Discounts as a Barrier to Entry
By Enrique Ide, Juan-Pablo Montero, and Nicolás Figueroa⇤
To what extent can an incumbent manufacturer use discount contracts to foreclose efficient entry? We show that o↵-list-price rebates that do not commit buyers to unconditional transfers —like
the rebates in EU Commission v. Michelin II, for instance— cannot be anticompetitive. This is true even in the presence of cost
uncertainty, scale economies, or intense downstream competition,
all three market settings where exclusion has been shown to emerge
with exclusive dealing contracts. The di↵erence stems from the fact
that, unlike exclusive dealing provisions, rebates do not contractually commit retailers to exclusivity when signing the contract.
JEL: L42, K21, L12, D86
Following many real-world examples, suppose we observe a dominant manufacturer o↵ering the following rebate contract to a retail buyer who is considering
buying a few units from an alternative small supplier: “As long as you buy exclusively from me, you get 10 percent o↵ the list price on all units you purchase;
otherwise, you pay the full list price for as many units as you want.” How can
the antitrust authority be sure that such a contract is not o↵ered to monopolize
the market? Despite its similarity to an exclusive dealing arrangement, which has
been shown to e↵ectively foreclose the entry of a more efficient rival in a variety
of market settings, one main objective of this paper is to explain why in those
same settings the rebate contract above cannot be anticompetitive.
One of the most controversial issues in antitrust and competition policy is
indeed the potential exclusionary e↵ects of exclusive dealing arrangements, discount contracts (e.g., rebates), and related vertical practices. This controversy
dates back at least to United States v. United Shoe Machinery (1922) and Standard Fashion v. Magrane Houston (1922) and has remained very much alive
since then, as illustrated by recent rulings regarding rebates on both sides of the
Atlantic; for example, EU Commission v. Michelin II (2003), EU Commission
⇤ Ide: Graduate School of Business, Stanford University, 655 Knight Way, Stanford, CA 94305
(email: [email protected]). Montero: Department of Economics, Pontificia Universidad Católica de
Chile, Vicuña Mackenna 4860, Santiago, Chile (email: [email protected]). Figueroa: Department of
Economics, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Santiago, Chile (email:
[email protected]). We thank two anonymous referees, Larry Samuelson (the Editor), Matı́as Covarrubias, Liliane Giardino-Karlinger, Laurent Linnemer, Cristóbal Otero, Sebastián Otero, Joaquı́n Poblete,
Patrick Rey, Dick Schmalensee, Ilya Segal, Andy Skrzypacz, Ali Yurukoglu, and audiences at CRESSE
2013, CREST-Paris, IIOC 2014, LAMES 2013, MIT, Paris School of Economics, PUC-Rio, TOI-Chile
2013, the 2013 IO Workshop at the University of Salento, and the Universities of Miami and Torcuato
di Tella for very good comments and discussions. The three authors thank the ICSI Institute (Milenio
P05-004F and Basal FBO-16) for financial support and Montero also thanks the Tinker Foundation
for support during his visit to Stanford CLAS and Economics. The authors declare that they have no
relevant or material financial interests that relate to the research described in this paper.
1
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v. British Airways (2003), AMD v. Intel (2005), Allied Orthopedic v. Tyco
Healthcare Group LP (2010), and ZF Meritor v. Eaton (2012).
What makes these rulings controversial is that these practices can arise without
an exclusionary motive and, more importantly, be efficient. Exclusivity, either de
jure through explicit provisions or de facto through rebate schemes, may foster
relationship-specific investments between manufacturers and retailers by solving
hold-up and free-riding problems (Marvel 1982; Spiegel 1994; Segal and Whinston
2000a). Rebates may also be used in a bilateral monopoly setting to avoid double
marginalization when demand is known to both sides, and as a screening device
when demand is known only to downstream retailers (Kolay, Sha↵er, and Ordover
2004), or simply to stimulate retailers’ sale e↵orts (Conlon and Mortimer 2014).1
According to the so-called Chicago critique (Posner 1976; Bork 1978), efficiency
gains are all that matter when evaluating these contracts, because a downstream
retailer would never sign an exclusive that reduces competition unless fully compensated for doing so, which the incumbent manufacturer cannot a↵ord if the
entrant is more efficient. We know now, however, that the Chicago critique fails
to hold in a variety of settings; namely, when the entrant’s cost is unknown to
both the incumbent and the buyer (Aghion and Bolton 1987; Spier and Whinston
1995; Choné and Linnemer 2015), when scale economies require the entrant to
serve more than one buyer (Rasmusen, Ramseyer, and Wiley 1991; Segal and
Whinston 2000b; Spector 2011), and when buyers are not local monopolies (e.g.,
retailers that sell in completely separate markets) but rather downstream competitors (Simpson and Wickelgren 2007; Abito and Wright 2009; Asker and Bar-Isaac
2014).
Although these post-Chicago models have focused on the exclusionary potential of exclusive dealing contracts, there appears to be a growing consensus that
discounts conditional on exclusivity, like the one in our opening example, may
also be used for similar exclusionary purposes (e.g., Rey et al. 2005; Beard, Ford,
and Kaserman 2007; Motta 2009; NY Attorney General in State of New York v.
Intel 2009). This is particularly important since the anticompetitive potential of
exclusives is greatly diminished if, as the legal practice in common law countries
seems to suggest, they cannot rely on penalties above expected damages (Masten and Snyder 1989; Simpson and Wicklegren 2007). This alleged exclusionary
equivalence, also witnessed in court rulings, is supported by claims that can be
organized around the same post-Chicago ideas:
Claim 1 - Rebates as entry fees
An incumbent may use rebates to impose a penalty on new entrants, analogous to a liquidated
damages clause in the rent-shifting model of Aghion and Bolton (1987). The buyer will buy from
the rival supplier only if the latter o↵ers a price lower than that charged by the incumbent minus
the rebate. Thus, the rebate plays the role of an entry fee designed to extract the efficiency gains
of new entrants, which in the presence of imperfect information, leads to some exclusion.
1 Exclusives can also be the result of fierce competition between two or more incumbent suppliers that
need to screen consumers (Calzolari and Denicolo 2013).
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DISCOUNTS AS A BARRIER TO ENTRY
3
Claim 2 - Discriminatory rebates and demand foreclosure
In the presence of multiple buyers and scale economies, the incumbent may use rebates to lock-in
a subset of buyers to prevent the rival from reaching the minimum viable scale of operation,
forcing all remaining buyers to buy from him at the monopoly price. This strategy is profitable
as the cost dispensed on rebates is more than o↵set by the extra revenues from monopolizing
the remaining buyers, and is analogous to the divide-and-conquer strategy implemented with
exclusives in the naked-exclusion models of Rasmusen, Ramseyer, and Wiley (1991) and Segal
and Whinston (2000b).
Claim 3 - Upstream exclusion and downstream competition
In the presence of intense downstream competition, as in the models of Simpson and Wickelgren
(2007) and Asker and Bar-Isaac (2014), a dominant manufacturer may o↵er rebates to incentivize
retailers to not deal with a more efficient entrant, as increased upstream competition will permeate to the downstream market, dissipating industry profits in the form of lower prices to final
consumers.
It seems from these claims that rebates and exclusive dealing contracts may lead
to the same anticompetitive outcome, the only di↵erence being how the exclusivity
is implemented. Once signed, exclusive contracts implement the exclusivity by
requiring buyers to pay a penalty in case they also buy from an alternative source.
Rebates, by contrast, the argument goes, achieve the same result by forcing buyers
to forgo a discount in case they do not conform to the exclusivity. All-unit rebates
appear particularly well suited for this. The reason is that the incumbent only
needs to o↵er a small per-unit discount to have a huge impact on a retailer’s profit
when the entrant is rather small and/or the incumbent’s product is a must-stock
item. Indeed, using the language of the EU Commission (2009, parr 39), the
incumbent can use the “non-contestable” portion of the buyer’s demand (that
is to say, the number of units that would be purchased by the buyer from the
incumbent in any event) as leverage to decrease the price to be paid for the
“contestable” portion (that is to say, the number of units for which the buyer is
willing to find substitutes). Therefore, when the buyer decides to purchase the
contestable units elsewhere, she forgoes all the discounts, most importantly those
applied to the non-contestable units. This can be substantial, especially when
the contestable demand is small, as contentiously argued in some recent cases,
notably AMD v. Intel.
The contribution of this paper is to show, however, a much more fundamental
di↵erence between exclusives and rebates subtly hidden in the description above:
when the exclusivity is committed. Exclusives commit buyers to the exclusivity
ex-ante —at the time contract is signed and before the entrant shows up— by
forcing the retailer to pay a penalty in case of breach. Rebates, by contrast,
induce the exclusivity ex-post by rewarding buyers once purchasing decisions are
made. This di↵erence is so fundamental that rebate contracts like the one in
our opening example —including a list price and an o↵-list-price discount upon
compliance with the exclusivity— cannot be anticompetitive in any of the postChicago settings of claims 1, 2 and 3 where exclusion has been shown to emerge
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under exclusives.2 This non-exclusionary result is important because many rebate
contracts we observe in practice,3 and in particular those examined in recent
antitrust cases (e.g., EU Commission v. Michelin II, EU Commission v. British
Airways), share these characteristics.
The general nature of our result suggests the existence of a fundamental underlying principle. Because exclusion of an otherwise efficient rival is in itself
inefficient, if all relevant parties were to participate simultaneously in the bargaining process, and sufficiently complete contracts (e.g., nonlinear prices) are
available for the parties to sign, exclusion could not arise. One may call this a
generalized Chicago critique. The key to all post-Chicago models, which explains
why exclusion may emerge, is that some market participant, usually the entrant,
is momentarily absent from the bargaining table. This opens up an opportunity
for the incumbent to use his first-mover advantage to extract additional rents from
a third party, the entrant in claim 1, some retail buyers in 2, and final consumers
in claim 3, which sometimes requires the exclusion of a more efficient entrant.
However, as all market participants eventually take part in the bargaining process, the incumbent can sustain such an inefficient outcome only if buyers are
e↵ectively locked-in ex-ante, that is, before the entrant shows up. This explains
why exclusion is always possible with exclusives, provided that penalties can be
made arbitrarily large, but not with o↵-list-price rebates. The latter’s lack of
ex-ante commitment translates in that retailers make all their decisions, most importantly whether to become the incumbent’s exclusive distributor or not, only
after observing o↵ers from all parties. This ex-post flexibility on the buyer side
forces the incumbent to o↵er larger rewards to keep buyers aligned with the inefficient outcome ex-post, which makes any anticompetitive scheme unprofitable
to begin with.4
There is a way, however, in which the incumbent can restore the anticompetitive potential of rebates: to commit retailers ex-ante to make an unconditional
transfer in exchange for a generous rebate ex-post. Whether the transfer is made
up-front or later is not important, as long as retailers commit to it regardless of
what they do ex-post, even if they do not buy from the incumbent at all. Unconditional transfers act as an ex-ante commitment device because they restrict
the buyer’s flexibility ex-post by forcing her to take an action before observing all
o↵ers. Interestingly, and despite being the only way to restore the anticompetitive potential of rebates in these post-Chicago settings, such transfers have never
2 Following Asker and Bar-Isaac (2014), there may be cases in which the incumbent can use lumpsum rebates, as opposed to o↵-list-price rebates, to exclude a more efficient rival. This distinction is only
relevant when retailers are intense downstream competitors, as we discuss in Sections I.B and IV.
3 For example, rebates used by the chocolate and candy manufacturer Mars with retail vending operators, as documented by Conlon and Mortimer (2014).
4 A similar intuition explains (i) why exclusives lose their exclusionary grip if liquidated damages must
satisfy efficient breach (Masten and Snyder 1989, and Simpson and Wickelgren 2007), and (ii) the need
for an ex-ante commitment in the tying model of Whinston (1990). We go back to both intuitions and
their connection to the generalized Chicago critique in Section V.
VOL. VOL NO. ISSUE
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5
been mentioned nor documented in any of the antitrust cases listed above.5
While unconditional transfers must invariably appear in any o↵-list-price rebate
contract signed for anticompetitive reasons, they are not strictly necessary when
o↵ered for efficiency reasons; for example, to deter inefficient entry. The intuition
is that in the latter case, rebates actually increase social surplus so the incumbent
does not need to sustain an inefficient outcome ex-post. The need for an ex-ante
commitment is also absent in some exclusionary bundling models that analyze
bundled discounts for otherwise completely unrelated products (e.g., Nalebu↵
2004, 2005; Greenlee, Reitman, and Sibley 2008). The mechanism of action
behind these models however, is more closely related to a price discrimination
argument than to a post-Chicago one. We explain this, as well as inefficient
entry, in Section V.
The paper’s key message is that post-Chicago models not only rely on the incumbent having a first-mover advantage, but also on the incumbent using contractual arrangements that extract and distribute surplus before the entrant shows
up. Since rebates are by construction exercised ex-post (i.e., after buyers hear
from the entrant), the incumbent cannot use them to exploit his first-mover advantage, and as a result, they cannot be used to foreclose efficient entry unless
they involve unconditional transfers.
The rest of the paper is organized as follows. In Section I, we present the basic
ingredients of our model that form the basis of the rent-shifting model of Aghion
and Bolton (1987), the naked-exclusion models of Rasmusen, Ramseyer, and Wiley (1991) and Segal and Whinston (2000b), and the downstream-competition
models of Simpson and Wickelgren (2007) and Asker and Bar-Isaac (2014). Sections II, III, and IV analyze the three claims that motivated our analysis, taking
the model to each of these three post-Chicago setups, respectively. In Section V,
we highlight how the need for an ex-ante commitment crosses all three exclusionary settings, which is then contrasted with situations in which such commitment
is not necessary; for example, to prevent inefficient entry. We conclude in Section
VI with a summary of our results and a closer look at some of the antitrust cases
listed above.
I.
The Model
A.
Notation
Consider a unit mass of final consumers with reservation value v for a good
that can be supplied by two (risk-neutral) manufacturers. Manufacturer I is an
incumbent supplier that can produce the good at a constant marginal cost cI < v.
Manufacturer E, on the other hand, is a potential entrant that can produce the
good at constant marginal cost cE < v only after paying a fixed entry cost F 0.
Our only departure from the basic structure of existing post-Chicago models is
5 Limited liability and/or asymmetric information (e.g., Ide and Montero 2015) may severely restrict
the use of these unconditional transfers, however.
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that only a fraction  1 of final consumers see no di↵erence between I’s and
E’s products; the remaining fraction buy either I’s products or nothing at all. In
the language of the EU Commission (2009), is the contestable demand, which
in most antitrust cases is thought to be rather small. In addition, and given our
focus on the possibility of writing anticompetitive contracts, we assume that entry
is efficient; that is cE + F/ < cI , unless otherwise indicated.
Manufacturers do not supply directly to final consumers but indirectly through
(risk-neutral) retail buyers, who, for the sake of simplicity, have no costs other
than those of purchasing the good from one or both manufacturers. We will
consider cases of one retail buyer (as in the rent-shifting setup of claim 1); two independent retail buyers, each serving half of the market (as in the naked-exclusion
setup of claim 2); and two retail buyers competing intensely for final consumers
(as in the downstream-competition setup of claim 3). We denote these buyers by
B, or B1 and B2 if more than one.
The timing of the game in any of these settings is as follows. On date 1, I can
make a take-it-or-leave-it contract o↵er to B (or to B1 and B2). The form of
the contract o↵er is specified below, as we will consider both rebate and exclusive
contracts. On date 2, E has the opportunity to make a take-it-or-leave-it o↵er
to B in a take-for-pay contract for units.6 Then, on date 3, E decides whether
to enter or not. If he does not, his contract is automatically canceled; if he does
enter, he pays the entry cost F . Finally, on date 4, B buys according to the
existing contracts; otherwise, B is served through the spot (wholesale) market,
where I and E compete in (nonlinear) prices.
B.
Classes of Rebate Contracts
We consider two classes of rebates, all of which are granted upon compliance
with exclusivity.7 The first class includes rebates like the one in our opening
example. Written as (ri , Ri ), these rebate contracts include a list price ri at
which buyer Bi is free to buy from I as many units as she likes, and a discount
o↵-the-list-price Ri applied to all units purchased if she conforms to the exclusivity
requirement. Under these contractual arrangements, Bi will be paying ri for each
of I’s units if she decides to buy from both I and E, and ri Ri if she decides to
buy exclusively from I. Good examples of these o↵-list-price rebates, which are
the most common of all, are found in EU Commission v. Michelin II and EU
Commission v. British Airways.
Rebates in the second class di↵er from those in the first in that the discount
is not established per unit but on a lump-sum basis. Written as (ri , Li ), under
these contractual arrangements Bi pays the list price ri , regardless of how much
she buys, and receives the lump-sum transfer Li if, in addition, she buys exclu6 Notice that in this linear world E’s optimal choice is to sell either
or nothing. Also, since E may
be relatively small, it is important to point out that our main results do not change if the bargaining
power between E and B is more evenly split.
7 In this simple linear setting, there is no reason to grant rebates for anything less than exclusivity.
VOL. VOL NO. ISSUE
DISCOUNTS AS A BARRIER TO ENTRY
7
sively from I. Note that the distinction between o↵-list-price and lump-sum is
immaterial in claims 1 and 2. When buyers are local monopolies that face a fixed
and certain demand, only the rebate’s total matters to them, whether Li or Ri qi ,
where qi is the fixed quantity to be purchased by buyer Bi (1 in claim 1 and 1/2
in claim 2). This changes when buyers compete, as in claim 3, because qi is no
longer fixed but depends on prices in the downstream market, which, in equilibrium, depend on the marginal costs internalized by buyers when setting these
prices. Thus, a buyer that decides to conform to the exclusivity under either
contractual arrangement faces a marginal cost of ri Ri under an o↵-list-price
rebate or ri under a lump-sum rebate. Perhaps the best examples of lump-sum
rebates are Intel’s, as documented in AMD v. Intel.
The natural benchmark to a rebate contract in these post-Chicago models is an
exclusive dealing arrangement. Although exclusives vary from setting to setting,
they all can be written as (ti , wi , Di ), where ti is a lump-sum payment from I to
Bi on date 1 in exchange for the exclusivity, wi is the wholesale price, which can
be nonlinear, and Di is the penalty (i.e., liquidated damages) that Bi must pay
I for breaching the exclusivity that was agreed to on date 1.
II.
Claim 1 - Rebates as Entry Fees
The first of the three claims has its roots in Aghion and Bolton (1987), the
first of the post-Chicago models to generate inefficient foreclosure with exclusive
contracts. In this setup there is a single buyer B and entry costs play no role, so
for simplicity we let F = 0. Neither B nor I knows cE at the time of contracting;
they only know that cE is distributed according to the cumulative distribution
function G(·) over the support [0, cI ], which ensures that entry is socially efficient
for any possible realization of cE . As usual, G/g is non-decreasing and (1 G)/g is
non-increasing, where g(·) = G0 (·). This rent-shifting setup also assumes that E’s
o↵er on date 2, if any, does not lead I and B to renegotiate a contract that they
had signed on date 1. This would be the case if B is the only one informed about
E’s contract o↵er (and cE is still unknown to I and possibly, but not necessarily,
to B).8 In any case, the value of cE becomes publicly known at the opening of
the spot on date 4.
To characterize the contracts I will o↵er in equilibrium we need first to compute
agents’ outside options, i.e., agents’ payo↵s when B is not supplied by a contract
but served in the spot market. Since entry is efficient, E will enter and o↵er his
units at a price slightly below cI , while I will o↵er a nonlinear schedule with a
list price cI and a fixed fee of (1
)(v cI ) conditional upon purchasing at
least one unit. Therefore, payo↵s in this no-contract benchmark are, respectively,
N C = (c
N C = (v
⇡IN C = (1
)(v cI ), ⇡E
cE ) and ⇡B
cI ).9
I
8 Dewatripont (1988) was the first to show that contracts as commitment devices are renegotiationproof if asymmetric information is introduced at the renegotiation stage.
9 Note that payo↵s do not change if, in the absence of a contract between I and B, E o↵ers a contract
to B on date 2.
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MONTH YEAR
Exclusive Contracts
To fully appreciate how rebates perform in this rent-shifting setup, we need
to understand first how exclusives work. Suppose that I o↵ers B the exclusive
contract (t = 0, w, D),10 where w  v is the wholesale price and D is the penalty
that B must pay if she decides to buy from E. If B signs the contract, E can still
persuade her to buy units from him on date 2 if D is not set too high. Since B
will charge v to final consumers in any event, the price wE that E needs to o↵er
B to persuade her to switch must satisfy
(1)
v
(1
)w
D
wE
v
w
or wE  w D/ . Thus, E will enter only if the most he can charge, w D/ ,
is enough to cover his cost cE .
Since B will end up paying w for each unit regardless of entry, the exclusivedealing program that I solves is
(2)
max ⇡IED (w, D) = [(1
)(w cI ) + D] G(w D/ )+(w cI ) [1 G(w D/ )]
w,D
N C , which immediately implies
subject to B’s participation constraint v w ⇡B
that w < v. In the case of entry, which happens with probability G(w D/ ), I
sells 1
units at price w and receives compensation D for the units that B
buys elsewhere; otherwise, I is the only one selling at price w.
N C in (2) and solving
Using B’s participation constraint to replace w = v ⇡B
for D leads to the well known Aghion and Bolton (1987) exclusionary outcome
(3)
w⇤
D⇤
⌘ c̃E = cI
G(c̃E )
g(c̃E )
N C . If we substitute this latter and D ⇤ = (w ⇤
and w⇤ = v ⇡B
c̃E ) into (2), we
ED
⇤
obtain ⇡I (w , D⇤ ) = (1
)(v cI ) + (cI c̃E )G(c̃E ), which is greater than
⇡IN C since c̃E < cI . As first shown by Aghion and Bolton (1987) for = 1, these
exclusive contracts are not only profitable for both I and B to sign, but they have
anticompetitive implications in that they block the entry of some efficient rivals.
In his e↵ort to extract rents from potential entrants, I is ready to foreclose those
with costs cE 2 [c̃E , cI ].
As discussed by Masten and Snyder (1989) and more recently by Simpson and
Wickelgren (2007), the exclusionary potential of these exclusives is subject to the
possibility of using penalties that can be enforced in court. If, as the legal practice
in common law countries seems to suggest, an exclusive contract cannot rely on
penalties above the expected damages that I will experience in the event that B
10 Assuming t = 0 results in no loss of generality, even if
Aghion and Bolton (1987).
< 1. The proof is analogous to the one in
VOL. VOL NO. ISSUE
DISCOUNTS AS A BARRIER TO ENTRY
9
breaches the contract, i.e., D  (w cI ), then the exclusive loses its exclusionary
grip altogether, i.e., w D/
cI . Given this legal constraint, it is natural that
attention has shifted toward alternative vertical practices with apparently similar
exclusionary potential, such as rebates.
B.
Rebate Contracts
According to claim 1, I should have no problems replicating the anticompetitive
outcome (3) with the o↵-list-price rebate (r, R), or with the lump-sum rebate
(r, L), for that matter. To see the validity of this claim, suppose that I o↵ers B
on date 1 the contract (r, R), with r  v. If B accepts, then E’s o↵er wE must
satisfy the following
(4)
v
(1
)r
wE
v
r + R,
or wE  r R/ , in order to induce B to buy from him on date 2.11 In antitrust circles, the term r R/ is commonly known as the e↵ective price of the
contestable demand, which is the price that E must compete with, and is lower
than r R because when B buys from E not only forgoes the discounts on the
marginal units, but on all units.
Since E will enter only if his cost cE is below the e↵ective price, I’s expected
payo↵ in case B accepts the rebate contract is equal to
(5)
⇡IR (r, R) = (1
)(r
cI )G(r
R/ ) + (r
R
cI ) [1
G(r
R/ )]
where the first term is the profit from selling 1
units at price r  v, which
happens when there is entry and the rebate R is not granted, and the second
term is the profit from selling all the units at price r R, which happens with
probability 1 G(r R/ ).12
An apparent similarity to the exclusive program above is hard to overlook. In
fact, one can arrive at (5) from (2) by simply relabeling w as r R and D as
(1 )R. This suggests that the exclusivity clause could be made equally costly to
break under either contract: in one case by paying the penalty D and in the other
by giving up an equivalent amount (1
)R in rebates for the remaining units.
There is, however, a fundamental di↵erence between the two schemes, so that a
contract (r, R) not only fails to deliver the exclusive’s anticompetitive outcome
(3), but any anticompetitive outcome at all.
11 In principle, one can think also of contracts with r > v, in which case the left-hand-side of (4) reduces
to (v wE ). It can be shown, however, that these contracts are strictly dominated by contracts with
r  v. If r is increased above v and R is increased accordingly to keep I’s profit r R cI unchanged,
the e↵ective price r R/ also remains unchanged. But if an arbitrarily small possibility exists that B
might buy from E, then I is strictly worse o↵ because B will not buy units from I above her reservation
price.
12 Note that in this particular setting we need
< 1 for a rebate contract to make any sense. If = 1,
I’s problem degenerates to the choice of a single price r R.
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PROPOSITION 1: In the rent-shifting setup of cost uncertainty, it is never profitable for I to o↵er an anticompetitive rebate contract (r, R), that is, a contract
where r R/ < cI .
PROOF:
Rearranging (5) and using x ⌘ r
(6)
⇡IR (r, x) = (1
R/ yields
)(r
cI ) + (x
cI )[1
G(x)]
Since r  v, otherwise B does not buy from I whenever there is entry, writing
an anticompetitive (r, R) contract with x < cI leaves I with strictly less than
⇡IN C = (1
)(v cI ). ⌅
Intuitively, while it is true that both rebates and exclusives give B the flexibility
to not purchase from I at all (which explains the ex-post participation restrictions
r  v and w  v), the two di↵er on how much is required from B ex-ante, i.e.,
at the time of signing the contract on date 1. In an exclusive contract, B has
already committed to an action, to pay the penalty D in case she breaches the
exclusivity, at the time E approaches B with an o↵er and the latter decides from
whom to buy and how much. There is no such ex-ante commitment in a rebate
contract, which explains this striking result.
To see it more formally, let us use the alleged exclusionary equivalence between
rebates and exclusives to relabel w as r R and D as (1
)R, so that r R/ =
w D/ = x and r = w + D/(1
). Equation (6) in Proposition 1 can then be
rewritten as
✓
◆
D
R
⇡I = (1
) w+
cI + (x cI )[1 G(x)]
1
Now, if the penalty is committed ex-ante, as in the exclusive dealing contract, the
term D/(1
) is sunk from B’s purchasing-decision perspective, which explains
why ex-post we only require w  v, not w +D/(1
)  v. However, if there is no
such commitment, as in the rebate contract, then the second term enters directly
in B’s purchasing decision, so the relevant ex-post restriction is not longer w  v
(which holds trivially in program (2) because the ex-ante participation constraint
N C ) but rather
already requires w  v ⇡B
r=w+
D
1
v
This severely limits the amount of surplus I can extract from B, rendering unprofitable any anticompetitive rebate o↵er x = r R/  cI .
It is important to emphasize that the result in Proposition 1 is robust to alternative discount contracts, as long as they do not involve an ex-ante commitment.
Consider, for example, a two-part-tari↵ contract in which x is the unit price and
T is a conditional fixed-fee, that is, a fee that is paid only upon purchasing one
VOL. VOL NO. ISSUE
DISCOUNTS AS A BARRIER TO ENTRY
11
or more units. This contract faces the same problem as the rebate contract, in
that T cannot be increased to o↵set the loss from setting x < cI . Any attempt
to increase T will stop B from buying from I. It is not surprising that these
two discount contracts are perfectly equivalent in this setting because they share
the same principle:13 it is not possible to foreclose the entry of a more efficient
rival E unless B is forced to take some action before E shows up that e↵ectively
reduces B’s purchasing-flexibility ex-post. As we will see next, the same principle
explains why these rebate contracts also fail to exclude in the naked-exclusion
and downstream-competition setups.
III.
Claim 2 - Discriminatory Rebates and Demand Foreclosure
The second claim is inspired by the divide-and-conquer strategy in the nakedexclusion models of Rasmusen, Ramseyer, and Wiley (1991) and Segal and Whinston (2000b). Key assumptions here are the presence of multiple buyers and scale
economies, requiring E to serve a sufficient number of buyers to achieve the minimum viable scale of operation. As a result, a contract signed by any buyer creates
a negative externality on all remaining buyers by reducing the probability of entry.
According to claim 2, I can likewise foreclose E’s entry by o↵ering rebates to lock
up a critical number of buyers, enough to make it impossible for E to achieve
such a minimum viable scale. This, in turn, allows I to exploit all remaining
(unlucky) buyers.
We analyze this claim with the simplest possible setting. Consider two retail
buyers, B1 and B2, each serving half of the final consumers in completely separate
markets, so again, they will charge v to final consumers. In addition, assume that
E’s fixed entry cost F is greater than the most he could obtain if dealing with
just one buyer, i.e., F > (v cE )/2, which necessarily forces him to deal with
both buyers to cover F . In addition, the standard naked-exclusion environment
involves no uncertainty over cE , though results do not change if we keep cE
unknown (see the online Appendix for a formal treatment of this case).
Before looking at the work of exclusive contracts and explaining why rebates
fail to replicate them, keep in mind that the equilibrium outcome when I does
not o↵er any contract or when both buyers reject I’s o↵ers follows the same
logic of the previous section. Hence, the no-contract payo↵s are, respectively,
N C = (c
N C = (v
⇡IN C = (1
)(v cI ), ⇡E
cE ) F , and ⇡Bi
cI )/2 for i = 1, 2
I
(for more details see the online Appendix).
13 To see the equivalence, notice that x = r
R/ and T = (r x)(1
)  (v x)(1
). Rebates
can be superior to two-part tari↵s in other contexts, however; for example, when there is asymmetric
information (Kolay, Sha↵er, and Ordover 2004; Ide and Montero 2015).
12
THE AMERICAN ECONOMIC REVIEW
A.
MONTH YEAR
Exclusive Contracts
We now show that the exclusionary results of existing naked-exclusion models
still apply in our slightly di↵erent structure,14 but we will be brief because these
results are well known. Following existing models, suppose that on date 1, I o↵ers
buyers exclusive contracts (ti , wi , Di ), where ti is transfer on date 1 from I to Bi
in exchange for the buyer’s promise to never buy from E (i.e., Di ! 1), and
the terms of trade wi are to be specified on date 4. It is well known that this
setting accepts multiple equilibria, as a buyer’s best response is to take any o↵er
deemed exclusionary if she conjectures that the other buyer will take her too.
To avoid such coordination failures, we will follow the literature and assume that
buyers can communicate with each other but cannot sign binding agreements. The
equilibrium concept we adopt, then, is perfect coalition-proof Nash equilibrium
(see Bernheim, Peleg, and Whinston 1987).
PROPOSITION 2: In the naked-exclusion setup of multiple buyers and scale
economies, it is profitable for I to deter E’s entry with a pair of exclusive contracts
N C + ✏ and t = ✏ with ✏ ! 0, or vice versa.
with compensations t1 = ⇡B1
2
PROOF:
It is clearly an equilibrium for both B1 and B2 to accept these exclusives. B1
gets nothing if she rejects her o↵er and B2 does not because in that case E does
not enter and I charges v in the spot. Similarly, B2 gets zero if she rejects her
o↵er and B1 does not because again E does not enter. Finally, it is easy to see
that foreclosure is a profitable strategy for I in that ⇡IED (t1 , t2 ) = v cI t1 t2 =
(1
/2)(v cI ) > (1
)(v cI ) = ⇡IN C .15 ⌅
The proposition shows that it pays I to induce one buyer to sign an exclusive for
N C , because by doing so he can fully exploit
slightly more than her outside option ⇡B1
16
the other buyer.
Note also that despite B2 and B1 would be, on aggregate,
N C + ⇡ N C ),
better o↵ if they both reject their o↵ers (and obtain a total of ⇡B1
B2
the absence of binding agreements rules out such coordination, i.e., B2 cannot
credibly commit to any compensation that would induce B1 to reject her o↵er in
the first place.
14 The models of Rasmusen, Ramseyer, and Wiley (1991) and Segal and Whinston (2000b) assume a
downward sloping demand, which creates an additional efficiency loss from linear monopoly pricing. As
first noticed by Innes and Sexton (1994), this loss is not needed for (naked) exclusion to exist and is
most reasonable to assume it away for wholesale markets, as firms can always use nonlinear prices. Our
inelastic demand model intends to capture this in the simplest possible way. See also Section V.C
15 Notice that in the proof, we made use of the coalition-proof Nash equilibrium concept by ruling
out Nash equilibria that do not survive the coalition-proof refinement; for instance, the Nash equilibria
N C ) for i = 1, 2.
where both buyers accept any pair of o↵ers with ti 2 (0, ⇡Bi
16 This result is robust to di↵erent buyers’ outside options. Suppose that each buyer’s outside option
is half the social surplus that E brings to market, (v cE ) F . It still pays I to compensate the lucky
buyer, provided that v is not too close to cI .
VOL. VOL NO. ISSUE
DISCOUNTS AS A BARRIER TO ENTRY
B.
13
Rebate Contracts
Proposition 2 shows that exclusives can foreclose entry when used as part of a
divide-and-conquer strategy, so the question here is whether a pair of discriminatory rebate o↵ers (r1 , R1 ) and (r2 , R2 ) can be used in a similar way. Following
claim 2, the idea would be for I to de facto lock-in one of the retailers, say B1,
by o↵ering her such an attractive (r1 , R1 ) rebate, that E would find it impossible
to induce her to switch without making a loss. Therefore, E would refrain from
entering, as he will be unable to achieve its minimum viable scale of operation,
allowing I to fully exploit the remaining buyer B2 with an o↵er (r2 = v, R2 = 2✏)
with ✏ ! 0.17
More formally, for the rebate contract (r1 , R1 ) to e↵ectively lock-in B1, it must
be true that such contract is strictly preferred by B1 to the best deal E could
possibly o↵er her. If wE1 is the price in that best deal, then this “no switching”
constraint is
(7)
(v
r1 + R1 )/2 > (v
wE1 )/2 + (1
)(v
r1 )/2,
that is, B1’s payo↵ from buying all 1/2 units from I must be greater than the
payo↵ from buying /2 units from E at price wE1 and (1
)/2 units from I at
price r1 . If such condition is satisfied, then entry is foreclosed and B2 exploited,
which would result in a profit to I equal to
(8)
⇡IR = (r1
R1
cI )/2 + (v
cI
2✏)/2
From here, we can immediately deduce that I would optimally set r1 = v, since
any contract (r1 < v, R1 ) satisfying (7) is strictly dominated by the still anticompetitive rebate (r10 = r1 + ", R10 = R1 + ") that allows I to pocket an extra
"(1
)/2.
This result follows the often-raised leverage argument against all-unit rebates
(see EU Commission 2009, parr 39). Because B1 will buy (1
)/2 units from I
in any event, by increasing r1 (and adjusting R1 accordingly), I can increase at
no cost the implicit penalty faced by B1 in case she decides to forgo the rebate.
Since this leverage e↵ect is greater the smaller the value of , it is not surprising
then the great deal of attention and controversy around the estimation of such
parameter.
Based on this leverage argument, it appears that all-unit rebates enjoy of a
huge anticompetitive potential, and should therefore have no problem replicating
the exclusionary outcome of Proposition 2. Indeed, comparing ⇡IR (r1 = r2 =
v, R1 , R2 ) = v cI R1 /2 R2 /2 to ⇡IED = v cI t1 t2 , one is tempted
to conclude that all that it is required is to set R1 = 2t1 and R2 = 2t2 . This is
17 Without loss of generality we can restrict attention to o↵ers that are not contingent on the action
of the other buyer. Such strategies add nothing here because I has all the bargaining power, as opposed,
for example, to Rey and Whinston (2013).
14
THE AMERICAN ECONOMIC REVIEW
MONTH YEAR
in fact the basis of the alleged exclusionary equivalence outlined in claim 2. As
already argued in Section II, however, the rebates’ lack of ex-ante commitment
prevents their anticompetitive use altogether.
PROPOSITION 3: In the naked-exclusion setup of multiple buyers and scale
economies, it is never profitable for I to o↵er a pair of discriminatory rebate
contracts (ri , Ri ) for i = 1, 2, to foreclose efficient entry.
PROOF:
Suppose that I o↵ers a pair of contracts (ri , Ri ) with the idea to lock up B1 and
exploit B2. We have already discussed that such o↵ers must be of the form (r1 =
v, R1 ) and (r2 = v, R2 = 2✏) with ✏ ! 0. If so, I’s profit is ⇡IR = v cI R1 /2;
and B1 is de facto locked-in if R1 /2 > (v wE1 )/2, for any potentially profitable
o↵er wE1 . Now, if E enters, his profit would be
⇡E = (wE1
cE )/2 + (wE2
cE )/2
F
But there is no ex-ante commitment that ties B2 to I, so given that B2 correctly
anticipates that she will get zero if E does not enter, E can persuade B2 to buy
units from him at virtually her reservation price, i.e., wE2 = v. In turn, this
allows E to set wE1 low enough just to satisfy
(9)
(wE1
cE )/2 + (v
cE )/2
F =0
Rearranging, we get
(v
wE1 )/2 = (v
cI ) + [ (cI
cE )
Hence, to block E’s entry, I needs to set R1 /2 > (v
But this implies that (✏ ! 0)
⇡IR = v
cI
R1 /2 < (1
)(v
cI )
[ (cI
F]
cI ) + [ (cI
cE )
cE )
F ].
F]
Since the no-contract benchmark guarantees I a payo↵ of ⇡IN C = (1
)(v cI ),
exclusion is profitable only if ⇡IR > ⇡IN C , that is, only if (cI cE ) F < 0, which
contradicts the efficient-entry assumption.18 ⌅
Rebates’ lack of ex-ante commitment opens up two important di↵erences when
compared to exclusive dealing contracts. The first is that to lock B1 with a
rebate, I needs R1 /2 > (v wE1 )/2, rather than t1 > (v cI )/2, as E has
always the possibility to make a countero↵er. And second, B2’s rebate contract
(r2 = v, R2 = 2✏) does not impose any exclusivity obligation, as opposed to the
exclusive dealing provision which it does upon transferring t2 = ✏.
18 Notice that since everything is executed ex-post the coalition-proof refinement becomes irrelevant.
E makes sure to eliminate any exclusionary outcome that relies on a buyers’ coordination failure.
VOL. VOL NO. ISSUE
DISCOUNTS AS A BARRIER TO ENTRY
15
These two di↵erences explain the result in Proposition 3: because B2 is not exante contractually committed to the exclusivity, but is nevertheless fully exploited
by I, E anticipates that if he enters, he can countero↵er B2’s rebate and appropriate B2’s entire surplus. Furthermore, because B1 has full flexibility ex-post to
decide whether to take I’s rebate or not (i.e., she has not taken any action before
E’s o↵er), E can use as much surplus from B2 as needed to persuade B1 not to
take the rebate. This, in turn, forces I to o↵er a larger reward to keep B1 aligned
with the exclusionary outcome, rendering unprofitable any anticompetitive rebate
scheme.
Similar to other post-Chicago models, to successfully implement an anticompetitive outcome in this setting, the contractual arrangement must necessarily
involve an ex-ante commitment. Otherwise, a more efficient entrant can always
distribute surplus among buyers to make sure he is not excluded from the market.19
IV.
Claim 3 - Upstream Exclusion and Downstream Competition
Up to now we have restricted attention to settings where retail buyers were
assumed to be local monopolies. However, this assumption does not fit well
with a number of relevant antitrust cases. In AMD v. Intel, for instance, the
market structure consisted of a dominant incumbent manufacturer, Intel, and a
small alternative supplier, AMD, selling microprocessors to original equipment
manufacturers (OEMs) such as IBM, Dell, and Lenovo, which in turn competed
intensely for final consumers. In this section, we look at the work of rebates in
such a setting by assuming that B1 and B2 are not local monopolies, but rather
undi↵erentiated Bertrand competitors, just like in Simpson and Wickelgren (2007)
and Asker and Bar-Isaac (2014).
An important observation, first made by Fumagalli and Motta (2006), is that
when retailers are intense downstream competitors, scale economies and externalities across buyers become irrelevant, since access to one retailer is all E needs
to reach the entire final market. Hence, we can work with F
0. In addition,
because the outcome of the downstream market is now endogenously determined,
we require of some extra notation and assumptions. First, we assume that if two
retailers o↵er the same price to final consumers but one has a lower marginal cost
than the other, then all final consumers buy from the retailer with the lower cost.
19 In a somewhat di↵erent naked-exclusion setting, where scale economies in production are replaced
by network externalities in demand, Karlinger and Motta (2012) find that rebates can sometimes lead to
foreclosure if E is only slightly more efficient than I. Although a complete discussion of the conditions
under which rebates could lead to exclusion is relegated to Section V, we can advance that what explains
their result is an exogenous restriction that reduces the set of “contracts” that E can o↵er to price
announcements with no purchasing obligations. In terms of the generalized Chicago critique principle
that we develop in Section V, this exogenous restriction violates a contract completeness assumption,
which is particularly problematic here because it is in the interest of E and each of the buyers, both
ex-ante and ex-post, to sign longer-term contracts that commit them to buy from E. This is why we
follow the post-Chicago models of Innes and Sexton (1994), Spector (2011) and Asker and Bar-Isaac
(2014), to name a few, and do not impose this restriction to the set of contracts that E can o↵er.
16
THE AMERICAN ECONOMIC REVIEW
MONTH YEAR
Second, and more importantly, we assume that retailers can price-discriminate
between final consumers in the non-contestable (the 1
segment) and those in
the contestable portion of demand (the segment). We will denote by pi,1 the
retail price that retailer i = 1, 2 charges for I’s products to final consumers in the
1
segment and by pi, the price she charges for both I’s and E’s products to
final consumers in the segment.
We adopt this particular price-discrimination assumption for several reasons.
The first one is tractability: characterizing the equilibrium downstream, conditional on manufacturers o↵ers, is much simpler when retailers compete separately
for contestable and non-contestable consumers. Second, as it will become clear
shortly, in this class of models the reason exclusion arises is because of intense
competition downstream. By allowing retailers to charge di↵erent prices in each
segment, we intensify downstream competition to the maximum extent making
this case the most favorable for exclusion.20 Third, the assumption is immaterial
to show that rebates fail to replicate exclusives.21 Finally, we believe that the
possibility that retailers can discriminate across demand segments may also have
some practical appeal. For example, consider one of the OEMs above buying chips
from both Intel and AMD. It would be easy for this OEM to price-discriminate
among contestable and non-contestable consumers —students and professionals,
for example— by placing the exact same Intel chip in machines that do not look
alike (similar to a damaged-good strategy).
A.
No-Contract and Exclusive Benchmarks
Downstream competition changes agents’ outside options in important ways,
and hence, the contracts they are willing to sign. To see how, consider first the
situation in which E does not enter the market. I will charge the pair of nonlinear
schedules {wI1 = v, TI1 = 0} and {wI2 = v, TI2 = 0}, where wIi is the unit-price
and TIi is a conditional fixed-fee, leaving retailers and final consumers with zero
surplus. Retailers will then split the market and charge v to contestable and
non-contestable consumers.22
The situation changes for the incumbent and final consumers, however, if E
enters the market. As we formally show in Appendix A, in equilibrium I will
o↵er the schedules {wI1 = cI , TI1 = (1
)(v cI )} and {wI2 = v, TI2 = 0} while
E will o↵er wE1 = wE2 = cI . Hence only one retailer will carry I’s products, B1,
but both will carry E’s products resulting in downstream retail prices equal to
p1,1 = p2,1 = v and p1, = p2, = cI .
20 Indeed, as we show in Appendix B, if we work with the alternative assumption that retailers set
di↵erent prices for I’s and E’s products, as opposed to di↵erent prices across demand segments, then there
exist equilibria where exclusion does not arise even with exclusives. The reason is that this alternative
pricing assumption softens competition downstream.
21 In Appendix B we also show that even if an exclusionary equilibrium with exclusives exists under
this alternative price-discrimination assumption, rebates still lack of any anticompetitive potential.
22 Notice that I can implement the same outcome but with only one retailer actually selling his units,
for example, with the pair of o↵ers {wI1 < v, TI1 = v wI1 } and wI2 = v.
VOL. VOL NO. ISSUE
DISCOUNTS AS A BARRIER TO ENTRY
17
Compared to the no-contract benchmark of the naked-exclusion setup, suppliers
N C = (c
are exactly as before (i.e., ⇡IN C = (1
)(v cI ) and ⇡E
cE ) F ), but
I
N
C
N
C
retailers are now strictly worse o↵ (i.e., ⇡B1 = ⇡B2 = 0). Any retailer’s surplus
is now in the hands of final consumers in the form of lower retail prices for the
contestable units. The reason for this surplus transfer is that here, retailers only
provide access to final consumers, and because of Bertrand competition in the
contestable segment, access can in principle be achieved with just one retailer.
By a↵ecting agents’ outside options, downstream competition also impacts the
exclusives I may o↵er. Consistent with Simpson and Wickelgren (2007) and Abito
and Wright (2008), it is possible to establish that:
PROPOSITION 4: In the downstream-competition setup of Bertrand retailers,
I can persuade both retailers to accept exclusive contracts (ti > 0, wi , Di ! 1)
for as low as ti = ✏ ! 0 for i = 1, 2.
PROOF:
Recall that an exclusive o↵er to retailer i = 1, 2 leaves the wholesale price w
—a price schedule consisting of a unit price wIi and, presumably, a conditional
fixed-fee TIi — unspecified until the beginning of date 4. Suppose then, I o↵ers
B1 and B2 exclusives with payments of t1 and t2 , respectively. By accepting the
exclusive, B1 gets just t1 , since any additional surplus can be extracted ex-post
by I, regardless of what happens with the other buyer. If, instead, B1 rejects the
exclusive, there are two cases to consider. The first is when B2 also rejects her
o↵er, in which case B1 gets zero, as indicated by the no-contract benchmark. The
second case is when B2 accepts her o↵er. Notice that the outcome of this subgame
is no di↵erent than the outcome in the no-contract benchmark above, because one
free retailer is all E needs to compete e↵ectively for the contestable portion of the
demand. In equilibrium of this subgame, E will o↵er the unit-price wE1 = cI to
B1 and nothing to B2 because the latter is locked up by the exclusive, while I will
o↵er the two-part tari↵s {wI1 = cI , TI1 = (1
)(v cI )} and {wI2 v, TI2 = 0}
to B1 and B2, respectively, leading to equilibrium retail prices p1,1 = p2,1 = v
and p1, = p2, = cI and to profits ⇡B1 = ⇡B2 = 0. Hence, since B1 gets 0
regardless of whether she accepts or rejects the contract, I only needs to o↵er
✏ ! 0 to induce both retailers to sign. This is clearly optimal for I, as he obtains
a monopoly profit of ⇡IED = v cI 2✏ > (1
)(v cI ) = ⇡IN C .⌅
The key to understand why exclusion emerges, even in the absence of cost uncertainty or scale economies, lies on the e↵ect of downstream Bertrand competition
on retailers’ profits. Intense downstream competition prevents retailers from appropriating any of the benefits of the additional upstream competition brought
forward by a more efficient supplier. All these benefits are fully passed-through
to final consumers in the the form of lower retail prices. This makes it virtually
costless for I to compensate retailers in exchange of exclusivity, allowing him to
increase his profit from ⇡IN C = (1
)(v cI ) to ⇡IED = v cI .
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THE AMERICAN ECONOMIC REVIEW
B.
MONTH YEAR
Rebate Contracts
If one were to interpret claim 3 as an extension of the logic of Proposition 4
to rebates, one would anticipate two implications. The first is that the rebates I
needs to o↵er to persuade retailers to stick to the exclusivity can be very small
because retailers are ready to accept any reward. And the second is what drives
E’s foreclosure is an attempt to prevent upstream competition from eroding industry profits with lower prices to final consumers. An important contribution
of this section is to show that how much of this interpretation is valid, if any,
depends crucially on contract characteristics.
As we anticipated in Section I, and in contrast to what we have seen in the previous two sections, when retailers compete downstream the distinction between
(ri , Ri ) and (ri , Li ) rebates matters a great deal. The reason is that now equilibrium retail prices depend on whether rebates are granted on an o↵-list price or a
lump-sum basis. We start our discussion with the most common rebate arrangement of all, an o↵-list-price rebate (ri , Ri ). Our first result goes against claim 3
in its entirety and follows exactly the results of Propositions 1 and 3:
PROPOSITION 5: In the downstream-competition setup of Bertrand retailers,
it is never profitable for I to o↵er a pair of discriminatory rebate contracts (ri , Ri )
for i = 1, 2, to deter E’s entry.
PROOF:
Consider first the case in which on date 1 I o↵ers the same rebate (r, R) to
both retailers and both accept (recall that both r and R are announced on date
1). If on date 2 both retailers decline to deal with E, they will both set retail
prices equal to r R, which is the relevant marginal cost under exclusivity (since
exclusivity requires retailers to carry only I’s products, regardless of how much
they sell). Suppose instead that E is able to convince one retailer, say B1, to
forgo the rebate in exchange for a wholesale price of wE1 for units. If so, B1
faces a marginal cost of r when selling I’s products and of wE1 when selling
E’s products, while B2 faces a marginal cost of r R when selling I’s units
(and infinity when selling E’s products). Then, the equilibrium retail price in
the non-contestable portion of the demand is r and in the contestable portion is
max{wE1 , r R}, which implies that E can induce B1 to accept his o↵er as long
as (max{wE1 , r R} wE1 ) 0, since B1 gets nothing if she rejects E’s o↵er.
Therefore, E will enter if and only if (r R cE ) F 0; so, to block entry, I
needs to set r and R such that r R < cE + F/ . But if so, I’s profit reduces to
r R cI < cE + F/
cI < 0.
The alternative case is when one retailer rejects I’s o↵er, or equivalently, when
I approaches just one retailer, say B2, with the o↵er (r2 , R2 ). Notice that in this
case, it is irrelevant whom E approaches, since the price he needs to o↵er any
retailer to e↵ectively enter the market and make positive sales must be lower than
r2 R2 , given Bertrand competition downstream. Therefore, to block E’s entry,
r2 and R2 must be such that r2 R2 < cE + F/ . But we already saw that doing
so is not profitable for I.⌅
VOL. VOL NO. ISSUE
DISCOUNTS AS A BARRIER TO ENTRY
19
While it is true that downstream competition greatly impacts retailers’ outside
options, so much that they are ready to become exclusive distributors for virtually
nothing, this is completely useless in the case of o↵-list price rebates due to their
lack of ex-ante commitment. This is intuitive: if the exclusivity is not contractually committed before E shows up, then I needs to set min {ri Ri , rj Rj } <
cE + F/ to block E’s entry. Otherwise, E could always approach the retailer
with the lowest ri Ri with a slightly better o↵er, granting himself access to
the contestable consumers at a margin large enough to cover his fixed cost F .
But min {ri Ri , rj Rj } < cE + F/ means I would be selling below cost. In
simple words, without an ex-ante commitment, any attempt by I to exclude a
more efficient rival using (ri , Ri ) contracts is equivalent to limit pricing, which is
clearly not profitable when the rival is more efficient.
The result in Proposition 5 raises an interesting contrast with the one in Asker
and Bar-Isaac (2014). In that paper, the authors show that if retailers are compensated with lump-sum rebates whenever they conform to the exclusivity and
irrespective of how much they sell, then, exclusion can arise under some parameter configurations. The reason is that since entry reduces industry profits along
with I’s, this latter is willing to transfer some or almost all of his profit reduction
back to the retailers in the form of lump-sum rebates as a way to induce them
to not facilitate entry. In other words, I uses these lump-sum rewards to make
B1 and B2 internalize the e↵ect on his profits of accommodating entry. However,
and as the next proposition formally shows, for all this to work it is crucial that
retailers do not internalize those rebates as reductions of their marginal costs.
PROPOSITION 6: In the downstream-competition setup of Bertrand retailers,
it is profitable for I to o↵er retailers a pair of discriminatory lump-sum rebates
(ri , Li ) for i = 1, 2, in exchange for exclusivity whenever
(10)
(v
cI )/2 > (cI
cE )
F
PROOF:
Following the contracts in Asker and Bar-Isaac (2014),23 suppose that on date
1, I o↵ers each retailer i = 1, 2 the contract (ri , Li ), where Li > 0 is the lump-sum
amount established on date 1 but to be paid at the end of date 4, provided the
retailer did not buy from E, and ri is a price schedule with a unit-price wIi and
a conditional fixed-fee TIi , both left to be specified at the beginning of date 4.
All retailers accept the o↵er since they have nothing to lose relative to their nocontract payo↵ of zero. Since E needs only one retailer to enter, he will, on date
2, approach the one with the lowest Li , say B2, with a price o↵er wE2 cE + F/
that commits B2 to buy at least units at that price on date 4. One necessary
condition for B2 to accept E’s o↵er is that wE2  cI , because she anticipates that
on date 4, I’s equilibrium response will be to channel all his sales through B1
23 Although results are the same, their model di↵ers slightly from ours in that they consider an infinitely repeated interaction between suppliers and retailers, which allows for the possibility of sustaining
relational contracts that do not need court enforcement.
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with the two-part tari↵ {wI1 = cI , TI1 = (1
)(v cI1 )}, while her equilibrium
response will be to price the contestable units slightly below cI . Since these
equilibrium prices report (cI wE2 ) to B2 ex-post, the second condition for B2
to accept E’s o↵er is (cI wE2 ) L2 . Anticipating all this, the best contract E
can o↵er B2 is wE2 = cE + F/ , which gives her (cI cE ) F L2 . In turn, this
latter implies that to block E’s entry, I will need to o↵er each retailer at least
L = (cI cE ) F on date 1. But this is worth doing only if the exclusionary
profit, v cI 2L, is greater than the profit under the no-contract benchmark,
⇡IN C = (1
)(v cI ); that is, only if condition (10) holds.⌅
Proposition 5 highlights the importance of distinguishing between o↵-list-price
and lump-sum rebates under intense downstream competition because of their
distinct e↵ect on equilibrium retail prices. The distinction is particularly evident
when we see what it takes E to induce retailers to switch. As indicated in the
proof of Proposition 5, it costs E virtually nothing to persuade either retailer to
forgo the rebate R and buy from him because they get nothing anyway if they
reject E’s o↵er. Conversely, it costs E exactly L to persuade a retailer to forgo the
lump-sum L. And in the latter case, as condition (10) indicates, E cannot a↵ord
to pay this much if his available surplus, (cI cE ) F , is less than the most I can
o↵er each retailer as compensation, (v cI )/2. There is a simple intuition for
these contrasting outcomes. If retailers internalize rebates as reductions of their
marginal costs, then, any surplus-transfer that I uses to persuade them to become
exclusive distributors gets dissipated away by downstream Bertrand competition,
which, in turn, eliminates their incentives not to deal with E.24
It is worth pointing out that lump-sum rebates still fall short of the anticompetitive potential of the exclusives in Proposition 4. Not surprisingly, the reason is
again connected to the (lack of) ex-ante commitment of discount contracts. Since
(ri , Li ) contracts, just as (ri , Ri ) contracts, do not include any ex-ante commitment on the buyer side, this opens up an opportunity for E to make countero↵ers.
This, in turn, forces I to give significant rewards to retailers in exchange for exclusivity, as opposed to virtually none when the contractual arrangement is an
exclusive dealing provision. But why is it that lump-sum rebates still have some
exclusionary potential if they do not include an ex-ante commitment? We now
discuss this as well as its connection to the results in Propositions 1, 3 and 5,
through the lens of what we call a generalized Chicago critique.
24 It is important to clarify that what makes a rebate “lump-sum” is not that it is paid retroactively
in a lump-sum manner, say, at the end of the accounting year, but rather that it has no e↵ect on retail
pricing at the margin because it is a fixed amount, independent of how much is actually purchased from
the incumbent. According to this interpretation, the rebates in AMD v. Intel appear to be lump-sum
while in EU Commission v. Michelin II they appear to be o↵-list-price, despite the latter’s nevertheless
being paid retroactively by the end of February the following year. Moreover, Michelin was paying rebates
that were increasing with sales not only because they were established as percentage of the list price but
also because the percentage discount was increasing as sales were crossing pre-agreed thresholds.
VOL. VOL NO. ISSUE
DISCOUNTS AS A BARRIER TO ENTRY
V.
A.
21
A Generalized Chicago Critique
The Need for an Ex-Ante Commitment
The key to assimilate the results of the three previous sections in a coherent
fashion, lies on understanding how post-Chicago models work. First, notice that
since exclusion of an efficient rival is in itself inefficient, if all relevant parties were
to participate simultaneously in the bargaining process, and sufficiently complete
contracts (e.g., nonlinear prices) are available for the parties to sign, exclusion
could not arise. One may call this a generalized Chicago critique. Post-Chicago
models depart from this, by letting one market participant be momentarily absent from the bargaining table, which opens up an opportunity for I to extract
additional rents from a third party, which sometimes may require the exclusion
of a more efficient entrant. In the setup of claim 1, for instance, it is E who is
initially absent, who also happens to be the target of I’s rent-extraction scheme.
E is also absent in the setup of claim 2, but this time, some retail buyers are the
ones being exploited. And in claim 3, both E and final consumers are absent, but
the latter are the ones being exploited with high retail prices.25
As originally-absent parties eventually take part in the bargaining process however, I can only sustain such an inefficient outcome if buyers are e↵ectively lockedin ex-ante; that is, before the absent parties show up. This explains why exclusion
is always possible with exclusives but not with o↵-list-price rebates. The rebates’
lack of ex-ante commitment allows retailers to make all their decisions, most
importantly whether to become I’s exclusive distributors or not, only after observing o↵ers from all remaining parties. It is precisely this ex-post flexibility on
the buyer side that renders I’s first-mover advantage irrelevant, as it forces him to
o↵er larger rewards to keep buyers aligned with the inefficient outcome ex-post,
making any anticompetitive scheme unprofitable to begin with.
This intuition is similar in spirit to the reasoning behind why exclusives lose
their exclusionary grip altogether when buyers are local monopolies, and liquidated damages must satisfy efficient breach (Masten and Snyder 1989, Simpson
and Wickelgren 2007). Because foreclosing an efficient rival leads to a deadweight
loss, all parties understand that the entrant will have enough surplus ex-post to
persuade buyers to switch, when they only need to pay expected damages. And
since this renders any ex-ante commitment irrelevant, these exclusives are, for
exclusionary purposes, analogous to rebate contracts.
The need for an ex-ante commitment is also present in the tying model of
Whinston (1990). Foreclosure in that model requires the incumbent to irreversibly
tie two unrelated products ex-ante, one from the market where he faces the entry
threat and one from a market where he is a monopolist, to credibly pre-commit
to be aggressive ex-post and deter entry. This, in turn, allows him to fully exploit
25 Strictly speaking, final consumers are also absent in the rent-shifting and naked-exclusion settings.
They are not, however, relevant parties in these models since they always end up paying v, irrespective
of whether contracts are in place or not.
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the (single) buyer that is present. If tying is not irreversible however, then this
strategy loses its exclusionary potential, since the entrant anticipates that in case
of entry, the incumbent will reverse the tying arrangement as this strategy gives
him lower profits than selling both goods separately.26
Is there anything I can do to restore the anticompetitive potential of rebate
contracts in all these settings? The previous discussion about the importance of
an ex-ante commitment gives us a hint: to add a clause in the rebate contract
that commits retailers to make an unconditional transfer in exchange for a generous reward ex-post.27 Whether the transfer is made up-front or later is not
important, as long as retailers commit to it regardless of their later actions, even
if they do not buy from I at all. The reason unconditional transfers act as an
ex-ante commitment device is because they restrict the buyer’s flexibility ex-post
by forcing her to take an action before observing all o↵ers. Interestingly, however,
these transfers have never been mentioned nor documented in any of the antitrust
cases listed above.
Applying these intuitions to the first two claims is straightforward. In both
cases, I finds himself unable to exploit the situation to his advantage with either
o↵-list-price or lump-sum rebates that do not include unconditional transfers, as
they fail to lock up buyers ex-ante. Things are slightly more involved when retailers compete downstream, because one of the two relevant parties that is originally
absent, final consumers, never directly participate in the bargaining process. Intense competition in the retail market, however, implicitly transforms retailers
into agents of final consumers, which indirectly brings them to the bargaining table. I can again avoid this intermediation by (i) locking up retailers ex-ante with
an unconditional transfer, something that is possible because E is also initially
absent; or alternatively, by (ii) softening downstream competition, which reduces
the indirect influence of final consumers in the bargaining outcome, and works
even if E is already present making o↵ers. This explains why (ri , Ri ) contracts
are no longer equivalent for exclusionary purposes to (ri , Li ) ones: only the latter
mute downstream competition.
While unconditional transfers are essential for writing anticompetitive (o↵-listprice) rebates in any of the post-Chicago models, they are not necessary for
writing rebates that preclude inefficient outcomes, for example, that deter inefficient entry. An ex-ante commitment is also absent in some exclusionary bundling
models. We elaborate on both cases below to better appreciate the scope of the
generalized Chicago critique.
26 Notice that the single buyer is the party being exploited here, which is why writing an exclusive
contract is useless to foreclose entry as opposed to tying: the incumbent needs to commit himself, rather
than locking up the buyer, to exploit a third party (here the same buyer). The fact that tying works but
exclusives do not explains why this model is not regarded as a post-Chicago model, even though its logic
connects well with the generalized Chicago critique.
27 The complete analysis of the anticompetitive e↵ects of rebates with unconditional transfers for claims
1 to 3 can be found in the online Appendix.
VOL. VOL NO. ISSUE
DISCOUNTS AS A BARRIER TO ENTRY
B.
23
Inefficient Entry
Building on the model of Section I, consider again the case of a single buyer
B and known entry costs, but suppose instead that E’s entry is inefficient, i.e.,
cI < cE + F/ < v. In the absence of contracts, there will be too much entry in
equilibrium. In fact, if I does not o↵er a contract to B on date 1 and E does not
enter on date 3, B gets charged v for every unit. Anticipating this, E and B have
all the incentives to write a contract on date 2, in which the latter commits to buy
units from the former at a unit price wE 2 (cE + F/ , v), leading to inefficient
entry.28 The exact value of wE will split the entry surplus (v cE ) F between
B and E according to their bargaining powers. This is equivalent to having B
subsidize a fraction of E’s fixed cost F , so as to increase competition in the spot
market.
Notice that even though (r, R) contracts were never profitable to foreclose an
efficient rival, here I can use them to block E’s inefficient entry as he is the
residual claimant of the increase in social surplus generated by eliminating this
distortion. I would o↵er B a rebate scheme with a list price r = v and a discount
R resulting in an e↵ective price x = v R/ such that (v x) = (v cE ) F ;
that is, an e↵ective price low enough to prevent E from making an o↵er that B
might accept. Under this rebate contract, I gets
✓
◆
F
⇡I = (1
)(v cI ) +
cE +
cI
which is greater than (1
)(v cI ), the payo↵ he would get otherwise. This
again can be viewed through the lens of the generalized Chicago critique: since
rebates eliminate a distortion, there is no need to lock up retailers ex-ante with
unconditional transfers for these rebate contracts to be profitable.
Interestingly, this form of using rebates to deter (inefficient) entry/expansion
conforms well to the developments in Barry Wright v. ITT Grinnell (1983). As
documented by Kobayashi (2005), ITT Grinnell, a manufacturer of pipe systems
for nuclear power plants, agreed to contribute to Barry Wright’s cost to develop a
full line of mechanical snubbers, an essential component in pipe systems, presumably to improve its negotiation power with Pacific, the existing dominant supplier
of mechanical snubbers. Pacific reacted to Grinnell’s deal with Barry Wright with
a rebate contract o↵er that Grinnell could not resist: it included discounts of 3025 percent o↵ list price if Grinnell would agree to purchase virtually everything
from Pacific.
28 This model of inefficient entry is originally due to Innes and Sexton (1994). Notice, however, that
this is not the only way to generate inefficient entry. A contestable demand < 1 will produce similar
inefficiencies if manufacturers are restricted to o↵ering linear prices in the spot, or if they supply to
Bertrand retailers who, in turn, are restricted to o↵ering linear prices. See the previous version of the
paper for more details.
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MONTH YEAR
Exclusionary Bundling
The need for an ex-ante commitment is also absent in some exclusionary bundling
models (e.g., Nalebu↵ 2004, 2005; Greenlee, Reitman, and Sibley 2008)29 that analyze bundled discounts for otherwise completely unrelated products; say, products A and B, where the incumbent is a monopolist in market A and faces competition in market B from a more efficient rival.30 Despite the similarity to our
non-contestable and contestable segments, our theory is di↵erent in that the exclusionary mechanism underlying these bundling models is more closely related
to a price-discrimination argument than to the post-Chicago ones behind claims
1, 2 and 3.
To put things in context, while post-Chicago models overcome the generalized Chicago critique by departing from the simultaneous bargaining assumption,
models in this price-discrimination theory of foreclosure do likewise by restricting
the set of available contracts parties can sign, usually to linear prices. Contract
incompleteness then, prevents the incumbent from appropriating the entire surplus in market A. As a result, exclusion arises as the incumbent has no other
way than to use market B to price discriminate and extract additional surplus
in market A. So exclusion in these models is not driven by a standard leverage
argument of extending monopoly power from market A to market B, but quite
the opposite, by the use of an otherwise competitive market to extract additional
surplus in a monopoly market.
While this incomplete surplus-extraction assumption may be reasonable when
the incumbent is dealing with a group of final consumers that have di↵erent
valuations for the two products, as in Nalebu↵ (2004, 2005), it has less support in wholesale markets where parties are free to sign contracts with nonlinear
schedules, which guarantees nearly full surplus extraction in the monopoly (i.e.,
non-contestable) segment. This notion of how wholesale markets operate is particularly stressed in Innes and Sexton (1994), but it is also adopted in many
other post-Chicago models (e.g., Aghion and Bolton 1987; Whinston 1990; Spector 2011; Asker and Bar-Isaac 2014, etc.). Our unit demand assumption captures
the same idea in the simplest possible way. Alternatively, we could have worked
with a downward sloping demand; but as we show in the online Appendix, none
of our results change as long as the incumbent is free to use nonlinear prices
(e.g., two-part tari↵s). Therefore, in order to make the case that a discount contract without unconditional transfers is anticompetitive, one necessarily requires
29 The
same is true with the exclusives in Calzolari and Denicolo (2015).
ex-ante commitment is also absent in the models of Ordover and Sha↵er (2013) and Chao and
Tan (2015). This is not entirely surprising because both models depart from the standard assumptions
in the literature of exclusive dealings and foreclosure. Ordover and Sha↵er (2013) is more of a predatory
pricing story with multiple periods, financially constrained firms and switching costs. Chao and Tan
(2015), on the other hand, is much closer to the literature on strategic investment in oligopoly. The
incumbent uses all-unit rebates to soften price competition, so while it is true that the entrant operates
below full capacity, his profits are actually higher, a fact at odds with all antitrust cases involving
foreclosure.
30 The
VOL. VOL NO. ISSUE
DISCOUNTS AS A BARRIER TO ENTRY
25
of incomplete surplus extraction in the non-contestable segment, an issue totally
absent in claims 1, 2 and 3.
Contrasting our theory to these bundling models also help us to highlight an
additional insight: the irrelevance of the relative size of the contestable demand,
, in our results. Unlike to what has been argued in some antitrust cases, a larger
non-contestable segment does not make exclusion with rebates any easier in any of
the post-Chicago settings, provided they also lack an ex-ante commitment. This
has important implications for antitrust policy. Any exclusionary theory that
builds upon the idea that the size of the contestable segment matters ( < 1) must
necessarily rely on a price-discrimination (and contract incompleteness) argument
of foreclosure.
VI.
Final Remarks
We opened the Introduction with the following question: How can the antitrust authority be sure that a rebate contract is not o↵ered to monopolize the
market? We have shown that a rebate contract under which an incumbent manufacturer promises a retail buyer a percentage o↵-list-price on all units purchased,
provided she buys exclusively from him, cannot be used to block an efficient
competitor. This non-exclusionary result is general enough as it applies to all
three post-Chicago settings where exclusion has been shown to emerge with exclusives; namely, the rent-shifting setting of uncertainty about the entrant’s cost,
the naked-exclusion setting of scale economies, and the downstream-competition
setting of Bertrand competitors.
Our goal has been to provide antitrust authorities with some logical consistency checks to better assess monopolization claims involving rebates. In that
regard, our analysis suggests that not only market conditions, such as degree of
dominance, extent of scale economies, or intensity of competition downstream,
are important, but also how these conditions interact with some specific features
of the discount contract in question. Thus, when it comes to revising some important recent cases, our results indicate that rebate contracts documented and
discussed in, for instance, EU Commission v. Michelin II and EU Commission
v. British Airways, could not have been used to block a more efficient competitor. We base our conclusion on the fact that these rebate schemes do not include
unconditional transfers from buyers to dominant suppliers, and were based on
o↵-list-price discounts. According to our theory, the reason these contracts are
observed in the first place cannot rest on exclusion, but presumably on the need
to support relationship-specific investments, exclude otherwise inefficient entry,
and/or stimulate retail e↵ort, to name a few.31
Our results do not allow us to be this conclusive for other cases, however. From
31 For example, in EU Commission v. Michelin II (see Decision 2002/405 of the EC for details of
the case and Case T-203/01 for the Court of First Instance’s upheld decision), in addition to the rebate
schemes, Michelin engaged in a series of transfers to “club members”, to contribute directly to the
investment and training of dealers and others of know-how, including priority access to training courses.
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reading ZF Meritor v. Eaton (US Court of Appeals for the Third Circuit Nos.
11-3301 and 11-3426, pp. 8-12), for example, it is not entirely clear whether the
rebates o↵ered by Eaton, the dominant supplier of heavy-duty (HD) truck transmissions in the North American market, to the four truck manufacturers buying
these transmissions, were o↵-list-price or lump-sum. According to our theory, the
distinction would be immaterial if truck manufacturers were selling their products to final consumers in completely separated markets, or if HD transmissions
were not that large a component in the production of a truck for the rebate to
a↵ect its retail pricing at the margin. If pricing is not a↵ected, and given that
the only transfers documented were from Eaton to the HD transmission buyers,
our theory would provide little support to Meritor’s anticompetitive claim. This
changes, however, when analyzing AMD v. Intel, and its European counterpart,
EU Commission v. Intel. Though no unconditional transfers from OEMs to Intel
are documented, the rebates used by Intel were apparently lump-sum, downstream competition was intense, and microprocessors are a considerable fraction
of the total cost of a computer, possibly a↵ecting its retail price. This makes
these rebate practices potentially exclusionary.
VOL. VOL NO. ISSUE
DISCOUNTS AS A BARRIER TO ENTRY
27
Appendix
A.
No-Contract Benchmark in Downstream Competition
If retailers are downstream competitors that buy from I and E in the spot,
then
LEMMA 1: There is an equilibrium in which I o↵ers the schedules {wI1 =
cI , TI1 = (1
)(v cI )} and {wI2 = v, TI2 = 0} while E o↵ers wE1 = wE2 = cI
with no fixed fee, which leads to retail prices p1,1 = p2,1 = v and p1, = p2, =
cI .
PROOF:
Suppose that I o↵ers the following pair of schedules to B1 and B2: {wI1 , TI1 =
(1
)(wI2 wI1 )} and wI2
wI1 , respectively. Regardless of E’s o↵ers, both
buyers are ready to sell I’s products because they anticipate that at least they
will be selling non-contestable units at price p1,1
= p2,1
= wI2 for nonnegative profits. If wI1
cE , E’s optimal reaction to I’s o↵ers is to set wE1 =
wE2 = wI1 (without loss of generality we can restrict attention to E o↵ering
linear prices), so retailers will be selling E’s units in the contestable segment at
price p1, = p2, = wI1 (obviously, if wI1 < cE + F/ , E’s optimal reaction is to
sell nothing). Consider now E’s o↵ers cI
wE2
wE1 (o↵ers above cI can be
ruled out since I can always undercut them with a pair of o↵ers {wI1 = cI , TI1 =
(1
)(wI2 wI1 ) + (wE1 cI )} and wI2
wI1 that induces B1 to charge
p1, = wE1 for the contestable units, so he can pocket an extra (wE1 cI )).
Since Bertrand competition ensures retail prices in the contestable segment be
equal to wE2 , I’s optimal reaction is to “abandon” the contestable segment and
set {wI1 = cI , TI1 = (1
)(v wI1 )} and wI2 = v. This latter together with E’s
optimal reaction completes the proof. ⌅
B.
Alternative Retail-Price Discrimination
We work here with the alternative assumption that retailers can set di↵erent
prices for I’s and E’s products, as opposed to di↵erent prices across segments of
demand. We will denote by pjI the retail price that retailer j = 1, 2 charges for
I’s products and by pjE  pjI the price she charges for E’s products. We explore
the e↵ects of this alternative price discrimination assumption on three results: (i)
the no-contract benchmark that is in Appendix A, (ii) Proposition 4, and (iii)
Proposition 5.
Retail Prices in the No-Contract Benchmark
LEMMA 2: There is an equilibrium for the no-contract benchmark game in
which I o↵ers the schedules {wI1 = cI , TI1 = (1
)(v cI )} to B1 and the
linear schedule {wI2 = v, TI2 = 0} to B2 while E charges the same linear schedule {wEj = ĉ, TEj = 0} to both retailers, where
ĉ ⌘ cI + (1
)(v
cI )
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MONTH YEAR
This leads to retail prices p1I = p2I = v and p1E = p2E = ĉ.
PROOF:
Given the schedules in the lemma, Bertrand retailers will set equilibrium prices
p1I = p2I = v and p1E = p2E = ĉ, which implies that B1 will carry both I’s and
E’s products, and B2 will carry only E’s products. The only deviation we need to
check is whether B1 would like to price I’s products not at v, but slightly below
ĉ, and serve the entire demand; clearly, he would not like to do that because
(1
)(v cI ) TI1 = ĉ cI TI1 . Notice that these retail prices result in
retailers’ profits being equal to ⇡B1 = ⇡B2 = 0 and suppliers’ profits being equal
to ⇡I = (1
)(v cI ) and ⇡E = (ĉ cE ) F . As for deviation incentives in the
upstream market (anticipating any e↵ect in the downstream market), notice first
there are no profitable deviations for E. Pricing above wEj = ĉ leads to no sales
because retailer B1 will sell I’s units slightly below any price above ĉ (which, in
turn, will be captured by I with an increase in TI1 ), and pricing below wEj = ĉ
only decreases his profit. I’s incentives to deviate are also null because any e↵ort
to compete for the last units would require him to lower the price slightly below
ĉ not only on those units, but also on the first 1
non-contestable units, which
we just saw is not profitable. ⌅
Exclusive Dealing Contracts
LEMMA 3: Under this alternative price discrimination assumption, there exist a
non-exclusionary equilibria, even when considering an exclusive dealing provision.
PROOF:
Consider the exclusives of Proposition 4. Suppose that I o↵ers B1 and B2
payments t1 and t2 , respectively, in exchange for exclusivity. If both retailers
accept the o↵ers, then, each gets exactly ti , no more (in the absence of entry, I
can extract all surplus ex-post). If, on the other hand, both reject their o↵ers,
then, each gets zero, the no-contract benchmark payo↵. The interesting case
arises when one retailer, say B2, accepts her o↵er and the other one rejects it.
We will first argue that the subgame with a free retailer (B1) and a captive
retailer (B2) accepts an equilibrium where I o↵ers
{wI1 = cI , TI1 = (1
)(v
cI )}
and
{wI2 = v , TI2 = 0}
To B1 and B2, respectively. While E o↵ers {wE1  cI , TE1 = (cI wE1 )} to
B1 (and nothing to B2 by construction). Given these o↵ers, the equilibrium
downstream results in prices pI1 = pI2 = pE1 = v, B2 making no sales, and B1
selling 1
and units of I’s and E’s products. To show that this is indeed
an equilibrium, notice, on the one hand, that since B1 is just indi↵erent between
purchasing 1
and units from I and E, and purchasing one unit from I, any
attempt by E of extracting more rents from B1 will lead him to earn zero profits.
Hence, E has no incentives to change his o↵er. On the other hand, conditional
on the contract o↵ered to B2, I has no incentives to change his o↵er to B1, since
any attempt to undercut E’s o↵er will give him less than (1
)(v cI ), as
VOL. VOL NO. ISSUE
DISCOUNTS AS A BARRIER TO ENTRY
29
B1 is already getting (v cI ) from selling E’s units. Hence, the only possible
deviation left to consider involves the following simultaneous deviations in B1
and B2’s o↵ers: {wI1 = v , TI1 = 0} and {wI2 < v , TI2 0}.
Computing I’s payo↵ from such simultaneous deviations is not straightforward
because there does not exist an equilibrium in pure strategies in the downstream
market. We can nevertheless show that I is not strictly better o↵ following
such deviation. Assume that I’s o↵er to B1 is still {wI1 = v , TI1 = 0} but that
I’s o↵er to B2 is not longer {wI2 < v , TI2 0} because I and B2 are now
vertically integrated. Since E o↵er to B1 is fixed, there are no benefits from
strategic separation à la Bonanno and Vickers (1988), so from I’s perspective the
vertical-integration outcome is weakly preferred to a two-part tari↵ o↵er to an
otherwise captive retailer. Consider then, the downstream pricing game between
the vertically integrated firm I B2, who can serve the entire market at marginal
cost cI , and retailer B1, who sells at most units at cost wE1 each. As formally
shown in the online Appendix, the equilibrium is in mixed strategies with the
vertically-integrated firm getting (1 )(v cI ), its residual monopoly profit. More
importantly, this is the payo↵ I gets in our proposed pure-strategy equilibrium,
hence, the simultaneous deviation is not a profitable.
From the above, it follows immediately that there exists a non-exclusionary
equilibrium. Since equilibrium payo↵s in this ensuing subgame (i.e., omitting
the exclusivity payments) are ⇡I = (1
)(v cI ), ⇡E = (cI cE ) F > 0,
⇡B1 = (v cI ) and ⇡B2 = 0, exclusion cannot arise as it requires I to o↵er each
retailer ti > (v cI ), which leads to an overall payo↵ of v cI t1 t2 which
is less than ⇡IN C = (1
)(v cI ). ⌅
It is important to highlight that the above argument does not allow us to
conclude that every equilibrium is non-exclusionary. The key to exclusion with
an exclusive dealing contract is retailers’ equilibrium profits in the subgame with
one free retailer. Without completely characterizing such equilibrium set, it is not
possible to discard an equilibrium (in mixed strategies) where I gets (1 )(v cI ),
but the free retailer, say B1, gets ⇡B1 = ⇡
¯B < (v cI )/2. In that case, exclusion
would arise. Both retailers would be ready to accept o↵ers t1 = t2 = ⇡
¯B in
exchange for the exclusivity, with I pocketing v cI 2¯
⇡B > (1 )(v cI ) = ⇡IN C .
Off-List-Price Rebates
LEMMA 4: Under this price discrimination assumption, it is never profitable
for I to o↵er a pair of discriminatory rebate contracts (ri , Ri ) for i = 1, 2, to
deter E’s entry.
PROOF:
Suppose that there exists an exclusionary equilibrium with exclusives under this
alternative price discrimination assumption (see previous section), we now prove
that even under this scenario rebates cannot be anticompetitive, which implies
that our result in Proposition 5 remains unchanged under this alternative price
30
THE AMERICAN ECONOMIC REVIEW
MONTH YEAR
discrimination assumption. Following the proof of Proposition 5, suppose that I
o↵ers both retailers the same rebate (r, R), which both accept. To enter, E will
approach one of them, say B1, if at all, with an o↵er of wE1 for units. Therefore,
an additional condition to block E’s entry is that B2 must find it profitable to
price I’s products slightly below wE1 , even if wE1 is as low as cE + F/ . B2’s
payo↵ of doing so is cE + F/
r + R, while of not doing so is (1
)R. This
latter is the result of selling 1
units at price r, which is the marginal cost faced
by B1 when selling I’s units. Thus, the new entry-deterrence condition reduces
to r
R  cE + F/ . But since I’s profit is
⇡IR = r
R
cI < r
R
cI < cE + F/
cI < 0
exclusion will not arise. The case of discriminatory rebates is straightforward.
⌅
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Online Appendix for:
Discounts as a Barrier to Entry
Enrique Ide, Juan-Pablo Montero and Nicolás Figueroa⇤
November 9, 2015
Abstract
There are seven sections in this online Appendix (all references are in the paper). The
first section considers rebate contracts with unconditional transfer in the rent-shifting setup
of Aghion and Bolton (1987). The section also considers the case of unconditional transfers
subject to an exogenous limit. Section 2 provides details of the no-contract payo↵s in the
naked-exclusion setting of Rasmusen et al. (1990) and Segal and Whinston (2000). Section
3 considers rebate contracts with unconditional transfers in the naked-exclusion setup. It
also explores how unconditional transfers can be used to extract rents rather than to exclude
efficient entrants. Section 4 extends the naked-exclusion analysis of rebates to a world of
unknown entry costs, both with and without unconditional transfers. Section 5 considers
rebates with unconditional transfers in the downstream-competition setup of Simpson and
Wickelgren (2007) and Asker and Bar-Isaac (2014). As in the previous section, it also covers
the use of unconditional transfers to extract rents. Section 6 analyze the downstream pricing
game between a retailer carrying only E’s products, and a vertically integrated firm selling
I’s product, when discrimination between di↵erent segments of demand is not possible.
Finally, Section 7 presents a model of a single buyer with a downward-sloping demand.
1
Unconditional transfers in rent shifting
We first show how unconditional transfers restore the full anticompetitive potential of rebate
contracts in the rent-shifting setup of Aghion and Bolton (1987), and then, look at how that
potential gets diminished as we impose an exogenous limit on those transfers.
⇤
Ide: Stanford GSB ([email protected]); Montero: PUC-Chile Economics ([email protected]); Figueroa: PUC-
Chile Economics ([email protected]).
1
1.1
Unconditional transfers in rebate contracts
A fundamental issue raised by Proposition 1 in the paper is what explains that I can write an
anticompetitive exclusive but not an anticompetitive rebate. Both contracts are signed on date
1 before E shows up, so the di↵erence lies in when exclusivity is committed. The exclusive
does so ex-ante by committing B on date 1 to pay a penalty in case the exclusivity is breached.
The rebate, by contrast, does so ex-post by rewarding B only after the exclusivity is observed.
Another way to put it is that under an exclusive, B does not benefit at all from the downward
pressure that penalty D exerts on E’s o↵er because the penalty goes directly to I, whereas under
a rebate, B is the one that directly benefits from E’s low o↵er (i.e., lower than cI ), and nothing
in the contract commits B to share all or part of this benefit with I. This, however, suggests
a way for I to amend the rebate contract (r, R) and restore its anticompetitive potential:
Lemma 1. I can implement the anticompetitive outcome of Aghion-Bolton (see expression (3)
in the paper) with the rebate contract (r, R, Z), where Z is an unconditional transfer from B to
I that is agreed at the time the rebate contract is signed on date 1. I has flexibility to write this
anticompetitive rebate from the one with the lowest unconditional transfer (r = v, R = (v c̃E ),
Z = (v
c̃E )
N C ), to the one with the highest (r = c̃ , R = 0, Z = v
⇡B
E
Proof. Since Z is unconditional, I’s problem is to maximize (x ⌘ r
E⇡I (r, R, Z) = Z + (1
subject to r  v and v
r+R
binding, replacing R = (r
Z
)(r
cI ) + (x
cI )[1
c̃E
N C ).
⇡B
R/ )
G(x)]
(1)
N C . Since this latter participation constraint will be
⇡B
x) in it and substituting Z = v
(1
)r
x
N C in (1) leads
⇡B
to
E⇡I (r, R, Z) = v
cI
NC
⇡B
+ (cI
x)G(x) = (1
)(v
cI ) + (cI
x)G(x)
and solving for x yields (3) in the paper. The rest of the proof follows from the fact that Z and
r  v are freely chosen from any combination that satisfies Z + (1
)r = v
c̃E
N C .⌅
⇡B
By committing B to make an unconditional transfer at the time the contract is signed, I
can now o↵er an exclusionary rebate with large rewards ex-post, i.e., r
R/ < cI , without
sacrificing profits, i.e., E⇡I (r, R, Z) > ⇡IN C .1 By trading ex-post rewards for ex-ante transfers,
I can now extract rents from those entrants that decide to enter (cE  c̃E ) while leaving B with
1
In a previous version of the paper, we show this is the only way to amend the (r, R) contract to obtain the
Aghion-Bolton outcome.
2
enough to sign the contract. As long as I and B have some uncertainty about cE , extracting
rents from E will not be perfect and some exclusion will necessarily occur.2
Whether the transfers required in Lemma 1 are feasible to implement in practice is not
obvious, which may explain why we rarely see them, if at all. Leaving aside cases that do not
require these transfers (see Section V in the paper), good reasons may restrict their use, such
as limited liability and asymmetric information. Since I has some flexibility to design these
rebates, as Lemma 1 indicates, we now show how the optimal contract (r, R, Z) varies as we
(exogenously) restrict the value of Z (see also Ide and Montero 2015).
1.2
Restricted unconditional transfers
In this section we look at the design of rebate contracts but under the assumption that unconditional transfers are subject to some exogenous limit. I’s problem can be written as (recall
that F = 0)
max E⇡I = (1
)(v
r,x,Z
cI ) + (x
cI )[1
x
NC
⇡B
G(x)] + Z
subject to r  v, Z  Z̄(✓) and
v
(1
)r
Z
where ✓ 2 [0, +1) is some parameter that captures how problematic is for I to demand an
unconditional transfer from B on date 1. For instance, ✓ could represent some limited liability
faced by retail buyers, or the presence of asymmetric information (Ide and Montero 2015), to
name a few. For generality we allow Z̄(✓) to take any value, even negative, which would make
it a slotting allowance. We assume that the larger the ✓ the more difficult is for I to demand
an unconditional transfer so that lim✓#0 Z̄(✓) = +1, lim✓"1 Z̄(✓) =
1, and Z 0 (✓) < 0.
Let us first define
where c̃E = cI
✓¯ :
¯ = (v
Z̄(✓)
c̃E )
NC
⇡B
✓¯ :
¯ = (v
Z̄(✓)
cE )
NC
⇡B
G(c̃E )/g(c̃E ) < cI , and cE = cI + [1
G(cE )]/g(cE ) 2 (cI , v).
It can be shown that the optimal rebate contract (r⇤ , x⇤ , Z ⇤ ) will depend on ✓ as follows:
2
This partial-exclusion result connects nicely with a recent paper by Calzolari and Denicolo (2015). In their
model, partial exclusion arises because it helps the incumbent to leave less information rents with privately
informed buyers. Here, it arises because it helps the incumbent to extract more entry rents.
3
1. If ✓  ✓¯ (i.e.
(v
c̃E )
N C  Z̄(✓)), then r ⇤  v, x⇤ = c̃ , Z ⇤  Z̄(✓) such that
⇡B
E
(1
2. If ✓¯ < ✓ < ✓¯ (i.e.
)r⇤ + Z ⇤ = v
c̃E
NC
⇡B
N C < Z̄(✓) < (v
N C ), then
(v cE ) ⇡B
c̃E ) ⇡B
⇢
✓
NC ◆
Z̄(✓) + ⇡B
⇤
⇤
r = v, x = v
, Z ⇤ = Z̄(✓)
⇣ ⌘
¯ ✓¯ given by Z̄(✓)
ˆ = (v cI ) ⇡ N C > 0
Notice moreover that there exists a unique ✓ˆ 2 ✓,
B
such that for all ✓
3. If ✓¯
✓ˆ the contract is not anticompetitive (x⇤
✓ (i.e. Z̄(✓)  (v
cE )
cI ).
N C ), then r ⇤ = v, x⇤ = c , Z ⇤ = Z̄(✓) .
⇡B
E
To demonstrate that these are indeed the optimal rebate contracts, write first the Langrangean of I’s problem
L = (1
)(v
cI ) + (x
cI )[1
G(x)] + Z + µ1 (v
(1
)r
x
Z
+ µ2 (Z̄(✓)
NC
⇡B
)
Z) + µ3 (v
r)
to obtain the first order conditions
@L
= (1
)(1 µ1 ) µ3 = 0
@r
@L
= [1 G(x) g(x)(x cI ) µ1 ] = 0
@x
@L
= 1 µ1 µ2 = 0
@Z
@L
NC
= v (1
)r
x Z ⇡B
0
and
@µ1
@L
= Z̄(✓) Z 0
and
@µ2
@L
=v r 0
and
@µ3
with µ1 , µ2 , µ3
(2)
(3)
(4)
@L
=0
@µ1
@L
µ2
=0
@µ2
@L
µ3
=0
@µ3
µ1
(5)
(6)
(7)
0. Depending on the value of ✓, we can now characterize three candidates for
the optimum.
1. Candidate 1: µ⇤1 > 0, and µ⇤2 = µ⇤3 = 0
Either from (2) or (3) we get µ⇤1 = 1 > 0. Moreover from (3):
cI ) = 0 =) x⇤ = c̃E
G(x) + g(x)(x
4
From (6) and (7) we get r⇤  v and Z ⇤  Z̄(✓). And from (5) we have
)r⇤ + Z ⇤ = v
(1
c̃E
NC
⇡B
Finally, since r⇤  v, this latter implies that
Z⇤ = v
NC
⇡B
c̃E
(1
)r⇤
(v
c̃E )
NC
⇡B
and since Z ⇤  Z̄(✓), this requires
Z̄(✓)
(v
c̃E )
NC
⇡B
2. Candidate 2: µ⇤1 , µ⇤2 , µ⇤3 > 0
(5), (6) and (7) imply that Z ⇤ = Z̄(✓), r⇤ = v, and
✓
NC ◆
Z̄(✓) + ⇡B
⇤
x =v
Moreover (2), (3) and (4) give µ⇤2 = 1
µ⇤1 = 1
µ⇤1 , µ⇤3 = (1
G(x⇤ )
g(x⇤ )(x⇤
(8)
)(1
µ⇤1 ) and
cI )
Finally, for this to be indeed the optimum we require that µ⇤1 2 (0, 1), but this is the case
whenever c̃E < x⇤ < cE . Combining the latter with (8) we get that
(v
cE )
NC
⇡B
< Z̄(✓) < (v
c̃E )
NC
⇡B
3. Candidate 3: µ⇤1 = 0, and µ⇤2 , µ⇤3 > 0
From (2) and (4) we get µ⇤3 = (1
1
) > 0, and µ⇤2 = 1 > 0. Moreover from (3)
G(x)
g(x)(x
cI ) = 0 =) x⇤ = cE
and from (6) and (7) we get r⇤ = v and Z ⇤ = Z̄(✓). Finally from (5) we get
v
(1
)r
x
Z
NC
⇡B
0 =) Z̄(✓)  (v
cE )
NC
⇡B
Depending on the value of Z̄(✓), one of these three candidates will characterize the optimum,
as stated in the lemma. The first candidate applies when the transfer restriction is not binding
¯ so I can achieve the Aghion-Bolton outcome. In the region where ✓ is above ✓¯ the
(✓  ✓),
transfer restriction becomes active and I must distort the e↵ective price upwards, leading to a
⇣ ⌘
¯ ✓¯ the rebate contract is not longer
less anticompetitive outcome. And when ✓ reaches ✓ˆ 2 ✓,
5
anticompetitive. Notice though, that at ✓ˆ the scheme still entails a positive, but substantially
small, unconditional transfer. A further tightening of the transfer restriction, for example to
Z̄(✓) = 0, forces I to raise the e↵ective price even further, which is when we arrive to our result
that (r, R) contracts in the rent-shifting model of Aghion-Bolton cannot be anticompetitive (see
Proposition 1 in the paper).
2
No-contract payo↵s in the naked-exclusion setup
In the absence of contracts, I and E compete in the spot market as in the rent-shifting setup.
In addition, it can be established that if I does not contract with B1 and B2 on date 1, then E
is indi↵erent between making o↵ers to either of the two buyers or going directly to the spot on
date 4. In either case, the no-contract payo↵s are: ⇡IN C = (1
N C = (v
and ⇡Bi
N C = (c
)(v cI ), ⇡E
I
cE ) F ,
cI )/2 for i = 1, 2.
To formally show that E cannot do better, notice that the maximum available surplus for a
“coalition” of E and the two buyers (EBB) is VEBB = (v
cE )
F . The question now is how
that surplus gets divided between E and the buyers. In particular, can a “sub-coalition” of E
and one of the buyers appropriate all of it by exploiting the other buyer? Suppose on date 2 E
makes simultaneous public price o↵ers wE1 and wE2 to both buyers B1 and B2, respectively
(sequential o↵ers do not change the result). There are three relevant cases to consider. The
first is when both o↵ers are slightly below cI . It is clearly an equilibrium for both buyers to
accept because E will enter in any case, which secures each buyer at least (v
cI )/2. The
second is when both o↵ers are above cI , in which case the equilibrium is for each of the two
buyers to reject, regardless of what the other does, because they know they can always get
(v
cI )/2. The most interesting case is when E o↵ers the pair wE1 < cI < wE2 such that
⇡E = (wE1
cE )/2 + (wE2
cE )/2
N C = (c
F > ⇡E
I
cE )
F , which would be E’s payo↵
in case both buyers accept, and
⇡E = (wE1
cE )/2 + (cI
cE )/2
F < 0,
which would be E’s payo↵ in case only B1 accepts but he enters anyway. If E could commit
not to renegotiate with B1, then B2 would, in equilibrium, accept any o↵er wE2  v
✏ to
make sure E enters. But the problem is that B2 anticipates that if she rejects, E and B1 will
renegotiate the terms of B1’s o↵er to make sure E enters, which in turn makes B2’s rejection a
best response. Anticipating buyers’ equilibrium responses, E gains nothing from making o↵ers
6
on date 2.3
3
Unconditional transfers in naked exclusion
We now considers rebate contracts with unconditional transfers in the naked-exclusion setup
of Rasmusen et al (1990) and Segal and Whinston (2000b). We first study how unconditional
transfers can be use to restore the full anticompetitive potential of rebates (i.e., can replicate
the work of exclusives) and then how they can also be used to extract rents.
3.1
Using unconditional transfers for exclusion
Unlike in Rasmusen et al (1990) and in Segal and Whinston (2000b), the reason I fails to
implement a divide-and-conquer strategy with rebates is because the unlucky buyer (B2) can
no longer be used to finance the rebates going to the lucky buyer (B1). Given that rebates do
not lock up buyers ex-ante, E has always the opportunity on date 2 to make counter o↵ers to
fight o↵ exclusion. Moreover, because B2 anticipates that she will be fully exploited by I, she
is ready to give up almost her entire surplus to E, forcing I to o↵er even larger rewards to lock
up B1, so large that they are not profitable. In other words, if I tries to exploit B2 in the first
place E will do likewise, so the same (ex-post) rents coming from B2 that I is using to pay for
the rebate to B1 are also used by E to induce B1 to switch. As with Lemma 1, however, there
is a way for I to go around this funding problem and deter E’s entry:
Proposition 2.
Lemma 3. In the naked-exclusion setup of multiple buyers and scale economies, I can profitably
deter E’s entry with a pair of discriminatory rebate contracts (ri , Ri , Zi ) for i = 1, 2, where Zi
is an unconditional transfer from Bi to I that is agreed at the time the rebate contract is signed
on date 1.
Proof. It is easy to see that I can replicate the exclusionary outcome of Proposition 2 in the
paper with no o↵er to B2 and a rebate o↵er (r1  v, R1 , Z1 ) to B1, the lucky buyer, that
satisfies both
(v
3
r1 + R1 )/2
Z1
NC
⇡B1
(9)
Note that if in fact E could commit not to renegotiate with B1, then buyers outside options would be smaller,
on average, which would make exclusion a bit cheaper for I.
7
and “the no-switching” condition (7) in the paper for wE1 = 2F/ + 2cE
v. To foreclose E’s
N C , I will set (9) to equality.⌅
entry at minimum cost, ⇡B1
Again, an unconditional transfer from the lucky buyer is what allows the incumbent to o↵er
very competitive rewards ex-post, making it impossible for E to persuade that buyer to switch.
But if such large unconditional transfers are feasible to implement, one can go further and ask
if I can do better than just deterring E’s entry.
3.2
Using unconditional transfers for rent extraction
We now establish the form of the most profitable rebate contracts I can o↵er when unconditional
transfers have no restrictions. Consistent with the exclusives in Innes and Sexton (1994) and
Spector (2011, section 4) that include breach penalties as vehicles to transfer rents, I can write a
pair of efficient discriminatory (ri , Ri , Zi ) contracts and pocket the entire social surplus, except
N C + ✏. By locking up just one retailer, B1, I can fully exploit B2 while simultaneously
for ⇡B1
extracting all of E’s efficiency rents. This non-exclusionary result, however, is clearly not robust.
For instance, rent extraction would be incomplete when cE is unknown, in which case, again,
there will be exclusion of some efficient entrants even when (ri , Ri , Zi ) contracts are at hand.
We leave that case for the next section.
When I is free to charge unconditional transfers to buyers on date 1, he will optimally let E
to come in with rebate o↵ers (r1 , x1 , Z1 ) and (r2 , x2 , Z2 ) that share the following characteristics:
(i) Z1 = ((1
)(v
r1 ) + (cI
cE )/2 + (x2
cE )/2
x1 ))/2
✏, (ii) Z2 = (v
F = 0, (iv) ri  v, (v) xi
(1
)r2
x2 )/2, (iii) (x1
cE , and (vi) x1 < 2cE
cI + 2F/ .
Note first that independently of whether Bj accepts or not, and irrespective of E’s entry,
if Bi accepts I’s o↵er she gets v
v
(1
)ri
xi
ri + R i = v
(1
)ri
xi ex-post, and therefore,
Zi ex-ante. If, however, Bi rejects I’s o↵er, her payo↵ does depend on
whether Bj accepts or not. If both buyers reject, we are in the no-contract benchmark where
NC =
each gets ⇡Bi
(v
cI )/2. But if Bi rejects and Bj accepts, her payo↵ will ultimately
depend on the o↵ers that E will present to each of them on date 2. Since xi
cE for both i and
j, E will find it optimal to slightly undercut the terms of the rebate contract of the “captive”
buyer (Bj in this case) with a price o↵er slightly below xi . As for E’s o↵er to the free buyer,
notice that Bj’s outside option depends on xi . In fact, if (xi
Bj’s outside option is (v
enter. But if (xi
cE )/2 + (cI
cE )/2
F
0,
cI )/2 because if she rejects E’s o↵er, E will still find it optimal to
cE )/2 + (cI
cE )/2
F < 0, Bj’s outside option is zero, because if she
8
rejects, E will not enter.
These outside options can be summarized in the following payo↵ matrix:
B1 \ B2
Accept
Accept
(v
Reject
Reject
cI )/2 + ✏, 0
(v
(v
cI )/2, 0
(v
cI )/2 + ✏, 0
cI )/2, (v
cI )/2
Notice that the reason B2 gets zero if she rejects and B1 accepts (upper-right corner) is because
(vi) leads to (x1
cE )/2 + (cI
cE )/2
F < 0. Similarly, the reason B1 gets (v
cI )/2 if
she rejects but B2 does not (lower-left corner) is because
(x2
cE ) + (cI
cE )/2
F = (cI
cE )/2
(x1
cE )/2 > 0
where the equality follows from (iii) and the inequality from x1 < cI , which follows directly from
the efficient-entry condition and (vi). Hence, in equilibrium both buyers accept the contract
(if B2 rejects makes no di↵erence), E enters and slightly undercuts both terms. Equilibrium
payo↵s, as of date 1, are ⇡I = v
cI + (cI
cE )
F
(v
cI )/2
✏, ⇡B1 = (v
cI )/2 + ✏,
N C , he cannot improve upon these
⇡B2 = 0 and ⇡E = 0. Since I must leave B1 with at least ⇡Bi
rebate contracts.
Finally, notice that characteristics (i)-(vi) accommodate to di↵erent rebate contracts; for
example, the pair (r1 = v, x1 = cE , Z1 = (cI
(v
cE )/2
cE )/2
✏) and (r2 = v, x2 = cE + 2F/ , Z2 =
F ). Note that Z2 < 0, an slotting allowance, but it is easy to construct another
example with Z2 > 0 by simply lowering the list price r2 accordingly. In any case, with any of
N C + ✏, which is what he needs to give up
these contracts I pockets all the social surplus but ⇡B1
in order to induce B1 to sign.
4
Unknown entry costs in naked exclusion
In this section we look at the work of rebate contracts in a naked-exclusion setup, both with and
without unconditional transfers, when cE is unknown on date 1; very much as in a rent-shifting
setting.
4.1
Rebates without unconditional transfers
To simplify matters assume that cE is distributed according to the cumulative function G(·)
over [v
2F/ , cI
F/ ], so that
(v
cE )/2
9
F  0 and 0 
(cI
cE )
F for any
realization of cE . That is, irrespective of cE : (i) E needs to serve both buyers to find it
profitable to enter the market, and (ii) entry is always efficient. This interval is well defined
because v
2F/ < cI
F/ , or (v
cI ) < F .
Consider the scenario where I o↵ers just one rebate contract, say (r1 , x1 ) to B1. E will find
it profitable to enter whenever
(x1
cE )/2 + (v
cE )/2
F
⌘
0 =)
v + x1
2
F
cE
as B2 gets charged v for each unit. Therefore, the probability of entry is G( ). Anticipating
this, I’s expected profit is
✓ ◆
✓ ◆
1
1
E⇡I =
G( )(1
)(r1 + v 2cI ) +
[1 G( )](r1 (1
2
2
✓
◆
✓

◆
v + r1
F
= (1
)
cI + [1 G( )]
cI
2
But I’s no-contract payo↵ is ⇡IN C = (1
✓
NC
E⇡I ⇡I = (1
) v
)(v
cI ), so
◆
v + r1
+ [1
2
G( )]
✓
) + x1 + v

cI
F
Since r1  v, a necessary condition for a rebate contract to be profitable (E⇡I
that
> cI
◆
⇡IN C
2cI )
0) is
F/ , that is, that entry must be inefficient.
Finally it is easy to see that I cannot improve upon o↵ering a second contract. If I goes on
and o↵ers B2 the contract (r2 , x2 ), in addition to the contract (r1 , x1 ) to B1, then he will end
up extracting (1
)r2 and r2
R2 from B2 in the event of entry and no-entry, respectively,
which is less than what he is currently extracting, (1
entry cuto↵ 0 ⌘ (x1 + x2 )/2
)v and v, respectively. Hence the new
F/ will have to be set even higher for I to be able to profit
from the contract.
4.2
Rebates with unconditional transfers
Assume again that cE distributes according to G(·) over [v
2F/ , cI
F/ ]. Suppose I o↵ers
only one contract, say (r1 , x1 , Z1 ) to B1. The main di↵erence with the previous section is
that now I can use the unconditional transfer Z1 to extract rents from B1. Since buyers can
communicate and coordinate their actions, I must leave B1with no less than her no-contract
payo↵, that is, (v
(1
)r1
x1 )/2
N C = (v
⇡B1
Z1
cI )/2. Since this constraint is
binding, we can rewrite I’s expected payo↵ as
E⇡I
= (2(v
= (v
cI )
cI )
G( )[v + x1
✓

G( )
cI
10
2cI ]) /2
◆
F
(v
(v
cI )/2
cI )/2
(10)
Di↵erentiating (10) with respect to
yields
✓

dE⇡I
0
=G( )
cI
d
which implies that
⇤
entrants, i.e., cE 2 [
⇤, c
< cI
I
F
◆
+ G( ) = 0
(11)
F/ . This optimal rebate contract blocks moderately efficient
F/ ].
This result follows the Aghion-Bolton logic for a single buyer. The only di↵erence is that
now the IB1 coalition acts as a monopoly over the EB2 coalition, since EB2 needs B1 to
enter. Because buyers cannot sign binding commitments ex-ante, I can get B1 to sign for not
N C . Notice again that I cannot do better by o↵ering a second
less than her outside option ⇡B1
NC.
contract since in any case must leave one of the buyers, say Bi, with no less than ⇡Bi
5
Unconditional transfers in downstream competition
We now considers rebate contracts with unconditional transfers in the downstream-competition
setup of Simpson and Wickelgren (2007) and Asker and Bar-Isaac (2014). We first study how
unconditional transfers can be use to restore the full anticompetitive potential of rebates and
then how they can also be used to extract rents.
5.1
Using unconditional transfers for exclusion
Following the results in the rent-shifting and naked-exclusion settings (Lemmas 1 and 2 above),
the obvious question is whether exclusion can again be restored by introducing unconditional
transfers. In rent-shifting and naked-exclusion, unconditional transfers were less a problem for
retailers because they could perfectly anticipate the monopoly rents available to pay for these
transfer ex-post. Here it is not that simple, because Bertrand competition tends to dissipate
those rents, so a retailer anticipating zero profits ex-post will not be willing to commit to any
payment up-front. As the next proposition establishes, however, I can get around this problem
with the proper contract design.
Lemma 4. In the downstream-competition setup of Bertrand competitors, I can profitably deter
E’s entry with a pair of discriminatory rebate contracts (ri , Ri , Zi ) for i = 1, 2.
Proof. This proof follows directly from the discussion at the end of the proof of Proposition
5 in the paper. Suppose on date 1 that I approaches B1 with a rebate o↵er (r1 , R1 , Z1 ) with
an e↵ective price r1
R1 slightly below E’s break-even price cE + F/ . At the same time, he
11
o↵ers B2 a uniform price in the form of a rebate o↵er (r2 , R2 , Z2 ) with R2 = 0 (Z1 and Z2
are defined shortly). It is clear that if B1 accepts the rebate and conforms to the exclusivity,
E will not enter because he cannot use B2 to compete with B1’s marginal cost of r1
R1 in
the contestable portion of the retail market (notice, as discussed in the proof of Proposition 7,
that the same applies if I o↵ers nothing to B2 on date 1). On the other hand, since retailers’
outside options are zero, by setting r1 = r2 = v, Z1 = R1
✏, and Z2 =
✏, I can induce
both buyers to sign for as little as ✏ ! 0. B2 is willing to sign because her outside option is
zero. And having observed B2 doing so (or at least the o↵er she received), B1 is also willing to
sign and commit to the unconditional transfer Z1 because she understands that B2’s contract
commits I to sell units to B2 at price r2 = v, nothing less, which is what allows B1 to pocket
R1 = Z1 +✏ ex-post. B1 also understands that any later e↵ort by I to renegotiate B2’s contract
to expropriate part or all of these ex-post rents is fruitless because she can always sell I’s units
for as low as r1
ex-ante payo↵ of v
R1 . Finally, it is easy to compute that with these rebates, I obtains an
cI
2✏ > (1
)(v
cI ) = ⇡IN C , which is exactly what he obtains with
the exclusives of Proposition 4.⌅
Notice that, due to the intense downstream competition, I just needs to make one of the
retailers (B1) aggressive enough in the contestable segment for E not to enter. But since this
benefits B1 greatly, as she can now charge v for all units, I uses the unconditional transfer
Z1 to extract this benefit while maintaining the large rebate R1 that prevents E’s entry in the
first place. In a way, this is similar to what happens in naked exclusion —where exclusion did
not require I to lock up both retailers— but for a very di↵erent reason. In naked exclusion
it was due to scale economies, while here it is due to the intense retail competition. Also, as
in rent shifting and naked exclusion, the introduction of unconditional transfers is what allows
I to o↵er large rewards ex-post; but unlike there, I needs here to purposefully soften retail
competition to make sure retailers can recover those unconditional payments later. The way to
do this is by o↵ering one retailer (B2) a rebate contract under such unfavorable conditions that
she cannot compete; a contract that she is nevertheless willing to take for virtually nothing,
precisely because of the intense retail competition.
5.2
Using unconditional transfers for rent extraction
In this section we look at I’s problem when unconditional transfers are not used to foreclose
E’s entry but to extract efficiency rents from him. Suppose B1 is the retailer approached
12
by E to enter the market. Consider the following pair of rebate contracts that I will o↵er
both buyers on date 1, respectively (r1 , R1 , Z1 ) and (r2 , R2 , Z2 , q2  q̄), where: (i) r1 = v,
(ii) r1
Z2 = (1
R1 = cE + F/ + 2✏, (iii) Z1 =
)R2
✏, and (vii) q2  1
R1
✏, (iv) r2 = v, (v) r2
, i.e., B2 can at most buy 1
R2 < r1
R1 , (vi)
from I.
To see why these pair of contracts allow I to extract E’s entry rents, notice first that the
worst entry scenario for E is to approach B1, which according to (v) is the buyer with the
higher e↵ective price r
R, with a price o↵er that slightly undercuts (ii), that is
wE1 = cE + F/ + ✏
This leaves E with virtually nothing as ✏ ! 0. B1 accepts this o↵er, since it is ✏ more attractive
than I’s current o↵er (ii), and becomes the only one selling to the contestable portion. This
latter is possible because the quantity limit (vii) prevents B2 from selling to the contestable
portion given that she is already selling 1
units to the non-contestable portion (note B1
cannot compete in the non-contestable portion because (iv) and the fact that she gave up the
rebate R1 once she accepted E’s o↵er). Thus, B1 pockets an ex-post rent equal to (v
wE1 ),
so her profit as of date 1, i.e., her profit net of the unconditional transfer (iii), is ✏(1+ ). On the
other hand, B2 sticks to the exclusivity selling at v to the non-contestable portion, obtaining
a profit net of the up-front (vi) equal to ✏. Hence, I gets the full social surplus using the list
prices in (i) and (iv) and letting ✏ ! 0.
Notice that this works for I as long as he can impose the quantity limit (vii) in B2’s contract,
which seems odd; if anything, rebate contracts typically include deeper discounts as buyers sell
more, as opposed to the radical price increase suggested by (vii). The problem is that in the
absence of that limit there is no way for I to extract rents from E because B1 and B2 will
always compete for the contestable units, preventing the retail price from going all the way to
v.
6
Downstream pricing: retailer vs. vertically integrated firm
In this section we prove the claim made in the Appendix of the paper, that when price discrimination between contestable and non-contestable consumers in the downstream market is not
possible, equilibrium profits of a vertically integrated firm selling I’s products (denoted BI ) are
always equal to (1
)(v
cI ), when facing a retailer carrying only E’s products (denoted BE )
that has marginal cost/wholesale price wE  cI .
13
First, it is clear that no pure strategy equilibrium exists. Indeed, let pE and pI be the
prices set by BE and BI respectively. If pE < pI , then BE has incentives to rise its price; if
wE < pI < pE , BE has the incentive to undercut I’s price; if pI  pE = wE , BI prefers to sell
units a price v, given that wE  cI < ĉ ⌘ cI + (1
1
)(v
cI ). And if pI = pE = v then
BI has incentives to undercut BE ’s price. Hence all possible candidates for a pure-strategy
equilibrium are discarded. By a similar argument, it is also easy to see that both firms must be
randomizing in equilibrium: if BI plays pI with probability 1 (w.p.1), then BE has incentives
to play pE = pI
✏ w.p.1, but if so, then BI undercuts E if pI > ĉ; or raises its price to v if
pI < ĉ.
Suppose then that BI and BE randomize according to the pdf (cdf) hI (·) and hE (·) (HI (·)
and HE (·)) respectively, denote
i
the support of Hi (·), and let ri ⌘ inf
i
and Ri ⌘ sup
i
(Ri  v, for i = I, E obviously). It is clear that in any equilibria, equilibrium profits must
⇤
satisfy ⇡B
I
(1
)(v
⇤ > 0. The first inequality comes from the fact that B can
cI ) and ⇡B
I
E
always charge pI = v and sell at least 1
units. The second, from the fact that BE can always
charge wE < pE < ĉ and prevent being undercut by BI , earning therefore (pE
⇤ = (1
We now claim that ⇡B
I
(1
)(v
But since (1
I.
⇤ >
cI ). Suppose not. Then it must be that ⇡B
I
cI ). This in turn means that:
⇤
⇡B
⌘ ⇡I (pI ) = (1
I
pI 2
)(v
wE ) > 0.
)(pI
)(pI
cI ) + [1
cI )  (1
HE (pI )] (pI
)(vI
cI ) > (1
)(v
cI ) , for all pI 2
cI ), this is only possible if 1
I
(12)
HE (pI ) > 0 for all
This latter is only possible if RI < RE or if RI = RE and hE (·) has an atom over RE .
The first case immediately leads to a contradiction, since it would imply that BE earns zero
⇤ > 0. The same is true if R = R = R, h (·)
profit when playing RE , which contradicts ⇡B
I
E
E
E
has an atom over R, but hI (·) is atomless at R. Finally, it neither can be that RI = RE = R
and both hE (·) and hI (·) have an atom at R, since either BI or BE (depending on the tie-rule)
will not play R, as playing R
✏ gives (almost) the same profit, but avoid the tie altogether,
strictly increasing profit. Hence 1
⇤ = (1
implies that ⇡B
I
7
)(v
HE (pI ) > 0 for all pI 2
I
leads to a contradiction, which
cI ).
Single buyer with downward sloping demand
In this section we show that our non-exclusionary results (Propositions 1, 3, and 5 in the
paper) do not depend on the full-surplus extraction over the non-contestable segment that
14
follows from the unit-demand model. Let then Q(p) be B’s downward sloping demand and
P (q) the corresponding inverse demand. Suppose also that E is sufficiently small that can
supply at most
< Q(cI ) units. The timing of the game is as before (see section II of the
paper).
Note first that to just block the entry of a rival of cost x, the discount contract must leave B
R
with at least 0 (P (q) x)dq, which is the most E can transfer B by pricing at cost his units.
A simple discount contract that implements this outcome is a two-part tari↵ (x, T ), where x
R Q(x)
is the unit price and T =
(P (q) x)dq is the (conditional) fixed fee equal to area ABD
in Figure 1 (market-share and quantity-discount contracts work equally well; see, for example,
Kolay et al. 2004). Thus, if I o↵ers the two-part-tari↵ contract (x, T ) with x < cI , his payo↵
will be
Z
Q(x)
(P (q)
when cE < x, since he will only be selling Q(x)
Z
when cE
cI )dq
units, and
Q(x)
(P (q)
cI )dq + (x
cI )
x, since he will now be selling Q(x) units.
Putting things together, I’s expected payo↵ reduces to
Z
Q(cI )
(P (q)
cI )dq +
Z
Q(x)
(P (q)
cI )dq + (x
cI )[1
G(x)]
(13)
Q(cI )
which di↵ers from expression (6) in the proof of Proposition 1 in the paper only in the extra
term in the middle, which captures the profit loss from selling additional units at an e↵ective
price x below cI . This loss adds to the loss of selling
not enter, which happens with probability 1
units below cost when x < cI and E does
G(x). As in (6), the first term in (13) captures
the residual monopoly profit, that is, I’s largest possible payo↵ when B has already
units in
her pocket.
Even without computing the best possible discount contract that I could o↵er B —which
would require to know agents’ outside options—, it is clear from (13) that I will never o↵er an
anticompetitive one, that is, one with an e↵ective price x < cI . Now, to see whether I will still
find it profitable to o↵er a two-part tari↵ (x, T ) with x = cI , we would need to know his outside
option. If, in the absence of a contract the competition in the spot market is in nonlinear
prices, then there is no reason for I to o↵er a discount contract because he can always secure
the residual monopoly profit in the spot anyway. If, however, competition is in linear prices, it
15
is then profitable for I to o↵er the discount contract because he would otherwise obtain strictly
R Q(cI )
less than
(P (q) cI )dq.4
Figure 1: Downward Sloping Demand
p
Q(p)
cI
x
A
B
D
Q(cI ) Q(x)
4
q
In the unit-demand case it was never profitable to o↵er the contract (r, R) because I always secured (1
)(v
cI ) in the spot regardless of whether competition was in linear or non-linear prices. The mixed strategy
equilibrium for the linear case can be found in an earlier version of the paper.
16
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17