EFFICIENCY, TECHNICAL CHANGE AND PRODUCTIVITY
IN THE EUROPEAN RAIL SECTOR:
A STOCHASTIC FRONTIER APPROACH
International Journal of Transport Economics, Vol. XXVII, No. 1, pp. 55-76.
Pedro Cantos Sánchez (Universitat de València)
Joaquín Maudos Villarroya (Universitat de València and IVIE)
Departamento de Análisis Económico
Universidad de Valencia
Edificio Departamental Oriental
Avenida de los Naranjos, s/n.
46022 Valencia (SPAIN)
Tel.: 34.6.382.82.46
Fax: 34.6.382.82.49
E-mail: Pedro.Cantos@uv.es
Abstract
This study estimates the levels of productivity, efficiency and technical change for a
number of European railway companies. Unlike most studies based on production
functions, we have estimated a stochastic frontier cost function. The results indicate that

Joaquín Maudos thanks the financial support of DGICYT PB94-1523
the principal source of productivity growth is technical progress, followed by gains in
efficiency (catching-up). It can also be seen that the most efficient companies are those
with a greater degree of financial and management independence.
2
1. Introduction
In recent years the international railway sector has been involved in a thoroughgoing
process of transformation as a consequence of two phenomena: the de-regulation of the
sector and the intense competition from other modes of transport. This last phenomenon
has caused the market share of the railways to be drastically reduced, in favour of other,
more competitive, modes of transport such as road transport and even air transport.
Criticisms were also levelled at the management of most railway companies, habitually
publicly owned monopoly enterprises, which in many cases acted as mere government
departments. Thus the companies were said to act with a high level of inefficiency,
because of their non-commercial character, and their limited concern for the efficient
management of their resources. One of the basic objectives of the current liberalisation
measures is to deal with this problem by installing the incentives necessary for the
companies to improve their levels of efficiency and productivity.
In this line, research carried out to analyse the evolution of the productivity and
efficiency of the sector in recent years constitute a first assessment of the measures
taken in the sector. Thus, although the most important measures were taken very
recently1, in the 1980s most of the European companies restructured their internal
organisation in order to give them greater autonomy, which would allow them to
1
See Kopicki and Thompson (1995) for a description of the most important deregulation and privatisation measures
carried out in the sector (United Kingdom and Sweden, or Argentina, Japan and New Zealand for other non-European
liberalization experiences).
3
improve the management of the companies and thus their productivity.
In transport economics relatively few studies have been concerned with assessing the
evolution of the productivity levels of the sector. The estimation of economies of scale
has been the basic aim of most of these studies2, since the existence of such economies
has been one of the basic arguments justifying the maintenance of regulation in the
sector.
The first studies by Caves, Christensen and Swanson (1980 and 1981) applied a simple
method to determine the changes in the productivity of North American railway
companies on the basis of a cost function. Later Bereskin (1996) broadened the sample
period analysed, showing that the de-regulation measures carried out since the
application of the Staggers Act in 1980 have achieved clear improvements in the
productivity levels of the sector.
A common aspect of many of the studies that analyse total factor productivity in railway
companies is the use either of accounting approaches or of index numbers 3 which, as
Grosskopf (1993) shows, offer biassed results for productivity in the presence of
inefficiency, or estimations of average production or cost functions rather than genuine
frontiers. For these reasons, studies like those by Perelman and Pestieau (1988) and
Compagnie, Gathon and Pestieau (1991) analysed the efficiency of companies with
2
For the case of North American companies, see the studies by Caves et al. (1981), Caves et al. (1985), Kim (1987),
Berndt et al. (1993), Friedlander et al. (1993); and for European companies, those by Filippini and Maggi (1992),
De Borger (1992), McGeehan (1993), and Preston and Nash (1996).
4
respect to their different operating environments by means of frontier approaches that
explicitly incorporate efficiency into the analysis as a source of productivity growth
distinct from technical progress.
Oum and Yu (1994) estimate the levels of efficiency for a sample of companies
applying the methodology of data envelopment analysis (DEA). This approach has
certain advantages over parametric approaches: it is not necessary to specify a
functional form, or to introduce any kind of assumption as to the distribution of
inefficiency.
The above-mentioned studies that use frontier approaches have used a determinist
approach, which assumes that any deviation from a firm’s frontier is explained
exclusively by its inefficient behaviour. This may cause estimations of inefficiency to be
upwardly biassed by capturing as inefficiency the effect of variables that are beyond the
control of the company (errors of measurement, bad luck, etc.).
Subsequently, Gathon and Pestieau (1995) estimated a stochastic frontier production
function, on the basis of which, as well as estimating the variations in the productivity
of the sector, they break them down into changes in the levels of efficiency of the
companies (catching-up) and of technical progress. The estimation of technical progress
is achieved by introducing a time trend, which interacts with the input variables
considered in the model. The problem with frontier production functions is the need to
3
See, for example, Tretheway et al. (1997).
5
aggregate in a single measurement the different outputs offered by the railway
companies, thus losing the possibility of capturing the advantages of specialisation in
the productive behaviour of the companies.
Gong and Sickles (1992) used a study by Monte Carlo to compare the results obtained
with DEA and stochastic frontier techniques in the measurement of efficiency levels of
a sample of companies. Their results indicate that on many occasions the stochastic
frontier approach represents individual efficiency and productivity levels better than
DEA approach.
For this reason, the approach chosen in this study is that of the stochastic frontier. It is
intended thus to solve one of the problems of determinist frontiers mentioned earlier,
namely that estimations of inefficiency may be upwardly biased. We will also estimate a
cost function instead of a production function. In a highly regulated context such as that
of the railways, it is to be expected that companies will attempt to minimise the costs of
a certain level of production. For this reason it is of greater interest to analyse cost
inefficiency (including both the technical and allocative types) than the technical
inefficiency analysed by the studies that estimate production functions. Furthermore, the
estimation of a cost function presents the added attraction of enabling the multi-product
nature of the production to be taken into account through the specification of different
outputs.
6
Our study aims, first, to assess the inefficiency levels of the principal European
companies and to specify their main determinants or explanatory factors. These
determinants include factors such as the degree of autonomy and financial
independence, numbers of passengers and tons carried per train, the traffic density
indices or the degree of electrification. Also we will evaluate the rate of technical
progress on the basis of the introduction of a trend, in a similar way to Gathon and
Pestieau (1995).
Another advantage of this type of specification is that it enables us to estimate
economies of scale and consequently to evaluate the inefficiencies produced by the
wrong scale of production at which the companies operate. Thus this approach allows us
to calculate the growth rate of total factor productivity distinguishing between the
movements of the cost function due to technical progress, changes in efficiency of
companies, and movements along the cost function due to the economies of scale.
Finally, one element that justifies the diversity of the results obtained by the literature is
the different specification of the variables representing output. Two types of variables
are normally used in these models. First, the variables passenger-Kms and Ton-Kms are
the series most used in the literature. The greatest disadvantage of these measurements
is that they can be strongly influenced by the regulatory system operating in each
country. For this reason, measurements of output such as passenger and freight train-Km
are more appropriate when attempting to assess the behaviour of companies in heavily
7
regulated environments. We have therefore opted for this latter modelisation of output.
The study is organised as follows. Section 2 describes the stochastic approach to the
analysis of efficiency. Section 3 describes the sample, and the variables used in our
model. Section 4 shows the indices of efficiency and their determinants for the various
companies. Section 5 analyses inefficiencies of scale, technical progress, and the growth
of total factor productivity. Finally, in Section 6 we will present the main conclusions of
the study.
2.- The stochastic frontier approach
Since the pioneering study by Farrell (1957), the measurement of inefficiency has run
parallel to the estimation of frontier functions. Inefficiency being defined as the
difference between the observed level of the variable being studied (production, costs,
profits, etc) and the maximum attainable, situated at the frontier, the measurement of
inefficiency necessarily requires the estimation of the frontier function.
The so-called X-inefficiencies are inefficiencies due to errors of management and/or
organisation. They include inefficiencies both of the technical type (the current level of
production could be produced with a smaller amount of input) and of the allocative type
(the proportion of inputs used is not the one that minimises costs given their relative
8
prices), and differ from inefficiencies of scale in that the latter arise from the choice of a
scale of production at which average costs are not minimised. This definition **is
equivalent to the original definition of X-inefficiency which captures the technological
and management differences that increase production costs4.
The stochastic frontier approach was introduced simultaneously by Aigner et al. (1977)
and Meeusen et al. (1977). This approach modifies the standard production (or cost)
function by assuming that inefficiency forms part of the error term. This composite error
term therefore includes an inefficiency component and a purely random component that
captures the effect of variables that are beyond the control of the production unit being
analysed (weather, bad luck, etc). Thus, the principal attraction of the stochastic frontier
approach as against determinist approaches such as DEA is that it isolates the influence
of factors other than inefficient behaviour, thus correcting the possible upward bias of
inefficiency of the determinist methods. Furthermore, unlike DEA, the estimation of
efficiency is less sensitive to the existence of outlayers in the sample5.
The basic stochastic cost frontier model posits that the observed costs of a firm deviate
from the cost frontier as a consequence of random fluctuations (vi) and of inefficiency
4
The original study by Liebenstein (1966) distinguishes between X-inefficiencies and allocative inefficiencies (the
social costs driving from the power of the market and from restrictions on trade that cause a bad allocation of
resources among firms, industries and nations). This last idea of allocative inefficiency differs from the commonly
used concept of allocative inefficiency as the increase in costs deriving from the choice of an incorrect proportion of
inputs given their relative prices. However, both technical and allocative inefficiency are frequently lumped together
in the generic concept of X-inefficiency. For this reason, throughout this study the term X-inefficiency will refer to
the total inefficiency, the more so as when estimating a cost function the cost inefficiency includes both the technical
and the allocative type.
5 A review of the different techniques for measuring efficiency can be found in Bauer (1990), Green (1993) and
Lovell (1993).
9
(ui). Thus, in the case of the cost frontier:
LnCi  LnC (Yi , Pi ,  )   i
 i  u i  vi
i  1, ..., N
(1)
where Ci are the observed costs of company i, Yi is the vector of outputs, Pi is the vector
of input prices,  is the vector of parameters to be estimated, and LnCi(Yi,Pi,) is the
logarithm of the predicted costs of a company that minimises production costs. The
random error term vi is assumed to be independent and identically distributed, and the
inefficiency term ui is assumed to be distributed independently of vi.
In
order to separate the two components, it is necessary to make distributional
assumptions regarding both components of the composite error term. Since the
inefficiencies can only increase costs above the frontier, it is necessary to specify
asymmetric distributions for the inefficiency term. It is usually assumed that vi is
distributed as a normal with zero mean and variance 2v, and ui as a half-normal (ui is
the absolute value of a variable that is distributed as a normal with mean zero and
variance 2u)
Under the assumption that both components of the error term are distributed
independently, the frontier function can be estimated by maximum likelihood,
inefficiency being estimated from the residuals of the regression. More specifically, the
individual estimations of inefficiency can be obtained by using the distribution of the
10
inefficiency term conditional on the estimate of the composed error term. Thus,
Jondrow et al. (1982) show that in the case of the half-normal distribution, the mean of
this conditional distribution adopts the following expression:
EF  exp  Eu i (u i  vi )
E u i (u i  vi )
    ( i  /  )  i  



(1  2 )   ( i  /  )  
(2)
where =u/v, 2=2u+2v, and  and  are respectively the density and distribution
functions of a random variable that is distributed as a normal. EF is the efficiency of
each company defined as the ratio between the minimum costs situated at the frontier
and the costs really incurred.
In order to analyse the economies of scale, the X-inefficiency and the technical progress
of the European railway sector, we will estimate a translogarithmic frontier cost
function because of its greater flexibility in relation to other specifications. Essentially,
the translog function is a quadratic approximation obtained as a Taylor approximation
around the point of approximation. Among its main advantages are the following: 1) it
does not impose any a priori restriction on the elasticity of substitution among inputs; 2)
it allows the estimation of the cost function to have the form of U; and 3) it allows
potential cost complementarities through its multi-product specification.
In the particular case of a cost function with two outputs and three inputs, the translog
function adopts the following specification6:
6
We impose the habitual restrictions of symmetry and linear homogeneity in the prices of the productive factors.
11
2
LnTC it   0    j LnY jit 
j 1
3
   l LnPlit 
l 1
1 2 2
 jk LnY jit Ln Ykit 
2 j 1 k 1
3
3
2
3
1

LnP
LnP

 jl LnY jit LnPlit 
 lm lit mit 
2 l 1 m 1
j 1 l 1
(3)
2
3
1
  T T   TT T 2    Tj T LnY jit    lT T LnPlit  u it  vit
2
j 1
l 1
where i refers to the company, t to the year, TCit= total costs, Yit is the vector of
outputs, Pit = is the vector of inputs, T=1,2,..,T, uit the inefficiency and vit the random
disturbance term. Note that a trend T has been introduced in order to reflect the
influence of technical progress. The more flexible alternative of introducing time effects
to interact with the other regressors presents the problem that it increases considerably
the number of parameters to be estimated, thus generating a high multicolinearity.
3. The data
The sample used covers the years 1970-1990. It was impossible to extend the period to
more recent years, as in 1991 the accounting systems of the companies changed
considerably. The data were obtained from the reports published by the UIC (Union
Internationale des Chemins de Fer). The total number of companies included in the
model was 15 (see table 1), although for some of the companies complete information
was not available for all years.
12
During the period of estimation no significant reforms occurred in the majority of the
companies. Only the Swedish company SJ carried out an effective process of separation
of infrastructure and services in 1988. In the late 1980s many companies undertook
processes of restructuring whereby it was intended to give them greater management
autonomy, by promoting the separation of the various railway services into
differentiated lines of business.
Operating costs, including labour costs, fuel and energy, and the consumption of
materials and purchases and external services, were taken as the dependent variable.
The introduction of physical capital costs causes substantial problems of homogeneity
in the sample, so they were not included.
As the variable representing the index of labour prices, we took the labour costs divided
by the total number of workers in the company (P1). To represent the price of fuel, the
costs deriving from energy and fuel were divided by the total train-Kms supplied by
each company (P2). Finally, the variable representing the price of materials and external
services was taken as the cost deriving from materials, supplies and external purchases
divided by the total train-Kms of each company (P3).
This approximation for the prices of inputs is similar to that proposed by Preston (1994)
and Preston and Nash (1996), given the impossibility of obtaining any more rigorous
indices. All these variables were expressed in constant dollars of 1990, by means of the
13
Purchasing Power Parity (PPP) indices obtained from information available in the
reports of the OECD.
It is habitual in estimations of the cost functions of transport companies to introduce
variables that represent the size of the network. However the introduction of this into
our study caused obvious econometric problems, given the high degree of correlation
between the variable representing the network (length of route) and the output
variables7. Since our study only proposed to analyse and break down the growth of total
factor productivity (TFP) without considering the optimum level of the network, we
opted not to include this variable. In this way we assume that the companies’ costs
change independently of the size of the companies’ network. We therefore estimate
economies of scale allowing variation in the size of network of each company.
As variables representing the different outputs of the railway companies we have used
the number of passenger train-Km (Y1) and freight train-Km (Y2). The reasons for
choosing this specification have already been given, and respond to the need to evaluate
the management of railway companies in closely regulated environments. During the
estimation period there were no significant reforms in the sector, except for that carried
out by the Swedish company SJ in 1989, consisting of the effective separation of
infrastructure from services8.
7
Oum and Waters (1998) point out the econometric problems that may be caused by the introduction of network
variables if they are not properly treated.
8 As is well known, some countries have recently begun important restructuring processes in this sector. The British
case stands out, in which as well as in 1995 separating the management of infrastructure and of services, a process of
privatisation of the whole sector was begun, which has now practically finished.
14
Table 1 gives a summary of the principal statistics of the variables for the sample as a
whole.
15
Table 1. Principal statistics of the sample (Averages for the period analysed).
Y1 (in thous.) Y2 (in thous.)
P1
P2
BR*
United Kingd.
340582,2
81192,1
15,85
0,71
CFF
Sweden
71572,2
28264,5
22,22
0,52
CH
Greece
13606,7
3236,0
11,51
0,88
CP
Portugal
27077,7
6510,3
9,99
1,38
DB
Germany
393556,3
199395,7
23,71
0,96
**
DSB
Denmark
39475,3
8134,3
14,76
0,82
FS
Italy
223832,3
59463,8
18,41
0,38
NS
Holland
96667,6
13788,8
19,30
0,59
NSB
Norwege
22671,8
10495,3
13,63
0,54
ÖBB
Austria
60679,4
35715,3
14,63
0,63
RENFE Spain
96569,3
44859,9
16,84
1,02
***
SJ
Sweden
59984,1
40451,9
15,13
0,48
SNCB
Belgium
68090,7
22291,4
22,25
0,90
SNCF
France
284255,9
198204,2
18,19
0,53
***
VR
Finland
23575,0
18405,3
10,92
0,69
* Except years 1970,1971 and 1972
** Except years 1973,1974, 1975 and 1976
*** Except the year 1990
Table 2. Efficiency levels (EF)
MED
MIN
BR
0,9127
0,8168
CFF
0,9243
0,8736
CH
0,8030
0,5853
CP
0,8471
0,6684
DB
0,8900
0,8311
DSB
0,9120
0,7496
FS
0,7507
0,4460
NS
0,9315
0,8897
NSB
0,8722
0,7290
OBB
0,8085
0,6451
RENFE
0,9057
0,8514
SJ
0,9394
0,8470
SNCB
0,7625
0,6408
SNCF
0,8850
0,7882
VR
0,8203
0,6406
Average
0,8649
0,9882
MAX
0,9846
0,9551
0,9647
0,9859
0,9675
0,9497
0,9557
0,9722
0,9618
0,9655
0,9528
0,9882
0,9161
0,9511
0,9511
0,4460
16
DESVT
0,0418
0,0238
0,1128
0,0935
0,0397
0,0483
0,1798
0,0264
0,0770
0,1083
0,0344
0,0356
0,0771
0,0563
0,0898
0,0982
P3
4,05
2,71
2,01
1,48
2,51
3,11
5,67
1,57
1,80
6,14
3,21
1,76
4,56
3,81
1,30
4. The level of efficiency.
Table 2 presents the average levels of efficiency (EF) of the companies included in the
estimation obtained on the basis of the estimation of the frontier cost function by
maximum likelihood, as appearing in table A.1 of the Appendix. The distribution
assumed for the inefficiency term was half-normal.
The ranking obtained in our study is hardly comparable with that obtained in others, due
both to the different technique used and to the different variables employed 9. At all
events, the order is not very different from that obtained by Oum and Yu (1994) when
they use the train-Km variables as indicators of output, and from that obtained by
Gathon and Pestieau (1995).
The average level of cost efficiency estimated for the aggregate of the countries of the
sample indicates that it would be possible to reduce costs by 13.5%, inefficiency which
includes both technical and allocative types. By countries, the Swedish company SJ and
the Dutch NS are those which present the lowest levels of inefficiency, in contrast to the
Italian company FS and the Belgian SNCB, the least efficient with inefficiency levels
around 25%.
In order to explain the differences in the efficiency levels of the different companies, we
9
Nor is it comparable with the studies that estimate production functions, as these only take into account technical
inefficiency.
17
undertook a further regression, introducing the variables that could potentially be
considered explanatory. The variables introduced were similar to those habitually
introduced by the literature:
- The number of passengers per train (PT) and the number of tons of freight per
train (TT), as indices of the utilisation of trains. This variable is expected to be
correlated negatively with the efficiency level when train-Kms are used as
indicators of output. This occurs because the costs per train-Km increase along with
the load factor of the trains (Oum and Yu, 1994).
- The number of passenger trains-Km per Km of track (TKPL) and the number of
passenger trains-Km per Km of track (TKPL), as indices of utilisation of the
infrastructure. In this case it is to be expected that a more intensive use of the
infrastructure will reduce the costs per train-Km.
- Electrified track as a percentage of the total (EL). In this case, it is highly
probable that a higher degree of electrification will improve efficiency levels, as it
is to be expected that this will reduce the levels of labour and energy consumption.
- The degree of financial autonomy (INGCO) defined as the ratio between the
company’s own income deriving from railway activity and the company’s costs.
This variable is expected to be highly correlated to the level of public subsidies
18
habitually used to finance the companies’ accounts. The literature on railway
companies (see Kopicki and Thompson, 1995) has found much evidence that the
companies with high levels of public subsidy are highly inefficient in their
behaviour.
- The degree of autonomy or independence in the management of the companies
(AUT). This variable was introduced on the basis of the indices provided by Gathon
and Pestieau (1991 and 1995). The degree of autonomy represents a large number
of factors, such as the mode of ownership, the degree of freedom in pricing and
level of service, the number of lines considered socially desirable, etc.. Like Oum
and Yu (1994) we also use this approximation despite the fact that this index is
only defined for one year. In any case, as we have already indicated, since the
reforms introduced during the period of estimation were not very substantial, and
since there is no alternative information available, we opted to introduce this index.
As noted by some authors (Vickers and Yarrow, 1988) the degree of autonomy and
independence in the management of companies is positively correlated with the
level of efficiency.
The explanatory equation of efficiency was estimated according to a Tobit model,
because the dependent variable was limited within the interval (0, 1). To make it easier
to interpret the results, the variables were expressed in logarithmic terms. The
dependent variable was defined as the logarithm of the inverse of the efficiency index.
19
Like Oum and Yu (1994), in order to keep the discussion in terms of levels of efficiency,
the sign of the parameters obtained was changed. The results appear in table 3.
It is observed that most of the estimates (if we except the variables LEL and LTKPL,
although these are not statistically significant) are of the expected sign and coincide
with those obtained in earlier studies (Oum and Yu (1994) and Gathon and Pestieau
(1995)). Thus, a greater degree of financial autonomy (LINGCO) and of management
(LAUT) will clearly increase the levels of efficiency. In particular, an increase of 1% in
the index of coverage by income of costs generates an increase of 0.046% in the
efficiency level. Also, an increase of 1% in the index representing the degree of
management autonomy of the company increases the efficiency level by 0.078%.
Table 3. Determinants of efficiency
Param.
Estimates
Stand. Error t-statistic
Const.
.5036
.2824
1.782
LINGCO
.0464
.0289
1.973
LAUT
.0778
.0374
2.077
LTKML
.0593
.0264
2.242
LTKPL
-.0250
.0215
-1.159
LPT
-.1243
.0275
-4.510
LTT
-.0690
.0371
-1.858
LEL
-.3025E-02
.1695E-02
-1.584
SIGMA
0.127
.5038E-02
25.295
Log of likelihood = 205.155
N = 306
Therefore, the companies such as SJ, NS, CFF or BR, which have traditionally been
known as the companies with greatest management autonomy, are clearly the most
20
efficient companies in the sample. It is observed that most of the estimators (if we
except the variables LEL and LTKPL, although these are not statistically significant) are
of the expected sign and coincide with those obtained in earlier studies (Oum and Yu
(1994) and Gathon and Pestieau (1995)). Thus, a greater degree of financial autonomy
(LINGCO) and of management (LAUT) will clearly increase the levels of efficiency. In
particular, an increase of 1% in the index of coverage by income of costs generates an
increase of 0.046% in the efficiency level. Also, an increase of 1% in the index
representing the degree of management autonomy of the company increases the
efficiency level by 0.078%.
Therefore, the companies such as SJ, NS, CFF or BR, which have traditionally been
known as the companies with greatest management autonomy, are clearly the most
efficient companies in the sample. Companies such as FS, SCNB, CH or ÖBB, much
more controlled and intervened by the public sector, reflect the lowest levels of
efficiency in the sample. Also, an increase in the occupation of trains increases the costs
per train-Km (although the estimate of the index of occupation for freight trains is not
statistically significant). An increase of 1% in the number of passengers per train (LPT)
would reduce the efficiency by about 0.12%. Also, an increase in the volume of freight
train-Kms, per Km of track (LTKML) increases the levels of efficiency, while the
number of passenger train-Kms per Km of track is not statistically significant.
Therefore, a more intensive use of the railway network by freight trains clearly reduces
the costs per train-Km, not so in the case of passenger trains. Finally, as we have
21
indicated, the degree of electrification (LEL) is not a statistically significant variable
either.
5. The sources of TFP growth: economies of scale, technical change, and efficiency
gains.
From a theoretical point of view, the concepts of economies of scale and of technical
change can only be applied in the context of a frontier function. To ignore the possible
existence of inefficient behaviour, a problem inherent in studies based on average
functions, not only constitutes a conceptual problem, but may also seriously influence
the estimations of economies of scale and of technical progress, so that these would be
clearly biased in the presence of inefficiency10. In the case of economies of scale, the
use of data corresponding to non-efficient companies may confuse inefficiencies of
scale with X-inefficiencies. If it is intended to estimate technical progress by estimating
average functions, the measurement of the technical progress is biased to the extent that
it confuses technical progress (movements of the frontier) with the fluctuations of
inefficiency that alter the distance from the frontier.
The estimation of economies of scale (ES) has been done on the basis of the sum of the
derivate of the cost function obtained for each of the outputs, i.e. as the sum of the
10
In some cases, this problem is unimportant when estimating economies of scale, though it usually affects
substantially the estimations of technical progress and economies of scope.
22
elasticities of each of the outputs11. Table 4 shows the information relating to both the
sum of the two elasticities and the elasticity relating to each of the outputs.
As can be observed, for most companies the elasticity of the level of passenger
train-Kms (ESP) is greater than the elasticity of freight train-Kms (ESF). This result
therefore indicates that that costs for these companies grow proportionately in greater
magnitude with an increase in the same proportion of passenger train-Kms than of
freight train-Kms. The sum of both elasticities (ES) indicates that, except for BR, DB,
FS and SNCF, the remaining companies present increasing economies of scale. These
exceptions are precisely the companies that present highest levels of out put, so that
according to these results they operate under diseconomies of scale.
Table 5 presents the annual rates of technical change (TC) obtained for the aggregate of
companies. This rate can be defined as the derived of costs with respect to the proxy
variable of technical progress, i.e. the trend. If the sign of the derived is negative,
technical progress exists, while if the sign is positive, there will be technical regress.
11
It is usual to express the index of economies of scale as the inverse of the sum of such elasticities.
will present the values relating to the elasticities in order to simplify the subsequent treatment.
23
However, we
Table 4. Economies of scale.
ES
ESP (Y1)
ESF (Y2)
Estim.
Stand. Err. Estim.
Stand. Err. Estim.
Stand. Err.
BR
1,1465
3,27E-02
0,7633
5,74E-02
0,3833
0,34957
CFF
0,9458
3,30E-02
0,5681
1,00E-01
0,3776
0,28673
CH
0,5527
0,42235
0,2916
1,2717
0,26107
0,4847
CP
0,6820
0,21332
0,4036
0,67148
0,2783
0,62664
DB
1,2924
4,43E-02
0,8554
0,10022
0,4370
0,20659
DSB
0,7319
0,1424
0,4345
0,39521
0,2974
0,21145
FS
1,0894
2,99E-02
0,6855
9,84E-02
0,4039
0,28455
NS
0,9824
3,55E-02
0,4839
0,1798
0,4984
9,82E-02
NSB
0,7233
0,15585
0,4327
0,19359
0,2906
0,63216
OBB
0,8486
5,42E-02
0,6165
8,12E-02
0,2321
0,64132
RENFE
0,9543
2,52E-02
0,6583
8,28E-02
0,2960
0,39867
SJ
0,9678
2,76E-02
0,6072
0,11175
0,3606
0,42801
SNCB
0,8754
5,00E-02
0,5527
0,11686
0,3226
0,21197
SNCF
1,2000
4,19E-02
0,8456
9,61E-02
0,3544
0,28773
VR
0,7345
0,14056
0,5105
0,11917
0,2239
1,2906
It can thus be observed that in most cases the estimate of technical progress presents a
negative sign, thus indicating that an improvement has occurred in the level of
productivity thanks to technological change. In any case, for many of the companies it
cannot be accepted that there have been increases in productivity thanks to technical
progress. The companies with greatest technical progress, according to these results,
were DB, SNCF, VR, RENFE, ÖBB and SJ.
Having estimated the movement of the cost function due to technical change (TC), the
movements throughout the cost function due to economies of scale (ES), and the

changes in companies’ efficiency ( EF ), the next step is to estimate the rate of growth of

total factor productivity ( FTP ) by means of the following expression:
24




FTP   TC  (1  ESP) Y1  (1  ESF ) Y2  EF

(4)

where Y1 and Y2 are respectively the annual rates of growth for the two outputs
(passenger and freight train-Kms).
Table 6 shows the annual growth rate of total factor productivity, as well as the
breakdown of this growth into the contributions of gains in efficiency, economies of
scale, and technical change. It is observed that, taking an average of all the companies,
productivity growth has taken place at an annual rate of 0.81%. This rate was due
mostly to technical change (0.45%), to a lesser degree due to gains in efficiency levels
(0.19%) and lastly due to changes of scale experienced by the companies (0.16%).
Observe, however, that in some companies the efficiency gains were a source of
productivity growth even more important than technical progress, the case of CP being
outstanding with a very high rate of efficiency gains.
These results can be compared (despite the difficulties noted earlier) with those obtained
by Gathon and Pestieau (1995), who also found that the greater part of the growth in the
level of productivity is explained by technical change and to a much lesser degree by
changes in efficiency. Also, the companies that experienced evident improvements in
their efficiency levels were, in this order, CP, FS, BR and ÖBB (companies that at the
start of the period were relatively inefficient). In particular, a negative correlation
(-0.54) is obtained between the levels of efficiency at the start of the period and their
25
rate of growth, and thus convergence occurs in the levels of efficiency.
26
Table 5. Technical change (TC).
Estim.
Stand. Error
BR
-7,88E-03
4,02E-03
CFF
-1,07E-03
3,58E-03
CH
6,33E-03
4,36E-03
CP
-2,66E-03
4,85E-03
DB
-2,00E-02
4,62E-03
DSB
6,36E-03
3,73E-03
FS
8,26E-04
4,42E-03
NS
1,19E-02
4,87E-03
NSB
-9,55E-04
3,53E-03
OBB
-1,20E-02
4,12E-03
RENFE
-1,46E-02
3,29E-03
SJ
-1,03E-02
4,20E-03
SNCB
-1,88E-03
3,17E-03
SNCF
-1,96E-02
4,28E-03
VR
-1,58E-02
3,86E-03
t-statistic
-1,967
-0,298
1,454
-0,548
-4,337
1,708
0,187
1,437
-0,271
-2,913
-4,434
-2,458
-0,592
-4,573
-4,087
Table 6. Growth of TFP and its determinants.
EF
BR
0,0067
CFF
0,0023
CH
0,004
CP
0,0121
DB
-0,006
DSB
-0,009
FS
0,0071
NS
-0,0009
NSB
0,0006
ÖBB
0,0045
RENFE
0,0001
SJ
-0,002
SNCB
0,0014
SNCF
-0,0002
VR
0,005
Average
0,0019
*
EST = (1  ESP) Y1 + (1ESF) Y2
EST*
-0,0129
0,0091
-0,0166
0,0096
-0,0024
0,0046
0,0033
-0,0004
0,0076
0,0160
0,0124
-0,0038
0,0019
-0,0060
-0,0051
0,0016
27
PT
0,0078
0,0010
-0,0063
0,0026
0,02
-0,0063
-0,0008
-0,0119
0,0009
0,012
0,0146
0,0103
0,0018
0,0196
0,0158
0,0045
TFP
0,0016
0,0125
-0,0189
0,0244
0,0115
-0,0107
0,0096
-0,0132
0,0091
0,0325
0,0271
0,0044
0,0052
0,0133
0,0156
0,0081
6. Conclusions
Most studies in the field of transport economics have concentrated on the estimation of
economies of scale and therefore the evaluation of the so-called efficiencies of scale.
There are very few studies that have evaluated the levels of X-efficiency of the
European railway companies, or that have estimated the growth in total factor
productivity (TFP) and its sources using frontier approaches which enable gains in
efficiency to be explicitly considered as an independent source of productivity growth.
Our study, unlike that of Gathon and Pestieau (1995) has estimated a function for the
companies’ operating costs using a stochastic frontier approach. Likewise in our
equation to represent the sources of growth of TFP we have included, as well as
technical progress and catching-up, the inefficiencies of scale, as a further element
explaining the increases in productivity of the companies.
Our results are similar to those obtained by the above-mentioned authors, since
technical progress, and not variations in the efficiency level, seems to be the factor
explaining in greater measure the growth of companies’ productivity, although in some
countries the gains in efficiency constitute a very important source of productivity
growth. Inefficiencies of scale are, on the other hand, the factors that least explains the
evolution of TFP. At any rate, the inefficiencies of scale obtained are relatively more
important than those normally obtained in other sectors. This is explained by the small
size of the sample and the great diversity of the companies, which gives much
28
importance to the differences of scale among them.
Our results also confirm the enormous importance of technical progress as an element
favouring productivity in a sector with the technical characteristics of the railways. For
this reason, policies of encouragement to invest and of R&D are vital aspects for this
sector. Thus it must be ensured that in a context like the present, of increasing
de-regulation and liberalisation in the sector, the levels of investment and technological
development are adequately safeguarded.
At present, there many private companies that operate as franchises (the UK in Europe,
Argentina, Chile or Brazil in America). In most cases, it has been demonstrated
(Kopicki and Thompson, 1995) that these companies tend to default on their plans for
investment and modernisation. This tendency may lead to important losses in
productivity and short term and long term results, very harmful in the long run, if the
levels of investment in infrastructure are not substantial enough. Thus, in the UK, the
levels of investment undertaken by the company owning the infrastructure, Railtrack,
can be considered, according to the Regulator’s office (Swift, 1997) clearly
unsatisfactory, which also explains the low levels of investment by the operators.
With regard to the determinants of the efficiency of the companies, our results are in
line with those obtained in other studies (Oum and Yu, 1994, and Gathon and Pestieau,
1995). Thus, the greater the freedom and independence of the managers of the
29
companies in decision making, the greater the efficiency of the companies. Furthermore,
the companies with the highest degree of autonomy of management and finance are
clearly the most efficient. It is therefore to be expected that the higher the level of
subsidy received by the companies, the more inefficient their behaviour will be.
30
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37
Table A.1. Appendix. Frontier Cost Function.
Estimates
t-ratio
0
14.479
485.544
1
0.42930
6.444
2
0.21985
4.056
21
0.58180E-01
1.158
13
-0.15758
-3.027
23
-0.23681
-5.828
1
0.53824
11.207
2
0.53030
13.328
11
0.16823E-01
0.728
22
-0.73708E-01
-2.251
12
0.12639
2.472
11
-0.29835E-01
-0.456
12
0.22941E-01
0.363
21
0.12666
2.346
22
-0.45153E-01
-0.839
T
-0.31295E-02
-0.589
TT
-0.65511E-03
-2.691
T1
0.11129E-01
3.304
T2
-0.16773E-01
-5.358
1T
0.87941E-02
38
Standard-Error
0.29821E-01
0.66622E-01
0.54202E-01
0.50221E-01
0.52053E-01
0.40631E-01
0.48027E-01
0.39788E-01
0.23120E-01
0.32751E-01
0.51122E-01
0.65443E-01
0.63189E-01
0.53988E-01
0.53837E-01
0.53105E-02
0.24349E-03
0.33689E-02
0.31302E-02
0.45981E-02
1.913
2T
-2.601
u/v
5.778
(v2 + u2)1/2
24.646
-0.97513E-02
0.37489E-02
4.6014
0.79641
0.21243
0.86193E-02
Number of obser. = 306
Log likelihood function: 217.2123
Variance components: v2 = 0.00204
u2 = 0.04309
39