = y2 because the upstream division of F does not
supply inputs to the upstream market. Then, proﬁts of U2 are given by
πU 2 = (p − c2 )y2 = (α1 (a + α2 c1 ) − 3d
− c2 )y2 ,2α21Solving the ﬁrst order condition for y2 yieldsŷ2 =α1 (a − 2α1 c2 + α2 c1 )
.
2α21
14

Substituting this into (13), we havep̂ =a + 2α1 c2 + α2 c1.
4α1

It is easy to check that p̂ > c2 under (12). In fact,p̂ − c2 =a + 2α1 c2 + α2 ca − 2α1 c2 + α2 c1− c2 =
> 0.
4α1
4α1

As ŷ2 is positive, the proﬁts of U2 are positive and U2 can supply inputs to the upstream
market. Substituting p̂ into eqs. (5) and (6) yields
x̂F

=

x̂1 =

5a + 2α1 c2 − 7α2 c1
a + α1 p̂ − 2α2 c1
=
> 0,
3
12
a − 2α1 c2 + α2 c1
a − 2α1 p̂ + α2 c1
=
> 0.
3
6

Finally, together with (1), equilibrium price in the downstream market isP̂ = a − x̂F − x̂2 =5a + 2α1 c2 + 5α2 c1.
12

It is easy to show that P̂ is larger than α1 p̂. In fact, per-units proﬁts (P̂ − α1 p̂) areP̂ − α1 p̂ =5a + 2α1 c2 + 5α2 c1
a − 2α1 c2 + α2 c1
a + 2α1 c2 + α2 c1
− α1
> 0,
=
12
4α1
6

where the sign comes from (12). This means that ﬁrm F can reap positive proﬁts and sell
outputs in the downstream market because equilibrium output x̂1 of ﬁrm 1 is positive.
Now, consider whether F has incentive to engage in a price squeeze. To do this, we compare
proﬁts πF of F in the OI game with proﬁts πsq under a price squeeze. In view of Lemma 2,
proﬁts πF∗ in the OI game are given by
πF∗ = (P̂ − α2 c1 )x̂F = (1
5a + 2α1 c2 + 5α2 c1
5a + 2α1 c2 − 7α2 c1− α2 c1 )=(5a + 2α1 c2 − 7α2 c1 )2 .
12
12
144

As F participates in the downstream market, F can manipulate downstream price to exclude
the downstream rival ﬁrm from that market. It follows from Lemma 2 that F can force proﬁts of
the rival equal to zero by setting the downstream price Psq equal to α1 p̂, which can be expressedas
Psq = α1 p̂ = α1 (a + 2α1 c2 + α2 c1
a + 2α1 c2 + α2 c1
.
)=
4α1
4

This is a downstream price under a price squeeze by F. It then follows from this and (1) that
output xsq of F under this strategy is
xsq = a − Psq = a −1
a + 2α1 c2 + α2 c1= (3a − 2α1 c2 − α2 c1 ),4415

where it should be noted that x2 = 0. Proﬁts under such a price squeeze can be described by
πsq = (Psq − α2 c1 )xsq = (=a + 2α1 c2 + α2 c1
1− α2 c1 ) (3a − 2α1 c2 − α2 c1 )4
4

1
(a + 2α1 c2 − 3α2 c1 )(3a − 2α1 c2 − α2 c1 ).
16

We can summarize our results as
Lemma 3. Proﬁts πsq under a price squeeze by the integrated duopolist F in OI game are given
by
πsq =

1
(a + 2α1 c2 − 3α2 c1 )(3a − 2α1 c2 − α2 c1 ).
16

In view of Lemma 2 and Lemma 3, it will be possible to show whether the integrated ﬁrm F
has incentives to monopolize a market by a price squeeze. It has been shown that the verticallyintegrated ﬁrm has no incentives to exclude a rival ﬁrm out of the market when an upstream
market is a monopoly. When we consider the OI game, it is not certain whether the same
results will be obtained. One of the upstream duopolists may be interested in driving the rival
out of the market because the duopolist can be a monopolist in the downstream market, whichis expected to provide the duopolist with the maximum proﬁts. We can be more precise in:
Proposition 3. The integrated duopolist in a Double Duopoly game has incentives to employ aprice squeeze to monopolize the market. Moreover, although a price squeeze is anti-competitivein its nature, such a strategy promotes market eﬃciency because it causes the downstream priceto go down.
1(5a + 2α1 c2 − 7α2 c1 )2 . The diﬀerence
Proof. Proﬁts πF∗ of F in the OI game are given by 144

between πsq and πF∗ is
πsq − πF∗

=
=
=

1
1
(a + 2α1 c2 − 3α2 c1 )(3a − 2α1 c2 − α2 c1 ) −(5a + 2α1 c2 − 7α2 c1 )2
16
144
1 2
(a + 8aα1 c2 − 20α21 c22 − 10aα2 c1 + 32α22 c21 − 11α22 c21 )72
1 2(a − 10aα2 c1 + 25α22 c21 + 8aα1 c2 − 20α21 c22 − 4α22 c21 ).
72

In what follows, consider whether the quadratic form above is positive. It is rewritten as(a2 − 10aα2 c1 + 25α22 c21 + 8aα1 c2 − 20α21 c22 − 4α22 c21 )> ((a − 5α2 c1 )2 + 8 × (4α1 c2 )α1 c2 − 20α21 c22 − 4α22 c21 )= ((a − 5α2 c1 )2 + 32α21 c22 − 20α21 c22 − 4α22 c21 )= ((a − 5α2 c1 )2 + 12α21 c22 − 4α22 c21 ) > 0,16

where the ﬁrst inequality comes from (12), and the second from (3) and c2 ≥ c1 . Then, proﬁts
under a price squeeze are larger than those in the OI game. It then follows that the integrated
duopolist F has incentives to employ a price squeeze.
Next, it follows from Lemma 2 and the discussion of price squeeze that the diﬀerence betweenP̂ and Psq is given byP̂ − Psq =a − 2α1 c2 + α2 c1
5a + 2α1 c2 + 5α2 c1 a + 2α1 c2 + α2 c1
−
=
> 0.
12
4
6

These results are in a stark contrast with those in the PM game where the upstream monopolist integrates a downstream ﬁrm. The partial monopolist does not have incentives tomonopolize a market. This is because additional proﬁts earned in the downstream market are
not compensated by proﬁts lost in the upstream market due to the exit of the downstream rival.
This in turn means that the monopolization of a market does not provide the ﬁrm with the
maximum proﬁts. This is the interesting feature of vertically related markets. On the other
hand, it follows from P roposition 2 that the integrated duopolist can make more proﬁts by
excluding a rival ﬁrm from a market. Thus, strategies such as a price squeeze which enable theintegrated ﬁrm to monopolize a market are adopted depending upon upstream market structure.
Thus, the upstream market structure matters.
However, it is interesting to note that such a strategy by the integrated duopolist brings abouta lower market price and hence promotes market eﬃciency. Even if the integration reduces thenumber of downstream ﬁrms, it enhances market welfare. Thus, this strategy is anti-competitivein the sense that a competitor is driven out of a market, but still promotes market eﬃciency.
The crucial assumption of the present model is that a > 4α1 c2 . In fact it is enough toassume that a > 3α1 c2 . This latter assumption will enable us to derive P roposition 3. Thus,
the assumption a > 4α1 c2 is too strong in the games which we have examined. However, whenother games such as games in which all ﬁrms are independent are taken up this assumption will
guarantee that all ﬁrms are viable. This is the reason why we have made the more restrictive
assumption. The second question to consider is, what happens when ineﬃcient ﬁrms are merged?
It will be shown that post-merger market prices go down. However, it is not certain that the
integrated ﬁrm has incentives to engage in a price squeeze. One of our results in Proposition 3
depends upon a combination of an upstream and a downstream ﬁrm.

17

4

Conclusions

In this paper, we have examined a price-squeeze strategy by the integrated ﬁrm in diﬀerent
settings and shown that the partial monopolist sets a price equal to the monopoly price under
this strategy. Thus, downstream price under such a strategy is higher than the price set in a
pre-merger game. The model has shown that the integrated monopolist does not have incentives
to monopolize the downstream market. Thus, it is not optimal for the partial monopolist to
exclude a rival ﬁrm from the market. In fact, proﬁts lost in the upstream market due to the
exclusion of a rival ﬁrm are not compensated by additional proﬁts earned in the downstream
market.
When the upstream market structure is a duopoly, the integrated duopolist has incentives
to drive a rival ﬁrm from the market by engaging in a price squeeze. However, this strategy
causes a decrease in market prices, which in turn advances welfare in the market. Althougha price squeeze is anti-competitive in its nature, a downstream market becomes more eﬃcientin the sense that market price becomes lower. Situations of “keiretsu” in Japanese industries
emerge in the one-sided integration game. These results are in a stark contrast with those in
the upstream monopoly game. Thus, upstream market structure matters when we consider theeﬀects of exclusionary strategies of the integrated ﬁrms in vertically related markets.
There are, however, several limitations to our model. Most importantly, it has been assumed
that the demand and production functions are linear. These enable us to simplify our analysis
and derive interesting and explicit results. When we try to broaden our model to include
non-linearity of important functions, diﬀerent results will be expected with possible loss of
explicit results. For example, if the production functions of downstream ﬁrms are concave,high production by a single ﬁrm results in decreasing average costs. Increased production of
outputs causes the integrated ﬁrm to reap more proﬁts. Thus, it is possible that an integrated
monopolist will employ a price squeeze in such circumstances. Similarly, when we assume a nonlinear demand function, diﬀerent results will be obtained. As these problems are of considerable
interest, their analysis will form the basis of our future research.

18

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