DOT-RSPA-DPB-50-79-31
Development of Hybrid Cost
Functions from Engineering and
Statistical Techniques:
The Case of Rail
TRANSPORTATION LIBRARY
NOV 1 10,90
N 0 R T H W EST £ R N y NIVERSITY
DECEMBER1979
FINAL REPORT
Under Contract: DOT-OS-70061
Document is available to the U.S. public through
The National Technical Information Service,
Springfield, Virginia 22161
U.S. DEPARTMENT OF TRANSPORTATION
Research & Special Programs Administration
Office of University Research
Washington, D.C. 20590
NOTICE
This document is disseminated under the
sponsorship of the Department of Trans¬
portation in the interest of information
exchange. The United States Government
assumes no liability for its contents or
use thereof.
Technical K
1. Report No.
DOT/RSPA/DPB-50/79/31
2. Government Accession No.
3. Recipient's Catalog No.
4. Title and Subtitle
DEVELOPMENT OF HYBRID COST FUNCTIONS FROM ENGINEERING
AND STATISTICAL TECHNIQUES: THE CASE OF RAIL - FIRST
YEAR FINAL REPORT
5. Report Date
JANUARY 1980
6. Performing Organi zation Code
7. Author's)
Andrew F. Daughety and Mark A. Turnquist
8. Performing Orgonization Report No.
9. Performing Organization Name and Address
Northwestern University
The Transportation Center
Evanston, IL 60201
10. Work Unit No. (TRAIS)
11. Contract or Grant No.
DOT-0S-70061
12. Sponsoring Agency Name and Address
U.S. Department of Transportation
Research and Special Programs Administration
Office of University Research
Washington, P.C. 20590
13. Type of Report ond Period Covered
First Year Final
Report
Id. Sponsoring Agency Code
DPB-50
15. Supplementary Notes
Technical Monitor: Joel Palley, RPD-32
16. Abstract
Meaningful economic analysis of public policy and resource allocation in the trans¬
portation industries requires empirical understanding of costs. Two traditional
methods of acquiring such knowledge are: 1) engineering techniques based on detailed
models of operations, and 2) statistical analysis based on expenditure and output
records of firms. The objective of this research is to develop a method for combin-
ining these approaches, in order to provide accurate, meaningful models of a cost
function for a railroad. The resulting hybrid cost function incorporates detailed
information on both operations and non-operations activities so as to provide a more
complete picture of the relation of costs to outputs produced and inputs purchased
and used by the rail firm. Thus hybrid models reflect costs associated with yard
and linehaul activity as well as marketing, planning and other non-operations ele¬
ments of the firm. This links together management, train crews, yard activity,
maintenance, financial considerations, etc. as elements of cost generation in the
firm.
A general methodology for integrating engineering and economic approaches to cost
analysis is developed and applied to data collected from a rail firm. A multi-
output, short-run variable cost function is estimated and examined. Concepts of
economies of size, configuration and density are clarified and used to examine
theoretical effects of regulation on firm maintenance policy.
17. Key Words
transportation
cost modelling
railroad cost analysis
scale economies
railroad operations analysis
18. Distribution Statement
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through the National Technical Informa¬
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21- No. of Pages
22. Pr ' ce
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Reproduction of completed page authorized
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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 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.
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 specified a relatively simple
ES-1
functional form for costs, and assert that the form is appropriate
without a test of that assertion. Few adjust for quality
of service, 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
strikingly different approach to the problem for a number of reasons.
1) Our analysis begins at the level of an individual
firm, and uses 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. We employ a
time series analysis for a single firm rather than
a cross-sectional analysis for a particular year.
2) The multiproduct nature of the firm is incorporated
into the analysis. Output will be characterized
both by the volume of freight hauled and by the
average speed of a shipment through the system.
Models with disaggregated volume (by commodity type)
have been estimated, as well as with aggregate data.
We explicitly recognize that speed of service is an
important determinant of rail costs, and include this
in our estimates.
ES-2
3) We use information about the underlying tech¬
nological production process, developed through
engineering process functions, to better specify
the nature of technology and to improve the
efficiency of our estimates.
In several respects the last point is particularly novel.
Historically, most econometric estimates of cost functions have ig¬
nored valuable information generated from an analysis of engineering
process functions to provide observations of service-related variables.
We have labeled this a "hybrid" approach for that reason, and we
believe that important new insights can be gained from applications of
this I technique to other modes, as well as in rail transport.
RESULTS ACHIEVED
A short-run variable cost function incorporating commodity flow
information, service characteristics, factor prices and a measure of
the plant quality was developed and estimated from data for a railroad.
Engineering models of linehaul and yard activity were used to provide
information on the average speed of a shipment through the system.
The models of the linehaul allowed for grade differences on linehaul
sections, track quality, trailing load of the train, amount of
available locomotive power and delays due to congestion on single-
track sections. The yard activity model predicted expected waiting time
in the yard. This'information, along with information on amounts of
ES-3
commodities moved, quality of plant and prices of factors such as
cars, locomotives, crews, non-crew labor and fuel were used to
provide an estimated cost model.
Three questions were examined, based on the estimated model:
1) Does engineering model information contribute
significantly to model performance and quality?
2) How do the various terms in the cost functions
influence predicted short-run costs?
3) Does the data support the use of some of the
simpler production models often resorted to in
earlier analyses?
First, a test of the value of the engineering information was
constructed and performed. The result was that the introduction of
engineering information significantly improved the model. This acts
as a test of the value of a hybrid approach to cost analysis and con¬
firms its superiority to traditional economic or engineering models.
Second, the impact on cost of changes in factor prices,
commodity flow level, speed and plant quality were examined. The
elasticities of cost with respect to factor prices reveal the some¬
what surprising result that the firm cannot easily substitute away
from non-crew labor into capital as the non-crew labor price rises.
This reflects the need for further computerization of the firm.
Increases in speed would reduce short-run variable costs. These would
ES-4
largely come about from improvements in linehaul condition, which
was also seen via the elasticity of cost with respect to plant
quality which was negative. Marginal commodity flow costs were
positive, as expected.
Third, the model structure admitted testing of various sub-cases
such as Cobb-Douglas production technology, which were rejected.
Thus, previous studies that have started out assuming such models, are
likely to be misspecified and produce biased results. This finding
is consistent with very recent literature in the area of cost and
production theory and estimation.
The project also provided two theoretical results. First, a
procedure for properly integrating economic and engineering models
was developed. The procedure provides the general form of the
cost function to be estimated and classifies the model variables as
to whether or not they are to be provided by engineering process
models. This is a general procedure which can be used in many other
analyses.
The second theoretical result concerns a clarification of economies
of scale. Economies (and diseconomies) of scale are discussed in terms
of economies (and diseconomies) of size, configuration and density.
These concepts are defined and used to relate firm-level maintenance
decisions to regulatory constraints on service abandonment.
ES-5
UTILIZATION OF RESULTS
Potential users of the research results include both government
agencies and private railroad firms. In fact, officials of railroad
"X", which cooperated with us on the empirical work in this project,
have already expressed a desire to use our results with respect to
marginal cost computations as a basis for submitting a proposed rate.
Certainly this is evidence of the immediate applicability of the
work to problems faced by rail firms. It is not, however, the only
way in which the results could be used by railroads. Significant
insights have also been gained with respect to cost elasticities
for various input factors. These elasticities have important
implications for corporate planning.
From a government perspective, this work provides an important
step toward operationalizing the concept of "incremental" costs as a
basis for policy-making and regulatory proceedings. This concept plays
a central role in the regulatory reform legislation currently in Congress.
By making the concept operational (specifying data requirements and
analysis procedures) regulatory proceedings and policy-making can
better incorporate economic principles.
CONCLUSION
A procedure for integrating economic and engineering approaches to
cost analysis was developed and employed to use data from a medium size
railroad to estimate a model relating short-run variable costs to
commodity flows, speed, factor prices and plant quality. Engineering
ES-6
models of yard and linehaul activity were used in a general cost
model incorporating financial and engineering data. The data
requirements for the study were not unreasonable] almost everything
that was needed was already maintained by the firm. The results of
the research are useful to both railroad firms and policy makers.
The cooperating railroad in this study is presently evaluating the
estimation results for possible inclusion in a rate proceeding.
Moreover, this study is a step toward providing an operational
definition of incremental costs.
ES-7
Acknowledgments
Major contributions to the inception and development phases of this
project were made by Ronald R. Braeutigam (at this writing, with Northwestern
University, on leave to California Institute of Technology),
Special thanks to the executives and staff of Railroad X who provided
access to data and help in understanding what we had stumbled upon. A num¬
ber of them put in no small amount of time digging out and assembling
information, without which this project would not have been possible. Thanks
also to William Delaney who, while with the Federal Railroad Administration,
was our contract monitor, and was without par.
We also thank Robert Baesemann, formerly of Northwestern University,
who was instrumental in starting the project and Greg Duncan, who spent a
short, but productive period beating on data requirements issues with us.
Joseph Swanson of Northwestern has been invaluable as a sounding board for
ideas and as a source of information on calculating prices of capital goods.
The Transportation Center contributed significantly with financial
support to augment the original contract from USDOT. Without this help we
would still be transcribing data.
Thanks to Ken Hurdle of the AAR for providing some very needed data,
and especial thanks to Melissa Bartlett, John Gray, Victor Liu and Hossain
Poorzahedy, who slaved over much of the data set helping piece it together.
Finally, thanks to all those who contributed to the typing of the various
papers and parts that have gone into this project, especially Theresa Bonk
who plowed through pages upon pages of manuscript, providing her usual
excellent typing and editing job. This work was sponsored by the U. S.
DOT, Research and Special Programs Administration, Office of University
Research under contract D0T-0S-70061.
i
TABLE OF CONTENTS
page
1. INTRODUCTION 1
1.1 Other Railroad Cost Estimates 1
1.2 A Time Series Estimate of a Railroad Cost Function 3
2. HYBRID ANALYSIS 5
2.1 Problems to be Solved 5
2.2 Technological Economies: Size, Configuration and
Density Economies of Scale 6
2.3 Hybrid Cost Models and the Use of Engineering
Information 11
2.4 Engineering Process Models of Linehaul Train
Movement and Yard Operations 17
3. DESCRIPTION OF THE DATA FOR RAILROAD X 46
3.1 General Overview of Data Needs 46
3.2 Flow 47
3.3 Prices of Variable Factors 49
3.4 Fixed Factor Levels 51
3.5 Data Used in the Engineering Analysis 52
3.6 Cost 55
4. THE MODEL TO BE ESTIMATED AND THE ESTIMATION RESULTS 56
4.1 The Model 56
4.2 Estimation Results 58
5. RESULTS AND IMPLICATIONS 64
5.1 Introduction 64
5.2 On the Statistical Evaluation of Results 64
5.3 A Test of the Hybrid Approach 65
5.4 Implications of the Model 65
6. SUMMARY AND DIRECTIONS FOR FURTHER RESEARCH 71
6.1 Summary of First Year's Results 71
6.2 Plans for the Second Year 73
BIBLIOGRAPHY 78
APPENDIX A - A MORE DISAGGREGATE COST FUNCTION A-l
APPENDIX B - PRODUCTION AND COST: THEORY AND EXAMPLES B-1
iii
LIST OF FIGURES
page
Figure
1 -
ECONOMIES OF SCALE
7
Figure
2 -
CLASSICAL LONG-RUN AVERAGE COST
9
Figure
3 -
SCALLOPED LONG-RUN AVERAGE COST
9
Figure
4 -
CONSTRUCTING A HYBRID PRODUCTION SURFACE
14
Figure
5 -
THE RESULTING HYBRID PRODUCTION SURFACE
14
Figure
6 -
LINEHAUL MODELS
19
Figure
7 -
CLASSIFICATION YARD MODELS
19
Figure
8 -
TRACTIVE FORCE AS A FUNCTION OF VELOCITY
21
Figure
9 -
EXAMPLE TRACK PROFILE
26
Figure
10
- TWO YARDS CONNECTED BY MAINLINE
29
Figure
11
- EXAMPLES OF ERLANG DISTRIBUTIONS FOR DIFFERENT
VALUES OF k
33
Figure
12
- CONFIGURATION AND MAINTENANCE POLICY
69
Figure
13
- DENSITY ECONOMIES AND MAINTENANCE POLICY
69
iv
CHAPTER 1
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 up¬
grade the quality of plant or whether, at an existing tariff, the revenues
for a service cover the incremental cost of providing the service. Cost data
may be used to argue for a change in tariffs. 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 profitable 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 im¬
proved 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 services
of a multiproduct firm, is subsidizing other services, and whether the provi¬
sion of service by one mode will eliminate another mode over a given route.
If regulators are interested in setting tariffs that allocate economic re¬
sources efficiently, they will require information about costs. Generally
speaking, then, regulators need cost information to determine how their poli¬
cies will affect market structure and economic performance. These comments
apply without exception 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
1
a single product, usually ton-miles. These studies typically have examined a
cross section of Class I railroads, using ICC data, to test whether there are
economies of scale in rail transport. The results have generally been mixed.
For example, Klein [50] used 1936 data to find economies of scale that were
statistically significant, though modest. On the other hand, estimates by
Borts [ 8 ] and Griliches [38] have suggested that, while there may be econo¬
mies of scale for smaller railroads, scale economies are not prevalent for
larger Class I railroads.
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 firms. They typically specify a relatively simple functional form
for costs, and assert that the form is appropriate without a test of that
assertion. They do not adjust for quality of service, and more importantly,
they do not account for the multiproduct nature of virtually every rail firm.
And, they do not attempt to adjust for the fact that some railroads operate
with a more complicated network than others.
Keeler [49] and Hasenkamp [42] 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 [ 9 ] and Friedlaender and Spady [ 32] develop
models that allow multiple outputs and do not enforce spearability of inputs
and outputs. Caves, Christensen and Swanson [13] have also used such tech¬
niques 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 [50] has
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. No
2
real use of service variables such as speed could be used since such data is
firm-specific and not usually published. While the above studies have repre¬
sented important advances in understanding of costs, more work is needed,
especially at the level of the individual firm.
1.2. A Time Series Estimate of a Hybrid Cost Function
Our own research on railroad transport costs represents a strikingly dif¬
ferent approach to the problem for a number of reasons.
1) Our analysis begins at the level of an individual firm,
and uses cost and production data obtained directly
from the firm rather than from the ICC. This has a num¬
ber 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 prob¬
lems arising from the use of ICC data, see, for example,
Friedlaender [30], Appendix A.) We employ a time series
analysis for a single firm rather than a cross sectional
analysis for a particular year.
2) The muZtiproduct nature of the firm is incorporated into
the analysis. Output will be characterized both by the
volume of freight hauled and by the average speed of a
shipment through the system. Models with disaggregated
volume (by commodity type) have been estimated, as well
as with aggregate data. We explicitly recognize that speed
of service is an important determinant of rail costs, and
include this in our estimates.
3
3) We use information about the underlying technological
production process, developed through engineering process
functions, to better specify the nature of technology and
to improve the efficiency of our estimates.
In several respects the last point is particularly novel. Historically,
most econometric estimates of cost functions have ignored valuable infor¬
mation generated from an analysis of engineering process functions to
provide observations of service related variables. We have labeled this
a "hybrid" approach for that reason, and we believe that important new
insights can be gained from applications of this technique to other modes,
as well as in rail transport.
The plan of this report is as follows. Chapter two represents the
extension of the theory of production and cost (reviewed in Appendix B)
to the problem of integrating (or hybridizing) economic and engineering
approaches. Here we examine concepts of economies of size, configuration
and density, a new view of scale economies that comes about from
economic/engineering insights gained in the project. These insights also
provide the general method for constructing a hybrid cost function
developed in the chapter. Finally, the chapter provides a complete
analysis of the engineering models to be used.
The third chapter provides an overview of the data used while the
fourth chapter presents the estimation results (other estimation results
on a disaggregate volume model are presented in Appendix A). Chapter
five analyzes the results and draws out implications for rail cost analysis.
Plans for second year activity on the project are discussed in Chapter 6.
4
CHAPTER 2
HYBRID ANALYSIS
2.1 Problems to be Solved
In Appendix B we review the general elements of production and
cost theory. The analysis contained therein is for a general firm.
When, however, we become more specific in the type of firm that we
wish to analyze, we can then refine and expand those notions. In
our case we will consider a rail firm, and though most (if not all)
of what we develop will be immediately transferable to other modes,
we will couch most of our discussion in terms of a railroad.
The purpose of this project has been to examine the feasibility
and value of linking economic and engineering approaches to cost
estimation and apply it in a case study of a railroad; chapters three
and four consider the case study. Before proceeding to the case
study two fundamental issues must be addressed.
1. Can one be more specific about the relationships
between long and short-run cost functions ? What
are the natural factors to consider as constant
in the short-run and how can one discuss economies
of scale?
2. How should engineering information be merged with
economic information: is there a way to structure
the integration so that a general procedure is
developed?
These questions are interrelated: the first question raises the
issue of what is the short-run cost function and how does it relate to the
long-run cost function; we shall see that insights from the engineering
perspective provide the clue. The second question concerns the integration or
5
hybridization of engineering information into the economic model of cost:
we shall see that insights from the economic perspective provide the pro¬
cedure.
2.2. Technological Economies: Size, Configuration and Density Economies
of Scale
Economies of scale have been of interest to economists and policy makers
for centuries. Adam Smith spent three chapters of his Wealth of Nations dis¬
cussing specialization and the division of labor in production. A number
of categorizations and definitions of economies of scale have been developed.
Generally, the view was that economies of scale existed if average
costs fell as output was expanded. This raises, however, the disturbing
problem of what is an average cost in a multi-product firm if we can not
refer to any one of the outputs as the output (or if no aggregation function
on output exists).
The result is that economies-of-scale definitions for multi-product
firms now deal with the technological description and not the cost function.
Note that this ignores pecuniary economies of scale [72] that are
associated with the ability of a large firm to affect the prices at which
it purchases inputs. This type of economy is ignored based on the usual
presumption of given factor prices (see section B.2), i.e. the firm faces
perfectly competitive factor markets.
Panzer and Willig [64] provide the following definition of economies
of scale.
A technology T with associated transformation func¬
tion T(z,x) has economies of scale at (x,z) if there
exist r > 1 and 6 > 1 such that
T(Arz, Ax) £ 0 for 1 _< A £ S
6
This is a local definition, i.e. for the point (x,z). The exponent r may
be a function of x and z. All that is required is that the point (Ax,A z)
be in the technology for a small neighborhood of A. Figure 1 shows a tech-
112 2
nology t and two points (x ,z ) and (x ,z ). There are no economies of
11 2 2
scale at (x , z ), while there are economies at (x , z ). It is this
definition that provides the measure of scale economies S in section B.2.2.3.
Figure 1 - ECONOMIES OF SCALE
When we become specific, however, about the application of the theory
it is important to differentiate between different mechanisms that give rise
to technological economies (as opposed to pecuniary economies) of scale. In
this section we will examine, and attempt to relate, concepts of scale econo¬
mies due to size, to configuration and to density.
Scale economies due to size come about from the distant and varied geo¬
graphical points that a transport system (such as a railroad or a modern motor
common carrier firm) connects. Measures such as average length of haul tend
7
to be used to reflect this type of size. Size, in this sense, is important
since it opens many markets to the firm and shippers who must send goods
long distances usually prefer to work with as few transport firms as pos¬
sible so as to expedite claims on loss and damage. Large size may, or may
not, be accompanied by intensive utilization (or high traffic density) of
the system. In between the issues of size and density are the firm's poli¬
cies on configuration and system maintenance.
For any system of a given size, there may be many ways to configure the
actual system itself. The same set of demand points can be serviced by a
minimally connected network (e.g. a tree network wherein each demand point
is connected to at most two other points, and for n "demand points there
are (n-1) "arcs" connecting the points) a hub-and-spoke network (e.g. a
yard connected directly to each demand point) or a completely interconnected
system (each demand point connected to each other demand point). There are,
of course, many other possible configurations of a system. Changes in con¬
figuration can occasion changes in operating policies (such as blocking
policies) and changes in capital utilization (such as the use of cars). Some
of the changes will result in economies of scale, some in diseconomies. It
is important to note that in general such changes are discrete in nature:
there may be severe lumpiness in such changes which may not be smoothable
by other input changes. To see this we consider Figures 2 and 3. Here we
have assumed one output. Figure 2 shows the classical treatment of the
long-run average cost curves (LRAC) as the envelope of the short-run aver-
f.
age cost curves. The short-run curves are labelled x 1 where this represents
different levels of the fixed inputs. Because the fixed inputs are assumed
to vary smoothly then LRAC is also smooth. Equation (B.ll) of section B.2,1
8
Average
Cost
LRAC
•> Z
Figure 2 - CLASSICAL LONG-RUN AVERAGE COST
Average
Cost
-> Z
Figure 3 - SCALLOPED LONG-RUN AVERAGE COST
shows the use of the envelope theorem to derive the long-run total cost
curve from the short-run curves.
If the fixed factors are lumpy then we still have an LRAC curve, but
it may be scalloped, as shown in Figure 3. Here there is no intermediate
value of x between x
H
and x (or below x ). The dotted curves
9
represent those portions of the short-run average cost curves that are not
part of LRAC. Notice also that if we had incorrectly assumed x^ to be con¬
tinuously variable then our estimated LRAC curve would almost always under¬
estimate long-run average costs, since only the tangencies between the short-
and long-run curves would not be underestimated. Clearly, as the possible
values of x^ becomes denser (i.e. as x^ can be varied in smaller jumps) then
the underestimation becomes less pronounced. Since, however, configuration
changes can imply significant land acquisitions or disposals (and other types
of lumpy inputs) we expect that such changes are quite discrete in nature
and that Figure 3 properly represents the long-run average cost curve.
Finally, for a given size and configuration there may be economies of
density. Stigler [77] observes that there may be times when inputs would
be more fully utilized, but are not, producing excess capacity:
"There may be some unavoidable 'excess capacity'
of some inputs. A railroad has a tunnel which is
essential for given traffic, but can handle twice
as much traffic. The emphasis here is on 'un¬
avoidable. ' If the railroad has unused locomo¬
tives, in the long run they can be sold or worn
out, and hence do not give rise to increasing
returns." (p. 153).
In the case of transport systems, especially rail, the density of traffic on
the line-haul portion may be low relative to the line-haul capacity. Keeler
[49] and Harris [41] have found significant economies of density for U.S.
railroads due to excess track capacity (in both these articles, economies of
scale are broken down into size and density only). Thus, for example, the
elimination of double tracking on some line-haul segment (a change in system
configuration) might result in increased traffic density (especially if the
10
original traffic density is light). Density economies can be realized by
increases in traffic density for a fixed configuration. Changes in con¬
figuration that result in increasing the density of traffic on a particular
piece of track (without increasing the flow through the end-points) would
appear to be appropriately labelled configuration economies rather than den¬
sity economies. On the other hand, changes in operating policies that re¬
sult in high traffic density for the same configuration should be associated
with economies of density.
In summary, then economies of size can come from changes in size hold¬
ing the nature of the output fixed (i.e. we ignore diversification of the
firm into other markets than that for transport services). Economies of con¬
figuration can come from changes in system configuration (number of yards,
location, interconnectedness, double vs. single line-haul tracking) while
economies of density arise for varying traffic levels within a configuration.
2.3. Hybrid Cost Models and the Use of Engineering Information
In Appendix B we explain how a cost function C(z,p) is derived from
the cost minimization problem:
(CMP) min p'x
s.t. T(z,x) £ 0
where x is an n-vector of inputs, z an m-vector of outputs, p an n-vector of
given prices and T(z,x) a transformation function. It is useful to charac¬
terize the output of transportation firms in general, and railroads in parti¬
cular, in terms of both the physical units of flow over the system (e.g. car-
miles of various commodities) and measures of the quality of service pro-
11
vided (e.g. average speed, reliability, loss and damage, etc.) In mathe¬
matical terms, let z = (y,T) where y is an m^-vector of flows and it is an
m^-vector of service characteristics with m^ + = m. Let T(z,x) = T(y,ir,x)
be of an arbitrary general form, presently unspecified.
Most previous cost analyses have fallen into one of two categories. On
one hand, economists have developed cost models of firms or industries that
have generally ignored models of the physical process associated with the
firms activities (an exception is [58])- The strength of the economists's
models lay in the recognition that non-operations activities (such as planning,
sales, etc.) contributed to output and cost. These things were, to some
degree, captured by the economist's model. On the other hand, engineering
models of the operations aspect of a firm provide excellent information, but
an incomplete picture of the firm. These process models (see [20], [21] for
examples in the transportation area) specify certain relationships between
inputs (x), flows (y) and characteristics (tt). For example, one of the process
models to be presented in section 2.4 relates trailing load (a y-variable)
to locomotive horsepower (an x-variable) and speed (a ir-variable). Thus,
another way to view the overall production process of the firm is to "layer"
physical relationships as constraints on some very general transformation
function. The transformation function is the "glue" that holds the system
together, including inputs and outputs that are not definable in process func¬
tions. Notationally, we then have the following description of technology.
T(y,TT,x) £ 0 (!)
g1(y,u,x) <^0 i = 1,...,I (2)
where some parts of the y, it and x vectors in the g* functions may not appear
12
at all; i.e., each g function is a process function and as such may not use
all the variables. It may also occur that some of the inequalities are
really equalities.
Suppose we dropped T(y,ir,x) and partitioned the x-vector into subsets,
x''", so that x = (x\x^,... jX1), each g^(y,7T,x) was a function of only one
element of the Output vector and one subset of the input vector,
g1(Zi,xi) =0 i = 1,...,I (3)
we would have the case of non-joint production (see [40]). We will not
assume this extreme case. Instead, possibly overlapping subvectors of y, tf
and x appear in the g1 functions and the entire vectors appear in T. So,
for convenience, let H(y,ïï,x) be defined as the set of y, tt and x values
that satisfy the joint conditions. We will then write the system as
H(y,ïï,x) £ 0 (4)
As a simple example, let us consider each vector to have only one element
and assume a transformation function:
T(Y,n,X) = Y + n + X-l (5)
and one process function:
g(Y,n,X) = n - .5 . (6)
This system is illustrated in Figure 4. The result is the trapazoidal
solid in Figure 5 which we will call H(Y,I1,X). Notice two points:
1) H(Y,n,X) is more refined than T(Y,II,X) since we have added the informa¬
tion contained in g(Y,]I,X); 2) H(Y,II,X) is more comprehensive than
g(Y,n,X) since we have not neglected relationships between variables not
addressed by g(Y,II,X).
13
Figure 4 - CONSTRUCTING A HYBRID
PRODUCTION SURFACE
14
Therefore, engineering relationships help us refine a very general
structure into a more specific structure, not by making functional assump¬
tions (Cobb-Douglas, CES, etc.) but by the use of physical principles to
restrict the possible relationships among the variables. As more and more
g1 functions are added, the technology becomes better defined.
What does this mean for the analysis at hand? Consider the cost func¬
tion dual to min {p'x| H(y ,tt ,x) 0} , namely C(y,ir,p). We can estimate
this cost function by assuming, for example, a flexible functional form that
approximates an arbitrary production function. The variables in the func¬
tion would be flows of goods, prices of inputs and characteristics of the
service provided. It is the last category especially to which we now turn.
The output of a rail firm is stochastic, just like many other firms.
This is especially true of the characteristics of service, such as speed.
While we would not expect to see significant variation from month-to-month
in total aggregate demand (flow), we very well could expect significant var¬
iation in such outputs as speed of a shipment through the system. The effect
of this on our model of costs is very significant. Let f(x,co) be a stochas¬
tic production function with random variable co. Thus the output Z:
Z = f (x,(o) (7)
is now a random variable. Let the distribution of co be G(co) with density
function g(co). In [16] it is shown that the profit-maximizing firm will, im¬
plicitly, solve the following cost minimization problem (see section B.2):
(CMP) min p'x
s. t. E(f (x,(o) ) = u
15
where p is given, u is expected output and the expectation is taken with
respect to G on its domain. Thus, the cost function is
C(E(Z) ,p)
where the expectation here is over the distribution on Z (which can be derived
from f(x,a>) and G). If, on the other hand, we had simply formed the cost
function on outputs and prices we would have
E(C(Z,p))
It can be shown [ 16] that these two cost functions are consistent (for unknown
G) if and only if f(x,ûj) is homogeneous of degree one in x (see Appendix B,
section B.1.2.1). Since we do not wish to make this an implicit
assumption in the analysis, we do not want to use Z as a vari¬
able in our cost function. Rather, we should use a model that predicts the
expected value of Z for the observed values of the non-random variables.
This is what an engineering process model can give us. Hence, for those
stochastic variables in the output vector (y,Tr) we will use a process model
to provide the expected value of the variable. Thus engineering models, in
fact, provide more useful information than the raw observations themselves.
In general, then, we see that the role of engineering models is two-fold:
1. They provide useful information on physical relation¬
ships between the model variables, thereby properly
restricting the model of production.
2. They provide the proper variables for inclusion in
cost models, especially when some of the output vari¬
ables are stochastic.
16
By using engineering process models to define variables in the cost model we
implicitly refine the technology to which the cost model is dual. There¬
fore the result is a more efficient estimate of a better specified cost
model.
This study has concentrated on speed of a shipment through the system
as the engineering input; thus the tt vector has one variable. Speed is a
stochastic variable since time of day, of month, season in the year, etc.
all contribute to significant variation in the time it takes a shipment to
pass through the yards and over the appropriate segments of the line-haul.
Therefore the engineering work will concentrate on process models that re¬
late speed to some of the inputs and other outputs.
What might we expect to see for a fixed configuration? At low density
there is little relationship between speed and short-run variable costs (due
to union work rules, only radical changes in speed over reasonably long dis¬
tances would affect costs). At high density, congestion effects act to reduce
speed and increase costs. Therefore, one would expect to see a negative rela¬
tionship between speed and short-run variable costs. Clearly, changes in
configuration change the density level at which this occurs. These engineer¬
ing relationships illustrate the importance of speed. With this in mind, we
turn next to formulating the process models for the hybrid cost function.
2.4. Engineering Process Models of Linehaul Train Movement and Yard Operations
2.4.1. Introduction
A basic premise of this research project is that models representing
the basic engineering processes involved in railroad operations can contri¬
bute significantly to the estimation of rail cost functions. The integra¬
tion of these process models with econometric estimation of cost functions
results in what may be termed a "hybrid" model. This section describes in
some detail the set of process models used in the project.
17
The models cover three important areas of railroad operations:
1) line-haul movement of trains;
2) line-haul delays due to interactions among trains;
3) classification yard operations.
The models presented here draw heavily on the work of previous research¬
ers. The major contribution of the current project is synthesis of these
component models into a workable system for use in cost function estimation.
The types of models involved, and their interactions, are illustrated
schematically in figures 6 and 7. The chapter is organized into three major
parts, describing the three models separately.
2. 4.2. Line-Haul Train Movement
The line-haul train movement model developed for this project is based
on six fundamental characteristics to insure flexibility in application.
These characteristics are as follows:
1. The basic form of the model is derived from physical relation¬
ships, with corrections for technical conditions separated
from the primary function. This allows adjustment for
further technical change without modification of the
basic form.
2. The model is applicable over a wide range of train types
and physical line configurations.
3. It uses a minimum number of variables.
4. The model is designed to use variables defined so as to
be commensurate with expected data available (e.g. trailing
load, rather than individual car weight statistics.)
5. The variables can be readily related to specific cost data.
6. It provides a base which is adaptable to experimentation
with differing types of operational and investment policies.
18
Train 1 Trailing Load
Locomotive
Train i Trailing Load
Figure 6. LINE-HAUL MODELS,
Delay
time
V
Delay
time
Yard
delay
Figure 7. CLASSIFICATION YARD MODELS
19
The first requirement of the model is to establish a relationship among dis¬
tance, time, weight of the train, and horsepower provided. Obviously time
and distance can be further reduced in part to the velocity over each track
segment.
The analysis is based on establishing the physical relationship at a
constant velocity. For simplicity, acceleration and deceleration will be
ignored. Thus
F = m ft = = °
where: F = force exerted by the train system
m = mass
dv/dt = acceleration
There are two basic forces at work on the train — the tractive force exerted
by the locomotive and the resistance of the train mass acting in the opposite
direction. Thus the basic balance equation is:
F = F - F =0
t *R (9)
where: Ffc = tractive force of the locomotive
F = resistance force of the train
K.
Tractive force of a locomotive is a function of its horsepower and the velo¬
city at which it is operated. Locomotive manufacturers publish graphs of
velocity versus tractive effort for all of their locomotive types. (See,
for example, [26].) These curves are essentially rectangular hyperbolas within
the velocity range of interest (10 mph - 65 mph). Exceptions to this rule
have the form shown in figure 8 and fit the normal form to the right of
point a.
20
a
Figure 8 - TRACTIVE EFFORT AS A FUNCTION OF VELOCITY
The value of a is usually no higher than 15 mph. For all locomotives Ft
can be defined as follows:
Ft = 375 HP (e/v) (10)
where: HP = locomotive horsepower
e = machine efficiency, (.825 for most North American applications)
v = velocity (miles per hour)
F = tractive force (pounds)
This equation describes the relationship between F and v indicated by manu¬
facturer data to within about 2% for both types of curves. This indicates
that curves of the type shown in figure 8 reflect distortions in the lower
velocity ranges rather than basic functional changes. Thus, the form of F
shown will be used for all locomotives operating in line haul service.
a
The resistive force, F , may be broken into two components: F , the
R R
c
resistance to motion by the locomotive, and FD the resistance to motion by
R
21
the cars. The equation used to represent both of these quantities is the
time-honored Davis Formula. A complete discussion of its component parts may
be found in references [17], [18], and [43]. For the present purposes, it
is sufficient to note that it is based on considerable empirical evidence
gathered over the last half century, and it has been shown to provide consis¬
tently useful results. It is based on a resultant resistance determined by
summing the components of journal, flange, air, track, and grade resistance
with appropriate constants. The form used in this work is that presented in
Hennes and Ekse [45] which is taken from Davis's earlier work [ 18]:
E * ¥2 + — + JV + — + 20s <ll>
ww wn
where: R = resistance to movement, in pounds per ton of train weight
w = average weight per axle in tons
V = speed, in miles per hour
n = number of axles per item of equipment
J,K = constants /for locomotives: J = 0.03, K = 0.0024 \
\for cars: J = 0.045,K = 0.0005 /
A = cross sectional area, in ft? I for locomotives: A = 120 \
\for cars: A = 87.5J
s = grade encountered, in % .
In the case of locomotives, a value of 32 can be used for w. A check
of mainline locomotive weights since the introduction of the General Motors
FT diesel in 1939 shows that axle loadings have remained remarkably consis¬
tent since that time [H]. While some roads own heavy duty locomotives with
axle loads as high as 35 tons and a few older units are as light as 30 tons,
22
this variation is of little consequence (less than 1% in most cases) when
viewed against the total tractive effort exerted by the locomotive. The 32
ton weight is also highly representative of the major builders standard models
as supplied to railroads not requiring extensive extra features. The number
of axles, n, is the total for the locomotive, not the number on each unit.
Likewise, in the horsepower-tractive effort equation the horsepower is the
total for a locomotive, not that of individual units.
Thus, after substitution of constants and performing the implied multi¬
plications, the total resistance for locomotives is as given in equation 12.
2
Fr = [2.05 + .03V + + 20s] 32n (12)
= 65.60n + .96Vn + . 29V^ + 640sn
The car resistance function is a bit more complex. The additional problems
are caused by the inability to find a constant value for w. However, it is
possible to designate two constant w's (one for loads, one for empties) for a
particular railroad's circumstances. This can be shown in the following way.
Examination of the resistance function indicates that three of the five
terms are inversely related to the weight of the equipment. Thus an empty
car will have a higher resistance per ton than will one which is loaded.
This reveals the physical reason behind a phenomena which is well known in
railroading: a given locomotive can pull a heavier train if it is composed
primarily of loaded rather than empty cars. This is reflected in the policy
which a railroad adopts in assigning tonnage ratings to its locomotives.
If it rates its locomotives for the 80 to 100 ton cars which are becoming
prevalent, or for a higher percentage of loaded rather than empty cars, then
it will assign relatively high tonnages. On the other hand, if it bases its
23
ratings on 50 to 70 ton cars, or a low percentage of loads, then lower
capabilities will be assumed. In many cases the lighter cars are assumed
as the more stringent conditions represent a more conservative view and
also allow for other factors, such as adverse weather conditions, recovery
of lost time when delays occur on the road and the fact that many cars are
not loaded to capacity [26]. In view of this, locomotive tonnage tables
published by General Motors for its locomotives [26] are in terms of 50-ton
cars with an assumed light weight of 20 tons and 50% empty cars. Thus the
general form of car resistance is (assuming n=4 in all cases):
FrSC= [ ^y/2 + + .045V + •°109V + 20s ] 4w
w w w e
e e e
1/2 ?
- 37.6wg + 50 + .18w^ + ,0436V + 80W s (13)
an^ Fr = 37.6wj ^ + 50 + ,18Vw^+ .0436V2 + 80w^s (14)
for empty and loaded cars respectively, where w^ and w^ are the
appropriate axle weights. If the conservative assumptions outlined
above are used, these two equations become:
pp 2
FDec = 149.22 + 1.26V + .043V + 560s (15)
K
F fc = 216.04 + 3.51V + .0436V2 + 1560s (16)
As noted earlier, the effects of loaded cars should be considered.
On most railroads, tonnage ratings are based on the assumption that 50%
of the cars in a train will be empty. This even split makes it possible
to consider resistance in terms of the sum of the resistance of one loaded
6C f C
and one empty car taken together. This quantity (F + FR ) will be
denoted a "car unit." Thus the trailing load for a particular locomotive
can be expressed as the net tractive effort (total tractive effort minus locomotive
24
resistance) divided by the resistance of a car unit with the resultant
multiplied by the weight of a car unit. Obviously, the proportion of
loads to empties may be changed by appropriately weighting the two
resistance and weight quantities of the car unit. Thus, trailing load is:
F — F
TL = [ Ac + Ac ] 4(We + V- (17)
r T r
R R
When expanded this becomes:
309HP/V - (65.6n + ,96Vn + .29V2 + 640sn)
TL = l 272 1/2 2 ^ 4 ^We+ Wf ^ '
37.6(w ' + w/' )+.18V(w + wr)+.037V + 80s(w + w_)+100
et et et
(13)
If standard values of w =7 and w. = 19.5 are assumed, as outlined
e t
previously, equation (18) becomes:
TL = 1Q6 j- 309HP/V - (65.6n + .96Vn + .29V2 + 640sn) j
(364.81 + 4.77V + .087V2 + 2120s)
Other values of w and w, may be substituted depending on the policy on
et
which tonnage ratings are based. This function is stated in terms of
only five quantities; velocity, locomotive horsepower, number of locomotive
axles, tonnage trailing the locomotive, and gradient. All of these items
represent readily available data for any particular railroad.
2.4.3 Application of the Model to Determine Running Time
To determine the running time for a given train between two yards, equa¬
tion (19) would be applied in two states. First, given the trailing load
and the most restrictive conditions the train will encounter (usually termed
the ruling grade), this relationship can be used to determine the horsepower
which must be assigned to the train. Then, given the horsepower and trail-
25
ing load, the resulting velocity for any other track segment can be deter¬
mined. Once this has been done for all segments, the total running time
for the entire route can be computed. Details of this procedure are illus¬
trated by the following example.
Consider a line whose profile is as shown in Figure 9 . The total line
is 50 miles long, and can be divided into four segments. The first seg¬
ment is 20 miles long with no grade; the second segment is the ruling grade,
with a 1.0% grade over a 5 mile section; the third segment is 10 miles long,
with a 0.5% grade; and the last section is 15 miles of level track.
o
E
Assume that the trailing load is 5000 tons, and that we wish to main¬
tain a velocity of 20 miles per hour on the ruling grade. Let us also
assume that the locomotives available are of the four-axle variety.
The first step is to find the required horsepower for the train. Equa¬
tion (19) can be rewritten as follows to solve for the value of HP:
HP = [(0.11+ .065s)V+ .00015V2 + (2.66X10"6)V3]TL+ [ ( . 21 + 2.07s) V + .003V2]n
+ .0009V3 . (2°)
Inserting values of s = 1.0, V = 20 and TL = 5000, we obtain the required
horsepower to be
HP = 8200
26
In any given situation, we may not be able to assign exactly this amount of
horsepower to the train. The actual assignment would reflect the numbers
and sizes of locomotives available. For example, if the available locomo¬
tives are all 3000 horsepower units, the assignment would be three of these
units, with a total of 9000 horsepower. For the purposes of this example
application of the model, we will assume that exactly 8200 horsepower are
assigned. Thus, we know that the velocity on segment BC will be 20 miles
per hour.
The second stage of the analysis involves finding the velocities on the
remaining track sections. For this purpose equation (19) can be written as
a cubic equation in V, as follows:
C(2.55 x 10-6)TL + .0009]v3 + (.003n + .00015TL)V2
+ [(.011 + .065s)TL + (.21 +2107s)n]v - HP = 0 (21)
The roots of this equation can be found using standard formulae. For
the values TL = 5000, n = 4, s =0, HP = 8200, this equation has one real
root and two conjugate imaginary roots. The real root is approximately V =74.
Thus, on the level track segments, the predicted attainable velocity is 74
miles per hour. In practice however, due to gearing, the locomotives may not
actually be able to run this fast. In addition, there may be speed restric¬
tions on the line, so that the expected velocity of the moving train would be
given by:
Va = min[Vt,Vg,VTL] (22)
where: = expected velocity
V = speed restriction on line imposed by timetable
V = maximum achievable locomotive speed
g
V = speed attainable with given trailing load .
1L
27
For the purposes of this example, let us assume a speed limit of 60 miles
per hour along the line, so = 60 for the level segments, AB and DE.
For the segment CD, we insert values S = 0.5, TL = 5000, n = 4,
HP = 8200 into equation (21). Once again, there is one real root and two
imaginary roots. The real root is approximately V = 50.
Table 1 summarizes the results of the computations, and indicates the
overall predicted running time.
Table 1. Summary results for example.
Segment
Length (mi.)
VTL (mph)
(mph)
Running Time (hrs.)
AB
20
74
60
0.33
BC
5
20
20
0.25
CD
10
50
50
0.20
DE
15
74
60
0.25
TOTAL
50
1.03
2.4.4. Modeling Delays Enroute
Total origin-destination time for a train is not generally composed of
running time alone. In fact, there is always some pre-departure time at
the origin yard and some post-arrival time at the destination which must be
recognized. However, in addition, there are often delays enroute due to
switching and/or interactions (meets or passes) between trains.
Trains are often delayed enroute due to passing or being passed by
other trains going in the same direction, or on single track mainline,
28
meeting trains going in the opposite direction. Detailed simulation models
are often used by railroads to evaluate train congestion. However, for the
purposes of this project, it is desirable to have a simpler, analytic model
which can be incorporated more readily into the specification of a produc¬
tion function for cost estimation. The model proposed here draws heavily on
work done by Petersen [66]. It is also similar to methods of analysis for
low density highway traffic with passing delays (see, for example, Haight [39]).
The situation to be considered is illustrated in Figure 10, showing
two yards, A and B, connected by a main line track segment which may be
either single or double track.
Classes of trains in each direction will be defined by different aver¬
age running times (or speeds). In practice, of course, each train will have
a somewhat different running time, as determined by the model in the pre¬
vious section. Some aggregation will normally take place, since we are
interested in identifying classes of trains. For example, we may have local
freights at an average of 20 miles per hour, regular through freights averag¬
ing 40 miles per hour, and special high priority trains averaging 60 miles
per hour.
If we consider ourselves to be located at A, we will assume that there
are K different inbound train (speed) classes, and L different outbound
Figure 10. TWO YARDS CONNECTED BY MAINLINE.
29
classes. We will also adopt the convention that outbound speeds are nega
tive, for algebraic convenience.
Define M . as the expected number of encounters (meets or overtakes)
13
between a single train of class i and all trains of class j, on its trip
between yards. If we assume that each encounter results in an average de¬
lay, D.j, to train i, we can write the expression for average transit time,
including delays, as follows:
W. = R. + S. + I D..M.. (23)
x 1 x ^ X] X]
where W = average total transit time for train class i
= average running time for train class i
=average delays enroute from all other occurences.
The values for R. are determined from the line-haul train movement model des-
x
cribed in Section 2.4.2.
In order to utilize the model in equation (23 ) . the expected number of
encounters between trains, , must be expressed in terms of quantities avail¬
able. These include traffic density of trains of different classes, their
speeds, and dispatching policies through time. As an example of the deriva¬
tion, let us consider the expected number of meets between trains going in
opposite directions.
If an outbound train of speed i leaves at time t=0, it will encounter
inbound trains already on the line at t=0, and those dispatched before the
outbound train arrives at the other end, at time t=W^. For inbound trains
of speed j, this will include all trains dispatched between t= -W and t=W .
J i
If we assume, as Petersen did, that train departures from either end
of the line are independent and uniformly distributed through time, the
30
expected number of meets is
M. . = N.(W. + W.) (24)
ij 3 1 3
where = rate of dispatching of train class j (trains/unit time).
In a similar manner, we can derive the result that train i will over¬
take trains of a slower class, j, that depart between t = -(W^ - VL) and
t=0. Furthermore, train i will be overtaken by faster trains, j, which
depart between t=0 and t = W. - W.. If we then let I be the set of inbound
13
train classes, 0^ of the set of outbound train classes of higher speed than
train i and 0 the set of outbound train classes of lower speed than i, we
s
can write equation (23) as
Since we have an equation of this type for each speed class, both in¬
bound and outbound, this defines a set of K+L simultaneous linear equations
that can be solved for the K+L unknowns, W^. Of course, if the line under
study is double track the term delay due to meets vanishes.
The model described above is essentially that developed by Petersen
[66], Several extensions of this basic model are possible. English [27]
has made modifications so that it reflects operations on high density lines
more accurately. These modifications account for multiple meets and delays
induced by signal systems in very high density operations. For lower density
operations typical of most lines, an extension is possible to account for
31
the fact that trains are often not dispatched at random times with a con¬
stant mean interdispatch time throughout the day. Such a modification is
described in the following section.
2,4.4.1 Extension to Other Dispatching Strategies
The model described above assumes that trains are dispatched indepen¬
dently, at random times with a constant rate throughout the day. The impli¬
cation of this assumption is that times between successive trains will be
exponentially distributed. Thus the probability density function is given
by f(t):
f(t) = Ae"Xt, t > 0 (26)
where t = interdispatch time
À = average rate of dispatching (=l/mean time between trains).
In many situations however, train dispatches can be scheduled somewhat
more regularly, and line-haul delays due to meets can be reduced. In such
cases, the times between trains must be characterized by a more general
probability distribution. A useful generalization is to characterize these
times by the Erlang-k distribution. By varying the value of k, a wide range
of distributions can be represented. When k=l, this distribution is equi¬
valent to the exponential model. As k increases, the variance of the dis¬
tribution decreases, reflecting more regular dispatches. In the limit, as
k 00 , the distribution becomes a spike at a given value, reflecting con¬
stant times between dispatches of trains. A few members of this family are
illustrated in Figure 11.
32
The general form of the probability density function for an Erlang-k
random variable is:
, ,, nk-1 -At
f(t) = (k-1)! ' (2?)
If interdispatch times of train class j are distributed Erlang-k, we
can derive the expected number of encounters of a train of class i with all
trains of class j as follows. If we denote the number of dispatches in a
given period as a random variable , the probability of observing X dis¬
patches in a period of length W is:
k-1 -AW,,IT.Xk+i
P(Y - X) - I' (jhki • (28)
1=0
This result arises from considering an Erlang-k random variable to be a sum
of k exponential random variables with common parameter, A. Thus, the prob¬
ability of observing exactly X occurrences of the event described by the
Erlang distribution is the probability of observing between kX and k(X+l)-l
fundamental exponentially distributed events. This is the probability repre¬
sented by the sum in equation (28).
Equation (28) defines the probability mass function for the number of
encounters of a single train i with all trains of class j. The expected num¬
ber of these encounters is then
X=0
If we define a function p(z,t) as
E(Y.) = Y XP(Y. = X)
y vtn J (29)
°° e~tti
p(z,t) = I 6 lt (30)
i=z
34
equation (28) can be rewritten as follows:
P(Yj = X) = p(Xk,XW) - p[(X+l)k, XW] . (31)
Equation (29) then becomes
CO 00
E(Y ) = I p(k,XW) - I Xp[(X+l)k,XW]
J X=0 X=0
CO
= I p(Xk,XW) . (32)
X=1
Values of the function p(z,t) are tabled (see, for example, Molinas [61].)
The value for E(Y_.) can be substituted into equation (23) to replace
the expression for given in equation (24) . The basic delay model thus
becomes
W = R + S + I D I p[Xk, (W + W )] . (33)
j J X=1 3
Equation (33) defines a set of K+L non-linear equations in the K+L
unknowns, W^.. These must be solved using iterative solution techniques, but
they can be used to provide a more general solution for line-haul delays.
2.4.5. Models of Yard Activity
According to data gathered by Reebie Associates [69] the average rail
car spends only 16% of its time actually moving in trains. An additional
56% is spent in classification yards. This underscores the importance of
35
representing classification yard activities if we are to reflect railroad
operations with any reasonable degree of accuracy.
While in a railyard, a car undergoes four basic operations:
1) inbound inspection
2) classification
3) assembly into outbound train
4) outbound inspection.
It is quite natural to think of these as a series of queues through which
the rail car passes. This perspective is adopted here.
Inbound and outbound inspections consume a relatively small amount of
time for each car, and the amounts of time required are not highly variable.
For these reasons, they are not analyzed in detail here. However, explicit
queuing models have been constructed for the remaining elements: classifica¬
tion and assembly. Average time in the yard is predicted as the sum of
delays due to classification and assembly, as shown in equation (34):
T = T + T (34)
yea w/
where = average time in yard
Tc = average delay for classification
Ta = average connection delay before assembly into outbound train.
2.4. 5- 1 Classification Delay
There are a number of different queuing models which could be suggested
for the classification operation. The major previous work along these lines
has been done by Petersen [67].
36
He suggests several possible models, including:
M/G/l: Poisson arrivals of cars on trains, a general
service time distribution, and one server;
M/M/s: Poisson arrivals, exponential service times,
and s servers;
M/D/s: Poisson arrivals, deterministic (constant) ser¬
vice times, and s servers.
It should be noted that Petersen considers the basic units of arrival
to the system to be trains, not individual cars, and thus he derives para¬
meters for service time to classify an entire inbound train. While this sim¬
plifies representation of some elements of the system, it leads to some con¬
fusion about the relationship of the output process of one queue to the
input of another. For this reason, the models developed here are based on
individual railcars.
As a result, we should recognize the fact that individual railcars
arrive in batches on trains. This fact dictates use of a more general batch
arrival model, denoted as:
(X)
M /G/1: Poisson arrivals in batches of size, X;
arbitrary service times; and 1 server.
In this case, X is a random variable corresponding to train length. A
solution for such a model, yielding average delay time, has been developed
by Gaver [34 ]. A concise summary of the results is available in Saaty [71 ].
Average delay (time in queue plus service) is given as follows:
37
T
c
2(1-p)
+ 1
/
(35)
where 6^ = average train length (cars)
6 £ - second moment of train length
p = Àô^/u = traffic intensity of system
A = arrival rate of trains (trains/hr.)
p = average service rate (cars/hr.)
2
a = variance of service time distribution,
The distribution of service times for classifying cars depends greatly
on the physical layout and operating plan of a particular yard. Probably
the most important distinction is between hump yards and flat yards. In
hump yards, the classification service is quite straight-forward. A switch
engine pushes the string of cars over the hump at essentially a constant
speed (generally 1.5- 2.0 miles per hour). As each car reaches the hump
crest, it is decoupled and rolls down into the classification bowl. The
only variations in time per car are due to variations in length of indivi¬
dual cars. Such variations are relatively minor, and a deterministic
service time distribution is an appropriate model. In this case, = 0
(X)
in equation ( 35), and the model can be denoted M / D / 1.
Models for flat yards are somewhat more complex, since the switching
operations for classifying trains are not as simple as hump operation. An
inbound train comprises a set of cuts (groups of cars with common origin
and destination that move together through the yard) that are to be sorted
into outbound "blocks" on classification tracks. Each cut will be switched
as a unit, and if successive cuts are to be placed in the same block, one
switching operation can handle multiple cuts. Thus, rigorous derivation of
38
a service time distribution for individual cars would require incorporation
of the distribution of number of cars per cut and the likelihood of succes¬
sive cuts having common block designations, as well as the distribution of
time to complete a particular classification switch.
Since good empirical data were not available on all of these character¬
istics, our approach has been somewhat less detailed. A sample of flat switching
operations were observed in one yard, and the total time required and number
of cars switched were recorded. For each of these observations an equivalent
"minutes/car" value was then computed. Finally, a gamma distribution was
fit to these values. The probability density function of a gamma distribu¬
tion with parameters a and 3 is as follows:
f(x) = xa 1 e 0 <_ x < <» . (36)
The mean value of the gamma random variable, x, is a/3 and the variance is
a/32.
Maximum likelihood estimates of a and 3 were computed as 1.3 and .28,
respectively, using the observed data. This corresponds to a mean switching
2
time of 4.6 minutes/car, and a variance of 16.6 minutes /car. Previously
reported estimates for average switch-engine-minutes/car have varied widely,
depending upon the number of cars per cut and the degree of congestion pre¬
sent in the yard facility. For example, Wright [84] reported estimates of
3.2 minutes/car for single car cuts, but average values below 2 minutes/car
for multiple car cuts. Martland and Rennicke [59] report average values from
3 to 10 minutes/car for different levels of workload in two yards on the
Boston and Maine. Thus, it appears that our estimate is well within the
range of plausible values.
39
Values for variance of the switching time have not been widely reported,
so it is difficult to verify our estimates based on previous results. How¬
ever, the wide range in previously reported average values tends to support
the contention that the process is highly variable. Thus, our finding that
the service time process is nearly exponential is not surprising (note: an
exponential service distribution would have a =1.0). In general, it appears
that our estimates of parameters of the service time distribution for flat
switching are quite consistent with earlier reported results.
As an example of the values produced by the model, data from one yard studied
show an average arrival rate of 5.33 trains/day, with an average length of
45.2 cars and a second moment of length equal to 2876. Combined with the
estimated service time parameters, these data result in an estimated utiliza¬
tion rate of .77. Substituting values into equation (35), we obtain a mean
delay for classification of 8.2 hours.
Available data for the yard under study did not include detail on time
spent waiting for classification, connection delay, etc. As a result, it is
difficult to verify this model directly. However, the predicted delay of
8.2 hours is well within the range of observed data presented by Folk [28],
Beckmann, et al [3], and Gentzel [35] for various terminal facilities.
Values for mean classification delay between 4.6 and 22.4 hours have been re¬
ported by these authors for different yards at different times.
2<4.5.2 Assembly into Outbound Trains
Once cars have been classified, they must wait for assembly and dispatch
of the appropriate outbound train before they leave the yard. Operationally.
40
we can think of this process as being one in which cars arrive on the classi¬
fication tracks, either singly or in small groups (cuts), and wait for the
designated outbound train to be "called." At this point, all the cars for
this train are assembled, and when made up, the train departs. In terms of
a queuing model, we may think of this as a batch-service system in which the
"server" is the outbound train. Service for a batch of cars begins when the
appropriate outbound train is called for assembly, and the service time is
the time between successive outbound trains on which a given cut of cars may
be dispatched. The delay time for connection with the outbound train is
then the waiting time in queue derived from such a queuing model.
It should be noted that this perspective on modeling the system places
principal emphasis on the outbound train schedule as the source of delay for
cars following classification. Delays in assembly due to insufficient num¬
bers of switch engines and crews are not considered directly. This effect is
only represented indirectly, in terms of late departures of outbound trains,
for example. The emphasis on schedule is in keeping with the findings of
several previous researchers, and has been recognized by a rail industry
task force on reliability studies [29].
The average delay for a simple batch-service queue of this type can be
derived easily. Let us assume that individual cars arrive randomly in
time (i.e. as a Poisson process) from the classification operation, and that
the outbound train takes all cars available at the time it is assembled.
This second assumption means that train length constraints on the outbound
trains are ignored, for the time being. We will return to this issue follow¬
ing the basic derivation.
41
Define a probability density function, g(t), 0 <_ t < 00 , which des¬
cribes the distribution of time intervals between successive outbound trains
for a given block of cars. If cars arrive randomly in time on the classifi¬
cation tracks, and the interval between two trains is a particular value, tQ,
the average delay time will be
Ta(to> = T * (37)
The expected number of cars arriving during any interval is proportional to
the length of that interval. That is, the expected number of cars arriving
in an interval of length tQ is
n(to) = k,to (38)
where k is the arrival rate. The total expected delay time for all cars in
an interval of length t will then be
S(t ) = n(t )• T (t ) . (39)
o o a o
The unconditional total expected wait time may be obtained by integrat¬
ing over the density function, g(t):
s = / S(t)g(t)dt . (40)
o
In like fashion, the unconditional expected number of cars in an interval is
00
n = / n(t)g(t)dt . (41)
o
Finally, the unconditional expected delay for cars is simply
42
2 J t g(t)dt E(t2)
T = - = ^ = . (42)
n k / t g(t)dt 2E(t)
o
If desired, equation (42) may be rewritten as
2
T = + —-— (43)
a 2 2E(t) V '
2
where cr_ is the variance in the time interval between successive departures.
2
Note that if departures are completely regular (a = 0) , the second term
vanishes, and the expected delay is one-half the interval between trains
(e.g., 12 hours for trains dispatched once per day.) On the other hand, if
dispatches occur very irregularly, the second term indicates that expected
delay to cars will increase.
Equation (42) is analagous to a result widely used in studies of urban
mass transit systems, expressing the mean waiting time of passengers at a
bus stop. Derivations of the result in the mass transit context can be
found in Welding [82], Osuna and Newell [631 or Kulash [51].
The derivation of equation (42) assumed that outbound train length is
unlimited, or in queuing terms, that that batch size is infinite. In prac¬
tical terms, this assumption is not really true, since there are limits to
the length of train which can be dispatched. Such limits can be the result
of mainline track configuration, power availability, etc. More sophisticated
batch-service queuing models can be constructed to reflect these constraints,
but for batch sizes in excess of 25-30, the numerical results are essentially
the same as for infinite batch size. (See Petersen [65].) Since train length
constraints would typically be well in excess of these values, use of a
simpler, infinite-batch-size model is appropriate.
43
Using the queuing models for classification delay and connection/
assembly delay described in the previous two sections, we can predict
total delay in the yard by simply summing the results from equations (35)
and (42), as indicated in equation (34).
2.4.6' Estimation of Average Shipment Velocity
Together, the line-haul train movement and classification yard models
provide the means for estimating the average speed of a shipment through the
system. This value will be used as a single index of service quality in
the cost model to be estimated.
From the running time and delay models described in section 2.4.2 and 2.4.4
average transit time for trains over each mainline track segment can be com¬
puted. Since we know the length of each segment, this can be converted to
an average velocity. By aggregating over track segments, an overall average
velocity of trains is determined. Let us denote this velocity, V .
The classification yard model predicts total delay (in hours) to cars
passing through a yard. We have denoted this delay by T , as shown in equa¬
tion (34). Since there is effectively no distance involved in this segment
of the trip, however, this time value is not directly expressable as a
velocity.
To obtain an overall velocity figure, we require one additional piece
of information, the average length of haul. We can then obtain average
velocity (miles per hour) by dividing average length of haul (miles) by aver¬
age total time in system (hours moving in trains and waiting in yard.) If we
denote average length of haul by L, overall average velocity, V, is computed as
shown in equation (44);
L + V T
a y
(44)
44
Equation (44) reflects two major simplifying assumptions which are
justified by the uncomplicated nature of operations on the railroad under
study. The first of these is that each shipment passes through one classi¬
fication yard. This is essentially accurate for the system examined in
the case study, but would require modification for more complicated rail
operations. Secondly, in equation (44) average time spent in trains is com¬
puted as L/V , rather than by observing which line segments would be crossed
SL
by a given shipment, summing those transit times, and then computing a
weighted average based on relative frequency of various origin-destination
pairs. Again, the simpler computation used in equation (44) is a reflec¬
tion of the simplicity of the rail network under study and the relatively
limited set of Origin-destination pairs. For this case, the simpler com¬
putation is quite adequate, but it would have to be modified in a more
complex setting.
45
CHAPTER 3
DESCRIPTION OF THE DATA FOR RAILROAD X
3.1. General Overview of Data Needs
The analysis of the previous chapter provides the following list of data
items needed by our model :
1) Flows of various commodity types -
(yl' ' " ' 'ym]L) '
2) Prices of input factors-
/ v v \ .
3) Levels of fixed factors -
t f f ^
(xl Vn^ ;
4) Levels of variable factors -
, v v v
(x, , • • • ) ;
T.
5) Engineering Data ;
6) Cost - C .
Each of these areas will be addressed in turn. To provide a specific cost func¬
tion for the discussion, chapter four will present a translog model of the cost
function
c(y,s,p1,p2,p3,p4,p5;Qk)
where 1) Y is total flow (loaded car-miles); 2) S is speed; 3) P is the price
l
of cars, fuel, crews, locomotives, and non-crew labor respectively; 4) QK is a
quality of plant representing the fixed factors. Appendix A presents the esti-
46
mation results for the translog model of C(Y^,,Y^,Y^,S,P ^,P^,P^,;QK)
where the Y^ are subaggregates with respect to commodity type. Observations
for both models are monthly observations from 1969 through 1977 (108 observations).
3.2 1 Flow
Historically, ton-miles has been used as the measure of output of a trans¬
port firm. This measure suffers in many ways:
1. Shippers do not often buy tons of a commodity
moved; they usually buy in car-loads, which can
vary in weight by the type of commodity.
2. A ton-mile can be misleading: is 100 ton-miles
the movement of 100 tons over 1 mile or the
movement of 1 ton over 100 miles or something
in between? These outputs are clearly not the
same.
3. The cost in providing service includes the move¬
ment of empty cars to be repositioned so as to
be available to make a revenue-generating move.
Thus empty car movements are an intermediate product
and not a final output of the firm.
Flow data from the firm was of two sorts:
1. Monthly listings by type-of-move (see below) by
seven-digit STCC (Standard Transportation Commodity
Code) of total loaded cars moved and total tons,
from 1969 to 1977.
47
2. Two years (1972, 1977) of records by type of move
(see below) of every move made on the system:
origination, destination, commodity. We will
refer to these as distance profiles.
Because shippers basically buy loaded cars rather than tons, we
decided to use this measure. Originally we planned to have as disaggregate
a model as possible, allowing for flows by line-haul segment and direction.
This became impossible to compute and counter-productive to the basic
goal of producing an integrated model. Since, in a translog model,
log(ab) = log a + log b, then if we included miles with a move we would
implicitly have:
log(loaded cars) + log(miles) = log(loaded car-miles).
Therefore we proceeded to use loaded car-miles as the output measure.
The construction of the loaded car-miles was based on using the distance
profile information to get an average distance traveled by commodity and
by type-of-move. There are four types of move: local (L), forwarded (F),
received (R), and intermediate (I). They are defined as follows:
L: origin on-line, destination on-line
F: origin on-line, destination off-line
R: origin off-line, destination on-line
I: origin off-line, destination off-line.
The model presented in the next chapter uses total loaded car-
miles to represent flow; a model using one possible disaggregation of
flow is presented in Appendix A. Many such representations of flow are
possible allowing not only for disaggregate commodity types but also
for distinctions such as unit train, etc. Issues of this type will be
48
investigated in the second phase of the work (see section 6.2).
3.3. Prices of Variable Factors
Prices were constructed for the following factors ;
1) Cars ;
2) Rail ;
3) Ties ;
4) Fuel ;
5) Locomotives ;
6) Train Crews ; and
7) Non-Crew Labor.
These will be discussed in turn below. It should be remembered that prices
are the marginal cost of another unit of the factor in question. As such
they should be constructed from national or regional market data. Since most
such data for capital items (cars, rail, ties, locomotives) is cost of pur¬
chase, we used the interest rate for the firm, which was provided by the
firm's main bank. The rate, while evidencing some fluctuation, appears to
have been reasonably stable during the period of estimation (1969-1977) given
the nature of the economy.
Thus, for capital items the following price construction was used to pro¬
vide monthly prices:
P.. = (r + Ô.) Unit Cost. /12
it t i,t
where r is the nominal interest rate for year t, is a depreciation rate,
49
unit cost i,t is the unit cost of factor i in year t (when data was found
indicating changes in costs during the year, the change was used to split
the year and associate unit costs with months). Of course, this leads to
some uniformity of some of the capital prices during a year.
3.3.1. Price of Cars
Lease information on cars from car-leasing concerns was unavailable for
most of the period studied. The firm's car leases were used to construct
yearly profiles of numbers and types of cars. Average costs [83] of new freight-
train cars installed in the years 1959-1978 by type of car were used in con¬
junction with the yearly car profiles to construct a unit-cost for a repre¬
sentative car, i.e. a car representing the mix of existing stock at each
point in time. A depreciation rate of six percent [78] was used.
3.3.2. Price of Rail and Ties
Data from the Association of American Railroads on per-ton rail costs and
per-tie tie costs was used. Turnover rates on the railroad under study
established depreciation rates of .02 and .03 respectively. It was found that
the prices were almost perfectly correlated (.987) and thus the price of ties
was dropped. Later analysis proved that the price of rail was correlated with
the prices of cars and locomotives (both > .9). Dropping the price of rail
removed some serious multicollinearity in the model in that almost all other
variables had much lower correlations.
3.3.3. Price of Fuel
Firm records provide monthly purchases of fuel. Price was taken as amount
paid over quantity purchased.
50
3.3.4. Locomotives
Firm records were again used to provide mo.nth-by-month profiles of types
of locomotives and numbers in use. The firm has only recently started rent¬
ing locomotives. Data from ICC Transport Statistics in the U.S., Table 37
(23 in 1974) was used to provide average costs, by type of locomotive. Again,
a composite locomotive unit cost was constructed based on the locomotive mix
at each point in time. Missing data (the ICC has stopped issuing the table
apparently) was filled by using the few leases the firm did have, which
happened to be in the missing data years. The depreciation rate was again .06
[78].
3.3.5. Labor: Crew and Non-Crew Costs
Firm records on monthly payments to various categories of labor were used.
Crews were taken as a unit and all other labor (executives on-down) were taken
as the unit "non-crew". Prices we computed by taking total hours paid for and
dividing by total hours actually worked. This is important since credit is
often given for time not actually worked. Per hour supplemental wage payments
were added into the wage rates to provide final wage rates (prices).
3.3.6. Deflation of Nominal Prices
The prices given above are nominal prices. They were deflated to 1969
by use of the AAR Charge-Out Indices [85]. Fuel was deflated by the fuel index,
cars and locomotives by the materials index and labor by the labor index.
3.4. Fixed Factor Levels
The fixed factor is the system configuration. In the case of railroad X,
51
a system configuration change occurred in 1976. The change was a simple one
involving the addition of a stretch of track which had previously been used
by another firm to ship goods onto railroad X's system. A number of pos¬
sible measures of the system configuration are possible. Since railroad X con¬
sists mainly of line-haul, we elected to measure the system quality and con¬
figuration via a vector of four variables representing the number of miles in
each of the Federal Railroad Administration's Track Classification categories
Clearly, for a given configuration, the elements of such a vector are corre¬
lated. Thus the vector was converted into a scalar measure:
„ , _ miles in category four
Track Quality = ; —
J total miles
where category four represents the best quality. Because the FRA classifica¬
tions have associated speed limits, and these speed limits affect the speed of
a shipment over the system, this index of system quality is a very effective
measure of the fixed factor (though clearly not the only one possible).
3,5. Data Used in the Engineering Analysis
The data used in estimation of the short-run cost functions for the rail¬
road under study include information on the physical characteristics of the
railroad's lines and yards and operating records of train and car move¬
ments .
3,5.1. Physical Characteristics of the Rail Plant
Detailed information on the nature of the rail plant is used in three speci¬
fic places in the model. First, the track profile (grades) and speed limits
52
have been used in computing running times for trains over various segments
of mainline track, as described in section 2.4.2 . Second, the number and
locations of passing sidings are used in the calculation of delays on single-
track lines, as described in section 2.4.4 . All of this information on the
physical nature of the plant was obtained from track charts and annotations
supplied by the Vice President - Operations of the railroad studied.
3.5.2. Operating Records of Train Movement
Records of train movements were sampled for each month of the study period
(January, 1969 - December, 1977) in order to estimate numbers of trains operated
over each major line segment, and characteristics of those trains including
number of cars, locomotive horsepower and total trailing load (in tons). The
number of trains operated is an important input to the calculation of line-haul
delays, as described in section 2.4.4. The horsepower and total trailing load
are important determinants of line-haul train velocity and are also used in
line-haul delay calculations. The number of trains and their lengths (in cars)
are important input for the classification yard delay models.
All of this information was obtained from dispatchers' records of train
movements. These are generally large sheets in which all movements of trains
on a given day over a particular section of track are noted. These sheets in¬
clude a good deal of information in addition to the data we required, and
constitute the most detailed records retained regarding train movements. Because
the records comprise many individual entries made manually by dispatchers
through the day, and because there is one sheet for each day's operation, ex¬
traction of the relevant information was a laborious, time-consuming process.
53
A sampling procedure was devised to allow extraction of a reasonable
statistical sample of this data for each month of the study period. This
sampling procedure involved taking detailed information on 8-12 trains per
month. This detailed data included origin and destination of run, total
horsepower, numbers of loaded and empty cars, trailing tons and delays en¬
countered. Sample trains were selected so as to cover all directions of
movement and various days of the week, in order to avoid obvious biases.
The sample data for each month was then aggregated to obtain character¬
istics of a "typical" train for that month over each line-haul segment. The
horsepower and trailing load for this typical train were then used to com¬
pute average line-haul velocity for a shipment during that month.
In addition to this sampling of detailed train movement information, ex¬
haustive samples of numbers and lengths of inbound trains to yard facilities
for classification were recorded for 15 days per month. This provided a suf¬
ficient sample to estimate the arrival rate of trains and the first two moments
of the train length distribution for each month. These three values were then
used in the formula (35 ) to estimate classification delay in yard operations.
The number and departure times of trains sampled from the train movement
records were also used to construct estimates of the mean and variance of
times between successive outbound trains. This is information needed to com¬
pute connection/assembly delay times in classification yards.
In summary, the bulk of the basic operating data for the engineering pro¬
cess models has come from two major sources. The first is track charts pro¬
viding the physical characteristics of the mainline track segments. The second
is dispatchers' records of train movement. These records include detail on
train characteristics and movement from which input values for the line-haul
and yard models can be derived.
54
3.6 Cost
Monthly records from 1969 through 1977 on operating costs were provided
by the firm. These records are used as the basis for ICC submissions. In
general, however, such costs do not include implicit capital costs on cars
and locomotives, i.e. the costs did not reflect the economic costs of the two
major short-run variable capital factors. An estimate of the missing costs
was made by using the car and locomotive prices and levels. These costs were
then added to the operating cost to provide short-run variable cost. Data on
credits, committed funds for leases, etc. were purposely excluded. In general,
one would expect that the main part of property-related taxes would be
assessed on the firm's plant (rather than equipment) and since this is
fixed in our model, we do not include such taxes in the short-run
variable costs. Income taxes are on profits, and thus are also
excluded from short-run variable costs. This is because the cost
function is homogeneous of degree one in prices (see section B.2.1).
Thus, the short-run variable costs cover such items as maintenance,fœl, crews
cars, locomotives, staff, supplies, etc.
Costs were deflated by using the AAR charge-out index (aggregate).
Autocorrelation analyses indicated that deflating the costs removed most of
the autocorrelation present in the cost observations. A weak yearly auto¬
correlation persisted, but was small enough that we felt it could be ignored
in the econometric estimation.
55
CHAPTER 4
THE MODEL TO BE ESTIMATED AND THE ESTIMATION RESULTS
4.1 The Model
The following model was estimated:
COST:
C = a. + a.. PCAR + a„. PFUEL + aon PCREW + a.. PLOCO + acn PMNGT
0 10 20 30 40 50
+ eio Y + Y10 s + 610 QK
+ | a11 (PCAR)2 + a12 PCAR» PFUEL + a±3 PCAR * PCREW + a14 PCAR« PLOCO
+ a15 PCAR- PMNGT
+ \ «22 (PFUEL)2 + a23 PFUEL» PCREW + a£4 PFUEL. PLOCO
+ a25 PFUEL - PMNGT
1 2
+ 2 a33 (PCREW) + a34 PCREW. PLOCO + a PCREW. PMNGT
1 2
+ - a.. (PLOCO) + a,_ PLOCO • PMNGT
2 44 45
+ ^ a55 (PMNGT)2
+ isu(ï)2+ ! ,U(S)! + i»uY.S
+ en PCAR.Y + 021 PFUEL* Y + 0^ PCREW.Y + 0 PLOCO. Y
+ 051 PMNGT , Y
+ aL1 PCAR. S + o21 PFUELS • S + PCREW. S + a PLOCO. S
+ o51 PMNGT » S
56
+ ôn(QK)2 + pnQK-Y + enQK-S
+ nL1PCAR'QK+ n21 PFUEL*QK + PCREW-QK
+ n, PLOCO-QK + n PMNGT • QK
41 51
where: C = Un (Cost/Average Cost)
PCAR = Ln (Price of Cars/Average Price of Cars)
PFUEL = Un (Price of Fuel/Average Price of Fuel)
PCREW = 2,n (Price of Crews/Average Price of Crews)
PLOCO = In (Price of Locos/Average Price of Locos)
PMNGT = Ln (Price of Non-crews/Average Price of Non-crews)
Y = Ln (Loaded Car-miles/Average Loaded Car-miles)
S = Un (Speed/Average Speed)
QK = £n (FRA Category Four Percentage/Average FRA Category
Four Percentage)
There are five prices, two outputs and one fixed factor thus resulting in
forty -five coefficients to be computed (see section B.2.4.1). To improve
the efficiency of the estimation process we will append the following
factor share equations:
FUEL:
XFUEL = a2Q + a12 PCAR + a22 PFUEL + a23 PCREW + a PLOCO
+ a25 PMNGT
+ e2i Y + S + n,lQK
CREWS:
XCREW = a3Q + a13 PCAR + a23 PFUEL + «33 PCREW
+ a., PLOCO + a,. PMNGT
43 53
+ 831 Y + S + "31 «K
LOCOS:
XLOCO = aAQ + a14 PCAR + ct^ PFUEL + PCREW
+ aA4 PLOCO + a5A PMNGT
+ 041 Y + °41 3 f n4l
57
NON-CREWS :
XMNGT = a50 + a15 PCAR + a25 PFUEL + a35 PCREW
+ a.c PLOCO+acc PMNGT
45 55
+ 6S1 Y + 051 S + %1 QK
where: XFUEL = Price of fuel* fuel purchased/cost
XCREW = Price of crew hour • crew hours purchased/cost
XLOCO = Price of loco hour* locomotive hours purchased/cost
XMNGT = Price of non-crew hour • non-crew hours purchased/cost
By purchased, we mean hours or amounts paid for. This is particularly impor¬
tant with respect to labor and locomotives since not all time paid for is used.
As will be observed from the cost function description, we have divided
all variables by their means, i.e. an observation is divided by the mean of
the observations before taking the logarithm. This is done mainly to protect
the proprietary nature of the data. By so transforming the variables we only
affect the intercept term, leaving the important coefficients undisturbed
This way, actual costs for the railroad under study are only predictable by
those with a proprietary interest while cost relationships are open to perusal
by all. In view of this, we will not be publishing the variables means since
they add nothing to understanding the cost functions, and only reveal proprietary
information.
4.2 Estimation Results
The above equations were estimated as a system of seemingly unrelated
equations [79] where we assumed an additive error structure. Since the fac¬
tor share equations are derived by differentiation of the cost function, the
58
error term in the cost function does not appear in the factor share equa¬
tions. As in [15] we assume the disturbances are joint normal and esti¬
mate the system using a maximum likelihood technique and thus the results
are invariant to which factor share equation is dropped. Table 2 provides
the estimated cost functions, while Tables 3, 4, 5, and 6 provide the esti¬
mated factor share equations for fuel, crews, locomotives and non-crew
labor respectively.
59
Table 2
Cost Function
VARIABLE
PCAR
PFUEL
PCREW
PLOCO
PMNGT
QK
Y
S
(PCAR)2
PCAR . PFUEL
PCAR « PCREW
PCAR • PLOCO
PCAR • PMNGT
(PFUEL)2
PFUEL- PCREW
PFUEL-PLOCO
PFUEL- PMNGT
(PCREW)2
PCREW-PLOCO
PCREW-PMNGT
(PLOCO)2
PLOCO-PMNGT
(PMNGT)2
COEFFICIENT
10
20
30
40
50
10
10
'10
11
12
13
14
15
22
>23
24
25
33
34
35
*44
45
55
ESTIMATE
.03997
.31748
.04767
.15185
.08354
.39945
-.92323
.08939
-.04843
-.02637
.00526
.00216
.04086
-.02167
,05928
-.01716
-.02293
-.02422
.10596
-.01861
-.07234
.03863
-.03794
.15617
STD■ ERROR
.01123
.00494
.00086
.00130
.00084
.00300
.13530
.07851
.05306
.02267
.00628
.00766
.00765
.01535
.01022
.00835
.00706
.00133
.01404
.00729
.01491
.00936
.01019
.02461
60
Table 2 (continued)
VARIABLE
COEFFICIENT
ESTIMATE
STD. ERROR
(Y)
(s)2
Y. S
PCAR • Y
PFUEL*Y
PCREW*Y
PLOCO-Y
PMNGT"Y
PCAR* S
PFUEL* S
PCREW* S
PLOCO.S
PMNGT* S
<QK)2
QK» Y
QK* S
PCAR-QK
PFUEL*QK
PCREW* QK
PLOCO* QK
PMNGT* QK
'il
"Il
11
11
21
31
41
51
'21
22
23
'24
25
11
U
11
•11
%
n2i
u
31
Î1
41
'51
.23904
- .07679
- .21683
.024151
.00451
.01064
- .01131
- .02800
.01480
- .00463
- .00381
.00367
- .01004
-12.84350
.81675
- .00451
- .21474
.02840
.05061
.00755
.12819
.49667
.14596
.20978
.04496
.00800
.01191
.00777
.02758
.01839
.00324
.00487
.00319
.01120
3.36250
.78897
.28319
.08022
.01436
.02123
.01399
.04957
61
Table 3
Fuel Function
VARIABLE
PCAR
PFUEL
PCREW
PLOCO
PMNGT
Y
S
QK
COEFFICIENT
20
*12
*22
X23
24
25
21
22
^21
a
a
cr.
ESTIMATE
.04767
.00503
.05928
-.01716
-.02293
-.02422
.00451
-.00463
.02840
STD■ ERROR
.00086
.00628
.01022
.00835
.00756
.01332
.00800
.00324
.01436
Table 4
Crews Function
VARIABLE
PCAR
PFUEL
PCREW
PLOCO
PMNGT
Y
S
QK
COEFFICIENT
30
13
a23
a33
a43
a53
631
°23
%1
ESTIMATE
.15185
.00216
-.01716
.10596
-.01861
-.07234
.01064
-.00381
.05061
STD. ERROR
.00128
.00766
.00835
.01404
.00729
.01491
.01191
.00487
.02123
62
Table 5
Locos Function
VARIA?LE
PCAR
PFUEL
PCREW
PLOCO
PMNGT
Y
S
QK
COEFFICIENT
40
14
24
34
44
54
41
24
'41
ESTIMATE
.08354
.04086
-.02293
-.01861
.03863
-.03794
-.01131
.00367
.00755
STD. ERROR
.00084
.00765
.00706
.00729
.00936
.01019
.00777
.00319
.01399
VARIABLE
PCAR
PFUEL
PCREW
PLOCO
PMNGT
Y
S
QK
Table 6
Non-Crews Function
COEFFICIENT
"50
«15
a25
a35
°45
«55
e51
°25
*51
ESTIMATE
.39945
-.02167
-.02422
-.07234
-.03794
.15617
-.02800
-.01004
.12819
STD, ERROR
.00300
.01535
.01332
.01491
.01019
.02461
.02758
.01120
.04957
63
CHAPTER 5
RESULTS AND IMPLICATIONS
5.1. Introduction
This section reviews the estimation results presented in Chapter 4.
Two major questions will be addressed.
1) Does engineering information add significantly to
the explanatory power of the cost function? This
is a test of the hybrid approach.
2) What implications can be drawn from the model? For
example, what are the impacts of the various factor
prices on cost?
5.2, On the Statistical Evaluation of Results
The tables in section 4.1 provide standard errors associated with the
coefficient estimates. The standard error provides a simple measure of the
quality of the coefficient estimate. While it has been traditional to report
t-values (which are simply coefficient estimates divided by the standard error)
in many regression analyses, such values can be very misleading in evaluating
a model of the type at hand. This is because basic variables in the model
appear in many terms in the form of composite variables. For example, speed
occurs as itself, in a second-order term and in cross-products with all the
other basic variables. Therefore to evaluate any of the basic variables, we
must pose hypotheses on the appropriate vector' of composite variables in the
model; t-tests are inappropriate for this purpose. Instead, a vector-based,
test (the log-likelihood ratio test) will be used; it is described in section
B.2.4.2.
64
5.3. A Test of the Hybrid Approach
As discussed in section 2.3, one would expect the coefficient of velo¬
city to be negative. This is due to the relationship between congestion,
short-run variable cost and velocity for a fixed configuration. This a priori
knowledge of the sign allows us to perform a one-tailed test on speed, which
appears to be marginally acceptable. If we perform a log-likelihood ratio
test, as discussed in section 5.2 above, by setting all coefficients of speed-
related terms to zero we find that the test statistic value is 21.49 with
eight degrees of freedom. This causes us to reject the hypothesis that
speed should not be in the model. Thus we conclude that the addition of
engineering information adds significantly to the model.
5.4. Implications of the Model
In general the model exhibits the expected properties. The first-
order price and flow terms are positive which implies that at the point
of means (where one has the greatest confidence in the predictions) the
partial derivative of cost with respect to a price, or flow, is positive.
This is to be expected. Further, as one would expect, the fixed factor
term has a negative coefficient. This is reasonable since we have estimated
a short-run variable cost function and thus, no price on the fixed factor is
included. Therefore improvements in the fixed factor should result in re¬
ductions in cost. For example, in the Cobb-Douglas case examined in section
B.2.3.4 we see that the coefficient on the fixed factor is negative, again
reflecting the lack of including a cost of making fixed-factor changes. While
one cannot derive the same result for the translog, it is reassuring to see
it come through so strongly.
Probably of more interest are the elasticities of cost with respect to
factor prices. These are easily found at the point of means since they are
65
the coefficients for the first-order price terms. For example, the elasticity
of cost with respect to the price of cars is 0.317. Notice that cost is most
responsive to non-crew labor wages, then car prices, then crew labor wages.
Fuel and locomotives appear to have much smaller effects.
It is not difficult to understand why cars contribute heavily to costs:
railroads, as regulated common carriers, must provide service to all who will
pay the tariff. This translates into the requirement to have ready access to
a number of cars of various types.
What, however, could be the reason for the disparity between non-crew
labor and crew labor? A reasonable explanation is the following one. Since
a crew is matched with a train (and not a specific number of cars), as crew
costs go up, the firm has the option of running longer trains to amortize the
crew cost. This is not true, however, of non-crew labor. Union restrictions,
regulatory requirements for information (resulting in a significant amount of
paperwork) and the fact that the firm under study is only partially computerized
all probably contribute to the inability of the firm to substitute away from
non-crew labor as its price rises. Thus, the relative effects of the two types
of labor on cost is quite reasonable.
A
The coefficient associated with QK (i.e. -.92323) is the elasticity of
cost with respect to the quality of plant measure. If we hold configuration
constant (and thus total mileage is fixed) then a one percent increase in the
number of miles in the top FRA track category will result in almost a one per¬
cent drop in short-run variable costs. It should be stressed that since the
short-run variable cost function does not include a price on QK, that the over¬
all reduction in total costs is unclear. In other words, this elasticity
neglects the cost of making the improvement to track.
The effect of such a change in plant can also be seen in the sign of the
velocity term. Since ^ is negative, then at the point of means this implies
66
the improvements in speed would reduce costs. Since the track categories
have effective speed limits associated with them (see section 3.4) this is
a very reasonable result.
Finally, cost appears to be increasing in flow as seen by the fact that
A
(and most of the other terms involving Y) is positive. Thus, flow marginal
cost is positive at the point of means. The cost elasticity of flow is
approximately .1, indicating that at the point of means the cost function is
relatively flat with respect to flow.
In terms of structure (e.g. separability, homotheticity, homogeneity) the
joint test for separability and homotheticity (see section B.2.4.2) was per¬
formed. The test statistic value was 33.796 with ten degrees of freedom, and
thus we reject the hypothesis. On the other hand, a review of the cross terms
between output and prices and those between output and the fixed factor seems
to lend some support to the notion that the function may be separable in inputs
and outputs. Again, the t-values can be very misleading, and this is certainly
not a test of the separability of the cost function; as is mentioned in sec¬
tion B.2.3.3 such test for the translog are complicated by the fact that the
conditions for separability imply specific functional structure to the sub-
aggregates thereby transforming the test into a joint test on separability and
a specific functional form for the subaggregate. Of course, since the test for
joint separability and homotheticity was rejected, it makes no sense to test
homogeneity or unitary elasticities of substitution.
The rejection of the test is consistent with other work in this area (see,
for example, [32]). It is also especially noteworthy that the second-order
price terms (own and cross) are generally very strong, thereby providing further
evidence that cost functions (and therefore production functions; see section
67
B.2.3.1 on the self-dual nature of Cobb-Douglas forms) that are Cobb-Douglas
are overly restrictive. This is important since these have been very popular
in analysis of transport cost and production.
What about density economies or diseconomies? From the discussion in
section 2.3 above we see that Railroad X is probably suffering from disecono¬
mies of density, at least during some parts of the year. Observation tends to
confirm this. During parts of the year the yard becomes significantly con¬
gested. As to whether or not a configuration shift would really improve things
is to be seen only by bringing in the cost of such a change (for example,
improved yard facilities). To the degree that better operations procedures
can be affected to smooth the congestion, the diseconomies will probably be
reduced.
Our empirical results also motivate the following theoretical
analysis of the relationship between density economies and maintenance
policy, and the effects of regulation. We define a maintenance policy as that
policy (level) of optimal maintenance activity given a fixed configuration and
output level. Thus, since for any given configuration various levels of out¬
put will be consistent with different levels of maintenance activity, one can
view the maintenance policy as giving rise to even "shorter-run" curves whose
envelope is the fixed configuration curve (see Figure 12). For any given main¬
tenance policy (which implies speed limits, among other things) density and
speed again trade-off as congestion becomes significant. Note that regulation
can act to provide incentives to not maintain in what would otherwise be an
optimal fashion. Consider Figure 13, which shows a firm that is regulated to
produce output on configuration B. Present output is at level Z. The "ap¬
propriate" maintenance policy is the small U-shaped curve on the left-side
68
FIGURE 12 - CONFIGURATION AND MAINTENANCE POLICY
FIGURE 13 - DENSITY ECONOMIES AND MAINTENANCE POLICY
69
of the cost curve for configuration B. Clearly, the firm would be better off
if it could shift to configuration A, which might involve some abandonment of
service. Since it cannot, an alternative is to simply pursue the maintenance
policy associated with configuration A: maintain some parts well and other
parts not at all (as if they didn't exist). This would appear to be a policy
of deferred maintenance, but in fact it is simply a result of the regulatory
induced incentives.
70
CHAPTER 6
SUMMARY AND DIRECTIONS FOR FURTHER RESEARCH
6.1. Summary of First Year's Results
As demonstrated by the discussion in the previous chapter, economic and
engineering principles can be brought together to provide a more complete cost
model than either can.produce by itself. This hybrid model incorporates both
the comprehensive picture of the rail firm that an economic approach provides,
and the depth of understanding of the production process that comes from de¬
tailed engineering analysis. The statistical test described in section 5.3,
in which the hybrid model is compared to a model with the engineering-related
variables omitted, provides strong empirical evidence of the value of the
hybrid approach. Not only is this approach theoretically appealing, but it
also produces a better explanatory model of costs from an empirical perspective.
In terms of theoretical development, the analysis included in Chapter 2
of the report has extended the traditional economic theory of cost and produc¬
tion to establish the appropriate role for engineering models in the study of
cost. In the first year's work on this project, we have concentrated on using
engineering models to predict one major service quality variable, average speed
of shipment through the system. However, the approach we have developed is
robust with respect to adding more service characteristics and network complex¬
ity. The procedure starts with a very general model of production which makes
a minimum of economic assumptions (e.g. it makes no assumption as to whether
or not there are returns-to-scale). This is important since we would like to
examine (i.e. test) such economic attributes rather than assume them. Engineer¬
ing models that reflect physical relationships among some of the input variables
and some of the outputs are then added. These engineering models may reflect
any such relationship which is physically meaningful to the production process.
They are not limited only to describing speed. The engineering models increas-
71
ingly restrict (and thereby further reveal) the model of production. Again,
it should be noted that the restrictions will reflect physical realities and
not economic assumptions that need to be tested. As more engineering rela¬
tionships are added (reflecting network considerations or service character¬
istics) the economic attributes of the model become more and more refined,
for the general production model becomes increasingly restricted by the en¬
gineering relationships and this, in turn, reveals more of the economic
relationships.
The formulation of the hybrid cost model implies certain data needs for
successful empirical analyses. We have been able to translate theoretical
data needs specified by the model into data requirements than can be fulfilled
by a firm using available information. Thus, the model allows us to specify
ways of combining available firm data correctly to produce measures of cost
that should be used in regulatory proceedings (e.g., "incremental" costs: see
section 6.2). Further, because the model allows for multiple outputs, includ¬
ing service characteristics as well as volumes of commodities, marginal costs
for particular commodities and services are computable.
It should also be noted that engineering considerations provide deeper
insight into the nature of factors that contribute to scale economies (and
diseconomies), and how size and density relate. This has been discussed in
some detail in section 5.4, with particular • emphasis on the interrelationships
of traffic density, maintenance policy, and regulatory restrictions on network
configuration.
The empirical work that has been done using railroad "X" as a case study
has also produced interesting and valuable results. The short-run elasticities
of cost with respect to various factor prices are quite illuminating. Short-
run variable cost is most responsive to non-crew labor wages, then car prices
72
and crew wages. Fuel and locomotives appear to have much smaller effects. The
high elasticity of cost with respect to non-crew wages is an interesting (and
somewhat surprising) result. It indicates the limited availability of oppor¬
tunities for substitution of other factors for non-crew labor as non-crew wages
rise.
The empirical work has also provided valuable insight with respect to
appropriate structure of cost functions. As described in section 5.4, a joint
test for separability and homotheticity (see section B.2.4.2) was performed,
and these properties were rejected. The rejection of the test is consistent
with other recent work in this area (see, for example, [32]). It is also
especially noteworthy that the second-order price terms (own and cross) are
generally very strong, thereby providing further evidence that cost functions
(and therefore production functions) that are Cobb-Douglas are overly restric¬
tive. This is important since these have traditionally been very popular in
analysis of transport cost and production..
In summary, the first year's research on this project has provided a num¬
ber of important theoretical and empirical results. The experience of this
first phase of the research has also raised several important issues for further
investigation. The focus of the second year's work, addressing some of these
issues, is described in the following section.
6.2. Plans for the Second Year
Beginning with the Railroad Revitalization and Regulatory Reform (4r) Act
of 1976, there was clear legislative recognition of the difficulties in using
traditional railroad accounting data in regulatory proceedings. Such proceed¬
ings rely heavily on measuring the "cost" of providing service, either as a
basis for setting rates, or more recently, as a basis for determining subsidies
for the continued operation of lines which would otherwise be abandoned. How-
73
ever, the accounting procedures prescribed by the Interstate Commerce Commission
(ICC) generally do not result in the collection of cost data in a form readily
usable (in a manner consistent with economic theory) in such proceedings. The
4R Act included attempts to deal with some of these problems, using new terms
such as "incremental cost" of providing service.
Except in rare instances, however, these new terms are not defined in the
Act, but are left to the ICC to determine. Several of these definitional
problems are being addressed in current proposed legislation. These terms will
heed to be defined and methods for actually applying them will have to be
established. It is precisely on this point that we wish to focus in the second
year of this research.
Specifically, we plan to address the question of defining and measuring
"incremental costs." This term is closely associated with the economic concept
of marginal cost, and reflects the change in cost incurred by a railroad as a
result of changing either the amount of service provided or the nature (quality)
of that service. The work done in this project during the first year provides
an excellent basis for estimating short-run marginal costs. This is because
the hybrid cost function technique that we have developed allows for multiple
commodity types and service characteristics. Thus, short-run marginal costs
with respect to the various components of the output vector can be found. With
suitable extensions, the hybrid cost technique will provide useful guidance in
developing procedures for implementing notions of "incremental costs" in regu¬
latory and public policy settings, including estimation of long-run marginal
costs. The three major areas in which our work to date requires extension
concern: 1) more complete definition and development of the output vector in
terms of commodities carried and service characteristics provided; 2) analysis
74
of the costs of producing services on more complicated networks; and 3) appro¬
priate measurement and pricing of fixed factor inputs, to allow construction
of long-run cost functions.
First, the question of what is a rail firm's output and how it should be
measured will be addressed. This will require us to examine problems of appro¬
priate disaggregation of commodity flows to reflect both the nature of the
goods carried and the network on which they move. This will also require an
expansion of the service characteristics considered. During the first year of
this work, we have concentrated on "speed of shipment" as the major service
characteristic. However, there are obviously a number of other characteristics
of importance such as transit time reliability and equipment availability.
Study of both transit time reliability and equipment availability require
a model of network operations which produces estimates of the distribution of
transit times by origin-destination pair, given an operating policy including
train schedules, blocking, etc., and which reflects the degree to which equip¬
ment is available for provision to a shipper when requested. Previous work on
service-differentiated demand models indicates that this is an important element
of service quality to many shippers. A key aspect of the ability of a rail¬
road to provide empty freight cars to shippers when requested is the effective¬
ness with which it distributes empty cars over its network. Thus, the second
major area for model extension in the second year is the development of models
of network operations which will allow us to focus on a broader range of service
characteristics and associated costs.
The third major area for model extension in the second year concerns measure¬
ment and pricing of fixed factors. For the empirical work in the first year, a
track quality index was constructed to measure the fixed factor representing
quality of plant. Measures of size and condition of the railroad's network
75
will be of importance in the second year also, and additional work will be done
to examine this area thoroughly from both theoretical and empirical perspectives.
The construction of appropriate prices for fixed factors is of particular
importance if we are to be able to derive long-run cost functions from the
estimated short-run functions. Because long-run functions are of central in¬
terest to government policy-makers and regulatory bodies, this is an important
step to make.
Once the structural form of the hybrid cost model has been extended, it
will be necessary to test the structure by acquiring data and estimating the
model parameters statistically. As in the first year of the study, our
approach will include working closely with a railroad as a case study. The
nature of the extensions to the model to be explored in the second year
necessitate working with a major railroad. A number of products will result:
1) first, an estimated model of a major rail firm's cost function
and a comparison of it with our present results for a small rail firm;
2) a technique for defining and computing theoretically défendable
incremental costs based on available data;
3) marginal cost functions that allow for multiple commodities, joint
use of facilities by different services, and various service
characteristics; and
4) finally, the resulting technique will provide necessary methodology
for both the government (for policy analysis) and the railroads
(for planning).
It should be emphasized that understanding costs of production is impor¬
tant both for policy analysis and regulatory review by the government and for
planning purposes in rail firms, and that the procedures we are developing
76
provide accurate, comprehensive cost analysis that can be conducted with readily
available data. Questions of returns-to-scale, the relationship of output and
input changes to changes in marginal and average costs, and the impact of various
possible capital investments on the costs of providing service, are problems
that the technique can address and answer.
77
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83
APPENDIX_A
A MORE DISAGGREGATE COST FUNCTION
This model is essentially the same as the model in section 5.1 except
for an expansion of the representation of flow. The flow vector has four
parts :
Y1 Unit-train Coal (STCC 11)
Y2 Low-value Bulks (STCC 1-10, 11, 12-18, 20)
Y3 High-value Bulks (STCC 21, 22, 26-29, 32, AO)
YA Manufactured (STCC 19, 23-25, 30, 31, 33-39, A1-A9)
The resulting model in the translog form with four factor share equations
was estimated and is reported on the next few pages.
A-l
Table A-l
Cost Function
Variable
Coefficient
Estimate
STD. ERROR
PCAR
PFUEL
PCREW
PLOCO
PMNGT
QK
Y1
Y2
Y3
Y4
S
(PCAR)2
PCAR* PFUEL
PCAR* PCREW
PCAR* PLOCO
PCAR* PMNGT
(PFUEL)2
PFUEL* PCREW
PFUEL* PLOCO
PFUEL* PMNGT
(PCREW)2
PCREW-PLOCO
PCREW -PMNGT
a
a
'10
' 20
'30
40
'50
10
10
20
30
40
10
' 11
' 12
13
1 14
15
22
23
24
25
33
34
35
.02582
.31857
.04779
.15206
.08281
.39878
-.63609
.04510
-.05418
.04139
.15028
.02412
-.07265
.01143
.01946
.03221
.09551
.06090
-.02410
-.02188
-.02635
.10297
-.01999
-.07835
.01781
.00498
.00107
.00129
.00076
.00308
.22069
.04299
.12626
.09210
.07852
.08172
.02836
.00893
.00995
.00894
.01954
.01336
.00941
.00760
.01547
.01515
.00795
.01547
A-2
Variable
Coefficient
Estimate
STD. ERROR
(PLOCO)
PLOCO* PMNGT
(PMNGT)2
(Yl)2
Y1-Y2
Y1«Y3
Y1-Y4
(Y2)2
Y2*Y3
Y2-Y4
(Y3) 2
Y3-Y4
(Y4) 2
(s)2
Yl* S
Y2*S
Y3* S
Y4*S
PCAR-Y1
PCAR*Y2
PCAR-Y3
PCAR*Y4
PFUEL-Y1
PFUEL-Y2
PFUEL-Y3
PFUEL-Y4
a
a
44
45
55
11
12
13
14
22
23
24
33
34
44
11
T11
T21
T31
t41
611
S12
013
e!4
e21
622
923
024
Y
.04923
-.03957
.13472
.06039
-.46422
.02897
.09933
.77209
.20011
-.31962
-.02631
-.08702
.14835
.07176
-.02381
.01726
-.15968
.07094
.02152
.03525
-.00362
-.02931
-.00134
-.00014
-.00024
.00662
.00918
.01001
.02592
.04627
.38545
.28206
.19583
.74816
.48582
.59987
.17141
.45818
.18496
.10457
.11996
.56547
.44404
.25793
.01553
.05266
.02659
.03089
.00334
.01168
.00572
.00708
A-3
Variable
PCREW-Y1
PCREW-Y2
PCREW-Y3
PCREW»Y4
PLOCO-Y1
PLOCO-Y2
PLOCO-Y3
PLOCO-Y4
PMNGT-Y1
PMNGT-Y2
PMNGT-Y3
PMNGT-Y4
PCAR-S
PFUEL-S
PCREW-S
PLOCO-S
PMNGT-S
(QK)2
QK-Y1
QK.-Y2
QK-Y3
QK-Y4
QK* S
PCAR-QK
PFUEL'QK
PCREW'QK
PLOCO*QK
PMNGT'QK
Coefficient
6 31
6 32
6 33
9 34
e41
942
943
044
051
952
953
954
Œ11
°"21
31
41
51
11
"11
21
31
41
'11
'11
21
31
'41
%1
Estimate
-.00148
-.00952
-.00061
.01533
-.00288
-.00810
.00688
-.01783
-.01581
-.01749
-.00241
.02519
.00645
-.00308
-.00161
.00139
-.00315
-4.76311
.52178
-1.10728
-.15783
.54583
.07494
-.16402
.02452
.04047
-.01672
.11574
STD■ ERROR
.00401
.01382
.00686
.00827
.00238
.00843
.00408
.00520
.00961
.03294
.01645
.01940
.01847
.00400
.00481
.00285
.01145
3.14795
.50587
1.59404
1.35530
.86997
.62061
.08197
.01789
.02134
.01282
.05106
A-4
Table A-2
Fuel Function
Variable
Coefficient
Estimate
STD. ERROR
PCAR
PFUEL
PCREW
PLOCO
PMNGT
Y1
Y2
Y3
Y4
S
QK
20
*12
X22
X23
X24
X25
321
322
323
324
7 2 i
"21
.04779
.01143
.06090
-.02410
-.02188
-.02635
-.00134
-.00014
-.00024
-.00662
-.00308
-.02452
.00107
.00893
.01336
.00941
.00760
.01547
.00334
.01168
.00572
.00708
.00400
.01789
A-5
Table A-3
Crews Function
Variable
PCAR
PFUEL
PCREW
PLOCO
PMNGT
Y1
Y2
Y3
Y4
S
QK
Coefficient
a
a
a
30
13
X23
X33
43
53
331
332
333
334
r31
^31
Efficient
.15206
.01946
-.02410
.01030
-.01999
-.07835
-.00148
-.00952
-.00061
.01533
-.00161
.04047
STD. ERROR
.00129
.00995
.00941
.01515
.00795
.01547
.00401
.01382
.00686
.00827
.00481
.02134
A-6
Table A-4
Locos Function
Variable
Coefficient
Estimate
STD. ERROR
PCAR
PFUEL
PCREW
PLOCO
PMNGT
Y1
Y2
Y3
Y4
S
QK
40
*14
*24
*34
44
54
41
42
943
944
J41
*41
a
a
0
.08281
.03221
-.02188
.01999
.04923
-.03957
-.00288
-.00810
-.00688
-.01783
-.00139
-.01672
.00076
.00894
.00760
.00795
.00918
.01001
.00238
.00843
.00408
.00520
.00285
.01282
A-7
Table A-5
Non-Crews Function
Variable
PCAR
PFUEL
PCREW
PLOCO
PMNGT
Y1
Y2
Y3
Y 4
S
QK
Coefficient
50
15
25
35
45
55
51
52
53
54
51
'51
Estimate
.39878
.00955
-.02635
-.07835
-.03957
.13471
-.01581
-.01749
-.00241
.02519
-.00315
.11574
STD. ERROR
.00308
.01954
.01547
.01547
.01001
.02592
.00961
.03294
.01940
.00308
.01145
.05106
A-8
APPENDIX B
PRODUCTION AND COST: THEORY AND EXAMPLES
B.l. Production
B.l.l Definitions and Assumed Properties of Production and Transformation
Functions
Let"'" x = (x^,...,xn)' be a non-negative n-vector of input levels
(factors) used by the firm to produce a single output Z. Examples of in¬
puts for a rail firm are fuel, various types of labor, locomotives, etc.
A production function f(x) is a mathematical model of the relationship
between x and Z and thus :
Z = f(x) . (B-l)
The above production function uses n inputs to produce one output; a
classic example of an aggregate output measure for transport firms is
total ton-miles of goods moved.
We define an isoquant of f to be the set of input levels that is
just sufficient to produce a given output Z:
Q(Z) = (xj Z = f(x)} .
Thus if x is such that f(x) > Z then x i Q(z) , i.e. the isoquants only re¬
flect efficient production.
2
We assume the following properties for f(x) :
1) f (0) = 0 ;
2) f(x) is continuous with continuous first and second
derivatives (unless explicitly stated otherwise) ;
3) if x1 > x then f(x^) >_ f(x) ;
4) f(x) is quasiconcave [57], i.e. f(Ax^+ (l-A)x^) >_ min [f(x^) ,f(x^)]
1 2
for all x , x ^.0.
B-l
The first property states that positive production requires at least some
positive inputs. The second condition imposes regularity on the function
while the third condition means that more inputs will not result in less being
produced. The fourth condition means that level sets of f (i.e. combina¬
tions of x that provide at least a specified output) are convex sets. This
in turn means that the isoquants are convex functions, i.e. they resemble
Figure B-la rather than Figure B-lb:
Figure B-l ISOQUANTS
Many firms, and in particular transport firms, produce a vector of out¬
puts rather than a single output. Transport firms, for example, move a
variety of commodities to and from various geographical points. Furthermore,
associated with the various commodity flows are characteristics of service
such as speed of delivery, schedule unreliability, loss and damage, etc.
Let the firm's output vector be the non-negative m-vector of flows and
characteristics z = (z,,...,z )'. A transformation function T(z,x) is a
I m
mathematical model of the relationship between the input vector x and the
output vector z. The vector z can be exactly produced from x if
B-2
T(z,x) = 0 (B-2)
which is the analogous statement to (B-l) above.^ Typical conditions on
T(z,x) are as follows:"'
1) T(z,x) <_ 0 Vx z '
2) T(z,0) <^0 => z = 0 ;
3) T(z,x) is continuous with continuous first and
second derivatives;
4) V^TCz.x) < 0, V^T(z,x) > 0 ;
5) V(z) = {x| T(z,x) <_ 0} is a closed, strictly
convex set.
The first condition simply defines what we mean by producing z from x. This
condition allows for both inefficient production (T(z,x) < 0) and efficient
production (T(z,x) = 0). Condition (2) is analogous to condition (1) for
production functions.^ Condition (3) is the regularity condition analogous
to condition (2) for production functions. Condition (4) which requires T
to be decreasing in z and increasing in x is a stronger condition than
condition (3) for production functions. Finally condition (5) is analogous
to condition (4) for production functions: it will guarantee that a unique
joint cost function exists (section B.2).
In what follows we examine various possible characteristics of pro¬
duction and transformation functions. The characteristics concern the way
in which inputs combine to produce output(s). The main considerations in
characterizing technology are as follows:
1. Does the technology exhibit economies (or diseconomies)
of scale of production for various levels of output?
B-3
2. Under what conditions can a vector of outputs be
aggregated into a scalar (e.g. ton-miles)?
3. Under what conditions can parts of the input vec¬
tor be aggregated? For example, must we repre¬
sent each and every type of labor used, car type
used, etc. or can we use models that have aggre¬
gate labor and capital inputs.
This report addresses some of the above questions in detail. Others will
be addressed more fully in the second year of the research.
B.1.2. Characterization of Production and Transformation Functions
Before proceeding to examine some of the characterizations of produc¬
tion and transformation functions we provide the following definition,
which will be of use later in this section:
Definition. Let H(u,v) = 0 be continuous and differentiable with
VH(u,v) ^ 0. The marginal rate of technical substitution
(MRTS) of v^ for v^ is
3H/3v
MRTS (u,v) =
J 8H/3v.
J
while the marginal rate of transformation (MRTr) of u^ for u. is
3H/3u.
MRTr..(u,v) = —
lj 3H/9Uj
Thus, in the case of the production function we will refer to MRTS (x) = f
i j i
where f^ = 9f(x)/8xk while in the case of the transformation function we may
B-4
also be interested in MRTr^Czjx) = (3T(z,x)/Sz^)/(3T(z,x)/3Zj). It is im¬
portant to note that we have assumed that VH / 0. While this is a stronger
condition than condition (3) on production functions, most production func¬
tions satisfy this requirement. In general Vf / 0 will hold for most of the
analyses; situations wherein this is not true will be noted.
B.. 1.2.1 Homogeneity and Almost Homogeneity
A production function f(x) is homogeneous of degree k (H.D.k) if
Xkf(x) = f (Xx) VX>0
where X is a scalar. The above condition states that multiplying all the
inputs by a positive scalar multiplies the output by a power of the scalar.
We observe the following classical categorization
1) k > 1 => increasing returns-to-scale ;
2) k = 1 => constant returns-to-scale ;
3) k < 1 => decreasing returns-to-scale .
H.D.k functions satisfy^ Euler's Theorem (see, e.g. M ):
kf(x) = x'»Vf(x),
Further it can be shown that the partial derivatives are H.D.(k-l). Thus
we see that:
f.(Xx) Xk ■'"f.(x) f (x)
MRTS..(Xx) = — = 1 = — = MRTS (x).
ij fj(Xx) X fj(x) fj(x) 2
Thus the marginal rate of technical substitution is unaffected by changes in
scale (i.e. it is H.D.O in x). This assumes Vf ^ 0. Since we typically will
B-5
take Vf > 0, then MRTS..(x) > 0. Production functions that have regions
ij
wherein MRTS^(x) < 0 are said to have non-economic regions since it will
not generally be profitable to operate in such a region.
Homogeneity of degree k for a production function provides the intui¬
tion for the following definition of almost homogeneity for the transforma¬
tion function, namely a transformation function is almost homogeneous of
degrees k^,k2 and k^ (AHD(k^,k2.k^)) if and only if:
k-i ko k V
T(X z ,X x) = X T(z,x) X > 0 .
Lau has shown that such functions satisfy a modified Euler's Theorem [53] in:
that T(z,x) is AHD(k^,k2,k^) if and only if :
k^z^*VzT(z,x) + k2x"*VxT(z,x) = k^T(z,x) .
In general, since T(z,x) = 0 for efficient production, we will also refer to
T(z,x) as AHD(k,l) where k = k^/k-j if it satisfies either of the above state¬
ments. It is also possible to show that MRTS..(z,x) and MRTr..(z,x) are inde-
ij ij
pendent of scale if T(z,x) is AHDCk^k^k^) .
B.l.2.2 Homotheticity
Homotheticity is a very important generalization of homogeneity. Many
of the more popular production functions are homethetic, and homotheticity
of the production function results in a very special structure for the cost
function. Homotheticity was initially developed by Shephard [73],
A production function f(x) is homothetic if there exist functions d(u)
and h(x) with u a scalar, d(u) monotonically non-decreasing and h(x) H.D.I
such that:
B-6
f(x) = d(h(x)) Vx .
In other words if f(x) can be written as a rescaling of a H.D.I function,
it is homothetic. All homogeneous functions are homothetic. The reverse
is not true; let d(u) = eU and h(x) = x (x of size one). The resulting
function is homothetic but not homogeneous of any degree.
O
If Vf £ 0 then f is homothetic if and only if MRTS..(x) is H.D.O.
V. _ 1J
in x i,j [52]- Thus independence of scale of the MRTS is a property
of homothetic functions. Geometrically, this means that isoquants are radial
expansions of the unit isoquant, i.e. Q(Z) can be geometrically constructed
by passing rays from the origin through Q(l). This is shown in Figure B-2.
Figure B-2; HOMOTHETIC ISOQUANTS
This again illustrates the rescaling concept behind homotheticity.
B-7
This intuitive notion underlies the definitions put forward by
Shephard [73 , Ch. 10] and Jacobsen [47] and an alternative, more general
notion in McFadden [33,Ch.I.1] (which is attributed to Hanoch). Shephard's
definition is a straightforward extension of the single output definition
9
to transformation functions that are input/output separable, i.e. we assume
T(z,x) = g(z) - f(x). Further let f(x) be homothetic and let g(z) have
properties (1), (2) and (3) of a production function with the added proper¬
ties of: (4) quasiconvexity (i.e. -g(z) is quasiconcave); (5) if z' >_ z
and z'^ z then g(z') > g(z) and (6) as z becomes arbitrarily large, so does
g(z) (i.e. g(z) unbounded for unbounded z). Notice that the function g(*)
acts as an aggregation function on z; such a function may not exist. The
basic notion of the definition is to place the homotheticity properties in
f(x) and use g(z) as a surrogate output measure.
McFadden's definition, on the other hand, does not require input/output
separability. A transformation will be input-homothetic if there exists a
function a(A,z), A a positive scalar, with a(A,z) increasing in A and a(0,z) =0
such that:
V(z) = a(|Iz|I,z/1 Iz|I) V(z/1 IzI I)
where V(z) = (x|t(z,x) _< 0} (the input requirements set of footnotes 3 and 4)
and I Iz|I is a norm of z, e.g.:
= (I <>
2 1/2
i=l 1
This definition is most easily understood by examining the single output case.
Here z = Z, ||z|| = Z and therefore z/||z|| =1. Then the definition reduces
to :
B-8
v(z) » a(z,i)v(i).
Thus a(Z,l) is the scaling effect on the unit isoquant represented by V(l).
In the multiple output case z/||z|| is a normalized output and a(||z|| , z/||z||)
acts as a scaling multiplier.
The two definitions will have somewhat different effects on the struc¬
ture of cost functions to be discussed in section B.2. The two conditions
are the same when T(z,x) is separable in inputs and outputs, which is one
type of separability to be discussed below.
B.l.2.3 Separability of the Production Function
The literature on separability (i.e. the ability to construct aggre¬
gate variables from disaggregate variables) is extensive; we will not attempt
to review it in depth here. Instead we will provide a very basic overview
of the area of separability which will be a primary focus for the second
year's work.
Issues of functional structure were addressed by Leontief [56] and
Sono [74]. An excellent overall reference is Blackorby, Primont and
Russell [ 6 ]. Two questions addressed by the literature are as follows.
1) Under what conditions can one rewrite the function
Z = f^
9 • • • 9
as
z = fCgj^
9 • • • 9
xi)'g2(xi+l
9 • • • 9
9 • • • 9
h(\
or perhaps as
Z - f( £ gj(x. i...»x, )) 5
i Ji 1
B-9
in other words, form subaggregates (for example a
labor variable to represent all different types of
labor) of non-overlapping subsets of variables?
2 ) Under what conditions can one separate inputs
from outputs in a transformation function, i.e.
when can we write T(z,x) = g(z) - f(x) ? Notice
that both g and f act as aggregation functions
with an aggregate input being just balanced by an
aggregate output. Many technologies are apparently
not separable this way (see, for example, [33, Ch. V.l],
[10] and [13]).
Two types of separability have dominated the literature: weak and strong
(see [6 ] for others). To define these, let P be a partition of the N
indices of the a function h(x) :
N = (l,. .. ,n)
Thus this partitions the x-vector into p parts, i.e.:
in correspondence with the partitioning of the indices. Separability will
be concerned with the effect of changes of a variable on the MRTS of other
1
P
with 1) N. H N. = 0
i 3
i ^ j (mutually exclusive)
2) U U ... U ^ = N (exhaustive) .
B-10
variables, i.e. we will examine when
3(h./h.)
3^L- = o
(Vh + 0)
)xk
where, as usual, the subscript on h refers to partial derivative. Now we
can define strong (S) and weak (W) separability
3(h./h )
h is S if ^— = 0
3xk
i e N
eN.
and V
h is W if
3(h./h.)
—J- = 0
3xk
ki N UN ;
u v
i,j e N
k i N
In words, h is strongly separable (S) in the partition P if when we pick
variables from two parts of the partition (part and part N^) and com¬
pute their MRTS, it is independent of changes in variables not in either
or N^. If this holds for all variables in all the parts of the partition
then h is S. Weak separability doesn't require us to have i and j tested
in different parts. In other words, weak separability tests each element
of the partition against the elements of other subvectors in the partition.
Thus, for example, the following function
a, a a
, / \ 12 n
h(x) = x •x ... x
12 n
is itself strongly separable and if we form two functions
B-ll
1 a9 am
, 1 f v 1 2 in
h (x) = X. *x0 ... x
l 2 m
o a ,. a , „ a
,2, . m+1 m+2 n
n (x) = x ,. •x ,„ ... x
m+1 m+2 n
then h is strongly separable in the partition {N^,^} with = {l,...,m}
and = {m+1 n} .
Goldman and Uzawa [36], have related the S and W conditions to cer¬
tain general functional forms. Berndt and Christensen [4 ] have (for homo-
thetic production functions) related S and W conditions to constraints on
elasticities of substitution. The elasticities of substitution attempts to
measure the sensitivity of the optimal input factor mix to changes in the
MRTS. For example, if production is a function of capital (K) and labor (L)
alone, i.e.
Z = f(K,L)
then a, the elasticity of substitution, is defined as [44]:
f /f d(K/L)
a = -
K/L d(MRTSKL)
Z fixed
When the production function has more than two factors a number of
possible measures can be constructed (see McFadden, Ch. IV.1 in [33]). If
f(x) is homothetic then the Allen Elasticity of Substitution [ 1 ] (called
AES) can be written as
n
I, V-
k-l '• k l(V2Bf)
o . = #
ij Vj lv2Bf|
I 2B
where |V f| is the determinant of the bordered Hessian matrix (see footnote
B-12
and |(V^Bf)^| is the determinant of the i,j cofactor of V^Bf. It will turn
out that the are computable from cost function information. Bendt and
Christensen [4 ] relate restrictions on the to issues of aggregation.
This work is extended to non-homothetic production functions by Russell [70].
B.1.3. Examples of Production and Transformation Functions
In this section we will provide some examples of production functions,
culminating with the most general forms currently in use.
B.1.3.1 Leontief Production
The Leontief (or fixed productions) production function is the following
Z = min( — ,,
where the a^ > 0. Thus inputs are used in fixed proportions (dictated by
the a^). Thus the isoquants (curves in the input space of constant output
level) are corner or L-shaped as shown in FigureB-3. There is no substitution
between factors: more of any factor will be wasted unless all factors are
constant proportions line
,2
"i Ai
Figure B-3: LEONTIEF PRODUCTION
B-13
increased proportionately. This function is H.D.I (i.e. constant returns-to-
scale). Furthermore, if one views the above as a process and there are other
processes available (i.e. other processes that entail different proportions)
then there is a possibility of substitution between processes, as shown in Figure B-4.
Figure B-4: SUBSTITUTION BETWEEN LEONTIEF PROCESSES
Thus this production function is not as limited as it seems at first glance.
It is the ability to substitute between processes that has made this func¬
tion so useful: linear programming models are based on Leontief production
processes (each column in an L.P. being a fixed proportions production pro¬
cess) .
It should be noted that this function is not differentiable. Thus,
for this case we rely upon our original condition (3) for production functions.
B. 1.3.2 Cobb-Douglas Production
This function has been extremely popular for a number of years. It is
written^ as follows:
n cxi
Z = A n x la = 1, A > 0 . (B-3)
i=l 1 1
B-14
The Cobb-Douglas production function is H.D.I. A more general version is
shown below (which is H.D.v):
n aj v
Z = (A II x ) £a. = 1, A > 0, v > 0 (B-4)
i-1 1 1
which is obviously homothetic. In this case any value of returns-to-scale
is possible if (B-4)is estimated (see [62]). For what follows, we continue
our analysis in the standard Cobb-Douglas,(B-3).
First, all factors are essential, i.e. if any one is zero then the
function ascribes zero output to the process under study. This is not
always a desirable result. Substitution between factors is possible (though
complete substitution is not since all factors are essential). The elas¬
ticity of substitution, cr^, is constant and equal to one for all i,j pairs.
Christensen and Greene [15] test for unitary elasticities of substitution
by constraining certain coefficient estimates in their cost function esti¬
mation. We will return to this later.
B.l.3.3 Arrow-Chenery-Minhas-Solow CES Function and the CET/CES Transfor¬
mation Function
Motivated by certain empirical evidence of the relationship between
the log of value added per labor unit and the log of the wage rate, Arrow,
Chenery. Minhas and Solow developed a general production function that was
H.D.I, had a constant elasticity of substitution (not necessarily equal to
one) and satisfied a simple model that explained (to some degree) the empiri¬
cal results. The production function is called the constant elasticity of
substitution production function (CES) and can be written as:^
B-15
^ p l/p A ,
z = A[ £ aixi ] > °» 1 > P
i=l
in which case = a = ^ for all i,j. Special cases are the Leontief
(when a -*■ 0), the Cobb-Douglas (a = 1) > and the perfect substitute case
(a -* + co ) where output is simply a weighted sum of inputs.
Powell and Gruen [68] were apparently the first to employ the CES
function as a multiple output function to form the CET (constant elasticity
of transformation) function. Joining the CET and CES functions, and assum¬
ing T(z,x) = g(z) - f(x) we have:
m . 1/b n . k/P
T(z,x) = ( Y ô.z.) - A( I a.x,)
i=l 1 i=l 1
where k is the degree of homogeneity of the CES function (see note 11).
Hasenkamp [42] has estimated such a function using cross-section data
on U.S. railroads for 1929 and 1936.
B.l.3.4 Flexible Functional Forms: The Translog and the Generalized Leontief
Within the last ten years a number of reasonably general production
functions have been developed. These are called flexible functional forms
(see, e.g. [33, Ch. II.1], [ 6 ]) • A general represèntation of such forms in the
following: ^
i|>(f(x)) = a00 + I ai0<t>i(xi) + 2 I I ai.(J)i(xi)(j) (x ) (B-5)
i i j J J J
where f(x) is the production function, the oc's are coefficients and ty(') and
<{k(') are suitable functions. To be more precise we have the following:
12
1) Transcendental Logarathmic Production Function [ 141
\Jj(u) = £n(u)
c}3 (u ) = Un u
B-16
and thus we have:
13
Anf(x)
2) Generalized Leontlef [22]
iJj(u) = u
(^(u) = AT
yielding:
f(x) = a
00 + \ ai0 + 2 II aij /Xi "S • <B"7>
i 1 j J
These functions can either be viewed as exact representations of technology
or as approximations (second order) to an arbitrary technology. Lau [54]
indicates that two notions of approximation have been used. McFadden [33,
Ch. II.2] has viewed the flexible form as a second order approximation if
first and second derivatives of the approximation are the same as the true
function at the point of approximation. Christensen» et al [ 14] have viewed
the flexible form as an approximation in the sense of a Taylor's series
expansion.
In both views, the notion of an approximation is a local notion. While
certain functional properties are globally inheritable by an approximation,
a significant caution must be observed. Simply put, the factors that con¬
tribute to a good approximation and the factors that cbntribute to a good
statistical estimation can be diametrically opposite to one-another. Approxi¬
mations which are locally good depend on tightly packed data. On the other
hand, good experimental design procedure usually calls for as great a dis-
B-17
persal of data points as is possible. This point is recognized in
[33, Ch. II.1].
One of the major justifications for using the flexible forms is that
many of the more standard production functions are sub-cases when restric¬
tions are placed on the form. For example, in the translog (transcendental
logarithmic) form, setting the second order coefficients to zero (i.e.
a„ = 0, i,j 1) yields the Cobb-Douglas case. The translog is also an
approximation to the CES production function and others (see [14]). A
more detailed review of such forms, their advantages and their failings is
given in [33, Ch. II.1].
As an example of estimating and testing a model, consider the trans¬
log production function above. It is easy to show that the restrictions
for the function to be H.D.I are (assuming a„ = see note 12):
*> I °i0 ■ ^
2) J a..=0 i = 1,...,n .
• il
3
To further restrict the form to be Cobb-Douglas (i.e. unit elasticities of
substitution), one sets a_ = 0 for all i,j. Thus a procedure would be as
follows .
1) Estimate the unrestricted function (with a.. = a..) .
13 3i
2) Estimate th'e model with the H.D.I restriction and
test the new model against the unrestricted model
(using a F-test or a likelihood ratio statistic).
3) If the restricted model can not be rejected, pro¬
ceed to the next set of restrictions; otherwise stop.
B-18
B.2. Cost
B.2.1. Definition of the Cost Minimization Problem and Related Cost
Functions: Long and Short Run Cost Functions
Let p = (p , ...,p )' be an n-vector of given factor prices, i.e. the
14
firm cannot affect p through individual firm actions. Moreover, we assume
p > 0. Finally we assume that the firm attempts to use the factors of pro¬
duction as efficiently as possible, i.e. for any specified output level, the
firm chooses the input vector that produces the required output at minimum
cost. Since cost is p'x = Ip^x-^ then the firms cost minimization problem
(CMP) is as follows:
(CMP) min p'*x
x
s.t. T(z,x)£0
where p and z are given. Here we have written the problem for a cost mini¬
mization over a transformation function. We shall continue with this form
*
with the understanding that the production function case is a subcase of (CMP).
The conditions on T(z,x) guarantee that the solution to (CMP) exists
and is unique: the objective function is linear and the set of x in the
constraints is convex. If we vary z, holding p fixed a function is traced
V * * /
out relating minimal cost C = ip/x^(z9p) (where x^(z,p) is the optimal
solution to (CMP) for given z and p) to z and p. This is called the cost
function (or long run cost function to indicate that all factors have been
allowed to adjust to optimality) and is written:
C(z,p) (B-8)
B-19
The conditions on T(z,x) imply that the following characteristics of
C(z,p) can be proved (see [73], [33, Ch. 1.1], or [81]):
V
1) C(z,p) is H.D.I in p, i.e. C(z,Xp) = XC(z,p) X > 0 ;
2) C(z,p) is monotonie nondecreasing in p:
p' >_ p => C(z,p") >_ C(z,p) ;
3) C(z,p) is concave in p, i.e.
C(z, <5p + (l-ô)p') >_ 6C(z,p) + (l-<S)C(z,p')
for all 6 such that 0 <_ 6 <_ 1 ;
4) C(z,p) is continuous in p (for p > 0) .
The first condition reflects the obvious result that if all prices increase
by the same proportion, so will the costs since the uniform price change
will not affect the choice of the factor levels (since relative prices didn't
change). The second condition is also straightforward: since C(z,p) repre¬
sents minimal costs, one should not be able to reduce costs by increasing
factor prices. The intuition for the third property takes more effort.
Let (p,x ) be the price and optimal quantity of inputs for some output level.
This results in a cost C . Now if, say, just p^ is increased slightly (to
p^), then a slight reduction in x^ will have to be made, thereby not changing
costs in proportion to the slight p^ change. Figure B-5 illustrates the
effect (see [33], [81]). Thus C(z,p) is concave. Continuity (property 4)
follows from the concavity of C(z,p) (concave functions are continuous, ex¬
cept possibly at the boundary).
B-20
cost
C
C(z,p)
Figure B-5: COST FUNCTION CONCAVITY
An observation is in order. The analysis above and that which will
follow rests on two important assumptions: (1) the firm faces fixed,
known factor prices; (2) the firm minimizes costs. The assumptions
cut two ways. On the one hand, we may develop cost functions for monopolists
as well as perfect competitors: no issues about the market(s) for the out¬
put were raised. Moreover, as long as the entity being studied is trying
to efficiently produce output we need not concern ourselves with problems
of whether the firm is profit-maximizing or regulated to provide "socially
optimal" output. However, it is important that the firm face reasonably
competitive factor markets, something that may not be true for very large
firms.
We can now define some standard related cost functions:
1) Marginal Cost: MC^(z,p) =
i
i = 1
,•••,m ;
2) For single output models
Average Cost: AC(Z,p)
C(Z,p)
Z
Z > 0.
B-21
Using Shephard's lemma [73], [81] the optimal factor demand equations
*
x^(z,p) are simply:
a 9C(z,p)
x.(z,p) = i = l,...,n .
3Pi
Some manipulation also shows that the elasticity of cost with respect to a
factor price is:
3C(z,p) p A P.
= x.(z,p) (B-9)
Sp.^ C(z,p) 1 C(z,p)
*
p x (z,p)
(B-10)
C(z,p)
which is simply the factor share, i.e. the percentage of cost spent on fac¬
tor i. Notice that the left-hand-side of (B-9) can also be written:
8 log C(z,p)
3 log p±
which will be especially useful in translog cost function studies (where
log C(z,p) is expressed in terms of log z^ and log p^).
If we restrict z to. be a single output Z then a graph of a typical
C(Z,p) can be drawn as shown in Figure (B-6). The cost function illustrated
reflects economies of scale (increasing returns-to-scale) for outputs up to
Z and diseconomies of scale for outputs greater than Z. The result is a
classical U-shaped average cost function with a minimum at Z = Z. This
will be the optimal size of the firm.
The above cost function represents the cost of producing output z
given factor prices p assuming all factors are free to adjust their levels
so as to minimize cost. This assumption is not always valid. Regulated common
B-22
Figure B-6: COST FUNCTION
carriers often cannot adjust their capital stock through abandonment, for
example, of service. There can be any number of reasons why, at least for
a short period of time, a firm can not optimally adjust certain factors of
production as it increases or decreases its output. This is an especially
important issue when we try to estimate the firms C(z,p) function, since
this means that some of the observations will lie on C(z,p) but some of them
will lie above C(z,p). Notice that no observations could lie below C(z,p)
by the definition of the function. Therefore if we attempt to pass a curve
through a scatter of points we are doomed to overestimate the cost function.
This was recognized by over a decade ago by Eads [24], Eads, Nerlove and
Raduchel [25] and Keeler [48] and has been employed by a number of investi¬
gators since then (see [49], [41], [60], [32], [12]).
B-23
To formulate the short-run cost function, we partition x into two
subvectors: xV and x^ (for variable and fixed factors):
x =
where xV is of dimension n^ >_ 1 and x^ is of dimension n-n^. Now (CMP) be¬
comes the short-run CMP (SRCMP):
(SRCMP) min p'»{
v.
' x i
f
s. t. T(z ,xV,x^) <_ 0 .
Since the partition of x induces a similar partition on p and since (p^)'x^
is fixed then SRCMP becomes the short-run variable cost minimization problem
(SRVCMP) min (pV)' xV
s.t. T(z,xV,xf) <_ 0 .
Again if we vary z the result is a cost function C(z,pV;xf). Note that
v
only p shows up in the cost function. The notation shows that the cost
function is conditioned on the values of the fixed variables xf. Total
short-run costs are equal to short-run variable cost plus short-run fixed
costs :
TC(z,pV; xf) = C(z,pV;xf) + (pf)'-Xf
Observe that short-run marginal costs MCi(z,pV;xf) can be calculated from
either TC(z,pV;xf) or C(z,pV;xf). Thus
MCi(z,pV;xf) = -K(«,P ;x I i = 1,... ,n
1
B-24
If there is a single output, average cost is well defined and we have the
definitions of short-run average cost, short-run average variable cost and
short-run average fixed cost:
TC(Z,pV;xf)
1) AC(Z,p ;x ) = Z > 0 ;
Z
C(Z,pV;xf)
2) AVC(Z,p ;x ) = Z > 0 ;
Z
t f\- f
vf (P ) x
3) AFC(Z,p ;x ) = Z > 0.
Z
Again, Shephard's lemma yields the short-run factor demand equations and the
short-run factor share equations for the variable factors (now with mul¬
tiple outputs):
v* v f. 9C(z,p ;x )
X± (z,p ;x ) = i = 1, • • • ,n^
3pi
V V*, V fv r, - v f.
v v f PiXi ^z,p ;x ^ 8 C^Z'P >x )
S (z,p ;x ) = ~—2— = i = l,...,n
C(z,p ;x ) 81og p1
where S^(z,pV;x^) denotes the share of costs attributable to variable factor i.
Finally, the short-run functions provide the long-run function:
C(z,p) = min(C(z,pV;xf) + (pf)'-xf) . (B-ll)
xf
Thus, estimating the short-run variable cost function provides esti¬
mates of the short-run marginal cost functions, the factor demand and factor
share equations. It should be noted that the estimated share equations from
the variable cost function will not be the same as the estimated share equa-
B-25
tions from the total cost function. In fact the following relationship
holds :
, C(z,pV;xf) , Slog TC(z,pV;xf)
„v. v f. >■»*•>/ v f. =
S (z,p ;x ) —- = S (z,p ;x ) - -
TC(z,p ;x ) 1 Slog P^
i = 1,•••,n^#
Thus, caution must be used in interpreting the estimated equations.
Furthermore, if p^ is known, the long-run cost function can be re¬
covered by solving the optimization problem in (B-ll) above. Thus by speci¬
fying a technology and a vector of prices, we can derive functional forms
(in some cases explicitly as will be shown below) that can be estimated.
B.2.2. Cost Functions and Implied Technology
In the previous section we defined cost functions for technologies
that were convex in their input factors (e.g. for production functions with
isoquants as depicted in Figure B-la). If the technologies are not convex
in the sense shown in Figure B-lb then the cost function will be derived
for the convexified technology. This is illustrated in Figure B-7.
B-26
This means that all the technologies that have the same convexification
have the same cost function. This is really not a problem, as McFadden [33,
Ch. 1.1] points out, since it can be shown that if the firm is facing given
input factor prices and minimizing costs, then the firm never would choose
an input mix that would place it in the non-convex region, i.e. it would act
as if it worked with the convexification anyway. Thus, while a number of
technologies can give rise to the same cost function, the convexified tech¬
nology is all we need care about since the firm (if it obeys our assumptions
on factor prices and cost minimization) would never be observed operating
in the non-convex region anyway.
Now consider instead what information you could draw from a cost
function C(z,p) . If you were given the function and told that it came from
a cost minimizing firm that faced fixed prices, you could form the follow¬
ing set:
£
V (z) = {x|p'x >_ C(z,p) for all p > 0}.
Geometrically, V (z) is illustrated in Figure (B-8),
A.
x,
'2
p'x for some value of p
p'x for some other value of p
x
1
Figure B-8: CONSTRUCTING V*(z)
B-27
i.e. it is the set of points lying above all the straight lines. Thus, if
we pick an output vector z and vary p and look at all the x that are north-
k
east of the lines p'x, we have V (z). There are an infinite number of such
lines and the result is a curve that looks very much like an isoquant and
the region to the northeast of it, as seen in Figure B-9 below.
Figure B-9
k
It can be shown that V (z) is always convex, irrespective of the technology
that gave rise to C(z,p). It also satisfies the properties that we require
k
of a technology (see [81j). In fact we will call V (z) the input require¬
ments set (see footnotes 3 and 4) of our implied technology. We now have
the following very important duality results(see, e.g. [81]).
*-28
2) If the original technology is convex in its inputs then the implied
technology will be identical to it.
3) If the original technology is not convex in its inputs then the im¬
plied technology will be identical with the convexification of the
original technology.
Therefore, a properly constructed cost function will provide all the infor¬
mation of interest about a technology (if the firm obeys our assumptions on
fixed factor prices and cost minimization).
Put more practically, we can estimate either a production (or trans¬
formation) function or a cost function and get what we want to know about the
underlying technology. We can use a cost function to inform us about the following.
1) Homotheticity .
2) Homogeneity .
3) Returns-to-scale .
4) Separability .
We will consider these in turn.
B.2.2.1 Homotheticity
A very useful and interesting result concerning the structure of the
cost function occurs if we let the production function f(x) be homothetic,
i.e.
Z = f(x) = d(h(x))
B-29
where d(*) is a monotonie increasing continuous function and h(*) is H.D.I.
It can be shown [73] that if f(x) satisfies the above, then there is an in¬
verse function to d (which we'll call s(*)) such that h(x) = s(Z). Now
s(Z) is simply a scalar so we have the following result:
C(Z,p) = min (p'x|f(x) = Z}
x
= min {p'x|h(x) = s(Z)}
x
= min (p^x|h(s(.*y) = l)
= s(Z)* min{p^w|h(w) = l} w = ^z)"
w
= s (Z)• &(p)
To explain: the first line is a statement of (CMP), the second line the
result of the transformation discussed above. In the third line we capi¬
talize upon h(x) being H.D.I and s(Z) being a scalar. Thus h(x) = s(Z)
means h(x/s(Z)) = 1, i.e. if we divide every element of x by s(Z) then the
output is 1. In the fourth line we change variables letting the vector w
be the vector x with every element divided by s(Z). To maintain the
equality we must multiply by s(Z). Finally in the fifth line we recognize
that the minimum on line four will be purely a function of p (since w will
be optimized out). Shephard has proved a more general version of the
above (and its converse) and thus we have the following theorem.
f(x) homothetic <==> C(Z,p) is multiplicatively
separable in z and p, i.e.:
C(Z,p) = s(Z) • S,(p)
B-30
Note that since C(Z,p) must be H.D.I in prices p we know that Up) is H.D.I.
Furthermore s(0) = 0, s(Z) > 0 if Z > 0, s(Z) is continuous, etc. from the
properties of the cost function.
This result can be extended to the transformation function case. Re¬
call that Shephard assumed T(z,x) = g(z) - f(x). In this case we have that
C(z,p) = g(z)*£(p), i.e. again, C(z,p) is multiplicatively separable if and
only if the separable transformation function T(z,x) is homothetic in x.
Notice also that g(z) is the aggregation function for the vector z.
If the transformation function is not separable then input homotheticity
gains us somewhat less. McFadden [33, Ch. 1.1] shows that in this case
C(z,p) = a(||z||,z/||z||)C(z/||z||,p)
where a(*,•) is the scaling function discussed in section p.1.2.2 above
and ||*|| is the norm function mentioned there also. What is important here
is that tests for homotheticity that rely upon the multiplicative separability
of C(z,p) are actually testing separability and homotheticity together; a
rejection may be a rejection of separability or homotheticity or both.
B.2.2.2 Homogeneity
Let f(x) by H.D.k. Then in a manner similar to that in B,2.2.1 we see
the following result.
C(Z,p) = min (p"x|f(x) = Z}
x
= min (p'x| f (x/Z^k) = i]
x
= Z ^min{p'w| f (w) = l} w = x/Z ^
w
= z1/k Up) •
B-31
Thus, in particular, if f(x) is H.D.I then C(Z,p) = Z &(p) an<3 vice-versa.
In other words if C(Z,p) is linear in Z then f(x) is H.D.I.
It should be noted that from the above result we have that if f(x) is
H.D.k then C(Z,p) is H.D.(l/k). Extending this to the multiple output case
provides motivation for the following definition:
C(z,p) is output homogeneous of degree r (O.H.D.r)
r V
if C(Xz,p) = X C(z,p) X > 0 k
It is with this definition in mind that we next consider economies of scale.
B.2.2.3 Economies of Scale
Baumol [ 2 ] has defined the notion of decreasing average ray cost for
multiproduct firms. A firm has decreasing average ray costs if:
C(Xz,p) < XC(z,p) X > 1 .
For example, if we consider the single output case we have:
C(XZ,p) < XC(Z,p) X > 1
U
C(XZ,p) C(Z,p) . ^
XZ Z A > 1
which simply is the condition of declining average costs (i.e. returns-to-
(or economies of) scale). Thus decreasing average ray costs should be asso¬
ciated with returns-to-scale, and they are (see Baumol [2]). Panzer and
Willig [64] extend this notion to provide a measure of scale economies
for multioutput firms. They show that the following measure captures
B-32
returns-to-scale^ in production:
S = C(z,p)/ I z± 1
9C(z,p)
i ^ "zi
Notice that if C(z,p) is 0.H.D.I then S = 1 since by Euler's Theorem the
numerator equals the denominator. Rearranging terms yields the following:
z 3C(z,p)
S 1/'| C(z,p) 3zi
_ 1 ,r 9&nC(z,p) .
'j 9£nz^
We shall see later that this function is particularly easy to calculate from
a translog cost function. The measure above is the sum of the elasticities
of cost with respect to output divided into one. The more inelastic the
cost function is to output, the greater the returns-to-scale.
B.2.2.4 Separability
Analysis of the separability of production and transformation functions
can be performed via the cost function. First, considering homothetic pro¬
duction functions Uzawa [80J has shown the following for the AES cr^ (see
section 2.1.2.3):
C(Z,p)C (Z,p)
a, = J
ij C1(Z,p)CJ(Z,p)
2r
where C^Z.p) = 3C(Z,p)/3pi and C^CZ.p) = 3 C(Z,p)/3p;.3pj •
Thus estimating the cost function provides estimates of the AES (the sub-"
scripts denote partial derivative with respect to price). Berndt and
Christensen [4 ] show that weak separability (i,j e N^, k i N^; see section
B.l.2.3) implies a =* a., (i,j e N , k i N ) which is true if and only if
XrC J K U U
B-33
C. (Z,p)C., (Z,p) - C.(Z,p)C (Z,p) = 0. These conditions provide for weak
J i J k
separability of the cost function. Further Lau has shown that the cost
function is weakly separable (strongly separable) with respect to a parti¬
tion P in prices if and only if the production function is homothetically
weakly separable (strongly separable) with respect to the partition P in in¬
puts [33, Ch. 1.3]. Again, this area is extensive; for further information
see [6 ], [33, Ch. 1.3], and [37] to name a few references.
B.2.3. Examples
As has been indicated above, there is a duality between production
and cost: technology descriptions give rise to cost functions (when prices
are incorporated) which give rise to implicit technologies.
In this section we provide cost functions for some of the production
functions in section B.l. Furthermore we discuss some of the flexible func¬
tional form cost functions: the Generalized Leontief, the Hall function
and the translog.
B.2.3.1 Cobb-Douglas Production and Cost
The Cobb-Douglas production function of section R.l.3.2 gives rise to
the following cost function:
n a -1 . n a./v
c(z,p) = (a n a. ) z n p. 1
i=l 1 i=l 1
where v is the returns-to-scale in (B-4) insection B.l.3.2. Notice that C(Z,p)
could be estimated by taking logarithms:
n
£n C(Z,p) = 01q + £ Y. in p, +g£nZ
i=l
B-34
n a. -1 a.
where ag = &n(A II ai ) ~ ~ and ® = "^v" For C(Z>P) t0 be H.D.I in
i=l
prices we would require the constraint = 1.
B.2.3.2 CES Production and Cost
Referring to section B.1.3.3, the dual cost function would be as fol¬
lows :
l/(l-a)
C(z,p) =7(1 (p • /a. ) )
A _i 1 i
z n 1—O
i=l
where a = is the elasticity of substitution. Notice that as ff + 0 (the
1-P
Leontief case) we get a cost function that is simply a weighted sum of prices
times the output level, i.e. for Leontief production
Z n
C(Z,p) = - I (p./a )
A i=l
where the a. correspond to the a. of sectionB .1.3.1.
x 1
B.2.3.3 Flexible Functional Forms
The above cost functions are examples of what are known as self-dual
technologies [46j. The coefficients of the production function appear in
the cost function and vice versa and the dual functions are members of the
same family. This is not in general the case with the flexible functional
form. A transcendental logarithmic production function may not give rise
to a translog cost function. The choice of which to use is thus a non-trivial
one since it is possible that, for example, estimating a translog cost func¬
tion and a translog production function could lead to different results
(see Burgess [10]).
B-35
C(Z,p) - Z ^ J aij Sti "u ■ <*ji •
In what follows we will briefly describe three cost functions: the
generalized Leontief (Diewert [22]), the generalized linear-generalized
Leontief joint cost function (Hall [40]) and the translog (Christensen,
Jorgensen and Lau [14]).
The generalized Leontief function resembles the production function
of section B.1.3.4 above. It is as follows:
n n
Notice that the cost function represents a Leontief production function if
> 0 and = 0 for i ^ j . This is the source of its name. While this
function is a second order approximation to any technology, it only admits
one output.
The Hall function is an extension of the Diewert function for mul¬
tiple outputs. It is:
c<z,p> " ii ii I ! ^ ^ ^ ^ •
Unfortunately, the Hall function assumes H.D.I production and has a large
2 2
number of parameters to estimate (n m ).
Finally the translog is as follows:
m n
LnC(z,p) = aQ + I ai0£nZi+ £ 3jQ An Pj
1—1 J ""1
mm n n
m n
+ .1 I Yi:j *nz. Anp
i=l j=l J J
aij " °jf 6ii " 6ji"
Observe that if all the second order terms are zero, the translog reduces to
B-36
the Cobb-Douglas. In fact, one could view the Cobb-Douglas as a first
order approximation to a function (cost or production) and the translog
as a second order approximation: both in logarithms.
A word is in order on factor demand equations. In general, most
studies that estimate a cost function estimate it simultaneously with the
factor demand equations. This provides increased efficiency in the esti¬
mation process. The factor demand equations for the Diewert form are
quite simple :
1/2
X (Z,p) = Z I a (p /p )
j=l 3 3
This is similarly true for the Hall function. On the other hand, the fac¬
tor demand equations are not simple for the translog: they are non-linear
in the parameters to be estimated. However, because the translog is ex¬
pressed in logarithms the factor share equations are linear:
Si(Z,p) = gi0 + I By p
J=1 J J
m
+ l Y in z .
j=l 3 3
Therefore in estimating the translog cost function we can append n-1 factor
share equations (since the cost function makes the n1"^ equation).
In summary the principle advantages and disadvantages of the three
flexible forms are as follows.
Generalized Leontief (Diewert [22]): Advantages- 1) Second order approxi¬
mation to a cost function
2) Low number of parameters
to be estimated;
B-37
3) Convenient form for
cost function and fac¬
tor demand equations ;
4) Allows easy test of non-
substitution (Leontief)
case ;
5) Zero level of variables
allowed ;
Disadvantages" 1) Assume homogeneous produc¬
tion ;
2) Assumes single output ;
3) It is separability-inflexible
(see below).
Generalized Linear-Generalized Leontief
(Hall [40]): Advantages- 1) Second-order approximation;
2) Multiple outputs;
3) Convenient form ;
4) Tests input/output separability;
Disadvantages- 1) Assumes constant returns-to-
scale;
2) Large number of parameters
to be estimated.
B-38
Translog (Chrlstensen, Jorgensen and Lau [14]):
Advantages- 1) Second order approxima¬
tion ;
2) Reduces to popular forms
(e.g. Cobb-Douglas, CES
as limiting case);
3) Reasonable number of para¬
meters ;
4) Convenient form for esti¬
mating economies of scale;
5) Multiple outputs allowed;
Disadvantages- 1) Zero levels of variables
not allowed (due to logarithms) ;
2) Factor demand equations
non-linear in parameters
(though factor share equa¬
tions are not) ;
3) It is separability-inflexible.
Separability-inflexibility for the translog (see [6 ] and [19]) and the
Diewert [6 ] means that imposing separability restrictions implicitly im¬
poses significant structure on the aggregation functions themselves. Thus,
not only is the separability of different variables being tested: the test
of separability will also be testing a specific structural form for the
aggregation functions. Thus rejection of the test may only reflect rejec¬
tion of the forms, not the separability itself.
B-39
B.2.3.4 An Example of Finding Long-Run and Short-Run Cost Functions
To help tie-together some of the notions discussed in the sections
above, this section presents a long- and short-run cost model. Assume a
firm uses capital (x^, labor (x2) and fuel (x3) to produce a single out¬
put (Z) following a Cobb-Douglas production function:
The long-run cost function C(Z,p) with p = (p^,p2>p3)' is found by solving
1 2 3
Z = A x^ x2 x3. •
(CMP):
(CMP)
minimize P^xi + P2X2 + P3X3
subject to
which yields
where
v = + a2 + a3
and
al a2 a3
aQ = (Aa3 a2 a3 )
1 2 3
Thus the factor demand equation for labor (x2) is:
«2 C(Z»P)
B-40
which means that the factor share equation for labor is
_ p2x2(Z,p) a2
s2(z,P)
C(z,p)
which could also have been found by taking logarithms of the cost function
and then computing 3log C(z,p)/31og p2>
The short-run cost function is found by fixing one or more of the vari¬
ables. If we fix capital (x^) at a given level x^ then (SRVCMP) becomes
(SRVCMP) min p2x2 + p^
_ox a2 a3
s.t. Ax^ x2 x^ = Z
which yields the short-run variable cost function:
1/ c9/u a./u _-a1/u
C(Z,p jx^ = 6qZ u p2 p3 x1
where
v t \ -
p = ^p2,p3
a, a, 1/u
S0 - u(Aa2 a3 )
u = a2 + a3
It can be readily shown that we can derive the long-run cost function:
C(Z,p) = min (C(Z,p;x1) + P-^) •
X],
The short-run factor demand function is :
ao C(Z,pV;x1)
¥*•'= V * if p, '
This means that the factor share function is as follows:
B-41
V - S
Si(Z,pV;x1) = ~ .
Calculating returns-to-scale on the long-run function yields the following
= C(Z,p) = C(Z,p) =
Z -8C(3zP) v C(Z'P)
which is the returns-to-scale parameter (see section B.l.3.2).
The system of equations to be estimated for the short-run variable
cost function is as follows:
log C(Z,pV;x1) = Y0 + Y1logZ + y2log p2 + Y3l°g P3 + Y4l°g *1 +
P2X2
= Yo + e<
C(Z,pV;x1) 2
where Yq»Y3jY2 and Y 3 are t0 be estimated, the are error terms and we
have used the factor share equation for labor since there are only two fac¬
tor share equations (labor and fuel, i.e. n=2) and thus n-l=l. The above
system can be estimated as a seemingly unrelated equations system [79] - To
enforce H.D.I in pV we would add the condition that y2 + Y 3 = !• Since
^ A A A A
1/Yj^ will estimate u, we could recover a2 and from y2 and Y3 ( means
estimated value).
B.2.4. The General Form of the Short-Run Variable Cost Function to be Esti¬
mated and the Associated Conditions to be Tested
In this report we present results of estimating a short-run variable
cost function (see chapter 4). In this section we will provide the basic
form for the cost model which will be made more specific later. We will
B-42
also present the conditions on the model for homogeneity in prices, joint
separability and homotheticity, homogeneity of degree k and unitary elasti¬
cities of substitution in inputs.
B.2.4.1 The Translog Short-Run Variable Cost Model
We have chosen to use the translog model for the reasons described
above in section B.2.3.3. The model is as follows (£n means natural logarithm)
n i
^ m 1
£n C(z,pV;x ) = otn + Y a.Jlnz. + Y 3 .M£npY
0 .L, iO l ,L iO l
i=l i=l
n-n.
V f
+ Y Y .n£n x.
.u , iO l
i=l
^ m m
+ — Y Y a. .£n z. £n z.
2 lii jii 12 1 J
ni ni
+ \ \ 1 8.,£n pY £n pT
2 i=l j=l 1 J
n-n^ n-n^
r J 1 Y-£n xf
2 i=l j=l 1 J
m
+ I I £n pY £n z.
i=l j=l ^ 1 J
m n-n
r r 1 zx f
+ Y Y ô. . £n z. £n x.
• 1 • i !! 1 J
i=l j=l J J
n n-n
+ I I ÔPX £n PV £n xf
i=l j=l 1 J
B-43
» with a. . = a .., = B .. and Y . = y ... This results in a total of
ij ji 13 ji ij Ji
m(m+3) + n(n+3) + mn + i
parameters to be estimated.
til
The factor share equation for the i variable factor is as follows:
n m
v. fv _ „ , r „ v . r XVZ0
3I(z'PV;x > = eio + I 3i/n pj + 6ij*n zj
3 3
n-n-
+ J 6PX In xf .
j-1 « 3
B.2.4.2 Constraints
v
1) Homogeneity of degree 1 in p :
nl
(a) lml* iO = 1
"l
(b) I B,, = 0 j = 1 n
i=l 1J L
nl
(c) l 6^ =0 j = 1 m
i=l iJ
px
r» r
(d) J 5 =0 j = 1,...,n-n
i=l 1
2) Separability of T(z,x) and Input Homotheticity
dz zx
^. = 6. =0 v . .
ij ij ij
B-44
k
3) Homogeneity of degree k (k given) in production (X Z = f(Xx))
If
and almost homogeneity of degree (k,l) (T(X z,Xx) = 0);
m nl
(a) k I a + I yl0 - 1
1=1 J=1
n-n.
m z
(b) k £ a + I 6 X = 0 i = l,...,m
i-1 13 j=l *3
n-n,
^ zx
(c) I Y.., + k I 5. . = 0 j=l,...,n-n
i=l 13 i=l 1
m n-n^
(d) k £ <5PZ + £ 6PX =0 i = l,...,n1 .
j=l 13 j=l i3
4) Cobb-Douglas Production (i.e. unitary elasticities of input sub¬
stitution) :
zx pz px
a.. = 3.. = y.. = 6 =6 =6 =0 V. . .
ij U iJ ij ij ij i.J
The above are necessary and sufficient. Conditions similar to (1),
(2) and (3) are discussed in detail in [75]. The standard procedure would
be the following (see, e.g. [79]).
1) Estimate the unrestricted model, form the estimated co-
A
variance matrix, Π.
u
2) Estimate the model subject to some restrictions and form
A
the estimated covariance matrix 0. .
K
3) If the estimation procedure is a maximum likelihood pro¬
cedure then form the following log likelihood ratio:
ifij
Q Un
l"J
where Q is the number of observations. This statistic is
B-45
2
asymptotically y distributed with degrees of freedom equal
to the number of independent restrictions in R(|*| means
determinant).
Values of the statistic greater than a pre-set critical value
on Type I error means rejection of the hypothesis implicit
in the restrictions.
B-46
Notes for Chapter 2
1. Vectors are lower case letters, elements of vectors are lower
case letters with subscripts, individual scalars (such as
aggregate total output) are upper case letters (unless otherwise
stated). Sets and matrices will also be upper case letters. All
vectors are column vectors unless otherwise noted. A prime on a
vector or matrix denotes transpose^ superscripts on vectors are
used to refer to different vectors.
2. Weaker properties are possible; see [73].
3. The level sets are sometimes referred to as input requirement sets'.
V(Z) = {x|f(x) > Z) (see, e.g. [33, Ch. I.l], [8l]
4. There are a number of important related concepts in the literature.
Let V(z) = {x|T(z,x) < 0}, i.e. x can produce z as represented by
T(z,x) < 0. Thus V(z) is the input requirements set analogous to
note 3 above. The distance function D(z,x) is
D(z,x) = max {X > o|^ x e V(z)} .
D(z,x) is the amount by which a vector of inputs must be scaled down
so as to just produce z. Thus, efficient production occurs when
D(z,x) = 1 and therefore the relation between D(z,x) and T(z,x) is
seen to be the following identity [40]:
T(z- 5(^0 x) * °-
B-47
The concept of a distance function is also used in [73], [33, Ch. 1.1],
[ 33» Ch. II.1]; it first appeared in [73].
While the distance function is now becoming a more standard way of
representing multiple output/multiple input production, we will continue
to employ T(z,x) so as to readily address separability issues (see [40]) >
V means gradient, i.e. Vg(x) = (3g/3x^,...,3g/3xn)". Vv means gradient
2
only with respect to the variables in the subscript. V means the
Hessian matrix of second derivatives, i.e.:
V g(x) =
32g(x)/3x2 32g(x) 73x^X2
3 g(x)/3xn3x1
3 g(x)/3x13xn
32g(x)/3x2
2B
Finally the bordered Hessian is denoted V g(x):
V2Bg(x) =
Vg(x)
Vg(x)
V2g(x)
which is an (n+1) x (n+1) matrix.
A slightly weaker condition that would allow making some outputs from
other outputs is possible, but of limited interest in the current context.
When this is of interest, one could respecify T(z,x) as an explicit pro¬
duction function in terms of one of the outputs (see [33, Ch. II.1]) or
define a function in terms of a non-producible input (see [33, Ch. 1.3]).
B-48
A _
7. If u = (u^ uy)' and v = (v^,...,v^)' then u«v = \ is
called the dot (or inner) product of u and v. We shall employ the
shorthand throughout the report.
8. Here homotheticity is, in fact, a slightly weakened version of the
above. See[33, Ch. 1.3], [33, Ch. III.3].
9. Here again, for convenience, we have taken slightly stronger properties
than are in Shephard (see [73, p. 255]).
10. The capital Greek letter pi(II) represents product, just as \ repre-
n
sents sum. Thus II i = l*2»3...n.
i=l
™ 1c
11. This function is H.D.I. By taking a monotonie transformation h(u) = u
of f we could have
Z = h(f(x))
which would be H.D.k, allowing for increasing or decreasing returns.
12. Jin means logarithm to the base e, (Naperian logarithms). It should be
recalled that
log(ab) = loga + logb
log(a/b) = loga - logb
log a*3 = bloga
13. In general, symmetry is enforced for the a_^, i.e. a„ =
14. McFadden [33 £h. 1.1] considers weaker conditions, i.e. some prices that
are zero (free factors).
A
15. The measure used here is referred to as S in [64].
B-49
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