GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.221

Market Size and Vertical Structure in the Railway Industry

Noriaki Matsushima and Fumitoshi Mizutani

October 2011

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

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

Market Size and Vertical Structure in the Railway Industry∗
Noriaki Matsushima†
Institute of Social and Economic Research, Osaka University
Fumitoshi Mizutani
Graduate School of Business Administration, Kobe University
October 7, 2011

Abstract
We provide a theoretical framework to discuss the relation between market size andvertical structure in the railway industry. The framework is based on a simple downstream
monopoly model with two input suppliers, labor forces and the rail infrastructure ﬁrm. The
operation of the downstream ﬁrm (i.e., the train operating ﬁrm) generates costs on the rail
infrastructure ﬁrm. We show that the downstream ﬁrm with a larger market size is more
likely to integrate with the rail infrastructure ﬁrm. This is consistent with the phenomenon
in the railway industry.

JEL classiﬁcation: L22, L13, R32
Key words: vertical integration, railway industry, market size, vertical coordination∗

The authors gratefully acknowledge ﬁnancial support from Grant-in-Aid for Encouragement of Young Scientists and for Basic Research from the Japanese Ministry of Education, Science and Culture. Any errors are
the responsibility of the authors.
†
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.jp1

1

Introduction

Vertically integrated and separated ﬁrms coexist in many industries, a typical case being therail industry. An important issue in the rail industry is whether to pursue a vertical separation policy, whereby a rail company is divided into two organizations, one responsible forrail operation and the other for rail infrastructure. There have been several variations in thevertical separation policy, including functional accounting separation (e.g. France in late 90s),
organizational separation of rail operations and infrastructure (e.g. the UK, Netherlands), or
organizational separation involving a holding company (e.g. Germany). While vertical separation has been carried out in some countries, massive horizontal separation of former state
railways has been adopted in others (e.g. the UK and Japan). One important aim of organizational reforms in the rail industry is to create a competitive environment, with a verticalseparation policy viewed as one option for stimulating competition.
There are quite clear policy diﬀerences between Europe and Japan. While vertical separation is a common policy in the European Union, vertical integration is still the structureof choice in the Japanese rail industry, notwithstanding the increasing incidence of verticalseparation in local areas in Japan, such as in the cases of Aomori Railway and Sanriku Railway. While there are many examples of vertically separated railways in Europe, theoreticalstudies describing behavior resulting from this policy are few. We therefore provide an analytic
framework to investigate this problem.
This paper investigates what factors determine the organization structure of railway companies. We construct a simple model including three players: a train operating company (ﬁrm
T ), a rail infrastructure company (ﬁrm R), and a labor union (group L). Firm R and group L
respectively supply an essential input (e.g., the usage of rails) and labor force to ﬁrm T . Using
those factors, ﬁrm T supplies some units of the travel service to consumers who enjoy the travel
service. By comparing their proﬁts in the two situations of vertical integration and verticalseparation, ﬁrms R and T determine whether or not they should be integrated vertically. We
take into account the following factor: the operation by ﬁrm T generates costs on ﬁrm R.

2

Those costs include track maintenance, electrical systems maintenance, and the depreciationof infrastructure. Firm R must pay a daily eﬀort to reduce those track-related costs. Thisfactor is important in ordinal track maintenance activities in the railway industry.
We show that ﬁrms R and T decide to integrate vertically if and only if the market sizeis large or the parameter related to the eﬀort cost is small. The intuition behind the result is
as follows. The quantity supplied by ﬁrm T depends on the input prices set by group L andﬁrm R. Because the inputs supplied by group L and ﬁrm R are perfect complements, the sumof the input prices aﬀects the quantity supplied by ﬁrm T . A higher input price set by an
input supplier reduces the quantity supplied by ﬁrm T . Anticipating the shrink in quantity,
another input supplier sets a lower input price. That is, the strategic interaction between thetwo suppliers is strategic substitution. Vertical separation is a credible commitment to set
a higher input price of ﬁrm R. The rent-shifting from group L to ﬁrm R is beneﬁcial fromthe viewpoint of ﬁrms R and T . This eﬀect depends on the market size. An increase in themarket size enhances the quantity supplied by ﬁrm T . The increase in the quantity enlarges
the importance of the reduction in per unit costs caused by the operation of ﬁrm T . That is,the vertical coordination between ﬁrms R and T becomes more important. To enhance thebeneﬁt from the eﬀort for per unit cost reductions, ﬁrm R lowers its input price to increase thequantity supplied by ﬁrm T . This implies that the rent-shifting eﬀect of vertical separation
becomes weak as the market size increases. The negative eﬀect of coordination failure caused
by vertical separation dominates the positive eﬀect of the rent-shifting if the market size islarge. Therefore, ﬁrms R and T vertically integrate if the market size is large. The result
captures the conjecture suggested by Williamson (1985, p. 94): A larger ﬁrm will also be more
likely to integrate if economies of scale in the “upstream” process result in lower costs for thelarge ﬁrm’s own-production compared to a small ﬁrm.
The key feature of this model is that more than one input is required for the ﬁnal productof the downstream monopolist.1 This feature is consistent with the examples mentioned above.
1

This setting is related to models with complementary suppliers (Economides and Salop (1992), Nalebuﬀ(2000), Baldwin and Woodard (2007), Casadesus-Masanell et al. (2007), and Maruyama and Minamikawa(2009)). Those papers discuss how mergers among complementary suppliers appear and/or how those mergers

3The model can be also applied to other industries. For instance, in the aircraft industry
two major ﬁrms, Airbus and Boeing, rely heavily on ﬁrm-speciﬁc inputs (e.g., engines, wings,
horizontal stabilizers) produced by independent manufacturers, and then sell their aircraft toairline companies, which are ﬁnal customers (Beelaerts van Blokland et al. (2008)).
Several researchers have investigated how the structure of vertical organizations is determined in competitive environments (Bonanno and Vickers (1988), Gal-Or (1999), Choi andYi (2000), Chen (2001, 2005), Lin (2006), Arya et al. (2008), Matsushima (2009)). Althoughthese papers consider downstream competition to derive results for vertical separation, we showthat vertical separation is proﬁtable even with only one downstream ﬁrm. Two exceptions are
Laussel (2008) and Matsushima and Mizuno (2009) who explicitly incorporate complementary
inputs in attempts to examine why vertical integration does not occur. Besides several diﬀerences in the setup, the present paper diﬀers from Laussel (2008) and Matsushima and Mizuno(2009) as our focus is primarily on the relation among vertical separation, market size, andthe diﬃculty to reduce operation costs. The last factor is not incorporated into the models in
Laussel (2008) and Matsushima and Mizuno (2009).
In a broad sense, since Coase’s seminal work (1937), researchers have discussed the problem
of vertical integration/separation with a transaction-cost-based approach. The related papers
mainly deal with well-known hold-up problems that illustrate the underinvestment hypothesis(e.g., Grout (1984) and Tirole (1986)). Coase (1937) suggested that transaction costs might be
avoided or reduced via other organizational structures, and Klein et al. (1978) and Williamson(1979) suggested vertical integration as an organizational response. The focus of this approachhas been on comparing costs internal to a transaction, between organizing the transaction
within a ﬁrm or through the market.2 Complementary to the transaction-cost based approach,change equilibrium outcomes. Such complementary suppliers provide their products directly to consumers.
This setting is quite diﬀerent from ours. Note that the meaning of the term ‘vertical integration’ in these
papers is diﬀerent from that in our paper. Although a merger among complementary suppliers is called ‘vertical
integration’ in these papers, in our model the term indicates a merger between an upstream and a downstreamﬁrm.
2

Using the property rights approach to address the question of whether vertical integration can escape thehold-up problem, Grossman and Hart (1986) and Hart and Moore (1990) considered how a particular ownership4

this paper incorporating multiple inputs into the standard models with vertical relations.
The remainder of the paper is organized as follows: Section 2 explains the basic modeland shows the main result. Section 3 extends the basic model. Section 4 provides concluding
remarks.

2

Model

We explain the market structure in this paper.
There are three players related to train service provision to consumers. Firm T (Train
operating company) supplies some units of the train operation service to consumers. Firm R(Rail infrastructure company) supplies an essential input (call it product r), in this case track
related to infrastructure, to ﬁrm T . Group L (Labor union) supplies labor (call it product l)
to ﬁrm T . Consumers enjoy the train travel service. Inverse demand for the service is given as
follows:p = α − βq,(1)where p is the price, q is the quantity supplied by ﬁrm T , and α and β are positive constants.3
Firm T needs one unit of product l and one unit of product r to produce one unit of train
travel service. Firm R and group L individually oﬀer per unit wholesale price wr and per unit
wage of labor wl to ﬁrm T . The following ﬁgure depicts the market structure in this model.
[Figure 1 here]
Firm T generates a cost for ﬁrm R. The cost is positively correlated to the quantity supplied
by ﬁrm T , q. We assume that the cost is Cq where C is a positive constant.

structure aﬀects the parties’ exposure to hold-ups. Che and Sákovics (2008) provided an excellent brief survey ofthe hold-up problem. The topic of vertical foreclosure is also related to the problem of vertical integration. The
vertical foreclosure issue primarily concerns the relation between vertical integration and the competitiveness ofdownstream ﬁrms (e.g., Ordover et al. (1990) and Hart and Tirole (1990)). See also O’Brien and Shaﬀer (1992),
McAfee and Schwartz (1994), Gaudet and Long (1996), Ma (1997), Riordan (1998), and Choi and Yi (2000).
Rey and Tirole (2007) provide an excellent survey of the literature.
3

In Section 3.1, we use a more generalized demand function.

5

Let u brieﬂy mention the cost structure in the railway industry. The role of a train operatingcompany (ﬁrm T ) is to provide service by running trains. Although the train operating company
holds rolling stock and employees such as engineers and conductors, this company borrows rail
tracks from a rail infrastructure company (ﬁrm R), to which it pays infrastructure charges.
The rail infrastructure company holds rail infrastructure, maintaining tracks and electricalsystems. Therefore, while the main costs of the train operating company are for labor, energy,
administration, rolling stock, and infrastructure rental, the main costs of the rail infrastructurecompany are for track maintenance, electrical systems maintenance, and the depreciation of
infrastructure. We believe that this accurately summarizes the cost structure of this industry.
Firm R has an ability to reduce C through its eﬀort. The constant C changes from c to c−e
if ﬁrm R pays its eﬀort e and incurs the eﬀort cost γe2 where c and γ are positive constants.
Assumption 1 To secure that the equilibrium price p is positive, we assume that βγ ≥ 1/3.
Given the quantity supplied by ﬁrm T is q, the consumer surplus is given as follows:
CS ≡

∫ q

(α − βm) dm − (α − βq)q =

0

βq 2
.
2

The social surplus is the sum of the consumer surplus and the total proﬁts in this market.
We investigate the incentive of ﬁrms R and T to integrate vertically. To do so, we consider
two vertical structures: (1) Firms R and T are vertically separated; (2) Firms R and T arevertically integrated.

Vertical separation

When ﬁrms R and T are vertically separated, the objective functionsof the three players are given as follows:
πL = wl q, πT = (p − wl − wr )q, πR = wr q − (c − e)q − γe2 .
In standard oligopoly models with labor unions, each downstream ﬁrm negotiates with its
labor union, which maximizes the product of its wage level and number of employees (forexamples, see Horn and Wolinsky (1988a,1988b), Davidson (1988), Dowrick (1989), Mumford

6and Dowrick (1994), Naylor (2002), Lommerud et al. (2003), and Lommerud et al. (2009)).
The model used here is the standard one concerning the objective function of labor union.
In this case, the game runs as follows:
1. Group L and ﬁrm R oﬀer per unit wage level (wl ) and per unit wholesale price (i.e.,
infrastructure charge) (wr ) to ﬁrm T .
2. Given the wage level and the wholesale price, ﬁrm R sets its eﬀort level e and ﬁrm T sets
the amount of service q simultaneously.
Note that the timing of the game implies that ﬁrm R’s eﬀort does not have a nature ofinvestment (credible commitment). If it has a nature of investment, the timing of the eﬀort
choice should be earlier than that of wage setting. Firm R’s cost generated by ﬁrm T is related
to the operation of ﬁrm T . Firm R has to pay its daily eﬀort to reduce this cost. To capture
the property of the cost-reducing eﬀorts, we have assumed that the eﬀort level e is determined
after the wage and the wholesale price are determined. We discuss how the timing structureaﬀects the decision of vertical integration/separation in Section 3.2.

Vertical integration

When ﬁrms R and T are vertically integrated (call them ﬁrm I), theobjective functions of the two players are given as:
πL = wl q, πI = (p − wl )q − (c − e)q − γe2 ,where I indicates the integrated ﬁrm. πI is the sum of πT and πR in the case of verticalseparation. The game runs as follows:
1. Group L oﬀers per unit wage level (wl ) to I.
2. Given the wage level, I sets its eﬀort level e and the amount of service q simultaneously.

2.1

Vertical separation

We solve the game by backward induction.

7

In the second stage, the proﬁts of ﬁrms T and R are given as:
πT = (α − βq − wl − wr )q,
πR = wr q − (c − e)q − γe2 .
The ﬁrst-order conditions of the ﬁrms lead to the quantity supplied by ﬁrm T and the eﬀort
level of ﬁrm R given the wage level and the wholesale price:
α − wl − wr
∂πT




,


 ∂q = α − 2βq − wl − wr = 0,
 q(wl , wr ) =2β→α − wl − wr


∂π
R


.

 e(wl , wr ) == q − 2γe = 0,4βγ∂e
In the ﬁrst stage, anticipating the outcome in the second stage, group L and ﬁrm R maximize
their objectives:
πL = wl q(wl , wr ) =wl (α − wl − wr ),2βπR = wr q(wl , wr ) − (c − e(wl , wr ))q(wl , wr ) − γ(e(wl , wr ))2=(α − wl − wr )(α − wl + (8βγ − 1)wr − 8βγc).
16β 2 γTheir reaction functions are given by the following functions and summarized as Figure 2.
wl (wr ) =α − wr(4βγ − 1)α + 4βγc − (4βγ − 1)wl
, wr (wl ) =.
28βγ − 1
[Figure 2 here]

When the market size increases (i.e., a decrease in β), the reaction function of R moves downward (see Figure 2). To understand the reason, we can rewrite the proﬁt of ﬁrm R as follows:
πR = (wr − c)q(wl , wr ) + {e(wl , wr )q(wl , wr ) − γ(e(wl , wr ))2 }=8β(wr − c)(α − wl − wr ) (α − wl − wr )2+.
16β 2 γ16β 2 γThe ﬁrst term is the direct net proﬁt through its input. The second term is the ‘gain’ from thereduction of the negative externalities caused by ﬁrm T . This equation shows that an increase in
the market size (i.e., a decrease in β) enhances the relative importance of the gain from the cost8

reduction (the second term). The second term is positively correlated to the quantity supplied
by ﬁrm T . Note that, the marginal gain from the cost reduction is ∂πR /∂e = q(wl , wr ) − 2γe.
Because ∂q(wl , wr )/∂wr < 0, the positive correlation between the second term and q(wl , wr )implies that an increase in the wholesale price set by ﬁrm R, wr , monotonically decreases thesecond term. The second term is more important for the proﬁtability of ﬁrm R as the market
size increases. Therefore, the wholesale price of ﬁrm R monotonically decreases in the market
size (i.e., a decrease in β).
Solving the simultaneous equations (the reaction functions), we havewl∗ =4βγ(α − c)α−c2γ(α − c)(4βγ − 1)α + 8βγc ∗, wr∗ =, e =, q∗ =.
12βγ − 112βγ − 112βγ − 112βγ − 1

The equilibrium proﬁts, the consumer surplus, and the social surplus are given as follows:
8βγ 2 (α − c)2(8βγ − 1)γ(α − c)2
4βγ 2 (α − c)2
∗
∗
,
π
=
,
π
=
,
R
T
(12βγ − 1)2(12βγ − 1)2(12βγ − 1)22βγ 2 (α − c)2(22βγ − 1)γ(α − c)2∗CS ∗ =,
SW=.
(12βγ − 1)2(12βγ − 1)2πL∗ =As a result, the sum of proﬁts πR and πT is as follows:∗
πR+ πT∗ =

2.2

(α − c)2 γ.
12βγ − 1

Vertical integration

We solve the game by backward induction.
In the second stage, the proﬁt of ﬁrm I and its ﬁrst-order conditions are expressed asfollows:
πI = (α − βq − wl − (c − e))q − γe2 ,∂πI= α − 2βq − wl − (c − e) = 0,∂q∂πI
= q − 2γe = 0.
∂e
The ﬁrst-order conditions lead toq(wl ) =2γ(α − c − wl )α − c − wl
, e(wl ) =.
4βγ − 14βγ − 19

In the ﬁrst stage, anticipating the outcome in the second stage, group L maximizes its
objective:
πL = wl q(wl ) =2wl γ(α − c − wl ).
4βγ − 1

The maximization problems lead towl∗∗ =

α − c ∗∗α−cγ(α − c), e =, q ∗∗ =.
22(4βγ − 1)4βγ − 1The equilibrium proﬁts, the consumer surplus, and the social surplus are given as
πL∗∗ =

2.3(14βγ − 3)γ(α − c)2γ(α − c)2γ(α − c)2
βγ 2 (α − c)2
∗∗
,
SW
=
.
, πI∗∗ =, CS ∗∗ =2(4βγ − 1)4(4βγ − 1)2(4βγ − 1)24(4βγ − 1)2Comparison

We ﬁrst compare the several values of cases of vertical separation and vertical integration,
which are derived in the previous subsections. Simple calculus leads to the following relations:wl∗ < wl∗∗ , e∗ < e∗∗ , q ∗ < q ∗∗ , πL∗ < πL∗∗ , SW ∗ < SW ∗∗ .
Vertical separation causes the standard double marginalization problem. As a result, thisreduces the quantity supplied by ﬁrm T (i.e., the consumer surplus) and the social surplus.
Therefore, vertical integration is preferable from the viewpoint of both consumer and social
welfare. However, it does not occur for a larger parameter range as the following equation shows.
When we compare the proﬁt diﬀerences between vertical separation and vertical integration:∗πI∗∗ − (πR+ πT∗ ) =γ(α − c)2(3 − 4βγ)γ(α − c)2γ(α − c)2−=.
4(4βγ − 1)12βγ − 14(4βγ − 1)(12βγ − 1)This result is summarized as the following proposition:
Proposition 1 Under Assumption 1, vertical integration is preferable for ﬁrms T and R if
and only if βγ < 3/4.
Vertical separation induces group L to lower its wage. The quantity supplied by ﬁrm T isq(wl , wr ) which is decreasing in wl and wr . Firm R sets its wholesale price at a positive level(wr > 0), although wr = 0 when ﬁrm R is integrated by ﬁrm T . Because of the higher value
10

of wr in the case of vertical separation, the monopoly power of group L for ﬁrm T is weaker
than that in the case of vertical integration. This means that a portion of the monopoly proﬁtof group L is transferred to ﬁrm R through ‘competition’ between the complementary input
suppliers (Cournot (1838) and Sonnenschein (1968)). The rent-shifting from group L to ﬁrm R
is beneﬁcial from the viewpoint of ﬁrms R and T . However, this eﬀect depends on the market
size. As explained earlier, as the market size becomes larger (i.e., β becomes smaller), thewholesale price of ﬁrm R, wR , becomes lower. This lower wholesale price allows group L to set
a higher wage, wL . This means that the rent-shifting eﬀect is weak if the market size is large.
Therefore, the negative eﬀect of coordination failure caused by vertical separation dominates
the positive eﬀect of the rent-shifting if the market size is large (i.e., β is small).

3

Extensions

We extend the basic model to two directions. First, we generalize the demand function. Second,
we change the timing structure of the basic models. We consider models in which ﬁrm R engages
in cost-reducing activities in the ﬁrst stage, although in the previous section, it carries out those
activities in the ﬁnal stage.

3.1

General function

First, we discuss the level of robustness in the outcome derived in the previous subsections.
We assume that the inverse demand function is given as P (Q/s) where Q is the quantity
supplied and s represents the market size. The proﬁts of ﬁrms T and R are given as follows:
πT = (P (Q/s) − wR − wL )Q, πR = wR Q − (c − e)Q − I(e).
The ﬁrst-order conditions are given as∂πT= P (Q/s) − wR − wL + P ′ (Q/s)Q/s = 0,∂Q
∂πR
= Q − I ′ (e) = 0.
∂e
We arrange ∂πT /∂Q as follows:∂πT= P (Q̃) − wR − wL + P ′ (Q̃)Q̃ = 0,∂Q11

where Q̃ ≡ Q/s. The value of Q̃ is derived by the above equation. This does not depend on
the value of s but depend only on the sum of wR and wL . We can write the optimal quantity
supplied by ﬁrm T as follows:
Q = sQ̃(wL , wR ).
Using the equation ∂πT /∂Q = 0, we show how an increase of wj changes Q(wL , wR ) (j = L, R).
Given the values of s and wi , the simple comparative statics of ∂πT /∂Q = 0 with respect to Q
and wj (i = L, R and i ̸= j) leads todQ(wL , wR )
1
=
< 0,
dwj
SOCT
where SOCT represents the second-order condition of πT .
The objectives of group L and ﬁrm R are given as
πL = swL Q̃(wL , wR ),πR = swR Q̃(wL , wR ) − (c − e(sQ̃(wL , wR )))sQ̃(wL , wR ) − I(e(sQ̃(wL , wR ))).
The ﬁrst-order conditions are given as[]∂πL∂ Q̃(wL , wR )= s Q̃(wL , wR ) + wL
= 0,
∂wL
∂wL
[
]
∂πR
∂ Q̃(wL , wR )= s Q̃(wL , wR ) + (wR − c + e(sQ̃(wL , wR )))
= 0.
∂wR
∂wR
First, we ﬁnd that wL (wR ) derived by the equation ∂πL /∂wL = 0 does not depend on thevalue of s. Second, we show that the value of ∂πR /∂wR between the brackets monotonicallydecreases in s. Diﬀerentiating it with respect to s, we obtain:()∂ ∂πR 1∂ Q̃(wL , wR )= e′ (sQ̃(wL , wR ))Q̃(wL , wR )·< 0.
∂s ∂wR s∂wRThis is because e′ (sQ̃(wL , wR ))) is an increasing function (see ∂πR /∂e) and ∂ Q̃(wL , wR )/∂wR
is negative. The two properties are consistent with Figure 2.

3.2

Does the timing of cost-reducing investments matter?

We consider the models in which ﬁrm R engages in cost-reducing activities in the ﬁrst stage
although it does in those activities in the ﬁnal production stage in the previous section. Except
12

the timing structures, the structures of the games are the same with those in the previous
section. We explain the timing of the games.
Assumption 2 To secure that the equilibrium price p is positive, we assume that βγ ≥ 1/12.
In the case of vertical separation, the game runs as follows:
1. Firm R sets its eﬀort level e.
2. Observing the eﬀort level, group L and ﬁrm R oﬀer per unit wage level (wl ) and per unit
wholesale price (wr ) to ﬁrm T .
3. Given the eﬀort level, the wage level, and the wholesale price, ﬁrm T sets the amount ofservice q.
Note that the timing of the game implies that ﬁrm R’s eﬀort has a nature of investment(credible commitment).
In the case of vertical integration, the game runs as follows:
1. Firm I sets its eﬀort level e.
2. Observing the eﬀort level, group L oﬀers per unit wage level (wl ) to ﬁrm I.
3. Given the eﬀort and the wage levels, ﬁrm I sets the amount of service q.
The calculus of the two cases lead to the following proposition (the calculus is summarized
in the Appendix):
Proposition 2 Under Assumption 2, vertical separation is preferable for ﬁrms T and R for
any βγ.
Under vertical separation, ﬁrm R has full bargaining power over ﬁrm T . This means thatﬁrm R internalizes its eﬀort to reduce its marginal cost.4 Vertical integration does not greatly
inﬂuence coordination problems between ﬁrms R and T . Reducing the rent shifting from ﬁrm
4Note that ﬁrm R cannot fully internalize its eﬀort. This is because group L also has full bargaining powerover ﬁrm T . Group L deprives of some rents generated by ﬁrm R through its wage setting.

13

T to group L is more important from the viewpoint of the ﬁrms. Therefore, vertical separationis always preferable.
This result is quite diﬀerent from that in the previous section. In this section, the eﬀort
cost of ﬁrm R is sunk before the wage and the wholesale price are determined. In the previous
section, however, cost is incurred when ﬁrm T sets its quantity supplied after the procurement
conditions are determined. This implies that a coordination problem exists between ﬁrms R
and T . The discussion in this section clariﬁes that, in the vertical structure discussed here, thekey factor of vertical integration is not the sunk-investment but the problem of coordination.

4

Concluding remarks

Vertically integrated and separated ﬁrms coexist in many industries, including railway. We haveprovided an analytic framework to investigate this problem. We show that the downstreamﬁrm which has the larger market size is more likely to integrate with the rail infrastructure ﬁrm.
This is consistent with the phenomenon in the Japanese railway industry. The result captures
the conjecture suggested by Williamson (1985, p. 94): A larger ﬁrm will also be more likely
to integrate if economies of scale in the “upstream” process result in lower costs for the large
ﬁrm’s own-production compared to a small ﬁrm. Because many empirical studies infer thatthe hypothesis of transaction cost economics holds (Lafontaine and Slade (2007)), we believethat our prediction can be applied to many economic environments.
As Proposition 1 states, if market size increases, vertical integration is preferable, as shownin current empirical ﬁndings of Mizutani and Uranishi (2011) regarding the rail industry. By
applying the total cost function to 30 railway organizations from 23 OECD countries for the 14
years between 1994 and 2007, Mizutani and Uranishi (2011) evaluate whether or not verticalseparation caused a reduction in costs. The authors show that vertical separation eﬀects withlower train density tend to reduce the total cost of a railway organization but as train density
increases, vertical separation causes total costs to increase. This result means that vertical
integration is preferable in the case of high train density while vertical separation is preferable
in the opposite case. In their empirical analysis, train density is used but the measure (i.e.

14train density) is highly related to market size. The deﬁnition of train density is measured by
how many trains run in a given railway network. If the market size is larger, then more train
services are required in the market. Therefore, if rail organizations have a large market, itmeans that the rail organizations have larger train density. Thus, current empirical results
support our theoretical implication.
As mentioned above, while vertical separation is a common policy in the European Union,
vertical integration is still the structure of choice in the Japanese rail industry. Among verticalseparation options, there are many variations. Within the largely vertically separated European rail industry, opinions have been voiced against the separation policy. Keeping in mind
these various circumstances, we will provide theoretical background that may prove helpful inevaluating vertical separation policy in the rail industry.
In conclusion, our theoretical results show that policy related to vertical separation depends
on market size, with vertical integration appropriate in rail organizations with a large market
and vertical separation preferable in organizations with a small market. Based on our theoretical results, the European Commission’s policy, which is that vertical separation policy should
be applied everywhere, is not correct.
There are several ways to extend the monopoly model discussed here. Considering anoligopoly model is a natural extension of the current model. For instance, we can consider
a situation in which two rail operation companies use the rails owned by two infrastructureﬁrms. In Japan, some vertically integrated railway companies use their own rails cooperatively
by operating their own trains from a terminal station owned by one of the companies to anotherterminal station owned by the other company. We think that an extended model can capture
the essence of the Japanese and the EU railway industries and should be considered for future
research.

15

Appendix
We solve the two cases discussed in Section 3.2.
Vertical separation

We solve the game by backward induction.

In the third stage, the proﬁt of ﬁrm T is given as:
πT = (α − βq − wl − wr )q.
The ﬁrst-order condition of the ﬁrm leads to the quantity supplied by ﬁrm T given the eﬀort
and the wage levels and the wholesale price:∂πT
α − wl − wr
= α − 2βq − wl − wr = 0 → q(wl , wr ) =.
∂q
2β
In the second stage, anticipating the outcome in the third stage, group L and ﬁrm R
maximize their objectives:
πL = wl q(wl , wr ) =wl (α − wl − wr ),2βπR = (wr − (c − e))q(wl , wr ) − γe2 =(wr − (c − e))(α − wl − wr )− γe2 .
2β

Their reaction functions are given by the following functions.
wl (wr ) =α − wrα + c − e − wl
, wr (wl ) =.
2
2

Solving the simultaneous equations (the reaction functions), we havewl (e) =α−c+e
α + 2(c − e)α−c+e
, wr (e) =, q(e) =.
33
6β

In the ﬁrst stage, anticipating the outcome in the second and the third stages, ﬁrm R
maximizes its objective:
πR = (wr (e) − (c − e))q(e) − γe2 =(α − c + e)2− γe2 .
18β

The ﬁrst-order condition leads to
e∗ =

α−c
6βγ(α − c)(6βγ − 1)α + 12βγc, wl∗ =
, wr∗ =.
18βγ − 118βγ − 118βγ − 116

The equilibrium proﬁts, the consumer surplus, and the social surplus are given as18βγ 2 (α − c)2γ(α − c)2
9βγ 2 (α − c)2
∗
∗
,
π
=
,
π
=
,
R
T
(18βγ − 1)218βγ − 1(18βγ − 1)2
9βγ 2 (α − c)2γ(99βγ − 2)(α − c)2CS ∗ =, SW ∗ =.
22(18βγ − 1)2(18βγ − 1)2πL∗ =The sum of proﬁts πR and πT is∗
πR+ πT∗ =

Vertical integration(27βγ − 1)γ(α − c)2.
(18βγ − 1)2

We solve the game by backward induction.

In the third stage, the proﬁt of ﬁrm I and its ﬁrst-order condition are
πI = (α − βq − wl − (c − e))q − γe2 ,∂πI= α − 2βq − wl − (c − e) = 0.
∂q

The ﬁrst-order conditions lead toq(wl , e) =α − c + e − wl.
2β

In the second stage, anticipating the outcome in the third stage, group L maximizes its
objective:
πL = wl q(wl , e) =wl (α − c + e − wl ).
2β

The maximization problems lead towl (e) =

α−c+e
α−c+e
, q(e) =.
2
4β

In the ﬁrst stage, the proﬁt of ﬁrm I and its ﬁrst-order condition are
πI = (α − βq(e) − wl 8e) − (c − e))q(e) − γe2 =(α − c + e)2− γe2 ,
16β

∂πI
α−c+e
=
− 2γe = 0.
∂e
8βThe ﬁrst-order conditions lead to
e∗∗ =

α−c
8βγ(α − c), wl∗∗ =.
16βγ − 116βγ − 1

17

The equilibrium proﬁts, the consumer surplus, and the social surplus are given as32βγ 2 (α − c)2γ(α − c)2
∗∗
,
π
=
,
I
(16βγ − 1)216βγ − 18βγ 2 (α − c)2γ(56βγ − 1)(α − c)2CS ∗∗ =, SW ∗∗ =.
2(16βγ − 1)(16βγ − 1)2

πL∗∗ =

We ﬁrst compare the several values derived in the previous subsections. Simple calculus
lead to the following relations:wl∗ < wl∗∗ , e∗ < e∗∗ , q ∗ < q ∗∗ , πL∗ < πL∗∗ , SW ∗ < SW ∗∗ .
The mechanism behind those inequalities is similar to that in the previous section.

Comparison

The diﬀerence between the proﬁts in the two cases is∗
πI∗∗ − (πR+ πT∗ ) =γ(α − c)2 (27βγ − 1)γ(α − c)2(7 − 108βγ)βγ 2 (α − c)2−< 0.
=16βγ − 1(18βγ − 1)2(18βγ − 1)2 (16βγ − 1)This result is summarized in Proposition 2 in the main text.

18

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21

Labor union

Rail

wl

wr
Train
q

Consumers
Figure 1: The market structure

22

wr

wl (wr )wr (wl )wl0
Figure 2: The reaction functions of L and R.

23