2002. 24
Privately Owned Railways' Cost Function,
Organization Size and Ownership

Fumitoshi Mizutani

Privately Owned Railways’ Cost Function,
Organization Size and Ownership*by

Fumitoshi MIZUTANI
Graduate School of Business Administration
Kobe University
2-1 Rokkodai, Nada-ku
Kobe 657-8501
JAPAN(Tel): +81-78-803-6905(Fax): +81-78-881-8100(E-mail):toshi@rokkodai.kobe-u.ac.jp[Abstract]
This paper aims to find the optimal size of an urban private rail organization according
to its supply size, as well as to evaluate cost difference by ownership, "optimal size"
being defined as the size which attains minimum average cost. When restructuring an
overly large monopolized railway organization into discrete parts, policy makers need toknow how big those parts should be. The methodology of this analysis is as follows.
First, privately owned rail companies are carefully selected, according to the type of rail
service they offer and the technology they use. Explanatory variables which affect thecost of rail service are explored. Second, keeping in mind previous cost studies of theurban passenger rail industry, we estimate both the variable and the total cost function
with the translog cost function and compare these results. Third, based on the average
cost function, conditions are pinpointed which attain minimum average cost. Finally,
based on estimated results, we calculate the size of an urban private rail company andthe ownership effects on cost. We conclude that optimal size is about 194 million
vehicle-km per year, with a network of 85km length. In terms of total costs, public
railways have higher costs than private railways. But there is no cost difference invariable costs.
[Key Words]
Optimal size, Privately owned railways, Translog cost function, Japanese railway companies[JEL Classification]*

L3, L9, R4

Portions of this research were supported by the Science Research Fund of the Ministry of Education.

11 INTRODUCTION
An important policy issue in the rail industry these days is the privatization of formerly
government-owned railway organizations.

Publicly owned railways have been widely perceived to beinefficient, and the movement toward privatization has met with little resistance, with citizens seeming torecognize the debilitating lack of competition in the industry and the absence of incentives to improve.
However, despite an apparent consensus that privatization should be carried out, to do so is not simple.
Hard decisions must be made as to what sorts of private railway companies should be formed from theold ones.

How should overly large railway organizations be divided?

organizations be?
encouraged?

How big should the resulting

How should they be managed for maximum efficiency?

How can competition be

These questions faced policy makers responsible for privatizing and dividing the Japan

National Railway (JNR) in 1987.

In the case of Japan, the huge former national railway was divided

into six regional passenger companies.

Other countries have chosen different ways of dividing theirunwieldy national railways, with British Railways, for example, being divided into 25 franchisingregions.

What is the correct way to subdivide a national railway?

Policy makers facing similar

choices in the future would benefit from knowing the optimal size of a railway, and that question is whatmost concerns us in this paper.

Preston (1996) concluded that the railway of optimal size, or the size

which results in the lowest operating costs, would have a network of around 4,000km and run 120
million train-km per annum.

Preston’s pioneering results might prove a benchmark for policy makers,

but his research is based on samples only of European railways.

A larger sample would help us obtain

more definitive results.
First, perhaps we need to show whether the private sector is really more efficient than the public.
Different studies give different answers.

While some empirical evidence suggests that in the railway

industry, the private sector is more efficient than the public (Miyajima and Lee, 1984, and Mizutani,1994), other research indicates there is little difference between the two sectors (Caves and Christensen,1980).

There is even a study, conducted by the U.S. Accounting Office in 1981, suggesting that in atleast one measure the public sector is the more efficient (Tittenbrun, 1995).

In sum, regarding the rail

industry, three studies show that private is more efficient, two studies show that there is not muchdifference, and one study shows that public is more efficient than private (Mizutani, 2000).

Answeringthe question of which is more efficient is complicated in the case of the Japanese railway privatization,
because the railways resulting from the breakup were not fully privatized at the beginning of theprivatization process, and in fact remain only partially privatized to this day.

Partial privatization has

not been unique to Japan, however, and we may look at the case of the privatization of the German
railway (DB) as well, where railway stock is still held by the German government.

To our growing listof questions, we need to add the question of how efficiency might differ among partially and fully
privatized organizations.

2The purpose of this paper is to address some of these questions.

The main goal will be todetermine the optimal size of a railway operation and to find any cost differences related to ownership.
The structure of the paper is as follows after the introduction.

In the first section, I give an overview ofthe private rail industry in Japan, some background to aid in understanding the rail market and thesample selection for the analysis found in the following section.
passenger transport and the organization of the rail industry.
the cost models will be explained.
here are presented.

I also explain the mode share of

In the second section, the methodology for

After a summary of previous cost studies, the cost models to be used

In this section, I address both total and variable cost function.

estimate cost functions.

Third, I will

After the assessment of cost functions, I will construct the long-run total costfunction from variable cost functions.

Fourth, by using these results, I also explain the methods for

obtaining minimum average costs and cost differences by ownership.

I calculate the optimalorganization size for attaining minimum average costs and analyze cost differences by ownership.
Finally, I summarize my results in a conclusion.

2 AN OVERVIEW OF THE PRIVATE RAIL INDUSTRY IN JAPAN
2.1 Mode Share of Passenger Transport
Although auto transport has increased dramatically since World War II, rail transport still plays
a very large role in transportation in Japan.

Especially when compared with the situation in otherindustrial countries, such as the U.K., Germany, and France, where rail share has declined to just a few
percent, rail share accounted for 27.3% of Japanese transportation in 1998.

For commuters in large

metropolitan areas such as Tokyo and Osaka, rail was the dominant transportation mode.

Table 1shows the trends in passenger transport and its mode share in Japan.
Table 1 Trends in Passenger Transport and Mode Share
Year
Passenger kilometer (million)
Rail
Auto
Air
Maritime
1950
105,468
9,000
2,628
1955
136,11227,500
225
1,996
1960
184,340
55,531
737
2,670
1965
255,484
120,756
2,952
3,402
1970
288,815
284,229
9,319
4,814
1975
323,800
360,868
19,1486,895
1980
314,542431,669
29,6886,132
1985
330,101
489,260
33,1195,752
1990
385,364
853,060
51,6236,275
1995
400,084
917,419
65,012
5,527
1996
402,156
931,72169,0495,635
1997395,278
944,972
73,243
5,368
1998
388,917954,807
75,9884,620(Source): Ministry of Transport (ed.) (1999), p.11

3

Rail
90.1
82.1
75.8
66.8
49.2
45.6
40.2
38.5
29.8
28.8
28.6
27.9
27.3

Mode Share (%)
Auto
Air
7.7
0.0
16.6
0.1
22.8
0.3
31.6
0.8
48.4
1.6
50.8
2.7
55.2
3.8
57.0
3.9
65.7
4.0
66.1
4.7
66.1
4.9
66.6
5.2
67.1
5.3

Maritime
2.2
1.2
1.1
0.9
0.8
1.0
0.8
0.7
0.5
0.4
0.4
0.4
0.3

There are several reasons why rail transport is still so important in Japan.

The most obvious isthat urban areas are so densely populated that rail operators can provide rail service efficiently and railusers can consume services more frequently.

Because rail organizations in Japan are expected to be

self-supporting through rail fare revenues and independent of government subsidy, this dense urban
structure is important to the railways’ continued well-being.

Second, it could be argued that rail shareremains high because private railways have always done their work very well, providing what is
popularly regarded as high quality service.

In fact, the example of the successful private railway

companies was the trigger for the privatization of the Japan National Railway (JNR) in 1987.

These

companies have not only managed their rail businesses well, but have ensured their survival, indeed theirprosperity, by diversifying their businesses--operating feeder bus services, developing real estate such as
housing and office complexes along their lines, building department stores at terminals, and opening
their own travel agencies (Killeen and Shoji, 1997). These activities have the demonstrable effect ofincreasing rail ridership and the more abstract one of enhancing and propagating the company name.
Despite the success of the private rail industry, however, rail share is still less than that of the auto, whichis the dominant transportation mode in Japan as it is in other industrialized countries.

Outside theTokyo and Osaka metropolitan areas, rail companies in Japan struggle to maintain their dwindling shareof the transportation market.

2.2 The Organization of the Rail Industry
The rail industry in Japan can be distinguished by the existence of a multitude of rail operators,
most of which are privately owned.

According to Mizutani (1999a), as of September 1, 1996, there

were 188 organizations defined as rail operators.

Of these, 160 were heavy and light rail operators, andthe others were monorails, automated guideway transit systems, and cable cars.
Table 2 Classification of the Passenger Rail Industry in Japan
Operators Number of
Legal
Ownership
Operators
Classification
Large
15
PrivatePrivateprivateCorporation

Service Type

Main Service Areas

Urban

Large metropolitan
areas (e.g. Tokyo,
Osaka, etc.)
Large metropolitan
areas (e.g. Tokyo,
Osaka, etc.)
From large to small
urban areas
From large to small
urban areas
6 regions in Japan

Mediumprivate6

PrivateCorporation

Private

Urban

Small
privateQuasiprivate
JR

67

PrivateUrban

Private-Public

Urban

Eidan

1

Partly public
or Public
PublicInter-city and
Urban
Urban

Public13

PrivateCorporation
PrivateCorporation
SpecialCorporation
SpecialCorporation
PublicPublic

Urban

36
6

4

Tokyo
Tokyo, Osaka, Nagoya

Organization
and 6 other large cities(Note): This table was modified by the author based on the table in Mizutani (1999a, p.265).

Of the 160 heavy and light rail operators, passenger rail services are most numerous, at 144.
Freight rail operators accounted for only 16 of the total. In Japan, passenger rail operators are classified
in four ways: according to their legal classification, their ownership, their transport type, and their main
service areas. First, there are three legal categories: private corporations, public organizations, andspecial corporations.

Private corporations are organizations legally considered to be private companies,

whose shares are held by the private sector.

It is worth noting that companies whose shares are partly

held by the public sector are included in this category.

A public organization is a department of thegovernment such as the department of transportation of a local government.

A special corporation isdefined as an organization which is set up and regulated by special law (Uekusa, 1991).
are considered special
and private-public.

corporations.

JR and Eidan

Second, there are three categories of ownership: private, public,

Most Japanese rail operators are privately owned, with public ownership being

limited to only 13 operators.

The recently “privatized” JRs are not fully private because most of theirshares are still held by the public sector (Mizutani and Nakamura, 1997; Mizutani, 1999b).

As forclassification according to transportation type, the most common type is the urban railway serving a large
metropolitan area.

The major 15 private railways, those that have been private since they were

established long ago, fall into this group, and contrast with the newly private companies, the JRs, whichhave both urban and intercity rail services.
Trends in passenger transport according to type of rail organization are summarized in Table 3,
which shows that the overall volume of passenger transport on the JRs is greatest, accounting for 62.4%
in 1998.

However, private railways’ share is still significantly high, accounting for 26.7% in 1998.

Their good performance might be a result of their entrepreneurial behavior.[1]In contrast to the JRs,

which provide both intercity and rail services, large private railways continue to provide highly essential
service in large metropolitan areas.
Table 3 Trends in Passenger Transport and Share by Type of Rail Organization
Year
Passenger kilometer (billions)Share by Organization (%)
Large
Other
JR
Eidan Public
Large
Other
JR
Eidan
Private Private
Private Private
1965
60.4
7.6
174.0
4.5
8.8
23.6
3.0
68.1
1.8
1970
75.8
8.4
189.7
8.5
6.1
26.2
2.9
65.7
2.9
1975
82.6
8.2
215.3
10.5
7.3
25.5
2.5
66.5
3.2
1980
91.6
8.8
193.1
11.7
9.3
29.1
2.8
61.4
3.7
1985
97.5
9.3
197.5
14.0
11.8
29.5
2.8
59.8
4.2
1990
110.5
8.1
235.5
16.0
15.3
28.7
2.1
61.1
4.2
1991
112.7
8.4
247.0
16.1
15.8
28.2
2.1
61.8
4.0
1992
112.2
8.6
249.6
16.1
15.9
27.9
2.1
62.0
4.0
1993
111.8
8.6
250.0
16.1
16.3
27.8
2.1
62.1
4.0
1994
111.0
8.7
244.4
15.9
16.4
28.0
2.2
61.7
4.0

5

Public
3.4
2.1
2.3
3.0
3.6
4.0
4.0
4.0
4.0
4.1

1995
109.9
8.7
249.0
15.8
16.7
27.5
2.2
62.2
3.9
4.2
1996
108.7
9.2
251.7
15.9
16.7
27.0
2.3
62.6
4.0
4.2
1997
105.5
9.5
247.7
15.9
16.7
26.7
2.4
62.7
4.0
4.2
1998
103.8
9.4
242.8
16.0
16.9
26.7
2.4
62.4
4.1
4.3
(Note): Monorails, automated guideway transit systems, and cable cars are included in the category of
Other Private, but their numbers are small.
(Source): Ministry of Transport (ed.) (1999), p.13

3 Cost Models
3.1 Previous Cost Studies Regarding the Passenger Rail Industry
Over forty studies have been undertaken, with various goals.

Published papers include

estimations of cost function or overviews of the industry (Borts, 1960; Meyer et. al.,1975; Brown et.al.,
1979); a study of cost allocation (Griliches, 1972); examinations of economies of scale and density(Harris, 1977; Keeler, 1974; Braeutigam et al., 1984; Caves et al., 1981a, 1985; Preston and Nash, 1993;
Savage, 1997); an examination of economies of scope (Kim, 1987); and a study of capital adjustment(Friedlaender et.al, 1993).

In the 1980s and 1990s, there were several studies related to regulation

issues: regulation effects (Caves et. al, 1981b; Friedlaender and Spady, 1981; Vellturo et.al, 1992); an
estimation of mergers’ effect (Levin et.al, 1979; Harris and Winston, 1983; Berndt et.al, 1993).

It hasalso been popular to measure productivity growth (Caves et.al, 1980; Dodgson, 1993; McGeehan, 1993;
Loizides and Giahalis, 1995; Hensher et. al, 1995).

Some studies concern methodology and model

specifications (Hasenkamp, 1976; Braeutigam et.al, 1982; De Borger, 1991, 1992)
Regardless of the purpose of these studies, the functional forms of the cost function used aremostly the log-linear and translog cost functions, the latter being widely used in recent studies.

The

disadvantage of this function is that, because it includes both first-order and second-order variables, alarge number of observations are required before it can be useful for assessing policy change.

However,

the translog cost function is considered a more flexible and therefore superior functional form because
the scale measure varies with output level, the elasticity of substitution is not one, and the function is not
homothetic.

Therefore, I judge this form to be appropriate for the purpose of my study here, which isthe calculation of the optimal size of a railway.
Previous cost studies cover mostly freight railways or both passenger and freight railways, andstudies on passenger railways are very limited.

Analysis of passenger transport as discrete from freighttransport is necessary, as current state railway reforms tend to separate their passenger divisions fromtheir freight operations (e.g. British Railway and JNR).

Several researchers have analyzed passenger

transport, such as Viton (1980), Marumo (1984), Miyajima and Lee (1984), Mizutani (1994), Nakamura(1994), and Savage (1997).

For example, Viton (1980), employing the translog cost function, evaluates

economies of network density in North American rail transit systems.

6

More recently, Savage (1997)

analyzes scale economies in U.S. urban railways, taking into account service quality variables considered
important factors affecting the cost structure of urban transit firms.
There are not many studies about ownership effects or costs of private railways.
(1984) study is an analysis of urban private railways in Japan.

Marumo’s

He analyzes the relationship between

average cost and several explanatory variables by using log-linear cost models.

Miyajima and Lee’s(1984) study is an efficiency comparison between the private and public sectors.

Although the authors

do not construct a cost function, they compare several performance measures, including operating costs,
and they conclude that private railways are more efficient than public railways.
More recent cost studies on private railways have been carried out by Filippini and Maggi (1992,
1993), Mizutani (1994), and Nakamura (1994).

It is worth noting that observations used by Filippini

and Maggi are not for passenger transport only but for both passenger and freight transport.

The mainfocus of Filippini and Maggi’s (1992, 1993) study is to evaluate economies of network density, scale andnetwork structure in Swiss railways and the factors’ effect on efficiency.

What distinguishes their study

from other cost studies in the rail industry is that they analyze “privately” owned rail operators.

The

methodology itself is not different from other studies, but they employ a translog cost model and estimate
both total and variable cost functions, and in this way detect the presence of economies of network
density and scale.

However, the “privately” owned railways in these studies cannot be considered truly

private railways, being owned partly by the federal state, the cantons, and the communes.

In fact,

private shareholders of these companies account for only 1 to 52% of all shareholders.

Also

questionable is their finding that ownership does not significantly correlate with cost efficiency.
On the other hand, Mizutani (1994) and Nakamura (1994) have studied railways which areclearly privately owned.

The main purpose of Mizutani’s study is to evaluate the cost difference

between privately and publicly owned railways.

After estimating the translog and the log-linear costfunctions, he evaluates the private-public cost difference while controlling output and network factors,
and concludes that public railways have about 10 to 20% higher total costs than private railways.
Nakamura’s study aims to examine economies of scale and network density.

He constructs a translog

cost function and concludes that there are both economies of scale and network density.

While

Mizutani’s study has provided empirical evidence that the private sector is more efficient than the public,
there remains the problem that the effect due to technological difference in the cost function has not been
removed.

As for Nakamura’s study, there is the worry that results may have been affected by the factthat some private railways receive subsidies, which may affect their behavior.

3.2 Economic Theory of Rail System
Long-run and Short-run Cost Function
Based on previous studies, I will summarize the basic idea of this study.

I will begin byexplaining the general microeconomic theory on cost, largely following the theoretical background of7

five studies: Friedlaender and Spady (1981); Braeutigam, et. al (1984); Caves, et. al (1985); Friedlaender,
et. al (1993); and Savage (1996).

This study aims to find the optimal size of a railway organization, thesize which attains minimum average cost in terms of output and network size.
If a firm minimizes all kinds of input factors under a given output level (Q) and given input
factor prices (w), then the long-run total cost function is expressed as follows:
C = C (Q, w),
where

(1)

C: long-run total cost
Q: output
w: vector of input factor prices.

However, if there are certain fixed input factors, the firm cannot minimize total cost by adjusting such
fixed input factors in the short-run.

In this case, the firm is considered to minimize the variable costs

under the condition of having the fixed input factors.

In this case, the short-run total cost is expressed

as follows:
CS = CV (Q, wv, K) + wf K ,
where

(2)

S

C : short-run total cost
CV: variable cost
wv: vector of input factor prices of variable input
wf: vector of input factor prices of fixed input
K: vector of fixed input.

Long-run total costs are always less than short-run total costs, except at that output level for which theassumed fixed input is appropriate to long-run cost minimization.

However, at the long-run equilibrium,

short-run marginal costs with respect to output must equal long-run marginal costs (∂CV / ∂Q = ∂C/ ∂Q)(Friedlander and Spady, 1981).

Furthermore, at the optimal choice of the fixed input factor, Ki , thefollowing condition must hold:∂CS / ∂Ki = ∂CV (Q, wv, K) /∂Ki + wfi = 0.

(3)

To solve the equation (3) for Ki, we can obtain the optimal size of the fixed input factor:
Ki*= K (Q, wv, wfi ).

(4)

By substituting equation (4) into equation (2), we can obtain the transformed log-run total cost function:
C = CV (Q, wv, K*(Q, wv, wfi )) + wf K*(Q, wv, wfi ).

(5)

An Approach to Constructing Cost Function
The research methodology of previous cost studies on the rail industry could be divided into
three groups.

One category of methods uses direct estimation of the long-run total cost function (I willcall it the “total-cost approach”).

Caves et. al use this approach in their 1985 study, which assumes thata firm can quickly adjust fixed input to optimal size.

Their approach is aparently correct because theestimation results are very similar to the indirect estimation based on variable cost function.

8

However,

in the case where the estimation result of the long-run marginal costs is not the same as for the short-run
marginal costs, then this approach is not appropriate.
The second category uses the variable cost function.

This approach is assuming that there is

disequilibrium with respect to the stock of way and structure and that a variable cost function is more
appropriate.

This approach is divided into two further categories by a difference in the recognition of

network factors.

The first approach (the “variable cost approach”) considers as fixed input network

variables such as route length.

This approach adheres more closely to economic theory, and typicalstudies using this approach are Savage (1996) and Braeutigam, et. al (1984).
On the other hand, there are studies which consider network variables to be different from thefixed input factor (the “ variable-cost with network approach”).
Friedlaender and Spady (1981), and Friedlaender, et. al (1993).

Studies of this type include

Friedlaender and Spady (1981) argue

that network factors such as route length are different from the fixed input factor because the network
variables show a tendency different from the property satisfied by fixed input factor.

Variables Used for Cost Models in the Rail Industry
Using previous cost studies on rail industries, I will obtain the long-run total cost function.

Iwill approach the problem using the three methods outlined above: the total-cost approach, thevariable-cost approach, and the variable-cost with network approach.

The basic structure of the costmodels used here is a function of output measure (Q), vector of input factor prices (w), and vector of
network factor (N)[2].

In this study, I use pooled data so that the technology index variable (T) is

included in the cost function.

I assume that the technology is identical among rail organizations for agiven year but varies from year to year.[3]This cost function is specified as a single output cost functionof passenger service, and the components of input factor are of four kinds: labor, energy, material, andcapital costs.

And as for network variables, three kinds of network variables such as line length, station

spacing and number of lines are included.
However, rail service as output could in actuality differ among railway companies because
output components might differ.

Furthermore, the service output provision of each railway could vary

based on demand conditions because the railway companies in this study are all privately owned and they
could use output strategically rather than exogenously.

In fact, Savage (1997) specifies many kinds of

output characteristics variables such as average journey length, peak-base ratio, and load factor.
variables could indeed be important in urban railway cost study estimations.

These

Therefore, in this study, I

include the conditions of service output provision as a function of passenger demand, as well as qualityof service.

The total cost function, variable cost function, and service output provision function are

expressed as follows:
C = C (Q, w; T, N)(6)

CV = CV (Q, wv, K; T, N)(7)9Q = g (Qd, Z),
where

(8)

C: long-run total costs
CV: short-run variable costs
Q: rail service output measure
Qd: rail service demand measure
w: vector of input factor pricewv: vector of input factor prices of variable input
K: vector of fixed input
T: technology
N: vector of network factor
Z: vector of service quality condition.

3.3 Econometric Cost Models
The functional form of the cost function is specified as translog cost model as follows. I specify
three kinds of cost functions because I have applied three separate approaches.
The first model is a variable cost model (VC1), which specifies the variable cost function.
this approach, I assume that the network length is the fixed input defined in economic theory.
other hand, the second model is a variable cost with network model (VC2).

On theIn this model, the network

variables are different from fixed input as Friedlaender and Spady (1981) argue.
model, the total assets of rail facilities are considered the fixed input.

In

Therefore, in thisThe long-run total cost functions(TC1and TC2) are obtained from these variable cost functions as the transformed total cost function.
will explain the method later.

Finally, the last model (TC3) is the long-run total cost model.

model includes the network variables included by Caves et al (1985).
(VC1 model):This

The models are as follows:ln CV = α0 + δQ lnQ + Σi βi lnwi + Σmγm lnNm + 1/2 δQQ (lnQ)2 +Σi δQi (lnQ)(lnwi) + Σm δQm (lnQ)(lnNm) + 1/2 ΣjΣi βij (lnwi) (lnwj) +ΣmΣi βim (lnwi)(lnNm) + 1/2 ΣnΣm γmn (lnNm)(lnNn) +γU lnNU + τ lnT,

(9)

(VC2 model): ln C = α0 + δQ lnQ + Σi βi lnwi + γK lnK + Σmγm lnNm + 1/2 δQQ (lnQ)2 +V

Σi δQi (lnQ)(lnwi) + δQK (lnQ)(lnK) + Σm δQm (lnQ)(lnNm) +1/2 ΣjΣi βij (lnwi) (lnwj) + Σi βiK (lnwi)(lnK) + ΣmΣi βim (lnwi)(lnNm) +1/2 γKK (lnK)2+ Σm γKm (lnK)(lnNm) + 1/2 ΣnΣm γmn (lnNm)(lnNn) +γU lnNU + τ lnT,
(TC3 model):(10)

lnC = α0 + δQ lnQ + Σi βi lnwi + Σmγm lnNm + 1/2 δQQ (lnQ)2 +Σi δQi (lnQ)(lnwi) + Σm δQm (lnQ)(lnNm) + 1/2 ΣjΣi βij (lnwi) (lnwj) +ΣmΣi βim (lnwi)(lnNm) + 1/2 ΣnΣm γmn (lnNm)(lnNn) +10IγU lnNU + τ lnT,
where

(11)

C: total costs
CV : variable costs
Q: rail service output measure (vehicle-km)
wi: input factor price (i (or j) = L (labor), E (energy), M (material and repair) for
variable costs) (i (or j)= L (labor), E (energy), M (material and repair), F (capital)
for total costs)
K: fixed input
Nm: network factor (m (or n) = N (line length), S (station spacing), R (number oflines)).
NU: underground length
T: technology index (which varies by year only).
In this model, we also impose the restriction on input factor prices such that Σi βi = 1, Σi δQi = 0, Σi βij = 0,Σi βim = 0, βij = βji, γmn = γnm.

Furthermore, we apply Shepherd’s Lemma from equation- (9) to (11) andobtain the input share equations:
Si = βi + Σj βij lnwj+ δQi lnQ + Σm βim lnNm

(12)

for VC1,

Si = βi + Σj βij lnwj+ δQi lnQ +βiK lnK + Σm βim lnNm

for VC2,

Si = βi + Σj βij lnwj+ δQi lnQ + Σm βim lnNm,

for TC3,

where

(14)

Si : input i's share of total costs.

The service output provision function is specified as the log-linear function as follows:
lnQ = δ0 + δQd lnQD + Σi ζi lnZi,
where

(13)

(15)

QD : rail service consumption (passenger-km)
Zi: quality of service conditions (i = T (trip length), P (peak ratio),
L (average load factor)).

As for the estimation method, I apply the FIML (Full Information Maximum Likelihood) for the
total cost function, three input share equations and the output provision equation, which I estimate
simultaneously.

For the estimation, I will divide all observations of each variable by the sample mean.

4 Estimation of Cost Function of Privately Owned Railways
4.1 Sample Selection and Definition of Variables
The observations used here, from which I estimate total cost function, are from a pooled data setof 59 privately owned railway companies for six time periods, the fiscal years 1970, 75, 80, 85, 90 and
95, a total of 354 observations.

A privately owned railway company is defined as a rail company inwhich privately owned stock shareholders account for more than fifty percent of all shareholders, and the11main data source is each year’s Annual Rail Statistics (Tetsudo Tokei Nenpo), edited by the Ministry ofTransport.
The variables used here are defined as follows and shown in Table 4.
are the sum of labor, energy, material (including repair) and capital costs.
sum of costs except capital costs.

First, total costs (TC)And variable costs (VC) are

Among these costs, energy costs consist of electricity and light oil

expenditures, with depreciation expenditures being considered capital costs because these are mainly
consumed for rail facilities and rolling stock.
Service output (Q) is measured as annual vehicle kilometer.

According to Small (1992), there

are two kinds of output measures: final outputs (or demand-oriented measures) and intermediate outputs(or technical output measures).

Examples of the former are passenger trips, passenger-miles, or revenue

passengers, while examples of the latter are vehicle-miles, vehicle-hours, or seat-miles.

The mainpurpose of this study is related to the purely technical efficiency issue of a company’s production,
making it appropriate to choose intermediate outputs.

Although seat-kilometer is desirable as an output

measure, unavailability of data compels me to use the measure of vehicle kilometer instead.

However,in this study, in order to control differences in demand conditions, I also use four kinds of variables:
passenger kilometer (QD), trip length (ZT), peak ratio (ZP), and load factor (ZL).
as average travel length per passenger.

Trip length is measured

Peak ratio is measured as the ratio of rail pass users to total

passengers, assuming that rail pass users are commuters.

The load factor is calculated as the percentage

of number of passengers to designated capacity of a train vehicle.
As for input factor prices, labor price (wL) is defined as the annual average salary per person, as
reported in the Annual Rail Statistics.

While it would seem that in a competitive market, wage levels

should not be different, in this sample selection, larger private railways have higher wages.

I cannotfind a concrete theoretical justification for this difference, but it may result somehow from a difference inthe level of skill required of workers at larger railways.

These firms pay higher wages and have moregenerous fringe benefits perhaps because they require various kinds of skills and fast, intelligent reaction
to unexpected accidents.

For example, in a larger system, operators have to operate many kinds oftrains such as super express, express, semi-express, and regular trains, and are sometimes required tooperate on rail tracks outside their own network.

Furthermore, engineers of train schedules must

accommodate many different kinds of trains in their diagram.

Thus, larger systems require more

complicated skills than smaller systems, perhaps accounting for the higher wage level.
(wE) is obtained by dividing electricity expenditures by electricity consumption.

Energy priceHowever, some rail

operators owning diesel vehicles use oils, so that I translate oil consumption into the equivalent
electricity consumption[4].

Material price (wM) is obtained by dividing material and repair expendituresby the aggregate material index[5].

Aggregate material index is defined as weighted average ofroute-km and amount of rolling stock because main material expenditures are considered as expenses
related to both tracks and rolling stock.

Capital price (wF) is defined as the price index of capital times

12the sum of the depreciation rate and the interest rate of short-term government bonds.[6]Thedepreciation rate is obtained by dividing depreciation expenditures by fixed assets of rail division.

As

for network factors, three kinds of network characteristic variables are defined.
length (NN) is used, which is measured by route kilometer per line.

First, the average line

Second, it is also important toinclude station spacing (NS) in order to distinguish characteristics of each urban railway system.
spacing is defined as the average route kilometer between stations.
for network characteristics.
technological difference.

StationThird, number of lines (NR) is used

Furthermore, underground length (NU) is also used to control the

Finally, fixed (capital) input (K) for VC is defined as fixed capital related torail facilities.

Table 4 The Definition of and Statistics on Used Variables
variabledefinition
mean
TC
VC
Q
wL

wE
wM
wF
NN

standard
minimum maximum
deviation
13,985,300 25,871,700
19,579 167,991,000

total cost sum of labor, energy, material
and repair, and depreciation
costs (thousand yen)variable sum of labor, energy, andcost
material and repair costs(thousand yen)service annual vehicle kilometer
output(thousand)
labor
price

11,163,300 20,479,400

average annual salary perperson(yen/person)

energy price electricity expenditure per kwh(yen/kwh)
material
pricematerial and repair expenditure
per aggregate input index(yen/input index)
capital price sum of the depreciation rate andthe interest rate of government's
short-term bonds (%)
average route kilometer per line
route length (km)14,863

135,253,000

28,28554,187

13

347,517

5,580,690

1,715,746

1,503,960

17,155,500

12.926

6.664

0.837

44.716

12,791

11,647561

72,59115.371

5.116

6.400

51.19418.76712.3594.200

78.500NS

stationspacingaverage kilometers between
stations (km)

1.421

0.573

0.465

3.580

NR

number of total number of lines which alines
railway company operates

3.853

4.947

1.000

27.000NU

undergroun underground kilometerd length (km)

2.959

6.280

0.000

39.03713

K
QD

fixed input fixed capital related to rail
facilities(million yen)demand of annual passenger kilometer
rail service (thousand)

209,959433,767101

3,307,068

1,710,820

3,267,431

112

15,136,400

ZT

trip length average trip length perpassenger (km)

9.200

4.206

2.815

27.570ZP

peak ratio ratio of rail pass users to total
passengers

0.584

0.117

0.192

0.781

ZL

load factor percentage of number ofpassengers to designatedcapacity of a vehicle (%)[continued]

36.55816.0705.122

153.518

T

technology technology index measured by
accident rate

96.002

5.913

83.212100.000

0.61050.11510.33940.8421SL

For total cost function
labor share ratio of labor costs to total costs

SEenergyshare

ratio of energy costs to total
costs

0.06550.02520.00770.1424SMmaterial
share

ratio of material costs to total
costs

0.19120.06500.07350.3774

capital share ratio of capital costs to total
costs

0.13280.07400.00750.37200.69990.09280.46780.8905SFSL

For variable cost function
labor share ratio of labor costs to total costs

SEenergyshare

ratio of energy costs to total
costs

0.07680.03170.01020.1828SMmaterial
share

ratio of material costs to total
costs

0.22340.08030.07730.47394.2 Estimation Results of Cost Function
The estimation method is the FIML (Full Information Maximum Likelihood) method for the
total cost function with input share equations and the output provision function, which are shown inequation (3) to equation (5).

A summary of estimation results is shown in Tables 5 and 6.

Thegoodness-of-fit in the regression of these variable and total cost functions is acceptably high and the14

first-order coefficients in these functions seem to show a reasonable sign.
almost all of the required properties.

The estimated result meets

First, as for the symmetry and homogeneity conditions, because I

impose restrictions on the cost model, symmetry and homogeneity in input factor prices are satisfied.
Second, the monotonicity conditions are met in output and input factor prices.

Finally, the concavity

condition in input factor prices was also satisfied at around the sample mean because the Hessian matrix
holds at negative semidefinite.

Strictly speaking, the concavity condition does not satisfy globally, but I

conclude that this function is acceptable because it satisfies in 80.8% of observations for VC1, 81.4% for
VC2 and 86.4% for TC3.

In sum, the estimation results seem reasonable enough to justify thefollowing policy analysis.

However, there are some differences in the results between the variable cost

and the total cost function.

Especially, the coefficient of the first-order output measure (δQ) in the total

cost function is significantly larger than that of variable cost functions.

The different results might

account for rail organizations failing to choose the optimal size of capital in the short-run.

Therefore,

based on a comparison of the coefficients of output measures and other statistics, in the next section wewill evaluate which results are reasonable.
Table 5 Estimation Results of the Translog Cost Function: Coefficients and Standard Error
Model
VC1
VC2
TC3
Model
VC1
VC2
TC3
VC function
VC
VC function
DirectVC
Directandfunction and estimation ofandfunction and estimation of
K=network
K=fixed
K=networkTC
K=fixedTC
lengthcapitallengthcapital16.1587 *** 16.1468 *** 16.4150 *** βMM 0.1212 *** 0.1193 *** 0.1089 ***α0(0.0299)(0.0311)(0.0328)(0.0056)(0.0057)(0.0049)0.7307 *** 0.7561 *** 0.8161 *** βMF- 0.0067
δQ(0.0203)(0.0479)(0.0223)(0.0046)
0.6794 *** 0.6775 *** 0.5625 *** βFF- 0.0318 ***βL(0.0066)(0.0071)(0.0080)(0.0107)0.1010 *** 0.1012 *** 0.0827 *** βLK- 0.0093
βE(0.0017)(0.0018)(0.0014)(0.0060)
0.2196 *** 0.2213 *** 0.1737 *** βLN - 0.0035- 0.0059
0.0266 **βM(0.0059)(0.0090)(0.0062)(0.0057)(0.0096)(0.0109)0.1811 *** βLS - 0.0426 *** - 0.0398 *** - 0.0654 ***βF(0.0102)(0.0050)(0.0104)(0.0132)******
0.02450.01660.02570.0083βLRγK(0.0070)(0.0361)(0.0075)(0.0094)
0.2733 *** 0.2614 *** 0.1685 *** βEK0.0012γN(0.0519)(0.0538)(0.0571)(0.0027)- 0.2743 *** - 0.2710 *** - 0.1786 *** βEN - 0.0096 *** - 0.0092 *** - 0.0028
γS(0.0590)(0.0030)(0.0608)(0.0633)(0.0032)(0.0025)
***
***
***
***
***
0.26700.01050.25620.17080.01020.0051 *βES
γR(0.0036)(0.0356)(0.0386)(0.0348)(0.0037)(0.0031)0.00150.00110.0029 ** βER - 0.0105 *** - 0.0102 *** - 0.0045 **γU(0.0014)(0.0024)(0.0015)(0.0013)(0.0026)(0.0020)
***
***
***
- 1.4991-1.5372- 1.2151
0.0081τβMK(0.1617)(0.1662)(0.1534)(0.0054)*0.1120 *** 0.1366 *** 0.1182 *** βMN 0.0131
0.0151 *0.0278 ***
δQQ

15

δQL
δQE
δQM
δQF
δQK
δQN

δQS
δQR
βLL
βLE
βLM
βLF
βEE
βEM
βEF

(0.0111)(0.0388)(0.0132)
0.0195 *** 0.0298 *** - 0.0079(0.0040)(0.0084)(0.0051)0.0146 *** 0.0132 *** 0.0088 ***(0.0015)(0.0034)(0.0012)- 0.0341 *** - 0.0430 *** - 0.0423 ***(0.0035)(0.0073)(0.0031)0.0415 ***(0.0037)- 0.0299(0.0307)- 0.0921 *** - 0.0616- 0.0955 ***(0.0180)(0.0438)(0.0194)(cont.)- 0.1062 *** - 0.0669- 0.0739 ***(0.0226)(0.0508)(0.0240)- 0.0846 *** - 0.0588- 0.0938 ***(0.0201)(0.0364)(0.0224)0.1402 *** 0.1389 *** 0.0901 ***(0.0073)(0.0077)(0.0161)******- 0.0249- 0.0253- 0.0273 ***(0.0033)(0.0033)(0.0036)- 0.1153 *** - 0.1136 *** - 0.0983 ***(0.0059)(0.0061)(0.0068)
0.0355 ***(0.0119)- 0.0308 *** - 0.0310 *** 0.0282 ***(0.0023)(0.0024)(0.0020)- 0.0059 *** - 0.0057 ** - 0.0038 **(0.0022)(0.0024)(0.0018)
0.0029(0.0029)

βMS
βMR
βFN
βFS
βFR
γKK

(0.0076)(0.0081)(0.0065)
0.0321 *** 0.0296 *** 0.0148 *(0.0090)(0.0091)(0.0077)
0.0350 *** 0.0359 *** 0.0434 ***(0.0062)(0.0064)(0.0052)- 0.0517 ***(0.0075)
0.0455 ***(0.0092)- 0.0472 ***(0.0069)
0.0352(0.0275)

γKN

-

γKS

-

γKR

-

γNN

0.1826(0.0560)
0.0913(0.0536)
0.0976(0.0454)
0.1247(0.1530)
0.1423(0.0562)0.0010(0.0521)

γNS
γNR
γSS
γSR
γRR

- 0.0255(0.0303)- 0.0348(0.0377)- 0.0235(0.0284)*** 0.1765 ***(0.0598)*
0.0924(0.0575)** 0.0826*(0.0466)
0.1292(0.1533)** 0.1399 **(0.0582)- 0.0030(0.0547)0.1607 ***(0.0564)0.0439(0.0549)
0.1026 *(0.0528)0.1336(0.1489)
0.1107 *(0.0616)0.0017(0.0539)(Note)(1) Log of likelihood: 34.173 for VC1, 34.805 for VC2, 58.393 for TC3.
(2) Pseudo R2: 0.9861 for VC1, 0.9861 for VC2, 0.9885 for TC1.
(3) Number of observations: 354 for VC1, VC2 and TC3.
(4) Significant at 1 percent (***), 5 percent (**) and 10 percent (*).

Table 6 Estimation Results of the Output Provision: Coefficients and Standard Error
Model
VC1
VC2
TC3
VC function and
VC function and
Direct estimation of TC
K=network length
K=fixed capital
***
***
***
0.25980.25980.2580δ0(0.0245)(0.0250)(0.0251)
***
***
***
0.92700.92710.9257
δQd(0.0072)(0.0075)(0.0076)- 0.0406- 0.0420- 0.0367
ζT(0.0318)(0.0329)(0.0310)
***
***
***
- 0.1996- 0.1994- 0.2054
ζP(0.0450)(0.0450)(0.0455)

16

ζL

***
- 0.6150- 0.6151(0.0320)(0.0333)
R2
0.9924
0.9924Significant at 1 percent (***), 5 percent (**) and 10 percent (*).

***

***

- 0.6055(0.0329)
0.9923

4.3 Transformation of Variable Cost Function to Long-run Total Cost Function
In this section, I will explain the method of construction of the long-run cost function from theshort-run variable cost function.

The main framework used here is based on a study by Braeutigam,

Daughety and Turnquist (1984).
As explained in section 3.2, according to microeconomic theory, at the optimal choice of the
fixed input equation- (3) must hold.

By solving equation- (3) for fixed input, we can obtain the optimal

amount of fixed input from equation- (4).

However, because in this equation there are other variables

such as output and input factor prices, we cannot decide upon only one value of fixed input.
Braeutigam et al. (1984) find the optimal size of fixed input by holding other variables except for fixed
input at sample mean points.
capital of rail facilities.

To simplify, I assume that the fixed input is one, which is the total fixed

Following their method, we can get the following equation from the specifictranslog cost function for the VC2 model:
CS = exp[α0 + γK lnK+ (1/2)γKK (lnK)2] + wf K.

(16)

From the condition shown in equation - (16), we can get the following result:[(γK + γKK lnK)/K] exp [α0 + γK lnK+ (1/2)γKK(lnK)2] + wf = 0
From equation- (17), we can find the optimal size of capital input, K*.
*(17)
According to my calculation, theoptimal size is K = 1.6024 K, when the capital at the sample mean is K.
of total fixed capital of rail facilities is included.
measure of fixed input.

In the VC1 model, a variable

But in this case, I choose line length as a proxy

Therefore, I calculate the same way by assuming that line length (NN) is capital

input (K)
Next, we will obtain the transformed long-run total cost function based on the optimal size ofcapital input, K*.

The transformation method is such that the parameters of variables which have across-term with capital input are recalculated by substituting the optimal size of capital input.
results of the transformed total cost function are summarized in Table 7.

The

If we compare the transformed

total cost function with the directly estimated total cost function, the overall results seem to be similar but
strictly speaking, the first-order coefficients of output measure could be different.

When I apply thet-test for the difference in the coefficients of output measure (δQ), I get a result showing a difference.
Furthermore, from the results of the variable cost function, we get a result showing that rail organizations
did not choose the optimal size of railway facilities.
cost function should be used.
difference.

Therefore, I conclude that the transformed total

Between the transformed total cost functions, there is not much

However, the log of likelihood in the variable cost function suggests that VC2 is slightly

better than VC1.

Therefore, I decided to use the TC2 function for policy evaluation.

17

Table 7 Comparison of Long-run Total Cost Function: Coefficients and Standard Error
Model
TC1
TC2
TC3
Model
TC1
TC2
TC3
Transformed Transformed
Transformed Transformed
DirectDirectTC function TC function estimation of
TC function TC function estimation ofused by VC1 used by VC2
used by VC1 used by VC2
TC
TC
15.9542 *** 16.1429 *** 16.4150 *** βMM
0.1212 *** 0.1193 *** 0.1089 ***α0(0.1136)(0.0352)(0.0056)(0.0328)(0.0057)(0.0049)0.8705 *** 0.7420 *** 0.8161 *** βMF- 0.0067
δQ(0.0394)(0.0505)(0.0223)(0.0046)0.6847 *** 0.6731 *** 0.5625 ***- 0.0318 ***βL
βFF(0.0162)(0.0080)(0.0080)(0.0107)(cont.)
βE
βM
βF

γK
γN
γS
γR
γU
τ
δQQ
δQL
δQE
δQM
δQF
δQK
δQN
δQS
δQR
βLL
βLE

0.1156 *** 0.1017 *** 0.0827 ***(0.0050)(0.0024)(0.0014)0.1997 *** 0.2251 *** 0.1737 ***(0.0137)(0.0071)(0.0057)
0.1811 ***(0.0050)
-

βLK

-

-

βLN

-

βLS

0.2494 *** 0.1685(0.0546)(0.0571)***- 0.4129 *** - 0.2874- 0.1786(0.1000)(0.0632)(0.0633)
0.1187 *** 0.2451 *** 0.1708(0.0830)(0.0439)(0.0348)0.00150.00110.0029(0.0014)(0.0015)(0.0013)- 1.4991 *** -1.5372 *** - 1.2151(0.1617)(0.1662)(0.1534)
0.1120 *** 0.1366 *** 0.1182(0.0111)(0.0388)(0.0132)
0.0195 *** 0.0298 *** - 0.0079(0.0040)(0.0084)(0.0051)0.0146 *** 0.0132 *** 0.0088(0.0015)(0.0034)(0.0012)- 0.0341 *** - 0.0430 *** - 0.0423(0.0035)(0.0073)(0.0031)0.0415(0.0037)
-

***

βEK

- 0.0426(0.0102)- 0.0245(0.0070)-- 0.0059(0.0096)*** - 0.0398 ***(0.0104)*** - 0.0257 ***(0.0075)
-

***

βEN

***

βES

**

βER

***

βMK

***

βMN

-

-

- 0.0616- 0.0955(0.0438)(0.0194)- 0.1062 *** - 0.0669- 0.0739(0.0226)(0.0508)(0.0240)- 0.0846 *** - 0.0588- 0.0938(0.0201)(0.0364)(0.0224)0.1402 *** 0.1389 *** 0.0901(0.0073)(0.0077)(0.0161)- 0.0249 *** - 0.0253 *** - 0.0273

βLR

0.0266 **(0.0109)- 0.0654 ***(0.0132)
0.0083(0.0094)-- 0.0092 ***(0.0032)0.0105 *** 0.0102 ***(0.0036)(0.0037)- 0.0105 *** - 0.0102 ***(0.0024)(0.0026)-- 0.0028(0.0025)0.0051(0.0031)- 0.0045(0.0020)-0.0151 *(0.0081)0.0321 *** 0.0296 ***(0.0090)(0.0091)
0.0350 *** 0.0359 ***(0.0062)(0.0064)
-

-

-

***

βMR

***

βFN

***

βFS

-

-

βFR

-

-

***

γKK

-

-

0.0278(0.0065)0.0148(0.0077)
0.0434(0.0052)- 0.0517(0.0075)
0.0455(0.0092)- 0.0472(0.0069)
-

***

γKN

-

-

-

***

γKS

-

-

-

***

γKR

-

-

-

***

γNN

-

βMS

18

0.1765 ***0.1607

*
**

***
*
***
***
***
***

***

βLM
βLF
βEE
βEM
βEF

(0.0033)(0.0033)(0.0036)- 0.1153 *** - 0.1136 *** - 0.0983(0.0059)(0.0061)(0.0068)
0.0355(0.0119)- 0.0308 *** - 0.0310 *** 0.0282(0.0023)(0.0024)(0.0020)- 0.0059 *** - 0.0057 ** - 0.0038(0.0022)(0.0024)(0.0018)
0.0029(0.0029)

***

γNS

***

γNR

***

γSS

**

γSR
γRR

(0.0598)(0.0564)0.09240.0439(0.0575)(0.0549)
0.0826 *0.1026(0.0466)(0.0528)
0.12470.12920.1336(0.1530)(0.1533)(0.1489)
0.1423 ** 0.1399 ** 0.1107(0.0562)(0.0582)(0.0616)0.0010- 0.0030
0.0017(0.0521)(0.0547)(0.0539)-**(Note)(1) Significant at 1 percent (***), 5 percent (**) and 10 percent (*).
4.4 Economies of Scale
We can discern several important implications relevant to policy, for example the existence ofeconomies of scale, an issue which has been explored in many previous studies.

A method for the

calculation of scale economies can be found in Jara-Diaz and Cortes (1996), who present a theoreticalbackground of the measures of scale economies, including the attributes of outputs.

I follow theanalytical method of Jara-Diaz and Cortes in this study, whereby there are five variables related to thecalculation of the measure of scale economies: output measure, average trip length, peak ratio, loadfactor, average route length, station spacing and number of lines.
and the return to scale (RTS) as follows.

I define the return to density (RTD)

RTD is measured by the inverse of the coefficient of output

measure and attributes, and RTS is defined as the inverse of the sum of the coefficients of variables usedin RTD and the three network variables of line length, station spacing, and number of lines.

Next,

according to their method, I assign the value of one as the weight of output measure and network
variables, and the weight of average trip length should be zero.

I assume that load factor and peak ratio

are the same and 0.1851 from the relationship between passenger-km and vehicle-km.[7]Finally,[8]

passenger-km as output measure and attributes are modified by the coefficient of vehicle-km.

Using

this procedure, I obtain the results that RTD at the sample mean is 1.736 and RTS is 1.277.[9]Following the method of Jara-Diaz and Cortes, I obtain the result that there exist both economies of scaleand network density at the sample mean, which suggests that the size attaining the minimum average cost
700
600
500
400
300
200
100

19
0
0.00

2.00

4.00

6.00

8.00

10.00

Output index

12.00

14.00

16.00

18.00

could be larger than the sample mean in this analysis. Finally, when I draw the average cost curve basedon the results of cost function (TC2), the shape is certainly U-shaped and the size is larger than thesample mean, as Figure 1 shows.
Figure 1 The Shape of Average Cost Function(Note)

Output Index is calculated by dividing numbers of output by sample mean.

5 Estimation of Minimum Average Cost and Ownership Difference
5.1 Minimum Average Cost
In this section, I will estimate the optimal size of a private rail company based on thetransformed total cost functions.
average total cost.

Optimal size used here denotes an organization size with the lowest

From previously expressed equations, I define the average cost function as follows.

In this case, as I will use the transformed total cost function from the variable cost function, I will attach
an asterisk to the coefficients in order to distinguish the coefficients of the directly estimated total costfunction.
AC = TC / Q= (1 / Q) EXP[α0∗ + δQ∗ lnQ + Σi βi∗ lnwi + Σmγm∗ lnNm + 1/2 δQQ (lnQ)2 +Σi δQi (lnQ)(lnwi) + Σm δQm (lnQ)(lnNm) + 1/2 ΣjΣi βij (lnwi) (lnwj) +ΣmΣi βim (lnwi)(lnNm) + 1/2 ΣnΣm γmn (lnNm)(lnNn) + γU lnNU + τ lnT].
From the average cost function, I will find the point with the minimum average cost.

(18)

The main focus of

this analysis is to find the minimum average variable cost in terms of output measure (Q) and network
factors (Nm), which are related to policy issues.

Because the organization size is concerned with

network length, here I take two network factors: average line length (NN) and number of lines (NR).
Differentiating the average cost function by these three measures--service output (Q), average line length(NN), and number of lines (NR)-- the following result can be obtained from the first order condition forthe minimum average cost:∂(AC)/∂Q = ∂(TC/Q)/∂Q= (1 /Q2) [EXP (G)][1 – (δQ∗+ δQQ lnQ + Σi δQi lnwi + ΣmδQm lnNm)] = 0∂(AC)/∂NN = ∂(TC/Q)/∂NN= (1 /QNN) [EXP (G)][ γN∗ + δQN lnQ + Σi βiN lnwi + ΣmγNm lnNm] = 0

(19)
(20)

∂(AC)/∂NR = ∂(TC/Q)/∂NR
where= (1 /QNR) [EXP (G)][γR∗ + δQR lnQ + Σi βiR lnwi + ΣmγRm lnNm] = 0,
G = [α0∗ + δQ∗ lnQ + Σi βi∗ lnwi + Σmγm∗ lnNm + 1/2 δQQ (lnQ)2 +

20

(21)

Σi δQi (lnQ)(lnwi) + Σm δQm (lnQ)(lnNm) + 1/2 ΣjΣi βij (lnwi) (lnwj) +ΣmΣi βim (lnwi)(lnNm) + 1/2 ΣnΣm γmn (lnNm)(lnNn) + γU lnNU + τ lnT.
Because EXP(G) ≠ 0, Q ≠ 0, NN ≠ 0, and NR ≠ 0, the following results are obtained from equations (19)to (21).
δQ∗ + δQQ lnQ + Σi δQi lnwi + Σm δQm lnNm = 1(22)

γN∗ + δQN lnQ + Σi βiN lnwi + Σm γNm lnNm = 0
γR∗ + δQR lnQ + Σi βiR lnwi + Σm γRm lnNm = 0.

(23)
(24)

By solving the three equations from (22) to (24) for Q, NN, and NR , we can obtain the optimal level of
output (Q*), line length (NN*) and number of lines (NR*), which attain the minimum average costs.
However, it is worth noting that we cannot decide the unique values of these three variables from thesegeneral equations because other variables such as input factor prices are not fixed.

However, if we takesome fixed numbers for other variables, we can obtain unique numbers for output, line length, andnumber of lines.

5.2 Cost Difference by Ownership
The estimated result of the total cost function of privately owned railways will be used for theestimation of cost differences by ownership.

I will consider two kinds of organizations: privately

owned railways and publicly owned railways.

The profile of the typical rail company of these kinds oforganizations is expressed as the combination of actual total cost (TC), output measure (Q), input factor
price vector (w), and network characteristics vector (N) for a given technology level (or year) (T).
natural to compare these variables, as we estimated the total cost function using them.
i

i

i

i

combination as (TC ; Q , w , N ) for ownership type i.

It is

I will express the

For example, a private and a public railway’s

characteristics are expressed as (TCpr; Qpr, wpr, Npr) and (TCpu; Qpu, wpu, Npu) individually.
The simplest method of estimating cost difference by ownership would be to compare the actual
total cost of a typical rail company in these three categories, that is to compare TCprand TCpu.

However,

this method is not appropriate because other conditions such as the size of output, input factor price level,and network characteristics are not the same (Mizutani, 1994).

Therefore, I will use the transformed

total cost function in order to control the other conditions.

For example, when we substitute the

combination of explanatory variables (Qi, wi, Ni) for each type of organization in given year (T) into theequation, we can obtain the predicted total cost of the privately owned railway.

The important point isthat the predicted total cost of the private railway is obtained by holding the same conditions in output
level, input factor prices, and network characteristics as for railways of different ownership.

21

TChpr = f (Q, w, T, N)(25)

Where TChpr : predicted (or hypothetical) total cost by substituting (Q, w, N) into thetransformed total cost function (TC2)).
Therefore, by substituting Q, w, and N of a typical rail company of each ownership in given
year (T), we can obtain the hypothetical private railway’s total cost.

For example, the hypotheticalprivate railway’s total cost with the same conditions as a public railway and a partially private railway is
obtained by substituting (Qpu, wpu, Npu) into the equation – (25).

Therefore, we can compare thehypothetical private railway’s cost, TCh , with the actual public railway’s cost, TCpu.
pr

I will define theindex (R) for the cost comparison by ownership as follows:
RTC = TC pu / TChpr(26)

Where RTC: index for total cost comparison by ownership
TC pu : actual total cost of a public railway company
TChpr : total cost of a hypothetical private rail company(a hypothetical private is divided into three levels: more efficient, average, lessefficient).
From this index, we can estimate how much more productive efficiency the private ownership has,compared with other types of ownership.

In this case I also compare the variable cost between the two

sectors by a similar method:
RVC = VC pu / VChpr(27)

Where RVC: index for variable cost comparison by ownership
VC pu : actual variable cost of a public railway company
VChpr : variable cost of a hypothetical private rail company(a hypothetical private is divided into three levels: more efficient, average, lessefficient).

5.3 Estimation Results
Minimum Average Cost
In this section, I will calculate the optimal size of a private rail company in terms of service
output, line length and number of lines.

In order to obtain optimal size, we can employ the previous

equations from (22) to (24), which are the first order conditions for the minimum average cost.

In thisanalysis, I assume that other variables, such as input factor prices except for service output, line lengthand number of lines, are fixed at the level of sample mean.

As a result, because these explanatory

variables in total cost function are standardized by dividing by the sample mean, terms which include

22

other variables except for output, line length, and number of lines disappear.
When these three equations are solved based on TC2, the optimal levels for output (Q*) and thetotal line length are individually 194,253 thousands vehicle-km and 85.1 km.

Compared with the caseof the directly estimated total cost function (TC3), the output level and the network length are larger(output: 124,818; total line length: 66.6, based on the TC3 model). Based on my calculations, theoptimal network has an average line length (NN*) of 4km and number of lines (NR*) of 21. In this case,the average costs are 287 yen per vehicle-km.

The network size is quite similar to large Japanese

private railway companies, which are considered efficient.
Are these numbers reasonable?

Preston (1996) reports that the optimal railway size forminimizing operating costs might have a network of around 4,000 km and run 120 million train-km perannum, based on calculations obtained from European state railways.

The studies deal with different

types of railways: my study is based on privately owned urban railways, while Preston’s results comefrom state railways in Europe.

Preston shows that the optimal size in terms of train-km is 120 million

train-km, and my results, when translated into the same output measure (i.e. train-km), show optimal size
to be about half that, at 65 million train-km (= 194 million vehicle-km / 3 cars per train).

However, interms of network size, I show an optimal total length of about 85 km, a result much smaller thanPreston’s.

Presumably, the difference reflects the fact that my study focuses on urban railways, and thatthese urban railways are much more densely operated in Japan than in other countries.

My resultsuggests that large private railways in Japan are the right size, which may partially explain their success.
If I evaluate the size of the newly privatized JRs according to my results, I find that JR Shikoku
is clearly smaller than the optimally sized railway, while JR East and JR West are too big.

Other JRcompanies such as JR Hokkaido and JR Kyushu seem to have too large a network relative to their output
size.

Cost Differences by Ownership
In this section, I will estimate cost differences by ownership.

According to the methodology

explained in the previous section, I will evaluate cost differences between the two kinds of railorganizations: private and public.

A public railway is a railway organization which has from its

beginning been owned only by the public sector.

For the purpose of this study, a public railway is the

sample average of ten public subway systems.

Because these public railways are subway systems, inorder to avoid bias by technological difference, I included the effect of underground in the cost function.
Therefore, in this analysis, the technological difference is subtracted.

The cost difference issummarized in Table 7.
Estimated cost difference by ownership is as follows.
compared with average private.
1.222.

First, the ratio in variable cost is 0.852,

However, compared with more efficient private, the ratio becomes

According to these results, there is not much difference between private and public rail

23

operators in terms of variable costs.

However, the total costs of public railways are certainly higher

than those of private railways because the ratio is almost one even if the public is compared with lessefficient private.

On average, the public ownership still costs 39% more than private ownership interms of total costs.
These results are slightly different from those found in other studies (e.g. Miyajima and Lee,
1984; Mizutani, 1994; Mizutani, 1999a).

While, most previous studies show that in both variable andtotal costs, private railways are more efficient than public, this study shows no difference or a slightadvantage in variable costs for public railways.

There are three main reasons.

First, smaller private

railways are not efficient because they are regional monopolies and fare regulation protects their business.
Second, public railways are relatively new so that their new technology saves on operating cost. Third,
the privatization of JNR and recent governmental budget constraints have decreased wasteful operating
expenditures.

However, as total costs of public railways are still higher than private, it may be thatpublicly owned organizations are overcapitalizing.
Table 7 Ownership Difference between Private and Public operators
Type of private

Less efficient private(Upper 10%)

Variable Cost
(RVC)
VC2
VC function with K=fixed capital

Total Cost
(RTC)
TC2
Transformed TC function used by VC2

0.594

0.970

Average private
0.852
More efficient private(Lower 10%)1.222
(Note):(1) Each type of private is defined as follows:
Average private: the estimated point on the line of cost function
Less efficient private: the upper limit of 10% confidence in the cost functionMore efficient private: the lower limit of 10% confidence in the cost function

1.391
1.996

6 CONCLUSION
This paper has aimed to estimate the cost function of privately owned passenger railways, tocalculate minimum average cost by using the cost function, and to evaluate the cost differences between
private and public railways.

This analysis is relevant to the work of policy makers as they restructurestate railways into private organizations of an optimal size for achieving efficiency.

We have seen thatprivate ownership indeed does produce rail service more cheaply than public ownership.

As for theorganization size, we have attained the specific following results: the optimal size of a railway
organization, that which attains minimum average costs, is the organization with 194 million vehicle-km
annually and a total network of about 85km, which is a route-km per line of 4 km, with 21 lines.

24

In thiscase, the average costs are 287 yen per vehicle-km in 1990 values.

Second, in terms of total costs, thecosts of public railways are certainly higher than for private railways.

On average, the costs of full

public ownership are 39.1% higher than those for full private ownership.

However, in terms of variable

costs, there are no differences between two sectors.

Therefore, ownership is one factor showing costdifference, but could not be the absolute factor.

Last, based on my calculations, the variable costfunction should be used because railway organizations fail to optimize their facilities in the short-run.

NOTES[1] As for the entrepreneurial behavior of private railways, see, for example, Van de Velde, D., F.
Mizutani, J. Preston and S. Hulten (1998).
[2] In this study, the cost function is long-run cost function.
the long-run cost function.

In theory, the capital stock is not fixed in

However, because the data set of the empirical cost study is pooled data,

the cost function necessarily includes such network variables as capital stock in order to control forheterogeneous network conditions.
[3] Strictly speaking, each railway organization might use differing technology, especially for
maintaining safety.
given year.

However, I assume that production technology is the same among railways per

Therefore, technology changes by time only.

[4] Based on data from a previous study (Pushkarev, Zupan and Cumella, 1982), I used the following
formula to translate oil consumption to equivalent consumption:
ELCe = 10526.9 OIL
where

ELCe : equivalent electricity consumption (kwh)
OIL : actual light oil consumption (kl)[5] Most material expenditure is related to rail tracks, including electric wires and rolling stock.

In

general, the expenditure is price times quantity, so that if we know the “quantity” of material, then we
can know the “price.”

In this study, I assume that the main quantity of materials is related to rail

track and rolling stock.

And I assume that quantity in each category depends on the expenditure ofmaterial in that category, as there is data on material expenditure regarding track and rolling stock foreach rail company.

I use these expenditures as weight to construct the aggregate index of materials.

I define the aggregate material index (AMI) as follows:
AMI = w1RKM + w2 TV
Where RKM: total route kilometer
TV: total number of train car
w1, w2: weight (defined as expenditure of tracks and rolling stock).
[6] Capital price (wF) is defined as follows:
wF = PI (r + δ )where

PI : price index of capital (base year : 1990)25

r : interest rate of short-term government bonds
δ : depreciation.
[7] According to Jara-Diaz and Cortes (1996), the weight of vehicle-km is smaller than that ofpassenger-km by γ.
variables.

I assume that this difference is shown in the difference of coefficients of these

As a result, I obtain the number as 0.1851 (= 0.9271 – 0.7420).

Therefore, this is used

for the weight of load factor.
[8] All coefficients of passenger-km and attributes are multiplied by 0.7420.
[9] The calculation results are as follows:
RTD = [δQδQd + 0.1851 δQ (ζP + ζL)]-1
RTS = [δQδQd + 0.1851 δQ (ζP + ζL) + γN + γS + γR ]-1.

[2002.9.18 628]
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29

Discussion Paper
No.
2001･1

AuthorTakashi Hatakeda
Nobuyuki Isagawa

2001･2Michiyasu Nakajima
Takeshi Mizuguchi
Katsuhiko Kokubu
Yasushi Onishi
Koji Okubayashi

2001･3
2001･4

2001･5
2001･6
2001･7

2001･8

2001･9Katsuhiko Kokubu
Akihiro Noda
Yasushi Onishi
Tomomi Shinabe
Kunio Miyashita
Akinobu Sakashita
KOKUBU,
Katsuhiko
NASHIOKA, ErikoDAIKUHARA, Rie
KOKUBU,
Katsuhiko
KURASAKA,
TomokoShuzo Murata

2001･11

Makoto Yano
Fumio Dei
Mahito Okura

2001･122001･13

Mahito Okura
Shuzo Murata

2001･14

Koji Okubayashi
Toshinori Takashina

2001･15
2001･16

Takuji Hara
Shuzo Murata

2001･17

Norio Kambayashi

2001･18
2001･19
2001･20

Yuichi Fukuta
Kazuhiro Tanaka
Kentaro Nobeoka
Kazuhiro Tanaka

2001･21

Mahito Okura

2001･22
2001･23

Kazuhiro Tanaka
Mahito Okura

2001･24

Katsuhiko Kokubu
Akihiro Noda
Yasushi Onishi
Tomomi Shinabe
Akira Higashida

2001･10

Title
Stock Price Behavior Surrounding Repurchase
Announcements:
Evidence from Japan
Material Flow Cost Accounting of IMU (October 2000)

Date
1／2001(in Japanese)

1／2001

Japanese Manufacturers Without Factories:
Cases of Sony, Matsushita, Misumi, People
Determinants of Environmental Report Publicationin Japanese Companies

1／2001

Logistics Strategy of Japanese Port Management
Issues and Method of Organizational Culture Studies:
Functionalism Approach
Environmental Accounting in Japanese Companies as ofNovember 2000

(in Japanese)

2／20013／2001(in Japanese)

3／20012／2001

Corporate Environmental Accounting: A Japanese
Perspective

The Comparative Study of Japanese & U.S. Companies(6) The Health & Long Term Care-The Management
Education
A Trade Model with Vertical Production Chain andCompetition Policy in the Downstream Sector
Risk Classification: A Study of the Tradeoff between
Quantity and Quality of Insured
Is the Separating Equilibrium Really Desirable?
The Comparative Study of Japanese & U.S. Companies(5) The Conflict Among Companies
Research Report on the relations between there-employment process and former job career at thecompulsory retirement of workers of a firm
The Social Shaping of Medicines
The Comparative Study of Japanese & U.S. Companies(7) Toward Restructuring Japanese Corporations
Cultural influences on IT usage among workers:a UK-Japanese comparison
A Test for Rational Bubbles in Stock Prices
Toward the Effective Reform of Japanese Corporate
Governance
Structural Changes in Top Management―The Effect ofCorporate Governance Reform in Japan
In the Separating Equilibrium Really Desirable ?
―revised version of No.2001･12―
Introduction to Corporate Governance
The Role of Some Insurance Contracts in a Subjective
Equilibrium Model with Observable Effort
Environmental Disclosure of Japanese companies

3／2001(in Japanese)

4／2001

12／2000(in Japanese)

5／2001(in Japanese)(in Japanese)

5／20016／2001(in Japanese)

7／2001(in Japanese)(in Japanese)

7／20017／20017／2001(in Japanese)

7／20017／2001(in Japanese)

7／2001(in Japanese)

8／2001(in Japanese)(in Japanese)

8／20018／2001(in Japanese)

8／2001

Discussion Paper
No.
2001･25

2001･26
2001･27

AuthorKatsuhiko Kokubu
Tomomi Shinabe
Akira Higashida
Yasushi Onishi
Akihiro Noda
Katsuhiko Kokubu
Eriko Nashioka
Masakazu Takagi

2001・28 Mahito Okura
2001・29 Atsushi Takao

2001・30 Mahito Okura
2001・31 Masakazu Takagi
2001・32 Norio Kambayashi
Harry Scarbrough2001・33 Fumitoshi Mizutani
Shuji Uranishi
2001・34 Mahito Okura

2001・35 Tsutomu Harada
2001・36 Nobuyuki Isagawa

2001・37 Nobuyuki Isagawa
2002・1

Nobuyuki Isagawa

2002・2

Nobuyuki Isagawa

2002・3Mahito Okura

2002・4

Elmer Sterken
Ichiro Tokutsu
Mahito Okura
Mahito Okura
Nobuyuki Isagawa
Koji Okubayashi
Toshinori Takashina

2002・5
2002・6
2002・7
2002・8

2002・9Hajime Shimizu

2002・10 Nobuyuki Isagawa
2002･11

Nobuyuki Isagawa

Title
Environmental Reporting of Japanese companies(in Japanese)

Date
8／2001

Environmental Accounting of Japanese companies(in Japanese)

8／2001Elementary Study of East Asian Corporate andManagement System
Vertical Differentiation in the Insurance Market
A Study on the Penetration Process of Modern Insurance
by Evolutionary Economics―Why do the Modern
Insurance and the Premodern Mutual Society coexist?―
An Essay in the Economics of Post-loss Minimisation:
An Analysis of the Effectiveness of the Insurance Law
and Clauses
Co-operation between Hanshin(Osaka-Kobe) Area andSouth-east Asia-Finding business opportunities of mutual
benefits
Cultural influences on IT use amongst factory managers：
A UK-Japanese comparison
The Post Office vs. Parcel Delivery Companies：
Competition Effects on Costs and Productivity
An Essay in the Economics of Post-loss Minimisation:
An Analysis of the Effectiveness of the Insurance Law
and Clauses―revised version of No.2001・30―
An Examination of IT Paradox in Japan
Open-Market Repurchase Announcements, Actual
Repurchases, and Stock Price Behavior in Inefficient
markets
Corporate Financial Strategy and Stock Price Behavior ina Noise Trader Model with Limited Arbitrage
Cross Holding of Shares, Unwinding Cross Holding of
Shares, and Managerial
Entrenchment
A Theory of Stock Price Behavior following Repurchase
Announcements
An Equilibrium Analysis of the Insurance Market
with Vertical Differentiation
What are the determinants of the number of bankrelations of Japanese firms?
Review Article：Adverse Selection in Insurance Market
Welfare Effect of Firm Size in Insurance Market
Does Investment Horison Affect Investment Strategy ?
Research Report of Careers and Employment Process ofWorkers over 60 years old :a Case of a Japnaese
Manufacturing Company
On the Existence of Tax Equilibrium : Incomplete Market
with Capital Income Tax
Cost of Financial Distress and Debt Restructuring :Debt Forgiveness and Equity-for-Debt Swap
Open-Market Repurchase Announcements, Actual
Repurchases, and Stock Price Behavior in Inefficient
Markets―revised version of No.2001･36―

9／2001(in Japanese)(in Japanese)

9／20019／20019／2001(in Japanese)

9／2001

10／2001
10／2001
11／2001(in Japanese)

11／2001
12／2001
12／2001(in Japanese)

1／ 2002(in Japanese)

1／ 20022／20023／2002(in Japanese)(in Japanese)(in Japnaese)

3／20023／20023／20024／2002(in Japnaese)

4／2002(in Japnaese)

4／20025／2002

Discussion Paper
No.
2002･12

AuthorKenji Kutsuna
Richard Smith

2002・13 Kunio Miyashita
2002・14 Hajime Shimizu
2002・15 Hajime Shimizu
2002・16 Koji Okubayashi
2002・17 Fumitoshi Mizutani
Shuji Uranishi
2002・18 Kazuhisa Otogawa
2002・19 Atsuo Takenaka
2002・20 Tsuneo Nakano

2002・21 Tsuneo Nakano
2002・22 Fumitoshi Mizutani
Shuji Uranishi
2002・23 Atsushi Takao
Mahito Okura
2002・24 Fumitoshi Mizutani

Title
Why Does Book Building Drive Out Auction Methodsof IPO Issuance?
Evidence and Implications from Japan
International Logistics and Modal Choice
Incomplete markets with capital income tax and public
goods
On the possibility of Pareto improving of tax-equilibrium
China-Japan Comparison of Work Organization
The Post Office vs. Parcel Delivery Companies:
Competition Effects on Costs and Productivity―revised version of No.2001･33―
Earnings Forecast and Earnings Management of JapaneseInitial Public Offerings Firms
The Classification of Foreign R&D Units
The Dutch East India Company and its Corporate
Governance―revised version of No.2000･40―
The British East India Company and its Corporate
Governance
Privatization Effects on TFP Growth and Capital
Adjustments
Some Notes on the Privatization of Postal Life Insurancein JapanPrivately Owned Railways’ Cost Function, Organization
Size and OwnershipDate
5／2002(in Japnaese)(in Japnaese)

6／20026／20026／20027／20027／20028／2002(in Japanese)(in Japanese)

8／20028／2002(in Japanese)

8／20028／2002(in Japanese)

9／20029／2002