GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 275

Regulated input price, vertical separation, and leadership in
free entry markets

Toshihiro Matsumura and Noriaki Matsushima

August 2012GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Regulated input price, vertical separation, and leadership in
free entry markets∗
Toshihiro Matsumura†Institute of Social Science, The University of Tokyo
Noriaki Matsushima‡
Institute of Social and Economic Research, Osaka University
August 9, 2012

JEL classification numbers: L51, L13
Key words: network industry, access charge, manipulation of accounting, regulation∗

The authors gratefully acknowledge ﬁnancial support from a Grant-in-Aid for Basic Research from theJapanese Ministry of Education, Science and Culture. Needless to say, we are responsible for any remaining
errors.
†Toshihiro Matsumura, Institute of Social Science, The University of Tokyo, Hongo 7-3-1, Bunkyo, Tokyo
113-0033, Japan. E-mail: matsumur@iss.u-tokyo.ac.jp‡
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.osakau.ac.jp

Abstract
We examine incentives of bottleneck facility holders to manipulate access charge accounting in free entry downstream markets. We consider the situation wherein one ﬁrm
holds an upstream bottleneck facility and new entrants use it at the regulated price (accessfee) to provide ﬁnal products. The bottleneck facility holder aﬀects the regulated input
price. We investigate how vertical separation aﬀects the incentive for manipulation andthe resulting input price. We ﬁnd that the results depend on whether the incumbent is the
Stackelberg leader in the product market. If the incumbent cannot take leadership in the
product market and faces Cournot competition, vertical separation reduces the incentive
for manipulation and the resulting input price. The opposite result is derived when theincumbent can take leadership in the product market.

1

Introduction

Competition between a ﬁrm holding essential facilities (usually a dominant ﬁrm) and ﬁrms
not holding them has been widely observed in many industries such as overnight delivery,
telecommunications, electric power, and natural gas distribution. In particular, dominantﬁrms in Japan are not vertically separated in the telecommunications, electric power, andgas distribution industries. Dominant ﬁrms were previously legal monopolists, and they havecompeted against new entrants since the liberalization of these markets. In these network
industries, new entrants pay an access fee to the dominant ﬁrms. This access fee aﬀects the
new entrants’ marginal costs and thus signiﬁcantly aﬀects the performance of the markets.
The access fee is typically regulated by the government. Considerable literature on thedesirable and/or existing rules for levying the access charge, such as the eﬃcient component
pricing rule (ECPR), the total element long-run incremental cost (TELRIC), and a historicalcost approach, exists;1 moreover, charges based on the costs of essential facilities are currently
levied all over the world. For example, in Japan, TELRIC has been adopted by traditional
local telecommunications, while the historical cost approach has been adopted in the opticalﬁber networks, electric power, and natural gas distribution industries. In the EU, although
the rules are diﬀerent for diﬀerent countries, cost-based approaches have been adopted in mostindustries and in most countries.
In practice, however, ﬁrms often inﬂuence the regulated input price (access charge) by
manipulating the accounting costs, whereas in principle, the rule for calculating the accesscharge is determined by a neutral government and there is no room for manipulation.2 For
instance, using a dataset on state campaign contributions by telecommunications companies in
1

For a general description of the access charge, see Vogelsang (2003) and Rey and Tirole (2007). See also

Laﬀont and Tirole (1994), Economides and White (1995), Hausman (1997), Armstrong and Vickers (1998), Ida(2001), Sappington (2006), and Higgins and Mukherjee (2010). Further, see Hori and Mizuno (2006), Mizuno
and Shinkai (2006), Ida and Anbashi (2008), and Gautier and Mitra (2008) for the dynamic eﬃciency of accesscharge rules.
2
For discussions on these activities in the general context of regulation policies, see Viscusi et al. (2005).

1

the United States, de Figueiredo and Edwards (2007) ﬁnd that there is a correlation between
the relative levels of contributions by ﬁrms and the level of local loop prices (regulated accessprices). In Japan, integrated gas companies insisted that the costs of vaporizers and pumps ofliquidated natural gas (LNG) tanks should be included in the access charge accounting for gas
distribution pipeline networks. They made substantial eﬀorts to rationalize their argumentin front of the Gas Energy Committee of the Advisory Committee for Natural Resource andEnergy, the agency that determines the rules for access charge accounting. These companies
succeeded in persuading the committee to adopt a resolution to include these costs in theaccess charge and maintained that resolution for more than 15 years. Similar instances havebeen widely observed in the telecommunications and electric power distribution industries inJapan. In this paper, we consider a situation wherein the access charge is inﬂuenced by theﬁrms holding the bottleneck facilities.3
Another important policy topic is whether dominant ﬁrms should be allowed to holdessential facilities. Vertical separation is always an important policy issue in this ﬁeld, andlegal and ownership unbundling has been widely observed, especially in the EU. It is often
insisted that vertical separation is beneﬁcial because it deters manipulation by verticallyintegrated ﬁrms.
In this paper, we discuss the means by which vertical separation aﬀects the regulated input
price. First, we formulate a model in which n ﬁrms (ﬁrm 1, ﬁrm 2,. . ., ﬁrm n) compete in a
ﬁnal product market and one monopoly input supplier, ﬁrm 0, holds an essential facility. The
number of ﬁrms is endogenously determined (free entry market). Firm 1 is the incumbent
ﬁrm (formerly legal monopolist). We compare two situations: ﬁrms 0 and 1 are verticallyintegrated (vertical integration case) and no ﬁrm is vertically integrated (vertical separationcase). We also consider the two cases concerning the leadership of the incumbent ﬁrm, ﬁrm 1.
One is the Stackelberg model in which ﬁrm 1 can commit to the output level before the entry
3

Baron (2001) emphasizes the importance of those nonmarket strategies including lobbying, self-regulation,and corporate social responsibility.

2

of new entrants (ﬁrm 2,. . . , ﬁrm n), and the other is Cournot model in which all downstreamﬁrms (ﬁrm 1, ﬁrm 2,. . ., ﬁrm n) produce independently.
Whether the existing incumbent ﬁrm takes leadership in the downstream market is an
important matter in deregulated industries. In some cases, the historically incumbent ﬁrms
have dominant positions (Weisman (1995), Sarmento and Brandão (2007)), implying thatthose incumbent ﬁrms take leadership in the downstream markets; thus, considering Stackelberg competition is reasonable for researchers. In other cases, such incumbent ﬁrms do
not have signiﬁcant market shares (Beard et al. (2001)), implying that considering Cournot
competition is reasonable for researchers. The diﬀerence between the market positions ofhistorically incumbent ﬁrms crucially depends on how to deregulate those former regulated
markets, entry conditions, technological properties, and so on. We therefore think that comparing the two competition modes (Cournot and Stackelberg) is important in the context of
deregulated markets.
We ﬁnd that the eﬀect of vertical separation crucially depends on whether the incumbent
ﬁrm takes leadership in the product market. If the incumbent cannot take leadership at the
product market, vertical separation reduces the incentive for manipulation of accounting andthus reduces the resulting regulated input price. On the contrary, if the incumbent can takeleadership at the product market, vertical separation increases the incentive for manipulationand thus raises the input price. Thus, to reduce the manipulation and to stimulate new entries
by vertical separation, eliminating the leadership position by the incumbent is important.
We explain why Cournot and Stackelberg yield opposite policy implications for the verticalseparation policy. First, consider the Cournot competition (where the incumbent, ﬁrm 1,
cannot take leadership). Under vertical integration, an increase in the input price reduces
other ﬁrms’ outputs and thus increases ﬁrm 1’s output. This production substitution increases
the proﬁt of ﬁrm 1, and thus ﬁrm 0, the holder of bottleneck facility, has a stronger incentive
to manipulate the input price because ﬁrm 0 maximizes the joint proﬁts of ﬁrms 0 and 1
under vertical integration. This eﬀect disappears under vertical separation. This is why
3vertical integration stimulates the manipulation by ﬁrm 0. Next, consider the Stackelberg
competition (where the incumbent, ﬁrm 1, can take leadership). As discussed in Etro (2006),
the incumbent produces aggressively, regardless of the input price. Thus, in contrast to
the Cournot case, the strategic eﬀect of raising the input price disappears under vertical
integration. On the other hand, an increase in the input price reduces the output and the
proﬁt of ﬁrm 1, but ﬁrm 0 does not take this reduction into account under vertical separation.
Thus, ﬁrm 0 has a stronger incentive to raise the input price. This is why vertical separation
stimulates the manipulation under Stackelberg competition.
We review related papers. There are two research lines that are closely related to our paper.
One is the literature on strategic commitment in free entry markets. Etro (2006, 2007) clearlyshows that the incumbent leader engages in aggressive strategic behavior to restrict new entries
in free entry markets in a general context. He derives many important policy implicationsin the context of competition policies.4 We introduce this strategic behavior in the context
of vertical separation/integration policy and regulated input prices. The entry restrictions
in markets with a bottleneck facility have been signiﬁcantly weakened. We observe the newentry of private ﬁrms in these markets such as the electric power, natural gas distribution, andtelecommunication markets in many European and Asian countries. Therefore, the analysis
of free entry market in this context is quite important.
The other research line is the literature on raising rivals’ costs. In the literature on vertically related industries, many works have already pointed out that vertically integrated ﬁrms
have strong incentives to raise their rivals’ costs. Economides (1998) investigates Cournot
competition in a downstream market and shows that the vertically integrated ﬁrm has anincentive to exclude the rivals through a prohibitive access charge. Mandy and Sappington(2007) show that the vertically integrated ﬁrm has an incentive to engage in cost-increasing
sabotage regardless of whether there is Cournot or Bertrand competition in the downstream
4

The literature on strategic behavior in free entry markets and discussions of policy implications has becomerich and diverse. See, among others, Brandão and Castro (2007), Davidson and Mukherjee (2007), Etro (2008,2011), Ino and Matsumura (2010, 2012), Marjit and Mukherjee (2008), and Mukherjee and Mukherjee (2008).

4

market, but has a weak (strong) incentive to engage in demand-reducing sabotage when there
is Bertrand (Cournot) competition. The discussions on increasing the rivals’ costs and costincreasing sabotage in this context are also provided by Beard et al. (2001) (see also the papers
cited by them). Bustos and Galetovic (2009) discuss a case in which a bottleneck monopoly
endogenously determines its vertical structure (integration or separation) and the degree ofsabotage. Our paper diﬀers in two crucial ways from these works. Under vertical separation,
the upstream ﬁrm has no incentive to engage in sabotage activities because these activities
never increase the revenue and thus the proﬁts of the upstream ﬁrm. On the contrary, raising
the regulated input price raises the proﬁt of the upstream ﬁrm, and thus the upstream ﬁrm
has an incentive to raise the input price even under vertical separation. Another and more
important diﬀerence is that all works mentioned above assume that the number of ﬁrms isgiven exogenously, whereas we consider free entry markets.
The remainder of this paper is organized as follows. Section 2 formulates the model.
Section 3 presents the result under Cournot competition. Section 4 presents the result under
Stackelberg competition. Section 5 checks the robustness of the result in Section 4. Section 6
concludes the paper.

2

The model without leadership

In this section, we formulate a model wherein downstream ﬁrms face Cournot competitionin the ﬁnal product market. We consider two cases: (1) A vertically integrated ﬁrm holding
a bottleneck facility exists (vertical integration); (2) The government undertakes verticalseparation of the ﬁrm holding the bottleneck facility (vertical separation).
We consider an oligopoly model. Firm 0 holds a bottleneck facility. Firm 1 is the incumbent ﬁrm and ﬁrm i (i = 2, . . . , n) is a new entrant. The number of new entrants is
endogenously determined by the free entry condition discussed below. Firm i (i = 1, . . . , n)uses the bottleneck facility held by ﬁrm 0 and pays f yi to ﬁrm 0, where f is the input price(per unit access fee of the bottleneck facility) and yi is ﬁrm i’s output. We assume that the5marginal production cost is constant, common among ﬁrms, and normalized to zero (except
for the access fee).
In the ﬁrst stage, the input price (access fee) f is determined. Let M (f ) denote ﬁrm 0’s
costs for manipulation of the access charge accounting. We implicitly assume that the input
price is zero if ﬁrm 0 does not make any eﬀort. That is, the input price is regulated but canbe aﬀected by ﬁrm 0’s eﬀorts.
In the second stage, given the input price, f , new entrants determine whether to enter themarket.
In the third stage, ﬁrms engage in Cournot competition. Firm i (i = 1, . . . , n) independently produces perfectly homogeneous products for which the market demand function is
given by p(Y ), where Y is the total output of the ﬁrms (price as a function of quantity). We
assume that p < 0. We also assume that p yi + p < 0 for all yi ≤ Y so as to ensure that thereaction curve at the Cournot competition stage is downward sloping (strategic substitute).
The proﬁts of ﬁrms 0 and i are given as
π0 = fn

yj − M (f ),πi = (p(Y ) − f )yi − K (i = 1, . . . , n),

(1)

j=1

where K > 0 is the entry cost. We assume that M  (f ) ≥ 0 and that M  (f ) > 0 in orderto ensure the concavity of ﬁrm 0’s payoﬀ function. We assume that M  (0) = 0 in order toensure interior solutions in the ﬁrst stage. We also assume that the number of new entrants
is strictly positive under the Cournot competition.

3

Equilibrium outcome without leadership by the incumbent

3.1

Vertical integration

First, we consider the vertical integration case, that is, ﬁrms 0 and 1 are integrated verticallyand maximize their joint proﬁts.
Consider the third stage. From (1), the ﬁrst-order conditions of the ﬁrms in the third6

stage are given byp + p y1 = 0,p + p  yi − f= 0, (i = 2, 3, ..., n).

(2)
(3)

In the second stage, the free entry condition (zero-proﬁt condition) must hold for i = 2, ..., n:(p(Y ) − f )yi = K.

(4)

These three conditions in (2), (3), and (4) yield the equilibrium outcome given the regulated input price f . We investigate how a change in the input price aﬀects the equilibriumoutcome. The total diﬀerentials of these three conditions are given as(p y1 + 2p )dy1 +(n − 1)(p y1 + p )dyi +yi (p y1 + p )dn = 0,+((n − 1)p yi + np )dyi +yi (p yi + p )dn = df,(p yi + p )dy1
p yi dy1+(n − 2)p yi dyi+yi2 p dn= yi df,(5)

where we use p − f = −p yi in (3) to derive the last equation in (5). These equations in (5)
yield
dy1
df
dyi
df
dn
df

2(p + p y1 )= −  ,p (2p + p yi )p yi= −  (i = 2, ..., n),p (2p + p yi )4p + 2p y1 + (n − 1)p yi=.
p yi (2p + p yi )

(6)
(7)
(8)

Because y1 + (n − 1)y2 = Y , we havedy1
dyi dn2dY=+ (n − 1)+yi = 
< 0 (i = 2, ..., n).
df
df
df
df
2p + p yi(9)Note that we have assumed that p yi + p < 0 (condition for strategic substitute) and p < 0.
In the ﬁrst stage, ﬁrm 0 chooses f so as to maximize π0 + π1 . The ﬁrst-order condition ofﬁrm 0 is2f
yi (2p + p y1 )d(π0 + π1 )=Y + 
+− M  (f ) = 0,df
2p + p yi
2p + p yi
where i = 1.
7

(10)

3.2

Vertical separation

Second, we consider the vertical separation case, that is, ﬁrm 0 maximizes π0 with respect tof and ﬁrm 1 maximizes π1 with respect to y1 .
Consider the third stage. The ﬁrst-order conditions of the downstream ﬁrms in the third
stage are given byp(Y ) + p (Y )yi − f = 0, (i = 1, 2, ..., n).

(11)

In the second stage, due to the free entry condition, equation (4) holds.
These two conditions in (4) and (11) yield the equilibrium outcome given the regulated
input price f . We investigate how a change in the input price aﬀects the equilibrium outcome.
The total diﬀerentials of these two conditions in (4) and (11) are given as(np yi + (n + 1)p )dyi +yi (p yi + p )dn = df
(n − 1)p yi dyi+yi2 p dn= yi df.

(12)

From the equations in (12), we have
dyi
df
dn
df

p yi= −  ,p (2p + p yi )2p + np yi=.
p yi (2p + p yi )

(13)
(14)

From these results, we havedY
dyi dn2=n+yi = 
< 0.
df
df
df
2p + p yi(15)

Firm 0 maximizes π0 with respect to f . The ﬁrst-order condition is
dY
dπ0
2f
=Y +f
− M  (f ) = Y + − M  (f ) = 0.
df
df
2p + p yi

3.3

(16)

Comparison

We now investigate how vertical separation aﬀects the equilibrium outcomes under Cournot
competition. Let the superscripts “IC” and “SC” denote the equilibrium outcomes under8

vertical integration and under vertical separation, respectively (I, S, and C indicate Integration, Separation, and Cournot competition, respectively).
Lemma 1 Consider the equilibrium outcomes in the second stage given f . Then, for all f ,(i) yiIC (f ) = yiSC (f ) (i = 2, 3, ..., n), (ii) Y IC (f ) = Y SC (f ) and pIC (f ) = pSC (f ), and (iii)
Y SC (f ) is decreasing in f .
Proof See the Appendix.
Lemma 1 states that the type of vertical structure does not aﬀect the total output and thus
does not aﬀect the consumer surplus at the free entry market. Because the type of verticalstructure does not aﬀect the total output under a given input price f , it does not aﬀect ﬁrm
0’s proﬁt. However, the type of vertical structure aﬀects the proﬁt of ﬁrm 1. Thus, it aﬀects
the equilibrium level of f .
Proposition 1 Without the leadership by firm 1, vertical separation reduces the input priceand increases consumer surplus.
Proof See the Appendix.
Proposition 1 states that vertical separation reduces the incentive of ﬁrm 0 for manipulation and thus reduces the resulting input price. An increase in the input price improves thecompetitive advantage of ﬁrm 1 and increases the joint proﬁts of ﬁrms 0 and 1. Such a strategic incentive disappears when the two ﬁrms are separated. This is why vertical separation
reduces the input price. From Lemma 1(iii), this reduction in the input price increases thetotal output, which beneﬁts consumers.
In the next section, we show that this result does not hold if the incumbent ﬁrm, ﬁrm 1,
can take Stackelberg leadership in the production market.

4

Equilibrium outcomes with leadership by the incumbent

In this section, we consider the situation in which the incumbent ﬁrm, ﬁrm 1, is the Stackelberg
leader. We follow the standard model formulation of leadership in free entry markets discussed9

in Etro (2006, 2007). In the ﬁrst stage, the input price f is determined. In the second stage,ﬁrm 1 sets its quantity y1 . In the third stage, given f and y1 , new entrants determine whether
to enter the market. Finally, those entrants simultaneously choose their outputs.
Consider the third and the fourth stages. The ﬁrst-order conditions of entrants and the freeentry condition are the same as in the previous section. Given f and y1 , these two conditions
yield the equilibrium number of new entrants and the output of each new entrant. We discuss
how y1 aﬀects these two variables. Let the superscripts “IL” and “SL” denote the equilibriumoutcomes under vertical integration and under vertical separation, respectively (I, S, and L
indicate Integration, Separation, and the Leadership by the incumbent, respectively).
Lemma 2 Consider the subgame given f and y1 . Regardless of f and y1 , as long as n > 0,(i) yiIC (f ) = yiSC (f ) = yiIL (f ) = yiSL (f ) (i = 2, 3, ..., n), (ii) Y IC (f ) = Y SC (f ) = Y IL (f ) =Y SL (f ) and pIC (f ) = pSC (f ) = pIL (f ) = pSL (f ), and (iii) dn/dy1 = −1/y2 .
Proof See the Appendix.
Lemma 2(ii) implies that as long as n > 0, the total output and the price do not dependon y1 . Thus, ﬁrm 1 serves as a price taker as long as n > 0. Because the price must be
higher than f when n > 0, ﬁrm 1 chooses y1 to deter the entry of new entrants, resulting in
n = 0 under both vertical integration and vertical separation. Firm 1 chooses y1 such thatp(y1 ) = f + (K/y2∗ ), where y2∗ = y2IC = y2SC . That is, y1 is equal to Y IC (f ) = Y SC (f ). This
result has already been shown by Etro (2006) for the situation in which f = 0. Under these
conditions, we obtain the following lemma.
Lemma 3 Suppose that firm 1 is the Stackelberg leader. Then, no entry occurs under bothvertical separation and vertical integration.
We now discuss how vertical separation aﬀects the equilibrium input price. Under vertical10integration, ﬁrm 0 chooses f to maximize
π0 + π1 = f y1 (f ) − M (f ) + (p(y1 (f )) − f )y1 (f )= p(Y IC (f ))Y IC (f ) − M (f ),note that y1 (f ) = Y IC (f ). Under vertical separation, it chooses f to maximize
π0 = f Y IC (f ) − M (f ).
From the equations, we have the following proposition.
Proposition 2 Under the leadership by firm 1, vertical separation increases the input priceand reduces the consumer surplus.
Proof See the Appendix.
This result contrasts with the result in the Cournot case (Proposition 1). Without theleadership by the incumbent in the production market, vertical separation reduces the incentive for manipulation, which reduces the input price. On the contrary, with the leadershipby the incumbent, vertical separation stimulates the incentive of ﬁrm 0 for manipulation andthus increases the resulting input price. As discussed in Etro (2006), the incumbent produces
aggressively, regardless of the input price. In contrast to the Cournot case, the strategic effect of raising the input price disappears under vertical integration. On the other hand, anincrease in the input price reduces the output and the proﬁt of ﬁrm 1, although ﬁrm 0 does
not take this reduction into account under vertical separation. Thus, ﬁrm 0 has a stronger
incentive to raise the input price. This is why vertical separation stimulates the manipulation
under Stackelberg competition.
The two propositions indicate that the market structure is a critical determinant of how
vertical separation aﬀects the input price. If the incumbent occupies the leadership position and adopts its strategic behavior before new entry, vertical separation is harmful for
consumers.

11Some readers may suspect that the diﬀerence between Propositions 1 and 2 comes fromthe diﬀerence between the properties of the solutions, not from whether the incumbent is the
Stackelberg leader. No entry occurs in the Stackelberg case (complete entry deterrence, corner
solution), although entry does occur (interior solution) in the Cournot case. This diﬀerence
may be crucial. If we replace the assumption of constant marginal costs with that of increasing
marginal costs, entry occurs even in the Stackelberg case. In the next section, we will checkthe robustness of our results by considering increasing marginal costs.

5

Robustness check: Increasing marginal costs case with leadership

In the previous section, in the Stackelberg case, we have shown that the corner solution (noentry) appears in equilibrium if the marginal costs of the ﬁrms are constant. In this section,
we consider increasing marginal costs and investigate the situation in which entry occurs evenunder the leadership by the incumbent. Let c(yi ) be the common production cost function ofﬁrm i (i = 1, 2, . . . , n). We assume that c > 0, c > 0, and c is large in order to ensure aninterior solution.
Consider the third and the fourth stages. The ﬁrst-order condition and the zero-proﬁt
condition for each new entrant are, respectively,(p(Y ) − f ) + p (Y )yi − c (yi ) = 0,(17)(p(Y ) − f )yi − c(yi ) = K.

(18)

From almost the same calculations as in previous sections, we can show that Lemma 2 also
holds under increasing marginal costs. We again obtain that the total quantity supplied byﬁrms, Y , does not depend on y1 . Let Ỹ (f ) denote the total quantity in this game. Firm
1 serves as the price taker as long as n > 0. Contrary to the previous case, complete entry
deterrence (n = 0) is not optimal for ﬁrm 1 because the marginal cost is increasing in quantity.
Firm 1 chooses y1 such that p(Ỹ (f )) is equal to its marginal cost. That is, under both vertical
12

integration and vertical separation, ﬁrm 1 chooses y1 such that p(Ỹ (f )) = c (y1 )+f . The typeof vertical structure again does not aﬀect the optimal output of ﬁrm 1. Some readers may
think that ﬁrm 1’s marginal cost is not c (y1 ) + f but c (y1 ) under vertical integration because
it maximizes the joint proﬁts of ﬁrms 0 and 1 and because f y1 is just a transfer within the
integrated ﬁrm. However, an increase in y1 reduces the total output of new entrants by thesame amount, thus reducing the revenue of ﬁrm 0 from the rivals. This means that f is themarginal opportunity cost generated by increasing y1 under vertical integration (Sappington(2005)).
In the ﬁrst stage, ﬁrm 0 chooses f . Under vertical integration, ﬁrm 0 maximizes
π0 + π1 = f Ỹ (f ) − M (f ) + (p(Ỹ (f )) − f )y1 (f ) − c(y1 (f )).
Under vertical separation, it maximizes
π0 = f Ỹ (f ) − M (f ).
Comparing the two ﬁrst-order conditions yields the following proposition.
Proposition 3 Suppose that the marginal cost of each firm is increasing in quantity. Consider
an interior solution (n > 0). Under the leadership by firm 1, vertical separation increases the
input price if p < 0. It does not change the input price if p = 0. It reduces the input priceif p > 0.
Proof See the Appendix.
In the increasing marginal cost case, vertical separation can increase the input price andthen reduce the consumer surplus. Vertical separation increases the input price if the demandis strictly concave. However, the result is weaker than that in the constant marginal costcase because this result does not hold if the demand is linear or strictly convex. If the inverse
demand is strictly concave, one unit increase in f increases the price at the product market byless than one unit. Thus, an increase in f decreases the ﬁrm 1’s proﬁt (p − f )y1 − c(y1 ). Firm
0 does not care about this eﬀect on ﬁrm 1’s proﬁt under vertical separation, implying that13

vertical separation raises f if the inverse demand is strictly concave. However, if the inverse
demand is strictly convex, one unit increase in f increases the price at the product market
by more than one unit. Thus, an increase in f increases the ﬁrm 1’s proﬁt (p − f )y1 − c(y1 ).
Firm 0 does not care about this eﬀect on ﬁrm 1’s proﬁt under vertical separation, implyingthat vertical separation reduces f if the inverse demand is strictly convex, in contrast to theconcave demand function case.

6

Concluding Remarks

We have considered the situation wherein one ﬁrm holds an upstream bottleneck facility
and new entrants use it at the regulated price (access fee) to provide ﬁnal products. The
bottleneck facility holder aﬀects the regulated input price through the manipulation of accesscharge accounting. We have investigated how vertical separation aﬀects the regulated input
price in a free entry market. We discuss two cases: with and without the leadership by theincumbent ﬁrm in the product market. We ﬁnd that vertical separation lowers the regulated
input price without the leadership by the incumbent ﬁrm and that it raises the regulated input
price with the leadership by the incumbent ﬁrm. Thus, to stimulate new entry by verticalseparation, eliminating the leadership position by the incumbent is important.
We have assumed that the bottleneck facility holder can aﬀect the regulated input priceby manipulating the costs of the incumbent ﬁrm. We do not explicitly formulate a formal
model analyzing the manipulating process. This important topic remains for future research.

Appendix
Proof of Lemma 1

If f = 0, there is no diﬀerence between the two cases. Thus, yiIC (0) =yiSC (0) and Y IC (0) = Y SC (0). Because (7) and (9) are, respectively, equal to (13) and (15),yiIC (f ) = yiSC (f ) (i = 2, 3, ..., n) and Y IC (f ) = Y SC (f ) for all f . This implies Lemmas 1(i)and (ii). (9) implies Lemma 1(iii).

Q.E.D.
14

Proof of Proposition 1

Comparing (10) and (16), we have that (10) is larger than (16)given f because 2p + p yi < 0 for i = 1, 2, . . . , n. Therefore, (10) yields a higher f than (16).
This implies Proposition 1.
Proof of Lemma 2

Q.E.D.

The ﬁrst-order condition and the zero-proﬁt condition for each newentrant are common under both vertical integration and vertical separation. Thus, the equation yiIL (f ) = yiSL (f ) must hold. The total diﬀerentials of these two conditions (the ﬁrst-ordercondition and the zero-proﬁt condition for each new entrant) are given as((n − 1)p yi + np )dyi +yi (p yi + p )dn = −(p yi + p )dy1(n − 2)p yi dyi+yi2 p dn= −p yi dy1 .

(19)

From these equations, we have dyi /dy1 = 0 (i = 2, 3, ..., n). Thus, the equation yiIL (f ) =yiSL (f ) does not depend on y1 . For any y1 = y1IC (f ), the equation yiIC (f ) = yiIL (f ) must holdbecause the two conditions (the ﬁrst-order condition and the zero-proﬁt condition for each newentrant) in the Stackelberg case are exactly the same as those in the Cournot case. From thisequation (yiIC (f ) = yiIL (f )) and Lemma 1 (yiIC (f ) = yiSC (f )), the equation yiIL (f ) = yiSC (f )holds. These imply Lemma 2(i).
From the zero-proﬁt condition, we have pj (f ) = f + K/yij for j = IC, SC, IL, SL. Lemma
2(i) implies Lemma 2(ii).
y1j + (n − 1)y2j is ﬁxed at Y j (f ) for j = IC, SC, IL, SL. This implies that dy1 + dny2 = 0.
From the equation, we have Lemma 2(iii).
Proof of Proposition 2

Q.E.D.

For expositional simplicity, we denote Y IC (f ) by Y (f ). Undervertical integration, the ﬁrst-order condition for ﬁrm 0 isd(π0 + π1 )= Y  (f )(p Y (f ) + p) − M  (f ) = 0.
df

(20)

Under vertical separation, the ﬁrst-order condition for ﬁrm 0 is
dπ0
= Y (f ) + f Y  (f ) − M  (f ) = 0.
df
15

(21)

The diﬀerence between d(π0 + π1 )/df in (20) and dπ0 /df in (21) isd(π0 + π1 ) dπ0−
dfdf= −Y (f ) + Y  (f )(p Y (f ) + p − f )= −Y (f ) + p Y  (f )(Y (f ) − yi (f ))2p (Y (f ) − yi (f ))2p + p yi (f )yi (f )(2p + p Y (f ))< 0,= −2p + p yi (f )= −Y (f ) +where we use the ﬁrst-order condition for each new entrant, p − f = −p yi , to derive thesecond line and equation (9), Y  (f ) = 2/(2p + p yi (f )), to derive the third line. Becausep + p y < 0 for all y and p < 0, the last line is negative. Therefore, given f , d(π0 + π1 )/df
in (20) is smaller than dπ0 /df in (21), implying that (20) yields a smaller input price, f , than(21).

Q.E.D.

Proof of Proposition 3

Because the type of vertical structure does not aﬀect π0 (f ),vertical separation increases (decreases, does not aﬀect) the input price if and only if ∂π1 /∂f >(<, =) 0.
The total diﬀerentials of the three conditions (the ﬁrst-order condition for ﬁrm 1, theﬁrst-order condition for each new entrant, and the zero-proﬁt condition for each new entrant(ﬁrm i)) are given as+yi p dn= df(p − c (y1 ))dy1 +(n − 1)p dyi





+((n − 1)p yi + np − c (yi ))dyi +yi (p yi + p )dn = df,(p yi + p )dy1
p yi dy1+(n − 2)p yi dyi+yi2 p dn= yi df.

(22)

From the equations, we have Ỹ  (f ):Ỹ  (f ) =

dyi
n
dy1
2p − c (yi )+ (n − 1)+ yi =  
.
df
df
df
p (2p + p yi − c (yi ))The diﬀerence between the diﬀerential of π0 +π1 with respect to f under vertical integrationand that of π0 with respect to f under vertical separation is(p (Ỹ (f ))Ỹ  (f ) − 1)y1 .
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Using the result of Ỹ  (f ), we have(p (Ỹ (f ))Ỹ  (f ) − 1)y1 =−p y1.
2p + p yi − c (yi )Because the denominator is negative (the second-order condition of ﬁrm i), the fraction isnegative (positive, zero) if and only if p < (>, 0). This implies Proposition 3.

Q.E.D.

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