Upstream incentives to encourage downstream competition in a
vertically separated industry∗
Joel Sandonis†
December 2012
(preliminary)
Abstract
In this paper we show that under observable contracts, different frameworks exist (linear
vs. two-part tariff contracts, upstream monopolist vs alternative supplier, no entry vs.
free entry downstream) where more competition in the downstream market increases the
upstream firm’s profits, which contradicts some previous results in the literature. This result
complements Caprice (2005), who presents a similar result for the case of secret contracts
and an alternative supplier.
Key words: vertically separated industries, downstream competition, wholesale price,
two-part tariff contracts
JEL codes: L11, L13 and L14.
∗
The support from Ministerio de Ciencia e Innovación and FEDER funds under SEJ 2007-62656 and the
IVIE are gratefully acknowledged. This work was partially written while the author was visiting the Economics
Department at Boston University.
†
University of Alicante. Economics Department, Campus de Sant Vicent del Raspeig, E-03071, Alicante,
Spain. E-mail address: [email protected].
1
1
Introduction
Consider an upstream monopolist selling an input to a downstream oligopolistic industry through
secret two-part tariff contracts. In this setting, a well known opportunistic problem arises: once
the upstream firm has contracted with a downstream firm, it has an incentive to renegotiate
other contracts in order to increase its profits at that firm’s expense. This opportunistic problem
reduces the upstream firm’s profits because downstream firms, anticipating the lack of commitment of the input producer are willing to pay less for the input. For example, under the so called
passive beliefs and Cournot competition downstream, the equilibrium wholesale price equals the
marginal cost of producing the input and the fixed fee is used to extract the entire downstream
firm’s profits. In this setting, it is very intuitive that the upstream firm’s profits are decreasing in the number of downstream competitors simply because downstream competition erodes
the downstream industry profits. This relationship between downstream competition and the
upstream firm’s profits is a key factor for example to discuss questions about the incentives
for vertical integration and market foreclosure (see Rey and Tirole’s (2007) survey). However,
as Caprice (2005) shows, this negative relationship could turn out to be positive if we add an
alternative less efficient supplier of the input. In this new framework, an interesting trade-off
arises because now the dominant supplier wants the outside option of downstream firms (what
they can guarantee if buying the output to the alternative supplier) to be as small as possible
and this outside option is decreasing in the number of downstream competitors. This author
obtains that if the alternative supplier is efficient enough (in other words, if upstream competition is strong enough) and up to a threshold value for the number of downstream firms, the
profit-reducing effect is offset by this new rent-shifting effect and then the dominant upstream
firm’s profit becomes increasing in the number of downstream firms.
2
In the present paper we want to show that Caprice’s result is, in fact, a more general result
and holds also under observable contracts in different settings concerning the contract type and
the existence or not of competition upstream. We start by reproducing Caprice’s (2005) result
in his same framework but assuming that the two-part tariff contracts are observable.1 In this
setting, the equilibrium wholesale price is not constant in the number of downstream firms.
In particular, a higher number of downstream firms reduces both the industry profits and the
outside options of downstream firms and so it leads the upstream firm to increase the wholesale
price in order to increase the industry profits. For an efficient enough alternative supplier, this
effect offsets the profit reducing effect of competition and leads to the result. Second, we show
that the existence of an alternative supply of the input is not a necessary condition to reproduce
Caprice’s (2005) result. In particular, we show that the result also holds in a setting where the
upstream firm uses linear wholesale prices no matter if there is an alternative supplier or not
(notice that under a two-part tariff contract and no alternative supply, the upstream firm would
always implement the monopoly solution regardless of the number of downstream firms). The
intuition is that the optimal wholesale price is constant in the number of downstream firms in
this setting and then, the wholesale revenues of the upstream monopolist are increasing in the
number of downstream competitors simply because in a Cournot setting, total output increases
with the number of firms in the market. Finally, we show that if there is costly free entry in
the downstream market, the upstream firm would find profitable to subsidize entry by reducing
the wholesale price (compared with the one we obtained in the setting without entry) in order
to encourage the entry of new firms in the downstream industry.
1
Concerning the discussion about the observability of contracts, this should be of course an empirical question.
For example we can think of an industry with many small downstream firms where negotiating the contracts one
by one would be very costly and time consuming and so the upstream firm prefers simply to post a two-part tariff
contract.
3
Apart from Caprice (2005), Chemla (2003) has investigated also the relationship between the
upstream market profit and the degree of downstream competition in a related framework with
secret contracts and bargaining between upstream and downstream firms. The upstream firm
faces now a trade off between a negative rent dissipation effect and a positive bargaining effect
of increased downstream competition. When downstream firms have high bargaining power, it
is shown that the upstream firm’s profits increase with the number of downstream firms. To the
best of my knowledge, mine is the first paper dealing with this issue in the context of observable
contracts.
The rest of the paper is organized as follows. In the next section, we analyze the model with
an alternative supplier and observable contracts. Section 3 proves the main result for the case
of linear wholesale prices with and without the possibility of entry in the downstream market.
Finally, Section 4 concludes. The proofs are relegated to the Appendix.
2
Observable two-part tariff contracts with an alternative supply.
Consider an upstream firm that produces an input at cost c. A number n of downstream firms
transform this input into a final homogenous good on a one-to-one basis, without additional
costs. Downstream firms may alternatively obtain the input from a competitive supply at cost
c0 The competitive supply is less efficient than the upstream firm so we assume c < c0 . For
simplicity we consider this second supplier as exogenous (it could be a competitive international
source of the input). The inverse demand for the final good is given by P = a − Q, where Q is
the total amount produced, a > c0 > c and downstream firms compete in quantities.
The game is modelled according to the following timing: first, the supplier offers a observable
two-part tariff contract (F, w) to downstream firms, where F specifies a fixed fee and w a linear
4
wholesale price. Second, downstream firms decide whether or not to accept the contract. The
ones that accept, pay F to the upstream firm. Finally, they compete à la Cournot, with the
marginal costs inherited from the second stage. In particular, the firms that accept the contract
have a marginal cost w and the firms that do not accept the contract buy the input from the
alternative supply and have a marginal cost c0 .
Assume that k firms have accepted a supply contract (F, w). Firms that have not accepted
the contract produce at the third stage in equilibrium:
qN (k, w) =
⎧
⎪
⎪
⎨
a−c0 (k+1)+wk
n+1
if
⎪
⎪
⎩ 0
w>
−a+c0 (k+1)
k
otherwise.
On the other hand, the firms that accept the contract produce in equilibrium:
⎧
0
0
⎪
⎪
⎨ a+c (n−k)−w(n−k+1)
if
w > −a+ck(k+1)
n+1
q(k, w) =
⎪
⎪
⎩ a−w
otherwise.
k+1
Observe that, if w is sufficiently low, the firms that do not accept the contract are driven out of
the market. In that case, the firms that accept the contract produce the Cournot output when
there are only k active firms in the market. Profits of non-accepting and accepting firms are
given, respectively, by ΠN (k, w) = (qN (k, w))2 and Π(k, w) = (q(k, w))2 .
In the second stage, downstream firms accept the contract offered by the upstream firm
whenever F ≤ Π(k, w) − ΠN (k − 1, w). Obviously, as the upstream firm maximizes profits, in
order for k firms to accept the contract,2 it will choose F to bind their participation constraint,
that is, such that F = Π(k, w) − ΠN (k − 1, w). But this implies that the problem of choosing
the optimal contract (F, w) is equivalent to that of choosing (k, w). Then, in the first stage, the
2
As
∂(Π(k,w)−ΠN (k−1,w))
∂k
< 0, this is the only equilibrium in the acceptance stage.
5
upstream solves the following problem:
M ax k (Π(k, w) − ΠN (k − 1, w) + (w − c)q(k, w))
(1)
k,w
s.t. 1 ≤ k ≤ n and w ≤ c0 .
This problem has been already solved in the literature. Erutku and Richelle (2007) solve
an equivalent problem for the case of a research laboratory licensing a cost-reducing innovation
to a n-firms homogeneous goods Cournot oligopoly through observable two-part tariff licensing
contracts. Making use of this previously existing result, we know first, that regardless of the
number of downstream firms, the upstream firm finds profitable to sell the input to all of them.3
Second, if we replace k by n and plug the corresponding profit expressions in the maximization
problem of the upstream firm we get:
⎧
µ³
´2
´
´¶
³ 0
³
⎪
⎪
a−c n+w(n−1) 2
a−w
a−w
⎪
if
−
+ (w − c) n+1
⎨ n
n+1
n+1
M ax
µ ³
´2
´¶
³
w
⎪
⎪
a−w
a−w
⎪
if
+ (w − c) n+1
⎩ n
n+1
c0 ≥ w ≥
w<
−a+c0 n
n−1
−a+c0 n
n−1 .
s.t.w ≤ c0 .
Direct resolution of this problem leads to the following optimal wholesale price:
w∗ (n) =
(n−1)(2c0 n+c−a)+2c
2(1−n+n2 )
if c0 <
a−c+(a+c)n2
2n2
and wM (n) =
−a+c+(a+c)n
2n
otherwise4 .
The intuition for this result is as follows: concerning the optimality of selling to all firms, we
know that with a fixed fee contract, the input would be sold to only a subset of firms in order to
protect industry profits from competition (Kamien and Tauman (1986)). With a two-part tariff
contract however, the upstream firm can always sell the input to more firms without affecting
3
See the proof of this result in the Appendix.
4
This optimal contract was obtained in Faulí-Oller and Sandonís (2012). They use the same framework to
discuss the profitability and welfare effects of downstream mergers.
6
the level of competition, by choosing an appropriate (higher) wholesale price. In other words,
the upstream firm can always use the wholesale price to control for the level of competition
downstream.5
Concerning the equilibrium contract, notice that the maximization program of the upstream
firm can be seen as maximizing total industry profits minus the outside option of downstream
firms (that is, what they can get when being supplied by the alternative supplier) and then
the optimal wholesale price trades-off two conflicting incentives. On the one hand, maximizing
industry profits requires a high wholesale price; on the other hand, reducing the outside option
of downstream firms asks for a low wholesale price. Observe that whenever c0 ≥
a−c+(a+c)n2
,
2n2
the outside option becomes zero and thus the upstream firm obtains the full monopoly profits.
In this case, as n increases the wholesale price is adjusted upwards in order to implement the
monopoly price in the final market. On the other hand, it can be checked that w∗ is an increasing
∗
function of n ( dwdn(n) > 0) and tends to c0 as n tends to infinity.6
We next compute the equilibrium profits of the upstream firm just by plugging the optimal
contract into its profit function. They are given by:
½ n((a−c)((a−c)(1+n2 )−2(a−2c0 +c))+4(a−c0 )(c0 −c)n3 ) if c0 ≤
4(1+n+n3 +n4 )
ΠU (n) =
¡ a−c0 ¢2
2
Now we have to check the sign of
∂ΠU (n)
∂n .
increasing in n for c0 > b
c(n), where b
c(n) =
a+c
2
or c0 >
a+c
2
and n <
otherwise.
q
a−c
2c0 −a−c
It is direct to see that the upstream profits are
c(1+n)3 +a(n−1)(1+(−4+n)n)
,
2n(4+(n−1)n)
and b
c(n) <
a+c
. So we
2
find that more competition downstream is good for the upstream firm whenever the alternative
5
This argument is also used in Sen and Tauman (2007) to prove that with an auction plus royalty contract, a
cost reducing innovation would be sold to all firms by an outsider patentee, and also by Faulí-Oller, González and
Sandonís (2012) to show that the same result holds for the case of differentiated goods and for both an outsider
and an insider patentee.
6
This holds for any n ≥ 2. Observe that, if c0 <
a + 3c
, w∗ (1) = c > w∗ (2). Notice also that the restriction
4
that the wholesale price can not be higher than c0 is never binding in equilibrium.
7
supplier is inefficient enough (or, in other words, when there is strong market power upstream).
The intuition for the result is the following: an increase in the number of downstream firms
reduces both total industry profits (which hurts the upstream firm) and the outside option of
downstream firms (which benefits the supplier). So the upstream firm has now flexibility to
increase the wholesale price in order to help increase the total industry profit. This positive
effect on the upstream firm’s profit is larger the higher is c0 or, in other words, the stronger the
market power of the dominant supplier. So as a result, for large enough values of c0 we have that
the positive effect produced by the increase in w offsets the negative effect of a more competitive
downstream market on the upstream firm’s profits, leading to our result. It is interesting to notice
that whereas in Caprice’s (2005) setting with secret contracts, the upstream profits increase with
n when the alternative supplier is efficient enough, in our context with observable contracts, it
is for large values of c0 when we get the result.
In the following section we show that indeed the existence of an alternative supply of the
input is not a necessary condition to get a positive relationship between the degree of competition
downstream and the upstream firm’s profits. The same result also arises in a framework without
an alternative supply (so the upstream firm is a pure monopolist in the upstream sector) and
where the input is sold through linear contracts.
3
Observable linear contracts and no alternative supply
In this section we assume that the upstream firm is a monopolist in the upstream sector, and
that the monopolist offers an observable linear wholesale price to downstream firms. More
specifically, the game is modelled according to the following timing: first, the supplier offers
an observable linear wholesale price w to downstream firms. Second, downstream firms decide
simultaneously whether or not to accept the contract. Finally, the ones that accept compete
8
à la Cournot, with the marginal costs w inherited from the previous stage. The firms that do
not accept the contract exit the market because there is no alternative source of the input.
We solve by backward induction. If the n downstream firms accept the contract, they produce
in equilibrium in the third stage: qD (n, w) =
a−w
b(n+1)
and get profits ΠD (n, w) = (qD (n, w))2 .
Observe that whenever w < a, downstream firms produce positive amounts and get positive
profits in equilibrium. So in the first stage the upstream firm chooses w to maximize:
M ax
n(w − c)
w
s.t.
a−w
b(n + 1)
c < w < a.
The solution to this problem gives us w∗ =
a+c
2 ,
which is lower than a, increasing in the
marginal cost of the upstream firm and independent of the number of firms in the industry. Observe that the profit function of the upstream firm can be written as ΠU (n, w) =
The term
n
n−1
n (w−c)(a−w)
.
n−1
b
works as a scale factor that shifts the (concave) profit function vertically as n
increases, without changing the value of w where the function reaches its maximum. Then, the
revenues (and profits) of the upstream monopolist increase with n simply because in a Cournot
setting, total output increases with the number of firms in the market. Observe that the upstream firm´s profits when using w∗ are given by ΠU (n, w∗ ) =
and tend to the monopoly profits ΠM =
(a−c)2
4b
n(a−c)2
4b(n+1)
which increases with n
as n tends to infinity. On the other hand, each
downstream firm’s profits are given by ΠD (n, w∗ ) =
(a−c)2
4b(n+1)2
which tends to zero as n tends
to infinity. Finally, the equilibrium price and total output in the market are given respectively
by p∗ (n) =
2a+n(a+c)
2(n+1)
and Q∗ (n) =
n(a−c)
2b(n+1)
and they tend to the monopoly values as n tends
9
to infinity7 . This simple model shows that an upstream monopolist selling an essential input8
through a linear wholesale price to a set of downstream firms competing in quantities has an
incentive to sell to all downstream firms. In other words, market foreclosure of downstream
firms is not an issue in this context.
3.1
Observable linear contracts with an alternative supplier
Suppose now that there is an alternative, less efficient input supplier that sells the input at a price
c0 > c. Now, if c < w∗ ≤ c0 this model is equivalent to the one in the previous subsection where no
outside option exists for downstream firms. Whenever w∗ > c0 , however, the optimal wholesale
price would be equal to c0 , in order to satisfy the participation constraint of the downstream
firms. In this case, the equilibrium upstream profits are given by ΠU (n, c0 ) =
n(c´−c)(a−c‘)
b(n+1)
which
are again an increasing function of n.
3.2
Observable linear contracts with free entry
Let’s assume now that there is free entry downstream at a fixed cost K. In order to study the
effect that w has on the number of firms in this setting, let’s assume that the contract offered by
the upstream firm is a long term contract. The equilibrium number of firms in the downstream
market for a given w is given by the number of firms n such that the profits of downstream firms
7
Observe that there is a double marginalization problem in this setting. The upstream monopolist charges a
monopoly margin over its marginal cost and the downstream firms margin goes from the monopoly margin when
there is only one of them to zero as n tends to infinity. In the latter case, the upstream firm implements the
monopoly outcome because the downstream market becomes competitive and no double margin exists any longer.
8
An input produced by a dominant firm is essential if it cannot be cheaply duplicated by users who are denied
access to it. Examples of inputs that have been deemed essential by antitrust authorities include a stadium, a
railroad bridge or station, a harbor, a power transmission or a local telecommunications network, an operating
system software and a computer reservation system (Rey and Tirole (2007)).
10
become zero, that is, such that
(a−w)2
b(n+1)2
= K, which gives us n(w) =
a−w
√
bK
− 1 (let us consider for
simplicity n as a continuous variable).
In this setting, a change in the wholesale price w has two different effects on the upstream
firm’s profits. It affects the per unit upstream revenue but also the number of downstream firms
active in the market.
We next write the upstream firm’s profits and then replace n by n(w), the equilibrium
number of firms:
√
a−w
√
ΠU (w) = n(w)(w − c) b(n(w)+1)
= (w − c)( a−w
− 1) K
K
Finally, maximizing the previous expression with respect to w we get:
w∗ =
√
a+c− K
.
2
And the equilibrium profits are given by:
ΠU (w∗ ) =
√
(a−c− K)2
,
4
which is a decreasing function of F.
As a conclusion we can see that a drop in the fixed cost of entry K would increase the
upstream profits. But a lower K is associated with a higher number of firms in the downstream
market. In other words, in this context we also obtain a positive relationship between the
upstream profits and the number of downstream firms. On the other hand, it can be seen that
compared with the case of an exogenous n, in the model with free entry the upstream firm
chooses a lower wholesale price w, with the aim to stimulate the entry of new firms in the
market.
4
Conclusions
In this paper we extend Caprice’s (2005) paper, where it is shown that the upstream profits
can be increasing in the number of downstream firms in a context with secret contracts and an
alternative inefficient supplier of the input. In particular, we show that the same result arises
11
under observable contracts with or without an alternative supplier and for different types of
supply contracts. So, in other words, neither non-observability of contracts nor the existence of
an alternative supplier are necessary conditions in order to get Caprice’s result.
The paper has some policy conclusions. As Caprice (2005) points out, we have shown that the
incentives of upstream firms to foreclose downstream firms is less important than the previous
literature had suggested. As we have seen, there are several different frameworks where an
upstream firm has an incentive to encourage competition downstream.
5
Appendix
Proof of the result that the upstream firm sells the input to all firms:
Let π(k, w) represent the upstream firm profits if it sells the input to k firms and sets a
wholesale price w ≤ c.
−
π(k, w) = (P − cu ) ((n − k)qN (k, w) + kq(k, w)) − k (qN (k − 1, w))2 − (n − k) (qN (k, w))2 (2)
−(c − cu )(n − k)qN (k, w).
We define the wholesale price w1 that solves nq(n, w1 ) = (n− k)qN (k, w) + kq(k, w). Observe
that if the upstream sells to n firms with the wholesale price w1 , the first term in the expression
(2) will also appear in π(n, w1 ). Then the difference in profits is given by:
π(n, w1 ) − π(k, w) = k (qN (k − 1, w))2 + (n − k) (qN (k, w))2 +
+(c − cu )(n − k)qN (k, w) − n (qN (n − 1, w1 ))2 .
In order to prove the result we have to check that the previous expression is non-negative in the
following three different regions:
when c ≥ w > c +
−a + c
, where qN (k, w) > 0 and qN (k − 1, w) > 0,
k
12
when
−a+ck
−1+k
when w ≤
<w ≤c+
−a+ck
−1+k ,
If c ≥ w > c +
−a + c
, where qN (k, w) = 0 and qN (k − 1, w) > 0 and
k
where qN (k, w) = 0 and qN (k − 1, w) = 0.
−a + c
, we have that w ≤ w1 =
k
c(n−k)+kw
n
≤ c and
π(n, w1 ) − π(k, w) > k (qN (k − 1, w))2 + (n − k) (qN (k, w))2 − n (qN (n − 1, w1 ))2 = (3)
=
If
−a+ck
−1+k
<w ≤c+
(n − k)k(c − w)2
≥ 0.
n(1 + n)
−a + c
, we have that w < w1 =
k
a(n−k)+k(n+1)w
(k+1)n
< c and qN (k, w) = 0.
We have to distinguish two cases:
If
c(1+k)n2 −a(k+n2 )
k(n2 −1)
< w ≤ c+
−a + c
, we have that qN (n − 1, w1 ) > 0. To sign the difference
k
in profits we obtain that
kqN (k − 1, w) − nqN (n − 1, w1 ) ≥ 0.
This implies that
π(n, w1 ) − π(k, w) = k (qN (k − 1, w))2 − n (qN (n − 1, w1 ))2 > 0.
If
−a+ck
−1+k
<w≤
c(1+k)n2 −a(k+n2 )
,
k(n2 −1)
then w1 ≤
−a+cn
−1+n
and, therefore, qN (n − 1, w1 ) = 0. Then,
π(n, w1 ) − π(k, w) = k (qN (k − 1, w))2 > 0.
If w ≤
−a+ck
−1+k ,
we have that w1 =
a(n−k)+k(n+1)w
(k+1)n
≤
−a+cn
−1+n
and, therefore, qN (n − 1, w1 ) = 0.
As we have also that qN (k − 1, w) = 0, then
π(n, w1 ) − π(k, w) = 0.
6
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13
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14