Upstream incentives to encourage downstream competition
under bilateral oligopoly∗
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 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 because downstream competition erodes the downstream industry profits. This relationship between downstream competition and the upstream
firm’s profits is important 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, this author shows that the upstream
firm faces now the previous commitment problem but also the threat of losing sales to a rival
supplier. The dominant supplier wants the outside option of downstream firms to be as small
as possible and this outside option is decreasing in the number of downstream competitors. He
obtains that if upstream competition is strong enough (in other words, if the alternative supplier is efficient 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 increases with 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 (for example
we can imagine a situation where the upstream firm posts a two-part tariff). In this setting,
the wholesale price is not constant in the number of downstream firms. In particular, a higher
number of downstream firms leads the upstream firm to increase the wholesale price in order
to reduce the outside option. For an efficient enough alternative supplier, this effect offsets the
profit reducing effect of competition and leads to the result. Next, 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 will 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 revenues of the upstream monopolist increase with more downstream competition simply
because in a Cournot setting, total output increases with the number of firms in the market.
And, finally, we show that under costly free entry in the downstream market and regardless of
the number of downstream firms currently in the market, the upstream firm would always be
willing to subsidize entry by reducing the wholesale price in order to encourage the entry of new
firms in the downstream industry.
Chemla (2003) has investigated also the relationship between the upstream market profit and
the degree of downstream competition in a related framework where there is bargaining between
upstream and downstream firms. The upstream firm faces a trade off between a negative rent
3
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.
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 < a. 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 > 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
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 .
4
Assume that k firms have accepted a supply contract (F, w). Firms that have not accepted
the contract produce 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,1 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
upstream solves the following problem:
M ax k (Π(k, w) − ΠN (k − 1, w) + (w − c)q(k, w))
k,w
(1)
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
1
As
∂(Π(k,w)−ΠN (k−1,w))
∂k
< 0, this is the only equilibrium in the acceptance stage.
5
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.2
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
n+1
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
otherwise3 .
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
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.4
Concerning the equilibrium contract, the optimal wholesale price trade-offs two conflicting
incentives. On the one hand, maximizing industry profits requires a high wholesale price; on
2
See the proof of this result in the Appendix.
3
This optimal contract has been obtained before in Faulí-Oller and Sandonís (2012). They use the same
framework to discuss the profitability and welfare effects of downstream mergers.
4
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
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.5
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 .
c(n), where b
c(n) =
increasing in n for c0 > b
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 may be good for the upstream firm. The intuition for
the result is the following: an increase in the number of downstream firms reduces both total
industry profits and the outside option of downstream firms. So the upstream firm has more
flexibility now 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.
5
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
In the following section we show that 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
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
a−w
b(n + 1)
s.t.w < a.
The solution to this problem gives us w∗ =
a+c
2 ,
which is lower than a, increasing in the mar-
ginal cost of the upstream firm and independent of the number of firms in the industry. Observe
8
n (w−c)(a−w)
.
n−1
b
that the profit function of the upstream firm can be written as ΠU (n, w) =
term
n
n−1
The
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
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
are given by: ΠU (n, w∗ ) =
ΠM =
(a−c)2
4b
n(a−c)2
4b(n+1)
which increases with n and tend to the monopoly profits
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 to infinity6 . This simple model shows that
an upstream monopolist selling an essential input7 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,
6
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.
7
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)).
9
the optimal wholesale price would be equal to c0 . In this case, the upstream profits are given by
n(c´−c)(a−c‘)
b(n+1)
ΠU (n, c0 ) =
3.2
which are again an increasing function of n.
Observable linear contracts with free entry
Let’s assume now that there is free entry downstream at a fixed cost F. Suppose that
F ≤
(a−c)2
4bn2
so that, anticipating the wholesale price w∗ =
a+c
2
(a−c)2
4b(n+1)2
<
only (n − 1) firms would enter
the downstream market8 . The question now is whether the upstream firm could be interested
in reducing the wholesale price in order to allow for the nth downstream firm to enter the
market. In order to answer that question we have to compare the profits of the upstream firm
if it sells the input to the (n − 1) firms with the optimal wholesale price w∗ with its profits
if it reduces the wholesale price so that the nth firm enters the market. In the former case,
the upstream profits are given by ΠU (n − 1, w∗ ) =
(n−1)(a−c)2
;
4bn
firm has to choose w’ such that ΠD (n, w0 ) = F , that is,
In the latter case, the upstream
(a−w’)2
b(n+1)2
= F. Solving for w0 we find
√
w0 = a − (n + 1) bF . By substitution we find that the upstream profits in this case would be
ΠU (n, w0 ) =
√
n(a−c) bF −n(n+1)bF
.
b
Finally, we want to analyze the sign of ΠU (n, w0 )−ΠU (n−1, w∗ ). We get the following result:
For
(a−c)2
4b(n+1)2
<F ≤
(a−c)2
4bn2 ,
ΠU (n, w0 ) − ΠU (n − 1, w∗ ) ≥ 0.
In other words, the upstream firm is always interested in reducing the wholesale price to
promote the entry of one more firm. So again, more competition downstream increases the
upstream firm’s profits in this context with free entry.
8
Observe that
(a−c)2
4b(n+1)2
is the Cournot profit of each downstream firm when there are n in the market under
the optimal wholesale price w∗ .
Similarly,
(a−c)2
4bn2
is the Cournot profit under the optimal contract when there are (n − 1) firms in the market.
10
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
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 many different frameworks where
an upstream firm has an incentive to encourage entry 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
11
(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:
−a + c
, where qN (k, w) > 0 and qN (k − 1, w) > 0,
k
−a + c
, where qN (k, w) = 0 and qN (k − 1, w) > 0 and
<w ≤c+
k
when c ≥ w > c +
when
−a+ck
−1+k
when w ≤
−a+ck
−1+k ,
If c ≥ w > c +
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.
12
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