DEVELOPMENT OF HYBRIO COST FUNCTIONS FROM
ENGINEERING AND STATISTICAL TECHNIQUES: THE CASE OF RAIL
Final Report to the
U.S. Department of Transportation
Under Contract
DOT-0S-70061
October, 1984
Northwestern University
Principal Investigator: Andrew F. Daughety
Faculty Associate: Ronald R. Braeutigam
Cornell University
Principal Investigator: Mark A. Turnquist
Acknowledgements
We especially thank the executives and staff of the railroad studied
herein. Without their help, both in terms of data and insights into rail
operations, this report would have been impossible.
Thanks also to William Jordan and Steven Lanning who helped with data
development and computer runs.
EXECUTIVE SUMMARY
Introduction
An understanding of the nature of costs of production is important in
every regulated industry, both for individual firms and their regulators. At
the most basic level a firm will require cost data for corporate planning. For
example, a firm may wish to know what size plant to build, whether to upgrade
the quality of plant or whether, at an existing tariff, the revenues for a
service cover the incremental cost of providing the service.
Regulators and other policy makers also have many reasons to seek improved
information about costs. When examined correctly, cost data can be used to
determine whether there are in fact economies of scale in production, and
whether regulation is a necessary tool of social control in a given industry.
Regulators often ask whether a service is being subsidized by other service of
a multiproduct firm, is subsidizing other services, and whether the provision
of service by one mode will eliminate another mode over a given route.
Problem Studied
Previous railroad cost studies typically have examined a cross section of
Class I railroads, using ICC data, and most have assumed a single product,
usually total ton-miles. Several aspects of these studies have served to limit
the inferences that can be drawn. They rely on data from the ICC accounts
rather than on raw data from the firm. With few exceptions, they have speci¬
fied a relatively simple functional form for costs, and assert that the form is
appropriate without a test of that assertion. Few adjust for quality of ser¬
vice, and more importantly, many do not account for the multiproduct nature of
virtually every rail firm. Finally, they do not attempt to adjust for the fact
that some railroads operate with a more complicated network than others.
Our own research on railroad transport costs represents a very different
approach to the problem. In an earlier report (Daughety and Turnquist, 1979)
we developed a notion of "hybrid" analysis that reflected some crucial differ¬
ences from the previous work.
ES-1
1) Our analysis focused at the level of an individual firm, and used
cost and production data obtained directly from the firm ratner than
from the ICC. This has a number of important advantages, including
the avoidance of arbitrary cost allocations of the sort often found
in the ICC accounts. We employed a time series analysis for a single
firm rather than a cross-sectional analysis for a particular year.
2) The multi-product nature of the firm was incorporated into the analy¬
sis. Models were estimated with disaggregated volume (by commodity
type) as well as with aggregate data. Output was characterized both
by the volume of freight hauled and by the average speed of a ship¬
ment through the system. We explicitly recognized that speed of
service is an important determinant of rail costs, and included this
in our estimates.
3) We used information about the underlying technological production
process, developed through engineering process functions, to improve
both the specification of technology and the efficiency of our
estimates.
In several respects the last point was particularly novel. Historically,
most econometric estimates of cost function! have ignored valuable information
on service-related variables which may be generated by engineering process
functions. We have labeled our method a "hybrid" approach because it included
such information.
This report builds on the first phase of the project in a number of im¬
portant ways.
1) We have again focused our attention at the level of incividual firm.
This time, we have workeo witn aata from a majGr class I railroac
with a complex network; the Phase I effort purposely examined a small
railroad with a simple network. Thus, we nave developeG techniques
that address a wide range ot existing firms. An important byproduct
is that we can use the two case studies to examine tne cross-secticr
analyses discussed above.
2) Again we address the multi-product nature of the firm oy i.ic.„dmc
quality variable (average speed of service) in the econometric model
of the firm's costs. The econometric results include estimated
short-run and long-run functions, thus allowing a direct comparison
with results from the cross-section analyses discussed above.
ES-2
3) We have expanded significantly the project's analysis of railroad
operations. In our Phase I report engineering process functions were
used to improve the econometric analysis. In this report we show how
economic theory can be used to extend the operations/engineering
analysis. Taken together, the two reports clearly show the advan¬
tages and potential of joint economic/engineering analysis of firm
activities.
Results Achieved in Phase II
A short-run variable cost function was estimated using monthly data on 1)
operating costs; 2) carloads moved; 3) average speed of service; 4) the prices
of fuel, equipment, and labor; 5) a measure of track capital called "effective
track." The long-run cost function was derived from the short-run function.
Analysis of the estimation results indicated the following:
1) The firm faces significant economies of density; i.e. given the fixed
configuration, at fixed speed-of-service increases in aggregate car¬
loads moved will result in reductions in average costs per carload.
Coupled with the Phase I results, this indicates that both large and
small railroads can have significant density economies.
2) The major short-run factors of production (fuel, labor and equipment)
are inelastic substitutes for one-another. Thus, each factor is a
substitute for the others, but only to a small degree.
Comparison with the cross-section cost models indicates two sources of
error in this literature:
1) Often such models do not control for systematic differences among
firms, leading to biases in estimated coefficients. Moreover,
cross-section analyses that do not control for firm differences
cannot separate economies due to changes in firm size and configura¬
tion from economies due to more intensive configuration use (i.e.
economies of density).
2) In general, cross-section studies have not used properly constructed
quality-of-service measures. We find that eliminating the speed-of-
service quality variable is not only a specification error in the
model; such elimination tends to bias downward the estimate of
returns-to-scale.
ES-3
We also developed a simple, but accurate, model of rail operations that
estimates system operating costs to within 15% of actual values. The model
provides a rail firm with a convenient tool for operations cost analysis be¬
cause it is easy to set up and inexpensive to solve. Moreover, we showed how
to use the model to generate an origin-destination specific marginal operating
cost prediction equation. This was another example of our hybrid analysis.
Economic theory was used to formulate the estimation problem, and engineering
analysis was used to provide the details on specific origin-destination move¬
ments. Together, the two methods produced a valid marginal cost function.
ES-4
Table of Contents
Page
1. Introduction 1
1.1 Other Railroad Cost Estimates 1
1.2 Time Series Analysis at the Level of the Individual Firm ... 2
2. Estimating a Long-Run Cost Function for a Railroad Firm 6
2.1 Basis for the Procedure Used 6
2.2 Formulation of the Short-Run Cost Model 9
2.3 Firm Selection and Data Development 14
2.4 Implicit Exogeniety of Speed-of-Service 20
2.5 Estimation Results for the Short-Run Variable Cost Function. . 23
2.6 Construction of the Long-Run Cost Function 24
2.7 The Long-Run Function and Returns-to-Density 30
2.8 Summary 33
3. A Network Model of Operating Cost 35
3.1 Problem Formulation 35
3.2 Network Representation 37
3.3 Costs on Linehaul Links 39
3.4 Costs on Yard Links 42
3.5 An Algorithm for Solving the Network Problem 47
3.6 Testing the Model 49
4. Estimation of an Origin-Destination Short-Run Marginal Operating
Cost Function 54
4.1 Procedure 54
4.2 Estimation Results 58
4.3 The Contribution of Economic Theory to Operational Analysis. . 63
4.4 Summary and Implications 64
5. Conclusions and Implications 65
5.1 Conclusions and Implications from the Cost Model of the Firm . 65
5.2 Conclusions and Implications from the Operations Model .... 66
5.3 Use of the Research Products 68
Bibliography 71
i
List of Figures
Page
1-1 Structure of operating cost analysis 5
2-1 Hypothetical cost data and the long-run cost function 7
2-2 The firm's average cost curve 32
3-1 Representation of a terminal area in the network 38
3-2 Yard engine-hours versus cars handled for three large yards 45
3-3 Yard engine-hours versus cars handled for three small yards 46
3-4 Predicted versus observed loaded car-miles 50
3-5 Predicted versus observed yard engine-hours 52
3-6 Predicted versus reported operating expenses 53
5-1 Use of the research products 69
A-l Listing of LKCOST subroutine to implement the formulae given
in Chapter 3 A-5
A-2 Example input data set A-8
i i
List of Tables
Page
2-1 Operating expenses for material and retirement accounts to relay
one main line track mile with new rail of same weight and welded
lengths of rail replaced 20
2-2 Short-run variable cost function estimates 23
2-3 Autocorrelation coefficients 25
2-4 Long-run cost function 27
2-5 Own and cross factor demand price elasticities 29
4-1 Estimation results - marginal operating cost (share) 60
4-2 Estimation results - fuel (share) 60
4-3 Estimation results - labor (share) 61
4-4 Estimation results - equipment (share) 61
A-l Example of link-by-link output from RAILNET A-13
A-2 Example of origin-destination marginal operating costs from RAILNET. A-14
iii
CHAPTER 1
INTRODUCTI ON
An understanding of the nature of costs of production is important in
every regulated industry, both for individual firms and their regulators. At
the most basic level a firm will require cost data for corporate planning. For
example, a firm may wish to know what size plant to build, whether to upgrade
the quality of plant or whether, at an existing tariff, the revenues for a ser¬
vice cover the incremental cost of providing the service. Cost data may be
used to justify tariff changes. A firm may want to know how a change in the
level of output of one service affects the costs of providing another service,
and it may rely in part on cost data to determine whether it would be profit¬
able to discontinue a service, introduce a new service, or attempt to merge
with another firm.
Regulators and other policy makers also have many reasons to seek improved
cost information. When examined correctly, cost data can be used to determine
whether there are in fact economies of scale in production, and whether regula¬
tion is a necessary tool of social control in a given industry. Evaluation of
proposed tariffs requires accurate and appropriate cost information. Regula¬
tors often ask whether a service is being subsidized by other services of a
multiproduct firm, is subsidizing other services, and whether the provision of
service by one mode will eliminate another mode over a given route. Another
example of current interest is the evaluation of seasonal or "peak-period"
pricing policies. In general, regulators need cost information to determine
how their policies will affect market structure and economic performance.
These comments certainly apply to the railroad industry.
1.1 Other Railroad Cost Estimates
A number of studies have examined costs in the railroad industry. The
early work in this area attempted to characterize the output of railroads as a
single product, usually ton-miles. These studies examined a cross-section of
Class I railroads, using ICC data, to test whether there were economies of
scale in rail transport. The results were quite mixed. For example, Klein
-1-
-2-
(1947) used 1936 data to find economies of scale that were statistically signi¬
ficant, though modest. On the other hand, estimates by Borts (1960) and Gri-
liches (1972) suggested that, while there may have been economies of scale for
smaller railroads, scale economies were not prevalent for larger Class I rail¬
roads .
Several aspects of these studies limited the inferences that could be
drawn. They relied on data from the ICC accounts rather than on raw data from
the firms. They specified a relatively simple functional form for costs, and
asserted that the form was appropriate without a test of that assertion. They
did not adjust for quality of service, and more importantly, they did not ac¬
count for the multiproduct nature of virtually every rail firm. Finally, they
did not attempt to adjust for the fact that some railroads operate with a more
complicated network than others.
Keeler (1974), Harris (1977) and Sammon (1978) have emphasized the differ¬
ences between economies of firm size and economies of density. Returns to size
are associated with a single firm serving a larger geographical area and more
markets. Returns to density are associated with moving more traffic on a given
network. This distinction has also been emphasized in the report by Daughety
and Turnquist (1979), but the earlier econometric studies which used a very
simple model form could not make this distinction.
Keeler (1974) and Hasenkamp (1976) used approaches grounded in production
theory to examine multi-product aspects of railroad activities, distinguishing
between freight and passenger activities. Using more sophisticated analysis,
Brown, Caves and Christensen (1975) and Friedlaender and Spady (1979) have de¬
veloped models that allow multiple outputs and relax several other assumptions
of structural form. Caves, Christensen and Swanson (1980) have also used such
techniques to examine productivity growth in U.S. railroads. In all these
cases, cross-section data drawn from ICC reports or based on Klein's work
(1947) have been used. Thus railroads with rates-of-return varying between
-10% and +40%, facing different geography, having different mixes of equipment,
customers and managerial perspectives were mixed together in the estimation
process. Service variables such as speed could not be used, because such data
are firm-specific and are not usually published. The above studies have repre¬
sented important advances in the understanding of costs, but more work is
needed, especially at the level of the individual firm.
-3-
1.2 Time Series Analysis at the Level of the Individual Firm
Our own research on railroad transport costs represents a very different
approach to the problem. In an earlier report (Oaughety and Turnquist, 1979),
we developed a notion of "hybrid" analysis. We used information about the
underlying technological production process, developed through engineering
analysis, to improve the specification of technology and the efficiency of our
statistical estimates of cost function coefficients. This approach reflects
some crucial differences from earlier literature.
First, our analysis focused on the level of an individual firm, and used
cost and production data obtained directly from the firm rather than from the
ICC. This has a number of important advantages, including the avoidance of
arbitrary cost allocations of the sort often found in the ICC accounts. (For a
discussion of the kinds of problems arising from the use of ICC data, see, for
example, Friedlaender (1969), Appendix A.) We employed a time series analysis
for a single firm rather than a cross-sectional analysis for a particular year.
Second, the multi-product nature of the firm was incorporated into the
analysis. Models were estimated with disaggregated volume (by commodity type),
as well as with aggregate data. Output was characterized both by the volume of
freight hauled and by the average speed of a shipment through the system. We
explicitly recognized that speed of service is an important determinant of rail
costs, and included this measure in our estimates.
This report builds on the first phase of the project in three important
ways.
1) We have again focused our attention at the level of individual firm.
This time, we have worked with data from a large railroad with a com¬
plex network; the Phase I effort purposely examined a small railroad
with a simple network. Thus, we have developed techniques that ad¬
dress a wide range of existing firms. An important by-product is
that we can use the two case studies to examine the cross-section
analyses discussed above.
2) Again we address the multi-product nature of the firm by including a
quality variable (average speed of service) in the econometric model
of the firm's costs. The econometric results include estimated
short-run and long-run functions, thus allowing a direct comparison
with results from the cross-section literature.
-4-
3) We have expanded significantly the project's analysis of railroad
operations with some very exciting results. The basic structure of
the analysis is illustrated in Figure 1-1. Information on the net¬
work configuration, the traffic volume (demand), resources available
and maintenance activities are used to support a network model to
predict traffic flows on links in the network and associated operat¬
ing costs. The information provided by this network model, together
with the input data form the basis for statistical estimation of a
function to predict marginal operating costs specific by origin and
destination of traffic flows. In our Phase I report (Daughety and
Turnquist, 1979), engineering process functions were used to improve
the econometric analysis. In this report we show how economic theory
can be used to extend the operations/engineering analysis. Taken to¬
gether, the two reports clearly show the advantages and potential of
joint economic/engineering analysis of firm activities.
The report proceeds in the following way. Chapter 2 presents the analysis
and results of estimating a short-run variable cost function for the subject
railroad. We also demonstrate how to construct the long-run function from the
short-run function. We then examine the long-run results.
Chapter 3 develops a network-based model of rail firm operations, reflect¬
ing yard and linehaul activity. The development of the model, data require¬
ments and the results of some sample runs are presented and discussed. Chapter
4 uses economic theory to extend the model in Chapter 3 to develop a function
for predicting marginal operating costs for specific origin-destination pairs
based on prices of inputs such as fuel, labor and equipment, and the quantity
of goods being shipped. Finally, Chapter 5 summarizes the results of the
research conducted in both phases.
NETWORK MODEL
AND
STATISTICAL ANALYSIS!
ORIGIN-DESTINATION
MARGINAL COSTS
Figure 1-1. Structure of operating cost analysis.
CHAPTER 2
ESTIMATING SHORT-RUN AND LONG-RUN COST FUNCTIONS FOR A RAILROAD FIRM
In this chapter we discuss the formulation and estimation of short-run and
long-run cost functions for a railroad firm.1 The procedure involves: 1) esti¬
mating the short-run variable cost function as a function of outputs, variable
factor prices and a fixed factor; 2) adding a short-run fixed cost; and 3) op¬
timizing over the fixed factor to derive the long-run function. The estimated
short-run and long-run functions are described and discussed.
Before entering into a discussion of the technical detail involved in con¬
structing a cost model, it is important to clarify the type of model we will
construct. One may divide statistically estimated models into two types: fore¬
casting and explanatory. Models of the first type are constructed to provide
estimates of costs without attention to the precise role of any particular var¬
iable in the model; the purpose of the model is to predict well. Explanatory
models are more concerned with the linkages among various variables and the
causes of cost generation. The objective is not forecasting, but insight into
the nature of the cost generation process and the sensitivity of that process
to specific input variables. The model we have constructed is of the second
type, since our focus is on trying to understand the production technology of a
rail system.
2.1 Basis for the Procedure Used
From economic theory we know that the long-run costs of a firm are a func¬
tion of the output levels the firm produces and the prices it pays for the
factors of production:
c = c(z, p)
where c is cost, z is a vector of outputs and p is a vector of input prices.
If one obtained from a firm monthly observations of costs incurred and levels
of output produced, one would probably see something like Figure 2-1 (here we
assume one output). This would reflect the fact that while the firm would
Daughety and Turnquist (1979) discuss long-run and short-run cost func¬
tions, and their relation to one another. This discussion is in Appendix
B of that report, especially on pp. 8-19 through B-26.
-6-
-7-
. • •
• • y
. * • . •
_ . . . •
S ^
°
. v
r
V
/
/
/
/
• /
/
. /
. • /
• • . • */
• • . . /
/*
Output, z
Figure 2-1. Hypothetical cost data and the long-run cost function.
-8-
prefer to be on Its long-run cost curve (the dashed line), changes In output
level can cause the firm to Incur short-run costs 1n excess of long-run costs
simply due to its inability to adjust all the factors of production instantane¬
ously. This is especially true in the case of a railroad, because changes in
its fixed plant (track, etc.) cannot be made rapidly. In other words, the
points above the long-run curve represent points on the family of short-run
curves whose envelope is the long-run curve.
Stated mathematically, let x be the vector of inputs used by the firm to
produce the vector of outputs z. Assume that some of the Inputs are variable
(the vector xv) and some are not as variable (i.e. fixed: x^) with x = (xv,
f / V
x ). The input price vector p is partitioned in a similar manner: p ■ (p ,
pf). The short-run variable cost function is:
c'U. pv; xf)
i.e. short-run variable costs (cv) are a function of the vector of outputs (z),
the vector of prices associated with the variable factors (pv) and the levels
of the fixed factors (x^). Short-run total costs are simply short-run variable
costs plus short-run fixed costs:
v, v f, , f f
c (z, p ; x ) + p x .
Long-run costs are found by optimizing over the fixed factors:
c(z, p) = min[cv(z, pv; xf) + pf xf].
f
x
This suggests the following procedure for estimating a long-run cost function:
1) Estimate the family of short-run variable cost functions,
c\ pv; Xf).
2) Compute a price for the fixed factors and use it to construct short-
run fixed costs, p^ x^.
3) Combine the two short-run functions and find the level of x^ which
minimizes total short-run cost; i.e. solve:
min[cV(z, pV; xf) + <pf xfj.
xf
This yields the optimal level of the fixed factor x^ :
f* fx v f.
x = x (z, p , p ).
-9-
4) Plug this into the short-run cost functions to obtain a long-run cost
function:
i '
c(z, p) = cv[z, pv; xf(z, pv, pf)] + pf Xf(z, pv, pf).
Obviously, this is a lot of work. Why not just estimate the long-run
function directly? Figure 2-1 shows that if the firm cannot adjust all its
factors of production in each time interval (e.g. within the month) then the
resulting observations will be on or above (never below) the long-run cost
function. If all we see is a scatter of points, passing a line through these
points will overestimate the location of the long-run cost function, biasing
the estimation results. The procedure outlined in steps 1-4 above was first
proposed by Keeler (1967, 1974) and Eads, Nerlove and Raduchel (1969) and
avoids the problem of overestimation.
2.2 Formulation of the Short-Run Cost Model
Our approach has been to use the same general cost modelling approach as
developed in the first phase of this work. Specifically, short-run variable
costs were modelled as
cV(y, s, pj, p2, p3, p4, p5; k) (2-1)
where y is quantity of goods moved (in carloads), s is the average speed of
service through the system, pi is the price of the ith variable input factor
(fuel, crew labor, non-crew labor, locomotives and cars) and k is the fixed
factor. Each of these will be discussed below. For reasons to be explained
later, this model was reduced to the following one:
c (y, s, Pp » P|_a» Plo' ^ (2~2)
where pp is the price of fuel, pLA is the price of labor and pLQ is the price
of locomotives. As we will explain in Section 2.3, (2-2) contains all the in¬
formation that would have appeared in (2-1).
In order to estimate (2-2) a functional form was assumed. We have used
the translog form, developed by Christensen, Jorgenson and Lau (1973). The
translog model has a number of important advantages.
1) The form is sufficiently general so as not to restrict the results of
the estimation process. The translog can be viewed as an approxima¬
tion to a general cost function.
-10-
2) The translog 1s the form used by a number of other studies (see
Daughety and Turnqulst, 1979), thus allowing direct comparisons.
3) The logarithmic form allows easy computation of important results of
the analysis, as we will see below.
The translog representation of (2-2) will be the following2:
6 ,66
C = I a X. + I I I a XX. (2-3)
1-1 1 1-1 j-1 lJ j
where
C = in (short-run variable cost/average of short-run variable costs)
= Jin (variable i/average of observations for variable i)
with the following correspondences:
1 Flow (in carloads/month)
2 Speed (in miles/day/car)
3 Fuel Price (in dollars/gallon)
4 Wages (in dollars/hour)
5 Locomotive Rental Price (in dollars/month)
6 Fixed Factor (miles of effective track)3
Thus, Xj and X2 represent the outputs, X3 to X5 the input prices, and X6
the fixed factor. The X^ are formed by taking an appropriate observation,
dividing by the sample mean for the variable, and then taking the logarithm of
the resulting ratio. This accomplishes the following.
1) It centers the estimation at the "point-of-means" thereby placing the
tightest part of the model's prediction confidence interval in the
middle of the data base.
Upper case variables used 1n connection with the estimation will represent
the logarithic, standardized variable from the original model.
Additional discussion of this factor is presented in Section 2.3.
-11-
2) It protects the proprietary nature of the data; by not providing the
sample mean we can still provide a complete analysis of the cost
function without revealing proprietary information on absolute levels
of cost.
We also estimate factor share equations simultaneously to improve the
efficiency of the estimation (see Christensen and Greene (1976)):
m1 = ai0 * jlj Vj 1 " 3'4 (2"4)
where
m. = the share of cost associated with variable factor i (i = 3,4).
Note that we use only two such equations, since total factor shares sum to one
and thus only two of the three are needed (the third is redundant). The factor
share equations are derived using Shephard's Lemma (see Daughety and Turnquist,
1979; p. B-22) which, for our problem, is:
m = — i = 3,4 (2-5)
i ax.
l
. v p.
(In non-logarithmic terms, m. = ( )(—), which becomes (2-5).)
i 3p. v
Ki c
Economic theory dictates that a cost function should have the following
properties (see Varian, 1978).
3C
1) It should be monotonically non-decreasing in output: — >0, — > 0.
8y 8s
2) It should be concave in prices.
3) It should be linearly homogeneous in prices; i.e., if we multiply all
prices by a constant, cost should be multiplied by the same constant.
The third requirement is the most straightforward to satisfy. To maintain
price-homogeneity, we restrict the parameters to satisfy the following
conditions :
-12-
Moreover, since the cost function and the marginal cost function are assumed to
be continuous functions, the cross-partials should obey symmetry; i.e.,
Thus, we will restrict the problem to satisfy symmetry:
2 2
rc _ 3 C
3X.3X. " 3X.3X.
l J J 1
ot.. s ct,, i,j - 1,...,6. (2-7)
ij ji
The first two requirements (output monotonicity and price concavity) are
not easily enforced. The first requirement, monotonicity, is an inequality
condition which places nonlinear restrictions on the parameters. The second
condition is presently unenforceable in any meaningful manner. Lau (1978) has
provided a non-linear method for restricting the a.. so that the translog cost
10
function is concave in prices; unfortunately, this does not really restrict the
underlying cost function to be concave in prices.
An important difference between this study and our Phase I study is the
assumption underlying the speed variable s (X2 in the translog representa¬
tion). In the Phase I study we dealt with a medium-to-small railroad that was
a bridge-line between two carriers. In that case it made sense to assume that
the average speed-of-service, s, was exogenous.
In the present case, the railroad studied is a major railroad which pre¬
sumably sets the speed of service so as to maximize profits. Thus, s should be
an endogenous variable set by the firm so as to equate the marginal revenue
with respect to speed (mr ) to the marginal cost with respect to speed (MC =
9r ^
^1). This restriction is not linear in the parameters of (2-3). However a
slight manipulation leads to a linear restriction. We note that:
mc i.*-
S C 3x„
= °20 + a2jXj* (2-8)
Thus, if there is evidence that the firm endogenously sets the speed of service
s, we will append equation (2-8) to the system to be estimated.
-13-
Therefore, our system of equations to be estimated is (2-3), (2-4) and
(2-8) subject to (2-6) and (2-7). Before passing on to a discussion of the
data to be used we need to account for one other problem: autocorrelation. We
will be using-monthly data, and thus observations in any given month may re¬
flect some of the same environmental aspects as affected the previous month's
observations. Let us pose our system to be estimated with error terms (e..) as
follows (t represents observation in month t):
rl - Y „ *t A 1 v v yt»t . t
" t01 lili A j J 1
MCs V •*> *♦ 4<2-9'
"5 = aio♦ j. v)♦ 1 ■3-4-
J ^
We will assume that the error terms are first-order autocorrelated; i.e.
this month's error term is affected by last month's. A representation of this
is the following:
4
= I P^e1:-1 + uj i - 1,... ,4 (2-10)
1 j=l
where the u^ are uncorrelated (and, we will assume jointly normally distribu¬
ted) and the p.. are called the autocorrelation coefficients (see Theil, 1971).
This is a standard autocorrelation assumption made in cases where autocorrela¬
tion is handled explicitly.
Thus, our statistical problem is to estimate the system (2-9) subject to
the constraints (2-6) and (2-7) and the assumption on the error process
(2-10). Once this system is estimated, we can then apply standard techniques
★
of numerical analysis to find k , the optimal level of the fixed factor, and
derive the long-run cost function.
-14-
2.3 Firm Selection and Data Development
Several criteria were used in considering potential railroad candidates
for this study. We desired a large Class I railroad with a management willing
to work with us 1n providing the data required for analysis. While the firm in
question completed some merger activity during the period studied, the compan¬
ies were already owned and operated as subsidiaries, so that data would be
added across the firms to maintain a consistent reporting base.
One obvious requirement for the analysis was that we obtain a complete set
of data, reported on a consistent basis, for all of the variables discussed in
the theoretical section. We were able to obtain such a set of data on a month¬
ly basis for the 35 months between January, 1976 and November, 1978.
One of the additional points we considered in our development was the pos¬
sibility of including explicitly a variable to represent technological change
in the system. If technology were adjusting during the time interval of data
collection, we would have needed to specify a time-varying cost model. How¬
ever, because the time horizon was only 35 months, the issue of technological
change was not a major point to be included in the model.
The following sections discuss the sources and nature of data used in the
analysis.
Operating Costs and Revenues
We obtained operating cost data directly from the company's records.
Since these data do not include implicit capital costs on cars and locomotives,
an estimate of these additional costs was made using the car and locomotive
prices and the numbers of cars and locomotives. These additional costs plus
operating costs yield the short-run variable cost data required for our esti¬
mation. Thus, the short-run variable cost includes maintenance, fuel, labor,
cars, locomotives, staff and supplies.
Operating revenues, reported on a monthly basis, were also obtained
directly from the firm's records.
Labor
At the beginning of our study, we intended to include two kinds of labor
in the analysis, crew labor and noncrew labor. We computed a factor price for
noncrew labor by dividing the total number of noncrew hours actually worked
-15-
into the sum of the noncrew wage bill, payroll taxes, and medical insurance
paid by the firm. We also computed a crew labor price by dividing the straight
time actually worked for crews into the sum of total wage payments to crews,
payroll taxes.for crew labor, and medical insurance payments.
We found that the prices for crew and noncrew labor had a correlation co¬
efficient in excess of 0.95. Therefore, we merged the information to get a
single price of labor, calculated by taking the total wage bill, payroll tax
and insurance payments for both crew and noncrew labor, and then dividing that
sum by the total hours worked in the two categories. We also generated a labor
share by dividing the total wage, payroll tax and medical insurance payments by
the total short-run variable cost of the firm.
We investigated whether labor should be treated as a variable or fixed
factor, and found overwhelming institutional evidence that it should be treated
as variable. The basis for this is the existence of "extra boards" of workers
who have no fixed assignment, nor fixed salary. These people may be assigned
to any job for which they are qualified, or not assigned at all if they are not
needed. This allows the firm to redistribute labor where needed in the firm.
This spatial and temporal redistribution mechanism leads to great flexibility
in the use of labor. Further, if the labor requirements of the firm are re¬
duced to such a level that redistribution of labor by extra boards cannot fully
employ the labor, then the firm can (and often does) furlough unneeded
laborers.
Locomotives and Cars
For each time period, we obtained data on the number of cars owned and
leased, by type of car. We also found the number and types of locomotives, by
type, both owned and leased. We obtained data on the prices of cars and loco¬
motives from Economic ABZ's of the Railroad Industry (1980), Welty (1978) and
from ICC Transport Statistics. These prices for physical units of capital were
converted to equivalent rental prices using the interest rate on equipment
obligations of the firm and depreciation rates from Swanson (1968).
We then found that the factor prices for locomotives and cars over the 35
month period had a correlation coefficient in excess of 0.95. Therefore, we
used the locomotive factor price to represent the two categories. Also, the
-16-
omitted share equation (as discussed earlier) was the one for locomotives and
cars; thus we did not compute a factor share for these inputs. This means that
the share attributed to the locomotive price is the "equipment share."
Speed
Data enabling us to calculate average speed on the system were obtained
directly from the railroad's operating records. We gathered data on the total
loaded car-miles during each month for the whole system, and then divided this
by the total loaded car-days on-line for that month. This calculation yielded
a monthly average velocity, in terms of miles per day for cars on the system.
This approach provides a convenient way to estimate average speed-of-service.
It might seem surprising to some readers that we are not including terms
reflecting variability in the speed of service as a reliability measure. There
is a subtle, but important point here. Firms can employ inputs to attempt to
control the quality of output (e.g. control for the purpose of maintaining a
selected speed of service with low variability). Such decisions are reflected
in the firm's use of inputs and thus the choice of degree of reliability is
endogeneous to the firm and 1s incorporated in the setting of the expected
speed. In other words, there is an optimal level of reliability that is
adopted by an expected profit-maximizing firm, which is a function of the
chosen expected speed of service. This is especially fortuitous since data on
reliability are difficult to accumulate.
Flow
From company data we obtained information on all movements in the system,
by origin-destination and by two-digit Standard Transportation Commodity Code,
on a monthly basis. The output is measured in total carloads moved for each
month.
Fixed Factor
It is not an easy task to characterize the fixed factors of a system as
complex as a railroad. The fixed factors would include track, switches, land
and buildings, to name the most obvious of the elements. We have employed a
measure of miles of track to represent the level of the fixed factor in our
-17-
arialysis. We have done so because track represents the largest component of
the system that can be regarded as fixed within a monthly horizon but which can
be varied (at least to some extent) over a 35 month horizon. Moreover, invest¬
ment in track appeared to be the main component of a general vector of fixed
factors that was adjusting during the period studied.
In our study we observed that the total track-miles in place did not vary
significantly over the 35 month period. However, the firm was investing in
track in amounts significantly greater than would be required to maintain a
constant quality of track in the face of normal depreciation. Thus, it was
apparent that the firm was varying not the quantity of track in place over the
three year period, but rather was improving track quality through track
investment.
Thus, we constructed a measure of effective track in the following way.
Consider the disposition of investment in track during period t. The amount of
gross investment is U. 1^ is considered to have three possible uses: (1) ex¬
pansion or contraction of the system (generally small in our case), (2) cover¬
age of depreciation of existing track, and (3) quality improvements in the
existing system. Thus, we define:
Then the uses of It (gross investment in track-miles) are summarized as:
The first term on the right hand side of the equation indicates the amount by
which actual track-miles change during period t. The second term shows how
much investment (in track-miles) would be required to offset normal deprecia¬
tion (wear and tear) on existing track. The last term represents improvement
in track quality above normal requirements to cover depreciation during period
t.
We observe that the above equation can be rewritten, using the fact that
kt - kt i = Akt, where kt denotes the effective track at time t. Then the
following relationship can be stated:
T = the number of miles of track at the end of period t
6 = depreciation rate of existing track
Ak^ = improvements in track during period t.
(2-11)
(2-12)
-18-
(We reemphasize that the third term on the right hand side is essentially
negligible 1n our actual case study.) Because this is a difference equation
one of the must be chosen arbitrarily. The equation then defines the re¬
maining values relative to this one. This was done by letting = T^ at the
end of 1976.
The following calculation illustrates the procedure more clearly. The
numbers employed are purely illustrative, and bear no particular relationship
to the data obtained from the actual firm studied.
Given ;
(1) actual track-miles at the beginning of the year = 8000 miles;
(2) annual depreciation rate = 0.03;
(3) number of track-miles laid during each of the twelve months of the
year in order:
(100, 150, 200, 50, 50, 100, 50, 50, 150, 50, 50, 200)
reflecting a total of 1200 miles laid during the year;
(4) number of actual miles in place at the end of the year = 8000 miles.
Calculation:
The improvements in track over the year can be determined using (2-12):
4k , = 1200 - 0.03(8000) = 960
year
Then the improvement in the quality of track from the first month of the year
would be
Ak - (-122.)(960) = 80 miles.
month 1 1200
For the second month, we would have
Ak .u o = (T^H96°) ■ 120 miles.
month 2 1200
The sequence is repeated for each month. If we initialize actual and effective
track to be 8000 miles at the beginning of the year, then by the end of the
year effective track will be 8960 miles, although actual track is only 8000
miles. The difference of 960 miles represents a real improvement in the qual¬
ity of track in the system. Obviously, this number will change as a function
of which k value is specified a priori. However, relative performance is
preserved.
-19-
The Price of the Fixed Factor (Track)
An extensive study of track costs was undertaken by Danzig, et _al_ (1976).
Table 2-1 displays some of the cost data. Thus, the net cost per mile of track
is in the range of about $55,000-360,000 per mile, assuming 25% tie replacement
and some ballast replacement. We note that the ballast cost includes labor ex¬
penditures which we would in principle exclude because we desire capital costs,
exclusive of labor. However, since total ballast costs are minimal, we view
this as a minor problem, especially since we have used a broad range of track
costs in our derivations of long-run cost functions from the estimated short-
run cost function.
Maintenance Costs Implicitly Included
The method used to derive the factor prices for locomotives, cars and the
fixed factor implicitly includes maintenance costs. This follows from the fact
that the monthly "rental-price-equivalent" that we form is a price for the ser¬
vices of a new item each month, not for an unmaintained item. Thus, there is
no need to separately include such maintenance costs.
2.4 Implicit Exogeniety of Speed-of-Service
Contrary to our expectations, our statistical evidence suggests that aver¬
age speed-of-service is exogenously determined in the short-run. , This was dis¬
covered when we estimated the marginal revenue with respect to speed-of-service
and discovered that it was often negative (in many cases substantially
negative).
Our procedure for imputing the marginal revenue with respect to speed-of-
service was to assume that total revenue (TR) is a function of speed (s) and
flow (y). Differentiating totally we have that:
dTR = MR ds + MR dy
s y
where MR$ is the marginal revenue with respect to speed, MR^ is the marginal
revenue with respect to flow and ds and dy are the infinitesmals of speed and
flow. Solving for MR^ we have:
-20-
Table 2-1
Operating Expenses for Material and Retirement Accounts to Relay One Main Line
Track Mile with New Rail of Same Weight and Welded Lengths of Rail Replaced
136 pounds per yard 119 pounds per yard
Item Continuous Welded Rail Continuous Welded Rail
New Rail Only $62,235 $54,455
New Rail plus Plates, 82,510 73,415
Angle Bars, Anchors,
Spikes, Coating, etc.
Salvage Value of Old Rail 41,970 37,305
Net Cost per Mile $40,540 $36,110
Tie Replacement Rate
20* - $12,000 - $12,000
25* - 16,000 - 16,000
30* - 20,000 - 20,000
Ballast (Resurface 3 - 3,000 » 3,000
inches ballast, single
line track, including
labor cost)
Source: J. Danzig, J. Rugg, J. Williams and W. Hay, Procedures for Analyzing
the Economic Costs of Railroad Roadway for Pricing Purposes, U.S.
Department of Transportation, Washington, D.C., 1976.
-21-
dTR - MR dy
MR = i_ .
s ds
Let dTR be approximated by the change in operating revenues from month to
month:
dTR1 = OR1 - OR1"1
where OR*" is the operating revenue for the firm in month t. Furthermore, let
dy = y - y and ds* = s* - s*~*. Finally, since the study period (January
1976-November 1978) predates liberalization of ICC rules on contract rate-
making, marginal and average revenue with respect to y are the same; i.e. MR =
t t -7
ARy. Let us approximate AR^ as:
Thus, our approximation for MR* becomes:
(OR1 - OR1'1) - 5!L (yl - y1"1)
K 1—fcr • (2'I3)
S - S
MR* was computed via (2-13) using the monthly data described above.
Sixty-five percent of the computed values were negative. If s were endogenous
to the firm this would not happen; marginal revenue should exceed zero since
otherwise reducing s would contribute to revenues and presumably reduce costs,
thereby increasing profits.
The implication of this is that these data will not support treating speed
as an endogenous variable. Thus, our model assumes that speed is exogenous to
the firm in the short-run. The exogeneity of speed-of-service implies that the
speed equation (2-8) should be dropped from the system (2-9). The resulting
system to be estimated has three equations and twenty coefficients.
-22-
2.5 Estimation Results for the Short-Run Variable Cost Function
The system of equations (2-9), minus (2-8), was estimated subject to the
error structure assumption (2-10), homogeneity 1n prices (2-6) and symmetry
(2-7) via full information maximum likelihood using the WYMER program, on
Northwestern's CDC/Cyber computer. The homogeneity restrictions were satis¬
fied by normalizing the cost variable and all prices to the price of loco¬
motives. Thus, values of coefficients associated with the price of locomotives
are implied by the regression and the standard errors of these coefficients are
computed by approximation.** Table 2-2 provides the estimated coefficients for
the cost function and the standard errors. Mnemonics for the prices have been
used (I.e. PFUEL is Pp, etc.).
The first-order coefficient estimates for flow, capital and the three
price terms are significant (at the .001 level) and of the correct sign. Thus,
the cost function is non-decreasing in outputs as required, and is increasing
in prices; it is homogeneous in price since this restriction was enforced dur¬
ing the estimation. The function 1s not concave since the own second-order
coefficients are significant and positive (i.e. a^, a44, and ^5).
The first-order price terms are the elasticities of cost with respect to
price at the point of means. Thus, for example, a one percent change in the
wage rate will increase costs by slightly over .5 percent. The coefficient on
the price of locomotives represents the impact of both locomotives and cars.
Increases in the amount of effective track (K) reduce short-run variable
costs at the point of means; a one percent increase in effective track miles
Implies a 0.2771 percent reduction in short-run variable costs. The negative
sign is expected because the cost of the improvement is not included in
short-run variable costs.
The first-order speed-of-service parameter (c^Q) is insignificant but a
test5 of the hypothesis that the speed variables should be dropped is soundly
rejected at the .005 level. Thus, even though the first-order speed term is
insignificant, the speed variable itself is very important for proper model
specification.
Assume a ■ N(a, z) and let h( a) be some function of the coefficients. If
we expand h(a) around a point a0, in a first-order approximation, then
Var(h(a)) = (Vh)'Z^i provides the squared standard error for the function
h( •).
This is a joint test that c2Q = al2 - a22 = a23 = a24 = a25 » o26 = 0,
performed as discussed in Daughety and Turnquist (1979), p. 65. The
resulting x value was 37.005 with five degrees of freedom.
-23-
Table 2-2
Short-Run
Variable Cost
Function Estimates
Variable
Coefficient
Estimate
Standard Error
Y (flow)
a10
0.3984
0.0694
S (speed)
«20
-0.0659
0.0746
PFUEL
a3 0
0.1902
0.0600
PLABOR
a40
0.5253
0.0547
PLOCO
«50
0.2845
0.0248
K (effective track-miles)
a60
-0.2771
0.0887
Y-Y
«11
4.1260
1.5776
Y-S
a12
-2.6510
1.3848
Y-PFUEL
«13
-0.0167
0.0069
Y-PLABOR
«14
0.0090
0.0258
Y-PLOCO
«15
0.0077
0.0264
Y-K
«16
-2.8813
1.0598
S-S
«2 2
-0.3681
1.8906
S-PFUEL
a23
0.0404
0.0069
S-PLABOR
«24
0.0067
0.0298
S-PLOCO
a25
-0.0471
0.0306
S-K
«2 6
2.9855
1.2109
PFUEL-PFUEL
«3 3
0.0623
0.0104
PFUEL-PLABOR
«34
-0.0489
0.0083
PFUEL-PLOCO
a35
-0.0134
0.0061
PFUEL -K
«36
0.0668
0.1557
PLABOR-PLABOR
«44
0.0860
0.0176
PLABOR-PLOCO
«4 5
-0.0371
0.0165
PLABOR -K
«46
0.1166
0.2213
PLOCO-PLOCO
«55
0.0505
0.0189
PLOCO -K
«56
-0.1834
0.3739
K-K
«66
2.5447
3.0202
-24-
The first-order flow term (Y) is significant and positive. A one percent
change in flow will generate a 0.3984 percent increase in short-run variable
v v MC
costs. Thus, 91nc = (X—) (-^—) = . y - 0.3984; i.e. short-run marginal costs
31 ny v' 3y AVC
c y
are below short-run average variable costs at the point of means. Thus at the
point of approximation, short-run average variable costs are falling with
respect to flow.
Since we view output as reflecting both a physical flow of goods y and a
quality measure s (average speed-of-service), simplification of the cost func¬
tion would result if y and s were jointly separable from the inputs (repre¬
sented by p and k). The hypothesis that y and s are separable from the inputs
was tested (see Daughety and Turnquist (1979), p. B-33 for a discussion of this
test) and rejected at the .005 level.
The estimation was performed under the assumption of first-order auto¬
correlation as discussed earlier and embodied in (2-10). Table 2-3 provides
the estimated coefficients, with their standard errors in parentheses. There
appears to have been a strong effect on the fuel share equation from all three
equations (i.e., the second row coefficients are all significant). Moreover,
there is evidence of first-order autocorrelation in the labor share equation
from the previous months labor share error term (i.e., p_- is significant). By
J J
allowing for these autocorrelations in the estimation process, we have cor¬
rected for their potential effects on the standard errors of the equation
coefficients.
2.6 Construction of the Long-Run Cost Function
Let pk be the price of a mile of effective track (see Section 2.3). At
the point of means for all the variables except k we have the following equa¬
tion for short-run total costs (i.e. variable plus fixed):
c(y, s, p; k) = c exp(a6Q *n(k/k) + <^6( j^(k/k))2 ) + pkk
-25-
Table 2-3
Autocorrelation Coefficients
1
-0.3272 1.2913 -0.8601
(0.1780) (1.1318) (0.4796)
0.0858* 0.7681* 0.1567*
(0.0220) (0.1668) (0.0472)
0.0393 0.5866 0.6031*
(.0738) (0.4335) (0.1698)
Table entry: p.^.
( a. . )
1J
i : 1 Cost Function
2 Fuel Share Function
3 Labor Share Function
* indicates those coefficients that are significant at the 5% level.
-26-
where a (-) over the variable represents the mean of the observations for the
variable. A one-d1mensional search technique (Wilde, 1964) was used to find
k*, the optimal level of the fixed factor. As discussed in Section 2.3, there
is considerable uncertainty concerning the proper cost of a mile of effective
track. Moreover, the appropriate depreciation rate and cost of capital are
also uncertain. We chose to vary the cost of a mile of track from $40,000 to
$130,000, the depreciation rate from 3 percent to 9 percent and the cost of
capital from 8 percent to 12 percent. The results are very encouraging: k* is
reasonably insensitive to such changes. In what follows we will use the
assumed values of track cost of $58,000/effective track-mile, depreciation of 3
percent and cost of capital of 10 percent. These values appear to be reason¬
able estimates of the appropriate numbers, based both on previous studies and
discussions with railroad management.
Under these conditions we find that k* = 1.079 R. At the point of means
the optimal plant size (level of fixed factor) is 1.079 times the average for
the period studied. Since the configuration is fixed this implies that the
firm should increase the quality of the existing track by approximately 8
percent. Using this value of k*, we can substitute into the short-run cost
function and derive the long-run cost which is presented in Table 2-4. Notice
that the first order terms have new coefficients, which are:
"iO = ai0 + 1 =
Moreover, a constant term a00 = a60 *n(k*/R) + a66(in(k*/k) )2 now appears. It
should also be noted that since we are simply substituting k* into cv(y,s,p;k),
the resulting function is net of fixed costs. This will not affect any of the
derivatives of the cost function, and thus Table 2-4 contains all the relevant
information associated with the long-run cost function.
-27-
Table 2-4
Long-Run Cost Function
Variable
Coefficient
Estimate
Standard Error
Y (flow)
a10
.1793
0.2852
S (speed)
°20
.1611
0.1051
PFUEL
a30
.1953
0.0574
PLABOR
P
-T
O
.5342
0.0669
PLOCO
a50
.2705
0.0698
Y-Y
aU
4.1260
1.5776
Y *S
CM
«r
-2.6510
1.3848
Y-PFUEL
a13
-0.0167
0.0069
Y-PLABOR
a14
0.0090
0.0258
Y-PLOCO
0115
0.0077
0.0264
S-S
a22
-0.3681
1.8906
S-PFUEL
a2 3
0.0404
0.0069
S-PLABOR
a2l+
0.0067
0.0298
S-PLOCO
a25
-0.0471
0.0306
PFUEL-PFUEL
a3 3
0.0623
0.0104
PFUEL-PLABOR
a34
-0.0489
0.0083
PFUEL-PLOCO
a35
-0.0134
0.0061
PLABOR-PLABOR
a41+
0.0860
0.0176
PLABOR-PLOCO
a4 5
-0.0371
0.0165
PLOCO-PLOCO
a5 5
0.0505
0.0189
CONSTANT
P
o
o
-0.0064
0.2283
-28-
As indicated in Table 2-4, the first-order coefficient on flow (o10) is
insignificant. This appears to be a direct result of the large negative co¬
efficient on the flow/capital cross-term in the short-run function (a16 in
Table 2-2) coupled with a substantial covariance between a16 and the main
determinants of k* (a60 and a66). Because the estimate of returns-to-scale (or
in this case, density) at the point of means depends on this coefficient (see
Daughety and Turnquist, 1979; p. B-32), this is a somewhat disappointing re¬
sult. We will return to this later, after discussing some of the other terms
in the function, and will employ an alternative approach to clarify issues of
returns-to-density.
The other first-order terms have the expected signs, including speed-of-
service which is now significant at approximately the .06 level. The co¬
efficient on the first-order speed variable is positive, as is expected since
higher quality service should incur higher costs.
Table 2-5 displays estimates of the factor demand and price elasticities
(own and cross). The own and cross-elasticities are computed as follows ( <5..
' J
is one if i = j and zero otherwise):
a. .
e = —— + a - 6 i,j = 3,4,5.
1J a. J 1J
1
For example, the own price elasticity of labor is -.3048, meaning that a
one percent increase in the wage rate generates a .3048 percent decrease in
demand for labor by the firm. On the other hand, a one percent increase in the
price of locomotives results in a .397 percent increase in demand for labor:
locomotives, labor and fuel are substitutes (at the point of means) for one
another, though the degree of substitution appears to be low (indicated by the
small magnitudes of the coefficients).
-29-
Table 2-5
Own and Cross Factor Demand Price Elasticities
QUANTITY
PRICE Fuel Labor Locomotives
Fuel -.4857 .2838 .2019
Labor .1038 -.3048 .2011
Locomotives .1458 .3970 -.5428
_ . . . X change in quantity
Table entry: ———2 : : -
% change in price
-30-
2.7 The Long-Run Function and Returns-to-Oensity
As Keeler (1974), Harris (1977) and Sammon (1978) have observed, one can
distinguish at least two types of scale economies in the railroad industry. If
the size of the firm in terms of geographic points served and configuration of
network can adjust, one is measuring econom1es-of-s1ze (and configuration; see
Daughety and Turnquist, 1979; pp. 6-11). If the network configuration 1s held
fixed then economies-of-scale are referred to as economies-of-density. This
separation is crucial since having one type of scale economy does not preclude
or imply the other. Thus, rail cost models must be structured to measure the
two effects separately. Unfortunately, most of the cross-section studies have
not done this; railroads reflecting different sizes and configurations are
mixed together in the estimations. Mundlak (1961) has shown that time series-
cross section studies can be biased if variables are not introduced to control
for firm differences. Caves, Christensen and Thetheway (1981), in a study of
airlines, found that introduction of dummy variables to distinguish each firm
in their sample resulted in findings of returns-to-scale, while deletion of the
dummy variables indicated constant returns-to-scale.
While not able to reject conclusively a finding of constant returns-to-
scale for railroads, Friedlaender and Spady (1979) indicate that the data tend
to support increasing returns-to-scale. Caves, Christensen and Swanson (1980)
find constant returns-to-scale. In both cases it is unclear whether these
results reflect size/configuration issues or density issues since firm differ¬
ences are not controlled. Thus the resulting scale economy estimate reflects
both types of economies. Friedlaender and Spady do, however, introduce tech¬
nological variables such as length-of-haul as a proxy for network size, which
may account for the difference between their results and those of Caves,
Christensen and Swanson.
As indicated above, the standard error on the first-order flow term is
very large and thus the usual procedure for examining the cost function for
returns-to-scale (specifically returns-to-density, since the size of the firm
and configuration of its network is fixed) by computing l/a10 seems question¬
able at best. Consider instead the following procedure. Form the average
total short-run cost function (for s and the price vector at their mean
values):
-31-
1 ^11?
AC (y, s, p; k) = - [c exp (<*10«n(y/y) + -L. ( ln(y/y) )c
+ ago «n(k/K) + ^ (*n(k/R))2) + P)(k]
and find values y* and k* that simultaneously minimize AC(y, s, p; k); i.e.
find the bottom of the average cost curve. Figure 2-2 displays the result of
this exercise (the dotted lines are approximate extrapolations). The horizon¬
tal axis represents flow on the fixed configuration in units of .1 MES (minimum
efficient scale: where average cost first becomes a minimum). Computations
show that y is approximately .4 MES. The range over which the average cost
function is at its minimum is from MES to 1.1 MES. Extrapolation indicates
that for y > 1.1 MES, average cost rises rapidly.
Three short-run average cost curves have been drawn in at k* = 1.079, 3.0
and 3.6 R, respectively. The last two are not overly realistic, since it is
hard to imagine the present network being improved to three times its present
quality. Rather, the point of the figure is to illustrate the fact that the
firm does face significant economies-of-density over a wide range of output;
average cost at y = y is 1.5 times the average cost at y = MES. This result is
consistent with Friedlaender and Spady's work on rail cost functions where they
found that exhausting economies-of-scale was not "just around the corner"
(Friedlaender and Spady, 1979, p. 263). These results do appear to conflict
with those of Caves, Christensen and Swanson (1980), because they found con¬
stant returns-to-scale.
A second reason for the difference between this study and the cross-
section results mentioned above is the inclusion in this model of the quality-
of-service variable, s. Caves, Christensen and Swanson (1980) did not include
a quality-of-service variable; Friedlaender and Spady introduce technological
variables, but these would at best be surrogates for a quality-of-service
measure. This is an advantage of firm-level analysis: such data are generally
not available at the aggregate level which cross-section studies must use.
-32-
\
\
\
\
X
/
k* - 3.0 k k*s 3.6ÎL/
\ \ , y
\ >-
y
Observed average output level
0 .2 .4 .6 .8 1.0 1.2
Output, in units of minimum efficient scale (MES)
Figure 2-2. The firm's average cost curve.
-33-
The effect of dropping this measure is very interesting. As mentioned
above, dropping s as a variable is a specification error; the x2-value on the
hypothesis test of setting the speed-related coefficient to zero was over twice
the table value at the .005 level. Proceeding, however, without the speed var¬
iable results in a value for k* in excess of the one computed above, slightly
higher average costs and a lower estimate of returns-to-density. Thus, failing
to include a properly constructed quality-of-service measure alters the esti¬
mated cost function in the direction of constant return-to-scale (i.e.
density).
Thus, since we are working at the level of an individual firm, our results
do not suffer from interfirm biases or specification error due to failing to
include quality of service. The railroad under study appears to have very
significant returns-to-density; computed in the traditional manner we find them
to be 1/aio " 5.6. While the standard error on this number is obviously quite
large (since the standard error on a10 is large relative to the magnitude of
aio). the existence of substantial returns-to-density are also supported by the
alternative (optimization) procedure.
The finding of substantial economies-of-density replicates a similar re¬
sult in our previous study of a smaller railroad (Daughety and Turnquist; 1979;
pp. 68-70). It is risky to attempt extrapolation to the entire industry on the
basis of two samples, but the fact that we have obtained very similar results
from two very different railroads suggests that density economies are not an
isolated phenomenon.
2 .8 Summary
The analysis described in this Chapter has yielded statistical estimates
of both short-run and long-run cost functions for the railroad under study.
This type of result is of greatest use to regulators and policy makers, because
it focuses attention on matters of general concern, not details. The estimated
models provide insight into economies-of-density, elasticities of demand for
the factors of production, and elasticities of substitution among input fac¬
tors. These results are important because they summarize important economic
characteristics of the production process.
-34-
The analysis performed here differs substantially from more traditional
econometric analyses of railroad costs. We have undertaken an analysis of
time-series data from a single firm, rather than data for a single time period
from many firms. This has allowed us to use data on service quality attributes
to improve model specification. It has also allowed us to be precise about the
type of scale economies found. These are economies-of-density over a fixed
network configuration, and should not be confused with the separate concept of
economies-of-size, which relates to the geographic extent of the markets
served.
It is also important to recognize that the firm-level data collected to
support the analysis can be used in a different way to support a complementary
type of analysis. This analysis focuses on the more detailed operating char¬
acteristics of the railroad, and gives additional insights into the costs asso¬
ciated with specific origin-destination movements over the railroad's network.
These results are more likely to be directly useful to railroad management in
determining prices for certain services, in identifying areas where costs are
higher than they should be, and in evaluating the effects on costs of changes
in operating policies or facilities.
The next Chapter describes a model of operating costs on a network. This
model represents another facet of the "hybrid" analysis of costs using both
engineering and statistical techniques, which is at the heart of this research.
CHAPTER 3
A NETWORK MODEL OF OPERATING COST
Economic theory specifies that a cost function should be the result of
solving a problem which may be stated generally as follows:
(PO) min p'x
s.t. f(z,x) = 0
where: p = vector of input prices
x = vector of input quantities purchased for use
z = vector of outputs produced.
The function f(z,x) provides the information on technological constraints
which determine how x is used to produce z. In general, f(z,x) is nonlinear,
implying that the cost function is the result of a nonlinear optimization. In
this Chapter, we describe a particular formulation of such a nonlinear program
which is very useful for studying the nature of operating costs on a railroad
network.
3.1 Problem Formulation
The basic structure of the problem may be specified as follows:
inputs: road locomotives
yard locomotives
cars
fuel
road crews
yard crews
maintenance of way
outputs: carloads of all commodities by origin-destination pair.
A fundamental assumption of the model is that the costs of the various
inputs can be related to flows of traffic over the links of the rail network.
The basic unit of flow is a carload, and links are of two types - linehaul
links and yard links.
-35-
-36-
The flow variables in the model may be defined as follows:
f9^ = carloads on link ij enroute to destination q
f.. = total carloads on link ij (= If?i).
•J q 'J
The problem may then be formulated as finding a minimum cost assignment of a
given set of origin-destination demands to the available network. In creating
such a formulation, a change in the structure of the optimization problem is
accomplished. Problem (PO) has a linear objective function which is minimized
subject to nonlinear constraints. By introducing the link flow variables, the
problem becomes one with a nonlinear objective function, and linear con¬
straints. This change is important in achieving an efficient solution
procedure.
The problem may be written as follows:
(PI) min G ( f ) = I I c. .(f..) -f..
i j J J J
S-1' f >j " ? fji * 21q V i>q
J J
ftJ - £ ■ o » ij
V?j>o vij, ,
where: c..(f..) = unit cost of flow on link ij, as a function of the volume
J J on that link,
z^ = volume of total demand from origin i to destination q.
The objective function simply indicates that we wish to choose a set of
link flows on the network (assign the demands to the network) in such a way as
to minimize the total cost incurred. The cij(f^j) cost coefficients include
the costs of all the inputs required to cross a particular link.
-37-
The first set of constraints state that for each destination, the flow out
of any node i minus the flow into 1 enroute to that destination must equal the
flow which originates at i and is destined for q. Thus, these constraints
simply ensure conservation of flow in the network.
The second set of constraints defines total carloads on a link (f^j) in
terms of the carloads destined for specific destinations (f^.), and the last
set simply insures that flows on the network are positive.
3.2 Network Representation
The railroad network is represented by a set of nodes connected by links.
At a conceptual level, the nodes are yards, terminal facilities and junctions,
and the links connecting them are main line and branch line tracks. However,
the detailed representation of the system is different from the conceptual
model in two ways.
First, the network has been aggregated. Each terminal in the analysis
network represents a collection of points in the actual network. The actual
system under study has more than a thousand points at which traffic can origi¬
nate or terminate. In concept, each of these points is a node in the network.
For analysis purposes, this set of points has been aggregated into 27 major
terminal areas which represent the origin and destination points for traffic in
the model. Network links were also aggregated along with the nodes.
Second, the analysis network uses a set of links and nodes to represent
each of the 27 terminals. This is because the network flow algorithms used in
the analysis require impedance to flow (cost) to occur only on links. Since
there are substantial costs associated with the movement of cars through yards
in the terminal areas, these terminals cannot be simply represented as nodes.
The general method of representing terminal areas is shown in Figure 3-1.
The terminal is represented by three nodes and three one-way links, in addition
to the links connecting this terminal to others. Link 1-2 represents the
actual yard, and has a positive cost associated with flow through it. Node 3
is used as the actual origin/destination for traffic originating or terminating
Figure 3-1. Representation of a terminal area in the network.
-39-
at this terminal area. Thus, originating traffic must traverse links 3-1 and
1-2 before departing. This forces it through the yard link 1-2. Likewise,
terminating traffic must enter at node 1, and traverse links 1-2 and 2-3 before
reaching its destination. Through traffic simply crosses link 1-2. As a re¬
sult, all traffic handled by this terminal area passes through the yard link,
and is included in the determination of congestion levels in the yard.
In summary, there are three types of links connecting the various nodes in
the network. There are linehaul links which connect the terminal areas; there
are yard links representing classification yards; and there are "dummy" links
within the terminal areas to connect the actual origin/destination nodes to the
rest of the network. The construction of unit cost functions for the linehaul
and yard links is described in the following section.
3.3 Costs on Linehaul Links
For each linehaul link in the network a unit cost function c..(f..), must
' J 'J
be computed. This unit cost must include the costs of five basic inputs:
1. train crews
2. fuel
3. locomotive ownership and maintenance
4. car-hire, car ownership and maintenance
5. maintenance of way.
Train crew costs are directly related to train-miles. To establish a unit
crew cost on a carload basis, we have used information on average train length
and the ratio of empty to loaded car-miles. Train-miles on link ij are com¬
puted as follows:
(1 + E)M
TRM" = ATI fu (3_1)
ij ATL ij
where: TRMij = train-miles on link ij
M. . = length of link ij (miles)
TJ
E = empty-to-loaded car-mile ratio
ATL = average train length (cars)
f.. = carloads moved on link ij.
ij
-40-
Since system-wide average figures for E and ATL are used, some variability
among individual links due to specialized operations is lost. For predictions
of aggregate system performance from month to month, we have found that this
loss of information is not important. However, if the model were to be used
for detailed examination of marginal costs of movement on a specific route,
appropriate link-specific data could be substituted for the system-wide average
figures.
Thus, if we denote train crew wage costs per train-mile as pw, the wage
costs per carload may be expressed as ;
Pw(1 + E)M1i , ,
c (f ) - — H . 3-2
11 j ij ALT
Note that expression (3-2) is independent of f„. This reflects an
assumption that the unit wage costs per train-mile, p^, are constant, or that
total wage costs are linear in volume. Thus, the model does not include de¬
tailed information on the effects of changes in the degree of linehaul conges¬
tion which could affect the wage cost per train-mile due to changes in running
speed. Effectively, the model assumes that the overall level of congestion
present in the data used to calibrate pw will remain unchanged as the model is
operated.
A railroad wishing to use this model for a specialized study of a specific
movement might decide to relax this assumption, and use more detailed data to
estimate pw as a function of f^, but for our purposes this was unnecessary.
The crew cost p^ was estimated for each month by simply dividing the total
train crew costs by total train-miles operated.
Fuel costs may be related directly to car-miles, since fuel consumption is
generally proportional to the amount of work performed. Observations on fuel
consumption per car-mile over a four-year period (1976-79) on the railroad
under study indicated that nearly all the monthly observations were within 10%
of the mean. Thus, if we denote the fuel consumption rate per loaded car-mile
as ru and the price of fuel per gallon as pF, the fuel cost per carload on
link ij is simply:
c2..(ft.) - (3-3)
-41-
Locomotive ownership and maintenance costs are computed using locomotive
prices, depreciation rates and a utilization rate, expressed as locomotive-
months per loaded car-mile. This utilization rate is computed by dividing
locomotives owned or leased by total loaded car-miles produced in each month.
The effective price of a locomotive-month is computed using the replace¬
ment cost in each month, the current interest rate, and a depreciation rate.
The calculation is as follows:
PL 55 (rt+d) V12 (3'4)
where: p£ = ownership cost of a locomotive for month t ($)
rt = annual interest rate available in month t
d = annual depreciation rate
Ut ~ price of a new locomotive in month t ($).
We have assumed a normal depreciation rate of 6% per year. This deprecia¬
tion rate assumes a normal rate of maintenance on the locomotive. Since the
price reflects a new locomotive, implicit maintenance costs are included in the
total effective cost of a locomotive-month. The price of a locomotive used in
the network model for linehaul links reflects only road locomotives, and not
yard locomotives. The price used in the econometric estimation in Chapter 2 is
a composite price including both types. In the network model, the cost of yard
locomotives are included in the yard links, but are separated from the locomo¬
tives used in road service.
If we denote the effective rental price of a locomotive-month as pL> drop¬
ping the superscript t, and the utilization rate as u, the locomotive costs per
carload are as follows:
c,.. (f.. ) = p, uM. .. (3-5)
3ij ij' L "tj
To determine car-hire and ownership costs, we must recognize the distinc¬
tion between system and foreign cars, and must also include two types of car
utilization factors. The first of these is the empty-to-loaded car-mile ratio
(E) discussed previously. The second relates to the time required to move cars
through the system, and can be expressed as the ratio of total car-days-on-line
to loaded car-miles. Let us denote this second utilization rate as T.
If a proportion a of the total cars-on-line on an average day are foreign,
the daily proportion of the per diem charge for foreign cars is b^, and the
-42-
daily ownership and maintenance cost for system cars is p , the effective cost
per car-day is on average:
*0 + (1-a)lV
For foreign cars, there is a mileage charge in addition to the daily
charge, bQ. Denote the per-mile cost as . If a proportion e of total car-
miles are made by foreign cars, the average cost per loaded car-mile is:
(1 + E) 8 br
Combining the daily costs and the mileage costs using the utilization
rate, T, the total unit car-hire and ownership cost per carload on link ij is:
C4ij(fij) = {[cb0 + T + (1 + E)Ébl}Wij ' (3_6)
The last category of linehaul costs, maintenance of way, is also included
by constructing a unit cost per loaded car-mile. This unit cost is determined
by taking total maintenance of way and structures expenditures, less deprecia¬
tion on non-track structures, and dividing by total loaded car-miles. This has
been done on a yearly, rather than a monthly, basis both because the data are
more accessible and because maintenance of way expenditures tend to be program¬
med on an annual basis. Thus, annual values are much more reliable than
monthly values. If we define this unit cost to be pm, the average unit cost of
way and structures maintenance is:
WV = PmV (3"7>
The total unit cost for linehaul links is then:
Cij^fij^ ~ ckij(fij) (3_8)
where the cfcij(f..) terms are given by equations (3-2), (3-3), (3-5), (3-6) and
(3-7).
3.4 Costs on Yard Links
Unit costs to be included for the yard links in the network reflect yard
locomotive ownership and maintenance costs, fuel, and yard labor costs. Note
that the car-hire, ownership and maintenance costs for the time that cars spend
in yards have been included implicitly in the linehaul links through the utili¬
zation factor T (total car-days on-line per loaded car-mile). Thus, these
-43-
costs are not included in the yard links to avoid double-counting. The costs
of input resources used in the yards are essentially proportional to the number
of yard engine-hours operated. As a result, the construction of Mic¬
tions for the. yard links involves two steps. First, the cost per yard engine-
hour is determined. This is assumed to be the same for all yards in the system
in any given month. Second, the relationship between carloads moving through
the yard and the number of yard engine-hours required must be determined. This
relationship will depend upon the physical and operating characteristics of
each yard, and thus will tend to be yard-specific.
The ownership and maintenance costs for yard locomotives are related to
the number of yard engine-hours in a manner similar to that used to relate road
locomotive ownership costs to car-miles. The two basic pieces of information
are the monthly cost of owning and maintaining a yard locomotive and the utili¬
zation rate of these locomotives, expressed as hours per locomotive per month.
The locomotive price is determined as described in Section 2.3 but using the
replacement purchase cost for yard locomotives only, not the composite cost
used in the econometric estimation. Available data on the number of yard
engines owned or leased and the total number of yard engine-hours produced in
each month have been used to calculate the utilization level for these locomo¬
tives. If the price of yard locomotives is denoted p^ and the utilization
level (hours/locomotive/month) is denoted y, the locomotive cost per yard
engine-hour is simply p^/y.
Fuel costs are also computed quite simply, by computing a fuel consumption
rate (gallons/yard engine-hour) and multiplying by the price of fuel. The fuel
consumption rate, r^, has been estimated by dividing gallons of fuel used in
yards for each month by the number of yard engine-hours operated. Note that
this consumption rate is time-based, rather than distance-based as in the case
of road locomotives, because of the different nature of the two operations.
The fuel price used is the same as for the road locomotives, pp.
Yard labor costs per engine-hour are computed by dividing total yard wages
paid in each month by the number of engine-hours operated. This includes both
engine crews and other yard employees. The result is an effective yard wage
rate. This wage rate may be denoted py.
-44-
The yard operating cost per engine-hour is then:
Cy = Py/Y + pFr2 + V (3~9)
The relationship between yard engine-hours and traffic volume has been
determined empirically using monthly data for individual terminals from the
period 1976-1979. Seven of the 27 terminals in the network are very large
yards, generally handling in excess of 25,000 cars per month, and two of these
are hump yards. The other 20 are smaller facilities, generally handling less
than 15,000 cars per month.
Figure 3-2 shows a sample of observed data on yard engine-hours and cars
handled for three of the large yards. Figure 3-3 shows comparable data for
three of the smaller yards. These figures illustrate two basic facts about the
data. First, there is substantial variation in the number of yard engine-hours
operated at each yard which is not explained by variation in the number of cars
handled, especially for some of tbe smaller yards. Second, the large yards
cover a very wide range of volumes, and appear to exhibit some degree of non-
linearity, indicating congestion effects.
In an attempt to find an acceptable simple model to relate yard engine-
hours to cars handled, a variety of polynomial regression models have been
tried. For the large yards, quadratic models of the form:
engine-hours = a + b(cars)2
provided the best results, and a test of the hypothesis that a and b are the
same for all yards resulted in rejection. Thus, seven separate quadratic
models have been estimated, one for each yard. For the smaller yards, tests of
including quadratic terms failed, and the selected model is linear. A single
linear model is used for all these yards. Separate models for each individual
small yard could have been estimated, but the quality of the data for several
of them was suspect, and their overall impact on the network 1s relatively
small. Thus, a single equation was used for all of them.
These simple models form the second component of the yard link-cost
functions. When multiplied by the cost per yard engine-hour from (3-9), the
result is a total operating cost function for each yard, expressed in terms of
traffic volume.
18
16
14
12
10
8
6
4
0
X
X
X
° o
o_ o
O Cb°
<9
x - Yard A
o- Yard 8
■ - Yard C
*
20
30 40 50 60
Cars handled (thousands)
70
80
90
Figure 3-2. Yard engine-hours versus cars handled for three large yards.
18
X
X
16
X X
o
xx
X X
14
o
o
12
XX o
o °
° o
o
■
A
x-Yard
D
10
■
o - Yard
E
■
■
■ - Yard
F
■
■
■ ■
8
■
6
0
\
1 1 a
V
0* 5 6
7 8 9
Cars handled (thousands)
10 II
12
1
Figure 3-3.
Yard engine-hours versus cars handled for
three small yards.
-47-
In fact, for reasons discussed in detail in the next section, our primary
interest is in the marginal cûst functions. These marginal cost functions will
be linearly increasing functions of volume for the large yards, and constant
for the small yards.
As illustrated in Figures 3-2 and 3-3, there is substantial variation in
yard engine-hours at each yard not explained by variation in the number of cars
handled. Thus, simple statistical models of the type estimated here are by no
means a complete picture of yard activities. However, they have proven useful
in estimating aggregate yard engine-hours for the system on a monthly basis,
and hence total yard operating costs. These results are discussed more fully
in Section 3.6. Before proceeding to that, the method for solving the problem
formulated in Section 3.1 will be discussed.
3.5 An Algorithm for Solving the Network Problem
The network problem formulated in Section 3.1 is a nonlinear programming
problem. The nonlinearity stems from the unit cost functions on some of the
yard links, which are functions of the traffic volume handled. When multiplied
by the flow volume, as in the objective function of problem (PI), the resulting
objective (total cost) is nonlinear. In concept, problem (PI) could be solved
by any of several nonlinear programming (NLP) methods, but the size of the
problem eliminates most of these from serious consideration. Note, however,
that the constraints are linear; the nonlinearity is solely in the objective
function. This suggests that a promising approach might be to use successive
linear approximations to the objective function because the problem to be
solved at each step is then simply a linear programming (LP) problem. An
efficient algorithm for large network problems has been developed by LeBlanc et
al. (1975) based on this idea.
A linear approximation to the objective function can be obtained using a
first-order Taylor series expansion. Let f be a current set of link flows
feasible for problems (PI) (one which satisfies the constraints). The value of
the objective function for another set of link flows, y, can be approximated as
fol lows :
G(y) » G(ft) + CVG(f) t]'(y-f1) (3-10)
r
where VG(f) f denotes the gradient of the objective function evaluated at the
t f
point f .
-48-
We are interested in G(y) as a function of y, so the expression for G(y)
may be rewritten by grouping the terms differently:
G(y) - {G(ft) - [VG(f) t]' • f1} + CVG(f) t]' -y. (3-11)
fi
The term in braces is a constant, independent of y, so this expression is a
linear function of y. The linear approximation of the NLP problem can then be
written as follows:
(P2) min[VG(f) J'y
r
S.t. I yjj - I yj, ■ 2jt| «t.,
J J
y,j - Iy?j = o v tj
, y?j > o » ij,q
Since the term in braces is independent of y, it is ignored for the pur¬
pose of writing and solving the LP. Once an optimal solution to (P2) is found,
we wish to search between ft and the solution to (P2) for a point which mini¬
mizes G. This is a one-dimensional search problem, which can be solved very
efficiently by any of a number of methods (Wilde, 1964). This search yields a
new feasible solution, ft+*. We can be sure that the new solution 1s feasible
since the feasible region is convex.
It can be shown that the following result holds:
1 im fl = f*
t+oo
where f* is the optimal solution to problem (PI). Thus, we can be certain that
the iterative procedure converges to the desired solution. This technique of
iteratively solving LP problems and one-dimensional searches is known as the
Frank-Wolfe algorithm (Frank and Wolfe, 1956).
The attractiveness of this method for solving problem (PI) stems from the
fact that the LP problem which must be solved at each step has a very special
structure. It is simply a multicommodity transhipment problem, which can be
-49-
solved using a shortest-path algorithm. Thus, vrtiile the problem is quite
large, very fast and efficient methods are available to solve it.
At each iteration, we evaluate the marginal link cost functions (find the
gradient of the objective function) and use these values as link cost coeffi¬
cients for a shortest-path problem for each origin-destination pair. This is
the reason for the interest in the marginal link cost functions expressed in
the previous section. Note that for the linehaul links and for the small
yards, the total cost functions are linear in volume. Thus, the marginal costs
are constant. It is only for the major yards that the marginal costs are a
function of volume, reflecting congestion.
The Appendix to this report describes in more detail how to use the model
developed for solving the network flow problems. The next section describes
the results of testing this model using data from a sample of months in the
1976-1979 period.
3.6 Testing the Model
Tests of the model have been conducted using a sample of 12 months from
the 1976-1979 period. The sample of months allowed us to test the model and
illustrate its use. The sample months were selected to span as wide a range of
variation as possible. As a summary of the results of these tests, we wish to
focus on three important measures; loaded car-miles, yard engine-hours, and
total operating cost. For each of these measures, observed data from the
actual operations are available for comparison to the model predictions.
Loaded car-miles measures linehaul activity, and the degree to which the
model reflects total traffic flows on the network. Yard engine-hours measures
yard activity, and tests the degree to which the simple statistical models
described in Section 3.4 reflect actual resource requirements. Finally, total
operating cost is the measure we are most interested in predicting.
Figure 3-4 illustrates the test results for loaded car-miles (normalized
to protect proprietary data). These results appear to be quite satisfactory.
In general, the model underpredicts, but in most cases by less than 10%. This
underprediction was expected, since the model produces an "optimal" solution,
given all the input data for the entire month. This should be better than the
solution achievable by managers in the actual system, who must respond to con¬
ditions on a day-to-day basis with only partial information available.
-50-
1.2
Loaded car-miles observed (normalized)
Figure 3-4. Predicted versus observed loaded car-miles.
-51-
Figure 3-5 shows the predicted versus observed results for yard engine-
hours. In general, these results also appear to be satisfactory. Despite the
simple form of the estimated relationships between cars handled and yard
engine-hours for each yard, in the aggregate the predictions are typically
within 10% of the observed values. The tendency of the model to overpredict
the number of yard engine-hours by a small amount is probably due to the aggre¬
gation of yards that has been done in the network. The actual system has many
more than 27 yards, but these have been combined in the representation of the
network used in the model. As a result, the flow through some of the yard
nodes is much greater than the individual yards experience in the real system.
If the relationship between flow and yard engine-hours were linear, this aggre¬
gation of yards would not matter; however, since the relationship for at least
some of the yards is nonlinear, the predicted number of yard engine-hours for
the aggregated flow is greater than the sum of the engine-hours for the dis¬
aggregated flows would be. The overall effect of this aggregation error,
however, appears to be minor.
Finally, Figure 3-6 shows predicted versus reported operating expenses
(again, normalized). These results are quite satisfactory. The model under-
predicts, generally in the range of about 5-15%. the reason for this is the
same as for the tendency to underpredict loaded car-miles. The model provides
an "optimal" solution. This should always be better than the observed re¬
sults. However, the closeness of the model predictions to the observed results
indicates that the model is a useful predictive tool.
In summary, the model results appear quite acceptable. It has a tendency
to underpredict slightly the level of activity (and hence costs) on the line-
haul portion of the network, and to overpredict activity (and costs) in the
yards. Errors, however, are typically less than 10% in the aggregate measures,
and the predictions of total operating costs are generally within 15% of the
observed figures.
As a result, this network model provides a reasonable basis for the esti¬
mation of a marginal operating cost function. Such a function serves to sum¬
marize the information in the network model relating to the sensitivity of
operating costs to various input prices and traffic levels. The estimation of
this type of marginal cost function is described in Chapter 4.
-52-
Figure 3-5. Predicted versus observed yard engine-hours.
-53-
l.4r
Predicted = observed
/ 15% error
.6 .8 1.0 1.2
Reported operating expenses ($ normalized)
Figure 3-6. Predicted versus reported operating expenses.
CHAPTER 4
ESTIMATION OF AN ORIGIN-DESTINATION
SHORT-RUN MARGINAL OPERATING COST FUNCTION
In this Chapter we extend the discussion of Chapter 3 to provide an equa¬
tion for predicting marginal operating costs, by origin-destination (0-0) pair,
as a function of traffic volume, Input prices and fixed factors such as levels
of capital utilization. The marginal operating cost function Is estimated and
its use for predicting 0-D marginal operating costs 1s discussed.
4.1 Procedure
The operations model 1n Chapter 3 can be summarized as the non-linear
program (PI):
(PI) m1n G(f, p; k)
f
s.t. Af » z
Bf = 0
f >_ 0
where f Is the vector of flows within the network, p a vector of prices (such
as yard crew wages, per diem rates, fuel price, switching locomotive prices,
etc.), k is a vector of fixed factors (capital utilization rates such as loaded
car-miles/car-day, empty-to-loaded car-miles ratio, etc.), z 1s the vector of
flows from origins to destinations, and the matrices A and B describe the
network.
For a given network (I.e. fixed A and B matrices) the optimal value of
(PI) is a short-run variable operating cost. If p and z are varied, we trace
out the short-run variable operations cost function c°(z. p; k):
c°(z, p; k) » {min G(f, p; k) Af » z, Bf - 0, f _> 0}
Because z is a vector of 0-D flows, the gradient of c° with respect to the
vector z 1s the vector of short-run marginal operating costs; I.e., 1f z^an
-54-
-55-
element of z) is the flow from origin i to destination j then the marginal
operating cost for the 0-D pair (i,j) is:
MCij = 3c°(z, p; k)
3z..
ij
We know, however, that MC1J is the shadow price on the constraint in (PI)
associated with right-hand-side z^ . Thus, if we run (PI) for month t, using
p^, z^ and as data, (PI) provides an "observation" on (c0)^ and on the
vector of 0-0 marginal costs (MC^)t. If we do this for N months (t =
1,... ,N), we have a sequence of observations {(c°, MC00, p, z, k)* on the
short-run variable cost function, the marginal cost function and the relevant
variables.
Note that the number of observations is N times the number of 0-D pairs.
Thus, if there are 25 nodes that can be origins or destinations, the potential
number of "observations" is 600N, since z would be of size 600 (25 2-25). Of
course, this means that the z vector in c° is also very large (in the example,
it is of size 600). Let r be the vector of 0-D distances and define aggregate
output as the scalar y (loaded car-miles) with
y = z' r . (4-1)
Now let us express the short-run variable operating cost function as a
function of y instead of z; i.e., as c°(y, p; k). To estimate this function we
assume (as in Chapter 2) that the function form is a translog:
n . n n
C°=ya x + - I I a XX (4-2)
q-1 <° " 2 q-l 4=1 q *
where, for variable x , Xq = in(xq/xq) and C° = xn(c°/c°). Let Xj = y; i.e.,
Xj is the transformed aggregate loaded car-miles of flow on the network.
Differentiating (4-2) we have:
-56-
Since y = z'r, then dy = (dz)'r. In particular, if we associate the in¬
crement to flow with the 0-D pair (i,j) and set the rest of the dz vector to 0,
then :
dy ■ <d Vu
and hence:
- l. . ""'"a - y . MC'J (4-3)
3X1 c° rij c° rij
Thus, the substitution of y for the vector z maintains the 0-D aspect of the
marginal costs. The marginal cost function (or more precisely flow-elasticity-
of-cost function) for the 0-D pair (i,j) is:
urij n
. — = a, + I a X . (4-4)
o r 10 . , lî 1
C lj 4=1
We can estimate (4-4) using our sequence of observations. Since some of
the x.j are prices, we can augment (4-4) with factor share equations. If we
assume that to xn_^ are price terms (i.e. there are n-2 prices, one output
and one fixed factor making up n variables) then the factor share equations to
add (n-3 in total) are the following:
mq = °q0 + °q4X4 q = 2»---»n-2 (4~5)
where is the share of optimal cost represented by factor q (q = 2,...,n-2).
As before, we assume symmetry:
V = % q, 4 = 1,...,n (4-6)
Finally, since a cost function should be homogenous in prices, so should
the marginal cost function. Thus, we restrict the estimation to satisfy
-57-
n-1
K aQ0 = 1
q=2 H
n-1
I aSL = 0 q = 1,... ,n (4-7)
1=2 q
Unfortunately, (4-5) and (4-7) cannot both be satisfied directly. There
are n-3 equations in (4-5) from the n-2 price terms. (Adding the last factor
n-1
share equation is redundant, since l m =1.) On the other hand, (4-7) calls
q=2 q
for restrictions over all the price-related coefficients, even those in the
dropped factor share equation. There are two alternative ways to resolve this
problem.
1. Estimate (4-4), (4-5) and the cost function jointly, since all the
price coefficients appear in the cost function.
2. Estimate (4-4) and (4-5) with all prices expressed relative to the
last price, i.e. Xn Thus, for example, the factor share equations
become:
n-2
% " °q0 + °ql*l + J2 V(Xt"Xn-l) * V" 1
since *n[ (x fi ^(Vl^n-l^ = x i " xn-l *
The second procedure above incorporates the homogeneity restrictions. Each
equation has one fewer variable and the restrictions (4-7) must be solved after
the estimation to calculate the coefficients for X^.
Thus, the system we will estimate is as follows:
(X_ • MClj) = a + aXt+ I a (X^X1 ) + a Xfc + e1
\0 r.. 10 11 1 U i n-r In n 1
(4-8)
■ «,0 + °11X1 * % °,«(xt-x5-l» * Vn 0 ■ 2 "-2
-58-
subject to the symmetry conditions (4-6) and any appropriate autocorrelation
structure assumptions, such as (2-10).
Note that the second approach (the relative price approach) estimates a
system of (n-1) variables by (n-2) equations. The first approach in which one
estimates the full translog function with (4-4) and (4-5) results in a much
larger system since the cost function has all the second-order terms, as well
as first-order terms. For example, if there are 3 prices, one output and one
fixed factor, alternative 1 results in 5 equations and 29 variable, while
alternative 2 results in 4 equations and 15 variables.
4.2 Estimation Results
System (4-8), subject to (4-6), has been estimated for a three price, one
fixed factor, one aggregate output (i.e. n = 5) model using data from the
twelve month sample mentioned in Chapter 3: February 1976, December 1976,
February 1977, July 1977, December 1977, February 1978, April 1978, February
1979, March 1979, May 1979, August 1979 and November 1979. The months were
selected to maximize the variation in observed prices and flows. The inde¬
pendent variables are the following:
xi: aggregate loaded car-miles
x2: fuel price in dollars/gallon
X3: price for locomotives in dollars/locomotive-month
x4: price for yard labor 1n dollars/engine-hour
X5: loaded car-miles/total car-days on-line (= 1/T).
These variables constitute the "best" subset of the inputs to the operations
model in the sense that all the other input prices, etc., are highly correlated
with one of the above variables.
Variable X5 represents a car utilization rate which is not easily adjusted
and has been used as a fixed factor. Note that X5 is different from the speed
variable used in Chapter 2 as an output measure. While the units of the vari¬
ables are the same, X5 includes implicitly the empty-to-loaded car-miles ratio,
which is clearly a capital utilization measure. This ratio is also difficult
to adjust, at least in the short-run. Thus X5 appears to be a reasonable
choice for a fixed factor to represent the nature of operations.
-59-
Tables 4-1, 4-2 and 4-3 provide the estimation results for the marginal
operating cost function, the fuel share equation and the labor share equation.
Since wages for yard and road labor are very highly correlated, we have used
only the price of yard labor, but the share reflects both yard and road labor.
Table 4-4 provides the equipment share equation, which is computed via the
homogeneity restrictions. Standard errors are in parentheses below each
estimated coefficient in each table.
The constant term, in each equation is the first-order term in the under¬
lying cost function. Thus, for example, a10 in Table 4-1 is the first-order
output term in the underlying cost function.
Using the results from Table 4-1 we can express the marginal operating
cost function as:
,. o
= rij y~ ' ^'9398 + °'152 ^(y/^ + °'0061 ^(pF/PF)
+ 0.0361 MPLA/PLA) - 0.0422 MPl0/Pl0)
- 0.0217 Jin(k/k) . (4-9)
Thus, marginal operating cost for any given 0-0 pair (i,j) is simply the dis¬
tance between i and j (r..) times a function of y, c°, the prices and the fixed
" J
factor. At the point of means:
-o
MC1J = rij — (.9398). (4-10)
y
The coefficients on output, input prices and the fixed factor which appear
in Table 4-1 (and equation (4-9)) are the coefficients on second-order terms
involving output in the original cost function. After differentiating with
respect to output to construct the marginal cost function, they become linear
terms. Note that the sum of the terms on input prices ( ai2 + ai3 + am) is
zero. This is as specified by constraint (4-7). The fact that the coefficient
on locomotive price (PLg) is negative does not mean that marginal costs will
fall as locomotive prices rise, because a change in locomotive price will also
affect c° which is also in the equation.
-60-
Table 4-1
Estimation Results
Marginal Operating Cost (Share)
Variable
Coefficient
Estimate
Standard Error
Y
an
0.1520
0.0052
PFUEL
a12
0.0061
0.0008
PLABOR
al 3
0.0361
0.0011
PLOCO
a11
-0.0422
0.0016
K
Qtl 5
-0.0217
0.0026
CONSTANT
a10
0.9398
0.0004
Table 4-2
Estimation Results
Fuel (Share)
Variable
Coefficient
Estimate
Standard Error
Y
a21
0.0061
0.0008
PFUEL
a22
0.1113
0.0004
PLABOR
a2 3
-0.0417
0.0004
PLOCO
a24
-0.0696
0.0003
K
a25
0.0459
0.0004
CONSTANT
a20
0.1116
0.0001
-61-
Table 4-3
Estimation Results
Labor (Share)
Variable
Y
PFUEL
PLABOR
PLOCO
K
CONSTANT
Coefficient
°3 1
a32
°3 3
a3<*
a3 5
a3 0
Estimate
0.0361
-0.0417
0.1109
-0.0692
0.0289
0.2590
Standard Error
0.0011
0.0004
0.0006
0.0001
0.0005
0.0001
Table 4-4*
Estimation Results
Equipment (Share)
Variable
Coefficient
Estimate
Standard Error
Y
-0.0422
0.0011
PFUEL
a42
-0.0696
0.0007
PLABOR
aU 3
-0.0692
0.0010
PLOCO
0.1388
0.0011
K
a4 5
-0.0748
0.0001
CONSTANT
<*4 0
0.6294
0.0001
* All coefficient estimates and standard errors in this table are computed via
the homogeneity restrictions. For example, a40 = 1 - a20 - a30.
-62-
For example, suppose initially all variables are at the point of means, so
that the marginal cost for 0-D pair ij is given by (4-10). Now suppose that
rising interest rates cause the price of equipment (represented by pLQ) to in¬
crease by 10%. What is the effect on marginal costs?
To answer this, we must refer to equation (4-2) to recompute c°. Because
all variables are initially assumed to be at the point of means, and only p
changes, all except X4 will be zero. Thus, we have
£n(c /c ) - at,Q MPlq/Plo) + P|_0^L0^ "
From Table 4-4, we find that a40 = .6294, and = .1388. In this case pLq =
1.1 pLQ, so we have
*n(c°/c°) = .6294 in(l.l) + .1388[*n(1.1)]2
= .06124.
This implies that:
o , .06124. -o . -o
c = (e ) c = 1.063 c .
Returning to (4-9), all the terms inside the brackets except the constant
and the p^Q term are zero, so we have
MClj' = rij 1,Q^3c {.9398 - .0422 *n(1.1) }
-o
= r.. £- (.9948)
J y
Thus, MCij /MC1J = .9948/.9398 = 1.058, so marginal cost has risen about 5.8%
as a result of the 10% increase in pLQ. Similar analyses can be done to exam¬
ine the effects of other changes in prices or output level.
The constant terms in Tables 4-2, 4-3 and 4-4 represent the first-order
terms in the cost function, and thus sum to one as a result of the first con¬
straint in (4-7). At the point of means in the data, all terms in the factor
-63-
share equations except these constants go to zero, and hence at that point the
shares of total operating cost attributable to fuel, labor and equipment are
about 11%, 26% and 63%, respectively. Because the point of means average con¬
ditions over a period during which fuel prices especially were rising rapidly,
these shares are not very reflective of current conditions. To evaluate cur¬
rent conditions more carefully, the other terms in the factor share equations
(with coefficients indicated in Tables 4-2, 4-3 and 4-4) would have to be used
also.
As a final comment regarding interpretation of these results, some caution
should be exercised in applying them to very short or very specialized move¬
ments. Very short movements may require relatively more switching and local
train activity than normal, and thus the linear relationship to distance may be
inaccurate. At the other extreme, the costs unit train movements may not be
reflected well because they have not been separated from other train operations
in the analysis, even though their operating costs may be relatively different.
4.3 The Contribution of Economic Theory to Operational Analysis
It is worth noting that without the above structure, derived from economic
theory, the estimation of the marginal cost function would have been much
poorer. To see this, suppose that we had simply estimated (4-4) without the
factor share equation (4-5), the symmetry restriction (4-6) or the homogeneity
restriction (4-7). The result of such an estimation is shown below; with
standard errors in parentheses:
y MC
ij
— = 0.4029 + 0.0688 *n(y/y) - 0.0158 «n(pp/pF)
ij (0.0005) (0.0090) (0.0039)
- 0.0155 *n(k/R) .
(0.0033)
(4-11)
-64-
Comparlng (4-11) to Table 4-1 1s quite revealing. The estimated coeffi¬
cient on y and the estimated Intercept 1n (4-11) are both less than half of the
corresponding estimates in Table 4-1. Both the price of fuel and the price of
locomotives have experienced sign changes. We also observe that (4-11) does
not reflect homogeneity in prices; the price coefficients should add to zero.
Since we know that the results in Table 4-1 are theoretically justifiable, we
can see that dropping the theoretical restrictions (which yields (4-11)) can
lead to seriously flawed and inferior results.
4.4 Summary and Implications
We have described a procedure for predicting marginal operating costs for
specific origin-destination prices in a rail firm. Such estimates reflect an
optimization model, and thus they provide lower bounds on actual marginal
costs. Obviously, the more accurate the network representation and cost data
the better the estimates will be. In general, however, the procedure outlined
above provides a systematic and defensible way to compute 0-D specific marginal
operating costs and would be useful both in pricing decisions and in operations
analysis studies. As an example of the former, it is clear that prices should
be above the marginal operating cost; this procedure provides a floor and its
justification. An an example of its role in operations analysis studies, one
could use such a procedure to examine the impact of network changes on the
sensitivity of marginal operating costs to input prices (such as fuel) by esti¬
mating MC1J functions for each network specification and then comparing the
functions.
CHAPTER 5
CONCLUSIONS AND IMPLICATIONS
5.1 Conclusions and Implications from the Cost Model of the Firm
Probably the most striking conclusion of the results presented in Chapter
2 is the clear evidence of returns-to-density for the railroad under study.
When coupled with the results of Phase I (returns-to-density for the small
railroad), the implication is that returns-to-density is not an isolated
phenomenon.
This finding can be contrasted with those cross-section studies which have
ignored density issues and have either found constant returns-to-scale or
slightly increasing returns-to-scale. In general such studies have not con¬
trolled for differences between firms within samples. As discussed in Chapter
2, this can lead to biases in estimates. Without such control, economies-of-
size (i.e. changes in configuration and size of the firm) and economies-of-
density (configuration and size held fixed) are confused.
A second reason for the difference is inclusion of a properly constructed
quality-of-service measure. In general the cross-section studies have either
not included quality measures or they have had to use surrogates. The test on
inclusion of the speed term clearly indicates that dropping the variable leads
to a misspecification error. It also leads to a lower estimate of economies-
of-density. Thus, the lack of a properly constructed quality-of-service vari¬
able in earlier studies may also be partially responsible for the difference
between our results and the cross-section results.
We also found that the most variable factors of production (fuel, labor
and equipment) do not appear to substitute readily for one another, and all the
factor demand functions are inelastic, with demand for labor most inelastic.
Labor is the major share of short-run variable costs (53%).
It is also interesting to examine the share of fuel in costs (19%) and
compare it to the result from the smaller railroad studied in Phase I (6%).
While part of this difference may be due to the difference in the sizes of the
firms, part may also reflect the fact that the estimate for the larger railroad
reflects conditions in the mid-to-late seventies, while the estimate for the
-65-
-66-
small firm reflects the early-to-m1d seventies. Both firms were inelastic with
respect to fuel prices and, given the enormous fuel price change experienced,
one might expect fuel's share of cost to increase.
5.2 Conclusions and Implications from the Operations Model
One principal conclusion from the network model described in Chapter 3 is
that it is quite feasible to produce reasonable operating cost estimates from a
relatively simple network model. This is true even for network operations as
complex as those on the railroad under study, which is a major Class I system.
The predictions of two major measures of activity - loaded car-miles and yard
engine-hours - were typically within 10% of observed numbers for a sample of
months over a four-year study period. The resulting operating cost estimates
were all within 15% of the reported operating expenses for those months.
In general, the model underestimates operating costs by a small amount.
This is to be expected, because it is an optimization model, and predicts the
"optimal" (minimum cost) pattern of car flows given all the information on
movements for the month. In reality, decisions made on a day-to-day basis from
incomplete information are unlikely to be optimal in the same sense, even
though they may be the best decisions possible under the given circumstances.
Thus, in some ways, the model provides a standard against which observed re¬
sults can be compared. This presents a potential use of the model for manage¬
ment evaluation and control of operations.
The purposes of this network model should be made clear. In contrast to
some other railroad network models, it is not intended to address detailed
operational questions involving blocking, train dispatching, power utilization,
etc. Its purpose is to provide estimates of overall operating costs and
origin/destination-specific marginal costs which are sensitive to changes in:
1) the level of traffic on the system, 2) the prices of cars, locomotives,
labor, fuel, maintenance, etc., and 3) aggregate characteristics of operations
on the network such as average train length, empty-to-loaded car-miles, loaded
car-miles per car-day, locomotive utilization and use of system vs. foreign
cars. This objective is quite different from that of many previous railroad
network models which have focused on routing, scheduling and blocking policies,
or on detailed operations of specific elements of the network.
-67-
The model is Inexpensive to use, both 1n terms of analyst time and com¬
puter time. Setting up for a run from "scratch" requires a few hours of
analyst time to compute Input prices and the various utilization factors.
Subsequent runs to explore changes 1n various Inputs can be made very quickly.
A preprocessor program could make this setup even easier and faster if the
model were to be used regularly, but such a program has not been developed in
this project. The computer costs have been less than $10 per run on Cornell's
IBM 370/168 system.
A primary potential use of this model is investigation of the marginal
costs of moving traffic on specific 0-D pairs for pricing purposes. Because
the model is an optimization model, it produces a lower bound, or floor, under
the marginal costs of providing various services. The rates charged should be
at least this great. The capability to evaluate such marginal costs is of
great use to railroads, in both justifying rates for regulatory proceedings and
in negotiating contracts with shippers.
The statistical estimates of marginal cost described in Chapter 4 provide
a convenient method for surmiarizing a large amount of information from the
network model. The functional form expresses the marginal cost as a function
of distance, input prices, total volume being moved on the system and a network
parameter (used as a fixed factor). This model is an excellent example of the
power of the "hybrid" approach to cost analysis which has been the cornerstone
of this project. The means for generating individual observations on marginal
costs for specific movements is an engineering model; the means for summarizing
this data in a meaningful way makes substantial use of economic theory. It
should be emphasized that without the economic theory to provide a structure
within which to do the statistical analysis in Chapter 4, the results were much
less satisfactory. Together, the economic theory and the engineering models
provide a powerful tool for the analysis of railroad costs.
The resulting statistical model in Chapter 4 is a convenient form in which
to express the general effects of changes in traffic volume or various input
prices on marginal costs. This can be extremely useful, for example, in exam¬
ining the potential impacts of fuel price increases on marginal costs, and
hence the potential of such price changes to affect rates. It also provides a
means for examining the potential impact of improvements in car utilization, as
measured by changes in the number of loaded car-miles per car-day on-line, to
-68-
reduce marginal costs. Information on such issues is very important to both
railroad management and to Federal policy-makers and analysts.
While the operating analysis as it stands is useful, there are also some
enhancements which might improve it further. First, somewhat more detailed
analysis of yard operations could lead to improved specification of the rela¬
tionship between yard engine-hours and the number of cars handled in yards.
This, in turn, would improve the ability of the network model to reflect
changes in yard operation (which would affect costs) more accurately. Second,
it could be desirable to incorporate different types of train operation (e.g.
local, run-through, unit trains, etc.) in the network specification. This
would require definition of various classes of links in the network. Such
modifications would Improve the sensitivity of the network model to more de¬
tailed operational changes, but would make the solution of the optimization
problem more expensive. Further investigation is required to determine whether
the benefits would be worth the costs. Finally, better specification of the
output variable in the statistical cost function is desirable. Currently, the
variable used is total flow on the system. This reflects the fact that flows
other than that on the 0-D pair in question affect the marginal costs for any
given 0-D pair. However, it is also true that not _alj_ the flows on the network
affect every other flow. It would be desirable to reflect only those flows
which directly interact with the flow on a given 0-D pair. This presents some
difficult measurement problems, but is a useful area for further investigation.
5.3 Use of the Research Products
Figure 5-1 summarizes the potential uses of this research. On the left-
hand side of the diagram are the three major research products - the network
model, the short-run marginal operating cost function, and the firm-level cost
functions (short-run and long-run). In the middle of the diagram are the prin¬
cipal outputs of these models. On the right-hand side are three major kinds of
activities undertaken by railroad firms and/or government policy makers or reg¬
ulators. The arrows indicate how each of the activities depend on the results
produced by the three models.
Figure 5-1. Use of the research products.
-70-
For example, short-range planning activities of the railroad would use the
total operating cost predictions from the network model, but probably not the
information on factor substitution, for example, from the firm-level cost
function. On the other hand, pricing decisions rely on information from the
firm-level model as well as marginal operating cost information. Medium-range
to long-range planning that might be done either by railroads or government
would not rely much on details of network operation, but would make use of the
more general information in the firm-level analysis.
In closing, we wish to emphasize that the form of cost analysis described
in this report differs markedly from earlier studies. We have demonstrated
that cross-sectional econometric studies using aggregate data are prone to
misinterpretation because of model misspecification. On the other hand, we
have also demonstrated that economic theory is very important in making sense
of marginal cost estimates produced by engineering models. Together, the
econometric and engineering techniques complement each other very effectively.
Joint use of these methods can lead to more reliable estimates of cost than can
be obtained from either by itself. This provides the basis for improved public
policy decisions and for improved ability of railroad management to understand
the costs of providing services.
-71-
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Economic ABZ's of the Railroad Industry, Association of American Railroads,
Washington, D.C., 1980.
Frank, M. and P. Wolfe, "An Algorithm for Quadratic Programming," Naval
Research Logistics Quarterly, 1956, pp. 95-110.
Friedlaender, A.F., The Dilemma of Freight Transport Regulation, Brookings
Institution, Washington, D.C., 1969.
Friedlaender, A.F. and R. Spady, "Equity, Efficiency and Resource Allocation in
the Rail and Regulated Trucking Industries," CTS Report No. 79-4, Center for
Transportation Studies, Massachusetts Institute of Technology, 1979.
-72-
Griliches, Z., "Cost Allocation in Railroad Regulation," Bell Journal of
Economics and Management Science, 2» 1972, pp. 26-41.
Harris, R.G., "Economies of Traffic Density in the Rail Freight Industry," Bel 1
Journal of Economics, 2» 1977, pp. 556-564.
Hasenkamp, G., Specification and Estimation of Multiple Output Production
Functions, Springer-Verlag, New York, 1976.
Keeler, T., Railroad Cost Function: An Empirical Study, unpublished B.A.
thesis, Reed College, 1967.
Keeler, T., "Railroad Costs, Returns to Scale and Excess Capacity," Review of
Economics and Statistics, May 1974.
Klein, L.R., "The Use of Cross Section Data in Econometrics with Application to
a Study of Production of Railroad Services in the United States," National
Bureau of Economic Research (mimeo), 1947.
Lau, L.J., "Testing and Imposing Monotonicity, Convexity and Quasi-Convexity
Constraints," in M. Fuss and D. McFadden (eds.), Production Economics: A Dual
Approach to Theory and Applications, Volume I, North-Holland, New York, 1978.
LeBlanc, L.J., E.K. Morlok and W.P. Pierskalla, "Approach to Solving the Road
Network Equilibrium Traffic Problem," Transportation Research, 9, 1975, pp.
309-318.
Mundlak, Y., "Empricial Production Function Free of Management Bias," Journal
of Farm Economics, February 1961, pp. 44-56.
Sammon, J.P., "Returns to Traffic Flow Concentration in the Railroad Industry,"
Proceedings of the Transportation Research Forum, XIX, 1978, pp. 104-111.
Swanson, J.A., Railroad Investment Demand, 1948-1965: A Neoclassical Model,
unpublished Ph.D. dissertation, University of Wisconsin, 1968.
Theil, H., Principles of Econometrics, John Wiley & Sons, New York, 1971.
Varian, H., Microeconomic Analysis, W.W. Norton, New York, 1978.
Welty, G., "Budget Planning 1979," Railway Age, November 17, 1978, pp. 26-32.
Wilde, D.J., Optimum Seeking Methods, Prentice-Hall, Englewood Cliffs, N.J.,
1964.
APPENDIX
USING THE NETWORK OPERATING COST MODEL
RAILNET 1s a FORTRAN program designed to predict operating costs on rail¬
road networks by solving the network flow problem described in Chapter 3. The
current version has been designed for the IBM VM/370 operating system, using
the Conversational Monitor System (CMS). The program includes four major
modules: (1) a control module which handles much of the interaction with the
user and invokes other modules in response to the user's instructions, (2) a
data input module, (3) an analysis module which performs the actual computa¬
tions, and (4) an output module. The nature and capabilities of each of these
modules are described in the individual sections which follow. Detailed in¬
formation on the use of the various options within each module is provided in
Section A.5.
A.l Program Control
To initiate a session with RAILNET, the user logs on at a terminal,
attaches a data set (if one is to be used) and invokes the program. S/he is
then communicating with the program control module. The program will ask the
user whether or not an existing data set is to be read, what kind of terminal
is being used, and then give the user a menu of activities from which s/he may
select what to do next. This menu includes: examining (and possibly changing)
the data, writing a file to save the current data including changes made dur¬
ing this session, conducting an assignment of traffic to the network, restart¬
ing the program, or ending the session. Additional details on these various
options are described in Section A.7.
When the user responds, the control module may ask additional questions
to clarify exactly what the user wishes to do. When the instructions are
completely specified, the control module invokes one of the other modules to
begin the requested activity. Each activity includes a set of potential
tasks. In some cases, the sequence of tasks in an activity is relatively
fixed. In other cases, there is considerable room for user intervention. At
the completion of each task, the control module may request further instruc¬
tions from the user if the next task in the sequence is not predetermined.
A-l
A-2
At the completion of an activity, the control module repeats the menu,
and the user may select another activity. This process repeats until the user
terminates the session.
A.2 Data Input
Data input may be from a permanent file constructed prior to the session
or directly from the terminal. In either case, the data include three sets of
information, pertaining to system parameters, links and traffic volumes. The
parameter data specify values which pertain to all links, such as wage rates,
fuel prices, equipment prices, etc. The link data describe each link in the
network including origin node, destination node, length, capacity, and a set
of parameters for use in computing the link cost as a function of volume. The
volume data provide an origin-destination (0-D) flow table. Data input for¬
mats are described in Section A.5.2.
An important element of the program is the capability to modify the data
file during a terminal session. This makes the analysis truly interactive,
and is of great use for exploring the effects of changes in the network quick¬
ly and easily. Any element of the data file can be modified or deleted, and
new nodes or links can be defined.
If the user makes several changes in the data file (or creates a new data
file) at the terminal, this new data may be stored as a permanent file. This
is done using the "WRITE" option in the activity menu, as described in Section
A.7. The program than writes the data to an output file in a format suitable
for rereading at a later time.
A.3 Computation
The RAILNET program has the ability to assign traffic to links in the
rail network using the algorithm summarized in Chapter 3 (pp. 47-49). The
user must supply a FORTRAN subroutine to compute the unit cost of carload
movements over each network link. This allows the user to specify the form of
marginal cost function used, and to use different functions for different
classes of links. The specifications for this subroutine are described in
Section A.5.1.
A-3
A.4 Output
The output module of the program allows the user to obtain detailed link-
by- 1ink output in tabular form at the end of an assignment. It also produces
summary statistics on total volume moved, total loaded car-miles, and total
cost.
A.5 Detailed Instructions for Use of the Program
This section provides detailed instructions for developing the subroutine
to calculate link impedances, preparing data input, and using the major op¬
tions within the program. These topics are discussed individually in the
following subsections.
A.5.1 The Link Cost Routine
The user must provide a subroutine which will calculate the unit cost
of a carload movement over each link in the network. This user-provided rou¬
tine enhances the capability of RAILNET because several different link types
may be designated and the link cost calculated for each type using a different
formula. The subroutine may use values of up to 5 parameters for each link,
which are input as arguments to the subroutine. The subroutine must be writ¬
ten in FORTRAN and named LKCOST. The format which should be followed is:
SUBROUTINE LKCOST (LT,CAP,DIST,ARG,VOL,COST)
DIMENSION ARG(5)
RETURN
END
where: LT = link type (integer-valued)
CAP = link capacity (cars/time period) (real-valued)
DIST = length of the link (miles) (real-valued)
ARG = vector of link parameters (real-valued)
VOL = link volume (real-valued)
COST = resulting cost (real-valued)
LT, CAP, DIST, ARG and VOL are inputs to LKCOST. COST is the output. Link
type (LT) may be used as an indicator to determine how to use the parameter
vector to compute the cost.
A-4
As a simple example of a possible LKCOST subroutine, suppose we desire to
calculate costs using the following function:
c = {a + bv + dv2}m
where: c = cost ($/carload)
v = volume (carloads/unit time)
m = length of the link (miles)
a,b,d = parameters.
This cost function has four link-specific parameters: the length and val¬
ues of a, b and d. The link length is passed to subroutine LKCOST explicitly,
but a, b and d would have to be passed as arguments in the ARG array.
Under the assumption that ARG(l), ARG(2) and ARG(3) give values for a, b
and d, respectively, for a given link, the following subroutine would perform
the required computations:
SUBROUTINE LKCOST (LT,CAP,01 ST,ARG,VOL,COST)
DIMENSION ARG(5)
COST = (ARG(1) + ARG(2)*V0L + ARG(3)*V0L*V0L)*DIST
RETURN
END
Note that the link type parameter, LT, is not used in this routine. If
some links in the network were to have costs calculated according to this
formula, and others use a different cost function, the LT parameter could be
used to distinguish among them and branch to the appropriate section of the
LKCOST routine.
This sort of branching is shown more clearly in a second example, which
implements the cost calculations described in Chapter 3 of this report. The
listing of LKCOST is shown in Figure A-l. The first executable statement of
the subroutine tests for actual usability of the link whose cost is to be cal¬
culated. A link can be "removed" from the network by setting its capacity to
zero. In that case, LKCOST will return a very large unit cost for that link
($108) so that no traffic will be assigned to it.
A-5
SUBROUTINE LKC08T (LT, CAP, DIST, ARC, VOL. COST)
DIMENSION AR0(5)
C
C COMMON BLOCK /PARM/ CONTAINS ALL THE SYSTEM PARAMETERS
C
COMMON /PARM/ RCWAOE. YCWAOE. FUELP. RLFRAT, YLFRAT,
X RLOWN, RLUTIL, YLOWN, YLUTIL, COWN, PDD,
X PDM, ATL, ELCMR, FCOL. FCM, CDLCM, WSCLCM
C
C*#****»••*•••***»*****»*»*#***»**»«»»*****##**#*•********#**
c
C RCWAOE - ROAD CREW WAGE RATE
C YCWAOE - YARD CREW WAOE RATE
C FUELP - FUEL PRICE
C RLFRAT - ROAD LOCOMOTIVE FUEL CONSUMPTION RATE
C YLFRAT - YARD LOCOMOTIVE FUEL CONSUMPTION RATE
C RLOWN - ROAD LOCOMOTIVE OWNERSHIP COST
C RLUTIL ■ ROAD LOCOMOTIVE UTILIZATION RATE
C YLOWN ■ YARD LOCOMOTIVE OWNERSHIP COST
C YLUTIL - YARD LOCOMOTIVE UTILIZATION RATE
C COWN ■ SYSTEM CAR OWNERSHIP COST
C PDD - DAILY PER DIEM CHARGE ON FOREION CARS
C PDM - MILEAGE PER DIEM CHARGE ON FOREIGN CARS
C ATL - AVERAGE TRAIN LENOTH
C ELCMR - EMPTY-TO-LOADED CAR-MILE RATIO
C FCOL - PROPORTION OF CARS ON LINE WHICH ARE FOREIGN
C FCM - PROPORTION OF TOTAL CAR-MILES MADE BY FOREIGN CARS
C CDLCM - CAR-DAYS PER LOADED CAR-MILE
C WSCLCM - WAY AND STRUCTURES MAINTENANCE COST PER LOADED
C CAR-MILE
C
C****»*************"»****»***»»***»#*'»****************#********
C FIRST CHECK FOR EXISTENCE OF LINK
C
IF (CAP GT. 1) GO TO 10
C IF NO LINK, SET VERY HIGH COST AND RETURN
COST - 1. 0E8
RETURN
C
C TEST FOR LINK TYPE 1 (LINEHAUL LINKS)
C
10 IF (LT NE. 1) 00 TO 20
C COMPUTE ROAD CREW COST PER CAR-MILE
CI - RCWAOE • (1. ♦ ELCMR)/ATL
C COMPUTE FUEL COST PER CAR-MILE
C2 - FUELP » RLFRAT
C COMPUTE LOCOMOTIVE OWNERSHIP COST PER CAR-MILE
C3 - RLOWN * RLUTIL
C COMPUTE CAR OWNERSHIP COSTS PER CAR-MILE
C4 - (FCOL • PDD ♦ (1 - FCOL) # COWN) * CDLCM
X + (1. + ELCMR) » FCM » PDM
C GET TOTAL COST BY ADDING Cl - C4, PLUS W*S MAINTENANCE,
C AND MULTIPLYING BY THE LENGTH
COST - (Cl + C2 + C3 ♦ C4 + WSCLCM) * DIST
RETURN
C
Figure A-l. Listing of LKCOST subroutine to implement
the formulae given in Chapter 3.
A-6
C TEST FOR LINK TYPE 2 (YARD LINKS)
C
20 IF <LT NE. 2) 00 TO 30
C COMPUTE COST PER YARD ENOINE HOUR
CY - YLOWN/YLUTIL + FUELP#YLFRAT ♦ YCWACE
C COMPUTE MARGINAL YARD ENOINE HOURS PER CAR
MEH - 2. ♦ ARO(I) » VOL/(CAP»CAP)
C THEN OET MARGINAL COST
COST - CY # MEH
RETURN
C
C TEST FOR LINK TYPE 3 (LOCAL CONNECTORS)
C
30 IF (LT NE. 3) 00 TO 40
C COST IS LOCAL AREA SPECIFIC («/MILE « DISTANCE)
COST - ARO(l) • DIST
RETURN
C
C IF LINK TYPE NOT 1, 2 OR 3. THERE IS AN ERROR IN DATA
C
40 WRITE(6. 1000) LT
1000 FORMAT(' ##« ERROR IN LINK TYPE *«» LT - '.13)
RETURN
END
Figure A-l. Continued.
A-7
If the link exists, the test at statement 10 checks for a link of type 1
(Unehaul). If LT 1s 1, the next five statements are executed, representing
equations (3-2), (3-3), (3-5), (3-6) and (3-8) from Chapter 3. Equation (3-7)
is embedded implicitly in the calculation of COST, and the mileage, DIST, is
included at the end, rather than in each separate computation.
If LT is not 1, the test at statement 10 causes a jump to statement 20,
which checks for a link of type 2 (yards). If LT is 2, the cost/yard-engine
hour (CY) is computed, representing equation (3-9) from Chapter 3. The mar¬
ginal engine-hours per car (MEH) is then computed. Note that the user may set
a specific ARG(l) value for each yard, to represent different operating char¬
acteristics among the yards. COST is then computed as the product.
If LT is not 2, the test at statement 20 causes a jump to statement 30,
which checks for a link of type 3 (dummy connector, as described in section
3.2). The cost on such a link may be used to represent local switching and
movement costs to and from shipper docks. Again, ARG(l) is used to specify a
link-specific parameter.
If LT is some number other than 1, 2 or 3, an error message is written,
and no computation is done.
A.5.2 Preparing Data Input
A data file for RAILNET includes three separate sets of information,
describing system parameters, links and traffic volumes. A data file may be
prepared and stored on a disk so that the data can be used at a later time.
Data may also be entered within the program or changed to consider the results
of an altered network.
To prepare a data file, the structure shown in Figure A-2 must be fol¬
lowed. The first record in the data file must contain the word "PARAMS" in
columns 1-6. The next six records (lines) contain the system parameters, as
fol lows :
A-8
Figure A-2. Example input data set.
PARAMS
5.00
42.00
0.85
0.2
11.5
250.
5.0E-7
170.
18.3
15.00
7.00
0.04
67.2
0.78
0.60
0.52
0.21
LINKS
10
1, 2, 1, 57.0, 3000, 3000, 0., 0., 0., 0., 0.
1, 3, 1, 85.0, 3000, 0, 0., 0., 0., 0., 0.
2, 3, 1, 72.0, 3000, 0, 0., 0., 0., 0., 0.
3, 4, 2, 0.0, 1800, 0, 622.5, 0., 0., 0, 0.
4, 1, 1, 85.0, 3000, 0, 0., 0., 0., 0., 0.
4, 2, 1, 72.0, 3000, 0, 0., 0., 0., 0., 0.
4, 5, 3, 20.0, 2000, 0, 0.83, 0., 0., 0, 0.
4, 6, 1, 93.0, 3000, 0, 0., 0., 0., 0., 0.
5, 3, 3, 20.0, 2000, 0, 1.03, 0., 0., 0., 0.
6, 3, 1, 93.0, 3000, 0, 0., 0., 0., 0., 0.
END LINKS
0-D DATA
5
1, 2, 34, 12
1, 5, 52, 73
2, 5, 90, 81
2, 6, 112, 17
5, 6, 42, 15
END DATA
A-9
Record # Data Items
2 Road crew wage rate ($/train-mile)
Yard crew wage rate ($/yard engine-hour)
3 Fuel price (S/gal)
Road locomotive fuel consumption rate (gals/loaded car-mile)
Yard locomotive fuel consumption rate (gals/yard engine-hour)
4 Road locomotive ownership cost ($/time unit)
Road locomotive utilization rate (loco-time units/loaded
car-m1le)
Yard locomotive ownership cost ($/time unit)
Yard locomotive utilization rate (hours/locomotive/time unit)
5 Car ownership cost ($/car/time unit)
Daily per diem charge, avg ($/car)
Per diem mileage charge ($/car-mile)
6 Average train length (cars)
Empty-to-loaded car-mile ratio
Proportion of total cars-on-line which are foreign, average
Proportion of total car-miles made by foreign cars, average
Total car-days on line/loaded car-mile
7 Way and structures maintenance cost factor ($/loaded car-
mi le)
Within each record the items are not limited to specific columns, but must
be input in the order indicated. All data are read as floating point numbers,
with decimal points.
Immediately after the last parameter input record, a record with "LINKS"
in columns 1-5 must be included. The next record specifies the number of links
in the network (free-format). For purposes of data input, a two-way link be¬
tween a pair of nodes is considered as one link.
A-10
For each link, the following pieces of Information must be Input 1n order
(free-format, separated by commas): node number 1, node number 2, link type,
link length (miles), capacity 1n the 1 to 2 direction (cars/unit time), capa¬
city In the 2 to 1 direction (cars/unit time), and the five link parameters
which are user-specified. Note that because free-format Input is used, all
five parameters must be specified for each link, even if they are not used.
Specifying a zero capacity in one direction can be used to make a link one-way;
this 1s done for many of the links 1n Figure A-2.
Immediately after the last link record, a record with "END LINKS" in
columns 1-9 must be included, followed by a record with "0-D DATA" in columns
1-8. The next record specifies the number of interchange records included, and
1s free-format.
The format of the interchange records is yard r, yard s, carloads/unit
time from r to s, and from s to r (free-format, separated by commas). Thus,
one record specifies traffic volumes 1n both directions for a single pair of
points. This is important 1n determining the number of Interchange records to
be input. Note that there Is no need to input data for interchanges where
there are zero carloads 1n both directions. However, if one direction is non¬
zero and the other 1s zero, the zero must be entered to complete the record.
The format for entering network data from the terminal within the program
is identical to that just described for establishing a file, the LOOK option
is used and the program will prompt the user for each line of data input (in¬
cluding format instructions). The LOOK option also enables the user to change
network characteristics for analysis without changing the network data main¬
tained in the file. The program prompts the user for the particular changes
desired.
A.6 Keywords in RAILNET
RAILNET is an interactive system. As such, 1t pauses at certain points in
the program when input is required from the user. Prompting messages are dis¬
played which tell what appropriate user responses are. The use of keywords
allows a short response to signify the desires of the user. Prompting occurs
at delays for observation of results, for input of data, and for selection of
program options.
A-11
Four keywords are common throughout the program and have meanings which
are relatively obvious. <YES> signifies an affirmative response to the ques¬
tion being displayed and correspondingly, <N0> signifies a negative response.
The keyword <Ç0NT> is used to instruct the program to continue processing after
a pause in which some result is displayed. The use of <EXIT> will cause pro¬
cessing in one phase to end and the program will move to the next sequence,
return to the main program, or terminate as appropriate.
Additional keywords are used to select particular program options. The
display will show appropriate selections and provide a brief description of
each choice.
A.7 The Menu of Activities in RAILNET
After beginning execution of the program and inputting data, RAILNET pre¬
sents the user with a menu of activity options to determine what actions will
occur next. These options are as follows.
WRITE causes the current data set and results to be written on a disk file
for future use and comparison of results. This write operation occurs to
FORTRAN logical unit 2, so the file will normally be called FT02F001. It is
the responsibility of the user to rename and catalog this file as appropriate
if it is to be retained after the terminal session ends.
LOOK gives the user access to the data files. S/he can then examine the
files, add new data, delete data, or make changes to the existing file.
RESTART causes the program to start at the beginning, including rereading
of the data set. Note that the program option RESTART should be used between
traffic assignments performed in sequence. The ASSIGN option, to be explained
next, uses the last result as a starting point. In order to preclude erroneous
results, it is advisable to use RESTART when you have examined one assignment
action and wish to begin another.
ASSIGN is the principal option of RAILNET. Selection of this keyword
starts the assignment process. Immediately after selecting ASSIGN, the program
will prompt for the number of iterations to be performed. Zero iterations
results in an all-or-nothing assignment. Since equilibrium results of suffi¬
cient accuracy for most problems can be obtained with 10 or fewer iterations,
if the user specifies more than 10 iterations, the program requests confirma¬
tion before proceeding. This is done to avoid excessive computation as a
result of a simple typing error.
A-12
A.8 Output from RAILNET
The primary output from RAILNET is two tables. The first is a lihk-by-
link report of volume (carloads) arid marginal cost for crossing that link,
together with, summary statistics on carloads moved, loaded car-miles and total
operating cost. An example of such output is shown in Table A-l. The values
in this table are the results of using the example input data in Figure A-2
together with the LKCOST subroutine shown in Figure A-l.
The second output table is origin-to-destination marginal costs, for each
pair of points between which shipments occur. An example of this output is
shown in Table A-2, again corresponding to the sample problem setup using the
data in Figure A-2 and the LKCOST subroutine in Figure A-l.
A.9 Obtaining the Software
Copies of the RAILNET software are available either from the U.S. Depart¬
ment of Transportation or from the authors of this report. The addresses are
shown below:
Mr. Joel Pal ley, RRP-32
Federal Railroad Administration
U.S. Department of Transportation
400 Seventh Street, S.W.
Washington, D.C. 20590
Professor Mark A. Turnquist
Hoi lister Hall
Cornell University
Ithaca, NY 14853
Questions regarding the capabilities of the model, computer system
requirements, etc., should be addressed to Professor Turnquist.
A-13
ORIGIN DEST. MARGINAL
NODE NODE VOLUME COST ($)
1
2
34
54.56
1
3
52
81.36
2
1
12
54.56
2
3
202
68.91
3
4
482
11.31
4
1
73
81.36
4
2
98
68.91
4
5
157
16.60
4
6
154
89.01
5
3
196
20.60
6
3
32
89.01
TOTAL
CARLOADS MOVED
= 528
TOTAL
LOADED CAR
-MILES
= 59205
TOTAL
OPERATING
COST
= $59278.
Table A-l. Example of link-by-link output from RAILNET.
A-14
ORIGIN
DEST.
MARGINAL
NODE
NODE
CARLOADS
COST ($)
1
2
34
54.56
1
5
52
109.27
2
1
12
54.56
2
5
90
96.82
2
6
112
169.23
5
1
73
113.27
5
2
81
100.82
5
6
42
120.92
6
2
17
169.23
6
5
15
116.92
Table A-2. Example of origin-destination marginal operating
costs from RAILNET.
METRIC CONVERSION FACTORS
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LENGTH
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TEMPERATURE (gxact)
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tamparatura
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Tachnicol Report Documentation Pope
'■ "«port No.
DOT/OST/P-30/85/007
2. Co
i No.
Titlo and Subtiil.
Development of Hybrid Cost Functions From
Engineering and Statistical Techniques: The Case
of Rail - Phase II Final Report
3. Recipient % Coiolog No.
5. Report Data
October 1984
6. Performing Orgoni cotton Codo
7' Au,hor'') Andrew F. Daughety, Mark A. Turnquist,
Ronald R. Braeutigam
8. Performing Organization Report No.
Porforming Organization Name and Address
Northwestern University, Transportation Center,
Evanston, Illinois 60201
Cornell University, School of Civil & Environmental
Engineering, Ithaca, New York 14853
10. Work Unit No. (TRAIS)
11. Controct or Grant No.
D0T-0S-70061
12. Sponsoring Agoncy Noma and Addross
U.S. Department of Transportation
Research & Special Programs Administration
Office of University Research
Washington, D.C. 20590
13. Typo of Roporf ond Poriod Covorod
Final Report
9/79 - 7/81
14. Sponsoring Agoncy Codo
P-30
IS. Supplementary Notes
Technical Monitor: Joel Palley, RRP-32
16. Abstroct
Cost analysis is important in every transportation industry, to the firms
or agencies which provide service, to regulatory bodies, and to public policy
makers. In the past, railroad cost analyses have been of two types: 1)
statistical analyses of aggregate cross-section data from a variety of firms,
or 2) very detailed operations-oriented studies. The premise of the work
reported here is that a "hybrid" approach, using both economic theory and
statistical methods on the one hand, and engineering analysis of operations
on the other, can produce superior results.
This report covers Phase II of the project, which focused on analysis of
a major class I railroad. A short-run variable cost function was estimated
econometrically, and used as a basis for deriving the associated long-run
function. We also developed a simple, but relatively accurate, network model
to estimate operating costs. This model may be used to estimate origin-
destination specific marginal operating costs. Econometric analysis of
the output from the model leads to a theoretically justifiable equation for
predicting marginal operating costs, and their sensitivity to changes in
flows and input prices.
17. K*y Word*
transportation
cost analysis
railroad operations analysis
scale economies
19. Security Clesstf. (of tbls roporf)
Unclassified
18. Distribution Stotomont
Document is available to the U.S. public
through the National Techical Information
Service, Springfield, VA 22161
20. Security Clossif. (of this pog*)
Unclassified
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Form DOT F 1700.7 (8-72)
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N0 The Case of Rail Phase II FINAL REPORT
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