Discussion Paper No. 860

INFRASTRUCTURE UPGRADES
AND FORECLOSURE
WITH COEXISTENCE OF SERVICE-BASEDAND FACILITY-BASED FIRMS

Noriaki Matsushima
Keizo Mizuno

December 2012

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Infrastructure Upgrades and Foreclosure with Coexistence ofService-Based and Facility-Based Firms
Noriaki Matsushima∗
Institute of Social and Economic Research, Osaka University
Keizo Mizuno
School of Business Administration, Kwansei Gakuin University
December 9, 2012

Abstract
We investigate the incentives for facility-based ﬁrms to invest in infrastructure upgrades
and to foreclose service-based ﬁrms. We focus on asymmetric regulation regarding servicebased ﬁrms’ access to the infrastructure held by a facility-based ﬁrm. Spillovers from theinfrastructure upgrades made by a regulated facility-based ﬁrm on service-based ﬁrms playa key role in the incentives for making these upgrades. The spillover eﬀect can enhance
the incentives for the regulated facility-based ﬁrm to make upgrades if access prices are
not regulated. The existence of rival facility-based ﬁrms strengthens the incentives for aregulated facility-based ﬁrm to make infrastructure upgrades, especially when the spillover
eﬀect is signiﬁcant. Furthermore, if access prices are not regulated, the existence of rival
facility-based ﬁrms weakens the incentives for a regulated facility-based ﬁrm to foreclose
service-based ﬁrms.
Keywords: investment, access, service-based ﬁrm, facility-based ﬁrm.
JEL classification: L43, L51, L96.

∗
Corresponding author: Noriaki Matsushima, Institute of Social and Economic Research, Osaka University,
Mihogaoka 6-1, Ibaraki, Osaka 567-0047, Japan. Phone: +81-6-6879-8571. E-mail: nmatsush@iser.osaka-u.ac.jp

1

Introduction

High-quality networks that allow high-speed Internet access, such as Next Generation Networks,
are a driving force of economic growth in many countries and in the world as a whole.1 Diﬀerent
countries have adopted diﬀerent policy approaches for infrastructure upgrades (or broadband
penetration) and the promotion of competition in telecommunications. In the US, the deregulation of telecommunications is considered to be an eﬀective tool for incentivizing infrastructure
upgrades and promoting competition, whereas in Korea and Japan, the government intervenes
directly. The EU takes an intermediate stance, and its policy remains under debate.2 Some
intrinsic characteristics of these countries, such as technological progress in telecommunications
and political or bureaucratic systems, aﬀect whether deregulation or intervention is a better way
to enhance infrastructure upgrades.
The main purpose of this paper is to examine the incentives for facility-based ﬁrms to upgrade
their infrastructure and foreclose other ﬁrms. We take into account the coexistence of servicebased ﬁrms that need to pay an access charge to borrow the infrastructure held by a facilitybased ﬁrm (call this facility-based ﬁrm the incumbent) and facility-based ﬁrms that can serve
consumers using their own infrastructure (so-called bypass). Allowing these ﬁrms to coexist gives
policymakers the option to impose “asymmetric access regulation” on a dominant incumbent
ﬁrm. Access regulation is asymmetric if a dominant incumbent ﬁrm is obligated to provide
access to service-based ﬁrms but other facility-based ﬁrms are not required to do so.3
It is important to investigate the eﬀect of asymmetric access regulation in a real policy arena
because many countries, including the US, the EU countries, Korea, and Japan, have adopted
such regulations in their telecommunication sectors. For example, in the US, the local loop
unbundling under the 1996 Telecommunications Act obligates telephone companies to provide
1

See Crandal (2005) and Hatﬁeld et al. (2005) for information on the eﬀect of the penetration of broadband
networks on the economy as a whole.
2
For example, see Huigen and Cave (2008), Picot and Wernick (2007), and de Bijl and Peitz (2008) for the
regulatory approaches in several developed countries.
3
We use the term, “asymmetric access regulation,” to refer to the situation where access regulation is imposed
only on the dominant ﬁrm to facilitate the entry of independent service-based ﬁrms. This term is diﬀerent fromthat in Peitz (2005a,b), who investigates access regulation in the context of two-way access.

1

access, but it does not aﬀect cable TV companies, which also are facility-based ﬁrms. This isa typical example of asymmetric access regulation, and service-based and facility-based ﬁrms
coexist under this type of regulation. The EU, Korea, and Japan have also set such asymmetricaccess regulations.4
Under asymmetric access regulation, the dominant incumbent ﬁrm’s freedom to set its accessprice signiﬁcantly aﬀects its incentives to make infrastructure upgrades or foreclose servicebased ﬁrms. When access price regulation is imposed on the dominant incumbent ﬁrm, its
incentives to make infrastructure upgrades fall because these upgrades also beneﬁt the servicebased ﬁrms that use the infrastructure and are its rivals in the retail market. On the contrary,when the dominant incumbent ﬁrm freely sets the access price for service-based ﬁrms, it can
indirectly control the service-based ﬁrms in the retail market through an optimal combinationof infrastructure upgrades and the access price. In particular, the incumbent may accommodate
other ﬁrms as its de facto aﬃliates to earn proﬁts from the access charge.5
Furthermore, strategic interactions between the incumbent and other facility-based ﬁrms
through investments in infrastructure upgrades also play a key role in this setting. To fully
understand the eﬀect of access price regulation on the market, we need to know how asymmetric access regulation and access price regulation aﬀect the incentives of all facility-based ﬁrms,
including the dominant incumbent ﬁrm. We therefore analyze the performance of those regulations by considering the incentives of all facility-based ﬁrms to invest in infrastructure upgrades
and foreclose service-based ﬁrms.
From the analysis in this paper, we derive four main results, which can be explained byspillovers from the usage of the incumbent’s infrastructure. First, the spillover eﬀect weakens
the incumbent’s incentives to make infrastructure upgrades if access price regulation is imposed
4

Hoernig (2011) investigates the eﬀect of asymmetric access regulation on the market environment whenthe market consists of two service-based ﬁrms and two facility-based ﬁrms, but he does not take into accountinfrastructure upgrades. Asymmetric regulation in his model means that only one of the facility-based ﬁrms has an
obligation to allow service-based ﬁrms to use its facility, whereas under symmetric regulation, each facility-based
ﬁrm has that obligation.
5
Of course, the incumbent may foreclose service-based ﬁrms from the market by setting a signiﬁcantly highaccess price if it is allowed to set its access price freely.

2

on the incumbent. On the contrary, this eﬀect can enhance the incentives of the incumbent tomake infrastructure upgrades if access price regulation is not imposed. Second, the existenceof another facility-based ﬁrm strengthens the incumbent’s incentives to make infrastructure
upgrades, especially when the spillover eﬀect is large. Third, the existence of another facilitybased ﬁrm weakens the incumbent’s incentives to foreclose a service-based ﬁrm even when there isno access price regulation. Fourth, the incumbent has more incentives to invest in infrastructure
upgrades without access price regulation than with it. On the contrary, another facility-based
ﬁrm can have more incentives to make infrastructure upgrades with access price regulation. This
last result is explained by the relative magnitude of the spillover eﬀect and the strategic eﬀectof these investments.
In contrast to this paper, the existing literature on the relationship between access price regulation and infrastructure investment focuses on the duopoly consisting of a vertically integrated
incumbent and a service-based entrant ﬁrm.6 Foros (2004) and Kotakorpi (2006) examine theeﬀect of access price regulation on an incumbent’s incentive to invest in infrastructure upgrades
and to foreclose an entrant. Bourreau and Doğan (2004) demonstrate the incumbent’s strategic
incentive to delay a facility-based entry by providing attractive terms of access to its infrastructure.7 Although recent studies by Bourreau et al. (2011) and Bender and Götz (2011) explicitly
deal with an entrant’s incentives with respect to the make-or-buy decision, they do not consider
the eﬀect of asymmetric access regulation when two types of non-incumbent ﬁrms coexist in themarket.
Section 2 describes the framework of the model. Section 3 analyzes the model, comparesdiﬀerent regimes under asymmetric access regulation, and derives the results regarding a ﬁrm’s
incentives to make infrastructure upgrades or foreclose service-based ﬁrms. Section 4 gives
concluding remarks.
6
For a survey on the relationship between access regulation and infrastructure investment, see Guthrie (2006)
and Cambini and Jiang (2009).
7
See also Gans (2001) and Gans and King (2004) for a policy design that promotes infrastructure investment
in a duopoly.

3

2

Model

There are two vertically related sectors in the market: an upstream sector and a downstream
sector. These two sectors are required to supply services to consumers. We assume that there
are three ﬁrms of diﬀerent types: ﬁrm I, ﬁrm S, and ﬁrm F . The market structure we examinein this paper is asymmetric in the sense that the market positions of ﬁrms I and F are diﬀerent
from each other. Firm I, the incumbent, has infrastructure upstream and a production facility
downstream (i.e., it is a vertically integrated ﬁrm). Firm S, a service-based ﬁrm, is a potential
entrant that has only a production facility downstream. To serve consumers, ﬁrm S needs toaccess ﬁrm I’s infrastructure by paying usage access charge a to ﬁrm I. In other words, ﬁrm I
is obligated to give access to a potential entrant that requires it. Firm F , called a facility-based
ﬁrm, is a vertically integrated ﬁrm that owns its infrastructure but is not obligated to give accessto ﬁrm S. That is, the diﬀerence between ﬁrm F and ﬁrm I is that only ﬁrm I is required toprovide access to its infrastructure to ﬁrm S. In this sense, access regulation is asymmetric. This
assumption reﬂects asymmetric access regulation found in policies in the EU, the US, Korea,
and Japan. In our model, only the service-based ﬁrm (i.e., ﬁrm S) is assumed to be a potential
entrant. This assumption is justiﬁed by the fact that a service-based ﬁrm has no infrastructure
upstream, so it does not incur a large sunk cost for entry and exit.
One unit of input produced upstream produces one unit of output downstream. We assume
that the constant marginal access cost c that ﬁrm I must pay to give ﬁrm S access is the sameas its marginal production cost upstream and that the production cost downstream for each ﬁrm
is zero. For simplicity, we also assume that the production cost of ﬁrm F is exactly the same asthat of ﬁrm I.
The two vertically integrated ﬁrms, I and F , have the opportunity to make investments to
upgrade their own infrastructure. These investments have a demand-enhancing eﬀect because
they improve the quality of services sold downstream. A typical example would be an investment
in broadband technology, such as FTTH, that upgrades the speed or quality of information4

searches and thus would enhance consumer demand.
The utility function of a representative consumer is given as
U = vI q I + v F q F + v S q S −(qI + qF + qS )2+ m,
2

where vj represents the quality of service j (j = I, F, S), qj is the quantity supplied by ﬁrm j(j = I, F, S), and m is a numéraire good. The inverse demand function for each service is givenas (j = I, F, S)
pj = vj − Q,
where Q ≡ qI + qF + qS . The net consumer surplus is given as

CS =

Q2
(qI + qF + qS )2=.
2
2

We examine diﬀerent regimes based on each ﬁrm’s ability to upgrade its infrastructure anddetermine its access charge. In particular, we deal with three innovation regimes: (1) only ﬁrm I
is able to invest in upgrading its infrastructure, (2) only ﬁrm F is able to invest in upgrading its
infrastructure, and (3) both ﬁrms I and F are able to invest in upgrading their infrastructures.
As shown below, this step-by-step approach helps to illustrate the diﬀerent incentives for ﬁrms I
and F , which face diﬀerent access regulations, to invest in infrastructure. Then, we examine theeﬀect of access price regulation under each of three innovation regimes. To do so, we compare
the “access price regulation regime,” in which a regulator determines the level of access chargea, and the “no access price regulation regime,” in which ﬁrm I unilaterally determines the levelof a. Notice that even in a no access price regulation regime, ﬁrm I is obligated to provide
access to ﬁrm S, whereas ﬁrm F is not.
In sum, we have the following six regimes.

5

Access Innovation regime

ﬁrm I

ﬁrm F

ﬁrms I & F

Access price regulation R

RI

RF

RIF

No access price regulation N

NI

NF

NIF

Then, in each of six regimes, the qualities of the services and the investment costs are givenbyvI = α + δI xI , vF = α + δF xF , vS = α + sδI xI , I(xj ) =

γx2j
,
2

where α (> c) is a positive constant that can be interpreted as consumers’ willingness to pay
for a service supplied by an old infrastructure, xj (j = I, F ) is the investment level of ﬁrm j, s
is the degree of spillovers from ﬁrm I to ﬁrm S due to the required access, and γ (> 0) is theinvestment technology parameter of ﬁrms I and F . We use the same investment technology forﬁrms I and F for analytical simplicity. δj ∈ {0, 1} indicates whether or not ﬁrm j (= I, F ) isable to invest in infrastructure. Thus, we can represent diﬀerent innovation regimes in a compact
way. The degree of spillover, s, reﬂects the relative inferiority or superiority of ﬁrm S’s retailproduction technology as compared to that of ﬁrm I under the upgraded infrastructure.
In the following analysis, we restrict our attention to the case in which ﬁrm S’s retail production technology is inferior (or at best identical) to that of ﬁrm I, s ∈ [0, 1]. As will be shownin the analysis, however, the results derived for the case in which s > 1 are qualitatively similar
to those for the case in which s = 1. This restriction is justiﬁed by ﬁrm I’s better ability to obtain knowledge about the new infrastructure or by the regulators’ mandating access diﬀerently
between mature and new infrastructures.
Furthermore, we deal with ﬁrm I (an incumbent) and ﬁrm F (a facility-based ﬁrm) symmetrically in terms of their production and investment technologies. In a real policy arena, imposing
asymmetric access regulations is usually justiﬁed by an incumbent’s dominant position causedby its better technology. However, including these asymmetries between ﬁrms I and F makes
the analysis too complicated. We think that a beneﬁt of the symmetric treatment is that we
can clearly examine the eﬀect of imposing access obligations on an incumbent. We also think
6

that our results do not qualitatively change even if we take into account these technologicalasymmetries.
To preserve analytical tractability, we make the following assumptions.
Assumptions (i) a ≥ c, (ii) γ > 3/2.
Assumption (i) is set for practical reasons.8 In fact, it is rare for access charges to be set below
the marginal access cost in the real world. Assumption (ii) guarantees interior solutions not only
for ﬁrm I’s proﬁt-maximizing investment problem but also for the regulator’s welfare-maximizing
investment problem, as shown in the following analysis.
The market structure in this model is depicted in Figure 1.
[Figure 1 here]
The timing of the game is as follows. In each of the six regimes, the investment stage comesﬁrst, because investments are irreversible and the ability to commit to an access charge is limited.
In the second stage, given the level of investment, the access charge a is determined. In thethird stage, ﬁrm S decides whether to enter the market with access or not. In the fourth stage,
all active ﬁrms compete downstream in a Cournot fashion.

3

Analysis

3.1

Preliminary result: the outcome in the fourth stage

In each of the six scenarios, the proﬁt functions of the ﬁrms are given by
πI = (pI − c)qI + (a − c)qS − δI I(xI ),πF = (pF − c)qF − δF I(xF ),πS = (pS − a)qS .
8This assumption also appears in Foros (2004).

7(1)Here, we assume that ﬁrm S’s sunk cost of entry is zero.
If S enters in the third stage, there is a triopoly in the fourth stage. Solving the maximization
problems of the ﬁrms, we have
α − 2c + a + (3 − s) δI xI − δF xF,
4
α − 2c + a − (1 + s) δI xI + 3δF xF,
qF (xI , xF ; a) =4
α + 2c − 3a − (1 − 3s) δI xI − δF xF,
qS (xI , xF ; a) =43α − 2c − a + (1 + s)δI xI + δF xF.
Q(xI , xF ; a) =4
qI (xI , xF ; a) =

(2)
(3)
(4)
(5)

If the degree of spillover s is larger than 1/3, ﬁrm I’s investment increases ﬁrm S’s quantity
produced downstream. From (1) to (5), the equilibrium proﬁts in the subgame of the fourthstage are given byπI (xI , xF ; a) = qI (xI , xF ; a)2 + (a − c)qS (xI , xF ; a) − δI I(xI ),(6)πF (xI , xF ; a) = qF (xI , xF ; a)2 − δF I(xF ),(7)

πS (xI , xF ; a) = qS (xI , xF ; a)2 .

(8)

If ﬁrm S does not enter the market, there is a duopoly in the fourth stage. The equilibrium
quantities supplied by ﬁrms I and F are given byqI (xI , xF ) =α − c + 2δI xI − δF xF
α − c + 2δF xF − δI xI, qF (xI , xF ) =
.
3
3

(9)

From (9), the equilibrium proﬁts in the subgame are given byF (xI , xF ) = qF (xI , xF )2 − δF I(xF ).
πI (xI , xF ) = qI (xI , xF )2 − δI I(xI ), π

8

(10)

3.2

Regimes under access price regulation

We ﬁrst consider the three access price regulation regimes in which the access price for ﬁrm S
is set by a benevolent regulator. We verify that foreclosure does not occur in the third stageof each of the three regimes as long as the cost-based access charge (i.e., a = c) is set by theregulator. Indeed, we can ensure that in the second stage, the regulator sets the cost-based
access charge if α is suﬃciently large.9
We in turn investigate the decisions of ﬁrms I and F in the ﬁrst stage. From (6), (7), andthe fact that ﬁrm S ’s entry is accommodated under a = c, the net proﬁts of ﬁrms I and F aregiven byπI (xI , xF ; c) = qI (xI , xF ; c)2 − δI I(xI ),(11)

πF (xI , xF ; c) = qF (xI , xF ; c)2 − δF I(xF ).
The maximization problems of ﬁrms I and F lead to the reaction functions
δI (3 − s)(α − c − δF xF )∂πI (xI , xF ; c)
= 0 ⇒ xI (xF ) =,∂xI
8γ − (3 − s)2 δI
3δF (α − c − (1 + s)δI xI )∂πF (xI , xF ; c)
= 0 ⇒ xF (xI ) =.
∂xF
8γ − 9δF

(12)
(13)

The spillover eﬀect represented by s directly weakens the incentives for ﬁrm I to invest. The
reaction function of ﬁrm I moves downward as the degree of spillover increases (see (12)) because
the spillover eﬀect helps ﬁrm S, which is a competitor of ﬁrm I. The spillover eﬀect also
indirectly weakens the incentives for ﬁrm F to invest. The reaction function of ﬁrm F steepens
as s increases (see (13)) because the spillovers to ﬁrm S weaken the strategic position of ﬁrm
F in the downstream sector. Note that the spillover eﬀect does not always diminish ﬁrm F ’s
investment level.
9

In the case where α is not suﬃciently large but ﬁrm I still obtains positive proﬁts after investments, theoptimal access charge set by a regulator can be higher than the access cost when the degree of the spillover issmall. See Mizuno and Yoshino (2011) for this point.

9

Solving the simultaneous equations, we haveδI (3 − s)(2γ − 3δF )A,
16γ 2 − 2(9δF + (3 − s)2 δI )γ + 3(3 − s)(2 − s)δI δF
3δF (2γ − (3 − s)δI )A=,2
16γ − 2(9δF + (3 − s)2 δI )γ + 3(3 − s)(2 − s)δI δF

x∗I =

(14)

x∗F

(15)

where A ≡ α − c. Note that x∗I and x∗F hold for each of the three access price regulation regimes(i.e., RI, RF, and RIF ) because of the indicator δj ∈ {0, 1}, where j = I, F . We now compare
the three innovation regimes under access price regulation.
Proposition 1 In the access price regulation regimes in which a benevolent regulator sets theaccess charge (i.e., regimes RI, RF, and RIF), the following hold in equilibrium:(i) Foreclosure does not occur in any of the three innovation regimes.
≤ x∗RF. The inequality holds as an equality if and only if s = 0.
(ii) x∗RI
I
F≤ x∗RIF. The inequality holds as an equality if and only if s = 0.
(iii) x∗RIF
I
F(iv) ∂x∗RIF/∂s < 0 for any s ∈ [0, 1]. ∂x∗RIF/∂s > 0 if and only if
I
F
⎧

⎪
⎨ s > 3 − 2γ + 2 γ(γ − 2) if γ > 9/4,⎪
⎩ s≥0

if γ ≤ 9/4.

(16)

Proof. See Appendix.
[Table 1 here]
The result in (i) is natural. As long as α (i.e., consumers’ willingness to pay for a service supplied
using an old infrastructure) is suﬃciently large, the regulator has an incentive to set a cost-based
access charge (a∗ = c) to induce competition in the downstream sector and promote a so-called
level-playing ﬁeld.
The results in (ii) and (iii) stem from the existence of spillovers. Indeed, ﬁrm I’s investments
create spillovers, whereas ﬁrm F ’s do not. Then, because ﬁrm I is unwilling to help ﬁrm S

10through these spillovers, ﬁrm I has less incentive to invest in infrastructure than ﬁrm F if
s = 0.10
The result in (iv) represents the strategic interactions between ﬁrms I and F . Because thestrategic interactions between ﬁrms I and F are strategic substitutes (see (12) and (13)), thedirect negative eﬀect of spillovers on ﬁrm I’s incentives to invest enhances ﬁrm F ’s incentives toinvest. When the investment cost γ is smaller, the incentives for ﬁrms to invest are larger, which
implies that the negative spillover eﬀect on ﬁrm I is also larger. Therefore, when the degreeof spillovers s is large or when the investment technology is eﬃcient (γ is small), this strategic
eﬀect of spillovers on ﬁrm F dominates their indirect negative eﬀect (see (13)).
Table 1 summarizes the conditions for accommodating ﬁrm S’s entry, a vertically integrated
ﬁrm’s innovation incentive, and the equilibrium access charge in each of the three access priceregulation regimes, RI, RF, and RIF. The table also gives the result in Foros (2004), called regime
R, as a benchmark. This regime is similar to RI except that ﬁrm F exists in RI. Comparing
the results of regime RI and regime R gives the eﬀect of the existence of a facility-based ﬁrm,
which is stated in the following proposition.
Proposition 2 Suppose that a benevolent regulator sets the access charge. Firm I engages ina higher level of investment in regime RI than in regime R if s ≥ 5/7 or if γ < 2(3 − s)(2 −s)/(5 − 7s) and s < 5/7. Otherwise, ﬁrm I engages in a lower level of investment in regime RI
than in regime R.
Proof. See Appendix.
We next discuss how the degree of spillovers aﬀects the gain from investment in infrastructure.
The marginal gain and loss from investment are given by
 ∂qi
∂πI
+
= pI qI
∂xI
∂xI
i∈H

qI

−I  (xI ),(17)

quality improvement eﬀect

strategic eﬀect
It is obvious that x∗Rj
> x∗RIF(j = I, F ). This is because ﬁrms’ investments are strategic substitutes forjjeach other.
1011

where H is the set of I’s rival ﬁrms. The ﬁrst term represents the strategic eﬀect, i.e., themarginal gain from the decrease in the rival ﬁrms’ production, whereas the second term represents the quality improvement eﬀect, i.e., the marginal gain from the improvement of productquality, and the last term is the marginal cost of investment.
We can explain how the spillover eﬀect aﬀects the gain from investment through the strategic
eﬀect and the quality improvement eﬀect. In regime RI, H = {S, F }, whereas in regime R,
H = {S}. The number of rival ﬁrms, #H, is larger in regime RI. This implies that themarginal eﬀect of investment on the total quantity of rival ﬁrms is larger in regime RI because
this total quantity is larger when the number of rival ﬁrms is larger. Moreover, the spillovers
beneﬁt ﬁrm S but harm ﬁrm F because only ﬁrm S beneﬁts from ﬁrm I’s investment. Therefore,
the negative eﬀect of spillovers on ﬁrm I is weaker in regime RI because the negative eﬀect ofthese spillovers on ﬁrm F mitigates the direct negative eﬀect of spillovers on ﬁrm I in regime
RI. From these two factors, we can conclude that the strategic eﬀect is stronger in regime RI
and that the diﬀerence in the size of this eﬀect between the two regimes increases with thedegree of spillovers. On the other hand, the quality improvement eﬀect is stronger in regime
R because downstream competition in this regime is milder. Which of the strategic and thequality improvement eﬀects is stronger depends on the degree of spillovers.
The following equations help us to discuss how the two eﬀects depend on the degree of
spillovers.
M RIR =
M RI∗RI=

2(2 − s)A + 2(2 − s)2 xI(1 − 2s)(A + (2 − s)xI ) A + (2 − s)xI
+
=
,
9
3
9
(3 − s)A + (3 − s)2 xI(1 − s)(A + (3 − s)xI ) A + (3 − s)xI
+
=
,
8
4
8

where M RIR is ﬁrm I’s marginal gain from investment in regime R and M RI∗RI is ﬁrm I’s
marginal gain from investment in regime RI. The left panel of Figure 2 shows equilibrium
investments in regime RI and in regime R for s = 0.
[Figure 2 here]
12

When s = 0, the constant term is larger in regime R, and the slope of xI is steeper in regime
RI. The latter reﬂects the stronger strategic eﬀect in regime RI. Because the two lines intersect,
we ﬁnd that ﬁrm I has more incentive to invest in regime RI only if the investment cost is low.
The right panel of Figure 2 depicts the case where s = 1. In the panel, the marginal gain frominvestment in regime RI shifts upwards so that ﬁrm I’s incentive to invest in infrastructure isalways higher in regime RI than in regime R. This is because the strategic eﬀect in regime RI is
signiﬁcantly larger than that in regime R. The strategic eﬀect in regime R is negative because
ﬁrm I’s investment just helps its rival, ﬁrm S.

3.3

Regimes under no access price regulation

Next, we examine the three no access price regulation regimes, regimes NI, NF, and NIF, inwhich ﬁrm I can unilaterally determine the level of a. In these no access price regulation regimes,
the access charge a set by ﬁrm I can be an eﬀective tool for foreclosure. One interesting questionis how each innovation regime aﬀects the eﬀectiveness of a as a foreclosure tool.
In the second stage, if ﬁrm I anticipates ﬁrm S’s entry, its proﬁt is given byπI (xI , xF ; a) = qI (xI , xF ; a)2 + (a − c)qS (xI , xF ; a) − I(xI ).

(18)

The maximization problem of ﬁrm I leads to the optimal access chargea(xI , xF ) = c +3A + (1 + 5s)δI xI − 3δF xF
.
11

(19)

We discuss how the investment levels aﬀect the access charge, a(xI , xF ).
Lemma 1 The access charge a(xI , xF ) set by ﬁrm I increases with xI but decreases with xF .
Investment by ﬁrm F lowers the quantities supplied by its rivals (ﬁrms I and S), which inturn lowers the marginal gains of ﬁrm I from its own downstream proﬁt through the accesscharge (the ﬁrst term in (18)) and from its access proﬁt (the second term in (18)). Hence,13

a(xI , xF ) decreases with xF . On the other hand, investment by ﬁrm I always increases its own
quantity and can lower the quantity of ﬁrm S. The marginal gain of ﬁrm I therefore increases
because the access charge prevents competition, but its marginal gain from the access proﬁtfalls. The former eﬀect dominates the latter, and hence, a(xI , xF ) increases with xF . Therefore,
the lemma holds.
Under the access price in (19), the quantity supplied by ﬁrm S isqS (xI , xF ; a(xI , xF )) =A − (7 − 9s)δI xI − δF xF
.
22

(20)

Given xI and xF , if (20) is positive, ﬁrm S enters the market in the third stage, otherwise, itdoes not. Note that in regime NI (δI = 1 and δF = 0), when s ≥ 7/9, qS (xI , xF ; a(xI , xF ))in (20) is larger than zero, so that ﬁrm I certainly accommodates ﬁrm S’s entry for s ≥ 7/9
regardless of the level of xI .
Lemma 2 In regime NI, ﬁrm I accommodates ﬁrm S’s entry if s ≥ 7/9.
If the degree of spillovers is high, ﬁrm I uses ﬁrm S as its aﬃliate to compete with ﬁrm F.
Firm I gains a proﬁt from the access charge. This contrasts with Foros (2004), who shows thatﬁrm I always forecloses ﬁrm S if s ≤ 1. The contrasting results imply that the existence of a
facility-based ﬁrm mitigates the incentive for ﬁrm I to foreclose ﬁrm S.
We next explore whether ﬁrms I and F have incentives to foreclose ﬁrm S through theirinvestments in equilibrium. When qS (xI , xF ; a(xI , xF )) ≥ 0, the quantities supplied by ﬁrms I
and F are given by
7A + (17 − 3s)δI xI − 7δF xF.
22
7A − (5 + s)δI xI + 15δF xF.
qF (xI , xF ; a(xI , xF )) =22
qI (xI , xF ; a(xI , xF )) =

(21)
(22)

When qS (xI , xF ; a(xI , xF )) ≤ 0, ﬁrm S is foreclosed and the competition structure becomes a14

duopoly of ﬁrms I and F . The quantities are given asqI (xI , xF ) =α − c + 2δI xI − δF xF
α − c + 2δF xF − δI xI, qF (xI , xF ) =
.
3
3

(9’)

In the ﬁrst stage, ﬁrms I and F set their investment levels. Their maximization problems
are given by⎧2⎪⎪⎪ qI (xI , xF ; a(xI , xF )) + (a − c)qS (xI , xF ; a(xI , xF )) − δI I(xI )⎪⎨πI (xI , xF ; a(xI , xF )) =if qS (xI , xF ; a(xI , xF )) ≥ 0, (23)⎪⎪⎪⎪⎩ q (x , x )2 − δ I(x )if qS (xI , xF ; a(xI , xF )) ≤ 0.
I I
F
I
I
⎧
⎪
⎨ qF (xI , xF ; a(xI , xF ))2 − δF I(xF ) if qS (xI , xF ; a(xI , xF )) ≥ 0,
πF (xI , xF ; a(xI , xF )) =(24)⎪
⎩ qF (xI , xF )2 − δF I(xF )if qS (xI , xF ; a(xI , xF )) ≤ 0.
The ﬁrst-order conditions lead to the following reaction functions:∂πI (xI , xF ; a(xI , xF ))=0∂xI⎧δI (9 + s)(A − δF xF )⎪⎪⎨ 22γ − (25 − 14s + 9s2 )δ
I⇒ xI (xF ) =⎪⎪⎩ 4δI (A − δF xF )9γ − 8∂πF (xI , xF ; a(xI , xF ))=0∂xF⎧
15δF (7A − (5 + 3s)δI xI )⎪⎪⎨242γ − 225δF⇒ xF (xI ) =⎪ 4δF (A − δI xI )⎪⎩9γ − 8

if qS (xI , xF ; a(xI , xF )) ≥ 0,(25)
if qS (xI , xF ; a(xI , xF )) ≤ 0.

if qS (xI , xF ; a(xI , xF )) ≥ 0,
(26)
if qS (xI , xF ; a(xI , xF )) ≤ 0.

The strategic interaction between ﬁrms I and F in the ﬁrst stage is strategic substitution. Whenﬁrm S is active, that is, when qS ≥ 0, ﬁrm F is less aggressive due to the spillover eﬀect ofinvestment by ﬁrm I (see (26)). On the other hand, this spillover eﬀect enhances the incentives
for ﬁrm I to invest because of the access proﬁt from ﬁrm S (see (25)). This property of ﬁrm

15

I’s reaction function is quite diﬀerent from those in regimes RI and RIF (see (12)). From thiscomparison, we easily anticipate that the eﬀect of spillovers on the incentives of ﬁrms I and F in
the no access price regulation regimes is quite diﬀerent from that in the access price regulation
regimes.
Solving the simultaneous equations, we haveδI (9 + s)(11γ − 15δF )A,(27)
242γ 2 − (225δF + 11(25 − 14s + 9s2 )δI )γ + 15(15 − 11s + 6s2 )δI δF
15δF (7γ − (10 − 3s + 3s2 )δI )A
,
(28)
x∗F =
242γ 2 − (225δF + 11(25 − 14s + 9s2 )δI )γ + 15(15 − 11s + 6s2 )δI δF
2(3γ − (3 − s)(1 − s)δI )(11γ − 15δF )A, (29)a(x∗I , x∗F ) = c +242γ 2 − (225δF + 11(25 − 14s + 9s2 )δI )γ + 15(15 − 11s + 6s2 )δI δFx∗I =if qS (x∗I , x∗F ; a(x∗I , x∗F )) ≥ 0. Otherwise,x∗j =4δj A(3γ − 4δ−j ), j, −j = I, F ; j = −j.
27γ 2 − 24(δI + δF )γ + 16δI δF

The inequality qS (x∗I , x∗F ; a(x∗I , x∗F )) ≥ 0 holds if and only ifγ ≥ 4(1 − s)δI ≡ γ̄.

(30)

The inequality qS (x∗I , x∗F ; a(x∗I , x∗F )) ≤ 0 holds if and only ifγ ≤ 4(1 − s)δI ≡ γ̄.

From (30), we have the following proposition:
Proposition 3 In the no access price regulation regimes in which ﬁrm I freely sets the level ofaccess charge a (i.e., regimes NI, NF, and NIF), the following hold in equilibrium:(i) Firm S is foreclosed if and only if γ ≤ γ̄.
I ≤ x∗N F if and(ii) Suppose that ﬁrm S is accommodated in regime NI (γ > γ̄). Then, x∗NIF

16

only ifγ>15(40 − 113s + 63s2 )and s ≤ 6/11 or
22(6 − 11s)√
15(40 − 113s + 63s2 )74 + 786γ<
and 6/11 < s <.
22(6 − 11s)
105
I ≤On the other hand, suppose that ﬁrm S is foreclosed in regime NI (γ ≤ γ̄). Then, x∗NI
F if and only if γ < 60/23.
x∗NF
IF ≤ x∗N IF if(iii) Suppose that ﬁrm S is accommodated in regime NIF (γ > γ̄). Then, x∗NIF

and only ifs ≤ 6/11 or γ <15(1 − s)(1 − 3s)and 6/11 < s < 4/5.
6 − 11sOn the other hand, suppose that ﬁrm S is foreclosed in regime NIF (γ ≤ γ̄). Then,
IF = x∗N IF for any γ and s.
x∗NIFIF /∂s ≤ 0 for(iv) Suppose that ﬁrm S is accommodated in regime NIF (γ > γ̄). Then, ∂x∗N
F
IF /∂s > 0 if and only if
any s and γ. ∂x∗NI√

475429and s >γ<242√
943 + 475429
or γ ≥
.
242
943 +


(40 − γ)(11γ − 10)(242γ − 225) − 27(11γ − 10)3(11γ − 10)On the other hand, suppose that ﬁrm S is foreclosed in regime NIF (γ ≤ γ̄). Then, forIF /∂s = 0 and ∂x∗N IF /∂s ≤ 0.
any s and γ, ∂x∗N
F
I

Proof. See Appendix.
[Table 2 here]
Figure 3 graphically shows the conditions of this proposition.
17

[Figure 3 here]
Proposition 3(i) indicates that if the investments are not costly (γ is small) and the degreeof spillovers is small (s is small), ﬁrm S is foreclosed. The former enhances the innovationcompetition between ﬁrms I and F , and the investment levels of the two ﬁrms are large. If thedegree of spillovers is small, the quality diﬀerence between the two ﬁrms and ﬁrm S becomeslarge. This means that the competitive disadvantage of ﬁrm S is large if γ < γ̄. As a result,ﬁrm S is inactive if investments are not costly and the degree of spillovers is small.
The mechanism behind Proposition 3(ii) is complicated, and we separately consider twocases to explain it. First, consider the spillover eﬀect and the strategic eﬀect of investment in
the case where ﬁrm S is active (γ > γ̄). The spillover eﬀect increases the quantity supplied
by ﬁrm S, which implies that ﬁrm I’s gains from access revenue increase with the degree of
spillovers. Hence, ﬁrm I has an incentive to invest more. The white area on the right-hand sideof Figure 3.1 shows this eﬀect. The strategic eﬀect of investment, which is shown in (17) for
ﬁrm I’s investment, is stronger when only ﬁrm F invests than when only ﬁrm I invests because
investment by ﬁrm F lowers the supplies of both rival ﬁrms (ﬁrms I and S) whereas investment
by ﬁrm I lowers ﬁrm F ’s supply but increases ﬁrm S’s supply because of the spillover eﬀect.
An investing ﬁrm’s marginal gain from its investment becomes more sensitive to its investment
level because of this eﬀect, as discussed in the previous subsection (see the left panel in Figure2). Because the investment level is higher when the investment cost is lower (γ is smaller), thestrategic eﬀect is larger when the investment cost is lower. The right-hand edge of the shaded
area in Figure 3.1 reﬂects this property.
We return to the spillover eﬀect. When the degree of spillovers, s, is small enough, theforeclosure eﬀect of ﬁrm I’s investment is stronger (the coeﬃcient of xI in (20) reﬂects theforeclosure eﬀect). The negative eﬀect of ﬁrm I’s investment on ﬁrm S’s supply is signiﬁcantly
higher than that of ﬁrm F ’s investment (see (20)). This eﬀect induces ﬁrm I to invest more. In
Figure 3.1, it is shown in the white area in the left-hand side above the line of γ = γ̄.
Second, consider the case in which ﬁrm S is inactive when ﬁrm I invests (γ ≤ γ̄). The
18

spillover eﬀect does not exist in this case because ﬁrm S is foreclosed. The main eﬀect in thiscase is the strategic eﬀect, as shown in (17). As explained earlier, the strategic eﬀect is larger
when the investment cost is lower. Firm F ’s has two rival ﬁrms under NF but only one underNI, which implies that the strategic eﬀect is stronger under NF. Therefore, ﬁrm F has a stronger
incentive to invest than ﬁrm I when the investment cost is lower (γ is smaller). This is similar
to the logic behind Proposition 2 (see also Figure 2).
The mechanism behind Proposition 3(iii) is basically similar to that behind Proposition 3(ii).
When ﬁrm S is foreclosed, ﬁrms I and F symmetrically compete with each other. Because of
IF = x∗N IF when γ ≤ γ̄.
the symmetry, x∗N
IF

The mechanism behind Proposition 3(iv) is simple. As mentioned earlier, the spillover eﬀect
enhances the incentives for ﬁrm I to invest. The partial derivative reﬂects the eﬀect.
Comparison between regimes NI and N

As in the subsection on access price regulation

regimes, we state the eﬀect of the existence of a rival facility-based ﬁrm in a proposition bycomparing this case to the case with no such ﬁrm, called regime N, described in Foros (2004).
Proposition 4 Suppose that an incumbent unilaterally sets the access charge. Then, we havethe following.
(i) Firm I in regime NI forecloses ﬁrm S if and only if γ ≤ γ̄, whereas it always forecloses
ﬁrm S in regime N.
(ii) Suppose that ﬁrm S is accommodated (γ ≥ γ̄). Then, the incentives for ﬁrm I toinvest in infrastructure in regime NI are stronger than that in regime N if and only if γ <(9s2 − 10s + 16)/2(2 − s).
On the other hand, if ﬁrm S is foreclosed (γ ≤ γ̄), the incentives for ﬁrm I to invest in
infrastructure are stronger in regime NI than in regime N.
Proof. See Appendix.
The reason for (i) is related to Lemma 2. When ﬁrm F exists, ﬁrm I is able to use ﬁrm S as
its aﬃliate because ﬁrm I earns access charge proﬁts from ﬁrm S. This mitigates the incentives
19

for ﬁrm I to foreclose ﬁrm S.
The reason for (ii) (i.e., the investment comparison) is also explained by the mechanism
related to Proposition 2. In the case where ﬁrm S is accommodated in regime NI, there arethree active ﬁrms in regime NI, whereas there is only one in regime N. Firm I’s marginal gainfrom investment is less sensitive to the level of investment in regime N than in regime NI. We
can apply the same logic in the discussion of Proposition 2 to that of Proposition 4. Hence, ﬁrm
I has more incentives to invest in infrastructure in regime NI than in regime N if the investment
cost is low but not if the investment cost is high. The strategic eﬀect of spillovers is the samein regime NI, as is explained in Proposition 2. The quality improvement eﬀect in regime N
is signiﬁcantly higher than in regime NI because the diﬀerence in the number of active ﬁrms
between regime NI and regime N (3 and 1) is larger than that between regime RI and regime
R (3 and 2).
Similarly, when ﬁrm S is foreclosed in regime NI, ﬁrm I’s marginal gain from investment inregime NI is less sensitive to the level of investment than that in regime N because there are
two active ﬁrms in regime NI but only one in regime N. Since the investment cost is low in thecase where a service-based ﬁrm’s entry is deterred in regime NI (γ < γ̄), ﬁrm I always has more
incentives to invest in infrastructure in regime NI than in regime N in equilibrium.

3.4

The eﬀect of access price regulation

In this subsection, we examine the eﬀect of access price regulation on infrastructure upgrades
and foreclosure when there exists a non-incumbent facility-based ﬁrm. We obtain the following
results.
Proposition 5 (i) Firm I has more incentive to invest in infrastructure in regime NI than inregime RI except in the case where both the degree of spillovers, s, and the investment cost, γ,
are small.
(ii) Firm F has more (less) incentive to invest in infrastructure in regime NF than in regime
RF if γ is large (small).
20

(iii.1) Firm I has more incentive to invest in infrastructure in regime NIF than in regime
RIF except in the case where both s and γ are small.
(iii.2) Firm F has more (less) incentive to invest in infrastructure in regime NIF than inregime RIF if γ is large (small).
(iii.3) The total investment level is larger in regime NIF than in regime RIF if γ > 4(1 − s)or if γ and s are small enough.
The detail of the conditions is shown in Figure 4.
[Figure 4 here]
We provide an intuitive explanation for Proposition 5. Basically, the existence of ﬁrm S
encourages ﬁrm I to invest because of the revenue from the access charge. Whenever ﬁrm S
is accommodated, ﬁrm I has more incentive to invest under no access price regulation regimes.
When ﬁrm S is foreclosed, the strategic eﬀect of ﬁrm I’s investment is larger under access priceregulation regimes because the number of rival ﬁrms is larger under those regimes. This eﬀectunder access price regulation regimes is stronger when the investment cost and the degree of
spillovers are small, that is, when γ and s are small, as explained in the previous section.
This logic also applies to the incentives for ﬁrm F to invest. That is, ﬁrm F has more
incentive to invest under access price regulation regimes if γ is small, that is, the strategic eﬀect
of ﬁrm F ’s investment is well functioned. When both ﬁrms invest and ﬁrm S is accommodated,
the incentives for ﬁrm F to invest become gradually weaker as the degree of spillovers increases(see the fourth panel in Figure 4) because of spillover eﬀects and the strategic substitutabilityof the investments.
Comparison of social welfare Finally, we examine the eﬀect of access price regulation onsocial welfare. To do so, we restrict our attention to regimes RIF and NIF.11 Due to the11

The social welfare comparisons between regimes RI and NI or between regimes RF and NF are easily conjectured from the discussion below.

21mathematical complexity, we use Figure 5 to discuss social welfare under regimes RIF and NIF.
(See Appendix for the derivation of equilibrium proﬁts in both regimes).
[Figure 5 here]
Access price regulation always enhances consumer surplus because it maintains downstream
competition. It does not always enhance social surplus, however. In particular, it lowers social
surplus in the case where foreclosure would happen if no regulation were imposed. The regulation
weakens the incentives for ﬁrm I to invest because it is prohibited from earning access charge
revenue. As a result, ﬁrm F has an incentive to invest more, which distorts the total expenditures
on investment. To understand the distortion, we set a targeted total investment level, x̄ ≡xI + xF . To achieve the targeted level, the ﬁrms expend total investment costs, I(xI ) + I(xF ) =γ(x2I + x2F )/2. The total cost is smaller when the diﬀerence between xI and xF is smaller. Thus,
distorting the investment levels generates additional costs. Due to this distortion, access priceregulation diminishes the social surplus in the white region in Figure 5.
Interestingly, when the investment cost is small enough (γ is small enough), access priceregulation enhances the social surplus. Under the access price regulation regime RIF , thespillover eﬀect signiﬁcantly weakens the incentives for ﬁrm I to invest because of the strategic
substitutability of investments, which implies that ﬁrm F has a stronger incentive to invest. As
a result, the total investment level under regime RIF is signiﬁcantly larger than that underregime N IF . This beneﬁt dominates the distortion eﬀect when γ is small enough.

4

Concluding Remarks

In this paper, we have examined the incentives for ﬁrms to make infrastructure upgrades andforeclose service-based ﬁrms. We take into account the coexistence of service-based and facilitybased ﬁrms. Including both types of ﬁrms allows us to examine the structure of “asymmetricaccess regulation,” which is often imposed on incumbent ﬁrms in the real world.

22

From the analysis, we derived four main results, all of which can be explained by the spillover
eﬀect of investments when service-based ﬁrms use an incumbent’s infrastructure. First, thespillover eﬀect weakens the incumbent’s incentives to make infrastructure upgrades regardless ofwhether access price regulation is imposed on the incumbent or not. However, the spillover eﬀect
can also increase the incentives for the incumbent to make infrastructure upgrades if access priceregulation is not imposed. Second, the existence of a rival facility-based ﬁrm strengthens theincumbent’s incentives to make infrastructure upgrades, especially when the spillover eﬀect islarge. Third, when there is no access price regulation, the existence of a rival facility-based ﬁrm
weakens the incumbent’s incentives to foreclose a service-based ﬁrm. Fourth, the incumbent has
more incentives to invest in infrastructure upgrades without access price regulation than with it.
On the contrary, a rival facility-based ﬁrm can have more incentives to invest in infrastructure
upgrades under access price regulation. The last result is explained by the relative magnitudesof the spillover eﬀect and the strategic eﬀect of investments in infrastructure upgrades.

References[1] Bender, C. M. and Götz, G., 2011, “Coexistence of Service- and facility-Based Competition:
The Relevance of Access Prices for ’Make-or-Buy’ Decisions,” mimeo.
[2] Bourreau, M. and Doğan, P., 2004, “Service-Based vs. Facility-Based Competition in Local
Access Networks,” Information Economics and Policy 16, 287-306.
[3] Bourreau, M., Cambini, C. and Doğan, P., 2011, “Incentives to Migrate to Next Generation
Networks,” mimeo.
[4] Cambini, C. and Jiang, Y., 2009, “Broadband Investment and Regulation: A Literature
Review,” Telecommunications Policy 33, 559-574.

23

[5] Crandall, R., 2005, “Broadband Communications,” in S. Majumdar, I. Vogelsang, and
M. Cave eds., Handbook of Telecommunications Economics vol. 2, Amsterdam: Elsevier
Publishers.
[6] de Bijl, P. and Peitz, M., 2008, “Innovation, Convergence and the Role of Regulation in the
Netherlands and Beyond,” Telecommunications Policy 32, 744-754.
[7] Foros, Ø, 2004, “Strategic Investments with Spillovers, Vertical Integration and Foreclosurein the Broadband Access Market,” International Journal of Industrial Organization 22,
1–24.
[8] Gans, J., 2001, “Regulating Private Infrastructure Investment: Optimal Pricing for Accessto Essential Facilities,” Journal of Regulatory Economics 20, 167–198.
[9] Gans, J. and King, S., 2004, “Access Holidays and the Timing of Infrastructure Investment,”
Economic Record 80, 89-100.
[10] Guthrie, G., 2006, “Regulating Infrastructure: The Impact on Risk and Investment,” Journal of Economic Literature 44, 925–972.
[11] Hatﬁeld, D., Mitchell, B., and Srinagesh, P., 2005, “Emerging Network Technologies,” in S.
Majumdar, I. Vogelsang, and M. Cave eds., Handbook of Telecommunications Economics
vol. 2, Amsterdam: Elsevier Publishers.
[12] Hoernig, S., 2011, “Asymmetric Broadband Wholesale Regulation,” mimeo.
[13] Huigen, J. and Cave, M., 2008, “Regulation and the Promotion of Investment in Next
Generation Networks - A European Dilemma,” Telecommunications Policy 32, 713-721.
[14] Kotakorpi, K., 2006, “Access Price Regulation, Investment, and Entry in Telecommunications,” International Journal of Industrial Organization 24, 1013–1020.

24

[15] Mizuno, K. and Yoshino, Y., 2012, “Distorted Access Regulation with Strategic Investments: Regulatory Non-commitment and Spillovers Revisited,” Information Economics
and Policy 24, 120-131.
[16] Peitz, M., 2005a, “Asymmetric Access Price Regulation in Telecommunications Markets,”
European Economic Review 49, 341-358.
[17] Peitz, M., 2005b, “Asymmetric Regulation of Access and Price Discrimination in Telecommunications,” Journal of Regulatory Economics 28, 327-343.
[18] Picot, A. and Wernick, C., 2007, “The Role of Government in Broadband Access,” Telecommunications Policy 31, 660-674.

25

Appendix
Proof of Proposition 1
First, we prove (i). We examine the triopoly case in which ﬁrm S enters the market. Firm S’s
entry condition is qS (xI , xF ; a) ≥ 0 or α + 2c − 3a + (3s − 1) δI xI − δF xF ≥ 0. In that case, theregulator’s problem for setting the optimal access charge in the second stage is given by
M ax W (a; xI , xF ) ≡ CS + πI + πS + πF
as.t. a ≤α + 2c + (3s − 1) δI xI − δF xF≡ aR (xI , xF ) ,3

where W (a; xI , xF ) is social welfare in the triopoly. The interior solution of the problem is givenbya∗ (xI , xF ) = −α + 2c − (11s − 5) δI xI + 5δF xF .
Thus, as long as α is suﬃciently large, a∗ (xI , xF ) < c < aR (xI , xF ) because W (a; xI , xF ) is continuous and concave in a. Therefore, under Assumption (i), we have the cost-based access charge,a∗ (xI , xF ) = c, irrespective the innovation regime. Moreover, because W (aR (xI , xF ) ; xI , xF ) =W (xI , xF ), where W (xI , xF ) is social welfare under a duopoly, we can also verify that the regulator prefers accommodation to foreclosure in any of the three access price regulation regimes.
≤ x∗RF
andSecond, applying the indicator δj ∈ {0, 1} to (14) and (15) ensures that x∗RI
I
F
≤ x∗RIF, which proves (ii) and (iii). Similarly, a comparative static exercise ensures (iv).
x∗RIF
I
F


26

Proof of Proposition 2
We compare x∗RI
and x#RI
I . We have
x∗RI
− x#R
I
I
=
=

(3 − s) A
2 (2 − s) A
2 −8γ − (3 − s)9γ − 2 (2 − s)2AΓ (γ, s).
8γ − (3 − s)2 9γ − 2 (2 − s)2

where Γ (γ, s) ≡ − (5 − 7s) γ + 2 (3 − s) (2 − s). At s = 0, Γ (γ, 0) ≥ (<) 0 if and only ifγ ≤ (>) 2.4. Moreover, the threshold Γ (γ, s) = 0 is increasing in s on the (γ, s) plane fors ≤ 5/7. When 5/7 < s ≤ 1, Γ (γ, s) > 0 under Assumption (ii). Therefore, we obtain the claimof the proposition. 

Proof of Proposition 3(i) is already derived in the text. Thus, we show the proofs of (ii) to (iv).
I and x∗N F . We have(ii) We compare x∗N
I
F

I
F
− x∗N
x∗N
I
F

=
=

105A
(9 + s) A−
222γ − (9s − 14s + 25) 242γ − 225AΨ (γ, s).
(22γ − (9s2 − 14s + 25)) (242γ − 225)



where Ψ (γ, s) ≡ 22 (11s − 6) γ + 63s2 − 113s + 40 . There are two thresholds Ψ (γ, s) = 0 thatare decreasing in s on the (γ, s) plane for γ ≥ 3/2 and 0 ≤ s ≤ 1. See Figure 3.1. Then, we
obtain the claim of the proposition.
IF and x∗N IF . The same procedure as in (ii) applies to this(iii) We need to compare x∗NI

comparison. See Figure 3.2.
(iv) A comparative static exercise ensure the claim in the proposition. 

27

Proof of Proposition 4(i) In the text, we state that, in Regime NI, the necessary and suﬃcient condition for ﬁrm I toaccommodate (deter) ﬁrm S’s entry is γ ≥ (<) 4 (1 − s). Hence, we report the accommodation
condition when there is no active facility-based ﬁrm in the market.12
When ﬁrm F is not in the market, the equilibrium quantities in the fourth stage are givenby
α − 2c + a + (2 − s) xI,
3
α + c − 2a − (1 − 2s) xI,
qS (xI ; a) =3
α − c + xI
qI (xI ; a) =,2
qI (xI ; a) =under a duopoly,

under a monopoly.

In the third stage, ﬁrm S enters the market if and only if qS (xI ; a) ≥ 0, so that the necessary
and suﬃcient condition for ﬁrm S’s entry is given by
α + c − 2a − (1 − 2s) xI ≥ 0 or a ≤1(α + c − (1 − 2s) xI ) ≡ a (xI ) .
2(31)Then, in the second stage, ﬁrm I determines the level of access charge by taking its entrydeterrence eﬀect into account. If ﬁrm I would like to accommodate ﬁrm S’s entry, its problem
isM ax πI (xI ; a) s.t. c ≤ a ≤ a (xI ) ,awhere a (xI ) is given by (31). Then, the interior solution of this problem is given bya∗ (xI ) =1(3α + 8c + (5s + 1) xI ) .
11

(32)

I (xI ) and πI (xI ; a) is continuous and concave in a, ﬁrm I accommodates
Because πI (xI ; a) = π
ﬁrm S’s entry if and only if a∗ (xI ) ≤ a (xI ). Substituting (32) into the inequality a∗ (xI ) ≤12

See also Foros (2004).

28

a (xI ), we obtain the necessary and suﬃcient condition for ﬁrm I to accommodate ﬁrm S’s entryas
6 (s − 1) xI ≥ 0 or s ≥ 1.

(33)

Otherwise, i.e., s < 1, ﬁrm I deters ﬁrm S’s entry, irrespective of the level of xI .
Comparing (33) with the condition that γ ≥ (<) 4 (1 − s), we have the claim of the proposition.
I and xN . We have(ii) We compare x∗N
I
I

I
− xN
x∗N
I
I

=
=

5A
(9 + s) A−
222γ − (9s − 14s + 25) 10γ − (4s2 − 8s + 9)AΦ (γ, s).
2(22γ − (9s − 14s + 25)) (10γ − (4s2 − 8s + 9))
where Φ (γ, s) ≡ 10γ (s − 2) − 4s3 − 17s2 + 7s − 44 . At s = 0, Φ (γ, 0) ≥ (<) 0 if and only
if γ ≤ (>) 2.2. Moreover, the threshold Φ (γ, s) = 0 is increasing in s on the (γ, s) plane forγ ≥ 3/2 and 0 ≤ s ≤ 1. Therefore, we obtain the claim of the proposition. 

The Derivation of equilibrium proﬁts in regimes RIF and NIF
Regulation regime

Substituting (14), (15), and a = c into (2), (3), (4), and (5), we have theequilibrium quantities supplied by the ﬁrms:
2γ(2γ − 3)A,
16γ 2 − 2(9 + (3 − s)2 )γ + 3(3 − s)(2 − s)2γ(2γ − (3 − s))A,qF (x∗I , x∗F ; c) =2
16γ − 2(9 + (3 − s)2 )γ + 3(3 − s)(2 − s)(2γ − (3 − s)(1 − s))(2γ − 3)A,
qS (x∗I , x∗F ; c) =16γ 2 − 2(9 + (3 − s)2 )γ + 3(3 − s)(2 − s)(12γ 2 − 2(6 + (3 − s)(2 − s))γ + 3(3 − s)(1 − s))A.
Q(x∗I , x∗F ; c) =16γ 2 − 2(9 + (3 − s)2 )γ + 3(3 − s)(2 − s)qI (x∗I , x∗F ; c) =

29

(34)
(35)
(36)
(37)

Using (1), (14), (15), (34), (35), (36), and (37), we have the equilibrium proﬁts of the ﬁrms:
γ(2γ − 3)2 (8γ − (3 − s)2 )A2,
2[16γ 2 − 2(9 + (3 − s)2 )γ + 3(3 − s)(2 − s)]2γ(2γ − (3 − s))2 (8γ − 9)A2,
πF (x∗I , x∗F ; c) =2[16γ 2 − 2(9 + (3 − s)2 )γ + 3(3 − s)(2 − s)]2(2γ − (3 − s)(1 − s))2 (2γ − 3)2 A2.
πS (x∗I , x∗F ; c) =[16γ 2 − 2(9 + (3 − s)2 )γ + 3(3 − s)(2 − s)]2
πI (x∗I , x∗F ; c) =No access price regulation regime

(38)
(39)
(40)

When γ > γ̄, we substitute (27), (28), and (29) into(2), (3), (4), and (5) and obtain the equilibrium quantities supplied by the ﬁrms:
qI (x∗I , x∗F ; a∗ ) =(11γ − 15)(7γ − (1 − s)(1 − 3s))(α − c)242γ 2 − (225 + 11(25 − 14s + 9s2 ))γ + 15(15 − 11s + 6s2 )

,

(41)

11γ(7γ − (10 − 3s + 3s2 ))(α − c),(42)242γ 2 − (225 + 11(25 − 14s + 9s2 ))γ + 15(15 − 11s + 6s2 )(11γ − 15)(γ − 4(1 − s))(α − c),(43)qS (x∗I , x∗F ; a∗ ) =2242γ − (225 + 11(25 − 14s + 9s2 ))γ + 15(15 − 11s + 6s2 )(165γ 2 − (120 + 11(15 − 11s + 6s2 ))γ + 15(5 − 3s)(1 − s))(α − c)
. (44)Q(x∗I , x∗F ; a∗ ) =242γ 2 − (225 + 11(25 − 14s + 9s2 ))γ + 15(15 − 11s + 6s2 )qF (x∗I , x∗F ; a∗ ) =Using (1), (27), (28), (29), (41), (42), (43), and (44), we have the equilibrium proﬁts of theﬁrms:(11γ − 15)2 (5γ − 2(1 − s)2 )(22γ − (25 − 14s + 9s2 ))(α − c)2,
2[242γ 2 − (225 + 11(25 − 14s + 9s2 ))γ + 15(15 − 11s + 6s2 )]2γ(7γ − (10 − 3s + 3s2 ))2 (242γ − 225)(α − c)2,πF (x∗I , x∗F ; a∗ ) =2[242γ 2 − (225 + 11(25 − 14s + 9s2 ))γ + 15(15 − 11s + 6s2 )]2(11γ − 15)2 (γ − 4(1 − s))2 (α − c)2.
πS (x∗I , x∗F ; a∗ ) =[242γ 2 − (225 + 11(25 − 14s + 9s2 ))γ + 15(15 − 11s + 6s2 )]2πI (x∗I , x∗F ; a∗ ) =When γ ≤ γ̄, we have the equilibrium quantities supplied by the ﬁrms:x∗I , x∗F ) =qj (3γ(3γ − 4)(α − c)3γ(6γ − 8)(α − c), Q(x∗I , x.
∗F ) =
227γ − 48γ + 16
27γ 2 − 48γ + 16

30

(45)
(46)
(47)

The equilibrium proﬁts of the ﬁrms arex∗I , x∗F ) =πj (γ(9γ − 8)(3γ − 4)2 (α − c)2.
(27γ 2 − 48γ + 16)2

31

Facilities

FacilitiesaFirm I

Firm S
qI

qS

Firm F
qF

Consumers

Figure 1: The market structure32

M RI∗RI

M R, I 
γH xI

M R, I 
γH xI

γL xI

γL xI

M RI#R

M RI∗RI
M RI#Rx∗R
I (γL )

0xIx∗R
I (γL )#R
#R
xI (γH )xI (γL )
(γ
x∗R
H)
I

#R
#R
x∗R
(γL )
I (γH ) xI (γH ) x
I

0

Figure 2: The marginal revenue and the investment cost
Left-hand side: s = 0; Right-hand side: s = 1.

33

xI

γ

γ

5

5
I
F
x∗N
< x∗N
I
F

IF
IF
x∗N
< x∗N
I
F

γ = 4(1 − s)
3
60/23
3/2

γ = 4(1 − s)

3

s
0

1

1/2

1. The condition in Proposition 3(ii)

IF
IF
x∗N
= x∗N
I
F

3/2

0

s
1/2

2. The condition in Proposition 3(iii)

5.0

4.5

4.0

xNIFs  0I

3.5xNIFs  0
I

3.0

2.5

2.0

1.5
0.0

0.2

1

0.4

0.6

0.8

1.0

IF /∂s > 0
3. The condition that ∂x∗N
I

Figures 3: The conditions in Proposition 3

34

5.0

5.0

4.5

4.5

4.0

4.0

xNI
 xRI
I
I

3.5

xNF
 xRF
F
F

3.5

3.0

3.0

2.5

2.5

2.0

2.0

1.5

1.5
0.0

0.2

0.4

0.6

0.8

1.0

0.0

1: The condition in Proposition 5(i)

0.2

0.4

0.6

0.8

1.0

2: The condition in Proposition 5(ii)

5.0

5.0

4.5

4.5

4.0

4.0

xNIF
 xRIF
I
I

3.5

xNIF
 xRIF
F
F

3.5

3.0

3.0

2.5

2.5

2.0

2.0

1.5

1.5
0.0

0.2

0.4

0.6

0.8

1.0

0.0

3: The condition in Proposition 5(iii.1)

0.2

0.4

4.5

4.0

xNIF
xNIF xRIFxRIF
I
F
I
F

3.0

2.5

2.0

1.5
0.0

0.2

0.8

1.0

4: The condition in Proposition 5(iii.2)

5.0

3.5

0.6

0.4

0.6

0.8

1.0

5: The condition in Proposition 5(iii.3)
Figure 4: The conditions in Proposition 5.

35

5.0

4.5

4.0

3.5

3.0

SWR  SWN

2.5

SWR  SWN

2.0

1.5
0.0

0.2

0.4

0.6

0.8

1.0

Figure 5: The comparison of social welfare.
Note: SW R and SW N are social welfare in RIF and NIF.

36

Always

Always

AlwaysRegime RI

Regime RF

Regime RIF

x∗RIF
=
I

(3 − s)(2γ − 3)A
C
x∗RIF
=
F

3A
8γ − 93(2γ − (3 − s))A
C

xRF
F =

—

(3 − s)A
8γ − (3 − s)2

—

—

x∗F

2(2 − s)A
9γ − 2(2 − s)2

x∗RI
=
I

xR
I =

x∗I

a∗RIF = ca∗RF = ca∗RI = caR = ca∗Note: C ≡ 16γ 2 − 2(9 + (3 − s)2 )γ + 3(3 − s)(2 − s)

Table 1: Comparison of investment levels under the access price regulation regimes.

AlwaysRegime R

condition

Accommodation

γ ≥ 4(1 − s)

Alwaysγ ≥ 4(1 − s)Regime NI

Regime NF

Regime NIF

A2γ − 14A
9γ − 8(9 + s)(11γ − 15)A
C
4A
IF
x̃∗N
=
I
9γ − 4

—

I
x̃∗N
=
I

(9 + s)A
22γ − (9s2 − 14s + 25)

IF
x∗N
=
I

I
x∗N
=
I

x̃N
I =

x∗I

IF
x∗N
=
F

105A
242γ − 225

15(7γ − (3s2 − 3s + 10))A
C
4A
IF
x̃∗N
=
F
9γ − 4

F
xN
=
F

—

—

x∗F

a∗N IF = c +2(11γ − 15)(3γ − (3 − s)(1 − s))AC6(11γ − 15)A
242γ − 225

2(3γ − (3 − s)(1 − s))A
22γ − (9s2 − 14s + 25)(5γ − 2(2 − s)(1 − s))A
10γ − (4s2 − 8s + 9)a∗N F = c +a∗N I = c +aN = c +a∗Note: C  ≡ 242γ 2 − (99s2 − 154s + 500)γ + 15(6s2 − 11s + 15)

Table 2: Comparison of investment levels under the no access price regulation regimes.

Never

Regime N

condition

Accommodation