In this paper we analyze how the technology used by downstream firms can influence input and output market prices. We show via an example that both these prices increase under a decreasing returns technology while the countrary holds when the technology is constant.successive oligopolies, vertical integration, technology, foreclosure

Jean J., GABSZEWICZ

Skerdilajda, ZANAJ

English

Research Papers in Economics

Département des Sciences Économiques
de l'Université catholique de Louvain
A note on successive oligopolies and
vertical mergers
J.J.  Gabszewicz and Sk. Zanaj
Discussion  Paper   2007-36A note on successive oligopolies and vertical
mergers
Core Discussion Paper 2007/74
Jean J. Gabszewicz and Skerdilajda Zanajy
15/09/2007
Abstract
In this paper we analyze how the technology used by downstream rms
can in
uence input and output market prices. We show via an example
that both these prices increase under a decreasing returns technology while
the contrary holds when the technology is constant.
Keywords: successive oligopolies, vertical integration, tech-
nology, foreclosure
JEL classication: D43, L1, L22, L42
CORE, Universit e Catholique de Louvain.
yCore, Universit e Catholique de Louvain and Universit a di Siena.
11 Introduction
We provide in this paper a framework to analyse vertical integration and hor-
izontal mergers in a market context, based on early work of the authors ; see
Gabszewicz and Zanaj (2006). The analysis of collusion in successive markets
between downstream and upstream rms often relies on successive quantity
oligopolies. In these markets, rms select  a la Cournot the quantities of output
of the good they produce, the output of the upstream rms serving as an input
for the production of the output in the downstream market. Collusive agree-
ments reduce the total number of decision units operating in the downstream
and upstream markets and, thus, the corresponding number of oligopolists in
each of them. Collusive outcomes are the Cournot equilibria corresponding to
this reduced number of oligopolists in each market, which are then compared
with those arising when downstream and upstream rms act independently. The
link between upstream and downstream markets follows from the fact that the
downstream rms' unit cost appears as the unit revenue for the upstream ones :
the price paid for a unit of input for the rms in the former constitutes the unit
receipt for the rms in the latter. In this paper, we consider the simplest situa-
tion in which, before any collusion takes place, the downstream market consists
of two rival rms while the upstream one embodies three input suppliers. We
apply to this context the methodology proposed above to study the eects of
vertical integration on input and output prices.
We propose a model which makes explicit how the downstream and upstream
markets are linked to each other via the technology used by the downstream
rms to transform the input into the output. We consider two examples of
technology-linked markets. The rst corresponds to a decreasing returns tech-
nology while the other uses a constant returns technology. Our main nding is
that the nature of the technology plays a crucial role on the eects of mergers
concerning the input and output prices.
2 Successive oligopolies with decreasing returns
2.1 The context
Consider two downstream rms facing a linear demand (Q) = 1   Q in the
downstream market with Q denoting aggregate output: They share the same
technology f(z) to produce the output, namely
q = f(z) = z
1
2:
with z denoting the input used in the production of q: Consider also three
upstream rms each producing the input z at the same linear total cost Cj(sj) =
sj; j = 1;2;3: We assume that this situation gives rise to two games. The
players in the rst game are the two downstream rms with output strategiesqi, while the players in the second are the three upstream rms with input
strategies sj:
The prots of the ith downstream rm at the vector of strategies (qi;q i)
obtains as
i(qi;q i) = (1   qi   q i)qi   pq2
i ; i = 1;2
As a result of the strategic choice qi; each rm i sends the input quantity signal
zi(p) = q2
i to the upstream market. When aggregating these signals, we get
the demand function of input over which the upstream rms select their selling
strategies sj: The jth upstream rm's prot  j a the vector of strategies writes
as
 j(sj;s j) = p(sj;s j)sj   sj; (1)
with p(sj;s j) such that 3
k=1sk = 2
i=1zi(p).
Given a price p in the input market, the best reply of downstream rm i in
the downstream game obtains as
qi =
1   q i:
2p + 2
(2)
Clearly, these best replies depend on the upstream market price p and we may
compute the symmetric Nash equilibrium of the above game, contingent on the
price p: Dening qi = q for i = 1;2; re-expressing equation (2) and solving it in
q; we get the symmetric solution
q
i =
1
(3 + 2p)
(3)
so that we obtain
z
i (p) = z(p) =
1
(3 + 2p)
2; i = 1;2: (4)
The upstream rms then face a total demand
2 P
i=1
z
i (p) of input equal to 2z(p):
At a given vector of input strategies chosen by the upstream rms in the up-
stream game, the input price clearing the upstream market must satisfy
2
(3 + 2p)
2 = 3
j=1sj;
so that we get
p(sj;s j) =
s
2
43
j=1sj
 
3
2
: (5)
2Substituting (5) into (1), the payo function  j(sj;s j) of the upstream rm j
in the upstream game rewrites as
 j(sj;s j) = (
s
2
43
j=1sj
 
3
2
)sj   sj:
Notice that the prot function  j is concave in sj;j = 1;2;3; so that we can use
the rst order necessary and sucient conditions to characterize the symmetric
equilibrium. Accordingly, at the symmetric Nash equilibrium of the upstream
game, we obtain
s() =
25
54(2 + 3)
2:
Finally, the equilibrium price p() in the input market obtains as
p() =
6
5
 +
3
10
:
Consequently, substituting this equilibrium price into the equilibrium quan-
tities zof input bought by each downstream rm, as given by (4), we get
z() =
25
36(2 + 3)
2
so that, from (3), we obtain
q
i () = q() =
5
6(2 + 3)
:
Therefore, the resulting output price () in the downstream market obtains
as
() =
6 + 4
6 + 9
:
3 Vertical integration
Consider that one of the downstream rms vertically integrates with one of the
upstream rms. After this merger, we move from an initial situation comprising
globally ve rms to a new one, with two rms in the downstream market and
a duopoly in the upstream one. We assume complete foreclosure. Indeed, the
integrated entity now internalizes output production by using the input provided
at a marginal cost  by the upstream rm belonging to the new entity.
Let us rst consider the game played among downstream rms operating in
the downstream market after collusion takes place. The new sturcutre of the
market is an asymmetric Cournot. The payo of the integrated rm I is given
by
I(qI;qi) = (1   qI   qi)qI   q2
I;
3with i denoting the downstream rm i;i 6= I; not belonging to the integrated
entity which payos i(qi;qI) is given by
i(qi;qI) = (1   qi   qI)qi   pq2
i ): (6)
Since I is concave, we may use the rst order condition to get the best response
function of the integrated entity in the downstream market game as
qI =
1   qi
2 + 2
: (7)
As for the downstream rm i; i 6= I; its best reply in the downstream market is
conditional on the input price p realized in the upstream market, namely
qi =
1   qI
2 + 2p
(8)
Solving the system of equations (7) and (8), we get the output quantities as
qi =
2 + 1
4p + 4 + 4p + 3
;
qI =
2p + 1
4p + 4 + 4p + 3
:
Consequently, as expected, the downstream equilibrium is conditional on the
input price obtained in the upstream market as a result of supply and demand
in this market. There is only one downstream rm with input demand identical
to the total demand in the upstream market, namely, qi(p) = (
2+1
4p+4+4p+3)2:
As for the supply, it comes from the strategies sj; j 6= I; selected by the un-
integrated upstream rms in this market. Consider one of the two upstream
rms which do not belong to the entity. Its prot  j at the vector of strategies
(sj;s j) writes as
 j(sj;s j) = p(sj;s j)sj   sj;
with p(sj;s j) such that sj + s j = qi(p); namely
p(sj;s j) =
1
4 + 4
p
sj + s j
sj + s j
(2 + 1)   4   3

: (9)
Therefore, at the symmetric equilibrium in the upstream market, each uninte-
grated rm supplies a quantity s
j of input which obtain as1
s =
9
32(2 + 3)
2:
1To determine the optimal input supply per rm see footnote 9 in Gabszewicz and Zanaj
(2006).
4Substituting the expression of s
j in the demand function (9) we get the input
price p as
p =
60 + 322 + 21
4 + 4
:
Furthermore, substituting p in the expression of qi and qI, we obtain
q
i =
1
8(2 + 2)
;
q
I =
16 + 23
16( + 1)(2 + 3)
:
The price  of the output obtains as
 =
1
16
(2 + 1)(16 + 23)
( + 1)(2 + 3)
.
We have now the equilibrium quantities and price for both cases with, and
without, vertical integration. It can be shown from a direct comparison of
prices that this merger determines both an increase in the output price and an
increase in the input prices. Consider rst the input market. The merger causes
a decrease of both the number of demanders of input and suppliers of it. So,
at least theoretically, there would be a priori place for an increase or decrease
of the input price. Under a decreasing returns technology in the downstream
market, the change in the input demand resulting from the merger countervails
the change in the input supply always leading to an increase in the input price.
Concerning the output market, the merger reduces the production cost of the
rm in the entity. But, on the other hand, the merger increases the production
cost of the other rm, since it leads to a higher input price. It turns out that,
in this framework, the rst eect is dominated by the second, so that the output
price increases.
4 Constant returns
We consider exactly the same case as above, with the exception that the tech-
nology f(z) shared by the downstream rms is now given by
f(z) = z;  > 0
as in Salinger (1988) and Gaudet and Van Long (1996)(with  = 1 in the
latter case).We also assume that   : this assumption guarantees that the
marginal cost of producing the input does not exceed its marginal product in
the production of output2. The prots i(qi;q i) of the ith downstream rm
at the vector of strategies (qi;q i) now obtains as
2We also assume that   : this assumption guarantees that the marginal cost of pro-
ducing the input does not exceed its marginal product in the production of output.
5Figure 1: Output price with and with-
out mergers
Figure 2: Input prices with and without
mergers
6i(qi;q i) = (1   qi   q i)qi   pzi:
As a result of the strategic choice qi; each rm i sends an input quantity signal
zi(p) =
qi
 to the upstream market. Rewriting the prot function of each rm
in terms of input z, we may compute the symmetric Nash equilibrium of the
above game contingent on the input price p;namely
z(p) =
   p
32 ; (10)
so that
q =
   p
3
: (11)
Given the input strategies chosen by the upstream rms in the second stage
game, the input price clearing the upstream market must satisfy
2(   p)
32 = 3
j=1sj
so that, for this example, we get
p(sj;s j) =    23
2
3
j=1sj (12)
Substituting (12) into the payo function  j(sj;s j) we have
 j(sj;s j) =

  
32
2
3
j=1sj

sj   sj;
leading to the best response function
sj(s j) =
1
32 (   )   2
 j=1s j:
Accordingly, at the symmetric equilibrium of the second stage game, we obtain
s() =
(   )
62 :
Finally, the equilibrium price in the input market obtains as
p(;) =
1
4
 +
3
4
: (13)
Consequently, substituting this equilibrium price into the equilibrium quantities
z
i of input bought by each downstream rm, as given by (10), we get
z(;) =
1
42 (   );
so that
q
i (;) =
1
4
(   ):
7Accordingly, the resulting output price (m;n) in the downstream market
obtains as3
(m;n) =
1
12
(6 + 6):
4.1 Modelling collusion with constant returns technology
Assume again that two rms, one upstream and one downstream merge. Let
us rst consider the game played between the two rms operating in the down-
stream market after collusion takes place. The payo of the integrated rm I
is given by
I(qI;qi) = (1   qI   qi)qI   
qI

. As for the downstream rm i; not belonging to the integrated entity, it has as
payo
i(qi;qI) = (1   qi   qI)qi   p(
qi

): (14)
It is clear from the above payos that the main dierence between the collusive
and non collusive members in the downstream market comes from the fact that
the former pays its input at marginal cost  while the latter buys it at the input
price p: Since I is concave, we may use the rst order condition to get the best
reply function of the integrated entity and the unintegrated rm. solving the
resulting system, we get the equilibrium output quantities q
I(;;p) as
q
I(;;p) =
    + (p   )
3
and
q
i (;;p) =
   2p + 
3
: (15)
Consequently, as expected, the downstream equilibrium is conditional on the
input price obtained in the upstream market as a result of supply and demand
in this market. There is only one rm with total demand equal to q
i (p;) =
 2p+
3 : As for the supply, it comes from the strategies (sj;s j); selected by
the unintegrated upstream rms in this market. Consider the jth upstream rm
which does not belong to the entity. Its prot  j at the vector of strategies
(sj;s j) writes as
 j(sj;s j) = p(sj;s j)sj   sj;
with p(sj;s j) such that sj + s j = zi; namely
p(sj;s j) =
( + )   32 (sj + s j)
2
(16)
3Notice that, in order to have (m;n)  p(m;n); - the requirement needed to guarantee
the survival of rms in the downstream market -, no condition on  is required.
8. Accordingly, the payo of the j-th upstream rm writes as
 j(sj;s j) =

( + )   32 (sj + s j)
2

sj   sj:
Therefore , at the symmetric equilibrium in the upstream market, the uninte-
grated rm supplies a quantity s
jof input which obtains as
s
j(;) =
(   )
92 :
Substituting the expression of s
j in (16) we get the equilibrium input price
p(;) as
p(;) =
1
6
 +
5
6
: (17)
Substituting (17) in (15) we get the output supply q
i (;) and q
I(;) of each
unintegrated downstream rm, namely
q
i (;) =
2
9
(   );
q
I(;) =
7
18
(   ):
Hence, the resulting output price is given by
(;) =
1
18
(7 + 11):
It can be easily checked (see graph 3 and 4) that, when downstream rms
have a constant returns technology, and a vertical merger takes place, we get
exactly the reverse result as in the case of decreasing returns: in the case of a
merger, both the output and input prices decrease.
5 Concluding remarks
Starting from our previous work, we have analysed here the eects of vertical
mergers in the framework of successive oligopolies on input and output prices,
when these are determined by the market mechanism. Our analysis is based on
a comparison between the market outcomes with, and without, merger. The
interest of this approach, it seems to us, consists in showing how these eects
are related to the technology used in the production of nal goods. It is indeed
the latter which determines the demand function of the downstream rms in
the input market. When a merger forms, it creates a reduction in total input
demand since the integrated entity produces output internally at marginal cost.
Similarly, with complete foreclosure (as assumed), a merger reduces input sup-
ply since the entity does not participate in the upstream market anymore. The
question is to know whether a demand decrease, combined with a supply de-
crease would lead to higher, or smaller prices in the two markets concerned by
9the merger. Surprisingly, the answer to this question is not unequivocal since,
under decreasing returns to scale, both prices increase while the reverse is true
in the case of constant returns !
Mergers are expected to in
uence market conditions because they internalize
production costs reducing thereby the eects of double marginalization. Further-
more, they decrease competition in the input market by reducing the number
of competitors. Thus a decrease in the output price is a priori expected from
the reduction of double marginalisation, and this is indeed the case when down-
stream rms use a constant returns technology. By contrast, under decreasing
returns, the reverse consequence is observed : the output price increases. Why
is it so ? As usual, the reduction of double marginalisation increases the output
supply of the entity. However, it increases the cost of production of the down-
stream which did not adhere to the entity, to such an extent that the reduction
of its output level nally increases the output price: this decrease does not
compensate the eect on price of the entity output increase. As stated above,
mergers (under complete foreclosure) reduce the number of rms operating in
the upstream market, and thus, it must be anticipated that their existence re-
duces competition and increases input price. However, under constant returns,
it has the opposite eect. Why is it so ? The explanation lies in the behaviour
of the unintegrated downstream rm. Due to the merger, this rm considerably
reduces its production level in the output market and, accordingly, its demand in
the input market becomes smaller at any input price. This leads to a reduction
of the input price.
In this paper, we have only scratched a research territory which looks very
promising. The theory of successive markets allows us to treat various forms of
collusion based on the market mechanism, which provides a natural way to eval-
uate the protability of such collusive arrangements. Another potential avenue
for an alternative protability evaluation would consist in relying on strategic
market games. From this viewpoint, Gabszewicz and al. would constitute a
natural departure point for researchers interested in the eld.
References
[1] Gabszewicz J. and Zanaj S., " Competition in successive markets: entry and
mergers", CORE DP 2006/97.
[2] Gabszewicz J., Laussel D., van Ypersele T, and Zanaj S., "Market games
and successive markets", CORE DP 2007/10.
[3] Gaudet G. and Van Long N., "Vertical Integration, Foreclosure and Prof-
its in the Presence of Double Marginalisation", Journal of Economics and
Management Strategy, 1996, Vol. 5(3), 409-432
[4] Salinger M., "Vertical mergers and market foreclosure", The Quartely Jour-
nal of Economics, 1988, Vol. 103, 345-356
10Figure 3: Output price with and with-
out mergers
Figure 4: Input prices with and without
mergers
11Département des Sciences Économiques
de l'Université catholique de Louvain
Institut de Recherches Économiques et Sociales
Place Montesquieu, 3
1348 Louvain-la-Neuve, Belgique
 ISSN 1379-244X         D/2007/3082/036

Competition in successive markets: entry and mergers&quot;,

Market games and successive markets&quot;,

Vertical Integration, Foreclosure and Profits in the Presence of Double Marginalisation&quot;,

Vertical mergers and market foreclosure&quot;,

A note on successive oligopolies and vertical mergers

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A note on successive oligopolies and vertical mergers

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