ASSESSING THE MARGINAL INFRASTRUCTURE WEAR AND TEAR
COSTS FOR GREAT BRITAIN’S RAILWAY NETWORK
Phill Wheat
Institute for Transport Studies, University of Leeds
Andrew S.J. Smith
Institute for Transport Studies, University of Leeds
1. INTRODUCTION
Amongst other directives in the European Commission’s First Railway
Package was 2001/14/EC which sets rules on charges to be levied on
operators for the use of member states networks. Charges should be based
on the direct cost of running the service, that is, the price of access should
reflect the costs to the infrastructure manager and society as a whole from
running the additional service.
Railways in Great Britain have undergone major reform. Prior to 1993, the
railway in Great Britain was vertically integrated and public owned. In 1993 a
set of measures were implemented which resulted in the formation of
Railtrack plc now Network Rail, a private company responsible for the
management of the infrastructure. Subsequently passenger services have
been franchised to private operators and freight services privatised with the
freight market subject to full open access. This was in order to separate the
natural monopoly element, believed to be the infrastructure, from areas that
were susceptible to competition either in- or for- the market.
Given the vertical separation in the UK system, it is important to establish
appropriate charging systems to optimise the allocation of capacity to
operators and give incentives to operators to use appropriate rolling stock.
Infrastructure access charges in the UK consist of a two part tariff for
franchised passenger operators. This consists of a fixed charge, that does not
vary with usage and a variable charge. Freight and passenger open access
operators only pay the variable access charge and the variable charge is
based on the principle of marginal cost.
A key research need, both to support policy in Great Britain and more widely
in the European Union, is the need to value the marginal social cost of
network usage. A substantial component of this is the marginal cost to the
infrastructure manager; the additional maintenance and renewal cost resulting
from running an extra service. However, traditionally, both industry and
academic work on railway costs has focused on the characteristics of the
vertically integrated railway; but as a result of the restructuring there been a
strong need to examine the interaction between operations and infrastructure
for pricing purposes. In academia, innovative econometric work has emerged
following the seminal paper by Johansson and Nilsson (2002, reprinted in
2004).
©Association for European Transport and contributors 2006
In Great Britain marginal wear and tear costs are currently estimated using a
hybrid approach between cost allocation and engineering approaches (see for
example ORR, 2000 and Booz Allen & Hamilton, 1999 and 2005). These rely
heavily on engineering judgement. This paper contributes to the literature by
utilising the econometric approach using a cross section data set of 53
“Maintenance Delivery Units” for the Great British network in 2005/06. The
approach is based on the examination of past data and should provide a
useful objective benchmark for the results from other approaches. This data
has now become available following the move to take maintenance in-house
by the infrastructure manager. We estimate a double-log (Cobb Douglas) cost
function for Permanent Way maintenance cost and compute marginal cost for
different characteristics of MDUs.
The structure of the paper is as follows. Following this introduction, Section 2
briefly introduces the alternative approaches taken to estimate marginal costs
of infrastructure use. Section 3 outlines the data set available for the study
and also the particular methods used to estimate the cost function. Section 4
discusses the results in terms of the final models and implied elasticity and
marginal cost estimates and benchmarks these against results from other
studies. Section 5 concludes.
2. LITERATURE REVIEW
Estimation of marginal infrastructure costs can be characterised into two
groups: bottom-up approaches and top-down approaches (Link and Nilsson,
2005). Bottom-up approaches rely on engineering models and judgement to
determine the likely wear and tear impact of running an extra vehicle on
different components of the infrastructure network. Top down approaches use
data on costs of maintaining and/or renewing the infrastructure and estimate
what proportion of these costs are variable with traffic. The top down
approach may be implemented through two methods: estimation of an
infrastructure cost function using econometric techniques; and cost allocation
methods which allocate constituent parts of total cost to common cost drivers
and then use engineering judgement to determine the variabilities of these
categories with the cost driver.
In this paper we follow the econometric approach. For the purpose of this
shortened conference paper, we do not review the literature in this area; see
Link and Nilsson (2005) for such a review.
3. METHODOLOGY AND DATA
We now outline the methodology and data adopted for our study.
3.1. Cost function estimation
The aim of the exercise is to estimate the marginal cost of running more or
less traffic on a fixed network; that is we wish to calculate short run marginal
cost. The variable cost function relates total variable costs to output
©Association for European Transport and contributors 2006
variables, prices of variable inputs (labour, energy etc) and levels of fixed
inputs (route miles, number of switches and crossings etc.); see, for example,
NERA (2000).
In keeping with the majority of the literature we adopt a double log (Cobb
Douglas) functional form:
ln(C i ) = α + β1 ln(Q1i ) + Λ + β q ln(Q qi ) + γ 1 ln(V1i ) + Λ + γ v ln(Vvi ) + δ1 ln(P1i ) + δ p ln(Ppi )
i = 1,2, Κ , N
where
• C i is the cost per annum for firm or unit i (in our case, the units are
based on 53 regional, maintenance delivery units or MDUs);
• Q i is a vector of outputs for MDU i – here we consider output to be
train related measures of output, primarily because this is the stage of
production for which we wish to derive marginal costs;
• Vi is a vector of fixed input levels for MDU i – In the short run several
factors of production are fixed. These are assumed to be the
infrastructure. Therefore, measures that naturally fit in here are track
length, track quality, track capability and track age in a MDU;
• Pi is a vector of input prices; and
• N is the number of MDUs
The specification allows for non-constant marginal effects and it is using this
specification that we tested the alternative possibilities for the composition of
each category of variables.
However this form is restrictive since it assumes constant cost elasticities.
The Translog cost function incorporates additional second order interaction
terms which yields a less restrictive specification. Unfortunately it became
apparent that the data, both in terms of variability and degrees of freedom, did
not support the estimation of such a complex functional form. We do,
however, test the inclusion of second order terms for key variables. In
particular, we consider the evidence that the elasticity of cost with respect to
traffic density is not constant.
3.2. Cost Data
In this study we have a cross section of data provided by Network Rail for 53
Maintenance Delivery Units (MDUs) for 2005/06. However not all cost data is
available at this level of disaggregation, and some data is only available at the
area level (the 53 MDUs aggregate up into 18 maintenance areas). Table 1
shows the categories by which maintenance cost data was available.
60% of total maintenance expenditure is available at the MDU level. The
remaining expenditure (40% of the total maintenance budget) includes
maintenance of electrification and plant equipment and other expenditure and
can not be allocated to individual MDUs. Instead it is allocated to one of 18
©Association for European Transport and contributors 2006
Maintenance Areas or more aggregate levels. Given the small sample size we
do not analyse the maintenance area data in any great detail.
Discussions with the industry suggested that the Permanent Way expenditure
was the cost category that would be subject to substantial usage related
variability. However concerns were raised as to the consistency of cost
allocation between Permanent Way and General maintenance activity and so
it was decided to analyse the combined Permanent Way and General
expenditure categories together.
Table 1 Cost categories available for analysis
Cost Category
Proportion
Level at Which Coverage
of Total
Data is
Maintenance Available
Expenditure
Signalling
and 15
Maintenance
Includes the maintenance of signalling
telecoms
Delivery Unit
(signals, cables, signal boxes) and
telecoms (signal post telephones)
Permanent Way
34
Maintenance
Includes maintenance of track, ballast
Delivery Unit
and sleepers
MDU General
11
Maintenance
The remaining expenditure which is
Delivery Unit
incurred at the MDU level including
general
depot
costs
such
as
management and production, expenses,
vehicle hire and property costs.
Electrification and 7
Maintenance
Includes both maintenance of (i) contact
Plant
Area
systems (both overhead lines and 3rd Rail
Conductor) (ii) plant (pumping stations,
signal power supplies, points machines)
and (iii)
Area other
26
Maintenance
Includes area services, area track
Area
engineer and area overheads
Territory and HQ
7
Territory
and Not available for analysis
HQ
Source: Network Rail
3.3. Output data
Our data set includes data about both the number of train miles and number
of tonne miles per MDU. From the raw data, there are many potentially useful
transformations of this data which can be used and Table 2 highlights the
transformations that are used in subsequent analysis. Following the findings
of Gaudry and Quinet (2003), who found a difference between the impact of
additional gross tones resulting from heavier trains, as opposed as a result of
more trains of the same weight, we adopt both a measure of the density of
train miles per track-mile and the average weight of those trains1. We aim to
test for statistically different impacts from each factor. Where we can not
distinguish between there two separate effects we adopt the gross tonne
miles per track mile measure.
Data is available at three levels of disaggregation; from total traffic at the
highest level to intercity passenger traffic, other passenger and freight traffic
at the most granular level. Efforts have been made to investigate whether,
©Association for European Transport and contributors 2006
after accounting for the average weight of trains, there exists detectable
differences in the wear and tear impacts of different types of trains.
Table 2 Output data available for analysis
Variable
Units
Name
Total trains
Average trains TOT_TRAIN
per track mile
per MDU
Average weight Tonnes
per TOT_AWT
of trains
train
Total trains
Tonne
miles TOT_TON_T
per track mile
Total
Tonne
miles TOTPA_TON
passenger
per track mile
trains
Total
freight Tonnes miles TOTFR_TON
trains
per track mile
Mean
16324
St. Dev.
6536
Minimum
5694
Maximum
31625
293.0
84.5
122.1
502.0
4809570
2304830
1172371
9027768
3345735
2110686
355828
8032005
1463835
975525
26286
3652366
3.4. Infrastructure Capability/Quality Data
In order to control for the impact on cost from the levels of the fixed factors in
estimation of short run marginal cost, it is necessary to select variables which
reflect the quantity, capability and quality of the fixed infrastructure. Our data
set includes a wide variety of measures per MDU including variables in the
following categories:
• Track Length including length by track type
• Route Length
• Maximum Line speed capability of track
• Maximum Axle load capability of track
• Numbers of each signal type and Signalling equivalent units
• Electrified track length (and by type of electrification)
• Rail Age
Firstly, we drew on engineering expertise from within Network Rail and our
own understanding and research to determine which categories of variables
are relevant for the analysis of the Permanent Way cost category. Clearly the
length of track will have an impact on the maintenance requirements and
there are strong arguments as to why Continuously Welded Rail (CWR),
jointed track and switches and crossings (S&C) should be distinguished
between. S&C are relatively complex pieces of track and so the cost of
maintaining them should be relatively high. Jointed track, while cheaper to
install than CWR, is more susceptible to usage related damage and so should
also be distinguished in the analysis.
Route length relative to track length is a useful proxy for network
concentration and so models would benefit from its inclusion, however its
expected sign is ambiguous. On one hand a higher network concentration
may mean possessions are easier to obtain as, say a four track line can be
partially closed rather than a singled line that has to be completely closed and
a concentrated network may mean reduced costs in getting to/from site.
©Association for European Transport and contributors 2006
However there is also an argument that the extra safety and operational
measures required to work on a partially open line may outweigh these gains.
Both maximum line speed and maximum axle load measures are measures of
the capability of the track to allow the carriage of more demanding traffic.
These variables may act as proxies for two key cost drivers. The first is that a
higher line speed or axle load capability will reflect a greater quality of
construction of the track which is expected to require less maintenance. On
the other hand higher line speed and axle load capabilities will usually be
associated with usage by higher speed and axle load traffic. It is expected that
this traffic damages the track to a greater extent than other traffic. As such the
overall effect on cost of these variables is ambiguous. We do note that some
variables, notably CWR track vis-à-vis jointed track, should also pick up the
relative quality differences of track and thus their inclusion in any specification
may result in a more positive effect on costs of the capability variables2.
Apart from some minor issues, we do not expect signalling and electrification
variables to reasonably reflect underlying drivers of cost. Thus we do not
consider them as part of the specification.
Older rail age is expected to increase maintenance costs. The older the track
the more maintenance it requires to keep it to a given standard.
However, the selection of categories of variables is not the end of this variable
selection process; there exists many possible variables that are potential
measures for each cost category. The measures chosen are shown in Table
3, where the choice was made on statistical grounds (further details on the
process by which the particular measures were chosen are available from the
authors).
The infrastructure variables that were found to be statistically superior are
shown in Table 3.
3.5. Input Prices
We have data on the price of labour in each MDU and this is summarised in
Table 3. This data is not railway specific; instead it refers to the general level
of wages in the geographical areas of each MDU. This was derived through
National Statistics data. We do not have data on the price of materials and
machinery, however we assume that this is constant between MDUs and thus
its effect is absorbed within the constant term.
©Association for European Transport and contributors 2006
Table 3 Non-output variables chosen for
function
Name
Description
Infrastructure variables
TRA_LEN
Length of Track (in logs)
MAL_G25_P
Proportion of track length
with maximum axle load
greater than 25 Tonnes
MLS_G100_
Proportion of track length
with maximum line speed
greater than 100 mph
TRACWR_PR Proportion of track length
which is CWR
AGE_G30_P
Proportion of track length
with rail age greater than
30 years
Price variables
WAGE
Labour price index by
MDU (in logs)
possible inclusion in the Permanent Way cost
Mean3
St. Dev.
Minimum
Maximum
367
0.55799
153
0.21334
132
0
873
0.93397
0.1743
0.19006
0
0.68408
0.75918
0.11362
0.42322
0.91572
0.45211
0.12995
0.21732
0.73305
13
2.02219
10
17
3.6. Summary of specifications
The discussion above about the functional form and explanatory data leads to
a number of possible model formulations that need to be and are tested as
part of the analysis. Table 4 summarises these.
Table 4 Summary of specifications
Model
Output
Infrastructure data
Model I
Full set
Model II
Model III
Model IV
Model V
Model VI
Train miles per track mile and
AWT
Train miles per track mile and
AWT
Tonnage miles per track mile
Passenger tonnage miles per
track mile and Freight tonnage
miles per track mile
Tonnage miles per track mile
and the square of Tonnage miles
As in IV but with second order
interaction
terms
between
passenger and freight variables
Input
Prices
Wage
Full set minus AGE_G30_P
(see section 4.1 for
explanation)
Full set minus AGE_G30_
Full set minus AGE_G30_
Wage
Wage
Full set minus AGE_G30_
Wage
Full set minus AGE_G30_
Wage
Wage
4. RESULTS
The preferred models are shown in Table 5. We first describe each model and
benchmark our findings against those in the literature and then we proceed to
calculate estimates of marginal cost.
©Association for European Transport and contributors 2006
4.1. Models Estimated
Model I shows the analysis of total traffic measured by train density (number
of trains per track mile) and average weight of trains. First note that we find
three outlying observations for this set of models for which we include dummy
variables. The overall fit, R 2 , is 0.78, which is in line with other models taking
a double log functional form estimated by other researchers in the field. The
elasticity of cost with respect to train density ( ε Train ) is 0.202 and with respect
to average weight of trains (AWT) is 0.34. Both are significantly different from
zero at the 5% level however a Wald test of the null hypothesis that both
elasticities are equal can not be rejected at any reasonable significance level.
This indicates that we can not reject the hypothesis that a proportionate
increase in AWT holding the number of trains constant has the same effect on
costs as the same proportionate rise in the number of trains holding AWT
constant.
The elasticity of cost with respect to track length ( ε Track ) is 0.5110 which is
significantly different from zero even at the 0.1% level. Importantly it is also
significantly different from one even at the 0.1% level. We note that this
implies an estimate of economies of scale (EoS) of 1.96 which is slightly
greater than from other studies (see Table 7), where we define EoS (in line
with Oum and Waters 1997) as
EoS = 1 (ε Track )
However, like in all other studies, there are increasing EoS.
The Wage variable has a coefficient with the expected sign and is significantly
different from zero at the 5% level. The coefficient indicates that a 1% rise in
wages results in a 0.5% rise in cost. Of the three variables that proxy for track
quality, not all are significantly different from zero even at the 10% level.
However we included them within the specification, since removing them
resulted in a relatively large change in the coefficient on TOT_TRAIN with little
reduction in standard error. As such we kept these variables in the
specification to guard against omitted variable bias.
However, we are concerned about the sign of the coefficient on the age
variable. It seems counter intuitive that older track is less costly to maintain
once track quality and usage have been controlled for. We note that this result
is robust across a range of models. A genuine reason for this maybe that for
assets that are very close to the end of their lives, maintenance effort maybe
reduced since these assets are marked for renewal. However, another reason
might be the limitations of the age measure that we have adopted. In
particular we recognise that age itself is not the driving factor behind
maintenance requirement but instead the proportion of life expired is a more
appropriate measure. This however is a function of usage and track
capability/quality which are included in the specification. In addition, the age
variable maybe picking up other usage characteristics, such as the speed of
trains (lower speed trains tend to be on assets which are older). Thus for
©Association for European Transport and contributors 2006
these reasons and because of the counter-intuitive sign on its coefficient, it is
considered that the age measure is misleading and should be removed.
Model II is a re-estimation of model I excluding the age variable. We accept
that the fit of the model has fallen slightly to R 2 = 0.742 , however the overall
model is thought more robust than model I for the reasons outlined above.
Both the elasticity of cost with respect to train density and AWT have
increased to 0.302 and 0.399 respectively due to the removal of AGE_G30_P
that is negatively correlated with both TOT_TRAIN and TOT_AWT (-0.542
and -0.304 respectively). All coefficients are statistically significant from zero
at the 5 % level except some of the outlier dummies.
However, the Wald test for the null hypothesis of equivalence between the
coefficients on TOT_TRAIN and TOT_AWT could not be rejected at any
reasonable significance level. This seems to be a robust result, thus we reestimate the model with just tonne density (TOT_TON_T) rather than the two
output measures. Model III shows the results. The fit of the model has fallen
very slightly, R 2 = 0.740 , as expected and the estimates of the elasticity of
cost with respect to tonnage density is significant at the 0.1% level and is
0.331, which is to be expected as this is between the two estimates using
TOT_TRAIN and TOT_AWT in model II. Otherwise all other coefficients are in
line with those for model II.
In model IV we investigate the usefulness of distinguishing between
passenger and freight traffic using the gross tonne measure of output4. We
first note the fall in the fit of the regression, with R 2 = 0.691 . We also note that
the freight coefficient is not significantly different from zero at the 10% level5.
Given that passenger traffic accounts for a substantial proportion of total
traffic, it should be no surprise that the elasticity of cost with respect to
passenger traffic is greater than the corresponding elasticity for freight. This is
not the same as saying that the marginal cost of passenger tonne-kms is
greater than the marginal cost of freight tonne-kms. Whether this implies a
difference in marginal cost per tonne-km will be discussed in subsequent
sections. We do note that the elasticity of cost with respect to total tonnage
density is 0.249 which is smaller than in model III.
Given the restrictive nature of the Cobb Douglas functional form, Model V
introduces a second order term in total tonnage density, TOT_TONSQ (equal
to TOT_TON_T2)6. The fit of this model is superior to that in model III with
R 2 = 0.756 . Except for the coefficient on WAGE and MLS_G100_ all
coefficients are significantly different from zero at the 5% level. The elasticity
of cost with respect to tonnage density is no longer constant and given by:
∂ ln Cost
= β1 + 2β 2 TOT _ TON _ T where β1 and β 2 are the
∂TOT _ TON _ T
coefficients on TOT_TON_T and TOT_TONSQ respectively.
ε Tonnage =
Figure 1 plots the elasticity of cost with respect to tonnage density and the
associated 95% confidence interval. Evaluated at the mean tonnage density,
©Association for European Transport and contributors 2006
the elasticity is 0.239. The elasticity is falling with gross tonnage density and
this finding seems to be robust to inclusion/exclusion of other explanatory
variables. The implication is that at higher tonnages marginal cost is a smaller
proportion of average cost, other things equal. This observation is in line with
findings from Johansson and Nilsson (2004) and Andersson (2005) using
Swedish data, however Gaudry and Quinet (2003) find an increasing
relationship using French data.
Finally, Model VI generalises model V to examine the effects and interactions
between passenger and freight traffic. The overall fit of the model has
improved over that in model V, R 2 = 0.764 , however at the loss of three
degrees of freedom, which thus results in a fall in R 2 . An F-test of the null
hypothesis that all of the coefficients on the second order terms are zero is
rejected at any reasonable significance level. Thus we conclude that the
second order model provides some additional explanatory power over the
simple Cobb Douglas model. MLS_G100_ is not in the final specification as
when included its coefficient had a very low t statistic and the precision of
estimation of the other coefficients improved considerably following its
removal. All of the coefficients on the output variables are significantly
different from zero at the 5% level except the coefficient on TOTP_TONS, i.e.
the square of TOTPA_TON and TOTF_TONS i.e. the square of TOTFR_TON.
The cost elasticity for passenger traffic is given by:
∂ ln Cost
= β1 + 2β 2 TOTPA _ TON + β 3 TOTFR _ TON
∂TOTPA _ TON
= 3.9097 − 0.07105TOTPA _ TON − 0.19569TOTFR _ TON
ε Tonnage,Passenger =
And for freight traffic:
∂ ln Cost
= β 4 + 2β 5 TOTFR _ TON + β 3 TOTPA _ TON
∂TOTFR _ TON
= 2.2374 + 0.05627TOTFR _ TON − 0.19569TOTPA _ TON
ε Tonnage,Freight =
©Association for European Transport and contributors 2006
Figure 1 Elasticity of cost with respect to tonnage – Model V
Elastciity of cost with respect to tonne density
0.8
0.7
0.6
Elastciity
0.5
0.4
0.3
0.2
0.1
0
0
1000000
2000000
3000000
4000000
5000000
6000000
7000000
8000000
9000000 10000000
-0.1
Tonnage per track-km
-0.2
Elasticity
Lower CI
Upper CI
Figure 2 plots the elasticity of cost for passenger and freight tonnage density.
First note that some observations have a negative elasticity. This is
counterintuitive given it implies negative marginal cost. However for all of the
observations with elasticities less than zero, the 95% confidence interval
spans zero, thus we can not reject the null hypothesis that each observation
has a positive elasticity.
Holding freight tonnage constant, increasing passenger tonnage decreases
the passenger elasticity, which is in line with the observation for total tonnage.
However holding passenger tonnage constant, increasing freight tonnage
increases the freight elasticity. We also note the highly significant negative
coefficient on the interaction variable between passenger and freight tonnage.
This implies that there are economies of scope in infrastructure maintenance
costs in running both passenger and freight on the same network as opposed
to on two different networks.
We do, however, have concerns about the robustness of this model. First, as
noted above, some of the coefficients on key output variables are not
significantly different from zero at a reasonable level of statistical significance.
Second, when evaluated at the mean passenger and freight tonnage, the
passenger and freight cost elasticities are 0.064 and 0.096. It is surprising that
the freight elasticity is greater than the passenger elasticity and also that the
total tonnage cost elasticity is only 0.160, which is substantially smaller than in
all other models.
©Association for European Transport and contributors 2006
Figure 2 Elasticity of cost with respect to tonnage – Model VI
Elasticity of cost with respect to tonnage density
1
0.8
Elastcity
0.6
0.4
0.2
0
0
1000000 2000000 3000000 4000000 5000000 6000000 7000000 8000000 9000000
-0.2
Tonnage per track-km
Passenger
Freight
Log. (Freight)
Log. (Passenger)
Finally, we note a general difficulty associated with the analysis of data at the
MDU level. An MDU is defined as the collection of tracks for which a specific
maintenance depot is responsible. As such the tracks included within an MDU
will not have uniform traffic, quality and capability characteristics. This,
coupled with the limited number of data points and thus degrees of freedom,
means that it is very difficult to adequately capture the effects of traffic, quality
and capability in the econometric specification. Once better data is available,
either via formation of a panel dataset through collection of more years of data
or by disaggregating costs down to individual track sections, a more robust
cost function can be specified and estimated. These issues may also explain
some of the counter intuitive signs on the CWR variable for example.
©Association for European Transport and contributors 2006
Table 5 Preferred models
Model
Regressor
I
II
t-stat in brackets
III
IV
V
VI
Constant
8.0583
6.3244
6.3872
7.8743 -34.4368 -34.0127
(6.2638) (5.2147) (5.3203) (6.4783) (-1.8408) (-2.2114)
Output
TOT_TRAIN
TOT_AWT
0.20209 0.30189
(2.3174) (3.5247)
0.34117 0.39877
(2.8139) (3.0982)
TOT_TON_T
0.33077
(4.5723)
TOTPA_TON
5.8336
(2.3356)
0.17365
(3.1368)
0.074929
(1.5471)
TOTFR_TON
TOT_TONSQ
3.9097
(2.5391)
2.2374
(2.7283)
-0.1818
(-2.1979)
TOTP_TONS
-0.035524
(-.88760)
0.028137
(1.5695)
-0.19569
(-4.2511)
TOTF_TONS
TOTPF_TON
Infrastructure
TRA_LEN
MAL_G25_P
MLS_G100_
TRACWR_PR
AGE_G30_P
0.511
(6.8354)
-0.41785
(-2.4785)
-0.30947
(-1.6540)
0.49681
(1.5206)
-0.74702
(-2.7841)
0.51671
(6.4178)
-0.38337
(-2.1163)
-0.4295
(-2.1895)
0.81157
(2.14574)
0.53384
(7.0774)
-0.33452
(-2.0498)
-0.38442
(-2.1139)
0.76944
(2.3939)
0.49938
(5.9343)
-0.31764
(-1.3155)
-0.41203
(-1.9428)
0.89079
(2.4652)
0.48225 0.45786
(6.5289) (5.9042)
-0.48602 -0.63729
(-3.0237) (-2.8797_
-0.28955
(-1.6979)
0.84015 0.85925
(2.6658) (2.5697)
Prices
WAGE
0.48714 0.42502 0.40127 0.42352 0.23258 0.23591
(2.5206) (2.0549) (1.9857) (1.8770) (1.2099) (1.1892)
Dummies
DUM35
DUM42
DUM44
DUM51
R-Squared
R-Bar-Squared
Equation Log-Likelihood
DW-Statistic
0.33246 0.36044 0.35146 0.43436
(1.7644) (1.7780) (1.7499) (1.9776)
0.54256 0.47765 0.49019 0.47224 0.48788 0.57161
(2.8780) (2.3699) (2.4606) (2.1210) (2.5300) (2.9504)
-0.41782 -0.47956
(-2.2520) (-2.5461)
0.37498 0.29778 0.28928 0.27001
(2.0393) (1.5205) (1.4908) (1.2545)
0.78305
0.72484
25.0729
2.4826
0.74203
0.68061
20.4841
2.0387
Wald test for H0:
0.97117 0.41069
coefficient TOT_TRAIN= [.324]
[.522]
TOT_AWT
0.73951
0.68598
20.2262
2.055
0.69136
0.61787
15.7317
1.9529
0.7551
0.70434
21.9068
2.1188
0.76407
0.70077
22.8506
2.2457
F-Test for H0:
6.8946
coefficient all second [.001]
order = 0
F(3,41)
©Association for European Transport and contributors 2006
4.2. Marginal cost calculation
We now proceed to calculate the marginal usage costs. We first note the
relationship between the marginal usage cost for MDU i ( MC i ) and the usage
elasticity for that zone:
MC i
Ci
and AC i =
is the average cost
AC i
Tonne _ miles i
MC i = ε i ⋅ AC i since ε i =
of usage.
The estimate of MC i , MĈ i , is given by
MĈ i = εˆ i ⋅ AĈ i = εˆ i ⋅
Ĉ i
Tonne _ miles i
Following Munduch et al (2002), we note that from our double log formulation
and assuming that the residuals are distributed normal:
(
( )
)
ln(Ĉ i ) ~ N ln (C i ) − u i , σ 2 where ln Ĉ i is the fitted value of ln (C i ) from the
regression which is equivalent to the expectation of ln (C i ) conditional on the
vectors Q i , Vi , Pi , σ 2 is the variance of the error term and u i is the error term.
ˆ is distributed log-normal. As such the estimate of Ĉ is
It then follows that C
i
i
given by:
(
Ĉ i = exp ln (C i ) − û i + 0.5σˆ 2
)
Figures 3 and 4 show plots of the marginal cost estimates for Models I and II
and Models III-VI respectively. Apart from the freight marginal costs in model
IV, for all models marginal cost is falling with traffic density, but at a
decreasing rate. Thus there seems to be a levelling off of marginal cost at
higher tonnage density levels.
In addition, mean marginal costs are calculated as a weighted sum of each
zones marginal costs, weighted by tonnage-miles (or train miles in the case of
models I and II):
MĈ =
1
53
∑ Tonne _ miles
53
(
⋅ ∑ MĈ i ⋅ Tonne _ miles
i =1
)
i =1
Table 6 shows the mean marginal cost for each of the six models estimated.
We present for each model the mean marginal costs per gross tonne and per
train. We compute the gross tonnage estimate (trains estimate) for models I
©Association for European Transport and contributors 2006
and II (models III-VI) by multiplying (dividing) the trains estimate (tonnage
estimate) by AWT.
Figure 3 Marginal costs for Models I and II
1
0.9
Marginal cost per train mile £
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5000
10000
15000
20000
25000
30000
35000
Train miles per track mile
Model II
Model I
Log. (Model I)
Log. (Model II)
Figure 4 Marginal costs for Models III-VI
0.02
0.015
Marginal cost per tonne mile £
0.01
0.005
0
0
1000000
2000000
3000000
4000000
5000000
6000000
7000000
8000000
9000000
10000000
Gross tonnage miles per track mile
-0.005
-0.01
-0.015
-0.02
Model VI - Freight
Model VI - Passenger
Model V
Model IV - Freight
Model IV - Passenger
Model III
©Association for European Transport and contributors 2006
Table 6 Mean marginal cost estimates
IV
IV
VI
VI
(Passenger) (Freight) V
(Passenger) (Freight)
Model
I
II
III
£ Per Thousand
1.343
1.176
1.370
1.065
1.747
Gross
Tonne 1.046 1.563 1.712
Miles
£ Per Train Mile
0.3095 0.4627 0.5051
0.3016
0.9834
0.4055
0.2392
1.4603
4.3. Comparisons against other studies
Table 7 shows the results from other studies. It can be seen that the
estimates of the (average) elasticity of cost with respect to traffic density
range from 0.13 to 0.37. All the estimates from our models fit into this range.
Given that we adopt a narrow cost base i.e. focusing on Permanent Way,
which we would expect to contain the components of maintenance activity
that are most associated with traffic, we would expect our elasticity estimates
to be in the high part of the range observed in other studies. For models II and
III this is the case since the elasticity estimate is in excess of 0.3. For model I,
the elasticity estimate is only 0.2, however we feel that this is not a robust
model for reasons outlined above. The total elasticity in model IV may seem
slightly low given the expectations outlined above however we note that only
one study (Gaudry and Quinet, 2003) has found an average elasticity greater
than 0.27 and so 0.249 is at the high end of all the other studies’ estimates.
The elasticity for passenger is greater than freight as is expected as
passenger tonnage is substantially greater than freight tonnage.
The Booz Allen & Hamilton (2005) study utilised a cost allocation approach to
determining the variability of cost with respect to usage for Great Britain.
Overall for maintenance they found that 18% of cost was variable with usage,
however like with all of the cost studies it is necessary to consider the cost
base that was considered. This includes more items than our cost base. For
track maintenance specifically, the study estimated variability of 24% which is
more inline with our findings.
Models V and VI incorporate 2nd order traffic terms. Model V seems to give a
reasonably acceptable average elasticity of 0.239. We are surprised about the
result that the elasticity of cost with respect to traffic density is falling with
traffic density as this indicates that marginal cost is decreasing at a faster rate
than average cost. However, we do note that this is consistent with the
findings of Johansson and Nilsson (2004) and Andersson (2005) for Sweden,
but that Gaudry and Quinet (2003) found the opposite result. Model VI gives
us cause for concern both in terms of the average total elasticity and that the
passenger elasticity is smaller than the freight elasticity. Since passenger
tonnage-miles substantially exceed freight tonnage-miles on the network (see
Table 2), it seems difficult to believe this result.
©Association for European Transport and contributors 2006
Table 7 Results from other studies compared against the estimated models
Study (maintenance Country
Returns to
Elasticity of
Marginal Cost
costs only) / Model
track length
cost with
Estimates
estimated
respect to
(Average)
tonne-km
Euro per
Thousand
Gross Tonnekm
Johansson and
1.256
0.169
0.127
Nilsson (2004)
(average)
Sweden
Johansson and
1.575
0.167
0.239
Nilsson (2004)
Finland
(average)
Tervonen and
1.325
0.18
Idstrom (2004)
Finland
0.133-0.175
Munduch et al (2002) Austria
1.621, 1.449
0.27
0.55
Gaudry and Quinet
Not reported
(2003)
France
0.37 (average)
Andersson (2005)
Sweden
0.1944
0.293 (pooled
(average
OLS model)
pooled OLS
0.272 (random
model) 0.1837
effects model)
(average
Random effects
model)
Booz Allen &
UK
N/A
Proportion of
Hamilton (2005)
maintenance
cost variable
with traffic:
0.18; 0.24 for
track
maintenance
1.196
Model I
UK
1.957
0.202 (wrt
trains)
0.944
Model II
UK
1.935
0.302 (wrt
trains)
1.411
Model III
UK
1.873
0.331
1.545
Model IV
UK
2.002
0.174
Passenger
1.939
0.075 Freight
passenger
0.249 Total
1.698 freight
Model V
UK
2.002
0.239
(average)
1.978
Model VI
UK
2.184
0.064
Passenger
1.538
0.096 Freight
passenger
0.16 Total
2.522 freight
Only mean marginal costs are reported, however within studies there are
large variations in marginal costs for specific observations and even groups of
observations with specific characteristics. For example Munduch et al (2002)
found that the average marginal cost for secondary lines in Austria was 3.09
Euro per thousand gross tonne-km which is more than 5 times greater than
the average marginal cost! Thus it maybe of no great concern that the
average estimate is so large relative to those from other countries. However
further work is needed to benchmark these results and to understand the
reasons for differences in marginal costs observed.
©Association for European Transport and contributors 2006
5. CONCLUSIONS
In this paper we have outlined the estimation a cost function for Permanent
Way maintenance costs. This was motivated by the need to provide objective
benchmarks to existing cost allocation analyses for the UK and to utilise an
emerging rich data set following the taking of maintenance in-house by the
infrastructure manager, Network Rail.
We have tried to test hypotheses suggested in the literature. In particular we
can not find evidence to reject the use of tonne-kms as an output measure in
favour of using both train-kms and average weight of train s output measures.
We have found elasticity estimates that are broadly in line with other
econometric studies undertaken in Europe and also consistent with existing
cost allocation undertaken for the UK. Model V incorporated a variable
elasticity of cost with respect to tonnage density and the elasticity is found to
be a decreasing function of tonnage density. This seems counter intuitive as it
implies that marginal cost falls faster than average as tonnage density
increases. We do note that this finding is consistent with some but not all of
the literature e.g. Johansson and Nilsson (2002) and Andersson (2005).
Mean marginal cost estimates are between just 0.944 to 1.978 Euros per
Thousand Gross Tonne-km7, however these vary considerably between
observations. In general marginal costs are found to be falling with traffic
density with very low marginal costs associated with high tonnage density.
We conclude that the approach taken has produced results that are consistent
with econometric studies across Europe and cost allocation work undertaken
for the UK network. As such this method seems to be a promising line of
research. There exist a number of ways to take the analysis forward, including
the use of a more disaggregate data set should this develop and the use of
panel data techniques should more years of data become available. In
addition the analysis could be expanded by adopting a stochastic frontier, and
by applying more advanced approaches to deal with potentially endogenous
quality variables in the cost function.
©Association for European Transport and contributors 2006
REFERENCES
Andersson, M. (2005) Econometric Models of Railway Infrastructure Costs in
Sweden 1999-2002, Proceedings of the third Conference on Railroad
Industry Structure, Competition and Investment. Stockholm School of
Economics, Sweden.
Booz, Allen & Hamilton (1999). Railway infrastructure cost causation, Report
to the Office of the Rail Regulator, London.
Booz, Allen & Hamilton with TTCI UK (2005) Review of Variable Usage and
Electrification Asset Usage Charges: Final Report, Report to the Office of Rail
Regulation, London.
Gaudry, M., and Quinet, E (2003) Rail track wear-and-tear costs by traffic
class in France, Universite de Montreal, Publication AJD-66.
Johansson, P. and Nilsson, J. (2004) An economic analysis of track
maintenance costs, Transport Policy 11(3), pp. 277-286.
Link, H. and Nilsson, J. (2005) Infrastrucutre. In Nash, C. and Matthews, B.
Editors (2005) Measuring the Marginal Social Cost of Transport, Research in
Transportation Economics Volume 14. Elsevier, Amsterdam.
Munduch, G., Pfister, A., Sogner, L. and Stiassny, A. (2002) Estimating
Marginal Costs for the Austrian Railway System Working paper 78.
Department of Economics Working Paper Series. Vienna University of
Economics & B.A., Vienna.
NERA (2000) Review of Overseas Railway Efficiency Draft final report to the
Office of the Rail regulator. London
Office of the Rail Regulator (ORR) (2000) Periodic review of Railtrack's
access charges: Final conclusions Volume I, London Available at www.railreg.gov.uk
Tervonen, J., Idström, T. (2004) Marginal Rail Infrastructure Costs in Finland
1997-2002, Report from the Finnish Rail Administration. Available at
www.rhk.fi accessed 20/07/2005.
©Association for European Transport and contributors 2006
NOTES
1
We note that because we adopt a double log functional form (see below) the model using
these two measures is the same, all other things equal, as a model with say trains per trackkm and tonnage per track-km i.e. the coefficients from one model can be combined to give
the coefficients in the other model. The choice of this formulation is that the coefficients
estimated are directly of interest, in particular the coefficient on trains per track-km can be
interpreted as the elasticity of cost with respect to traffic density holding average weight of
trains constant.
2
It should be noted that although we have a variable AWT, this is a measure of the weight of
trains rather than an axle load weight measure. Thus we can not separately control for the
actual weight axle loads run on the network.
3
Mean, St Dev, minimum and maximum are for the variable before it has been subject to the
ln transformation, where applicable
4
Very little success was found using the number of trains and AWT
5
In addition there maybe a case for exclusion of the MAL_G25_P variable on the grounds of
an insignificant t-statistic, although doing this results in large changes in the values of other
coefficients and substantial reductions in the precision of estimation of other coefficients. As
such it is retained.
6
Note, we had little success in incorporating a second order term in track length and an
interaction term between track length and tonnage density and do not report these models
here.
7
Excluding model VI.
©Association for European Transport and contributors 2006