Foreclosure with Incomplete Information
LUCY WHITE
Harvard Business School
Baker Library 345, Soldiers Field Road
Boston, MA 02163, USA
[email protected]
and
Swiss Finance Institute and DEEP, Ecole des HEC
Université de Lausanne
Bureau 553, Internef, 1015 Dorigny
Lusanne, Switzerland
[email protected]
We investigate the robustness of the new foreclosure doctrine and its associated welfare implications to the introduction of incomplete information. In
particular, we let the upstream firm’s marginal cost be private information,
unknown to the downstream firms. The previous literature has argued that
vertical integration is harmful because it allows an upstream monopolist to
limit output to monopoly levels, whereas a disintegrated structure will “oversell,” producing more in equilibrium. By contrast, we find that with incomplete
information, high-cost firms will often “under-sell” in equilibrium, that is,
supply less than their monopoly output. Low-cost firms continue to over-sell,
so all types of firms have a reason to integrate downstream, but this is socially
harmful only for low-cost types. For high-cost firms vertical integration can be
Pareto-improving, resulting in higher output, profits, and consumer surplus.
1. Introduction
Foreclosure may be defined as a dominant firm’s practice of inhibiting
one or more (downstream) firms’ access to an essential input, which
it supplies. The Chicago School (Bork, 1978; Posner, 1979) argued that
such foreclosure cannot be profitable because the upstream firm cannot
extend its market power by monopolizing the downstream market.
Loosely speaking, there is only one monopoly profit to be earned in
Much of this work was completed although the author was at GREMAQ, Université
des Sciences Sociales, Toulouse and Nuffield College, Oxford. The author thanks an
anonymous referee and an Associate Editor for very useful comments; as well as Liam
Brunt, Paul Klemperer, Kai-Uwe Kühn, Catalina Martinez, Felix Ng, Patrick Rey, and
Jean Tirole, seminar participants in Oxford, M.I.T. and the 1999 Meeting of the European
Economic Association for helpful comments and suggestions.
C 2007, The Author(s)
C 2007 Blackwell Publishing
Journal Compilation Journal of Economics & Management Strategy, Volume 16, Number 2, Summer 2007, 507–535
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Journal of Economics & Management Strategy
any given industry, and an upstream monopoly position is already
sufficient to extract this: there is no need to interfere with competition
downstream.
More recently, a strand of the literature, which (for the sake of supplying a label) we will call the new foreclosure doctrine, has suggested that
this argument is flawed because when there is competition downstream
it can be difficult for the upstream firm to commit to limiting output to
monopoly levels. Then vertical integration by a dominant upstream firm
is harmful because it can lead to foreclosure of the unintegrated downstream firm(s), and thence reduced output. This more negative view
of vertical mergers has been based on a series of complete information
models (see, for example, the seminal paper by Hart and Tirole, 1990;
McAfee and Schwartz, 1994, and the recent survey by Rey and Tirole,
2005). In this paper, we examine the robustness of the results of this
literature to the introduction of incomplete information about upstream
firm costs. This exercise leads to rather different policy conclusions. We
show that whilst the new foreclosure doctrine continues to hold for
low-cost firms, vertically disintegrated high-cost firms have signaling
incentives to restrict output. These incentives do not arise in complete
information models. Moreover, these incentives disappear—leading to a
potential expansion of output—if vertical merger (or exclusive dealing)
is allowed. Thus, the effect of allowing such vertical restraints on the
output of high-cost types is ambiguous. In some cases, vertical merger
will increase profits, output, and welfare. Thus, the current analysis
suggests that antitrust authorities should adopt a less hostile approach
to such vertical restraints than that implied by the new foreclosure
doctrine.
Introducing incomplete information into a foreclosure model is
thus not just a simple robustness check, but yields different policy
conclusions. There are two other reasons for pursuing this line of
enquiry. The first reason is methodological. The reader may recall that
the results of the new foreclosure doctrine depend on the assumption
that downstream firms have passive conjectures about upstream firm sales
when the latter makes offers to them. That is to say that these results
depend on the assumption that when a downstream firm receives an
out-of-equilibrium offer from the upstream monopolist, it continues
to believe that its downstream rivals receive their on-the-equilibrium
path offers. By contrast, if downstream firms instead have symmetric
conjectures—that is, they believe that their downstream rivals always
receive exactly the same offer as themselves, whether or not the offer was
the expected one—then there is no need for firms to vertically integrate
because the monopoly outcome is sustainable under vertical separation.
Although passive conjectures seem generally more reasonable, one may
Foreclosure with Incomplete Information
509
feel a little uneasy about the choice because both forms of conjecture
constitute perfect Baysesian equilibria. It had been thought that introducing incomplete information might offer a solution to this dilemma.1
Unfortunately, we find that, at least in the particular game studied here,
the introduction of asymmetric information does not offer the promised
solution: one must still make an assumption about downstream firms’
off-the-equilibrium-path beliefs about the behavior of a given type of
upstream firm in selling to his rival. But, arguably, the introduction of
incomplete information does have the more limited benefit of making
what amounts to an assumption of passive conjectures feel somewhat
more natural by allowing downstream firms to update their beliefs if
the unexpected change in output is large enough.2
The second, more concrete, advantage of the introduction of incomplete information is that the situation where upstream firms have
more information about their costs than downstream firms is that
presumably this is frequently the empirically relevant case.3 This is
especially interesting as the majority of empirical studies show that
vertical integration is associated with lower prices (see Lafontaine
and Slade, 1997). This makes it difficult for antitrust policy makers
to give too much weight to the new theory because according to
the new foreclosure doctrine, vertical integration, when it forecloses
unintegrated downstream firms, should be associated with higher prices
1. See, e.g., McAfee and Schwartz, 1994, p. (225), who suggest that “In order to
make serious headway on these contracting problems it will be necessary to introduce
asymmetric information explicitly . . . .”
2. For an alternative approach to resolving the difficulties surrounding the choice of
conjectures in this and more general games, see Segal and Whinston (2003). Their work
is related to ours in that they use menus of contracts to allow a retailer to distinguish ex
post between suppliers according to what offers they have made to his rival. They do not
consider incomplete information per se, and their aim is different: they wish to establish
bounds on possible output levels and to consider how these converge as the number of
retailers becomes very large. They do not consider the issue of vertical integration or its
welfare effects. The problem of conjectures is circumvented by Riordan (1998) by simply
assuming that one (dominant) downstream firm moves first and his moves are observed
by the others. Chen (2001), on the other hand, examines the effects of vertical merger when
there are multiple upstream firms one of which must be (weakly) more efficient than the
others, and the offers made by upstream firms are public information.
3. For other (complete information) theories of foreclosure which focus on upstream
costs of supplying downstream firms, see Choi and Yi (2000) and Church and Gandal
(2000). In these papers, costs are perfectly known, but are endogenous to choices about
input specificity or compatibility made by upstream firms. Ma (1997) considers the case
where downstream firms write contracts with consumers to supply options on upstream
goods as in for example the market for health insurance. This line of literature, starting
with Ordover Saloner, and Salop (1990) is theoretically somewhat distinct from that
considered in this paper in that the anti-competitive effects of vertical merger arise even
when upstream firms can make public offers, but generally require the presence of a
competing upstream firm and a friction which means that an integrated firm will be
unable or reluctant to sell to unintegrated downstream firms.
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(Cooper et al., 2005). Of course, no one ever claimed that foreclosure
was the only reason for vertical integration; many and perhaps most
vertical mergers may be expected to result in efficiency enhancements
which reduce costs and hence prices. For example, vertical integration
might lower transaction costs (Williamson, 1975; 1985) or eliminate
double marginalization (Spengler, 1950) and this may explain why in
statistical studies vertical integration seems to reduce prices.4 Perhaps
more problematic for the emerging literature is the surprising result
that even if one restricts attention to studies of industries where the
government has intervened to outlaw specific vertical restraints—prima
facie a more favorable testing ground for new foreclosure theories—one
finds that the supposedly procompetitive government policy has tended
to raise prices (Lafontaine and Slade, 2005).
On the other hand, the traditional explanations of the efficiency
gains arising from vertical integration mentioned above face a theoretical difficulty, in that it is never entirely clear why these efficiencies cannot
be obtained through contractual means without vertical integration.
For example, offering two-part tariffs would seem to be a much less
costly solution to the problem of double marginalization than vertical
integration.5 In this paper, we demonstrate the existence of what can
be seen as another type of “efficiency enhancement” associated with
vertical integration: the removal of the need to engage in costly signaling.
We show that when there is incomplete information about costs, vertical
integration can easily result in high-cost types selling to downstream
firms at lower prices than they would if vertically separated. Vertically
integrated low-cost types will sell at higher retail prices. Thus it is
possible that, consistently with the empirical literature, prices will fall
“on average” if high-cost types occur sufficiently frequently compared
to low cost types. We provide a numerical example in the Appendix
where this is the case.
In finding that vertical integration can reduce prices and raise
welfare, our paper is related to Baake, Kamecke, and Normann (2004).
They consider a model very similar to that used here, but they allow
the upstream monopolist to publicly choose “capacity” before making
secret contract offers to downstream firms. By publicly committing to
a capacity which is very low, the upstream firm commits to having
high marginal costs, and thus to low output. Thus, the monopolist
underinvests in capacity as a way of solving his commitment problem;
4. For surveys of other reasons why vertical integration and other vertical restraints
might arise, see Perry (1987) and Katz (1987). More recent perspectives on theory and
policy are provided by Hovenkamp (2001), Klass and Salinger (1995), Rey and Tirole
(2005), and Riordan and Salop (1995) (see also the comment by Reiffen and Vita, 1995).
5. See Whinston (2003) for a recent critique of the transactions cost theory.
Foreclosure with Incomplete Information
511
the cost of this is that ex post he uses too little capacity and too much of
the variable input, that is, he does not minimize the cost of producing
his chosen output. This implies that, under some conditions, vertical
integration, which leads to monopoly with efficient production, can
be socially preferable to vertical separation. The broad logic behind
this policy conclusion—that the upstream firm pursues a “puppy dog”
strategy to gain some commitment not to overproduce—is similar to
that in the present paper.6 The specifics, however, are very different.
The Baake, Kamecke, and Normann result relies on the upstream firm
being able to publicly commit to having high marginal costs; whereas on
the contrary, the result in the present paper relies on the upstream firm’s
marginal cost being unknown. Once the upstream firm’s marginal cost
becomes common knowledge, the signaling incentives, which sustain
the commitment to low output, are destroyed, as we now explain.
The reason why introducing incomplete information about upstream costs makes such a difference to the standard model is relatively
simple and intuitive. In the complete information model, each of the
downstream firms constitutes an entirely separate market from the
point of view of the upstream firm. When downstream firms do not
observe each others’ orders, the upstream firm’s profit from selling
to one downstream firm is independent of how much it actually sells
to the other downstream firm(s) (it depends only on how much it is
expected to sell to it in equilibrium). Therefore the upstream firm has no
way of credibly signaling to one downstream how much it is selling to
the other(s), so downstream firms fear opportunistic behavior by the
upstream firm. The only way they can be sure that the quantities sold to
other firms will be as anticipated is if all quantities are best responses to
one another, so that there is no incentive to deviate from expectations.
This implies Cournot output when the strategic variable is quantity.7 By
contrast, when upstream costs are unknown, a low quantity offer may
be interpreted as indicating high costs, and hence low quantity offers
to other downstream firms as well. This creates signaling incentives
for all cost-types to reduce output that do not exist in the complete
information model.8 Under incomplete information, there is a tendency
for output to be distorted downwards under vertical separation, which
can be alleviated by vertical integration.
6. I thank an insightful referee for bringing this to my attention.
7. As remarked above, this argument depends on the assumption of passive conjectures,
and will be explained in Section 2 below. The case of price competition downstream has
been considered by O’Brien and Shaffer (1992), yielding similar results; though in this
case the analysis is considerably more sensitive to the choice of equilibrium concept (see
Rey and Vergé, 2004).
8. Incomplete information has been introduced into the related model of the durable
goods monopolist by Ausubel and Deneckere (1992).
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Of course, the problem with introducing incomplete information
to the model is as usual as that the possibility of signaling leads to a
multiplicity of equilibria, some of which are very inefficient, for example,
equilibria in which all firm types produce less than their monopoly
output. Clearly, some mechanism is required to eliminate equilibria that
are supported by “unreasonable” beliefs. Since this is a model of an
informed principal, we use the device suggested by Maskin and Tirole
(1992) to alleviate this problem. We will focus on the Cho–Kreps leastcost signaling equilibrium. In this equilibrium, the lowest cost types
have output that is undistorted relative to complete information case
(i.e., Cournot output). The high-cost types have output low enough that
the low-cost types do not want to imitate them—so if costs are similar
enough, high-cost types must produce less than their own monopoly
output. In consequence, policies that allow the monopolist to achieve
his monopoly level of output—such as vertical integration, exclusive
dealing, or nondiscrimination clauses9 —improve output, profits, and
welfare for high-cost types, and reduce welfare for low-cost types.
The plan of the paper is as follows. In Section 2, we begin by
setting out the complete information benchmark for those unfamiliar
with the new foreclosure doctrine.10 Section 3 sets out and introduces
our model of foreclosure with incomplete information about upstream
costs. We explain our approach in dealing with the problem of multiple
equilibria that arises. We will use the “market-by-market” assumption
to select off-the-equilibrium path conjectures about the actions of a given
type, and solve the signaling game as an informed principal problem
using the device suggested by Maskin and Tirole (1992). In Section
4, we discuss the properties of the least-cost separating equilibrium
of the signaling game, which is the one selected by this approach. In
Section 5, we close the model by requiring that beliefs be correct in
equilibrium, and offer some thoughts on welfare analysis. Section 6
concludes. In the Appendix, we provide some proofs, discussion of
alternative equilibria, and also a numerical example to show that—
contrary to the conclusion under complete information—welfare can
be improved by vertical merger.
9. Whether nondiscrimination clauses can fully solve the upstream firm’s commitment
problem depends on exactly how these clauses are applied (to a whole contract or to each
separate component of the contract) — see DeGraba and Postlewaite (1992), McAfee and
Schwartz (1994), DeGraba (1996) and Marx and Shaffer (2001). The usefulness of, and
potential difficulties in enforcing, exclusive dealing and resale price maintenance contracts
is investigated in Alexander and Reiffen (1995). For brevity, in this paper we will concern
ourselves with impact of vertical merger, but, subject to the caveats noted in these papers,
the extension of the results to other vertical restraints should be straightforward.
10. Readers familiar with the theory may wish to skip this section.
Foreclosure with Incomplete Information
513
FIGURE 1. THE STRUCTURE OF THE MARKET
2. The Complete Information Benchmark
The original complete information model where lack of output commitment yields incentives for vertical merger is due to Hart and Tirole
(1990). Here we adopt the simpler structure used by Rey and Tirole
(2005) in their recent survey of the literature.
There is one upstream firm, U, which produces a good with
constant marginal cost c. U does not sell his good directly to consumers,
but to downstream firms, who “process” (or package, retail, etc.) the upstream good on a one-to-one basis into an undifferentiated downstream
good, which they then sell to consumers. There are two downstream
firms, denoted D1 and D2 .11 We let the marginal cost of processing be
constant, and without loss of generality set it to zero. Let all the fixed
costs in the upstream and downstream firms be zero. Consumers have
inverse demand curve for the product P(Q), where Q is the total amount
offered for sale by the downstream firms D1 and D2 , whose sales are
denoted q1 and q2 , respectively. Figure 1 illustrates the structure of the
industry.
11. The model is easily generalized to the case of more than two downstream firms.
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The timing of the game is as follows. At t = 1, U makes simultaneous take-it-or-leave-it contract offers to D1 and D2 .12 The contracts
which U offers consist of tariff schedules Ti (qi ) for i = 1, 2, stating
the price which firm i must pay in exchange for receiving quantity
qi . Di ’s contract may refer only to the quantity, which Di will receive,
and is not allowed to depend on the quantity that Dj will receive. This
may be either because this information is unverifiable, or because such
contracts are outlawed by antitrust authorities.13 Neither downstream
firm observes the contract, which his rival is offered. However, each
may form conjectures about what his rival has been offered based on
the offer that he himself has received.
At t = 2, given his beliefs about the likely behavior of his rival, each
downstream firm i accepts or rejects U’s contract offer. If Di accepts the
offer, he chooses a quantity qi from the menu offered by U and pays the
corresponding tariff Ti (qi ). Notice however that because U has perfect
information about the downstream firms, it can actually dictate their
quantity choices, for example, by offering tariffs schedules of the form
“Ti (q) = Ti if q = qi , and ∞ otherwise.” Therefore we will henceforth
suppose that U makes offers of the form (Ti , qi ), where, because U makes
take-it-or-leave-it offers, Ti is set so as to extract all of Di ’s expected profit.
It follows that in making its offer to Di , U aims to set the preferred
quantity to maximize Di ’s expected profit net of its own costs of supply,
which is given by Ti − cqi = max qi P(qi + qje ) − cqi , where the superscript
e denotes firm i’s expectation of firm j’s output.
At t = 3, downstream firms take delivery of their orders, and
transform some or all of the intermediate good received into the final
downstream (consumer) good. They observe each other’s output, set
their prices, and sell to consumers. Thus, this stage of the game is as in
Kreps and Scheinkman (1983).
The perfect Bayesian equilibrium of this game evidently depends
very strongly on the conjectures made by downstream firm Di about
the offer that has been made to its rival Dj (i = j). Suppose first that
both downstream firms have symmetric conjectures, which is to say that
suppose whatever offer Ti (.) Di receives, it believes that Dj will receive
exactly the same offer, Tj (.) = Ti (.), even if Di ’s offer itself was an
unexpected one. Then the fact that Di does not actually observe Dj ’s offer
proves irrelevant: the outcome is the same as if such observation were
possible. This is because Di will refuse to pay more than T = qi × P(2qi )
12. For a consideration of the effect of allowing the downstream firms some bargaining
power, see Chemla (2003).
13. Note that in the complete information model, antitrust authorities would be wise to
prohibit such contracts, as they will allow the upstream firm to restrict output to monopoly
levels.
Foreclosure with Incomplete Information
515
for an offer of quantity qi . Therefore in choosing his offer to Di , U
maximizes qi P(2qi ) − cqi , and so chooses an offer qi = 12 qm , where qm =
argmax qP(q) − cq, is the industry monopoly quantity. Each downstream
firm receives half the monopoly quantity, so it is as if the upstream
monopolist could commit to selling only that amount.
However, there is no very good reason to suppose that downstream
firms do hold symmetric conjectures. If U makes an unexpected (i.e.,
out-of-equilibrium) offer to D1 , why should D1 suppose that U has made
the same deviation with D2 ? The problem is that from U’s point of
view, D1 and D2 constitute entirely separate markets (although of course
D1 and D2 themselves perceive a strong inter-dependency). U’s aim
is to maximize profits with each of D1 and D2 , and given that it has
constant marginal costs, U’s optimization program with D2 is unaffected
by how much he actually sells to D1 . (Observe that it only depends on
D2 ’s expectation of how much D1 receives, not on how much D1 actually
receives.) So if U is expected to maximize profits with D2 , there is no
reason for U to change D2 ’s offer when he changes the offer he makes
to D1 , and therefore no reason for D1 to suppose the contrary.
For this reason, the literature has concentrated on the use of socalled “passive conjectures.”14 D1 and D2 each have prior beliefs about
the quantities each of them will receive, and neither updates these
beliefs during the course of the game, even if they themselves receive
an unexpected offer. If downstream firms hold such beliefs, and U
maximizes its profits with each of them, it should sell q1 = BR(q2e ) =
argmaxq1 P(q1 + qe2 ) − cq1 to D1 and similarly q2 = BR(q1e ) to D2 ,
where BR(.) denotes the Cournot best response function. Imposing the
condition that beliefs must be correct in equilibrium, we find that in
equilibrium, U sells the “Cournot” quantity qc for a firm with marginal
cost c (where qc solves qc = BR(qc )) to each of the downstream firms.
Using the symbol π to denote downstream revenues, U’s total profit is
therefore given by the sum of the two firms’ Cournot profits 2(π c − cqc ),
which is less than the monopoly profit π m − cqm .
Thus, even though U is in a monopoly position and can make
take-it-or-leave-it offers to downstream firms, his inability to commit
himself makes it impossible for him to extract the whole monopoly
profit from the industry. This problem is sometimes called a Coasian
commitment problem, by analogy with Ronald Coase’s seminal analysis
14. McAfee and Schwartz (1994) also investigate the equilibrium of a similar game
under what they call wary beliefs. A firm that holds wary beliefs thinks that its rival received
an offer which is a best response to its own offer. That is, it thinks that the monopolist
is maximizing joint profits with its rival, given its own offer. In their game wary beliefs
and passive beliefs yield the same outcome, though the conditions for the existence of an
equilibrium are more stringent with wary beliefs.
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of the over-selling problem of a durable goods monopolist (Coase, 1972).
By contrast, if U were permitted to integrate vertically (i.e., in this
context, share ex post profits) with one of the downstream firms, say
D1 , U would internalize the loss of profit D1 suffers when U sells to
D2 . Writing an exclusive dealing contract with D1 (in which U agrees to
deal only with D1 ) serves the same purpose.15 Any of these measures
allows U to foreclose the downstream market by denying D2 access to
his product. Thus, foreclosure restores U’s monopoly power and results
in reduced output and consumer welfare. For this reason, it has been
argued that antitrust authorities should take a tough stance against
attempts to foreclose markets.
3. The Incomplete Information Model
We saw in the previous section that the new foreclosure doctrine relies
heavily on the assumption of passive conjectures. If conjectures were
symmetric, vertical integration would have no anticompetitive effects
and would merely reflect efficiency considerations, as originally argued
by the Chicago School (Bork, 1978; Posner, 1979). But the argument for
the use of passive conjectures, although persuasive, may nevertheless
leave one feeling a little uneasy. Suppose one were actually in the
position of D1 receiving an unexpected offer. Wouldn’t one ask oneself:
why didn’t U make the offer that seemed to maximize his profits with
me? In the complete information model, there can be no answer to this
question: U took a probability zero action. But in reality, one might think
that what one thought was U’s best offer turned out not to be U’s best
offer— perhaps his costs were higher than expected, for example. This
suggests that an incomplete information model might be a better tool
to capture the kind of phenomena which the new foreclosure literature
analyzes.16
This section introduces incomplete information into the model.
For simplicity and ease of comparison, the model we use is exactly as
before (Section 2) with one addition. We allow c, U’s marginal cost, to be
15. Note that the direction of the exclusive dealing contract is the opposite to that
usually studied in the literature: U deals exclusively with D1 , though D1 does not
necessarily deal exclusively with U. It has been argued that exclusive territories can play
the same role in geographically differentiated markets (McAfee–Schwartz 1994). For a
model incorporating exclusive dealing between a downstream monopolist and one of
two upstream rivals (i.e., the reverse set-up to that considered here) see O’Brien and
Shaffer (1997).
16. Interestingly, in their experimental study of foreclosure, Martin et al (2001) find
that downstream participants’ beliefs seem to be something of a mixture of passive and
symmetric conjectures. This will be reminiscent of our result that beliefs will adjust if the
cut in output is large enough and will be passive otherwise.
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517
initially unknown. At t = 0 in the timing given above, Nature makes a
move determining U’s type, which is revealed privately to U. There are
two possible types of upstream firms, high cost and low cost, denoted
UH and UL , respectively. We denote their respective utility functions by
UUH and UUL . Downstream firms D1 and D2 (about whom there remains
complete information) have utility functions UD1 and UD2 , respectively.
Nature selects a high-cost upstream firm with probability α, and a lowcost upstream firm with complementary probability 1–α. We denote D1 ’s
e
beliefs about what will be sold to D2 by q2H
if he is facing a high type and
e
q2L for a low type. The actual amounts sold are denoted q2H and q2L ,
respectively. An analogous notation is used for the quantities sold to D1
e
(q1H and q1L ), and for D2 ’s expectations about these quantities (q1H
and
e
q1L ). As in the previous section, downstream profit (gross of payments
to the upstream firm) is denoted by π(.,.).
As in the previous section, we will not model the integration
decision explicitly; we will simply treat the case where downstream
firms are unintegrated and assume that an integrated vertical structure
can achieve the vertical monopoly profit by pricing accordingly (because
in that case there is no commitment problem). But in order for this
approach to be valid, we now need to assume that if firms integrate (or
sign exclusive dealing contracts) then they must do it at a stage t = −1,
before upstream costs are known. Otherwise there might be information
contained in the integration or exclusive dealing offers, which would
complicate our analysis.
Although in this model (unintegrated) D1 does not care directly
about U’s cost, U’s cost affects the offer he can be expected to make
to D2 , so D1 ’s utility is indirectly dependent on U’s type. Therefore U
will wish to signal his type through his offer to D1 . This means that
we have a signaling game with a potential multiplicity of equilibria.
Some of these are very inefficient—for example, they may involve
both types of upstream firm producing more than the best-response
(Cournot) output or less than the high-cost monopoly output (in each
case, firms “pool” for fear of being thought to be low-cost for certain).
But such equilibria are plainly implausible, and can be sustained only
if D1 “threatens” to “punish” deviations from equilibrium behavior
by adopting particularly unfavorable beliefs about deviating firms’
types.
In fact, the problem of multiplicity of equilibria is in some sense
even worse than in standard signaling games, because it is compounded
by the multiplicity of equilibria in the complete information game—
described in the previous section—which results from the relative
freedom in choosing a firm’s beliefs about the upstream firm’s sales to
its rivals when it receives an off-the-equilibrium path offer. Sometimes
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Journal of Economics & Management Strategy
introducing a little uncertainty into a complete information game does
serve to reduce the set of equilibria (e.g., in the supply function literature,
Klemperer and Meyer (1989)) but that is not the case for the particular model studied here. Intuitively, the two forms of multiplicity are
“independent” of one another in that even if one solves the multiplicity
of equilibria in the signaling game by a suitable choice of refinement,
one is still free, given an assignment of type, to assign beliefs about
what that type will offer to the rival downstream firm in the event of an
out-of-equilibrium offer; and conversely, that even if one decides that
“passive beliefs” are appropriate in the complete information model, if
different types act differently, one still has to solve the signaling game
multiplicity problem to assign a type (and hence an offer to the rival) in
the event of an out-of-equilibrium action.
Thus our approach will essentially be to tackle these two multiplicity problems separately. With respect to the multiplicity problem that
arises due to conjectures even in the complete information model, we
will adopt the “market-by-market” approach described in the previous
section, which, when applied to the complete information problem,
implies passive conjectures. We do this partly because we think that,
because upstream profits do not depend on sales to downstream rivals,
it is reasonable to assume that upstream offers do not depend on this
either. But the main point is that if we wish to investigate how the
introduction of incomplete information affects the previous complete
information literature, it makes sense to adopt the same approach as that
literature in modeling the way that off-the-equilibrium-path conjectures
are formed. Note, however, that (as the opening paragraph of this section
suggests), and as we will show in Section 4 below, the adoption of the
market-by-market approach with incomplete information will not imply
beliefs that are entirely passive, but rather beliefs that will be updated
only if the offered output is sufficiently low.
The adoption of the market-by-market approach allows us to
proceed in the following discussion by taking the outcomes of UL and
UH ’s interactions with D2 as given and concentrating entirely on U’s
interaction with D1 . In Section 5, we will close the model by arguing
that a similar interaction occurs between D2 and the upstream firm on
the “other side” of the market, and considering what this implies for the
market as a whole. Before that, in the remainder of this section, we tackle
the problem of the multiplicity that arises from the incompleteness of
information about U’s type.
Perhaps the most elegant way of handling the problem of multiplicity in signaling games is to use the framework suggested by Maskin
and Tirole (1992). Rather than introducing an equilibrium refinement,
they instead argue that traditional signaling models do not exhaust the
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519
contracting opportunities open to the informed principal (i.e., in our
context, the upstream firm), because they assume that the informed
party offers a single contract to the uninformed party. Then, if the
informed party offers an unexpected contract, the uninformed party
must form beliefs about the informed party’s type, and the flexibility
the modeler has in assigning these beliefs can be used to support a large
set of equilibria. Maskin and Tirole show that if, instead, the informed
party is allowed to offer an incentive-compatible menu of contracts, this
multiplicity of equilibria will be much reduced, sometimes (depending
on parameter values) to a unique equilibrium. In the working paper
version of this paper (White, 2003), we show in detail how this method
can be used to restrict the set of equilibria in our game.17 Here we
spare the reader the technical details and merely provide an informal
discussion and intuition for the result.
The way that the Maskin–Tirole mechanism works in the context
of this model is as follows. Because U has two possible types, U offers
each downstream firm D1 a pair of contracts (TL , qL ), (TH , qH ) from which
U will choose after D1 has accepted. If the pair of contracts is incentive
compatible for U, then D1 knows which type will choose which contract
ex post, so there is no room for D1 to inflict punishment by beliefs.
Moreover, if D1 breaks even on each of the possible contracts in the
menu, given the type that will choose that contract, then D1 does not
care which type he faces. Consider maximizing each upstream firm
type’s utility from his own choice from such a menu subject to it being
profitable for D1 type-by-type: the resulting menu can be called the low
information intensity (LII) optimum for that type because (given that the
allocation is such that the downstream firms will break even with each
type of upstream firm separately) they have little incentive to gather
information about the upstream firm’s type. Clearly each upstream type
can guarantee itself at least as much utility as it gets in its LII optimum
(simply by offering its LII menu to D1 ), so this provides a lower bound
to each upstream type’s equilibrium utility. Now combine each type’s
own choice from its LII optimum to form the low information intensity
(LII) allocation, which, by construction, yields the same utility for each
type and is also incentive compatible.
What does the low information intensity allocation look like?
It turns out to be none other than the familiar Rothschild–Stiglitz–
Wilson or least-cost separating equilibrium. The reasoning relies on
the observations that (a) the low-cost (“bad”) type can always do
at least as well under asymmetric as under symmetric information
17. See also Segal and Whinston (2003) for another recent application of the Maskin–
Tirole framework., and Tirole (2006) for a detailed textbook exposition of the method.
520
Journal of Economics & Management Strategy
(b) if a downstream firm were to accept the low-cost type’s optimal
symmetric information contract and yet discover that he was in fact
facing the high-cost (“good”) type, he would not lose money, and
(c) the high-cost (“good”) type can do at least as well when the lowcost type chooses his preferred contract under symmetric information
as the high-cost type does when he offers his LII optimum, because
the low-cost type’s incentive constraint may bind more tightly in the
latter case.
The final step in the argument is to note that if this low information
intensity allocation is interim efficient, then no type can do any better than
this allocation; and, because we have just argued that each type can do
at least as well, the low information intensity allocation, or equivalently,
the least-cost separating equilibrium, is the unique equilibrium of the
game where upstream firms offer menus of contracts. One can show
that, provided that the probability α of the high type occurring is
less than some critical value α ∗ > 0, then the LII is indeed interim
efficient, and the least- cost separating (LCS) allocation is thus the unique
equilibrium. The derivation of α ∗ is given in Appendix A. Numerical
calculations suggest that, though strictly positive, α ∗ can be quite small
(see Appendix C). If instead this condition is not satisfied (i.e., if α >
α ∗ ) then the separating allocation is still a perfect Bayesian equilibrium,
but it is no longer unique. The set of equilibrium payoffs is then the
set of incentive-compatible, profitable-in-expectation allocations that
(weakly) Pareto dominate the low information intensity allocation.18
These “high information intensity” equilibria are discussed in Appendix
A; here in the text, we will focus on the low information intensity
optimum. There are several justifications for this.
18. To see this, note that when there exist allocations which Pareto dominate the
LCS allocation it may be possible for the upstream firm to make a move which will
give him strictly more than his low information intensity optimum. Specifically, when
the probability of encountering the high (“good”) type is large enough, it may be more
profitable to offer a high information intensity contract, where the downstream firm breaks
even not with each upstream type individually, but “on average” across types. Because
this allows the pooling of two participation constraints, it may increase the value of
the optimization program. Any incentive-compatible, profitable-in-expectation allocation
which Pareto dominates the low information intensity allocation can be sustained as an
equilibrium, by simply assuming that if the upstream firm offers the option contract
corresponding to the desired equilibrium then D1 does not update his beliefs from (α,1−α);
in which case D1 ’s best response is to accept the offered contract. Offering the contract is
in turn the best move for the upstream firm if D1 would otherwise “punish him by beliefs”
by putting probability zero on the high-type if he offered any contract Pareto-dominating
the expected contract. (Offering a contract other than the expected one is a probability zero
event, so, given that our equilibrium concept is perfect Bayesian, we can apply any off-theequilibrium-path beliefs here.) Clearly the Pareto-dominated low information intensity
allocation can be sustained as an equilibrium in the same way.
Foreclosure with Incomplete Information
521
Firstly, using further equilibrium refinements, such as Cho and
Kreps’ (1987) “intuitive criterion” in the familiar fashion also yields
the least-cost separating allocation as the unique equilibrium. This is
because in any equilibrium which is not profitable type-by-type, the
high-cost type can find a deviation that, if it leads the downstream firm
to believe that he is high cost, makes the high-cost firm better off but the
low-cost firm worse off. The intuitive criterion suggests that in such a
case, the downstream firm ought indeed to update his beliefs in such a
way that he places zero probability on the type that would not gain from
making this deviation—and this breaks the profitable in expectation
high information intensity equilibria.
Secondly, if the downstream firms can find out (at some cost to
themselves) the upstream firm’s marginal cost, then, as their name
suggests, the high information intensity equilibria may not be stable. The
reason is that downstream firms make a loss with the low-cost types and
a profit with the high-cost types, so they will have a strong incentive to
find out which type of firm has offered them a high information intensity
contract. If they do so, then they will accept such a contract only when
they face a high type, which will upset the equilibrium.
4. Properties of the Least-Cost
Separating Equilibrium
In the least-cost separating (or low information intensity) equilibrium,
the low-cost type receives his symmetric information payoff. The intuition behind this is that, because in a separating equilibrium, his
type will anyway be revealed (as the worst possible), the low-cost firm
might as well maximize profits rather than distort output in a failed
attempt at signaling his type.19 As we saw in Section 2 above, under
symmetric information about costs, the low-cost type simply offers a
contract, which is the best response to what D2 is expected to receive,
and a tariff to extract all the expected profit. Thus, the Coasian situation
of the low-cost firm is unaltered by the introduction of incomplete
information.
By contrast, (if the no-mimicking constraint in the separating
programme is binding) the high-cost type does not maximize his profit
with D1 given the expectation of q2H , so the Coasian situation of the highcost type is altered. Let us assume for the moment that this constraint
does bind. (We will show below under what conditions this will be the
19. For readers thinking in terms of the Cho–Kreps equilibrium selection device, this
property comes from the deletion of dominated strategies. Strategies where the low-cost
type D1 ’s type are revealed and yet he does not produce his best-response output are
dominated by producing his best-response output and having his type revealed.
522
Journal of Economics & Management Strategy
FIGURE 2. EQUILIBRIUM DISTORTION IN UH ’S OUTPUT TO AVOID
IMITATION BY UL
case in equilibrium.) If UH simply aimed to maximize joint profits with
D1 by producing his best-response quantity, the low-cost type would
e
imitate him, and so D1 could not be sure that q2H
was indeed the correct
expectation. So UH has a commitment not to sell “too much” to D1 ,
because if he did then D1 ’s beliefs would change adversely. UH must
distort his output downwards compared to his best response.20 The
distortion is just enough that the low-cost type is indifferent between
his best response to his own expected sales to D2 on the one hand;
and the advantage of having D1 believe that his rival will receive only
e
q2H
combined with the disadvantage of being able to selling only the
high type’s distorted output on the other.21 (See Figure 2.)
20. This must be true in any separating equilibrium, not only in the least-cost separating
equilibrium that we have selected. In this sense our results are robust to the equilibrium
refinement. I thank Paolo Garella for this remark.
21. Again, those familiar with the Cho–Kreps framework will recognise that this
property arises from the test of equilibrium domination. This ensures separation rather
Foreclosure with Incomplete Information
523
Notice that the introduction of incomplete information to the
model has allowed us to partially endogenize the passivity of conjectures. Although one can make an argument in favor of passive
conjectures, the choice between passive and symmetric conjectures in
the complete information model is necessarily somewhat arbitrary. In
the incomplete information model, this defect is partly remedied because each downstream firm will sometimes update his beliefs about
the offer received by his rival, depending on the offer that he himself
receives.22 That is to say, the updating of beliefs is regulated by equilibrium refinements. If the offer which a downstream firm receives is
sufficiently small that a low-cost firm would never want to make such an
offer (even if he were thereby believed to be high cost), the downstream
firm will indeed update his beliefs to suppose that he is facing a highcost firm, and that the quantity his rival receives will also be small. If the
quantity offer is not so small, then the downstream firm’s conjectures
will be genuinely passive: he will not be led to believe that he is facing
a high-cost firm.23
5. Results
To close the model, we need to consider what happens in the contract
game between U and D2 . Clearly if there is separating on one side of
the market, then there is an incentive to signal on the other side of the
market.24 So this game must also be solved for D2 on the other side
e
e
and q1L
. Evidently the
of the market, given arbitrary conjectures q1H
than pooling, as low-cost type would never want to produce less than this even if he were
thereby believed to be a high-cost type; thus the high-cost type can deviate to this output
and make a speech to this effect. Lower outputs than this are dominated for the high-cost
type.
22. Note that we retain the “market-by-market” assumption that U offers contracts
separately and unobservably to D1 and D2 . This assumption was used to justify passive
conjectures in Section 2. In a complete information framework, this argument is reasonable
in that U’s maximization problem with each downstream firm is independent of how large
a quantity he actually sells to the other downstream firm. But in an incomplete information
context, it can no longer serve to justify passive conjectures per se, because the quantities
sold in each market are nevertheless correlated because of their mutual dependence on
upstream costs. Therefore some quantity offers to D1 will lead him to update his beliefs
about the quantity offered to D2 (and vice versa).
23. Note, however, that the remedy is only partial. If downstream firms were to hold
completely symmetric beliefs, then the equilibrium output would be the monopoly output
as in the complete information model. This is because the upstream firm’s type then
becomes irrelevant information from D1 ’s point of view (he already “knows” sales to D2 )
so no signaling occurs. But we would argue (see Section 3 and the previous footnote) that
symmetric beliefs seem less palatable—and “market by market” beliefs less stark—in an
incomplete information context.
24. Thus pooling (i.e., both cost types producing the same quantity) on only one side of
the market is not an equilibrium. Pooling on both sides of the market is not an equilibrium
either because then quantity offers on one side, say D1 , do not signal anything about
524
Journal of Economics & Management Strategy
case will be analogous: the low-cost type will sell D2 his best response
to the expected output of D1 , and the high-cost type will reduce his
output to just enough that the low-cost type does not mimic. Finally,
to complete the equilibrium, we need to check that beliefs are correct
e
e
e
in equilibrium, that is, q1H = q1H
and q2H = q2H
, q1L = q1L
and q2L =
e
SI
e
q2L . Given that we saw above that q1L = q1L = BR(q2L | cL ) (where the
superscript SI denotes symmetric information), and the same will be
true for q2L , this means that q1L and q2L will be best responses to one
another, so the low-cost type will produce his Cournot output qLC on
each side of the market. For the high-cost type, the interesting case arises
when his output is distorted away from mutual best responses (highcost Cournot output qHC ) by signaling considerations. Therefore we
assume
A1: Costs are sufficiently close that the low-cost type’s incentive comC
C
C
patibility constraint (IC : UL →UH ) binds: π ( q H
, qH
) − cL q H
≥ π (q LC , q LC )
C
− cL q L .
A sufficient condition for this to occur is that the high-cost type’s
Cournot output is greater than the low-cost type’s monopoly output.
Then, if the H-type produces best responses to his own output, the
L-type can certainly increase profit by mimicking him, because by doing
so, the L-type moves closer to his own monopoly output. When A1
holds, the high-cost type’s output qSE (SE for separating equilibrium) is
distorted below his best response to his own output on each side, and
is defined by
π(q SE , q SE ) − c L q SE = π q LC , q LC − c L q LC .
Under A1, it is easy to show that the high type produces less under
incomplete than under complete information:
Proposition 1: In equilibrium, the low-cost type produces his type’s
Cournot output, and the high-cost type produces less than his own Cournot
output.
This proposition follows easily from A1, as the high-cost firm’s
output must be greater than his Cournot output in order to successfully
separate from the low-cost firm. Notice however that the signaling that
occurs in this model has a rather strange feature. It is costly on each
side of the market individually, because output is distorted away from
the best response. But overall, the need to signal is actually beneficial
quantity offers on the other side D2 . So firms do best by producing their best responses to
D2 ’s output, which differ according to cost.
Foreclosure with Incomplete Information
525
to the high-cost type because it allows him to commit to low output.
In particular, under the assumption of a quadratic profit function (i.e.,
linear demand), we can prove that:
Proposition 2: The low-cost type is just as well off, and the high-cost
type is strictly better off, under incomplete information compared to complete
information about costs.
Proof. See Appendix
This result implies that the upstream firm will have little ex
ante incentive to make his costs transparent to downstream firms. He
anticipates greater profits in a situation where his costs are private
information. This theoretically validates our investigation of incomplete
information about upstream costs.
The result demonstrated in proposition 2—that the high-cost firm
will be better off under incomplete information—would be obvious if 2
× qSE could be shown to be always above the high-cost type’s monopoly
output, qM
H . However, this need not necessarily be the case (see the proof
of the proposition above and the numerical example in the appendix).
It is clear that the low-type will always want to imitate the high-type
if the latter’s total production is more than qM
L . If the costs of the two
types are very close, then their monopoly outputs will also be close, so
that the low-type would still want to imitate the high-type if the latter
were believed to be selling 12 qM
H to Di . For the incentive compatibility
constraint to hold in this case, q1H + q2H will have to fall below the
monopoly output qM
H . Thus we have:
Proposition 3: If costs are sufficiently close, high-cost output in the
signaling game will be less than high-cost monopoly output.
By contrast, if the high-cost U is allowed to take measures to
‘foreclose’ D2 from the market, his production will be qM
H . This is because
the need to signal costs (and so, indirectly, sales to rivals) disappears
when downstream firms no longer fear expropriation.25 For example, if
D1 is vertically integrated with U, D1 knows that U will internalize the
effect on D1 ’s profits if he sells to D2 . Similarly, if U signs a contract to
deal exclusively with D1 , D1 does not care about the level of U’s costs
because he knows that U will not sell to D2 . Thus we have the following
corollary:
Corollary: When costs are sufficiently close, output, profits, and welfare
will improve when the high-cost type vertically integrates, or takes other
25. As noted above, we are assuming that the decision to vertically integrate or sign an
exclusive dealing contract must be taken before costs are known so that the negotiations
over integration or exclusive dealing do not themselves reveal information.
526
Journal of Economics & Management Strategy
measures (such as exclusive dealing), which would ordinarily foreclose the
market.
This result is important for three reasons. Firstly, as noted in the
introduction, it brings the foreclosure literature closer into line with the
available empirical evidence that downstream prices tend to be higher
on average with vertical separation. Our result shows that this fact is
not necessarily inconsistent with the existence of Coasian commitment
problems if there is incomplete information.
Secondly, it is interesting to note as an aside that the model gives no
support to the intuitive idea that the need to avoid opportunism could
give rise to contractual rigidity. McAfee and Schwartz (1994) suggest
that the lack of variation among franchise contracts might be attributable
to upstream firms’ need to signal to nonintegrated downstream firms
that they are not acting opportunistically. A complete investigation
of this idea is outside the scope of this paper. But consider a simple
two-period version of our signaling model, where U’s cost is either
H or L, independently drawn in each period. The repeated play of the
equilibrium we have highlighted for the one-shot game is an equilibrium
of the two-period repeated game.26 But in this game, it is easy to see
that the need to signal that there is no opportunism tends to result in
greater rather than less variation in contractual terms across periods.
If the upstream firm could contractually commit to D1 not to sell
opportunistically to D2 , then D1 ’s one-period contract would be either
qLM or qH
M according to U’s cost realization in that period. This compares
to sales of either 2qLC (more than qLM ) or 2qSE (which could easily be less
than qH
M ) when no such commitment is possible. Thus the need to signal
no opportunism induces a greater variation in contracts as conditions
vary.
Thirdly, and most importantly, let us return to the policy implications of the result. It would be desirable for antitrust authorities
to allow (indeed, encourage) measures, such as vertical integration
and exclusive dealing by high-cost types, at least in situations where
the separating output is below the monopoly output. By contrast,
foreclosure by the low-cost type should be prevented for the same
reasons as under complete information. This is the ideal policy, but
the informational requirements for implementing it may be formidable.
However, it may be that antitrust authorities are often in a better
position to demand cost audits of upstream firms than are downstream
firms.
26. I believe that this is the only perfect Bayesian equilibrium given the finite horizon,
the market-by-market assumption, the use of contract menus à la Maskin–Tirole in each
period with α < α ∗ , and appropriate restrictions on the ability to “punish by beliefs” across
periods.
527
Foreclosure with Incomplete Information
On the other hand, in situations where authorities are in no better
position than downstream firms to determine whether upstream costs
are high or low, it would be desirable to determine what happens
to expected welfare under vertical separation and integration. Clearly,
welfare will increase with integration if and only if the probability of
the high type is large enough. In particular, the probability of the high
type must be larger than the critical α ∗W defined by:
q HM
M
qL
∗
∗
αW
[ p(q ) − c H ] dq = − 1 − αW
[ p(q ) − c L ] dq
q SE
qC
Unfortunately, it is difficult to get general results about the welfare
changes from the move from vertical separation to vertical integration
without making specific assumptions about the demand function. The
problems are akin to those encountered in the literature on third degree
price discrimination: output shrinks in one market, and expands in
another, and the welfare effects depend on the form of the demand
function. In our case, the two “markets” occur probabilistically: with
some probability we have a low-cost firm and output shrinks, but with
complementary probability we have a high-cost firm and output may
rise. Rather than enter into these difficulties, we content ourselves with
providing a simple numerical example in the appendix to show that
welfare can increase with foreclosure, though in general the result is
ambiguous.
6. Conclusions
The last decade has seen a trend towards a tougher policy stance
on vertical restraints by antitrust authorities.27 This trend has been
associated with a more critical approach to the old Chicago School
arguments in the academic literature, and the so-called new foreclosure
doctrine forms a part of this. The papers expounding this doctrine show
that an upstream monopolist will generally find it difficult to commit to
supplying only the monopoly quantity to downstream firms, but that
vertical integration or exclusive dealing helps him to do so. Therefore
vertical integration or exclusive dealing are profitable but reduce output
and social welfare. In this paper, we have shown that this conclusion
is not robust to the introduction of incomplete information about the
monopolist’s cost. When downstream firms do not know the upstream
firm’s cost, a high-cost upstream firm has a strong incentive to signal to
them that his cost is high (and consequently that sales to their rivals will
27. See, for example, Klass and Salinger (1995) and more recently, the survey by
Riordan (2005).
528
Journal of Economics & Management Strategy
be low). To credibly signal his type, the high-cost firm may have to cut his
output below his desired monopoly output. Thus the signaling effect on
the unintegrated high-cost firm’s output can easily offset his Coasian
tendencies to oversupply. When that happens, allowing the high-cost
firm to vertically integrate or sign an exclusive dealing contract increases
output, profits and welfare, because the need to signal disappears along
with the fear of opportunism. By focusing exclusively on complete
information models, the reduction in output associated with signaling
under vertical separation have been overlooked, leading to one-sided
policy conclusions. We argue for a more balanced attitude on the part
of competition authorities, because foreclosure will be damaging for
low-cost firms but may be Pareto improving for high-cost firms.
Appendix A: The High Information
Intensity Equilibrium
One reason why foreclosure might be generally bad for welfare is that
before the monopoly output is reached, the distortion away from his
best response in the LII equilibrium becomes so costly that UH prefers
to offer a contract that is profitable only in expectation, and not typeby-type. Such a contract is known as a high information intensity
contract because of the fact that downstream firms make losses on some
cost-types means that their payoff is sensitive information about the
upstream firm’s cost type. The reasons why such an equilibrium might
not survive equilibrium refinements or investigation of upstream costs
by downstream firms are discussed in the text. Here we nevertheless
investigate such equilibria, since as we will see, they allow the highcost firm to expand his output relative to the low information intensity
equilibrium and so (if they occur) lead to the conclusion that foreclosure
will be more harmful than in LII equilibria.
To characterize the high information intensity equilibria that
might arise when α > α ∗ , we proceed in the following way. First
we find the L(R) function, which is the minimum loss, which D1
must bear if the low-cost firm is to get a rent R above what he
receives in the separating equilibrium. Setting up and solving this
program,
e
−L(R) = max(over T1L , q 1L ) : UD1 = max − T1L + π q 1L , q 2L
e
s.t. T1L − c L q 1L ≥ R + π1L q 1SI , q 2L
− c L q 1SI
we find, not very surprisingly, that the most efficient way to give the
low-cost firm a rent R is not to distort his quantity choice away from
e
BR(q2L
| cL ), but simply to increase the tariff TL by this amount R. This
means that L(R) = R.
Foreclosure with Incomplete Information
529
Now the high-cost type considers maximizing his payoff when he
has more slack on UL ’s IC and less on D1 ’s IR than in the LII program.
Program HII:
Max(over T1H , q 1H , R): UU H = T1H − c H q 1H
e
− T1H − (1 − α)L(R) ≥ 0
s.t. α π q 1H , q 2H
e
& UL (T1H , q 1H ) = T1H − c L q 1H ≤ R + π1L q 1SI , q 2L
− c L q 1SI
Setting up the Lagrangean with multipliers λ and µ on the first
and second constraints respectively, the expression for q1H is then given
by
Max (over T1H , q 1H , R):
e
L = T1H − c H q 1H + λ α π (q 1H , q 2H
− T1H ) − (1 − α)R
e
+ µ T1H − c L q 1H − R − π1L q 1SI , q 2L
+ c L q 1SI + ρ R,
∂ L/∂ T1H = 1 − λα + µ = 0,
(A1)
∂ L/∂ R1H = −λ(1 − α) − µ + ρ = 0,
(A2)
e
∂ L/∂q 1H = −c H + λαπ q 1H , q 2H
− µc L = 0,
(A3)
e
∂ L/∂λ = α π q 1H , q 2H
− T1H − (1 − α)R = 0,
(A4)
e
∂ L/∂µ = T1H − c L q 1H − R − π1L q 1SI , q 2L
+ c L q 1SI = 0,
(A5)
∂ L/∂ρ = R = 0
(A6)
or ρ = 0.
Equations (A1) and (A2) imply that if pooling is profitable (i.e., R >
0, ρ = 0) − λ(1 − α) = −(1 − λα) →λ = 1 and µ = −(1 − α). Substituting
into (A3)
e
∂ L/∂q 1H = −c H + απ q 1H , q 2H
(A7)
+ (1 − α)c L = 0,
which yields the optimal “pooling” output as the implicit solution
to
e
α π (q 1H , q 2H
− c H − (1 − α)(c H − c L ) = 0.
(A8)
e
Thus for α < 1, output is strictly below the best response to q2H
,
with the size of the distortion decreasing in α.
530
Journal of Economics & Management Strategy
One can find the critical α at which pooling is just profitable by
making the following line of reasoning. At the separating output qSE ,
imitation by the low-cost type may have forced the high-cost type to
produce qSE such that π (qSE , qSE ) − cH > 0. In this case, output could
profitably be expanded if it were not for imitation by the low-cost type.
But imitation by the low-cost type can be avoided by giving the low-cost
type more rent R on his own contract. So consider the following strategy
for the high type: increase output by ε units, and increase the tariff
T1H by εcH units, so that the change is just profitable. The downstream
firm D1 can now expect to make a profit of approximately ε(π (qSE ,
qSE ) − cH ) units when he faces the high-cost type, which happens with
probability α. This means that he will be willing to accept a loss of
α/(1 − α) × ε (π (qSE , qSE ) − cH ) on the low-cost type, which equals
the extra rent that can be given to the low-cost type to encourage him
not to imitate. This extra rent will be just sufficient if it covers the
rent that the low-cost type could earn by copying the high-cost type’s
increase in output ε(cH − cL ). Thus the critical α ∗ at which pooling occurs
solves
α ∗ = Min α such that α/(1 − α) × (π (q SE , q SE ) − c H ) ≥ (c H − c L ),
That is, α ∗ = (c H − c L )/(π (q SE , q SE ) − c L ).
If the two types have strictly different marginal costs, and assumption A1 holds (so the high type’s LII allocation is distorted)
then α ∗ > 0. This establishes Lemma 3 in the text. Another way
to reach the same conclusion is to use the implicit expression for
pooling output equation (A8) above. When the constraint R ≥ 0 just
ceases to bind, the optimal pooling and separating outputs coincide.
Substituting π (qSE , qSE ) as the derivative of downstream revenue in
equation (A8) therefore gives the same expression for the critical pooling
α∗.
Appendix B: Proof of Proposition 4
Assume that the total revenue function TR(Q) (sum of revenues from
sales to D1 and D2 ) is single-peaked and symmetric. The argument
will proceed by using symmetry to find the derivative of TR(.) at the
separating output, and from there to showing that high-cost profits must
C
be greater at qSE than at qH
.
Step 1: The derivative of TR(.) at qLC .
dR1
dR2
d
(TR(q 1 , q 2 )) =
+
dq 1
dq 1
dq 1
531
Foreclosure with Incomplete Information
At
q LC ,
dp
d R1
= p + q1
= c,
dq 1
dq 1
d R2
dp
q2.
= q2 ·
=
dq 1
dq 1
q1
d R1
−p .
dq 1
Now, in equilibrium, q1 = q2 = q. So consider the derivative of profit
when changing both outputs at the same time:
d 1
∂ 1
∂ 1
[ /2T R(2q ) − cq ] =
[ /2T R(2q )] dq 1 +
[ /2T R(2q )] dq 2 − c
dq
∂q 1
∂q
2
d R1
= 2×1/2 2
− p − c at q LC
dq
= 2c − p − c
= c − p.
Step 2: The derivative of TR(.) at qSE .
If the profit function is symmetric, then because the level of profit at
qSE is the same, the derivative must be of equal magnitude and opposite
sign:
d 1
[ /2T R(2q ) − cq ]|q =q SE = −c L + p q LC .
dq
Step 3: The derivative of high-cost profits.
Now consider the high-cost firm. Its profits when q1 = q2 = q are
− cH q. So the derivative of high-cost profits at any q are given
1
TR(2q)
2
by
d 1
d 1
/2TR(2q ) − c H =
/2TR(2q ) − c L + (c L − c H )
dq
dq
= derivative of UL ’s profits + (c L − c H )
Thus the derivative of high-cost profits at qSE are − cL + p(qCL ) +
(cL − cH ) = p(qLC ) − cH . By A1, qSE is to the left of qHC . If the profit function
is downward-sloping at qSE , (this is the case if p(qLC ) − cH < 0), we are
still to the left of the maximum, so qSE must yield more profit than qCH .
So suppose instead we are on an upward sloping part (p(qLC ) − cH > 0).
Because profit functions are symmetric, this will represent higher profits
as long as the function is flatter at qSE than at qHC (because this implies that
we are closer to the optimum). This is true if the sum of the gradients
is negative. By reasoning analogous to that above, the gradient at qHC
must be cH −p(qHC ). The sum is thus p(qLC ) − cH + cH − p(qHC ), which
is indeed always negative, because the price is higher with high-cost
Cournot output than low-cost Cournot output.
532
Journal of Economics & Management Strategy
Appendix C: A Simple Numerical Example
of Welfare-Enhancing Vertical Integration
In this section we provide a simple numerical example to show that
the output of the high-cost firm can easily fall below his own monopoly
output and that average welfare can therefore increase with foreclosure.
Suppose we have:
Linear Demand : p = a − b(q 1 + q 2 ),
with a = b = 1
Constant marginal costs c L = 0.1, c H = 0.2.
We saw above that the low type always produces his Cournot
output.28 Here, Cournot output for the low-type is q1L = 0.3 and the
low-type’s profit is 0.09 (=T1L − cL q1L ) from D1 and similarly from D2 .
This is the low-type’s symmetric information payoff. Thus the
least-cost separating program for the high-cost type simplifies to
Max(over T1H , q 1H ): T1H − c H q 1H
(I C : UL → UH ) UU L (T1L , q 1L ) = 0.09 ≥ T1H − c L q 1H = UU L (T1H , q 1H )
e
(IR: D1 ; H)
≥ 0.
UD1 (T1H , q 1H ) = −T1H + π q 1H , q 2H
Supposing that both constraints bind, we have T1H = π (q1H , qe2H ), and
0.09 = π(q1H , qe2H ) − cL q1H = (a − b(q1H + q2H ))q1H − cL q1H . Letting
qe2H = 0.15 gives q1H also equal to 0.15, so this is clearly an equilibrium,
because the high-type is maximizing in both markets subject to not being
imitated by the low-type.29
So, q1H = q2H = 0.15 is the high-type’s separating (LII) allocation.
Notice that the monopoly output for the high-cost type is 0.4 > 2 ×
0.15, so the high-cost type is producing strictly less than his monopoly
output in order to separate himself from the low-cost type. Clearly then,
output and welfare improve when the high-cost type is allowed to take
measures (e.g., vertical integration, exclusive dealing, and nondiscrimination clauses) which would allow him to achieve monopoly profits
and output.
28. Actually this continues to be true in the high information intensity equilibrium, as
we show in Appendix A.
29. Could there be an HII asymmetric equilibrium? The answer is no, because (the
symmetry of the downstream markets means that) in that instance there would always be
some slack on one of the constraints, so the high-cost type could do better by changing his
output. For example, if qe2H were lower, say 0.14, q1H would have to fall to 0.147 in order
to be incentive compatible for the low-type with D1 . Then considering the contract with
D2 , we would have:
π (0.147, 0.14) − cL q2H = (1 − (0.147 + 0.14))(0.14) − (0.1)(0.14) = 0.08582 < 0.09
Thus the incentive constraint in D2 ’s market is slack. So U should increase output in
this market. Anticipating this, D1 should not expect qe2H = 0.14.
Foreclosure with Incomplete Information
533
Of course, if the low-cost type is allowed to take such measures, welfare will fall. In case the antitrust authorities are unable to
determine whether the upstream firm has output below monopoly
levels due to signaling (i.e., high-cost) or above monopoly levels due
to lack of commitment (i.e., low-cost), it may be interesting to look
at what happens to expected welfare under vertical separation and
integration:
Under Vertical Separation we have
L-type LII profit = 0.09 × 2 = 0.18; L-type consumer surplus = 0.18
Consumer + Producer surplus with low-cost type: 0.36
H-type LII profit = 0.075 × 2 = 0.15; H-type consumer surplus = 0.045
Consumer + Producer surplus with high-cost type: 0.195.
Whereas if we had the monopoly output (e.g. through vertical
integration),
L-type monopoly profit = 0.2025; L-type consumer surplus = 0.10125
Consumer + Producer surplus with low-cost monopoly: 0.30375
H-type monopoly profit = 0.16; H-type consumer surplus = 0.08
Consumer + Producer surplus with low-cost monopoly: 0.24.
It is straightforward to show that for any α ≥ 5/9, expected
welfare will be higher under monopoly than under separation if the
LCS equilibrium is played. In particular, if α = 0.56, we have
Weighted average welfare under monopoly = 0.56(0.24) +
0.44(0.30375) = 0.26805
Weighted average welfare in LCS equilibrium = 0.56(0.195)+
0.44(0.36) = 0.2676.
Note however, that in this example, the critical α ∗ below which the
LCS is the unique equilibrium is only 2/11, so that the LCS is not the
only possible equilibrium outcome in the absence of integration.30
30. Calculations for the high information intensity optimum for the high-cost type (see
Appendix A) when α = 5/9 yield an output of 6/25 for the high-cost type (with the lowcost type continuing to produce its symmetric information output), resulting in aggregate
welfare of approximately 0.41. Interestingly, in this example, the high-cost type makes
higher profits in the HII allocation than in the LCS allocation, although there is no reason
to suppose that this is generally the case, because the HII is applied “market-by-market.”
It allows the high-cost type to profitably expand output with, say D1 , taking the profits
with D2 as given; but of course in equilibrium D2 will be willing to pay less for a given
output if he anticipates that U will offer the HII rather than the LCS allocation to D1 .
534
Journal of Economics & Management Strategy
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