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A joint initiative of Ludwig-Maximilians University’s Center for Economic Studies and the Ifo Institute for Economic Research
Area Conference on
Applied
Microeconomics
11 – 12 March 2011 CESifo Conference Centre, Munich
Leaving the Door Ajar:
Nonlinear Pricing by a Dominant Firm
Philippe Choné and Laurent Linnemer
CESifo GmbH
Poschingerstr. 5
81679 Munich
Germany
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Leaving the door ajar:
Nonlinear pricing by a dominant firm
Philippe Choné∗and Laurent Linnemer†
March 2011, preliminary: do not quote without authorization
Abstract
An incumbent firm and a single buyer may sign a supply contract before a competitor can negotiate with the buyer. The competitor’s relative efficiency and the
share of the demand he can address are random. When only the former characteristics is unknown at the time of the contract, the parties may rely on a two-part tariff.
When both characteristics are unknown, they use a truly nonlinear price schedule.
In both situations, uncertainty yields inefficient exclusion. Partial foreclosure, however, arises only under two-dimensional uncertainty. We link the shape of the tariff
to the distribution of the uncertainty.
Contents
1 Introduction
2
2 The model
6
3 Negotiation between the buyer and
3.1 Maximization of the joint surplus .
3.2 Examples . . . . . . . . . . . . . .
3.3 Implementable quantity functions .
the entrant
7
. . . . . . . . . . . . . . . . . . . . . . 8
. . . . . . . . . . . . . . . . . . . . . . 9
. . . . . . . . . . . . . . . . . . . . . . 11
4 Designing the price schedule
4.1 Comparing effective price and incremental cost
4.2 Virtual surplus and elasticity of entry . . . . .
4.3 The optimal quantity function . . . . . . . . . .
4.4 Constructing the optimal tariff . . . . . . . . .
5 The
5.1
5.2
5.3
5.4
shape of the optimal price schedule
Elasticity of entry constant in sE . . . .
Elasticity of entry increasing in sE . . .
Elasticity of entry decreasing in sE . . .
Elasticity of entry non monotonic in sE
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∗
CREST
CREST. corresponding author:
[email protected].
†
15, bd Gabriel Péri 92245 Malakoff cedex, France.
1
lau-
6 Disposal costs, buyer opportunism, and conditional tariffs
23
7 Discussion
24
7.1 The entrant’s bargaining power . . . . . . . . . . . . . . . . . . . . . . . . 25
7.2 The marginal cost of the incumbent is not constant . . . . . . . . . . . . . 25
7.3 Antitrust implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
A Inefficient entry never arises in equilibrium
28
B Derivation of the ptimal quantity function
29
C The buyer’s total purchases are not lower than his total demand
31
D Complete information
32
E Implementation
32
E.1 Flat portions of the boundary functions . . . . . . . . . . . . . . . . . . . 34
E.2 The shape of the boundary function and the buyer-entrant coalition’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
F Finite disposal costs
35
F.1 Proof of Proposition 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
F.2 Proof of lemma 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1
Introduction
Nonlinear pricing, an ubiquitous business conduct, keeps attracting the attention of competition agencies.1 The firms under scrutiny had market power, and antitrust enforcers
were concerned that quantitative rebates might have discouraged buyers from switching
part of their requirements towards as efficient competitors (for fear of losing rebates). As
a result, efficient competitors might have been driven out of the market or marginalized,
and, in any case, prevented from selling the efficient quantity.
The nonlinear pricing literature has considered monopoly or oligopoly situations.2
The monopolist’s pricing problem has first been studied when consumers differ through
a single unobserved characteristic (e.g. Mussa and Rosen (1978) and Maskin and Riley
(1984), then extended in a multidimensional settings (Armstrong (1996), Rochet and
Choné (1998) and Armstrong (1999)). Nonlinear pricing under oligopoly is studied in
Armstrong and Vickers (2001), Martimort and Stole (2009), and Armstrong and Vickers
(2010), mainly focusing on symmetric equilibria of simultaneous games.
The present article considers a dominant firm on a market where a smaller firm
is present and may challenge its position, at least to a certain extent. We build on
the generic incumbency model. The basic setting has three players: an incumbent, a
strategic entrant or a competitive fringe, and a buyer. First, the incumbent and the
1
Among recent antitrust cases, Virgin/British Airways (Commission Decision 2000/74/EC of 14 July
1999), Michelin II (Commission decision of 20 June 2001 COMP/E-2/36.041/PO), in 2006 ProkentTomra (COMP/E-1/38.113), and in 2009 Intel (COMP/C-3 /37.990 ).
2
See Wilson (1993) for an overview of the seminal papers.
2
buyer negotiate a sale contract. Next, the entrant and the buyer learn the characteristics
of the good produced by the entrant. Finally, the entrant and the buyer negotiate a sale
contract taken as given the contract established between the incumbent and the buyer.3
This model has been already used to study nonlinear pricing by a dominant firm under
complete information.4
As regards incomplete information, Aghion and Bolton (1987) remains, to the best of
our knowledge, the only model where the characteristics of the entrant (his cost or more
generally, the surplus he creates with the buyer) are unknown to the contracting parties
at the time of the negotiation.5 We enrich their model to study nonlinear pricing by
a dominant firm in a general environment, adding a second dimension of heterogeneity
on top of the buyer’s surplus. First, we allow the buyer to split her purchases between
suppliers. Second, we assume that only part of the buyer’s demand can be supplied by
the entrant, and this share is unknown to the buyer and incumbent when they agree on
a price schedule. Third, we allow the buyer to purchase more than her requirements and
assume that she can dispose of excess units at some cost. We now explain the last two
features of our setting in greater details.
As noticed by competition authorities, it is often unrealistic to assume that a buyer
can shift all of her requirements within a relevant time period from the dominant supplier
to a competitor. This can be due to demand-side or supply-side considerations. It may be
the case that competitors are capacity constrained and cannot serve all of the demand
of large customers. It may also be the case that the incumbent’s product is a “muststock” for retailers because only a fraction of final consumers is ready to experiment with
competing products (regardless of their price). In both cases, within a relevant time
horizon, the entrant can serve only a fraction of the buyer’s demand (“contestable share
of the demand”), which constitutes the maximum scale of entry. The incumbent faces
no competition for the complementary part of her requirements (“captive share of the
demand”).
The third extension of Aghion and Bolton (1987) involves disposal costs and buyer
opportunism. The buyer and the incumbent negotiate a price schedule that places competitive pressure on the entrant and forces him to sell at a low price. We show that
this “rent-shifting” strategy involves marginal prices below marginal costs, and may even
involve negative marginal prices. A negative marginal price allows the buyer to extract
rents from the entrant, but also gives her an ex post incentive to buy more than she
needs from the incumbent. While this opportunistic behavior is anticipated ex ante, it
constrains the choice of price schedules by the buyer-incumbent pair. The possibility
of buyer opportunism depends on the magnitude of disposal costs, as the buyer has to
get rid of units she does not consume nor resale. Buyer opportunism is maximal under
3
Such a framework where an incumbent chooses a strategy first and the entrant plays second has
been used to model incumbency and/or a dominant firm at least since Spence (1977, 1979), Dixit (1979,
1980).
4
In particular, Marx and Shaffer (1999) looks at two-part tariffs with a focus on below cost pricing,
Marx and Shaffer (2004) studies how equilibrium is affected when certain classes of tariffs are forbidden,
Marx and Shaffer (2007) focuses on the order of negotiation (Is it better for the buyer to negotiate first
with the incumbent or with the entrant?), and Marx and Shaffer (2010) shows how bargaining powers
affect profits and when break-up fees are used.
5
In Majumdar and Shaffer (2009), a dominant firm resorts to nonlinear pricing to discriminate a buyer
who is informed about the size of demand when the buyer also sells a good provided by a competitive
fringe.
3
free disposal and does not exist when disposal costs are infinite. Disposal costs can vary
substantially from one industry to the other.
The contributions of our analysis are threefold. First, we solve a multidimensional
screening problem where the number of instruments is smaller than the dimension of
unobserved heterogeneity, and we do so for general distributions of the heterogeneity.
Second, the generality of the analysis allows to assess the robustness of the AghionBolton framework and to reveal new properties, such as the curvature of optimal price
schedules and partial foreclosure. Third, we explain how conditioning the tariff on the
quantity purchased from the competing supplier (when feasible) allows to eliminate buyer
opportunism, and thus to achieve the same outcome as if disposal costs were infinite. We
now explain each of these contributions in more details.
First, we contribute to the nonlinear pricing literature. In our framework, the type of
the entrant has two components: the surplus he creates with the buyer and the maximum
scale of entry. However, to screen out the entrant’s types, the incumbent has only
one instrument, namely a price-quantity schedule. This configuration, which generates
extensive pooling, has received little attention.6 Here, the structure of the model makes
it possible to characterize the set of implementable allocations, and to construct the
solution with few restrictive assumptions on the distribution of the uncertainty. As
explained below, the equilibrium pattern of the pooling regions reflects the barriers to
entry and to expansion created by the optimal tariff.
Second we link the curvature of optimal price schedules and the form of inefficient
exclusion (partial versus full foreclosure) to the entrant’s bargaining power and to the
elasticity of entry. The former parameter is zero in the case of a competitive fringe and is,
in general, positive in the case of strategic entrants. The latter parameter expresses how
entry at a given scale is sensitive to the competitive pressure exerted by the incumbent.
It is an key statistics summarizing the two-dimensional distribution of the entrant’s
characteristics. Our findings can be summarized as follows.
When the size of the contestable demand is known to the buyer and the incumbent
(only the entrant surplus is unknown), the pricing problem involves a standard tradeoff
between rent extraction and efficiency, and is a mere reformulation of the Aghion-Bolton
analysis. Some efficient competitors are foreclosed, and the extent of inefficient exclusion
decreases with the entrant’s bargaining power and the elasticity of entry. The optimal
schedule is a two-part tariff, inefficient exclusion arises in the form of full foreclosure only:
no partial foreclosure is observed. These results readily extend under two-dimensional
uncertainty, provided that the two unknown parameters are statistically independent or,
equivalently, that the elasticity of entry remains constant with the size of the contestable
market. We now turn to cases where the elasticity of entry varies with the size of the
contestable market.
When the elasticity of entry increases with the size of the contestable share of the
demand, the optimal policy of the buyer-incumbent pair is to reduce the competitive pressure exerted on the entrant as the scale of entry increases, which cannot be achieved with
two-part tariffs. Optimal tariffs are shown to be concave for high quantities. Here again,
inefficient exclusion arises, in the form of full foreclosure only. Solving the efficiency-rent
6
A notable exception is Laffont, Maskin, and Rochet (1987) who solve an example with uniform
distributions.
4
tradeoff à la Aghion-Bolton separately for any given level of the contestable demand (i.e.
solving the “relaxed problem”) yields the optimal tariff.
When the elasticity of entry decreases in or is non monotonic with the contestable
demand, the solution to the relaxed problem is not incentive compatible as entrants
with a large contestable demand would mimic entrants with a smaller one at the relaxed
allocation. The incentive compatibility constraints translate into convex parts of the
tariff and into partial foreclosure at the optimum. Some efficient entrants sell a positive
quantity but are prevented from achieving the maximum scale of entry and serving all
of the contestable share of demand. When the elasticity of entry first decreases, then
increases as the maximum scale of entry rises, the buyer and the incumbent want to be
soft with entrants with small and large contestable markets and to be aggressive with
entrants with intermediate contestable markets. This tension generates highly nonlinear
tariffs that induce many entrants to choose the same quantity, as is the case with so-called
“retroactive rebates” challenged by European competition agencies in recent cases.
Finally, we contribute to the literature on market-share rebates.7 Specifically, we
allow the buyer and the incumbent to condition the negotiated price schedule on the
number of units purchased from the entrant, and we show that this instrument allows
them to overcome the problem of buyer opportunism. When the price-quantity schedule
only depends on the number of units purchased from the incumbent, placing competitive
pressure on the buyer may involve negative marginal prices, which, in turn, trigger buyer
opportunism if the buyer can freely dispose of excess units and, to a lesser extent, if
disposal costs are finite. Our analysis, however, straightforwardly extends to the case
where the tariff also depends on the number of units purchased from the entrant. We
show that the same quantity allocations are implementable under conditional rebates.
There is a huge difference, though: conditional tariffs allow to exert competitive pressure
on the buyer without resorting to negative marginal prices (i.e. to negative price for
marginal units sold by the incumbent). The competitive pressure can instead come from
the implicit price of marginal units sold by the entrant. Thus, conditional rebates, when
feasible, allow to eliminate buyer opportunism and to achieve the same outcome as under
infinite disposal costs.
The article is organized as follows. Section 2 introduces the model, and purely for
ease of exposition, impose the restrictive assumption that disposal costs are infinite.
Section 3 explains the negotiation between the buyer and the entrant. This negotiation
takes place under complete information. Section 4 focuses on the negotiation between the
buyer and the incumbent. This negotiation takes place under incomplete information.
It introduces the notion of virtual surplus and of elasticity of entry before characterizing
the construction of the optimal price schedule. Section 5 relates the shape of the optimal
negotiated price schedule to the primitive of the model. Section 6 introduces finite
disposal costs. It extends the previous results, discusses the opportunism of the buyer
and shows how a market-share tariff could overcome it. Section 7 we discuss some
extensions of the model as well as antitrust implications.
7
Inderst and Shaffer (2010) assume complete information and study a setting with a dominant firm,
a competitive fringe and two retailers. They show that market-share rebates are used by the dominant
firm to dampen (intra- and inter-brand) competition. Calzolari and Denicolo (2009) address the issue
in a duopoly setting (simultaneous game, symmetric firm).
5
2
The model
Two firms, E and I, compete to serve a single buyer B. For convenience, we call firm I
the incumbent and firm E and the entrant. The game under study, however, does not
involve a genuine entry decision8 but these terms are convenient to convey the idea that
firms are asymmetric.
Figure 1: Timing of the game
The timing, sketched in Figured 1, is as follows: in a first stage, the buyer and
the incumbent negotiate a price-quantity schedule T (qI ) (in Section 6 we examine the
possibility of market-share tariffs) before the entrant can negotiate with the buyer. Next,
the entrant and the buyer observe the entrant’s characteristics, (cE , sE , vE ). Then, in a
third stage, the buyer and the entrant agree on a price and a quantity, both knowing the
terms of the agreement between the buyer and the incumbent. This negotiation takes
place under complete information and is assumed to be efficient. Finally, in stage 4, the
buyer purchases from the incumbent.9
The buyer’s total demand is inelastic, and normalized to one: qE + qI ≤ 1. We
assume that firm E can address at most a fraction of the buyer’s demand, that we call
the “contestable” part of the demand and denote by sE : 0 ≤ sE ≤ 1. This embodies
two interpretations: a supply-side variant in which firm E has capacity sE ≤ 1 and a
demand-side variant where the buyer does not value units of good E in excess of sE . In
both cases, the buyer never purchases more than sE from the entrant: qE ≤ sE .
The buyer’s gross benefit are vE and vI per unit of goods E and I. Thus, having
purchased quantities qE ≤ sE and qI from the buyer and the incumbent, the buyer gross
profit is:
vE qE + vI qI , if qE + qI ≤ 1
(1)
V (qE , qI ) =
−∞
otherwise.
That is, we assume, here, that all purchases are consumed which amounts to assume
that disposal costs are infinite (failing to consume all of the purchased units is infinitely
costly). In Section 6 we relax this assumption and introduce finite disposal costs.
We note cE firm E’s constant marginal costs, and ωE = vE − cE ≥ 0 the unit surplus
generated by good E. Firm I’s constant marginal cost is noted cI , and ωI = vI − cI ≥ 0
8
In particular it does not feature a sunk entry cost incurred by firm E.
We assume that the buyer and the incumbent cannot renegotiate their agreement once uncertainty
is resolved. They commit to abide by the price-quantity schedule regardless of what they learn about
the demand for the new product and the entrant’s production cost.
9
6
is the unitary surplus of good I. In section ?? we consider more general cost functions
for the incumbent.
At the time of signing the contract the parameters sE and ωE are uncertain. The
production capacity of the entrant, sE , is distributed on [sE , s̄E ] according to a c.d.f.
G(.) (with a continuous density function g(.)) and the surplus created by entry, ωE , is
distributed on [ω E , ω̄E ] according to a c.d.f. F (.|sE ) which admits a continuous density
function denoted f (.|sE ). To focus on the most interesting case, we assume that ω E <
ωI < ω̄E . That is, the entrant can be, a priori, less or more efficient than the incumbent.
Efficiency benchmark. Total surplus is W (qE , qI ) = ωE qE + ωI qI if qE + qI ≤ 1 and
−∞ otherwise. The maximization program writes:
ωE qE + ωI qI
max
qE + qI ≤ 1
qE ≤ sE
(2)
This expression increases with the quantities purchased when qE +qI ≤ 1 hence efficiency
requires qE + qI = 1, which allows to write the total surplus as ωI + (ωE − ωI )qE and
the efficient quantity purchased from the entrant is
sE if ωE ≥ ωI
∗
qE
(sE , ωE ) =
(3)
0
otherwise.
Entry, if efficient, should occur at maximum scale. Hence the maximal value of total
surplus is ωI (1 − sE ) + ωE sE when ωE ≥ ωI and ωI when ωI ≥ ωE .
Second best. The negotiation between the buyer and the entrant (studied next in
Section 3) generates a surplus and the entrant gets a share of it and we denote ΠE this
profit. The buyer and the incumbent share ex-post:
ΠBI = W (qE , qI ) − ΠE .
Ex-ante, they maximize the expectation of ΠBI (over sE and ωE ) and they share it given
their outside options and bargaining powers.
3
Negotiation between the buyer and the entrant
In this section, we build, using backward induction, the first blocks of the nonlinear
pricing problem of the dominant firm. Once the price schedule T has been chosen and
once the type of the entrant is known, the buyer and the entrant negotiate under complete
information yielding an efficient outcome. That is, the parties maximize their joint
surplus. In subsection 3.1, we characterize the quantity bought to the the entrant and
how the surplus is split between the buyer and the entrant. Next, in subsection 3.2, we
provide some examples to show how this quantity choice depends on the shape of T .
Finally, in Subsection 3.3 we characterize all the implementable quantity functions.
7
3.1
Maximization of the joint surplus
We let UB (qE ) represent the buyer’s net utility after he has purchased qE units from the
entrant10
UB (qE ) = max V (qE , qI ) − T (qI ).
(4)
qI
The buyer and the entrant choose qE to maximize their joint surplus
SBE (cE , sE , vE ) = max UB (qE ) − cE qE .
qE ≤sE
(5)
The buyer and the entrant share SBE according to their respective bargaining power and
outside options. The entrant’s outside option is normalized to zero. As to the buyer, he
may source exclusively from the incumbent, so his outside option is UB (0). The surplus
created by the relationship between B and E is
∆SBE (cE , sE , vE ) = SBE (cE , sE , vE ) − UB (0).
Noting β ∈ (0, 1) the entrant’s bargaining power vis-à-vis the buyer, the entrant gets ΠE
and the buyer gets ΠB given by
ΠE =
0
+
β
∆SBE
ΠB = UB (0) + (1 − β) ∆SBE
The case β = 0 corresponds to a competitive fringe from which the buyer can purchase
any quantity at cE . On the opposite, the case β = 1 happens when the entrant has all
the bargaining power.
In Lemma C.1, we show, that in equilibrium the buyer always purchases such that
qE + qI ≥ 1. Indeed, as vI − cI ≥ 0, it is not profitable (even for commitment purposes)
for the buyer-incumbent pair to agree on a tarif that “refuses to sell” at the margin. As
the buyer never purchases more than 1, then qE + qI = 1. In consequence, the joint
surplus maximization of the buyer and the entrant writes:11
SBE (sE , ωE ) = vI + max (ωE − vI )qE − T (1 − qE ).
qE ≤sE
Moreover, the buyer’s outside option is UB (0) = vI − T (1), therefore the surplus from
the trade between the buyer and the entrant is
∆SBE (sE , ωE ) = max (ωE − vI )qE − T (1 − qE ) + T (1).
qE ≤sE
(6)
and we denote qE (sE , ωE ) the quantity which solves (6).
Two important properties of this program should be noted. First, taking the second
derivative of (ωE − vI )qE − T (1 − qE ) + T (1) with respect to qE leads to −T 00 (1 − qE ),
and therefore the program is convex (concave) if T is concave (convex). If T is concave,
then the program exhibits a corner solution: either qE = 0 (no production) or qE = sE
(production at full capacity). If T is convex, however, the solution can be interior and
10
We omit the price paid by the buyer to the entrant, which is sunk when deciding on the quantity to
be purchased from the incumbent. This price plays no role in the analysis.
11
From now on, we write SBE (sE , ωE ) rather than SBE (cE , sE , vE ).
8
characterized by the first order condition. In other words, an interior solution can exist
only in a range where the price schedule is convex.
Second, remark that (ωE − vI )qE − T (1 − qE ) + T (1) = [ωE − vI + pe (qE )] qE where
pe (qE ) =
T (1) − T (1 − qE )
qE
(7)
is mathematically the slope of the chord between (1, T (1)) and (1 − qE , T (1 − qE )).
Economically, pe (qE ) is the incumbent’s average price of the last qE units. That is, once
the buyer already plans to buy qI = 1 − qE from the incumbent, buying these last qE
units, still from the incumbent, costs the buyer: T (1) − T (1 − qE ) and dividing by qE
gives the average or unit price. We call pe the effective price of the last units as the
entrant compete against pe . Indeed, the buyer purchases qE from the entrant (and 1 − qE
from the incumbent) only if the entrant offers a larger surplus than the the incumbent
which is possible whenever: ωE ≥ vI − pe .
These two remarks are related as the effective price plays an important role in characterizing the solution of (6) when the program is convex. We now use three particular
types (linear, concave, and convex) of price schedule T to illustrate how the negotiation
between the buyer and the entrant works.
3.2
Examples
To begin with, Figure 2 depicts the choice of qE which results from the maximization
program (6) in the case where T is linear. The left-hand side of the picture shows a linear
T and the right-hand side the optimal choice of qE in the (sE , ωE ) space. This choice
takes a very simple form. Let pe denote the slope of T . If ωE is below the threshold
vI − pe it is optimal that the entrant does not produce: qE = 0. Indeed, buying qE from
the entrant creates a surplus ωE qE . Now, when T = pe qI buying the quantity qE from
the incumbent instead creates, for the buyer, a surplus (vI − pe )qE . Consequently, it is
profitable to buy from the entrant if and only if ωE qE > (vI − pe )qE or ωE > vI − pe .
On the other hand, if ωE is above the threshold, it is, then, optimal to buy as much as
possible from the entrant, hence qE = sE in that case. Suppose that pe = cI , then the
∗ . In other words, the
buyer-entrant pair has an incentive to pick the efficient quantity qE
first best outcome obtains when the contractual nonlinear tariff is the incumbent’s cost
function plus a constant margin.
Figure 3 helps to explain the solution of (6) when T is concave. First, the lefthand side plot shows a concave T and depicts the chord of T between (1, T (1)) and
(1 − qE , T (1 − qE )) the slope of which is pe . As the program in (6) is convex once
T is concave. Therefore the program exhibits a corner solution: either qE = 0 (no
production) or qE = sE (production at full capacity). If ωE > vI − pe (qE ) for some qE it
pays to produce at this level, but under the assumption that T is concave, pe is increasing
and therefore ωE > vI − pe (sE ) and it pays to produce at full capacity. Otherwise, if
ωE ≤ vI − pe (qE ) for all qE > 0, then it is better not to produce and qE = 0. The
intuition behind this result is the following: ωE is the surplus generated by the entrant
while for a given T , vI − pe (sE ) is the surplus offered by the incumbent for the last sE
units (that is when the buyer buys from 1 − sE up to 1). Thus, if the latter is larger
than the former, the buyer buys sE from the entrant instead of the incumbent, and viceversa. The comparison is made simple here because a concave T makes vI − pe (qE ) a
9
Figure 2: Entry under a linear T
decreasing function. The right-hand side of Figure 3 depicts the function vI − pe (sE ).
Figure 3: Entry under a concave T
If ωE ≤ vI − pe (sE ), then qE = 0 while if ωE > vI − pe (sE ), then qE = sE , hence the
vertical lines. The concave tarif case is, therefore, quite similar to the linear tarif case.
The only difference is that vI − pe is decreasing when T is concave but constant when T
is linear. We now turn to the convex case.
As already shown, when T is convex, the program (6) is concave and one expects an
interior solution characterized by the first order condition which writes: ωE − vI + T 0 (1 −
qE ) = 0. Figure 4 shows on the left-hand side a convex T and on the right-hand side the
quantity choice after the negotiation between B and E.
The right-hand side plot might appear hard to read at first sight. The important fact
is that the solution which should simply be characterized by the first order condition
ωE = vI − T 0 (1 − qE ) has to take into account the constraint qE ≤ sE . Therefore there
10
Figure 4: Entry under a convex T
are two cases: either the f.o.c. solution is lower than sE and this is the solution, or it
1
is larger than sE and in that case sE is the solution. Hence the isolines. Fix ωE = ωE
1
1
0
and let sE denote the unique solution of ωE = vI − T (1 − sE ) which is assumed to
1 ), then it is optimal from the
be in the relevant range. If the entrant’s type is (s1E , ωE
1
buyer-entrant’s point of view to choose qE = sE . Assume, now, that the entrant’s type
1 , the surplus generated by entry is larger but as the production
is (s1E , ωE ) with ωE > ωE
capacity are the same, qE = s1E remains the solution. That is, along the vertical lines,
1)
production is constant and at full capacity. Assume that the entrant’s type is (sE , ωE
1 is unchanged, it remains optimal to produce q = s1 ,
with sE > s1E . As the surplus ωE
E
E
the increased capacity is useless. That is, along the horizontal lines production is constant
but production is below full capacity.
3.3
Implementable quantity functions
In this section, we show that any implementable quantity functions qE (sE , ωE ) may
be represented by a boundary line in the (sE , ωE )-plan such that qE = sE above the
boundary and qE does not depend on sE below the boundary. Such boundary lines have
equations of the form ωE = Ψ(sE ), where Ψ is called a boundary function. We demonstrate below the existence of a one-to-one map between quantity functions qE (sE , ωE )
and boundary functions Ψ(sE ). To solve the two-dimensional problem, it turns out to be
convenient to work with boundary functions rather than directly with quantity functions.
Hereafter, we say that a quantity function qE (sE , ωE ) from [sE , s̄E ]×[ω E , ω̄E ] to [0, 1]
is implementable if it can be obtained as solution to (6), where T is a tariff.
Starting from a tariff T , we consider the profit function ∆SBE (sE , ωE ) given by
equation (6). By the envelope theorem, its derivative with respect to ωE is qE (sE , ωE ).
We have:
Z ωE
qE (sE , x) dx.
(8)
∆SBE (sE , ωE ) =
ωE
11
Figure 5: Implementable quantity function (isolines)
Since qE is nondecreasing in ωE , there exists, for any sE > 0, a threshold Ψ(sE ) such
that the solution to (6) is interior (qE (sE , ωE ) < sE ) for ωE < Ψ(sE ) and the solution is
qE (sE , ωE ) = sE for ωE > Ψ(sE ). We define the boundary function Ψ(sE ) associated to
the quantity function qE (sE , ωE ) by
Ψ(sE ) = inf{x ∈ [ω E , ω̄E ] | qE (x, sE ) = sE },
with the convention Ψ(sE ) = ω̄E when the above set is empty.
In the (sE , ωE )-space, the quantity isolines, i.e. the sets of pairs (sE , ωE ) such that
the quantity qE (sE , ωE ) is constant, are horizontal below the boundary Ψ, and vertical
above, as shown on Figure 5. Because the quantity function qE (sE , ωE ) is nondecreasing
in sE and constant below the boundary, we have:
qE (sE , ωE ) = min{ x ≤ sE | Ψ(y) ≥ ωE for all y ∈ [x, sE ]},
(9)
with the convention that qE = sE if the above set is empty. We have thus proved the
necessary part of Lemma 1. In Appendix E, we show the sufficient part, by recovering
the price schedule T from the boundary function Ψ (see equation 22)).
Lemma 1. A quantity function qE (., .) is admissible if and only if there exists a boundary
function Ψ(.) defined on [0, 1] such that (9) holds.
When crossing the boundary ωE = Ψ(sE ), the quantity qE is continuous at points
where Ψ is increasing, and is discontinuous at points where Ψ is decreasing. In other
words, the constraint qE ≤ sE is binding (slack) on decreasing (nondecreasing) parts
of the boundary. Increasing parts of the boundary function thus translate into partial
12
foreclosure: 0 < qE (sE , ωE ) < sE for some types located below the boundary. In such
regions, the constraint qE ≤ sE is slack: increasing sE does not allow the competitor to
enter at a larger scale and qE does not depend on sE .
For a two-part tariff, the boundary is horizontal. For a globally concave tariff, it
is monotonically nonincreasing, the quantity is sE above the boundary and zero below.
For a globally convex tariff, the boundary is monotonically nondecreasing, with equation
T 0 (1 − sE ) = vI − ωE . A convex kink in a tariff translates into an upward jump of the
boundary and two-dimensional pooling: two entrants whose type lies below the boundary,
at the right of the discontinuity, sell the same quantity while they are different along the
two dimensions sE and ωE .
The above observations could wrongly suggest a simple link between the variation
of the boundary and the curvature of the tariff. In fact, the link works in one direction
only. If the boundary is increasing, the constraint qE ≤ sE is slack, and the second-order
conditions of the buyer-entrant pair’s problem imply that the tariff must be locally convex
at 1 − sE . Similarly, flat parts of the boundary reflect linear portions of the tariff.12 On
the other hand, it may well happen that the constraint qE ≤ sE binds, and thus that the
boundary is decreasing, in a region where the tariff is locally linear or convex.
4
Designing the price schedule
The incumbent and the buyer maximize their expected joint surplus, which they share
according to their respective bargaining powers and outside options. Their expected joint
surplus, EΠBI , equals the total surplus minus the profit left to the entrant:
EΠBI = E {W (qE , qI ) − ΠE } ,
or, replacing ΠE with its value,
EΠBI = E {ωI + (ωE − ωI )qE − β∆SBE } .
(10)
The above expressions do not depend on the general level of the price quantity schedule
T . Indeed, replacing T by T + k, for any constant k, does not alter the entrant’s profit
(see (6)) and thus the solution of (10). The general level of the tariff is determined by
the relative bargaining powers of the buyer and the incumbent.
4.1
Comparing effective price and incremental cost
The buyer-incumbent’s pair does not maximize the surplus W , and thus has no incentive
to set pe = cI . But contrary to what could be expected from a dominant firm, the buyerincumbent pair always sets the effective price below (or at) the marginal cost. This below
cost pricing result holds regardless of the information structure, i.e. whether or not the
buyer and the incumbent know the entrant’s characteristics at the time of contracting.
It generalizes a result of Marx and Shaffer (1999) obtained under complete information
(but with a more general demand function for the buyer).
12
In such a situation, the buyer-entrant pair’s problem (6) has many solutions. Equation (9) selects
the lowest.
13
Proposition 1. Setting the effective price of contestable units at or below the incumbent’s
average incremental cost, pe (q) ≤ cI for all q, does not deteriorate the buyer-incumbent
pair’s expected payoff. Inefficient entrants are inactive in equilibrium.
Proof : In Appendix A, we consider any effective price schedule pe (.) and examine the
effect of replacing pe (.) with p̃e = min(pe , cI ). We show that this change does not
increase the entrant’s profit and does not decrease the total surplus, and hence does not
deteriorate the buyer-incumbent pair’s expected profit. We also show that the buyer
does not purchase from inefficient entrants under the new schedule p̃e .
4.2
Virtual surplus and elasticity of entry
Expanding (10), and using Proposition 1, the joint surplus of the buyer-incumbent pair
writes:
Z Z ω̄E
EΠBI =
{ωI + (ωE − ωI )qE − β∆SBE } dF (ωE |sE ) dG(sE ),
sE
ωE
where qE (sE , ωE ) and ∆SBE (sE , ωE ) are given by (6) and from which it comes (using
the envelope theorem) that ∂∆SBE (sE , ωE ) /∂ωE = qE (sE , ωE ). Moreover, types with
ωE = ω E are inefficient and get no rent. Integrating by parts for each sE , yields
Z Z ω̄E
EΠBI =
S v (qE ; sE , ωE ) dF (ωE |sE ) dG(sE ),
(11)
sE
ωE
where, following Jullien (2000), the buyer-incumbent coalition’s objective is expressed in
term of a “virtual surplus” S v (qE , sE , ωE ):
1 − F (ωE |sE )
v
S (qE , sE , ωE ) = ωI + ωE − ωI − β
qE .
f (ωE |sE )
The virtual surplus is the total surplus W (qE , 1−qE ) adjusted for the informational rents
βqE (1 − F (ωE |sE )) /f (ωE |sE ) induced by the self-selection constraints.
To solve (11), the problem of the buyer-incumbent pair, a natural approach is to
maximize S v separately for each possible value of the size of the contestable market.
This method amounts to solve a one-dimensional problem for each sE . This is what
we call the “relaxed problem” which solves for the classic efficiency-rent tradeoff. If the
solution to the relaxed problem is incentive compatible, then it is the solution of (10).
The solution to the relaxed problem, however, is not always incentive compatible.
Here it is convenient to introduce an elasticity of entry. For a given threshold, say
ωE , assume that all types ω > ωE enter, then 1 − F (ωE ) is the probability of entry, and
ε(ωE |sE ) =
ωE f (ωE |sE )
1 − F (ωE |sE )
is the elasticity of (the probability of) entry at ωE . We assume hereafter that this
elasticity is non decreasing in ωE for any given sE . It is, in particular, true when the
hazard rate (1 − F (ωE |sE )) /f (ωE |sE ) is non increasing in ωE (a usual assumption in the
nonlinear pricing literature). A case of interest is when the elasticity is constant in ωE ,
14
which happens when ωE conditional on sE follows a Pareto distribution: 1 − F (ωE |sE ) =
(ωE /ω E )−ε(sE ) .
Given that S v is linear in qE the solution to the relaxed problem takes a simple form.
R (s , ω ) = arg max
v
Defining qE
qE ≤sE S (qE , sE , ωE ), we have:
E
E
0
if ωE ≤ ω̂E (sE )
R
qE (sE , ωE ) =
sE otherwise.
where ω̂E (sE ) is the unique solution to
β
ω̂E (sE ) − ωI
=
.
ω̂E (sE )
ε(ω̂E (sE )|sE )
(12)
The threshold ω̂E (sE ) summarizes the tradeoff between efficiency and rent extraction
at a given level of sE . Hereafter, we call the curve ωE = ω̂E (sE ) the ERT line (Efficiency
Rent Tradeoff). Equation (12) shows an analogy with the textbook monopoly pricing
formula. Indeed, the buyer-incumbent pair acts like a monopoly towards the entrant
fixing (indirectly) ω̂E to extract more rent from entry (but with the drawback of reducing
the probability of entry).
In Equation (12), the entrant’s bargaining power, β, plays in the opposite direction
of ε. Intuitively, when ε decreases the buyer-incumbent pair can extract more rents from
the entrant, this is also true when β decreases. Yet, a lower ε leads to a larger ω̂E while
a lower β implies a lower ω̂E . Indeed, when β decreases, the buyer gets a larger share
of the surplus created by entry. Therefore, it becomes more profitable to induce more
entry.
4.3
The optimal quantity function
The following results characterize the optimal quantity function and its corresponding
boundary line. As shown in Section 3.3, an implementable quantity function is respectively constant in sE below its boundary line and constant in ωE above.
Proposition 2. Suppose the incumbent’s marginal cost is constant. Decreasing parts of
the optimal boundary are included in the ERT line. Increasing parts satisfy the following
property:
• If the boundary is increasing and not vertical at sE , we note S the horizontal pooling
interval with left extremity (sE , Ψ(sE )). Then
Z
SqvE (sE ; s, Ψ(sE ))f (Ψ(sE )|s)g(s) ds = 0.
(13)
S
• If the boundary is vertical at sE , we note A the pooling region at the right of the
vertical part. Then
ZZ
SqvE (sE ; s, ωE )f (ωE |s)g(s) dωE ds = 0.
(14)
A
Proof. See Appendix B.
15
Figure 6a: A non implementable
R
qE
SB
Figure 6b: Optimal boundary ωE
Consider an interval I = [s0E , s1E ] where Ψ is decreasing. Let Ψ̂ be the smallest
nonincreasing function larger than ω̂E on this interval: Ψ̂(sE ) = supsE ≤s≤s1 ω̂E (s). We
E
know from Lemma ?? that Ψ is larger than or equal to ω̂E . Therefore it is larger than
or equal to Ψ̂. Noting Ψ̄ = max[Ψ̂, Ψ(s1E )], we have: Ψ ≥ Ψ̄, with equality at s1E .
As Ψ̄ is nonincreasing and coincides with Ψ at s1E , replacing Ψ with Ψ̄ on I does not
change the quantity outside the region I ×[ω E , ω̄E ]. As the quantity function is discontinuous when crossing Ψ, the change from Ψ to Ψ̄ increases qE (sE , ωE ) by a non-infinitesimal
quantity if sE ∈ I and Ψ̄(sE ) < ωE < Ψ(sE ), and leaves qE (sE , ωE ) unchanged otherwise.
As Ψ̄ ≥ ω̂E , the quantity rise increases the objective (11), a contradiction.
It follows that we must have Ψ = Ψ̄ and hence Ψ = Ψ̂ on I. Now the fact that Ψ̂ is
decreasing implies that Ψ̂ = ω̂E (because Ψ̂ is constant on intervals where Ψ̂ > ω̂E ). We
conclude that the boundary, where it is decreasing, is included in the ERT line: Ψ = ω̂E
on I.
Next, consider a small interval [sE , sE + dsE ] on which the boundary is increasing and
not vertical, and replace Ψ with Ψ + dΨ on this interval. As the quantity is continuous
across the boundary, the change has a second order effect in the region [sE , sE + dsE ] ×
[ω E , ω̄E ]. The change, however, affects the (constant) value of qE on the pooling interval
with left extremity (sE , Ψ(sE )). Equation (13) reflects the corresponding variation in
the buyer-incumbent coalition’s expected profit (11).
Finally, consider a vertical portion that is included in an increasing part of the boundary. Formally, suppose that the boundary function Ψ is nondecreasing in the neighborhood of some point sE , and jumps upwards at sE . Suppose we make a small change in
sE , slightly moving leftwards and rightwards the vertical part of the boundary. Again
the changes in [sE , sE + dsE ] × [ω E , ω̄E ] have second-order effects. The moving of the
boundary, however, changes the (constant) value of qE on the pooling region at the right
of the vertical portion of the boundary. Equation (14) reflects the corresponding variation
in the buyer-incumbent coalition’s expected profit (11).
To construct the optimal boundary, we proceed as follows. We first draw the ERT line
16
ωE = ω̂E (sE ). We start with sE = 1 and then consider lower and lower values of sE . For
sE = 1, we know that Ψ(1) = ω̂E (1). If the ERT line decreases at sE = 1, the boundary
coincides with the ERT line, as long as it remains decreasing. When the ERT line starts
increasing (possibly at sE = 1), we know that there is horizontal pooling. Equation (13)
provides a candidate value for Ψ(sE ). This candidate is the solution provided that it
increases with sE and remains above ω̂E (sE ). If the candidate boundary hits the ERT
line at some value of sE , it must be on a decreasing part of that line and, from that value
on, the optimal boundary coincides with the ERT line (as long as ω̂E remains decreasing).
If the candidate boundary decreases with sE , we must have two-dimensional pooling, and
we apply the same procedure with equation (14).
4.4
Constructing the optimal tariff
Il faut passer à T en disant :
SB est défini q ∈ [1 − s, 1 − s]
0) récupérer le tarif lorsque ωE
I
1) la forme de T ne compte pas en dessous de 1 − s
2) pour qI ∈ [1 − s, 1] prolonger par une droite
5
The shape of the optimal price schedule
As shown by (12), ω̂E (and therefore the solution of the relaxed problem, and therefore
the solution of the problem) is related to ε(ωE |sE ) which capture the link between the
random variable sE and ωE (through F (ωE |sE )). To begin with, Lemma 2 makes this
relationship more precise.
Lemma 2. The elasticity of entry, ε(ωE |sE ), does not depend on sE if and only if the
random variables sE and ωE are independent.
If the elasticity of entry increases (decreases) with sE , then ωE first-order stochastically decreases (increases) with sE .
Proof. The elasticity of entry varies with sE in the same way as the hazard rate h given
by
f (ωE |sE )
h(ωE |sE ) =
.
1 − F (ωE |sE )
We have
Z ωE
h(x|sE ) dx = − ln[1 − F (ωE |sE )].
ωE
If the elasticity of entry does not depend on (increases with, decreases with) sE , the
same is true for the hazard rate, and hence also for the cdf F (ωE |sE ), which yields the
results.13
Consequently, we study respectively in the following sections the case where ε(ωE |sE )
does not depend on sE (independence of sE and ωE ), where ε(ωE |sE ) increases with sE
(ωE first-order stochastically decreases with sE ), and where ε(ωE |sE ) decreases with sE
(ωE first-order stochastically increases with sE ), finally we look at cases where ε(ωE |sE )
is not monotonic in sE .
13
The variable ωE first-order stochastically decreases (increases) with sE if and only if F (ωE |sE )
increases (decreases) with sE .
17
5.1
Elasticity of entry constant in sE
As shown in the following proposition, when sE and ωE are independent, the solution of
the relaxed problem is incentive compatible and is therefore the solution of the buyerincumbent’s problem.
Proposition 3. Suppose that the elasticity of entry, ε(ωE |sE ), does not depend on sE ,
then the second best can be achieved with a two-part tariff with slope: vI − ω̂E . The
equilibrium features inefficient exclusion but partial foreclosure is not present.
Proof. Independence implies that F (ωE |sE ) = F (ωE ) and f (ωE |sE ) = f (ωE ) therefore
the elasticity ε(ω̂E (sE )|sE ) is independent of sE and also ω̂E (sE ). It is straightforward
R (s , ω ) = q R (ω ) is
from (12) that ω̂E > ωI . The solution of the relaxed problem, qE
E
E
E
E
admissible according to Lemma 1 with a constant boundary function Ψ = ω̂E .
The second best tariff is obtained as follows. From ∂∆SBE (sE , ωE ) /∂ωE = qE (sE , ωE )
and using qE (sE , ωE ) = sE it comes that ∆SBE (sE , ωE ) = (ωE − ω̂E )sE . On the
other hand it comes directly from the definition of ∆SBE (sE , ωE ) that ∆SBE (sE , ωE ) =
(ωE − vI )sE + T SB (1) − T SB (1 − sE ) combining these two expressions gives:
T SB (1) − T SB (1 − sE ) = (vI − ω̂E )sE
which is, indeed, a two-part tariff.
To make sure that the competitor serves all of the contestable demand if ωE ≥ ω̂E
and is inactive otherwise, the buyer and the incumbent set the effective price at vI − ω̂E .
The smaller the elasticity of entry, ε, the larger the ERT threshold, ω̂E , and the smaller
the slope of the two-part tariff, and, therefore, the larger the competitive pressure put
on the entrant. Notice that the slope of the optimal price schedule is negative whenever
vI is lower than ω̂E . When vI < ω̂E and the entrant is active (ωE > ω̂E ), it sells qE = sE
and the buyer purchases only qI = 1 − sE from the incumbent whereas it would cost less
to purchase qI = 1 while still buying qE = sE from the entrant. The buyer, however,
cannot take advantage of the negative marginal price offered by the incumbent because
it would leave him with unconsumed units and the disposal cost is infinite (see Section 6
for finite disposal costs).
As pictured in Figure 7a, the tradeoff between efficiency and rent extraction results in
some efficient entrants being fully foreclosed in equilibrium. Inefficient foreclosure arises
due to incomplete information as in Aghion and Bolton (1987). The fraction of efficient
types that are inactive increases with the entrant’s bargaining power vis-à-vis the buyer
as ω̂E increases with β. In the limit case where the buyer has all the bargaining power
vis-à-vis the entrant (β = 0), there is no tradeoff, and hence no inefficient exclusion: the
ERT threshold coincides with the efficient threshold ωI .
A special case of Proposition 3 arises when one of the (or both) parameters, sE or
ωE , is known ex-ante to the buyer-incumbent pair. There is, however, a difference as
shown in the following corollary.
Corollary 1. If both sE and ωE are known or if sE is unknown and ωE is known,
then ε(ωE |sE ) = 0 and ω̂E = ωI . The equilibrium outcome is efficient, achieved with a
two-part tariff, and the buyer and the incumbent appropriate all of the surplus.
If sE is known and ωE is unknown, then ε(ωE |sE ) is constant, strictly positive, and
inefficient exclusion arises.
18
Figure 7a: Second best with ε(ωE |sE )
constant in sE
Figure 7b: Optimal price
(case vI > ω̂E )
schedule
The complete information benchmark, where all entrant’s characteristics (cE , sE , vE )
are known when the buyer and the incumbent sign the contract can be found in Marx
and Shaffer (2004) with more general assumptions on the demand. The entrant is active
if and only he is efficient and the quantity purchased from him is undistorted, compared
to the first best. Rent extraction is complete as the dominant position of the incumbent
can be used unrestrictedly. This is also true when only the size of the contestable market,
sE , is unknown as the two-part tariff exhibited in Proposition 3 does not depend on sE .
These two cases are reminiscent of first-degree price discrimination by a monopoly.
5.2
Elasticity of entry increasing in sE
From now on, we consider cases where the elasticity of entry is not constant with sE .
This implies that a two-part tariff is no longer optimal. In this section, we start with the
case where the elasticity increases with sE .
Proposition 4. Assume that ε(ωE |sE ) increases with sE . The effective price, pe (qE ),
increases with qE . The price schedule is concave in a neighborhood of qI = 1. It is globally
concave if ω̂E is concave or moderately convex in sE .
Proof. Under the assumption that ε(ωE |sE ) increases with sE . Then ω̂E decreases with
sE and the solution of the relaxed problem is implementable. The solution to the relaxed problem, associated to the boundary function Ψ(sE ) = ω̂E (sE ), is thus incentive
compatible, and hence is solution to the complete problem.
Following the same reasoning as in Section 5.1, we show that the buyer-entrant coalition’s profit variation, ∆SBE (sE , ωE ), given by (8), equals (ω− ω̂E (sE ))sE above the ERT
line and zero for below. Then we derive the tariff, given by equation (22) in appendix:
T (1) − T (1 − sE ) = (vI − ω̂E (sE ))sE ,
implying that the effective price pe (sE ) is set at vI − ω̂E (sE ).
19
To prove that T is concave in a neighborhood of 1, remark that T (qI ) = T (1) +
0 (1 − q )(q − 1) and
(vI − ω̂E (1 − qI ))(qI − 1), then T 0 (qI ) = (vI − ω̂E (1 − qI )) + ω̂E
I
I
00
0
00
00
0 (0) < 0 (assuming
T (qI ) = 2ω̂E (1 − qI ) − ω̂E (1 − qI )(qI − 1), therefore T (1) = 2ω̂E
00 (0) is not infinite).
ω̂E
As shown in Figure 9a the entrant is either inactive (qE = 0) or serves all the contestable demand (qE = sE ). For a given ωE the jump from zero to sE can never occur
if ωE is not large enough. The jump occurs when sE is large enough for intermediate
values of ωE . Finally, if ωE is large enough, qE = sE for any sE . On the other hand, for
a given sE , qE = 0 if ωE is small (below ω̂E (sE )) while qE = sE when ωE is large (above
ω̂E (sE )).
Some efficient entrants are foreclosed. As the elasticity of entry increases with sE ,
the ERT results in a lower ω̂E as sE increases. Consequently, the optimal effective price
pe = vI − ω̂E (qE ) increases with qE : the larger the contestable market-share, the lower
the competitive pressure. If vI < ω̂E (qE ), the effective price can even be negative. As
ω̂E (sE ) is decreasing, this can happen for small sE (that is, in that case, T (qI ) would be
first increasing and next decreasing when qI → 1).
Figure 8a: Second best with ε(ωE |sE )
increasing in sE
5.3
Figure 8b: Optimal price schedule
(case sE = 0 and vI >
ω̂E (0))
Elasticity of entry decreasing in sE
We now turn to the case where the elasticity of entry is decreasing with sE . A two-part
tariff cannot be optimal.
Proposition 5. Assume that ε(ωE |sE ) decreases with sE , then the optimal tariff is
convex. The equilibrium outcome exhibits inefficient exclusion, in the form of both full
and partial foreclosure.
Proof. When ε(ωE |sE ) decreases with sE , then ω̂E is monotonically increasing and thereSB is
fore it is not implementable. As seen in Section 4.3, the optimal boundary line ωE
20
Figure 9a: Second best with ε(ωE |sE )
increasing in sE
Figure 9b: Optimal price schedule
(case sE = 0 and vI <
ω̂E (0))
increasing and given by Proposition 2. To obtain the optimal tariff, we follow the lines of
Section 4.4. For all sE ∈ [sE , s̄E ], let s0E > sE then the solution of (6) is interior for the
SB (s )) and the solution is q = s . Therefore T 0 (1 − s ) = v − ω SB (s ) or
type (s0E , ωE
E
E
E
E
E
I
E
SB
T 0 (qI ) = vI − ωE
(1 − qI )
SB (1 − q )/∂q = (ω SB )0 (1 − q ) > 0 as ω SB is increasing, which proves that T is
now, ∂ωE
I
I
I
E
E
convex.
Figure 10a: Second
ε(ωE |sE )
in sE
best
with
decreasing
Figure 10b: Optimal price schedule
(sE = 0 and vI > ω̂E (1))
21
As depicted in Figure 10a, when the entrant’s type lies in the hatched triangle below
SB (0), the entrant produces a quantity
the boundary line and above the horizontal line ωE
strictly lower than sE . That is, entry is partially foreclosed. Here, the price schedule
plays the role of a barrier to expansion in addition to a barrier to entry. Some efficient
types of entrant are active but prevented to serve all the contestable demand (they face
a barrier to expansion). Otherwise, and as in sections 5.1 and 5.2, some efficient types
remain inactive (they face a barrier to entry).
SB (0)), when ω
For a given ωE , qE can be null for all sE (whenever ωE is lower than ωE
E
is intermediate, qE = sE when sE is small and qE is constant when sE is above a threshold
SB )−1 (ω ). Finally, if ω is large enough (formally above ω SB (s̄ )), then
(formally (ωE
E
E
E
E
qE = sE for all sE . Fixing sE and letting ωE increase from ω E to ω̄E , qE is null until ωE
SB (0), then q increases with ω , and finally q = s for all ω above ω SB (s ).
reaches ωE
E
E
E
E
E
E
E
A small market share of E can, therefore, reflect either a small sE (this is the case
for an efficient enough E who sells at full capacity) or a large sE with partial foreclosure
SB
(this is the case when E is sufficiently efficient to enter but not enough to break the ωE
line and sell at full capacity).
Qualitatively these situations are very different. In the first one, E is frustrated
because he had to abandon a fraction of his surplus to the buyer. However, depending
on the interpretation of sE , either he cannot produce more or the buyer is not interesting
in buying more from the entrant. In the second case (partial foreclosure), E is similarly
deprived of some surplus, but in addition he is also frustrated because he cannot sell all
the units that the buyer would like to acquire in the absence of T .
As ε(ωE |sE ) is decreasing, the buyer-incumbent pair would like to extract more surplus from a large entrant and hence put more competitive pressure on him. Yet, a large
entrant can always mimic a smaller one. To take an example, the efficiency rent tradeoff
would imply that for some ωE it would be optimal to ask a a type with a large sE to
sell nothing while a type with a small sE to sell at full capacity. Obviously this is not
possible with only T as an instrument.
In Figure 10b, the optimal price schedule is increasing because it is drawn under
SB (s ) for all s . If, however, ω SB becomes
the assumption that vI is larger than ωE
E
E
E
larger than vI for sE large enough, then the slope of T is negative for the small qI (as
qI = 1 − sE ).
5.4
Elasticity of entry non monotonic in sE
We consider a case where the entrant expected efficiency conditionally on the size of the
contestable demand has a u-inverted shape. When the size of the contestable market is
small or large, the entrant tends to be not very efficient. When the size of the contestable
demand, the entrant tends to be very efficient. We show that this pattern leads to a highly
nonlinear tariff with a change of curvature.
Specifically, we assume that sE uniformly distributed on [0, 1] and that, conditionally
on sE , ωE is exponentially distributed with
λ(sE ) =
α
1 + sE − s2E
where α > 0 is a given constant. The incumbent’s marginal costs are constant, the
22
virtual surplus is given by (??) and the ERT line by
ω̂E (sE ) = ωI +
6
β
β
= ωI + (1 + sE − s2E ).
λ(sE )
α
Disposal costs, buyer opportunism, and conditional tariffs
We have assumed so far that the buyer incurs an infinite cost if he does not consume
all of the purchased units. This section is devoted to the case where the disposal costs
are finite: We note γ the cost of purchased units in excess of consumption (disposal
costs). We treat γ as an exogenous parameter, which can take any value between zero
and infinity. When γ = 0, the buyer can freely dispose of units he does not need. When
γ = ∞, the buyer must consume all of the purchased units, otherwise his utility is −∞.
The buyer’s total consumption is inelastic, and normalized to one: xE + xI ≤ 1.
Thus, having purchased quantities qE and qI from the buyer and the incumbent, the
buyer chooses consumption levels so as to maximize
V (qE , qI ) =
max
(xE ,xI )∈X
vE xE + vI xI − γ(qE − xE ) − γ(qI − xI ),
(15)
where the set X is defined by the constraints xE ≤ qE , xI ≤ qI , and xE + xI ≤ 1: the
buyer cannot consume more than he has purchased and more than his total requirement,
normalized to one.
In the demand-side variant of the model, there is no capacity constraint, and the
contestable part of the market reflects the buyer’s preferences: the buyer does not value
units of good E in excess of sE . The buyer’s gross benefit is modified as follows:
V d (qE , qI ) =
max
(xE ,xI )∈X
vE min(xE , sE ) + vI xI − γ(qE − xE ) − γ(qI − xI ).
We first establish an optimality result that holds irrespective of the informational
structure, i.e. whether or not the buyer and the incumbent know (cE , sE , vE ) when
signing the contract. The proof can be found in Appendix F.
Proposition 6. The buyer and the incumbent are better off using a tariff with slope
greater than or equal to −γ.
Consequently, we may assume, without loss of generality, that the incumbent does not
buy more than its total requirements: qE + qI ≤ 1.
Combined Proposition 6 with Lemma C.1 in appendix, we conclude that the buyer
purchases the exact quantity necessary to meet its requirements: qE + qI = 1, and
consumes all of the purchased quantities: xE = qE , xI = qI . The expressions (6)
and (??), respectively for the surplus created by the trade between the buyer and the
entrant, ∆SBE , and the expected profit of the buyer-incumbent coalition, EΠBI , still
hold.
Throughout this section, we say that the entrant is super-efficient if ωE ≥ vI + γ. It
follows from Proposition 6 and from the buyer-entrant problem (6) that, in equilibrium,
whatever the informational structure, a super-efficient entrant serves all the contestable
demand: qE (sE , ωE ) = sE for all ωE > vI . Indeed, for such an entrant, the function
(ωE − vI )qE − T (1 − qE ) is nondecreasing on (0, sE ).
23
Proposition 6 prompts us to extend the notion of admissibility to the case with finite
disposal cost. We say that a quantity function is admissible if it can be obtained as
solution to (6), where T is a tariff satisfying T 0 ≥ −γ and pe (q) ≤ ceI (q) for all q.
Lemma 1 must be adapted as follows.
Lemma 3. A quantity function qE (., .) is admissible if and only if there exists a boundary
function Ψ(.) defined on [0, 1], with vI −ceI (sE ) ≤ Ψ(sE ) ≤ vI +γ for all sE , such that (9)
holds.
The new condition on the boundary, Ψ(sE ) ≤ vI + γ, expresses that super-efficient
entrants serve all of the contestable demand: qE = sE . The sufficient part of the lemma
is proved in Section F. The analysis with infinite disposal costs (Sections ?? and ??)
carries over without any change when there are no super-efficient entrants: ω̄E ≤ vI + γ.
Under perfect information, only super-efficient entrants earn a positive profit. The
equilibrium configuration can be obtained with a constant effective price, pe (q) = max(vI −
ωE , −γ). Super-efficient entrants earns β(ωE −vI −γ)sE . Other entrants earn zero profit.
Under one-dimensional uncertainty (ωE unknown), the expression (??) applies to
entrants that are not super-efficient. For those entrants, it must be replaced with sE .
If the incumbent’s cost function is affine or concave, the second-best quantity is sE for
ωE ≥ min(ω̂E , vI + γ), where ω̂E is given by (12), and zero otherwise. The second best
can be achieved with a two-part tariff, by setting the constant effective price schedule at
pe (q) = max(vI − ω̂E , −γ). Efficient entrants with vI − ceI (sE ) < ωE ≤ min(ω̂E , vI + γ)
are excluded.
Under two-dimensional uncertainty, the construction of the optimal quantity is modified as follows: First apply the procedure exposed in Section ??, then replace Ψ with
min(Ψ, vI + γ).
Proposition 6 extends to the case where the tariff T can be made conditional on the
quantity purchased from the entrant. Applying the same reasoning as in Appendix F to
each value of qE , one may show that the buyer and the incumbent are better off using
a tariff T (qE , qI ) such that the marginal price of an extra unit of good I, TqI (qE , qI ), is
greater than or equal to −γ, and that the buyer purchases the quantity necessary to meet
its requirement: qE + qI = 1. These properties, however, do not imply that the effective
price be greater than −γ and that super-efficient entrants serve all of the contestable
demand. Indeed, the effective price is now given by [T (0, 1) − T (qE , 1 − qE )]/qE , which
can be lower than −γ, and the objective of the buyer-entrant coalition, (ωE − vI )qE −
T (qE , 1 − qE ), is not necessarily monotonic for super-efficient entrants.
Proposition 7. Regardless of the informational structure, the expected profit of the
buyer-incumbent coalition is weakly lower when the disposal costs are finite than when
they are infinite. It is unchanged only when there exist no super-efficient entrants.
Conditioning the tariff on the quantity purchased from the entrant allows the buyer
and the incumbent to earn the same profit as if disposal costs were infinite.
7
Discussion
In this paper, we adopt a positive perspective and develop a framework to analyse the
pricing policy of a large firm contracting with a buyer under the threat of entry (or
24
expansion) by a smaller rival. We show how nonlinear pricing emerges in equilibrium
and how it builds barriers to entry. The strength of the entrant is uncertain.
7.1
The entrant’s bargaining power
7.2
The marginal cost of the incumbent is not constant
As nonlinear pricing may reflect economies of scale, we allow firm I’s marginal cost to
be nonincreasing. Firm I’s cost function, noted CI (qI ), is thus assumed to be a concave
function. We also assume that the marginal cost CI0 (qI ) ≤ vI for all qI .
7.3
Antitrust implications
Citer le DOJ et Shapiro
The as-efficient competitor test seems accepted, at least as a matter of principle. (Its
administrability for specific conducts is sometimes questioned.)
According to Proposition 1, a regulator or antitrust enforcer can, in theory, achieve
the first best outcome by prohibiting the incumbent from selling below cost, formally by
imposing the constraint
pe (q) ≥ ceI (q)
(16)
for all q. This condition expresses the as-efficient competitor rule advocated by the
European Commission. In practice, however, enforcing such a constraint is by no means
trivial. To our knowledge, the Intel decision contains the first and, to date, the sole
attempt to check (16) in an antitrust case. Yet, the decision insists that the test is
only implemented to “illustrate” the exclusionary effect and is not a legal requirement to
characterize the infringement. Furthermore, it should be noted that checking (16) at any
point of the tariff would require computing the entire incumbent’s cost function, which is
clearly unfeasible. Accordingly, the Commission has run the test for one single value of q,
namely q = 1−sE , where sE was a “realistic” value for the size of the contestable demand.
This approach thus requires to agree on the magnitude of the contestable demand as well
as on the incumbent’s incremental cost evaluated at this point. Both issues have proved
highly contentious in the Intel case, and are still to be discussed before the Tribunal of
First Instance.
In sum, while imposing condition (16), at least at a particular point, is not impossible,
it certainly entails huge administrative costs, to be borne by both dominant firms and
regulators. It is therefore crucial to assess the costs of laissez-faire.
Leaving aside predation-like scenarios and considering a purely static environment,
we have shown how nonlinear pricing can be used to deter efficient entry. The analysis
highlighted that inefficient exclusion depends on incomplete information: when the buyer
and the incumbent know the characteristics of the entrant, the contractual price schedule,
while violating the as-efficient competitor test, leads to the first best outcome. Under
laissez-faire and complete information, the effect of the contract is simply to shift (part
of) the rent from the efficient entrant towards the buyer and the incumbent.
Incomplete information, combined with the incumbency advantage, leads to inefficient
exclusion. When the informational asymmetry only concerns the entrant’s efficiency,
inefficient exclusion arises in the form of complete foreclosure, and the shape of the tariff
reflects that of the cost function. The tradeoff between efficiency and rent extraction
25
creates a wedge between incremental cost and incremental price, and this wedge leads to
the complete foreclosure of some efficient entrants.
When both the entrant’s efficiency and the size of the contestable market are unknown
at the time of contracting, inefficient exclusion can take the form of partial foreclosure,
meaning that the entrant may serve part of the contestable demand, while it could and
should (from an efficiency perspective) serve all of that demand. Partial foreclosure is
achieved by introducing non-convexities in the contractual price schedule, similar to those
induced by retroactive rebates challenged by European antitrust agencies.
We have set up a simple, static framework in which the complicated price schedules
observed in practice can be adopted absent any predation motivation, and have explained
how these schedules induce partial or complete foreclosure. We have thereby provided
a theoretical background in which the as-efficient competitor rule (16) is optimal. The
scenario does not involve any short-term sacrifice from the dominant firm, and hence
the antitrust authority, in this stylized environment, should not have the burden of
establishing a serious probability of recoupment.
On the other hand, the theoretical scenario we have suggested should certainly not be
taken literally in any particular antitrust case, not least because it does not encompass
specific efficiency gains from nonlinear pricing. From a general policy perspective, the
as-efficient competitor standard, as argued above, puts a heavy burden on firms and
regulators, because enforcement requires estimating the cost function and deciding on
a relevant magnitude for the size of contestable demand for each and every contract.
Antitrust agencies should be convinced, before initiating such cases, that the distortions
due to inefficient exclusion are likely to overcome the enforcement costs.
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27
Appendix
A
Inefficient entry never arises in equilibrium
In this section, we prove Proposition 1 under the assumption that disposal costs are
infinite.14 To this aim, we consider any tariff T and define T̃ as
T̃ (qI ) = max(T (qI ), T (1) − CI (1) + CI (qI )).
We show that the change from T to T̃ weakly enhances the buyer-incumbent coalition’s
expected profit. This transformation yields T̃ (1) = T (1) and amounts to replace pe (q)
with p̃e (q) = min(pe (q), ceI (q)). Thanks to Lemma C.1, we may assume, with no loss of
generality, that the original tariff satisfies T 0 ≤ vI .
The slope of T̃ does not exceed the monopoly price vI as T 0 ≤ vI and CI0 ≤ vI .
Since γ = ∞, we know that the buyer purchases qI = 1 − qE from the incumbent if he
has purchased qE from the entrant, and thus that the expressions (6) and (??) for the
coalitions’ profits are valid under T̃ . The buyer-entrant coalitions profit gain under T̃ is
given by
(17)
∆Π̃BE (sE , ωE ) = max (ωE − vI )qE − T̃ (1 − qE ) + T̃ (1).
qE ≤sE
Since p̃e ≤ pe , the tariff T̃ exerts a stronger competitive pressure on the entrant: ∆Π̃BE ≤
∆SBE . It follows that the change from T to T̃ does not increase the entrant’s profit.
Next, we show that, for any given (sE , ωE ), the change does not decrease the total
surplus. In other words, we prove that for any (sE , ωE ): S(q̃E (sE , ωE )) ≥ S(qE (sE , ωE )),
where qE and q̃E are solution to respectively (6) and (17), and the total surplus function
S is defined by equation (2). To this aim, we introduce the following functions:
∆S(qE ) = (ωE − vI )qE + CI (1) − CI (1 − qE )
u(qE ) = (ωE − vI )qE + T (1) − T (1 − qE )
ũ(qE ) = (ωE − vI )qE + T̃ (1) − T̃ (1 − qE ).
We have ũ = min(u, ∆S). We distinguish two cases:
• If pe (qE ) ≥ ceI (qE ), then p̃e (qE ) = ceI (qE ), and, by definition of q̃E : ∆S(qE ) =
ũ(qE ) ≤ ũ(q̃E ) ≤ ∆S(q̃E );
• If pe (qE ) ≤ ceI (qE ), then p̃e (qE ) = pe (qE ), ũ(qE ) = u(qE ), and hence ∆Π̃BE =
∆SBE . It follows that qE is also solution to problem (17). Conversely, q̃E is
solution to problem (6) as u(q̃E ) ≥ ũ(q̃E ). The change from T to T̃ leaves the
incentives of the buyer-entrant coalition unchanged.
In sum, the change from T to T̃ either does not change the incentives of the buyerentrant coalition or it does, but then the surplus S(qE ) is not reduced. As seen above,
the change does not increase the entrant’s profit. Hence, the change does not decrease
the buyer-incumbent coalition’s expected payoff.
14
In Appendix F, we show that Proposition 1 holds for finite γ as well.
28
Finally, for an inefficient entrant, the maximum of ∆S(qE ; ωE ) over qE ≤ sE is zero.
It follows that the maximum of ũ over qE ≤ sE is also zero. The buyer and the entrant
may choose q̃E = 0 under T̃ .
B
Derivation of the ptimal quantity function
For any implementable quantity function qE and any ωE ∈ [ω E , ω̄E ], the function of one
variable qE (., ωE ) is nondecreasing on [0, 1] and satisfies
x
for x2i ≤ x ≤ x2i+1
qE (x, ωE ) =
x2i+1 for x2i+1 ≤ x < x2i+2 ,
where (xi ) is a finite increasing sequence in [0, 1] whose first term is zero and last term is
one.15 Such a sequence defines a partition of [0, 1) into “even intervals” [x2i , x2i+1 ) and
“odd intervals” [x2i+1 , x2i+2 ). The quantity function qE (., ωE ) coincides with the identity
map on even intervals and is constant on odd intervals. It is continuous at interior “odd
extremities”, x2i+1 , and has upward jumps at interior “even extremities”, x2i . Even (odd)
extremities constitute the nonincreasing (nondecreasing) parts of the boundary line Ψ
associated with the quantity function qE .
We consider the set K of all functions r of one variable that satisfy the above properties for some partition of [0, 1] into even and odd intervals. Let a(.) be a continuous
function on [0, 1]. We consider the problem
Z 1
max
a(x)r(x) dx.
(18)
r∈K
0
Lemma 4 (First-order conditions for interior extremities). Consider a solution r∗ of
Problem (18). For any interior even extremity x∗2i , the function a equals zero at x∗2i and
is negative (positive) at the left (right) of x∗2i . For any interior odd extremity x∗2i+1 , the
function a is nonnegative at x∗2i+1 and satisfies
Z x∗
2i+2
a(x) dx = 0.
(19)
x∗2i+1
Proof Letting I(r) =
R1
a(x)r(x) dx, we have
Z
X Z x2i+1
X
I(r) =
xa(x) dx +
x2i+1
0
i
x2i
i
x2i+2
a(x) dx,
x2i+1
where the index i in the two sums goes from either i = 0 or i = 1 to either i = n − 1 or
i = n, in accordance with the conventions exposed in Footnote 15. Differentiating with
respect to an interior even extremity yields
∂I
= a(x2i ).[x2i−1 − x2i ].
∂x2i
15
For notational consistency, we adopt the following conventions. If qE (., ωE ) is constant (and equal
to zero) at the bottom of the interval [0, 1], we denote the first term of the sequence (xi ) by x1 = 0. If
qE (., ωE ) coincides with the identity map at the bottom, we denote the first term by x0 = 0. If qE (., ωE )
is constant at the top of the interval [0, 1], we denote the last term of the sequence (xi ) by x2n = 1. If
qE (., ωE ) coincides with the identity map at the top, we denote the last term by x2n+1 = 1.
29
The first-order condition therefore imposes a(x∗2i ) = 0. The second-order condition for a
maximum shows that a must be negative (positive) at the left (right) of x∗2i .
Differentiating with respect to an interior odd extremity yields
Z x2i+2
∂I
=
a(x) dx.
∂x2i+1
x2i+1
R x∗
The first-order condition therefore imposes x∗2i+2 a(x) dx. The second-order condition
2i+1
for a maximum imposes that a is nonnegative at x∗2i+1 .
Lemma 5. The problem (18) admits a unique solution r∗ characterized as follows:
• If a(1) > 0, then r∗ (x) = x at the top of the interval [0, 1]. There exists n such that
x∗2n+1 = 1.
• If a(1) < 0, then r∗ is constant at the top of the interval. There exists n such that
x∗2n = 1.
In both cases, the lower extremities are determined sequentially starting from the highest
one, using Lemma 4.
Proof If a(1) > 0, then it is easy to check that r∗ (x) = x at the top, namely on the
interval [x∗2n , x∗2n+1 ] with x∗2n being the highest zero of the function a and x∗2n+1 = 1.
If the function a admits no zero, it is everywhere positive and hence r∗ (x) = x on the
whole interval [0, 1].
If a(1) < 0, then r∗ is constant at the top, namely on the interval [x∗2n−1 , x∗2n ], with
R1
R1
a(x) dx = 0. If the integral y a(x) dx remains negative for all y, then
x∗2n = 1 and x∗
2n−1
r∗ is constant and equal to zero on the whole interval [0, 1].
We denote the virtual surplus per unit by sv :
sv (sE , ωE ) =
∂S v
1 − F (ωE |sE )
= ωE − ωI − β
.
∂qE
f (ωE |sE )
The buyer-incumbent pair maximizes
ZZ
sv (sE , ωE )qE (sE , ωE ) dF (ωE |sE ) dG(sE )
over all implementable quantity functions qE . A quantity function is implementable if
and only if for any ωE , the function of one variable qE (., ωE ) belongs to the above-defined
set K and even (odd) extremities x∗2i (ωE ) (x∗2i+1 (ωE )) are nonincreasing (nondecreasing)
in ωE .
Solving the problem separately for each ωE yields a candidate solution. Applying
Lemma 4 with the functions of one variable a(sE ) = sv (sE , ωE ), we find that for any
ωE the virtual surplus is zero at candidate even extremities: sv (x2i (ωE ), ωE ) = 0 and
is negative (positive) at the left (right) of these extremities. In other words, candidate
even extremities belong to decreasing parts of the ERT line.
Lemma 4 also implies that the virtual surplus is nonnegative at candidate odd extremities: sv (x2i+1 (ωE ), ωE ) ≥ 0. In other words, candidate odd extremities lie on or
above the ERT line. Unfortunately, the first-order condition (19) does not imply that
candidate odd extremities x2i+1 (ωE ) are nondecreasing with ωE : odd extremities could
decrease with ωE in some regions, generating two-dimensional pooling.
30
C
The buyer’s total purchases are not lower than his total
demand
This section presents an optimality result that holds irrespective of the information structure, i.e. whether the buyer and the incumbent know the entrant’s characteristics at the
time of contracting. It is in the buyer’s and incumbent’s common interest to agree on a
price schedule that induces the former to purchase at least 1 − qE units from the latter,
once he has purchased qE from the entrant, for any value of qE . Hence, in equilibrium,
the buyer’s total purchases are equal to, or exceed, his total demand.
Lemma C.1. Regardless of the disposal cost γ, the buyer and the incumbent are better
off using a tariff with slope T 0 smaller than or equal to vI .
Consequently, we may assume, with no loss of generality, that the buyer does not buy
less than its total requirements: qE + qI ≥ 1.
Proof: We start from any price schedule T . Let T̃ be defined by
T̃ (qI ) = inf T (q) + vI (qI − q).
q≤qI
(20)
The tariff T̃ is derived from the tariff T as follows. When the incumbent offer q units at
price T (q), she also offers to buy more units than q, say q 0 > q, at price T (q) + vI (q 0 − q).
The additional units are offered at the monopoly price vI . By construction, the slope of
T̃ is lower than or equal to vI .
Let ŨB (qE ) denote the buyer’s net utility after he has purchased qE units from the
entrant under the price schedule T̃
ŨB (qE ) = max V (qE , qI ) − T̃ (qI ).
qI
(21)
As T̃ ≤ T , we have: ŨB ≥ UB . Suppose that, under T̃ , it is optimal for the buyer to
purchase q̃I from the incumbent if he has purchased qE from the entrant. By construction
of T̃ , there exists qI ≤ q̃I such that T̃ (q̃I ) equals or is arbitrarily close to T (qI )+vI (q̃I −qI ).
We have:
ŨB (qE ) = V (qE , q̃I ) − T̃ (q̃I )
= V (qE , q̃I ) − T (qI ) − vI (q̃I − qI ) ≤ V (qE , qI ) − T (qI ),
which implies ŨB (qE ) ≤ UB (qE ), and hence ŨB (qE ) = UB (qE ) for all qE . In the above
inequality, we have used the property (coming from the envelope theorem) that the
derivative of V (qE , qI ) with respect to qI is lower than or equal to vI .16
Regardless of the entrant’s type (cE , sE , vE ), the buyer and the entrant agree on the
same quantity qE under T and T̃ , as the problem they face depends only on the functions
UB (.) and ŨB (.), which coincide. The entrant’s profit is the same under T and T̃ .
Now, for any given qE , suppose that the buyer, having purchased qE from the entrant,
chooses to purchase qI from the incumbent under the price schedule T . As T̃ (qI ) ≤ T (qI ),
16
More precisely, we have: VqI (qE , qI ) = −γ + µ = vI − ν, where µ and ν are the respective Lagrange
multipliers for the constraints xI ≤ qI and xE + xI ≤ 1 in the buyer’s problem (1). It follows that VqI
equals vI when qI < 1 − qE (as ν must be zero in this case) and VqI equals −γ when qI > 1 − qE (recall
that xE = qE , and hence xI must be smaller than qI if qI > 1 − qE , which yields µ = 0).
31
the buyer may always choose to purchase the same quantity from the incumbent under
T̃ as under T : q̃I = qI :
UB (qE ) = ŨB (qE ) = V (qE , qI ) − T (qI ) ≤ V (qE , qI ) − T̃ (qI ).
Yet, if qI < 1 − qE , then, by definition of T̃ , we have T̃ (1 − qE ) ≤ T (qI ) + vI (1 − qE − qI ).
As, from the envelope theorem, VqI = vI in this region, we have:
UB (qE ) = ŨB (qE ) = V (qE , qI ) − T (qI ) ≤ V (qE , 1 − qE ) − T̃ (1 − qE ),
implying that the buyer may choose to purchase q̃I = 1 − qE from the incumbent under
T̃ . As vI ≥ CI0 , the change from qI to q̃I does not decrease the total surplus:
V (qE , q̃I ) − cE qE − CI (q̃I ) ≥ V (qE , qI ) − cE qE − CI (qI ).
In sum, the change from T to T̃ does not alter the entrant’s profit and does not decrease
the total surplus. We conclude from (10) that the change does not decrease the expected
payoff of the buyer-incumbent coalition.
D
Complete information
We prove Lemma 1, and assume that the buyer and the incumbent know the entrant’s
type (cE , vE , sE ) when choosing the tariff T .
First, suppose firm that the entrant is not super-efficient: ωE ≤ vI . Let ε be an
arbitrarily small positive number. Setting the effective price of contestable units pe (q) =
(1 − ε)(vI − ωE ) + εceI (q), then by (6) the buyer and the entrant maximize
∆SBE = ε max (ωE − vI )qE + CI (1) − CI (1 − qE ).
qE ≤sE
∗ from the entrant, and the latter gets
The buyer thus purchases the efficient quantity qE
an arbitrarily small profit.
Finally, suppose that the entrant is super-efficient: ωE > vI . We know from Lemma ??
that the effective price can be assumed to be nonnegative. It follows from (6) that the surplus gain generated by the trade between the entrant and the buyer, ∆SBE (cE , sE , vE ),
is greater than or equal to (ωE − vI )sE . It then follows from (??) that the buyer’s and
incumbent’s objective satisfies:
ΠBI ≤ vI − CI (1) + [ωE − (vI − ceI (qE ))]qE − β(ωE − vI )sE .
The right-hand side of the above inequality is maximum when qE = sE . Now when
the buyer and the incumbent agree on a zero effective price, pe (.) = 0, we know from
equation (6) that any super-efficient entrant sells qE = sE to the buyer and earns β(ωE −
vI )sE , the best outcome the buyer and the incumbent can possibly achieve.
E
Implementation
We prove here the sufficient part of Lemma 1. Starting from any boundary function Ψ
defined on [0, 1] such that vI − ceI (sE ) ≤ Ψ(sE ) for all sE , we define the quantity function
32
qE (sE , ωE ) by equation (9), and the profit function ∆SBE (sE , ωE ) by equation (??). We
observe that the functions thus defined qE (sE , ωE ) and ∆SBE (sE , ωE ), are nondecreasing
in both arguments, and the latter function is convex in ωE . Next, we notice that the
expression
(ωE − vI )qE (sE , ωE ) − ∆SBE (sE , ωE )
is constant on qE -isolines. Indeed, both qE (., ωE ) and ∆SBE (., ωE ) are constant on horizontal isolines (located below the boundary Ψ). On vertical isolines (above the boundary), ∆SBE (sE , .) is linear with slope sE , guaranteing, again, that the above expression
is constant. We may therefore define T (q), up to an additive constant, by
T (1) − T (1 − q) = (vI − ωE )q + ∆SBE (sE , ωE ),
(22)
for any (sE , ωE ) such that q = qE (sE , ωE ). Equation (22) unambiguously defines T (1−q)
on the range of the quantity function qE (., .). This range contains zero. Indeed, as
ω E < vI − ceI (1) ≤ Ψ, entrants with types (sE , ωE ), with ωE < vI − ceI (1), are inactive
according to (9). In contrast, the range of the quantity function does contain any interval
(s1E , s2E ) on which Ψ > ω̄E and Equation (22) does not allow to define T (1 − q) on such
intervals. We explain in Section E.1 that we can construct T as being linear with slope
vI − ω̄E on such intervals.
Newt, we prove that the buyer and the entrant, facing the above defined tariff T ,
agree on the quantity qE (sE , ωE ):
∆SBE (sE , ωE ) ≥ (ωE − vI )q 0 + T (1) − T (1 − q 0 )
(23)
for any q 0 ≤ sE . To this aim, we first notice that the quantity function qE (., .) defined
from (9) is nondecreasing in both ωE and sE , as well as the gain function ∆SBE (., .)
defined from (??). The latter function is therefore convex with respect to ωE . Moreover,
the quantity function qE (., .) is continuous when crossing increasing part of the boundary
and jumps upwards when crossing decreasing parts of the boundary. Equation (23) is
equivalent to:
∆SBE (sE , ωE ) ≥ (ωE − vI )q 0 + TI (0, 1) − TI (q 0 , 1 − q 0 )
0
0
0
0
= (ωE − vI )qE (ωE
, s0E ) + ∆SBE (ωE
, s0E ) − (ωE
− vI )qE (ωE
, s0E )
0
0
0
= ∆SBE (ωE
, s0E ) + (ωE − ωE
)qE (ωE
, s0E ),
(24)
0 , s0 ) such that q 0 = q (ω 0 , s0 ). Since q (ω , q 0 ) = q 0 and
for any q 0 ≤ sE and any (ωE
E
E
E
E E
E
0
0
0
∆SBE (ωE , q ) = ∆SBE (ωE , s0E ), we may assume, without loss of generality, s0E = q 0 , and
hence s0E ≤ sE . Using the monotonicity and the convexity of ∆SBE (., .) respectively in
sE and in ωE , we get
∆SBE (sE , ωE ) ≥ ∆SBE (ωE , s0E )
0
0
0
≥ ∆SBE (ωE
, s0E ) + (ωE − ωE
)qE (ωE
, s0E ),
which yields (24), and hence (23).
Finally, we show that : pe ≤ ceI .
33
E.1
Flat portions of the boundary functions
0 , if and only if
Lemma 6. The boundary function Ψ is constant on [q0 , q1 ], Ψ(sE ) = ωE
the following properties hold:
0 and s ∈ [q , q ], s belongs to the set of solutions to the buyer-entrant
1. for ωE = ωE
0 1
E
E
coalition’s problem (6);
2. T is linear on [1 − q1 , 1 − q0 ].
0 and s ∈ [q , q ], we have
Proof Suppose Ψ is constant on [q0 , q1 ]. For ωE > ωE
0 1
E
(ωE − vI )sE + T (1) − T (1 − sE ) = ∆SBE (sE , ωE ).
0
The function ∆SBE is convex, and hence continuous, in ωE . Letting ωE tend to ωE
yields
0
0
(ωE
− vI )sE + T (1) − T (1 − sE ) = ∆SBE (ωE
, sE ).
(25)
0 , q (s , ω ), and hence ∆S
For ωE < ωE
E E
E
BE (sE , ωE ), do not depend on sE ∈ [q0 , q1 ].
Using again the continuity of ∆SBE in ωE , we conclude that the right-hand side of (25)
0.
does not depend on sE . It follows that T is linear on [1 − q1 , 1 − q0 ], with slope vI − ωE
Conversely, suppose that items 1 and 2 hold. Let t denote the slope of the tariff on
0 = v − t. For ω < v − t, the function (ω − v )q − T (1 − q )
[1 − q1 , 1 − q0 ] and let ωE
I
E
I
E
I E
E
decreases in qE on [q0 , q1 ], and hence the solution of problem (6) with sE ∈ [q0 , q1 ], must
0 , s is also solution
be interior. As sE is solution to (6) for sE ∈ [q0 , q1 ] and for ωE = ωE
E
for higher values of ωE (this is because the quantity function is nondecreasing in ωE ). It
0 for all s ∈ [q , q ].
follows that Ψ(sE ) = ωE
0 1
E
The same line of reasoning shows that any boundary function such that Ψ = ω̄E ≤ vI
on some interval [q0 , q1 ] is consistent with a tariff linear on [1 − q1 , 1 − q0 ]. Specifically,
suppose that Ψ(sE ) = ω̄E in [q0 , q1 ], we know that, for any ωE < ω̄E , qE (sE , ωE ), and
hence ∆SBE (sE , ωE ), do not depend on sE . The function ∆Π is convex in ωE , and hence
continuous. It follows that, for ωE = ω̄E , the right-hand side of (22) is constant in sE
and any value of the quantity function for ωE = ω̄E and sE ∈ [q0 , q1 ] must be consistent
with a linear tariff with slope vI − ω̄E .
E.2
The shape of the boundary function and the buyer-entrant coalition’s problem
Here we link the shape of the boundary function Ψ to that of the tariff T .
Lemma 7. If Ψ is nondecreasing at sE , the constraint qE ≤ sE in the buyer-entrant
coalition problem (6) with ωE = Ψ(sE ) is not binding. Moreover,
• if Ψ is continuous and increasing in the neighborhood of sE , then Ψ(sE ) − vI +
T 0 (1 − sE ) = 0 and the tariff is convex in the neighborhood of 1 − sE .
• If Ψ jump upwards at sE , T jumps downwards at 1 − sE (retroactive rebates).
• If Ψ is locally flat, the tariff is locally flat.
34
If Ψ is decreasing at sE , the constraint qE ≤ sE in the buyer-entrant coalition problem (6)
with ωE = Ψ(sE ) is binding.
We have:
∂∆SBE (sE , ωE )
= λ(sE , ωE )
∂sE
where λ(sE , ωE ) is the Lagrange multiplier associated to the constraint qE ≤ sE in (6).
Recall that
Z ωE
∆Π(sE , ωE ) =
qE (x, sE ) dx.
ωE
Suppose Ψ increases in some region. Then qE (sE , ωE ) is continuous when crossing the
boundary and it follows that
∂∆SBE (Ψ(sE ), sE )
=0
∂sE
and hence λ(Ψ(sE ), sE ) = 0. Thus the first order condition Ψ(sE ) − vI + T 0 (1 − sE ) = 0
holds, implying that T 0 increases in this region, T convex (this is only an implication).
Next, suppose that Ψ decreases in some region. Then qE (sE , ωE ) is discontinuous
and it follows that
∂∆SBE (sE , ωE )
= −Ψ0 (sE )[q + (Ψ(sE ), sE ) − q − (Ψ(sE ), sE )] > 0
∂sE
and hence λ > 0, the constraint qE ≤ sE is binding (but here T may be convex, Ψ concave
is only a sufficient condition for the constraint to be binding.). The distinction between
the case Ψ increasing and Ψ decreasing is purely related to the slackness of the capacity
constraint. When Ψ increases, the constraint qE ≤ sE is slack, when Ψ increases, it is
binding, Ψ(sE ) − vI + T 0 (1 − sE ) > 0.
F
F.1
Finite disposal costs
Proof of Proposition 6
Starting from any tariff T , we define T̂ as
T̂ (qI ) = inf T (q) + γ(q − qI ).
q≥qI
Starting from any quantity level q, the incumbent offers the incumbent the opportunity
to buy less units than q, qI ≤ q, in return for the payment T (q) + γ(q − qI ). This option
allows the buyer to avoid disposal costs, and is relevant only if γ is finite. The slope of
the new tariff is larger than or equal to −γ.
Let Û (qE ) the buyer’s net utility after he has purchased units qE units from the
entrant under the price schedule T̂ :
ÛB (qE ) = max V (qE , qI ) − T̂ (qI ).
qI
(26)
As T̂ ≤ T , we have: ÛB ≥ UB . Suppose that, under T̂ , it is optimal for the buyer to
purchase q̂I from the incumbent if he has purchased qE from the entrant. By construction
35
of T̂ , there exists qI ≥ q̂I such that T̂ (q̂I ) equals or is arbitrarily close to T (qI )+γ(qI − q̂I ).
Using the definition of V , we get:
ÛB (qE ) = V (q̂I , qE ) − T̂ (q̂I )
= V (q̂I , qE ) − γ(qI − q̂I ) − T (qI )
≤ V (qI , qE ) − T (qI ).
It follows that ÛB (qE ) ≤ UB (qE ), and hence ÛB (qE ) = UB (qE ). The buyer and the
entrant agree on the same quantity qE as their choice only depends on UB and ÛB ,
which coincide. The entrant’s profit, β∆SBE is the same under T and T̂ .
Suppose that the buyer has purchased qE from the entrant and let qI be the optimal
quantity purchased from the incumbent under tariff T . As T̂ (qI ) ≤ T (qI ), the buyer may
always choose to purchase the same quantity from the incumbent (q̂I = qI ) under the
tariffs T̂ and T :
UB (qE ) = ŨB (qE ) = V (qE , qI ) − T (qI ) ≤ V (qE , qI ) − T̂ (qI ).
Yet consider the special case where qI > 1−qE . We know from Footnote 16 that V (qE , qI )
is nonincreasing and linear in qI with slope −γ on (1−qE , qI ). By definition of T̂ (1−qE ),
we get
V (qE , qI ) − V (qE , 1 − qE ) = −γ[qI − (1 − qE )] ≤ T (qI ) − T̂ (1 − qE )
or
UB (qE ) = Û (qE ) = V (qE , qI ) − T (qI ) ≤ V (qE , 1 − qE ) − T̂ (1 − qE ).
It follows that the buyer may purchase q̂I = 1 − qE from the incumbent. The change
from qI to q̂I does not decrease the total surplus. On the contrary, it avoids production
and disposal costs:
V (qE , q̂I ) − cE qE − CI (q̂I ) ≥ V (qE , qI ) − cE qE − CI (qI ).
In sum, the change from T to T̂ does not alter the entrant’s profit and does not decrease
the total surplus. We conclude from (10) that the change does not decrease the expected
payoff of the buyer-incumbent coalition.
F.2
Proof of lemma 3
Conversely, assume that Ψ(sE ) ≤ vI + γ, and define the quantity function by (9), the
profit function ∆SBE (sE , ωE ) by equation (??) and the tariff by (22). Differentiating
the latter equation with respect to ωE below the boundary – a region where qE increases
with ωE – yields T 0 (q) = vI − ωE ≥ −γ.
36