Contracts as a barrier to entry in markets with
non-pivotal buyers
Özlem Bedre-Defolie
∗†
Gary Biglaiser‡
December 15, 2015
Abstract
We analyze whether an incumbent forecloses efficient entry using long-term contracts with breakup fees given that it can offer a spot price in the future, assuming
buyers are non-pivotal and heterogeneous either in their switching costs or entrant
match values. We show that long-term contracts are distortionary only if there is
entrant market power. With switching cost heterogeneity, the incumbent always forecloses the more efficient entrant using a sufficiently high breakup fee. With match
value heterogeneity, inefficient foreclosure occurs if the entrant is not very efficient.
∗
We would like to thank Felix Bierbrauer, Dominik Grafenhofer, Michal Grajek, Paul Heidhues, Martin
Hellwig, Bruno Julien, Simon Loertscher, Markus Reisinger, Andrew Rhodes, Tommaso Valletti and the
participants of the ANR-DFG Workshop on Competition and Bargaining in Vertical Chains (Dusseldorf,
June 2015), seminar at Max-Planck Institute (Munich), Berlin IO Day (September 2015), ICT Conference
(Paris, October 2015) for their comments.
†
European School of Management and Technology (ESMT), Berlin,[email protected].
‡
University of North Carolina, [email protected]
1
1
Introduction
Breakup fees, which are also known as early termination fees (ETFs), are widely used in
contracts for a variety of services including wireless telephone, cable and satellite TV, and
health club memberships. Many regulatory agencies have been concerned that ETFs hurt
consumers. The U.S. Federal Communications Commission’s (FCC) 2010 survey found that
the ETFs charged by wireless telephone service providers depend on the service plan and type
of phone; there might be no breakup fee or the breakup fee may be over $300.1 European
policy makers have also been concerned about ETFs included in long-term contracts for
wireless phone and cable services.2 In September 2013, the European Commission (EC)
adopted a proposal for a regulation which (among other things) gives consumers “right to
terminate any (telecom) contract after 6 months without penalty with a one-month notice
period; reimbursement due only for residual value of subsidized equipment/promotions, if
any”.3
Very recently the four major communication network providers of the U.S. are being
investigated by the FCC over ETFs of their long-term contracts for high speed data carriage
in a $20 billion market. Customers, who are subject to these contract terms, are firms
or organizations who need to transport large amount of data, including telecom/internet
service providers that do not own the infrastructure, banks, manufacturers, and schools.
The main complaint behind the investigation was that “New network builders struggle to
attract customers who are held hostage by AT&T and Verizon in lockup provisions that can
extend up to seven years in length,” as described by the CEO of Comptel trade group.4
We know very little about the implications of long-term contracts and breakup fees in
markets with non-pivotal buyers (see our summary of the literature below), in particular
when the firms are asymmetric in terms of their market power, like in the above example
of major network owners acting as an incumbent and new network builders acting as an
1
They often apply to contracts for post-paid, fixed-term mobile and broadband services, in particular
when the contract involves subsidized equipment, like a headset subsidy. The FCC 2010 survey also found
that 54% of consumers would have to pay ETFs, 28% would not have to pay, and 18% did not know whether
they would have to pay ETFs. Of those who know the level of ETFs, 56% reported that the ETFs exceed
$200. See Horrigan and Satterwhite [2010]. Also, see https://www.fcc.gov/encyclopedia/early-terminationfees for the replies of the service providers to the queries of the FCC.
2
Since 2009 the European Commission (EC) obliged member states to ensure that consumer contracts
for electronic communication services should have an initial commitment period that does not exceed 2
years under the condition that a 1-year-only option must also be available. The Article 30 of the Directive
2009/136/EC sets rules facilitating switching service providers.
3
See the EC Memo, 2013, p.24. In March 2013 the EC ”recommended that national regulators negotiate
or set maximum termination fees (for Internet Service Provider contracts) that are reasonable and do not
become a barrier to switching provider”. See the EC Service Study, 2013, p.327-329
4
See http://www.kansascity.com/news/business/article39487791
2
entrant. This paper analyzes whether the use of breakup fees by an incumbent might induce
an inefficient allocation of consumers and possibly foreclose efficient entry.
To capture the facts of the above markets where we see long-term contracts with breakup
fees, we consider a two-period model of entry under the following assumptions: buyers
are non-pivotal (one buyer cannot influence a seller’s demand) and are willing to buy one
unit of a good in each period, the incumbent can offer a long-term (two-period) contract
before entry, but cannot commit not to offer a spot price in the future, when competing
against a more efficient entrant. The incumbent’s long-term contract is a combination of
unit prices for today and tomorrow and, when it is allowed, a breakup fee, which is paid if a
consumer who signed the long-term contract does not buy from the incumbent tomorrow. We
consider two sources of consumer heterogeneity leading to differentiation between the firms
and downward sloping demand in period 2: 1) buyers’ costs of switching from the incumbent
to an entrant; 2) buyers’ lower utility (match value) from the entrant’s product relative to
the incumbent’s. Buyers are uncertain about their switching cost/match value ex-ante before
the decision of whether to sign the incumbent’s long-term contract. Switching costs might
arise from the costs of calling the current provider to cancel the contract, waiting for the
new provider to activate the service, calling the bank to change the automatic bill payment
details, etc. Consumers’ heterogenous beliefs about an entrant’s perceived quality relative
to the incumbent might be manifested by how good a match (or mismatch) an entrant’s
product is for a particular consumer, or how willing she is to try a new product. The
difference between these two interpretations is that if a consumer does not make a purchase
from an incumbent, then she faces no switching costs when contemplating a purchase from
an entrant in the future, while under the match value interpretation a consumer’s utility
from buying from an entrant will not depend on whether she made a prior purchase from
the incumbent. We assume no fixed costs of entry,5 and so an entrant can be a firm existing
in another market extending to a new market.
If there are competing entrants, for both sources of firm differentiation, then the equilibrium allocation is efficient: all consumers sign the incumbent’s long-term contract and switch
to a more efficient entrant if and only if the switching cost is smaller than the entrant’s cost
advantage compared to the incumbent. This is true even when the incumbent could have
foreclosed the more efficient entrants with a sufficiently high breakup fee. The incumbent
prefers to maximize the total expected consumer surplus (implements the efficient outcome)
because it captures the expected surplus from ex-ante homogenous consumers via the first
period unit price after leaving consumers their exogenous outside option. Consumers’ outside option of rejecting the incumbent’s long-term contract is exogenous, does not depend
5
Allowing fixed entry costs would make our full-foreclosure result stronger.
3
on the incumbent’s actions, since the competitive entrants price at their marginal cost. The
long-term contract is crucial in implementing the efficient outcome, since it enables the incumbent to commit to the highest price for its locked-in consumers and thereby not to exploit
its locked-in consumers in period 2.6
We next consider the model with a single entrant and switching costs. The main effect
of allowing for entrant market power is that consumers’ outside option of rejecting the
incumbent’s long-term contract becomes endogenous. This changes our analysis in two ways.
First, the entrant’s price in period 2 will depend on the incumbent’s long-term contract and
any potential spot contract it can offer. Second, the outside option for consumers can depend
on whether they think other consumers will accept or reject the incumbent’s long-term
contract (possible coordination failure among consumers). If breakup fees are not allowed,
the incumbent makes no sales in period 2 when the entrant’s efficiency advantage is large
relative to the highest switching cost; all consumers sign the incumbent’s first-period contract
and switch to the entrant in period 2. This is also efficient. With lower levels of the entrant’s
efficiency advantage, the incumbent makes sales in period 2 with a price above marginal cost.
The entrant’s best response is also price above its marginal cost, which lowers the consumers’
outside option to signing the incumbent’s long-term contract. We demonstrate conditions
under which there is either too much or too little switching in equilibrium relative to the
efficient level. The conditions compare the entrant’s cost advantage with the highest level of
switching costs.
When the incumbent is allowed to have a breakup fee in its long-term contract, we find
that it always forecloses the more efficient entrant by making it too costly for consumers to
switch to the entrant. The benefit of foreclosing the entrant is that the incumbent does not
have to compensate consumers for their expected costs of switching. The costs of foreclosure
are that the incumbent must compensate consumers for not buying from the more efficient
entrant. Consumers’ outside option to signing the incumbent’s long-term contract is the net
surplus of buying from the entrant which is determined by consumers’ expectation about
what the entrant’s price will be in equilibrium. When the breakup fee of the incumbent’s
long-term contract is very high, consumers anticipate that the entrant will not be able to
steal anyone from the incumbent’s customer base and so the entrant will set its price at the
incumbent’s second period spot price as a result of undifferentiated Bertrand competition
for consumers who did not sign the incumbent’s long-term contract. In equilibrium, the
incumbent’s second period spot price is equal to the second-period price specified in the
long-term contract, since a price below the long-term contract price induces losses from
6
We thank Felix Bierbrauer and Bruno Julien for pointing out this.
4
nearly all consumers and generates gains from measure zero of consumers.7 As a result,
using the second unit price of the long-term contract, the incumbent endogenously lowers
consumers’ outside option of not signing the contract to zero, and obtains twice the static
monopoly profit from foreclosing the entrant. This result is surprising, in particular, because
it does not depend on the efficiency advantage of the entrant. Intuitively, consumers who
did not sign the incumbent’s first contract are homogenous (as they do not face a switching
cost), and so undifferentiated asymmetric Bertrand competition between the entrant and the
incumbent implies that the entrant’s price is equal to the incumbent’s second period price
in equilibrium.8
Having non-pivotal buyers is critical for the previous results, since the expected second
period price does not depend on a buyer’s own behavior, but depends on what she believes
about the others’ choices.9 This implies that when there is entrant market power, buyers’
outside option to the incumbent’s long-term contract becomes endogenous (depends on the
terms of the long-term contract) and so the incumbent distorts the equilibrium outcome to
lower buyers’ outside options. This distortion is found to be a major factor making full
foreclosure profitable for the incumbent.
The above discussion was based on the firms being differentiated due to consumer heterogeneity in switching costs. When the differentiation is due to heterogeneity in mismatch
value with the entrant’s product, then consumers who did not sign the incumbent’s long-term
contract are not homogeneous as they were in the switching cost interpretation. Without
breakup fees, there are no inter-temporal effects; the incumbent acts as static monopoly in
period 1 and period 2 allocations are determined by differentiated duopoly competition.
When breakup fees are feasible, the long-term contract affects the second period equilibrium and the results qualitatively differ from the analysis of switching costs. The incumbent
forecloses the efficient entrant by using sufficiently high breakup fee if and only if the entrant’s efficiency advantage is sufficiently low compared to the highest level of mismatch to
the entrant’s product.10 These differences arise because, compared to the switching cost case,
1) the incumbent’s profit from accommodating the entrant is higher and 2) the incumbent’s
7
This is because consumers could switch between the incumbent’s plans at no cost. In other words, by
making it free to switch from the long-term contract to its spot contract, the incumbent enables itself to
commit not to undercut the second unit price of the long-term contract in period 2.
8
Note that switching costs do not matter for this result. If there were no switching costs, the incumbent
would still be able to foreclose the efficient entrant by creating switching costs endogenously via breakup
fees.
9
If there were one buyer, rather than mass one of buyers, the buyer’s outside option to the incumbent’s
contract would be exogenous, since, if she did not sign that contract, there would be undifferentiated Bertrand
competition for her and so the second period price would be the incumbent’s marginal cost.
10
If the entrant is very efficient, like in the switching costs case without breakup fees, it serves the entire
market in period 2 and this is efficient.
5
profit from foreclosure is lower, decreases in the entrant’s cost efficiency, and increases in
the highest mismatch to the entrant’s product. Intuitively, accommodating the entry is less
costly because consumers’ outside options to the incumbent’s long-term contract are lower
than the switching cost case, since consumers face mismatch value with the entrant’s product
even when they did not sign the incumbent’s long-term contract. On the other hand, it is
more costly to foreclose the entrant, because consumers’ outside option to the incumbent’s
long-term contract will be positive, and so higher than the foreclosure case with switching
costs: the entrant faces a downward sloping demand of consumers who did not sign the
incumbent’s long-term contract. The latter also implies that the incumbent’s foreclosure
profit is decreasing in the entrant’s efficiency advantage and increasing in the highest level
of mismatch value.
The paper proceeds as follows. In the next subsection we summarize our contribution to
the related literature. In Section 2, we present the switching cost model with competitive
entry in period 2. We then allow the entrant to have market power in Section 3. We present
the analysis of the mismatch value case in Section 4 and conclude in the final section. All
formal proofs are in the Appendix. We extend our main findings to more general distribution
of consumer heterogeneity parameter (available in our web Appendix).
1.1
Related Literature
One of our main contributions is to the literature on entry deterrence by exclusionary clauses
(see Aghion and Bolton [1987]; Spier and Whinston [1995]; Rasmussen et al. [1991]; Segal and
Whinston [2000]; and Choné and Linnemer [2015]). It is well established that an incumbent
might foreclose an efficient entrant by using breakup fees. In Aghion and Bolton [1987],
the coalition of the incumbent and the buyer shifts rent from the more efficient entrant
via breakup fees (liquidated damages) and this limits the extent of efficient entry in the
case where the entrant’s cost is uncertain. The findings of the previous literature apply to
situations where a breakup fee is used in a contract between a seller and a single or many
pivotal buyer(s); for instance, in contracts between upstream and downstream firms.
Our analysis is complementary to this literature in the sense that we look at potential
entry deterrence due to breakup fees used by the incumbent in markets with many nonpivotal buyers, like final product markets. To capture the facts of such environments, we
have three key assumptions that are different from Aghion and Bolton [1987]: 1) we look
at the case of infinitesimal (non-pivotal) buyers, 2) we allow the incumbent to sell one unit
in each period, offer a long-term contract in period 1 and a spot price in period 2 (no exante commitment to period 2 price), and 3) consumers face intrinsic uncertainty on their
6
switching costs or mismatch value (downward sloping demand in period 2).
The critical assumption of Aghion and Bolton [1987] is that the incumbent commits to the
breakup fee ex-ante (before the entrant makes its offer to the buyer). Without commitment
the incumbent would want to forgive some of the breakup fee to benefit from the entrant’s
offer. Spier and Whinston [1995] allow for renegotiation after the entry decision and show
that if the incumbent invests in cost reducing technology before the entry decision, the
incumbent may still have an incentive to block entry by over-investing in improving its
own technology: Sunk investments in cost reduction changes the entrant’s behavior in the
continuation of the game, even if there is renegotiation.
Our results are robust to allowing the incumbent to offer a spot contract in period 2.
The incumbent is able to commit to a maximum price in period 2 by the long-term contract,
but cannot commit not to lower the price of the second unit in period 2. However, we show
that this lack of commitment does not lower the incumbent’s profit from accommodating
entry since the incumbent is acting as a Stackelberg leader and just like in that model
with linear demand,11 the leader would not want to raise its output (lower its price in our
model). In the switching costs case, when the incumbent forecloses the entrant by a breakup
fee, it can perfectly control the second period price via its long-term contract’s second unit
price. However, in the match value case, the incumbent’s profit from foreclosure suffers from
the lack of commitment to the second period price, since this increases consumers outside
option. Hence, if we assumed full-commitment to the second period price, that would affect
the analysis only by increasing profits from foreclosure in the match value case and so make
foreclosure more likely in that case. Moreover, we do not require commitment to the level
of breakup fee in the long-term contract, since the incumbent does not have an incentive to
lower the breakup fee in period 2.
Very recently, Choné and Linnemer [2015] extend Aghion and Bolton [1987]’s setup by
allowing downward sloping buyer demand and analyzing the implications of non-linear tariffs
that might be conditional on the quantity purchased from the entrant. Similar to their
setup we also have downward sloping demand for consumers who signed the incumbent’s
long-term contract in switching costs interpretation and for all consumers in the match
value interpretation. Our setup of match value interpretation is closest to theirs. In our
setup allowing the incumbent to use non-linear tariffs would not make any difference since
each buyer is willing to buy one unit in each period. Our main result of the match value
interpretation foreshadows their finding that for low levels of net surplus from the entrant’s
product there is full-foreclosure. However, in the case of no foreclosure (when the entrant is
11
We show in the web Appendix that this is still be the case with non-linear demand, that is, more general
distribution of switching cost/mismatch value, if the density function is non-increasing.
7
very efficient) in our setup the entrant serves the entire market in period 2, whereas in their
setup there is partial foreclosure. The main reason of this difference is that they assume exante full-commitment by the incumbent, but we allow the incumbent to offer a spot contract
in period 2 and so the incumbent cannot commit not to lower its price when the entrant is
very efficient,
When a buyer’s decision to accept an exclusive deal incurs an externality on other buyers, coordination failure among buyers will lead to inefficient foreclosure of the entrant in
equilibrium (Rasmussen et al. [1991]; Segal and Whinston [2000]). Similar to this literature
we find that when foreclosure arises in equilibrium, this is due to coordination failure between consumers since consumers would be better-off if they all coordinate and reject the
incumbent’s long-term contract with a high breakup fee. On the other hand, different from
the previous literature, we have the unique equilibrium in the match value interpretation,
since it is individually rational to sign the incumbent’s long-term contract, regardless of what
consumers believe about the others’ behaviour.12
The literature on switching costs have considered different types of endogenous switching
costs, like loyalty discounts in Caminal and Matutes [1990], discounts to new customers
(subscription models) as in Chen [1997], or poaching in Fudenberg and Tirole [2000].13 The
key difference of our paper from this literature is that we consider ex-ante asymmetric firms
(the incumbent and entrant (s)) and analyze the impact of allowing the incumbent to increase
switching costs endogenously by breakup fees on the allocation of consumers between the
incumbent and the more efficient entrant. One would expect switching costs to make it
more difficult for the incumbent to deter efficient entry, since the incumbent would have
to compensate consumers for the expected costs of switching in case they terminate their
contract with the incumbent to buy from the more efficient entrant. Similarly, common
view suggests that the entrant’s cost advantage would make foreclosure more costly for the
incumbent. Surprisingly, we show that switching costs or the entrant’s efficiency advantage
do not affect the incumbent’s profit when there is full foreclosure, but they do affect the
incumbent’s profitability without foreclosure. We show that the incumbent’s maximum
profit without foreclosure is lower than its profit with full foreclosure.
Finally, our model has links to the durable goods literature (Coase [1972], Butz [1990]).
By offering long-term contracts to consumers, the incumbent bundles today’s consumption
with future consumption. Without breakup fees, consumers could switch to a cheaper con12
We have the unique symmetric equilibrium in the switching costs interpretation, that is, it is individually
rational to sign the incumbent’s contract if consumers believe that all the other consumers accept the
incumbent’s long-term contract or if consumers believe that all the others reject it.
13
See Klemperer [1995] and Farrell and Klemperer [2007] for excellent reviews of the switching costs
literature.
8
tract in period 2. The incumbent cannot commit not to offer a spot contract when facing
a more efficient entrant in period 2. In the spot market, the incumbent has an incentive
to undercut the second-period price of its long-term contract. One can view the offering
a long-term contract converts a non-durable good (consumption today) to a durable good.
The use of breakup fees in the long-term contract is a possible way for the incumbent to
solve the commitment problem when it blocks the efficient entry: A sufficiently high breakup
fee keeps the entrant out of the market and by making it free to switch from the incumbent’s long-term contract to its spot contract, the incumbent does not have an incentive to
undercut the second unit price of its long-term contract in period 2.
2
Benchmark: Competitive Entry
We consider a two-period model of entry. In the first period there is only one firm, the
incumbent (I), and in the second period the incumbent faces at least two symmetric entrants,
each of which offers exactly the same product as the incumbent. Let cI and cE denote the
marginal cost of the incumbent and each entrant (E), respectively. We assume that the
entrants are more efficient than the incumbent: cI > cE and define ∆ ≡ cI − cE as the cost
difference.14
There is mass 1 of consumers who are willing to buy one unit of the product in each period.
Consumers have switching cost s, which is uniformly distributed over [0, θ]. Following Chen
[1997], we assume that consumers learn their switching cost s at the beginning of period 2.
Firms never observe s. The timing of the contracting is the following:
Period 1 The incumbent offers a two-period contract, (pI1 , pI2 ), where pI1 is paid if a
buyer takes the contract for buying one unit in period 1 and pI2 is paid if the consumer
buys an additional unit from the incumbent in period 2. Consumers then decide whether to
accept the incumbent’s offer.
Period 2 Simultaneously, each entrant offers a price pE and the incumbent offers a spot
price pSI2 . Consumers then learn their switching cost. Consumers who signed the incumbent’s
first period contract decides whether to stay with that contract, switch to the incumbent’s
spot contract, or switch to the entrant’s contract. Consumers who did not sign the incumbent’s contract decide whether to buy the incumbent’s spot offer, the entrant’s offer, or
nothing. A consumer who did not sign a contract faces no switching cost.
14
All arguments would go through if we assumed an arbitrary number of entrants with different marginal
costs and that the second lowest marginal cost is below cI .
9
We assume that consumers’ valuation from the product is sufficiently high so that consumers will always buy the product in period 2 in equilibrium and look for a Subgame Perfect
Nash Equilibrium where the firms never price below their marginal costs.15
2.1
Equilibrium Analysis
In period 1, consumers are homogenous, so either all consumers sign the incumbent’s longterm contract or no one signs it.16 In sub-game where all consumers sign the incumbent’s
long-term contract ex-post efficiency requires that consumers switch to the entrants in period
2 if and only if their switching cost is lower than the entrant’s cost advantage: s < ∆c. When
amount of consumers switch to the entrants and
∆c < θ ex-post efficiency requires that ∆c
θ
θ−∆c
amount of consumers buy from the incumbent in period 2. When ∆c > θ ex-post
θ
efficiency requires that all consumers switch to the entrants.
In period 2, competitive entrants price at the marginal cost: p∗E = cE . If pI2 ≤ (θ +
cI + cE )/2 ≡ pS∗
I2 , then the incumbent does not want to undercut its long-term contract
price. This is the constraint that the long term contract must meet to induce the incumbent
not to offer a spot price that will be chosen (in the spirit of renegotiation proof contracts).
If a consumer does not sign the incumbent’s contract in period 1, she gets v − cE , which
constitutes her outside option when she decides whether to buy the incumbent’s contract
in period 1. If a consumer signs the incumbent’s contract in period 1, she stays with the
incumbent if her switching cost is above the price difference between the incumbent and
the entrant, s > pI2 − cE , and switches to an entrant otherwise. So a consumer’s expected
surplus from signing the incumbent’s contract is
Z
EUsignI = 2v − pI1 − pI2 P r(s > pI2 − cE ) − cE P r(s < pI2 − cE ) −
0
pI2 −cE
s
ds,
θ
(1)
which is the consumption value of two units less the first unit price (up-front fee of the
incumbent’s contract), the expected price of the second unit (the incumbent’s second period
price times the probability of buying from the incumbent plus the entrants’ price times the
probability of buying from an entrant), and the expected switching costs. Note that the
demand for entrants is always positive since we must have p∗I2 − cE > 0 in equilibrium
given that the incumbent is less efficient than the entrant and the incumbent does not offer
15
We focus on equilibria where firms never price below their marginal cost in order to avoid potential
multiplicity of equilibria in mixed strategies when the firms compete for consumers who did not sign the
incumbent’s contract (homogenous Bertrand competition). See Blume [2003] for more. We thank Paul
Heidhues for pointing us this.
16
This is because consumers do not change the state of period 2 and the entrants always charge marginal
cost in period 2.
10
a second period price below its marginal cost (pI2 ≥ cI ). The incumbent’s second period
demand is positive if and only if p∗I2 − cE ≤ θ, which we will verify at the end.
All consumers sign the incumbent’s contract in period 1 if and only if their expected
surplus from signing it is at least equal to their outside option:
EUsignI ≥ EUnosignI = v − cE .
(2)
Under the latter constraint the incumbent’s expected profit over the two periods is
ΠI = pI1 − cI + (pI2 − cI )P r(s > pI2 − cE ).
(3)
The incumbent sets (pI1 , pI2 ) by maximizing its profit subject to the consumers’ participation constraint, (2). At the optimal solution the incumbent sets the highest first-period
price satisfying the constraint:
p∗I1
Z
pI2 −cE
= v + (cE − pI2 )P r(s > pI2 − cE ) −
0
s
ds.
θ
(4)
Replacing the latter into the incumbent’s profit we can rewrite it
Z
ΠI = v − cI − ∆cP r(s > pI2 − cE ) −
0
pI2 −cE
s
ds.
θ
Given that s is uniformly distributed over [0, θ] the optimal period 2 price is at the marginal
cost: p∗I2 = cI . The incumbent’s second period demand is positive if and only if p∗I2 − cE ≤ θ
or ∆c ≤ θ. Hence, we obtain the equilibrium of the baseline model:
Proposition 1 If there are competitive entrants in period 2, there is a unique equilibrium
where all consumers buy the incumbent’s first period contract, the entrants set p∗E = cE and
the incumbent sets p∗I2 = pS∗
I2 = cI , which implement the efficient switching in period 2:
consumers switch to the entrants if and only if s ≤ ∆c. The consumers’ expected surplus is
U ∗ = v − cE .
Competitive entrants price at the marginal cost, so consumers’ outside option to signing
the incumbent’s first contract is exogenous (EUnosignI = v − cE ). Using the upfront fee, pI1 ,
the incumbent can capture all expected consumer surplus ex-ante after leaving consumers
their outside option. The incumbent therefore wants to maximize this surplus by inducing
efficient switching. This requires setting pI2 at its marginal cost. Hence, in equilibrium all
consumers buy the incumbent’s contract in period 1 and switch to the entrants if and only
if s ≤ ∆c. If all consumers switch to the entrants, ∆c ≥ θ, the total industry surplus is
11
2v − cI − cE − 2θ , which is the total consumption utility minus the total costs of the active
firms minus the expected switching costs. However, if ∆c ≤ θ, the incumbent sells to θ−∆c
θ
of consumers in period 2 and the entrant sells to ∆c
of
consumers.
In
this
case
the
total
θ
∆c2
industry surplus is 2(v − cI ) + 2θ . The incumbent captures the total surplus after leaving
consumers their outside option, v − cE .
Note that a long-term contract in period 1 is crucial in implementing the efficient outcome.
If the incumbent offered a spot contract in period 1, it would then be tempted to exploit its
market power over its customer base (consumers who signed the first contract) by setting
its second period price above its marginal cost. Consumers anticipate such an exploitative
behavior by the incumbent, so the incumbent would have to compensate consumers for the
expected high price in the future by lowering its first period price, like in the switching costs
literature when consumers are forward looking. As a result, the equilibrium outcome would
be inefficient due to too much switching and the incumbent would be worse off compared
to the benchmark equilibrium. Hence, the incumbent prefers to write a long-term contract
in period 1 even if it cannot commit not to offer a spot price in period 2. The long-term
contract enables the incumbent to commit to the highest period 2 price for its locked-in
consumers and so enables the incumbent to commit not to exploit its locked-in consumers
in period 2 and implements the efficient outcome.
Corollary 1 If there are competitive entrants in period 2, the long-term contract enables the
incumbent to implement the efficient outcome, which cannot be achieved by a spot contract.
Allowing the incumbent to offer a spot contract in period 2 is not critical for the equilibrium since the incumbent does not want to undercut its long-term contract price set to
marginal cost.
2.2
Breakup fees
As we discussed in the Introduction, breakup fees are common features of long-term contracts
for many services to consumers. We now analyze the equilibrium if the incumbent is allowed
to use a breakup fee d in its first-period contract such that d is paid if the consumer who
signed the contract does not buy from the incumbent in period 2. The analysis with breakup
fees have two differences from the previous analysis: 1) The switching decisions in period 2
will now depend on the breakup fee, 2) The incumbent can use breakup fees to earn revenues
from the switchers. Consider consumers who signed the incumbent’s first-period contract.
In period 2 these consumers switch to the entrants if and only if s < pI2 − d − cE , in which
case they have to pay the entrant’s price, cE , as well as the incumbent’s breakup fee, d. If
12
they stay with the incumbent, that is, when s > pI2 − d − cE , they buy at pI2 and do not
have to pay d. The expected surplus from signing the incumbent’s contract is now
Z
pI2 −d−cE
EUsignI = 2v − pI1 − pI2 P r(s > pI2 − d − cE ) − (cE + d)P r(s < pI2 − d − cE ) −
0
s
ds.
θ
All consumers sign the incumbent’s contract in period 1 if and only if EUsignI ≥ v − cE .
Under this constraint the incumbent’s expected profit over the two periods is
ΠI = pI1 − cI + (pI2 − cI )P r(s > pI2 − cE − d) + dP r(s < pI2 − d − cE ).
At the optimal solution the incumbent sets the highest first-period price satisfying the consumers’ participation constraint:
p∗I1
Z
= v + (cE − pI2 )P r(s > pI2 − d − cE ) − dP r(s < pI2 − d − cE ) −
0
pI2 −d−cE
s
ds.
θ
Replacing the latter into the incumbent’s profit, we can rewrite it as
Π∗I
Z
= v − cI − ∆cP r(s > pI2 − d − cE ) −
0
pI2 −d−cE
s
ds
θ
Observe that the incumbent’s equilibrium profit depends only on the difference between the
second period price and the breakup fee, pI2 − d. This is because the incumbent captures
ex-ante expected surplus of consumers via pI1 , the breakup fee revenues are fixed transfers
from consumers to the incumbent and so they wash out. Similar to the baseline model,
the incumbent can implement the efficient outcome described in Proposition 1 by setting
pI2 − d = cI . Indeed, this is the optimal thing to do if the incumbent does not foreclose
the entrants. Different from the baseline model, the incumbent can now foreclose the more
efficient entrants by setting pI2 − d < cE , since then the entrants cannot steal any consumer
from the incumbent’s long-term contract even if the entrants sell at marginal cost. Consumers
would be willing to sign such a contract if they are compensated for what they would get
from not signing it, that is, if the incumbent leaves them their outside option: v − cE . But
then the total industry profit would be 2(v − cI ) and the incumbent would capture this
after leaving consumers v − cE . Hence, the incumbent’s profit from foreclosing the efficient
entrants would be v − cI − ∆c. Comparing this with what the incumbent would earn under
2
the efficient outcome: Π∗I = v − cI − 2θ if ∆c ≥ θ and Π∗I = v − cI − ∆c + ∆c
if ∆c ≤ θ,
2θ
we demonstrate that the incumbent does not have an incentive to foreclose more efficient
entrants even if it can do so by using breakup fees.
13
Corollary 2 Suppose that the incumbent’s first period contract also has a breakup fee, d,
which is paid if a consumer who signed the first-period contract does not buy from the incumbent in period 2. In equilibrium,the incumbent implements the same (efficient) outcome as
the one described in Proposition 1. Hence, there is no inefficient foreclosure in equilibrium.
Intuitively, the incumbent has to leave the same surplus to consumers (v − cE ) and the
total industry surplus is lower under foreclosure (due to inefficiency).
It is common in many markets such as wireless phone and cable to use constant monthly
prices along with breakup fees. We now demonstrate that the incumbent can offer a contract
that has the same unit price in each period along with a breakup fee d which is paid if
the consumer who signed the contract does not buy from the incumbent in period 2 and
implement the same outcome as when the incumbent can offer different prices in each period.
Corollary 3 Suppose that the incumbent’s first period contract has a constant per period
unit price pI and a period 2 breakup fee d. In equilibrium the incumbent implements the
same (efficient) outcome as the one described in Proposition 1. The equilibrium prices are
p∗ −p∗
p∗ +p∗
p∗I = I1 2 I2 , d∗ = I1 2 I2 where (p∗I1 , p∗I2 ) are the equilibrium prices in the baseline model.
3
Single Entrant
In the benchmark we assumed that entrants are perfectly competitive and so they always
set their prices to their marginal costs. Now, we relax this assumption to allow for market
power of entrants. To highlight the differences from the benchmark we consider the extreme
case of a single entrant, which has cost cE < cI as before. The main effect of giving the
entrant market power is that consumers’ outside option of rejecting the incumbent’s longterm contract becomes endogenous. This is for two reasons. First, the entrant’s price in
period 2 will depend on the incumbent’s period 1 contract along with any potential spot
contract offered by the incumbent. With competitive entry, the entrants priced at marginal
cost and thus consumers’ outside option of rejecting the incumbent’s contract did not depend
on the incumbents’ actions. Second, consumers’ outside option can depend on whether they
think other consumers will accept or reject the incumbent’s first period offer.
We modify the second stage of the game such that the single entrant offers a price pE .
The other assumptions and notation of the model remain the same as before.
Proposition 2 If there is single entrant in period 2, in equilibrium all consumers buy the
incumbent’s first period contract.
14
• If ∆c ≥ 2θ, there is a unique equilibrium where all consumers switch to the entrant
and
θ
Π∗I = v − cI − , Π∗E = ∆c and EU ∗ = v − cI .
2
• If ∆c < 2θ, there is a unique equilibrium where both firms sell in period 2 and
Π∗I = v − cI +
Π∗E =
θ2 − 4∆cθ + ∆c2
,
6θ
θ + 2cE + cI
(∆c + θ)2
and EU ∗ = v −
≡ U < v − cI .
9θ
3
To prove the result, we first assume that all consumers believe that each of their fellow
consumers will accept the incumbent’s first period offer and derive equilibrium payoffs. Next,
we find whether this equilibrium is unique for all parameter values and beliefs of consumers
about how other consumers will react to the incumbent’s offer.
One key to the equilibrium is to find the entrant’s price in period 2. If no one signed the
incumbent’s contract, then all consumers are identical since they face no switching cost, and
thus Bertrand competition would ensue where the entrant would get all consumers at a price
of cI . If the consumers do sign the incumbent’s contract in period 1, then the entrant’s price
will depend on whether the incumbent is willing to offer a lower spot price in period 2 than
the stated price in the period 1 contract; along the equilibrium path, the incumbent does
not offer such a price. We find that there are two cases to analyze, one where a spot price
offered by the incumbent would keep a positive measure of consumers, when ∆c ≤ 2θ, and
one where the spot price would attract no consumers, when ∆c ≥ 2θ. In the latter case, a
spot price by the incumbent even at marginal cost, cI , can attract no consumers because the
entrant’s optimal price will not exceed cI − θ. This can be seen by noting that the entrant’s
best response to the incumbent’s second period price is
pE =
pI2 + cE
,
2
given pI2 ≤ v and if ∆c < 2θ. When pI2 = cI and ∆c = 2θ, the entrant sets a price of cI − θ
and attracts all consumers. Thus, as θ falls the entrant sets pE = cI − θ, which is higher
E
than pE = cI +c
, and still attracts all consumers.
2
By pinning down the continuation game in period 2, we go back to the first period contract
offered by the incumbent. Again, the analysis breaks down into two regimes depending on
how much more efficient the entrant is than the incumbent. In both settings, the incumbent
(efficiently) induces all consumers to sign the first period contract and sets the terms to
make consumers indifferent between signing it and not signing and buying from the entrant
15
in period 2. The incumbent needs to take into account the consumers’ expected switching
costs when offering such a contract. When ∆c is small, a non-signing consumer will obtain
v − pE , where the entrants price is greater than cI − θ. As it turns out, the incumbent’s
unconstrained second period price in the long-term contract is exactly the same as what its
spot price would be in period 2. This is due to the fact that the incumbent is acting as a
Stackelberg leader in period 2 and just like in that model with linear demand, the leader
would not want to raise its output (lower its price in our model). On the other hand, when
the entrant is very efficient, the incumbent finds it unprofitable to try to keep any consumers
in period 2, and sets its second period price in the long term contract equal to marginal
cost, cI , since any contract with a higher period 2 price is not credible and will have the
incumbent lowering its price to marginal cost. Now, the incumbent attracts consumers by
charging v − 2θ , which compensates consumers for their expected switching costs.
If all consumers reject the incumbent’s long-term contract, there will be Bertrand competition with no switching costs resulting in price equal to the incumbent’s cost. Hence,
consumers expect to receive v − cI if they all reject (or all other consumers reject) the longterm contract. To obtain uniqueness, we see if consumers have a credible outside option to
rejecting the incumbent’s long-term contract if the incumbent does not offer them at least
v − cI . When ∆c is large, v − cI is what a consumer’s expected utility is in the equilibrium
we derived when all consumers expect each other to accept the incumbent’s offer. When
∆c is small, consumers’ outside option is U < v − cI . To see that it is not credible for a
consumer to reject an offer of U if she thinks all other consumers will reject it, suppose that
there is such an equilibrium. If the incumbent offers U with a price that is strictly less than
v, which it does in our equilibrium, an individual consumer will then deviate and accept
the incumbent’s contract, since she gets a positive surplus in period 1 and the incumbent
will charge cI as a spot price in period 2 to compete for the mass of consumers, and the
defecting consumer will obtain the same second period utility as all consumers who rejected
the incumbent’s first period contract. That is, since the rejecting consumer does not change
the state of the market, it is optimal to deviate from the equilibrium. Thus, we cannot
have such an equilibrium. We also demonstrate in the Appendix that there cannot be an
equilibrium where only a subset of consumers accept the incumbent’s first period contract.
3.1
Discussion of the result
We have some further observations about the equilibrium. First, p∗I2 > cI when ∆c <
2θ. Intuitively, the incumbent sets p∗I2 above cI in order to raise the entrant’s equilibrium
price and thus lower the consumers’ outside option in period 1, that is, to leave less rent
16
to consumers. When there were competitive entrants (Section 2), the outside option of
consumers was exogenous and so the incumbent did not distort the total surplus to capture
more rent from ex-ante homogenous consumers. In general, the key difference between a
setup with entrant market power and a setup with perfectly competitive entrants is that in
the former consumers’ outside option in period 1 is endogenous: it depends on the entrant’s
price which depends on the initial contract of the incumbent and consumer expectations of
what other consumers will do regarding the incumbent’s contract.
Second, it is interesting to see that consumers do not have a credible incentive to all reject
the incumbent’s offer of more than U , but less than v − cI , which is their outside option if no
consumer sign the incumbent’s contract in period 1. This is because an individual consumer
can be tempted to sign the incumbent’s contract in period 1 if the price is less than v. The
consumer will have a positive first period utility and in the continuation game the incumbent
will price down to marginal cost because it will compete for the “market” with the entrant in
period 2. This is because the incumbent can offer spot contracts in period 2. Similar to the
naked exclusion literature (Rasmussen et al. [1991], Segal and Whinston [2000]), here there
is coordination failure among buyers; but unlike this literature our equilibrium is unique.
Third, efficiency requires that consumers switch to the entrant if and only if s < ∆c.
The previous result implies that if ∆c/2 > θ, in equilibrium all consumers switch to the
entrant and this is also efficient. However, if ∆c/2 < θ, in equilibrium consumers switch to
. This implies the following result:
the entrant if and only if s < p∗I2 − p∗E = θ+∆c
3
Corollary 4 If θ < ∆c/2, in equilibrium all consumers efficiently switch to the entrant. If
θ ∈ (∆c/2, 2∆c) there is too little switching, and if 2∆c < θ, there is too much switching.
To understand the intuition behind the potential inefficiency in equilibrium first note that
the difference between the incumbent’s and the entrant’s price, p∗I2 − p∗E = θ+∆c
, determines
3
the marginal consumer type in period 2 and the marginal type increases in ∆c in equilibrium.
When the entrant becomes more efficient (∆c increases), the entrant’s price decreases more
than the incumbent’s second unit price, so the marginal type increases. Alternatively, when
the incumbent becomes more inefficient (∆c increases), the incumbent’s second period price
increases more than the entrant’s price, so the marginal type increases. At 2∆c = θ, the
price difference (marginal type) is exactly the difference in cost and so we get efficiency. For
small θ, less than ∆c/2, all buyers efficiently switch. As θ grows from there up to 2∆c, the
price difference is less than the cost difference and hence there is too little switching. For
values of θ above 2∆c , the equilibrium price difference is larger than the cost difference and
there is too much switching.
17
Observe that θ measures both the mean and variance of the switching cost heterogeneity.
A higher θ implies a higher average switching cost as well as a higher variance of switching
costs (higher level of heterogeneity in switching costs). Intuitively, when θ is very small
relative to the efficiency difference, both the mean and variance of switching costs are low,
and so the incumbent cannot compete with the entrant in period 2 and cannot attract a
positive measure of consumers even when pricing at marginal cost. As θ grows, the incumbent
can compete with the entrant profitably in period 2. It is useful to observe that the two
firms’ prices are strategic complements and that both are increasing in the switching cost
parameter θ, with the incumbent’s price increasing at twice the rate as the entrant’s price.
To see the intuition consider an increase in θ. Keeping the marginal consumer type constant
the increase in θ increases the measure of consumers staying with the incumbent since there
will be more consumers with high switching costs. As a reaction the entrant raises its price
(strategic effect). On the other hand, the increase in θ (keeping the marginal consumer
unchanged) decreases the measure of consumers switching to the entrant, since there will be
fever infra-marginal low switching cost consumers given that the same measure of consumers
will be distributed across a larger range (higher heterogeneity in switching costs). As a
reaction the entrant lowers its price (own-demand effect). We show that the positive strategic
effect on the entrant’s price dominates the negative own-demand effect, and so the net effect
on the entrant’s price is positive and half of the increase in the incumbent’s price. Hence,
the price difference (marginal consumer type) increases in θ. When θ is in the intermediate
range, the incumbent can compete with the entrant, but the price difference is still small
enough relative to the difference in firm costs such that p∗I2 − p∗E < ∆c and there is too little
switching. As θ grows then the heterogeneity in switching cost dominates the cost difference
and the incumbent’s price exceeds the entrants price by more than ∆c. Thus, there will be
too much switching in equilibrium. We can show the following comparative static result:
Corollary 5 An increase in θ i) will have no effect on switching behavior if θ < ∆c/2; ii)
will efficiently induce more consumers to switch to the entrant if θ ∈ (∆c/2, 2∆c); iii) will
inefficiently induce more consumers to switch if 2∆c < θ.
Finally, the assumption of infinitesimal consumers is critical for the equilibrium outcome.
If there was one buyer who had uncertain switching cost s ex-ante and so downward sloping
demand ex-post, the outside option of the buyer to the long-term contract would again be
exogenous and equal to v − cI , since if the buyer did not sign the long-term contract, the
incumbent and the entrant compete for the buyer and asymmetric Bertrand competition
would result in price at the incumbent’s cost. In that case, the incumbent would like to
maximize the total industry surplus since it could not affect the buyer’s outside option via
18
the long-term contract (similar to the case of competitive entrants). To do so, the incumbent
would set p∗I2 to induce efficient switching in period 2 and capture the total surplus ex-ante
after leaving the buyer her outside option.
3.2
Breakup fees
Now, we allow the incumbent’s first-period contract to include a breakup fee d in period 1,
which has to be paid if a consumer who signed the incumbent’s long-term contract does not
purchase from the incumbent in period 2. The incumbent may also offer a spot price pSI2 in
period 2. The consumers who signed the incumbent’s contract in period 1 do not have to
pay d if they switch to the incumbent’s spot contract in period 2.17
Proposition 3 Suppose that the incumbent offers a two-period contract (pI1 , pI2 ) and a
breakup fee d in period 1 and a spot contract pSI2 in period 2, where it faces a more efficient
entrant. In the unique symmetric equilibrium all consumers buy the incumbent’s first period
contract, the incumbent sells to all consumers in period 2, and so the more efficient entrant
is fully foreclosed. The equilibrium prices, payoffs, and expected utilities are
∗
p∗I1 = p∗I2 = pS∗
I2 = pE = v
Π∗I = 2(v − cI ), Π∗E = 0, and EU ∗ = 0.
We will first argue that there exists an equilibrium when consumers believe that all of
the other consumers sign the incumbent’s first period offer. Next, we show that this is
the unique symmetric equilibrium for all parameter values.18 The key is to determine the
entrant’s equilibrium price, since this will determine consumers’ outside option to signing
the incumbent’s long-term contract.
If none of the consumers signed the incumbent’s long-term contract, undifferentiated
Bertrand competition between the entrant and the incumbent implies that the entrant sells
to all consumers at price cI . If all consumers signed the incumbent’s long-term contract,
the breakup fee of the incumbent’s contract affects the amount of consumers who switch
from the incumbent to the entrant in period 2. If a consumer stays with the incumbent, she
pays the incumbent’s lowest price for a unit consumption in period 2: min{pI2 , pSI2 }, but
17
We will show that it is indeed optimal for the incumbent to require its locked-in consumers to pay d only
if they do not buy from the incumbent in period 2. This will enable the incumbent to control the second
period equilibrium price by its first period contract price.
18
We focused on symmetric equilibria, where all ex-ante identical buyers choose the same strategy in period
1, they either all accept or all reject the incumbents first period contract. We do this because it is somewhat
implausable for a continuum of buyers to both coordinate with just the right proportion of them accepting
the incumbent’s offer both on and off-the-equilibrium path.
19
if she switches to the entrant she pays the entrant’s price plus her switching cost and the
breakup fee. Consumers could also decide not to buy in period 2 in which case they have
to pay the breakup fee to the incumbent (we verify the condition under which consumers
do not want to do this in equilibrium). On the equilibrium path, the incumbent sets the
second period price of the long-term contract equal or below the equilibrium level of its spot
contract so that it would not have an incentive to undercut itself in the continuation of the
game (in the spirit of renegotiation proof contracts). This would then imply that consumers
with switching costs below pI2 − d − pE switch to the entrant and the rest continues buying
from the incumbent under the condition that buying from the incumbent gives them some
non-negative surplus, pI2 − d ≤ v.
The incumbent can use breakup fees as a tool to affect consumers’ beliefs about what
the entrant’s price will be. Consider a subgame where the incumbent sets d > pI2 − cE .
Consumers then expect that the entrant cannot steal any consumer from the incumbent’s
long-term contract even if the entrant sells at its marginal cost. The entrant then prefers to
undercut the incumbent’s spot price to attract consumers who did not sign the incumbent’s
first-period contract (if any). If all or almost all consumers signed the first period contract,
the incumbent does not have an incentive to lower its spot price below pI2 , since then it
would lose positive margin from a measure 1 of consumers and would gain a positive margin
from consumers of measure zero. But then the incumbent could control the entrant’s price
perfectly via the choice of pI2 . It is optimal for the incumbent to reduce the consumers’
expected utility from their outside option to zero by offering pI2 = v, since then consumers
will expect the entrant’s price to be v and so be willing to sign the incumbent’s first period
contract as long as pI1 ≤ v. In the equilibrium of this subgame, the incumbent earns twice its
static monopoly profit by setting a sufficiently high breakup fee, d∗ > v − cE , and the prices
that capture all surplus from consumers, p∗I1 = p∗I2 = v . We then show that the incumbent
prefers to implement this foreclosure outcome rather than any subgame equilibrium where
the entrant has some positive sales in period 2.
The incumbent’s second option is to accommodate the entrant. A necessary condition to
have some positive sales by the entrant is that the incumbent’s first contract has a sufficiently
low breakup fee: d < pI2 − cE . But then consumers expect to switch to the entrant in period
2 with probability pI2 − d − pE , in which case they have to pay the entrant’s price as well as
the breakup fee. So the expected consumer surplus from signing the incumbent’s contract is
Z
EUsignI = 2v − pI1 − pI2 P r(s > pI2 − d − pE ) − (pE + d)P r(s < pI2 − d − pE ) −
0
20
pI2 −d−pE
s
ds,
θ
which is the two period consumption utility less the period 1 price, the expected price
in period 2, and the expected switching costs. The expected surplus from not signing the
incumbent’s contract is v −pE , given that consumers are infinitesimal and so do not influence
the equilibrium level of pE . Consumers sign the incumbent’s first period contract if and only
if their expected surplus from signing is greater than their outside option: EUsignI ≥ v − pE .
Under this constraint, the incumbent’s profit is the profit from period 1 sales, expected profit
from period 2 sales and breakup revenues to be collected from consumers who switch to the
entrant:
ΠI = pI1 − cI + (pI2 − cI )P r(s > pI2 − d − pE ) + dP r(s < pI2 − d − pE ).
It is optimal for the incumbent to set the highest first-period price satisfying the participation
constraint of consumers:
p∗I1 (pI2 )
Z
= v + (pE − pI2 )P r(s > pI2 − d − pE ) − dP r(s < pI2 − d − pE ) −
0
pI2 −d−pE
s
ds.
θ
After replacing the equality of the optimal upfront fee into the incumbent’s profit, its profit
becomes
Z pI2 −d−pE
s
ΠI = v − cI + (pE − cI )P r(s > pI2 − d − pE ) −
ds.
θ
0
First observe that the incumbent’s profit depends only on pI2 − d, since the second period
consumption behavior depends only on this difference (pE is a function of pI2 − d in the
equilibrium of period 2), the incumbent can capture the ex-ante expected consumer surplus
via the first-period price and this washes out the breakup revenues and period 2 sales revenues
from the incumbent’s profit function: the level of breakup fee matters only via its effect on
the second period consumption decisions. Hence, different levels of pI2 and d that lead to
the same difference pI2 − d will result in the same profit for the incumbent. Without loss of
generality we could set d = 0 in any subgame where the entrant has positive sales in period
2. But then the equilibrium outcome without breakup fees must be the equilibrium among
such subgames. In this equilibrium we have
p∗I2 − d∗ =
2θ + cE + 2cI
.
3
The previous analysis illustrates that if the incumbent accommodates the entrant, it
will have to compensate consumers ex-ante by lowering pI1 by the amount of breakup fees
and switching costs that consumers expect to pay in period 2 (when they switch to the
entrant). Hence, the incumbent cannot use breakup fees as a tool to shift rent from the
21
entrant. The incumbent’s second period profit must therefore be below static monopoly
profit, Πaccom
< v − cI , if it accommodates the entrant since accommodating the entrant will
I2
imply some differentiated competition in period 2 without shifting any rent from the entrant.
As a result, the two period profit of the incumbent from accommodating the entrant must
be below twice its static monopoly profit, Πaccom
< 2(v − cI ), and so the incumbent strictly
I
prefers to foreclose the entrant when consumers believe that the others sign the incumbent’s
first period contract.
Finally, we show the uniqueness of the symmetric equilibrium. Note that consumers
would be better off if they could coordinate and reject the incumbent’s first offer. To see
this suppose that all consumers reject the incumbent’s first contract. But then in period 2
the entrant wins the Bertrand competition by offering a price slightly below the incumbent’s
cost. Hence, consumers would get a surplus of v − cI if they could coordinate to reject
the incumbent’s offer. On the other hand, we show that rejecting the incumbent’s offer is
not credible for each consumer given that she believes the others to reject the incumbent’s
first offer. A consumer’s expected surplus from signing the incumbent’s first period contract
would be 2v − pI1 − cI since she expects cI to be the price in period 2 equilibrium. Each
consumer unilaterally prefers to sign the incumbent’s first period contract as long as pI1 ≤ v.
If the incumbent makes the equilibrium offer of Proposition 3 with a slightly lower first-period
price, p∗I1 < v, a consumer will be strictly better-off by taking this offer rather than rejecting
it. Since all consumers will have this incentive, their threat of rejecting that offer is not
credible. Hence, there exists no equilibrium where all consumers reject the incumbent’s
long-term contract.19
3.3
Discussion of the result with breakup fees
This result is surprising in particular because the incumbent always forecloses the more
efficient entrant and captures the total industry surplus under foreclosure, regardless of the
level of switching costs or efficiency advantage of the entrant.
Four things are crucial in obtaining our full-foreclosure result: 1) the entrant has market
power; 2) consumers are infinitesimal (non-pivotal), 3) the incumbent can use a break-up fee
in its long-term contract; 4) consumers who did not sign the incumbent’s contract do not
have intrinsic heterogeneity (as they do not face a switching cost).
When the entrant has market power, the entrant’s price depends on the incumbent’s
first-period contract. But then with infinitesimal consumers, as we discussed in Section 3.1,
consumers’ outside option (i.e., surplus from not signing the incumbent’s long-term contract)
19
A formal proof of Proposition 3 is available in our web Appendix
22
becomes endogenous. This is the key difference from the section where we had competitive
entrants and the incumbent implemented the efficient outcome in equilibrium. However,
with entrant market power and non-pivotal buyers, the incumbent distorts the equilibrium
prices to lower consumers’ expected outside option. In other words, the incumbent decreases
the size of the total pie to get a larger share of it. When the incumbent is allowed to use
a breakup fee in its first-period contract, the incumbent could guarantee to get twice of
its static monopoly profit by fully foreclosing the entrant. We show that this is greater
than the incumbent’s profit in the distorted outcome without breakup fees. Hence, the
incumbent prefers to foreclose the entrant with a sufficiently high breakup fee in its first
period contract. If the incumbent was not allowed to use a break-up fee, the more efficient
entrant would always have positive sales in period 2, as we illustrated in the analysis without
breakup fees.
The fourth crucial point implies undifferentiated asymmetric Bertrand competition between the entrant and the incumbent, which results in the spot market equilibrium prices
that are independent of the entrant’s cost.
It is important to note that switching costs do not matter for the equilibrium described
in Proposition 3. If there were no switching costs, the incumbent would still be able to
foreclose the efficient entrant by creating switching costs endogenously via breakup fees.
We have some other points that, we think, are important. First, the result does not
require ex-ante commitment to the second period price or commitment to the breakup fee
level since the incumbent is allowed to offer a spot price in period 2 and the incumbent does
not want to offer a lower breakup fee in period 2. In Aghion and Bolton [1987] the ex-ante
(before entry) commitment to the breakup fee is crucial to have inefficient foreclosure in
equilibrium: If the incumbent and the buyer were allowed to renegotiate their contract after
the entry decision is made, the incumbent would not be able to foreclose the entrant since
then in the case of entry the coalition of the incumbent and the buyer would be betteroff ex-post by setting a zero breakup fee in order to trade with the more efficient entrant.
Furthermore, ex-ante uncertainty on consumers’ intrinsic switching costs is not necessary to
have foreclosure of the more efficient entrant in equilibrium. This is another difference from
Aghion and Bolton [1987] where the uncertainty on the entrants’ cost level is necessary to
have inefficient foreclosure in equilibrium.
Second, offering a long-term contract converts a non-durable good (consumption today)
to a durable good (consumption today and tomorrow). The fact that the incumbent can
offer a spot price in the future generates a commitment problem for the incumbent similar to
the durable good monopolist’s commitment problem (Coase [1972]): after selling long-term
contracts in period 1, the incumbent might have an incentive to undercut the second period
23
price of the long-term contract when it competes against a more efficient entrant in period 2.
The incumbent overcomes this problem by committing to allow consumers to switch from its
long-term contract to its spot contract for free: This makes it too costly for the incumbent to
undercut the second unit price of the long-term contract in period 2 spot market, since such
undercutting would imply margin losses from all consumers (given that consumers could
switch from the long-term contract to the spot price freely). This commitment mechanism
is very similar to how price matching guarantees enable the durable good monopolist not to
undercut itself in the future (Butz [1990]). Hence, the incumbent would prefer to use this
commitment tool and make it free to switch from the long-term contract to its spot contract
if we allowed the incumbent choose whether to do that.
Finally, one might ask whether the result is dependent on the fact that the incumbent is
not using history based pricing. That is, suppose that the incumbent can offer a lower price
in period 2 to consumers who did not purchase the long term contract. This would give
the incumbent the incentive to price more aggressively in period 2 for any new customers.
From the incumbent’s point of view this is problematic, since it will raise the expected
consumer surplus from not signing the period 1 contract to v − cI due to the resulting
Bertrand competition in period 2. To prevent itself from competing aggressively in period 2
the incumbent can offer a most favored nations (customer) clause in its first period contract,
which will allow consumers who signed the first period contract to always obtain the lowest
price offered by the incumbent in future periods.
In the next section we will look at an alternative source of differentiation between the
firms by changing the interpretation of s from switching cost to mismatch value, that is,
the reduction in consumers’ valuation, if they buy the good from the entrant. The new
interpretation implies that consumers who did not sign the incumbent’s contract will have
to pay s if they buy from the entrant in period 2 and allows us to have downward sloping
demand also when consumers do not sign the incumbent’s first period contract.
4
Match value
Now assume that the utility from consuming the incumbent’s good is v as before, but the
value from consuming the entrant’s good is v − s such that consumers learn their draw
of s at the beginning of period 2 and s is uniformly distributed over (0, θ]. We interpret
s as the differentiation value of the incumbent’s product compared to the entrant’s and
assume that consumers do not know their match value to the entrant’s product before the
entrant appears. We assume that the consumption value is sufficiently high to ensure that
in the foreclosure scenario with breakup fees the incumbent’s optimal second period price is
24
below the consumption utility so that the entrant would not behave like a local monopoly
in equilibrium.
In the benchmark model, s was the switching cost paid by a consumer who signed the
incumbent’s contract in period 1 and switch to the entrant in period 2. Switching cost made
the incumbent’s product differentiated from the entrant’s for those consumers who signed the
incumbent’s first period contract. In that model if a consumer did not sign the incumbent’s
first period contract, she would not have to pay a switching cost if she buys from the entrant
in period 2, and so in that case the incumbent’s and the entrant’s products were homogenous.
However, here we interpret s as the incumbent’s differentiation value and so each consumer
loses s if she buys the entrant’s product rather than the incumbent’s product in period 2.
This is true even if the consumer did not sign the incumbent’s first period contract.
When the incumbent is not allowed to use a breakup fee, any consumer who signed
the incumbent’s first contract could buy at the lowest price of the incumbent or switch to
the entrant in period 2. This means that the incumbent and the entrant compete for all
consumers in period 2 (regardless of the amount of consumers who signed the incumbent’s
long-term contract). The solution to the differentiated duopoly competition determine the
second period equilibrium prices. Consumers with low match value relative to the price
difference of the incumbent and the entrant, s < pI2 − pE (given that in equilibrium we
have pSI2 = pI2 ), would buy from the entrant and the rest would buy from the incumbent. A
consumer’s outside option to the incumbent’s contract is now:
EUnosignI = v − pI2 P rob(s > pI2 −
p∗E )
Z
pI2 −p∗E
−
0
1
(s + p∗E ) ds.
θ
(5)
A consumer’s expected utility in case she signs the incumbent’s long-term contract is:
EUsignI = v − pI1 + EUnosignI .
Hence, the consumer gets the same expected surplus from period 2 consumption whether
she signs the incumbent’s first-period contract or not, and this does not depend on what
she believes about the other consumers’ behavior. In other words, the first period outcome
has no affect on the second period outcome. This implies that each consumer wants to sign
the incumbent’s contract if and only if pI1 ≤ v. At the optimal solution the incumbent sets
p∗I1 = v (as if it was static monopoly), p∗I2 = pS∗
I2 and the second period prices are given by
the equilibrium of the differentiated duopoly competition:
Proposition 4 In the match value interpretation, if the incumbent cannot use a breakup fee
in its long-term contract, there are no inter-temporal effects. In the unique equilibrium all
25
consumers sign the incumbent’s long-term contract.
• If ∆c ≥ 2θ, all consumers buy from the entrant in period 2. The equilibrium profits
and expected consumer utility are
Π∗I = v − cI , Π∗E = ∆c − θ,
θ
U ∗ = v − cI + .
2
• If ∆c < 2θ, some consumers switch to the entrant in period 2. The equilibrium profits
and expected consumer utility are
Π∗I
U∗
(2θ − ∆c)2 ∗
(θ + ∆c)2
= v − cI +
, ΠE =
,
18θ
9θ
θ + 2cE + cI (θ + ∆c)2
+
.
= v−
3
18θ
If the incumbent is allowed to have a breakup fee in its first contract, a nonzero breakup
fee generates inter-temporal effects. Suppose that the incumbent accommodates the entrant
to have some sales in period 2 by setting the net price of buying from the incumbent in
period 2 above the entrant’s marginal cost: pI2 − d ≥ cE . We first prove the existence of
an equilibrium where all consumers sign the incumbent’s long-term contract and then show
that this is the unique symmetric equilibrium.
If all consumers signed the incumbent’s long-term contract, the second period equilibrium
analysis is the same as the previous section since the incumbent’s first-period contract affects
the second period competition in the same way as the case of switching costs when all
consumers sign the incumbent’s first period contract.
Suppose that each consumer believes that the others sign the incumbent’s long-term
contract. A consumer’s expected utility from signing the incumbent’s long-term contract is
EUsignI = v − pI1 + v − pI2 P rob(s > pI2 −
p∗E
Z
− d) −
0
pI2 −p∗E −d
1
(s + p∗E + d) ds.
θ
The period 1 solution differs from the previous section’s analysis, since a consumer’s
outside option to the incumbent’s contract is now (as given by (5)) her expected surplus
from buying a unit in period 2 where she buys from the more efficient entrant and gets
v − s − p∗E if her s is low relative to the price difference between the firms and buys from
the incumbent and gets v − pI2 if her s is high relative to the price difference. This implies
that the incumbent’s equilibrium profit if it does not foreclose the entrant is higher than
the previous section, since now consumers’ outside options have a lower value given that
26
consumers have to pay s whenever they buy the entrant’s product, even when they did not
sign the incumbent’s first period contract.
If the incumbent forecloses the entrant by setting pI2 − d < cE , a consumer’s expected
utility from signing the incumbent’s first contract is
EUsignI = 2v − pI1 − pI2 ,
since she expects to buy from the incumbent in period 2 when she signs the incumbent’s
first period contract. A consumer’s expected utility if she does not sign the incumbent’s
first contract is again given by (5). Given pI2 − d < cE , the entrant knows that it cannot
attract any consumer from the incumbent’s first contract and therefore competes against the
incumbent for the consumers who did not sign the incumbent’s contract. In the spot market
the incumbent does not undercut its long-term contract’s second unit price, since lowering
price below pI2 would lead to a loss from measure 1 of consumers and some gains from
measure 0 of consumers. The entrant would not extract the entire surplus from consumers
who do not sign the incumbent’s first period contract (due to downward sloping demand),
so consumers’ outside option to signing the incumbent’s first contract (5) is above zero and
increasing in the entrant’s marginal cost. As a result, different from the previous section, the
incumbent’s profit from foreclosure is decreasing in the entrant’s cost advantage. Comparing
the incumbent’s foreclosure profit with the highest profit without foreclosure we prove our
main result:
Proposition 5 In the match value interpretation, if the incumbent is allowed to use a
breakup fee in its long-term contract, in the unique equilibrium all consumers sign the incumbent’s long-term contract.
• If ∆c ≥ 2θ, the entrant sells to all consumers in period 2. The equilibrium profits and
consumer utility are
Π∗I = v − cI , Π∗E = ∆c − θ,
U ∗ = v − cI + θ/2.
• If ∆c < 2θ, the incumbent forecloses the more efficient entrant. The equilibrium profits
and consumer utility are
Π∗I = v − cI + 2θ − ∆c, Π∗E = 0,
U ∗ = v − cE − 2θ.
27
When the entrant is not very efficient, the incumbent finds it profitable to block the
entry by compensating consumers for not buying from the more efficient entrant. It is too
costly for the incumbent to try to foreclose a very efficient entrant. In this case, even if
the incumbent uses a breakup fee, the first period outcome has no effect on the second
period since the incumbent cannot compete against the very efficient rival in period 2, so
sets the difference between the second period price and breakup fee at its marginal cost, and
compensates consumers for the breakup fee by lowering its first period price. In other words,
breakup fees do not make any difference for the equilibrium outcome when the entrant is very
efficient. The entrant serves the entire market in period 2 and captures its cost advantage
after compensating consumers for the costs of switching (like in the case without breakup
fees).
Note that the equilibrium is unique regardless of what consumers believe about the others’
behaviour. For instance, suppose each consumer believes that every other consumer does
not sign the incumbent’s contract. If ∆c < 2θ, each consumer expects the second period
equilibrium to be the differentiated duopoly outcome, like in the case without breakup fees,
Proposition 4. Hence, her expected surplus from not signing the incumbent’s contract is the
expected utility given in Proposition 4. Let pS∗
I2 denote the spot price of the incumbent at this
equilibrium. The expected utility from signing the incumbent’s contract with pI2 − d < cE
will be :
EUsignI = 2v − pI1 − min{pI2 , pS∗
I2 },
given that consumers expect not to switch to the entrant and switch to the incumbent’s spot
contract if it is cheaper. We show in the appendix that the expected utility from signing the
long-term contract is strictly greater than the expected utility from not signing it, and so
each consumer benefits from signing the incumbent’s contract even if she thinks that all the
others do not sign it. We also prove that this is the case when each consumer believes that
only 0 < α < 1 amount of consumers sign the incumbent’s long-term contract. Hence, even
if consumers would benefit from coordinating and not-signing the incumbent’s contract, they
unilaterally prefer to sign it, regardless of what they believe about the others’ behaviour.
This gives us the unique equilibrium where all consumers sign the incumbent’s long-term
contract.
5
Conclusions
We have investigated the effects of breakup or early termination fees in markets where buyers
are non-pivotal, like in the examples we give in the Introduction. We use a simple two-period
28
model of entry where the incumbent offers a long-term contract before facing an entrant,
but cannot commit not to offer a spot price in period 2, and consumers are willing to buy
one unit in each period. We find that when there are competitive entrants, the incumbent
does not foreclose the efficient entrants, even if it could do so by using breakup fees, and
the equilibrium outcome is efficient. Long-term contracts are necessary to implement the
efficient outcome since they enable the incumbent to commit not to exploit its locked-in
consumers in period 2. On the other hand, when an entrant has market power and consumer
heterogeneity is due to switching costs, then the incumbent will always foreclose the entrant
using breakup fees, no matter how much more efficient the entrant is than the incumbent.
Breakup fees enable the incumbent to control perfectly the consumers’ expected period 2
price: high breakup fee implies that consumers who signed the long-term contract will not
switch to the entrant and so for those consumers who did not sign the long-term contract
there will be homogenous asymmetric Bertrand competition between the incumbent and
the entrant. On the other hand, if consumer heterogeneity is due to different match values
for an entrant’s product, then the incumbent will foreclose the entrant using breakup fees
if the entrant’s efficiency advantage is sufficiently low. In this case, breakup fees do not
enable perfect control of expected second period price, since consumers who did not sign the
long-term contract are heterogenous and so period 2 demand is downward sloping. When
the entrant is sufficiently efficient, the incumbent allows the entrant serve the all market in
period 2 and this is efficient.
Our theoretical results suggest that in final product markets or intermediate good markets
with non-pivotal buyers if entry costs are very low, so potentially many firms compete to
enter, or when an entrant comes with a very efficient technology compared to the incumbent,
entry deterrence by breakup fees will be very unlikely. On the other hand, when an entrant’s
efficiency advantage is not very high, the incumbent might foreclose the entrant using a high
enough breakup fee. Regulating breakup fees might not mitigate potential entry deterrence
if the incumbent could raise consumers’ switching costs via other ways, like bundling services
with the long-term contract that consumers could not carry if they switch.
29
Appendices
A
A.1
Competitive entrants
Proof of Proposition 1
If ∆c < θ, the incumbent sets p∗I2 = cI and induces efficient switching: Consumers switch to
∗
the entrants if and only if s < ∆c. The incumbent’s second period demand is DI2
= θ−∆c
θ
∗
and the amount of consumers switching to the entrants is DE∗ = ∆c
.
Replacing
p
=
c
into
I
I2
θ
∗
the equality for pI1 , (4), determines the optimal up-front fee of the incumbent’s contract:
2
2
. In equilibrium the incumbent’s profit is Π∗I = v − cI − ∆c + (∆c)
and
p∗I1 = v − ∆c + (∆c)
2θ
2θ
the consumer surplus is v − cE .
If ∆c ≥ θ, the incumbent again sets p∗I2 = cI , but sells to nobody in period 2, that is,
all consumers switch to the entrants. The incumbent’s optimal first period price is then
1
1
p∗I1 = v − 2θ
. In equilibrium the incumbent’s profit is Π∗I = v − cI − 2θ
and the consumer
surplus is v − cE .
A.2
Proof of Corollary 3
As in the base model competitive entrants set p∗E = cE and the incumbent never sets a
spot price in period 2 that will be taken by any consumers. If a consumer does not sign
the incumbent’s contract in period 1, she gets v − cE . If a consumer signs the incumbent’s
contract in period 1, she expects to get
Z
pI −cE −d
EUsignI = 2v −pI −pI P r(s > pI −cE −d)−(cE +d)P r(s < pI −cE −d)−
0
s
ds. (6)
θ
The difference between (1) and (6) is that a consumer pays d if she breaks the contract and
this affects the probability that a consumer purchases from the entrant and the payment
that the consumer has to pay the incumbent for changing firms. All consumers sign the
incumbent’s contract in period 1 if and only if EUsignI ≥ v − cE . Under this constraint the
incumbent’s expected profit over the two periods is
Π = pI − cI + (pI − cI )P r(s > pI − cE − d) + dP r(s < pI − cE − d).
(7)
The differences between (3) and (7) are the probability that a consumer will switch will now
also depend on the breakup fee and the firm can use the fee as a revenue source.
First observe that the incumbent’s profit expression in (7) will be the same as in (3),
30
that is, the profit under the two-period contract, (pI1 , pI2 ), if it sets: pI − d = pI2 and
I2
, which in turn uniquely determine the unit price of the contract with breakup
d = pI1 −p
2
pI1 +pI2
fees: pI = 2 . But then the expected surplus from buying the incumbent’s contract (6)
is equivalent to the one in (1). Hence, there is one-to-one correspondence between these two
contract types. Using the equilibrium prices derived in Proposition 1 and the equivalence
I2
I2
, d = pI1 −p
, it is straightforward to show that the equilibrium prices
conditions, pI = pI1 +p
2
2
are
• If ∆c ≥ θ,
p∗I =
1
v − cI
1
v + cI
− , d∗ =
− , pS∗
≥ p∗I , p∗E = cE .
2
4θ
2
4θ I2
• If ∆c < θ,
p∗I
B
B.1
v + cE (∆c)2 ∗ v + cE (∆c)2
∗ ∗
+
,d =
+
− cI , pS∗
=
I2 ≥ pI , pE = cE .
2
4θ
2
4θ
Single entrant, switching costs
Proof of Proposition 2
To derive the equilibrium, we first assume that all consumers believe that each of their fellow
consumers will accept the incumbent’s first period offer and derive equilibrium payoffs. Next,
we find whether this equilibrium is unique for all parameter values and beliefs of consumers
about how other consumers will react to the incumbent’s offer.
Step 1: Equilibria where all consumers think each other consumer will accept the incumbent’s offer.
Period 2:
If nobody signed I’s contract in the first period, then the only equilibrium has the entrant
attracting all consumers at a price of cI . If all consumers signed I’s contract in the first period
and if the incumbent undercuts its long-term contract price, pSI2 < pI2 , then the incumbent’s
spot offer dominates its long-term contract price for consumers. Consumers with switching
costs lower than the difference between the incumbent’s spot price and the entrant’s price,
s < pSI2 − pE , switch to the entrant and the rest buy the incumbent’s spot contract. The
pS −p
entrant’s demand is then DE = I2 θ E and the incumbent’s demand for its spot contract is
θ−pS
I2 +pE
DI2 =
. The incumbent sets pSI2 , by maximizing its second period profit
θ
ΠI2 = (pSI2 − cI )
31
θ − pSI2 + pE
θ
θ+pE +cI
The best-reply of the incumbent to the entrant’s price is pS∗
if at that price
I2 (pE ) =
2
it has some positive demand: pS∗
I2 (pE ) − pE < θ. Otherwise, the incumbent cannot compete
S∗
against the entrant and sets pI2 = cI . The entrant sets pE by maximizing its profit
ΠE = (pE − cE )
pSI2 − pE
θ
pS +c
The best-reply of the entrant to the incumbent’s spot price is p∗E (pSI2 ) = I2 2 E if at this
price the entrant does not attract all consumers: pSI2 − p∗E (pSI2 ) < θ. Otherwise, the entrant
sets p∗E (pSI2 ) = pSI2 − θ and sells to all consumers.
The simultaneous solution to the best-replies determine the spot market equilibrium
prices and demands (in this sub-game where pSI2 < pI2 ). If ∆c < 2θ,
pS∗
I2 =
θ + 2cE + cI
2θ + cE + 2cI ∗
, pE =
,
3
3
2θ − ∆c ∗
θ + ∆c
∗
DI2
=
, DE =
.
3θ
3θ
(8)
If ∆c ≥ 2θ,
∗
pS∗
I2 = cI , pE = cI − θ,
∗
DI2
= 0, DE∗ = 1.
(9)
• Suppose ∆c < 2θ
– If pI2 > 2θ+cE3 +2cI , the previous equilibrium outcome, (8), prevails since then in
period 2 the incumbent sets pS∗
I2 < pI2 .
– If pI2 ≤ 2θ+cE3 +2cI , the incumbent does not undercut its long-term contract price,
pS∗
I2 ≥ pI2 , and so the spot contract is dominated by the long-term contract of the
incumbent. Consumers with switching costs s < pI2 − pE , switch to the entrant
and the rest buy from the incumbent at price pI2 . The entrant’s demand is then
E
DE = pI2 −p
. The entrant sets its best-reply to the incumbent’s first period
θ
−cE
∗
E
offer, pE (pI2 ) = pI2 +c
. The entrant sells to consumers of measure DE = pI22θ
2
and the rest of the consumers continue buying from the incumbent at the longI2 +cE
term contract price: DI2 = 2θ−p2θ
. Note that the incumbent’s demand will be
2θ+cE +2cI
positive: pI2 − cE < 2θ, since pI2 ≤
and ∆c < 2θ.
3
• Suppose now that ∆c ≥ 2θ. The entrant is very efficient and the incumbent cannot
sell in the second period spot market and the incumbent’s best-reply is pS∗
I2 = cI . The
32
entrant sells all consumers at p∗E = cI − θ.
Period 1:
First consider the case where ∆c < 2θ. In equilibrium, the incumbent should set pI2 ≤
2θ+cE +2cI
in order to induce the sub-game where in the second period the incumbent does
3
not have an incentive to undercut its long-term contract price, pS∗
I2 ≥ pI2 (in the spirit
of renegotiation-proof contracts). Given this in the continuation game, the entrant sets
E
. If a consumer signs the incumbent’s first period contract, she expects
p∗E (pI2 ) = pI2 +c
2
−cE
and expects to
to switch to the entrant with probability P r(s < pI2 − p∗E (pI2 )) = pI22θ
I2 +cE
. So a consumer’s
continue buying from the incumbent at price pI2 with probability 2θ−p2θ
expected surplus from signing the incumbent’s contract is the two-period expected surplus
from consumption minus the expected switching costs:
EUsignI
2θ − pI2 + cE
pI2 − cE
= 2v − pI1 − pI2
− p∗E (pI2 )
−
2θ
2θ
Z
pI2 −p∗E (pI2 )
0
s
ds.
θ
If a consumer does not sign the incumbent’s first-period contract, she expects that the
entrant will set p∗E (pI2 ) since she expects that the other consumers will sign the incumbent’s
offer and her decision of not signing the incumbent’s contract does not change the entrant’s
E
<
equilibrium price given that each consumer is infinitesimal. Observe that p∗E (pI2 ) = pI2 +c
2
pI2 since pI2 ≥ cI > cE . This implies that p∗E (pI2 ) < pS∗
I2 , and so a consumer would prefer to
buy from the entrant if she did not sign the incumbent’s first contract. A consumer’s surplus
from not signing the incumbent’s contract is therefore the net surplus of buying from the
entrant in period 2: EUnosignI = v − p∗E (pI2 ). Consumers buy the incumbent’s long-term
contract if and only if their expected surplus from buying is greater than their outside option:
EUsignI ≥ EUnosignI . Under the latter constraint the incumbent’s expected profit is
2θ − pI2 + cE
.
2θ
It is optimal for the incumbent to set the highest upfront fee satisfying the participation
constraint of consumers:
ΠI = pI1 − cI + (pI2 − cI )
2θ − pI2 + cE
(pI2 − p∗E (pI2 ))2
−
.
2θ
2θ
After replacing the equality of the optimal upfront fee into the incumbent’s profit, its
profit becomes a function of the second period unit price:
p∗I1 (pI2 ) = v + (p∗E (pI2 ) − pI2 )
Π∗I (pI2 ) = v − cI + (p∗E (pI2 ) − cI )
2θ − pI2 + cE
(pI2 − p∗E (pI2 ))2
−
.
2θ
2θ
33
The incumbent’s unconstrained optimal p∗I2 is the one that maximizes its profit:
p∗I2 =
2θ + cE + 2cI
.
3
Observe that p∗I2 = pS∗
I2 , so this satisfies the constraint ensuring that the incumbent does
∗
not want to undercut pI2 in the second period spot market. This is due to the fact, that
the incumbent is acting as a Stackelberg leader in period 2 and just like in that model, the
leader would not want to raise its output (lower its price in our model). The first-period
unit price is then p∗I1 = p∗I1 (p∗I2 ). Hence, we conclude that if ∆c < 2θ, the equilibrium prices
are
(∆c + θ)(5θ − ∆c)
,
18θ
2θ + cE + 2cI
,
= pS∗
I2 =
3
θ + 2cE + cI
=
.
3
p∗I1 = v −
p∗I2
p∗E
The incumbent sells all consumers in period 1. In period 2
DE∗ =
θ + cI − cE
2θ − p∗I2 + cE
=
2θ
3θ
amount of consumers switch to the entrant and the rest,
∗
DI2
=
2θ + cE − cI
2θ − p∗I2 + cE
=
,
2θ
3θ
continue buying from the incumbent at p∗I2 . The resulting equilibrium profits are
Π∗I
Π∗E
θ2 − 4∆cθ + ∆c2
= v − cI +
,
6θ
(∆c + θ)2
=
.
9θ
On the other hand, if ∆c ≥ 2θ, the entrant is very efficient compared to the incumbent
and so the incumbent cannot compete against the entrant in period 2; it does not find
it profitable to keep any consumers when the entrant’s costs are at least twice the highest
possible consumer switching cost. As a result, in equilibrium, the incumbent sets p∗I1 = v − 2θ ,
∗
p∗I2 = pS∗
I2 = cI , the entrant sets pE = cI − θ, all consumers buy the incumbent’s long-term
contract in period 1 and switch to the entrant in period 2. The resulting profits will be
Π∗I = v − cI − 2θ , Π∗E = ∆c. Observe that here the incumbent has to compensate consumers
34
for the expected switching costs, 2θ .
Step 2. Uniqueness. To see whether the equilibrium is unique, we need to see if consumers
have a credible outside option to all reject the incumbent’s offer and get a payoff of v − cI ,
which is the unique equilibrium utility if all reject the offer.
A consumer’s expected equilibrium payoff that we computed in step 1 was (when ∆c < 2θ)
U ≡v−
θ + 2cE + cI
.
3
If this value is less than v − cI , then consumers do not have a credible threat to reject the
incumbent’s offer of U from step 1. Thus, we have a unique equilibrium for 2θ > ∆c > θ/2.
When ∆c ≥ 2θ, a consumer’s expected equilibrium payoff that we computed in step 1 was
v − cI . Hence, we have a unique equilibrium for ∆c > θ/2.
Now, we argue that the equilibrium is unique for ∆c ≤ θ/2. The difference between
the argument for this case and for when ∆c > θ/2, is that U < v − cI in this case. The
distinction is that now the consumers outside option if they all say no to the incumbent’s
offer is higher than the equilibrium utility that the incumbent offered when all consumers
believe that all other consumers will accept the offer.
So, we need to check if the consumers can credibly reject an offer that gives them more
than U up to v − cI . Suppose that consumers coordinate to reject any offer from the
incumbent in period 1 that gives them less expected utility than U ∈ (U ,v − cI ]. Suppose
that the incumbent makes the offer of U , which will include a price of pI1 = p∗I1 < v. It
cannot be an equilibrium that all consumers will reject the offer. A consumer could deviate,
accept the offer, and improve her payoff. To see this, the continuation equilibrium has the
incumbent and the entrant both offering a price of cI , no matter what an individual consumer
does since she does not affect the state next period. The deviating consumer improves her
payoff by accepting the contract, since v − p∗I1 + v − cI > v − cI . Since all consumers will
have this incentive, their threat of rejecting U is not credible.
Finally, can an equilibrium where only some consumers accept the incumbent’s offer
exist? If a measure α consumers were to accept an incumbent’s contract in period 2, then
for spot prices that gives the incumbent positive second period sales and the entrant’s price
are
θ(1 + α)/α + cE + 2cI ∗
θ(2 − α)/α + 2cE + cI
pS∗
, pE =
.
I2 =
3
3
We should note that for small α no consumers will buy from the incumbent in period 2.20
∗
In this case, the equilibrium prices will be pS∗
I2 = cI and pE = cI − θ. This case corresponds to
the equilibrium outcome when ∆c > 2θ and so consumers do not have an incentive to deviate from this
equilibrium given that they get v − cI .
20
35
The consumers who reject the offer have a utility of
U (α) = v −
θ(2 − α)/α + 2cE + cI
.
3
It is straightforward to show that U (α) <U for any α < 1. But, this contradicts the fact
that consumers will reject offers that give them some utility U ∈ (U , v − cI ].
C
Single entrant, match value
C.1
Proof of Proposition 4
When the incumbent’s long-term contract cannot have a breakup fee, the incumbent and the
entrant are differentiated competitors for all consumers in period 2 and the solution to the
differentiated duopoly competition determine the equilibrium prices. This is true regardless
of who signed the incumbent’s first period contract given that consumers could switch from
the incumbent’s long-term contract to the entrant at no cost. The entrant’s demand will be
DE = P rob(s < pSI2 − pE ) and the incumbent’s demand will be DI2 = P rob(s > pSI2 − pE ).
Observe that these demands and so the solution of the second period equilibrium are the
same as those of the model with switching costs when the incumbent’s first contract cannot
have a breakup fee and all consumers signed the incumbent’s first contract, see Proposition
2 and its proof. We summarize below the steps of this solution:
pS +c
The entrant’s best-reply to the incumbent’s spot price is p∗E = I2 2 E if pSI2 ≤ min{2θ +
E
cE , v}.21 If pSI2 > v, the entrant will be local monopoly with price p∗E = v+c
and demand
2
v−cE
∗
S
DE = 2θ (this will not happen in equilibrium). If 2θ + cE < pI2 ≤ v, the entrant sells all
consumers at p∗E = pSI2 − θ. Similarly the incumbent’s best-reply to the entrant’s price is
θ+pE +cI
pS∗
if pSI2 ≤ min{θ + cI , v}. If pE > v, the incumbent will be the local monopoly
I2 =
2
v+cE
S∗
E
with price pS∗
and demand DI2
. If θ + cI < pE ≤ v, the incumbent sells all
= v−c
I2 =
2
2θ
consumers by setting pS∗
I2 = pE . If ∆c < 2θ, the interior solution to the duopoly competition
is
pS∗
I2 =
2θ + cE + 2cI ∗
θ + 2cE + cI
, pE =
3
3
2θ − ∆c ∗
θ + ∆c
S∗
DI2 =
, DE =
3θ
3θ
(10)
2θ+cE +2cI
We verify that v > pS∗
by Assumption 1 and so there is active competition
I2 =
3
between the incumbent and entrant. If ∆c ≥ 2θ, the equilibrium prices are pS∗
I2 = cI and
21
It cannot be the case that cE > pSI2 in equilibrium since the incumbent is less efficient than the entrant.
36
∗
p∗E = cI − θ, and the entrant serves all consumers: DI2
= 0, DE∗ = 1.
In equilibrium we must have pI2 ≤ pS∗
I2 , since otherwise the incumbent would have a
profitable deviation from pI2 to implement pS∗
I2 in period 2 and so pI2 would not be paid by
pI2 +cE
∗
.
any consumer. Given pI2 ≤ pS∗
I2 , the entrant’s best-reply will be pE =
2
We now look for an equilibrium where all consumers signed the incumbent’s first contract.
Suppose that each consumer expects every other consumer to sign the incumbent’s first
period contract. The solution of period 1 differs from the previous section analysis since
a consumer’s outside option to the incumbent’s contract is now her expected surplus from
buying a unit in period 2 where she buys from the more efficient entrant and gets v − s − p∗E
if her s is low relative the price difference between the firms and buys from the incumbent
and gets v − pI2 if her s is high relative to the price difference (given that we have pI2 = pS∗
I2
in equilibrium):
EUnosignI = v − pI2 P rob(s > pI2 −
p∗E )
Z
pI2 −p∗E
−
0
1
(s + p∗E ) ds,
θ
(pI2 − p∗E )2
= v − pI2 +
.
2θ
(11)
Observe that period 1 has no effect on period 2 competition, since a consumer’s expected
utility in case she signs the incumbent’s first contract is equal to
EUsignI = v − pI1 + v − pI2 P rob(s > pI2 −
p∗E )
Z
pI2 −pE
−
0
1
(s + p∗E ) ds
θ
= v − pI1 + EUnosignI .
Hence, a consumer wants to sign the incumbent’s contract if and only if pI1 ≤ v. At the
optimal solution the incumbent sets p∗I1 = v (as if it was static monopoly) and the second
period prices are given by the equilibrium of the differentiated duopoly competition, (10).
In equilibrium consumers get their expected outside option:
(p∗I2 − p∗E )2
2θ
2θ + cE + 2cI (θ + ∆c)2
= v−
+
.
3
2θ
∗
EUnosignI
= v − p∗I2 +
Observe that this is the unique equilibrium, since each consumer expects to get exactly the
same second period payoff as her outside option in case she signs the incumbent’s contract,
and so she prefers to sign the incumbent’s first contract as long as pI1 ≤ v. This is true
regardless of her beliefs about the others’ behaviour.
37
C.2
Proof of Proposition 5
If the incumbent is allowed to have a breakup fee in its first contract, a nonzero breakup
fee generates inter-temporal effects. The incumbent has two options: accommodate the
entrant by setting pI2 − d ≥ cE or foreclose the entrant by pI2 − d < cE . We now derive
the incumbent’s expected profit from each option to determine the incumbent’s equilibrium
strategy.
Period 2: If none of the consumers signed the incumbent’s first period contract, the second
period equilibrium prices and demands are given by the equilibrium of the differentiated
Bertrand competition between the incumbent and the entrant (like in the solution of the
case without breakup fees). If ∆c < 2θ, the equilibrium outcome is given by (10). If
S∗
S∗
= 0, DE∗ = 1.
= cI and p∗E = cI − θ, the entrant srves all consumers: DI2
∆c ≥ 2θ, pI2
Suppose that all consumers signed the incumbent’s first period contract where the incumbent sat pI2 − d ≥ cE allowing the entrant to have some sales in period 2. Given the period
1 contract, the entrant’s demand in period 2 is DE = P rob(s < min{pSI2 , pI2 } − pE − d) and
S
= P rob(s > min{pSI2 , pI2 } − pE − d). By solving
the incumbent’s period 2 demand is DI2
the problem of the incumbent and the entrant, similar to the proof of Proposition 3, we can
show that if ∆c < 2θ, the interior solution to the duopoly competition prevails:
pS∗
I2 − d =
2θ + cE + 2cI ∗
θ + 2cE + cI
, pE =
3
3
2θ
−
∆c
θ
+ ∆c
S∗
DI2
=
, DE∗ =
.
3θ
3θ
(12)
∗
If ∆c ≥ 2θ, the equilibrium prices will be pS∗
I2 − d = cI and pE = cI − θ, and the entrant
S∗
serves all consumers: DI2
= 0, DE∗ = 1.
In equilibrium we must have pI2 ≤ pS∗
I2 , since otherwise the incumbent would have a
profitable deviation from pI2 to implement pS∗
I2 in period 2 and so pI2 would not be paid
S∗
by any consumer. Given pI2 ≤ pI2 and pI2 − d ≥ cE , the entrant’s best-reply will be
E
(this can be shown by solving the entrant’s problem, like we did in the proof
p∗E = pI2 −d+c
2
of Proposition 3.)
Period 1: As usual we will first show that there exists an equilibrium where each consumer
signs the incumbent’s long-term contract and then show that this is the unique equilibrium
for any belief that each consumer has on the others’ behaviour.
Suppose that each consumer expects every other consumer to sign the incumbent’s first
period contract. A consumer’s expected utility if she does not sign the incumbent’s first
38
contract is the same expression as the case without breakup fees, (11):
EUnosignI
(pI2 − p∗E )2
= v − pI2 +
,
2θ
E
. A consumer’s expected utility if she signs
but here the entrant’s best-reply is p∗E = pI2 −d+c
2
the incumbent’s first contract is equal to
EUsignI = 2v − pI1 − pI2 P rob(s > pI2 −
p∗E
Z
pI2 −p∗E (pI2 )−d
− d) −
0
1
(s + p∗E + d) ds.
θ
The incumbent maximizes its profit subject to the consumers’ participation constraint,
EUsignI ≥ EUnosignI , the second period price equilibrium, pI2 ≤ pS∗
I2 , and the constraint
that ensures some positive sales by the entrant, pI2 − d < cE :
maxpI1 ,pI2 ,d ΠI = maxpI1 ,pI2 ,d [pI1 − cI + (pI2 − cI )P rob(s > pI2 − p∗E − d) + dP rob(s < pI2 − p∗E − d)]
st.(i)EUsignI ≥ EUnosignI , (ii)pI2 ≤ pS∗
I2 , (iii)pI2 − d < cE .
At the optimal solution the incumbent sets the highest pI1 satisfying the participation constraint, (i):
p∗I1
(pI2 − p∗E )2
− pI2 P rob(s > pI2 − p∗E − d) −
= v + pI2 −
2θ
Z
pI2 −p∗E −d
0
1
(s + p∗E + d) ds.
θ
Replacing the latter into the incumbent’s profit we rewrite its problem:
Z pI2 −p∗E −d
(pI2 − p∗E )2
1
∗
maxpI2 ,d [v − cI + pI2 −
− cI P rob(s > pI2 − pE − d) −
(s + p∗E ) ds]
2θ
θ
0
S∗
st.(ii)pI2 ≤ pI2 , (iii)pI2 − d < cE .
Given that p∗E =
pI2 −d+cE
2
the first-order conditions with respect to pI2 and d are, respectively
2pI2 = d + cI + cE + 2θ,
2d = pI2 − cI .
If ∆c < 2θ the solution to the previous first-order conditions gives us the equilibrium prices:
cI + 2cE + 4θ ∗ 2θ − ∆c
,d =
3
3
θ + 2cE + cI
=
3
p∗I2 =
p∗E
39
22
since the unconstrained optimal prices satisfy the second period price constraint: p∗I2 = pS∗
I2 ,
and the constraint ensuring some positive sales to the entrant: p∗I2 − d∗ = cE +2c3 I +2θ > cE
given that cI > cE and θ > 0. On the other hand, if ∆c ≥ 2θ, the incumbent cannot compete
∗
∗
against the entrant, so sets p∗I2 − d∗ = pS∗
I2 − d = cI and cI − d > cE (accommodating the
entrant), and the entrant reacts by setting p∗E = cI − θ. In this case, the entrant sells all
consumers in period 2 (this is efficient) and the incumbent captures its static monopoly
profit, Π∗I = v − cI , by collecting p∗I1 = v − d∗ upfront.23
Observe that the second period demands and the entrant price are the same as the
previous section where s referred to switching cost and the incumbent was not allowed to
use breakup fees (see Proposition 2 and its proof). The equilibrium profit of the incumbent
is different from the latter case since now consumers’ outside option (equilibrium utility) has
a lower value:
U∗ = v −
θ
θ + 2cE + cI
θ + 2cE + cI
− <U =v−
,
3
2
3
since consumers have to pay s whenever they buy the entrant’s product, regardless of the fact
that they did not sign the incumbent’s first contract. Hence, the incumbent’s equilibrium
profit if it does not foreclose the entrant is 2θ higher than the case with switching costs
(without breakup fees):
Π∗I = v − cI +
θ2 − 4∆cθ + ∆c2 θ
+ .
6θ
2
(13)
We now derive the incumbent’s highest profit if it forecloses the entrant to see whether/when
we could have foreclosure in equilibrium. If the incumbent sets pI2 − d < cE , the entrant
knows that it cannot attract any consumer from the incumbent’s first contract and therefore
competes for the consumers who did not sign the incumbent’s contract. The entrant’s bestpS +c
reply price is therefore p∗E = I2 2 E (under the condition that pSI2 ≤ v). The incumbent’s
optimal spot price is pS∗
I2 = pI2 since lowering price below pI2 would lead to a margin loss
from measure 1 of consumers and a market share gain from measure 0 of consumers. Hence,
pI2 +cE
∗
in equilibrium of the second period we have pS∗
.
I2 = pI2 and pE =
2
A consumer’s expected utility from signing the incumbent’s first contract is
EUsignI = 2v − pI1 − pI2 ,
cI +2cE +2θ
Recall that if ∆c < 2θ, pS∗
+ d, and if ∆c ≥ 2θ, pS∗
I2 =
I2 = cI .
3
In this case, the incumbent is indifferent between any level of d as long as d < ∆c, that is, breakup
fees are washed out in the incumbent’s profit since the incumbent has to compensate consumers ex-ante for
expected amount of breakup fees to convince them to sing the long-term contract.
22
23
40
since she expects to buy from the incumbent in period 2 when she signs the incumbent’s
first period contract. A consumer’s expected utility if she does not sign the incumbent’s first
contract is again given by (11):
EUnosignI
(pI2 − p∗E )2
,
= v − pI2 +
2θ
E
.
where the entrant’s best-reply is p∗E = pI2 +c
2
The incumbent maximizes its profit subject to the consumers’ participation constraint
and the foreclosure constraint:
maxpI1 ,pI2 ,d ΠI = maxpI1 ,pI2 ,d [pI1 − cI + pI2 − cI ]
st.(i)EUsignI ≥ EUnosignI , (ii)pI2 − d < cE .
At the optimal solution the incumbent sets the highest pI1 satisfying the participation constraint:
(pI2 − p∗E )2
.
p∗I1 = v −
2θ
Replacing the latter into the incumbent’s profit we rewrite its problem as:
(pI2 − p∗E )2
]
2θ
st.(ii)pI2 − d < cE .
maxpI2 ,d [v − 2cI + pI2 −
Given that p∗E =
rium price:
pI2 +cE
2
the first-order condition with respect to pI2 determines the equilib-
p∗I2 = 4θ + cE .
and the incumbent sets a sufficiently high breakup fee to foreclose the entrant: d∗ > 4θ. The
equilibrium price of the entrant and upfront fee of the incumbent are then respectively
p∗E = 2θ + cE , p∗I1 = v − 2θ.
The incumbent’s profit from foreclosure is
Π∗I = v − cI + 2θ − ∆c.
(14)
Comparing the latter profit with the incumbent’s highest profit without foreclosure (13) we
conclude that the incumbent prefers to foreclose the entrant if and only if ∆c < 2θ.
41
To sum up, if consumers believe that all the other consumers sign the incumbent’s contract, the following unique equilibrium prevails:
• If ∆c < 2θ, all consumers sign the incumbent’s long-term contract and continue buying
from the incumbent in period 2, that is, the incumbent forecloses the entrant. The
equilibrium prices, demands and payoffs are then
∗
p∗I1 = v − 2θ, d∗ > 4θ, p∗I2 = pS∗
I2 = 4θ + cE , pE = 2θ + cE
∗
∗
DI1
= DI2
= 1, DE∗ = 0
Π∗I = v − cI + 2θ − ∆c, Π∗E = 0.
(15)
• If ∆c ≥ 2θ, all consumers sign the incumbent’s long-term contract and switch to the
entrant in period 2. The equilibrium prices, demands and payoffs are then
∗ ∗
p∗I1 = v − d∗ , d∗ < ∆c, p∗I2 = pS∗
I2 = cI + d , pE = cI − θ
∗
∗
DI1
= 1, DI2
= 0, DE∗ = 1
Π∗I = v − cI , Π∗E = ∆c − θ.
(16)
Uniqueness:
We will now show that the previously characterized equilibrium is the unique equilibrium
for any belief of consumers about the others’ behavior. To do so suppose that each consumer
believes that α < 1 measure of consumers sign the incumbent’s first period contract and
1 − α > 0 of consumers do not sign the long-term contract.
Consider first the case of ∆c < 2θ. Given the incumbent sets pI2 − d < cE , consumers
believe that in the second period locked-in consumers will continue buying from the incumbent and there will be differentiated competition between the incumbent and the entrant for
the consumers who did not sign the incumbent’s long-term contract. So they believe that in
the second period the demands will be
DI2 = α + (1 − α)P r(s > pSI2 − pE )
DE = (1 − α)P r(s < pSI2 − pE ).
The incumbent maximizes its second period profit, ΠI2 = (pSI2 − cI )DI2 , and so its best-reply
spot price will be
pS∗
I2 =
cI + p E + θ
αθ
+
.
2(1 − α)
2
42
Similarly, the entrant maximizes its profit, ΠE = (pE − cE )DE , and so its best-reply will be
p∗E =
pI2 + cE
.
2
The simultaneous solutions to these best-replies imply that the second period prices will be
2αθ
2cI + cE + 2θ
+
,
3(1 − α)
3
αθ
cI + 2cE + θ
p∗E =
+
.
3(1 − α)
3
pS∗
I2 =
(17)
. However,
if the incumbent’s spot demand is positive at these prices, that is, if α < 2θ−∆c
3θ−∆c
2θ−∆c
if α ≥ 3θ−∆c , the incumbent prefers not to compete (for non-locked-in consumers) against
the entrant and not to undercut its long-term contract’s second period price: pS∗
I2 = pI2 . In
this case the foreclosure equilibrium prevails since consumers’ expected outside option will
be the same as the one when they all believe that the others sign the incumbent’s contract
and the rest of the analysis will be the same as above.
. In this case,
Now, we analyze whether the foreclosure equilibrium is unique if α < 2θ−∆c
3θ−∆c
a consumer’s expected surplus from not signing the incumbent’s long-term contract will be
EUnosignI = v − pS∗
I2 +
∗ 2
(pS∗
I2 − pE )
,
2θ
∗
where pS∗
I2 and pE are given by (17). Suppose that the incumbent offers the equilibrium
long-term contract, p∗I1 = v − 2θ, d∗ > 4θ, p∗I2 = 4θ + cE ,. If a consumer signs this contract,
S∗
she expects to pay pS∗
I2 in period 2 since 4θ +cE > pI2 and she can switch to the spot contract
of the incumbent at no cost. But then the consumer’s expected surplus from signing the
incumbent’s long-term contract will be
EUsignI = 2v − pI1 − pS∗
I2 .
Consumers sign the incumbent’s long-term contract if and only if EUsignI > EUnosignI or
∗ 2
(pS∗
I2 − pE )
2θ
2
(θ + ∆c)
(αθ)2
<v−
−
.
18θ
18θ(1 − α)2
v − pI1 >
⇐⇒ pI1
43
which is the case at the equilibrium price, p∗I1 = v − 2θ, if and only if
(αθ)2
(θ + ∆c)2
+
< 2θ.
18θ
18θ(1 − α)2
(αθ)2
18θ(1−α)2
(2θ−∆c)2
.
18θ
Observe that
goes to
is increasing in α and when α goes to the upper bound, this fraction
(αθ)2
(2θ − ∆c)2
.
=
3θ−∆c 18θ(1 − α)2
18θ
limα→ 2θ−∆c
The latter implies that, for α <
2θ−∆c
,
3θ−∆c
(θ + ∆c)2
(αθ)2
5θ2 − 2θ∆c + 2∆c2
(θ + ∆c)2 (2θ − ∆c)2
+
+
=
.
<
18θ
18θ(1 − α)2
18θ
18θ
18θ
2
2
Besides, observe that the fraction 5θ −2θ∆c+2∆c
is maximized at ∆c =
18θ
θ
value of this fraction is 4 . But then we prove that
θ
2
and the maximum
(αθ)2
θ
(θ + ∆c)2
+
< < 2θ.
2
18θ
18θ(1 − α)
4
and so EUsignI > EUnosignI if the incumbent offers the equilibrium long-term contract.
Hence, the foreclosure equilibrium we described in (15) is the unique equilibrium if ∆c < 2θ.
Finally, if ∆c ≥ 2θ and the incumbent offers p∗I1 = v − d∗ , d∗ < ∆c, p∗I2 = cI + d∗ ,
consumers anticipate that in the second period all consumers buy from the entrant at price
p∗E = cI − θ and expect to get disutility 2θ from buying the entrant’s product in period 2
(this is true regardless of what they believe about the others’ behaviour). Hence, consumers’
expected surplus from not signing the long-term contract is EUnosignI = v − cI + 2θ and the
expected surplus from signing the offered contract is
EUsignI = 2v − pI1 − p∗E −
θ
θ
= v − cI + d∗ + ,
2
2
Consumers are strictly better-off if they sign the incumbent’s long-term contract, EUsignI >
EUnosignI , if and only if d∗ > 0, which the incumbent can offer in the equilibrium contract, as
the incumbent is indifferent between any level of d∗ such that d∗ < ∆c. Hence, this concludes
that the efficient equilibrium we described in (16) is the unique equilibrium if ∆c ≥ 2θ and
d∗ > 0.
44
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