Short run cost functions for Class II railroads

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Faculty Working Papers

SHORT RUN COST FUNCTIONS FOR CLASS II RAILROADS

Alberta U. Cliamsy, llancy D. Tidhu
and John F, Due

#321

Transportation Research Paper Wo. 12

College of Commerce and Business Administration

University of Illinois at Urbana-Champaign

FACULTY WORKING PAPERS
College of Commerce and Business Administration
University of Illinois at Urbana-Champaign

July 29, 1976

SHORT RUN COST FlWCTIOigS FOR CLASS II RAILROADS

Alberta H. Cliamsy, llancy D. fidhu
and John F. Due

#321
Transportation Research Paper Uo. 12

SHORT RUN COST FUNCTIONS FOR CUSS II RAILROADS
Alberta H. Charney, Nancy D. Sldhu, and John F. IXie

There Is a substantial mileage of light traffic railroad lines In the
United States and yet little Is known of the cost behavior of these roads.
Knowledge of their cost functions is necessary to determine to what extent
these light density lines can be made more economically viable by a traffic
increase. The economic viability of a light traffic road depends to a
large extent upon the manner in which costs react to a change in the
volume of traffic. If total costs change very little with an increase in
volume (that is, marginal cost is very low), then improvement of traffic
will have a substantial impact upon average cost and therefore upon
profitability.

This paper seeks to determine the short run responsiveness of various
cost categories to volume changes. Specifically, this paper is concerned
with the short run cost functions of a sample of ten Class II railroads

for the period 1963-1973. Other work has previously been reported on the

2
long-run cost functions for this group.

The first part of this paper describes the statistical work used to

estimate the short run cost functions of these railroads and the estimates

of the average costs derived therefrom. Then, elasticities are derived to

^Class II railroads are those with annual gross revenues of less than
$5 million.

^Nancy D. Sldhu, Alberta H. Charney, and John F. Due, "Cost Functions
for Class II Railroads," Faculty Working Papers. No. 262, College of
Commerce and Business Administration, University of Illinois at Urbana-
Clifimpalgn, 1975.

I y^^r- '

Indicate the magnitude of the reductions in average costs that result from
an increase in traffic density.

I

The model. Traditionally, economists divide a firm's total costs in the
short run into a fixed component that does not vary with changes in output and
a variable component that does. This classification is, however, inadequate
when applied to railroads. The usual examples of fixed costs do exist in
railroading: depreciation of equipment, interest charges, some taxes (e.g.,
property taxes), some minimal administrative costs (necessary to hold the firm
together), and a normal rate of return on capital. However, the variable
cost component requires disaggregation Into three groups. The first, or
constant variable costs, are costs that can be eliminated if the railroad
la not operating (their variable dimension) but change very little if at all
when railroad output changes (their constant dimension). A major item in this
group is minimum track maintenance cost. The second group of variable costs
are more or less constant, in total, per train and, therefore, vary only
with incrementa of output large enough to require operation of more trains.
For example, the size of a train crew does not vary with the number of cars
but the man-hours vary with the number of trains operated per time period.
The third group of variable costs includes the usual variety that vary
directly with the level of traffic. These include part of the fuel costs,
any taxes related to traffic levels, per diem charges and switching costs.
If additional traffic comes from different sidings.

This brief classification scheme implies that, aside from the fixed
costs, total railroad costs can be expected to vary with volume but less

-3-

than proportionately. More tonnage requires more assembly time, etc.,
and if great enough, more trains. Because some costs are related to terminal
operations, Independent o£ the length of haul. It would also be expected
that increased length of haul would not increase costs proportionately.

Let us, therefore, begin with a function determining total costs (TC):

(1) TC ■ f (tons, average length of haul)
Dividing both sides by the unit of output ton-miles (T-M):

(2) AC = ^ - f<ton8, average len;sth of ha^l)^
We choose a linear specification of (2) to get:

(3) AC « a + bi average length of haul ^ u tons
1 X-M 2 x-M

- a + bi i + b, i ,
^ V -^ D

ti^ere V ■ volume, or ton-miles divided by average length of

haul, and

D - distance, or ton-miles divided by total tons.

Equation (3) represents one of the functional forms used in estimation of

short run statistical average cost functions in this paper.

The use of equation (3) assumes that average costs approach an asymptote,

a, as V and D increase. To avoid this restriction, we also ran each

regression presented in this paper in a semi- log form:

(3*) AC » a + b InV + b^lnD.

Results of these regressions are summarized below. It will become evident

that differences between the two models are not great over the observed

ranges of output.

It was expected that the signs of b and b. would be positive for the
inverse model and negative for the semi-log models indicating that average
costs per ton-mile decline as volume or distance or both increase.

Ihe data. Data for this study were compiled from annual reports filed
by ten railroads (listed on Table 1) with the Interstate Commerce Commission
for the years 1963 through 1973. We collected data and calculated figures
fox total tons, ton-miles, average length of haul, and the eighteen cost
items that are listed in Table 2 for each year for each of the ten roads.

To avoid biasing the results on account of inflation during the sample
time period, all items that required it were deflated. We tried to match
the index used as a deflator as closely as possible to the variable to be
deflated. For example, all cost components involving wages were deflated
by an index of "wages — excluding supplements," train fuel by an index of
"fuel costs," property taxes by a weighted average of relevant property tax
deflators. Further details on the deflating procedures are available
from the authors upon request.

Because of the number of observations on each road is relatively small,
we had hoped to pool the data to increase the efficiency of our estimates.

^Our heartfelt thanks go to Mr. Robert Byrne, formerly with the Rail
Services Planning Office, I.C.C, who arranged to make copies of the annual
reports available to us.

Association of American Railroads, Yearbook of Railroad Facts. 1974
edition, was the source of the wage, fuel, and "other materials and supplies"
deflators. Where appropriate we also used as deflators the wholesale price
index, the federal unemployment index, and the federal retirement index,
and property tax indices of the various states assessing railroad property
belonging to roads in our sample.

TABLE 1
RAILROADS IK THE SAMPLE

Line

..,N9t

Railroad Name

Abbr.

Median V

Median D

1

Arcade & Attica

ASA

17.193

4.837

2

Amador

AMAD

102.253

11.790

3

Apache

APAC

515.560

43.258

4

Bellefonte Central

BELC

59.303

5.611

5

Cadiz

CAD

27.147

10.000

6

City of Prlnevllle

CoP

359.799

18.340

7

Corinth & Counce

C&C

876.424

10.000

8

Hlllsboro & Northeastern

HNE

19.160

5.000

9

Mississippi Export

MISS

795.121

38.413

10

Pecos Valley Southern

PVS

48.736

13.925

*Distance la measured In miles; volume is measured in thousands of tons.

TABLE 2
RAILROAD COSTS USED AS DEPENDENT VARIABLES

Type of Cost Abbr,

I. Total Operating Costs C

A. Maintenance of Way Costs C

la

1. Roadway maintenance q.

2, Other maintenance of way coats Ci

B. Maintenance of Equipment Costs 0-.^

lb

1, Locomotive repairs Cii,

2, Equipment depreciation C,.

Ib2

3, Other maintenance of equipment costs Cii,

C. Transportation- Rail Line Costs C,

'Ic

1. Einployee compensation (of train crews) C,

Ici

2. Train fuel costs C,

^^2

3. Costs of loss, damage, casualties, and

personal injuries Ci

4c Other transportation coats d

lc4

D. Traffic, Administrative, and Miscellaneous Costs C, ^

Id

II. Other Expanses E

rtc

A, Equipment Rentals E

c

B. Elate of return calculated on railroad equipment E

C. Tax Payments E^
III. Total Railroad Costs (I plus II) ALL

■5-

Because we expected correlations to exist between the disturbances of
equations for different railroads, an attempt was made to utilize a generalized

least squares (GLS) estimator proposed by Zellner for situations involving

5
what he calls seemingly unrelated regressions. This estimator also has

the desirable small sample properties of unbiasedness and efficiency

6
relative to the ordinary least squares (OLS) estimator.

However, an unexpected computational snag developed that made it

impossible to use the GLS procedure. The many matrix manipulations

required for GLS resulted in computer roundoff errors which were large

enough to result in slightly negative numbers in the diagonal of the

variance- cover lance matrix- -an impossibility. Rather than push the data

farther than it could legitimately go, we therefore are presenting here

only the ordinary least squares (OLS) results for each railroad. Comparisons

with the few successful GLS runs indicate that the OLS parameter estimates

are substantially the same, as is to be expected because the OLS and GLS

estimators are both unbiased. However, the OLS standard errors are larger,

and therefore the OLS estimators are less efficient.

^The original article is A. Zellner, "An Efficient Method of Estimating
Seemingly Unrelated Regressions and Tests for Aggregation Bias." Journal
of the American Statistical Association, LVII: 2 (June 1962), pp. 348-368;
or see the discussion in Jan Knenta, Elements of Econometrics (New York:
the Macmillan Company, 1971), pp. 517-529.

Small sample properties of these estimators are worked out in
A. Zellner, "Estimators of Seemingly Unrelated Regressions: Some Exact
Finite Sample Results," Journal of the American Statistical Association.
LXIII: 4 (December 1968), pp. 1180-1200.

Three rallroeds in our 9aniple--the City of Prineville,
Corinth & Counce, and Hillsboro £e Northeastern- -showed little or no
variation in their distance figures ver the sample yetrs; so we had to
run their regressions in the following single variate form:

(4) AC = a + b ij, and

(4') AC « a + blnV.

We also ran the other seven roads using equations (4) and (4') for purposes

of comparison. We shall indicate where the differences between the two

versions are large.
The results.
Examination of the results shows that the overall quality of the

estimated cost functions varied widely. The best results of the group

are those for the Mississippi Export and the Pecos Valley Southern, and

the worst are for the Apache,

In the single variate analysis, most of the volume coefficients have

the correct sign. Sign reversals tended to occur mostly in the smallest

cost components like C,, rather than in the larger divisions like C , end

Ibo lb

only four o^ these ere significant!'" different from zero^ Model II
(equation (4)) is preferred over Model I (equation (4')) in slightly more
than ons-half of the cost equations. This preference is based largely on
sign reversals and significance of the b coefficients and the values of F
since the estimated amount of Model I serial correlation, shown by the
Durbin-Watson statistic in the last column, is almost always close to that
of the Model II version.

The equations incorporating both volume and distance show the following
differences from the single variable:

-7-

(1) Addition of the distance variable tends to increase slightly the
number of sign reversals of the volume coefficient bj^;

(2) At the same time, it also decreases the number cf significant
volume coefficients. This phenomenon appears more or less at random except
for equipment rentals (E ); here, b, was reduced to insignificance In four
of the seven roads we compared, and we can conclude that the blvariate
specification may well be incorrect in this case. On the other hand,

(3) Well over one-half of the distance, or average length of haul,
coefficients, the b2S, are of the wrong sign, and

(4) Only about one-eighth of the b^s are significantly different from
zero. The significant b.s also appeared at random, though about one- third
of them occurred with the various maintenance of way costs. This finding
leads to the contrary suspicion that for a few cost items, the blvariate
specification is the correct one.

Of more Interest are the descriptive uses to which these equations can
be put. We calculated the average cost per ton-mile of each road, for a
year in which it experienced its median volume level and median average
length of haul. These costs are displayed in Tables 3 and 4, calculated
from the tables for the Individual road. Several items are of note:

(1) The costs estimated in Table 4 are remarkably close to those
shown in Table 3 with the exception of the Arcade & Attica, for which the
single variate model estimates average total costs that are about two cents
per ton-mile less than those of the blvariate model. This difference is
due to the significant distance variable for Arcade & Attica. As expected,
most of the two cents difference is accounted for in the maintenance of way

TAflLi; 3

Estimated iv.edian Costs per Ton-Mile
for Sample Railroads--'/olu;Tfie Only *

Type of

(1)**

(6)

(8)

(5)

(3)

(7)

Cost

I.iodel

A&A

AI-'IAD

APAC

B5LC

CADIZ

CoP

1

"1

I

$.^935

$.0392

$.0201

:^.1494

0.1786

.;.o4i3

2

II

.ii-652

.0889

.0200

. 1479

.1758

. 0403

3

Clft

I

.1275

.0^^2

. 0034

.0270

. 0B26

.0102

u.

II

.1192

. Qkin

.0063

.0267

.0840

.0101

5

^la,

I

.0783

.0420

. 0066

.0155

.0732

.0081

6

II

.0725

.0420

.0065

.0154

.0750

.0080

7

^la2

I

.Q492

.0022

.0018

.0114

.0095

.0021

8

II

.Oi^.V?

.0022

.0018

.0113

.0090

.0021

Q

"lb

I

.0818

.0115

.0031

.0211

.0226

.0065

10

II

.0806

.0112

.0030

.0207

.0217

.0064

11

n
-Ibl

I

.0393

.0051

.0013

.0035

.0053

.0049

12

II

.0339

.0050

.0013

.0034

.0054

. 0049

13

"lb2

I

.0099

. 0038

.0002

.0116

.0058

.0010

1^

II

.0097

. 0035

.0002

.0112

.0058

.0010

15

Ot.3

I

.0308

.0024

.0015

.0060

.0115

.0005

16

II

.0303

.0024

.0015

.0060

.0105

.0005

17

Cl^

I

.1685

.0246

.0067

.0611

.0549

.0196

18

Ic

II

.1565

.0246

.0066

.0609

.0515

.01^4

19

^le^

I

.0993

.0122

.0023

.0293

.0252

.0112

20

XCJ

II

.0916

.0122

.0028

,0290

.0238

.0111

21

^ICg

I

.0110

.0027

.0010

.0018

.0056

.0012

22

II

.0102

.0027

.0010

.0018

.0057

.0012

2^^

Cl=3

I

.0092

.0013

.0012

.0033

.00004

.0012

II

. 0089

.0013

.0012

.0033

.00003

.0012

25

"Ic^

I

.0/1.90

.0085

.0017

.0267

.0241

.0060

26

II

. oi^SB

.0034

.0017

.0268

.0221

.0059

27
28

"Id

I

II

.1157
.1089

.0039
.0089

.0020
.0020

.0403
.0396

,0184
.0136

.0050
.0049

29

'2

I

.0212

.0153

.0051

.0146

.002?

.0039

30

""r

II

.0130

.0153

.0050

.0150

.0024

.0039

51

3_

I

.0268

.0053

.0011

.0091

.0205

.0043

32

c

II

.0252

.0053

.0011

.0039

.0191

. 0043

33

"t

I

.0368

.0081

.0032

.0175

.0166

.0023

3i^

II

.0353

.0081

.0032

.0171

.0157

.0027

35

ALL

I

.5763

.1184

.0294

.1905

.2185

.0523

36

II

.5^24

,1181

. 0292

.1888

.2131

.0518

(continued on

the next

page )

Type of

(10)

Cost

iiodel

I

1

,7.0220

2

II

.0220

Cla

I

.0066

k

II

.0066

I

=la,

I
II

. oo^^3
.00^3

7

Cla,

I

. 002'J-

R

li

.002^!-

9

-lb

I

.0027

10

II

.0023

11

=1^1

I

.0007

12

II

.0007

13

^lb2

I

.0013

14

II

.0013

15
16

=»3

I
II

.0008
.0008

17

°lc

I

.0091

J o

XNi*

II

.0090

19

= 10,

I

.0059

20

II

.0058

21

■^ICo

I

.0007

22

2

II

.0007

23

°lc~.

I

.0009

2i4-

■^ J>

II

. 0009

25

^Ic/^

I

.0016

26

II

.0016

2?

^Id

I

.0036

23

X -^

II

.0035

29

■^

"r

I

.0075

30

II

i007J^

31

Sp

I

.0012

32

c

II

.0012

33

2t

I

.0037

y^

II

.0037

35

ALL

I

.03^5

3'-

II

.0343

TA3LS 3 CONTINUSD

(2)

mn.

(9)

i iloS

(4)

J. 1689

0.0217

^.0386

.1677

.0212

.0853

.0^0^

.0067

.0265

. Qklv7

.0065

.0253

,0170

.0049

.0137

.0131

.0047

.0129

.026^!-

,0019

.0123

. 0266

.0018

.0125

.0123

.0027

.0105

.0125

. 0026

.0100

.0016

.0010

.0018

.0017

.0009

.0018

.0062

.0009

.0051

.0055

.0008

. 0048

.0051

.0008

.0033

.0053

.0008

.0032

.057^

.0100

. 0344

.0572

.0098

.0332

.0089

.0012

.0165

.0093

.0011

.0160

.0005

.0002

.0006

.0006

.0002

.0006

.0004

.0016

.0032

.0004

.0016

.0032

. 0476

.0070

.0141

.0470

.0069

.0135

.0553

.0022

.0173

.0533

.0022

.0167

.0315

.0075

.0039

.0312

.0076

.0037

.0233

.0009

.0088

.0267

.0009

.0086

.0123

.0022

.0037

.0124

,0022

.0084

.2410

.0323

.1101

.2379

.0319

.1061

*The median costs v;ere estimated using each road's median volume
level as the value of the independent variable in the equations of
\ppendix Tables 1 through 11.

**The numbers above the columns indicate the ordering of the railroads by
median volume.

TABLE 4

Estimated Median Costs per Ton-i-lile for
Saraple Rp.ilroads- -Volume and Distance*

1
2

3
if.

5
6

n
I

8

Tyt)e of
Cost

'1
'la

'la.

'lag

'lb

'Ibi
'Ibo

lb.

Hod el

I

II

I
II

I
II

I
II

I
II

I
II

I
II

I
II

'^Ic

I

II

-ici

I
II

-1=2

I
II

=103

I
II

;ic4

I
II

I

II

^^r.

I
II

c

I
II

^^t

I
II

ALL

I

II

(1)**

^>.5027

.1366
.1327
.0860
.0830
.0^07
.0498

.0853

.OB53
.0^1-26

.0i^25
.0093
.0093

.0315

.0317

.1670
.1578
.0977
.0917
.0107
.0101
.00'33
. 0086
. O^i-97
.0^75

.1133
.1089
.0230

.0213
.0265
.025^^-
.0384
.0361

.5887
.5673

(5)

Am AD

(6)

(4)

.11^

B2I

.0902

0.0188

.0889

.0190

. 0441

.0077

.0441

.0078

.0418

.0060

.0419

.0060

.0023

.0017

.0022

.0017

.0123
.0113

.0057

.0050
.0039
.0038
.0026
.0025

.0246
.0246
.0120
.0122
. 0026

.0027
.0014
.0013
.0037
.0035

.0091
.0089
.0149

.0153
.CO53
.005s
.0080
.0081

.1139
.1131

. 0029
.0029
.0013
.0013
.0002
. 0002
.0013
.0014

.0063
.0064
.0027
.0027
.0010
.0010
.0010
.0011

.0015
.0016

.0020
.0020

.0051
.0050
.0011
.0010

.0031
.0031

.0280
.0282

.;5.1493

.1480

.0269

.0267

.0155
.0154

.0114
.0113

.0211
.0207

.0035
.0034
.0116
.0112
.0060
.0061

.0611
.0609
.0293
.0290
.0013
.0018

.0033
.0033

.0267
.0265

.0403
.0396
.0146
.0150
.0091
.0039

.0175
.0170

.1905
.1838

(2)

CAJia

M787
.175^
.0818
.0833
.0721
.0741
.0096
.0091

.0227
.0217
.0052
.0052
.0055
.0055
.0120
.0110

.0562

.0524

.0252

.0235

.0055

.0056

,00005

.00004

.0254

.0232

.0130
.0181
.0028

.0025
.0207
.0190
.0170
.0160

.2191
.2129

(7)

$.0216

.0211
.0067
.0064
. 0048
.0046
.0019
.0013

.0027
.0026
.0010
.0009
.0009
.0008
.0008
.0008

.0100

.0099

.0012
.0012
.0002
.0002
.0016
.0016
.0070
.0069

.0022
.0022
.0074

.0075
.0009
.0009
.0022
.0021

.0321
.0315

(3)

^.0900
.0367
.0272
.0260
.0145
.0137
.0127
.0123

.0104
.0100
.0018
.0018
.0052
.0049
.0033
. 0032

.034-'^

.f^.'35
.0166
.0162
.0006
.0006
.0030
.0030
.0143
.0137

.0178
.0172
.00-17
.0035
. 0088
.0086
.0088
.0085

.1114
.1074

*The median costs were estiiaated using each road's median levels of
volume and distance as the values of the independent variables in the equa-
tions of Appendix Tables 12 through IS.

**The numbers Above the columns Indicate the ordering of the railroads by median
volume .

-8-

category, Cj^^ (with about one cent due to different estimates of roadway
maintenance (Cj^^^ ) alone), and most of the rest shows up in equipment
maintenance, Cj^j^, particularly locomotive repairs, C,. .

(2) If the roads are ordered according to median volxime level, and
their average total costs are compared, the expected pattern (AC falls as
V rises) appears. We calculated the Spearman rank correlation coefficient
for these two variables and found it to be significantly different from
zero even at the .001 level. This relation is also displayed graphically
in Figures 3 and 4.

(3) If the ten roads are again ordered by median volume, a tendency
exists for the ratio of average operating costs to average total costs
(C^/ALL) to fall as median volume rises. The Spearman rank correlation
coefficient is significant at the .01 level for this comparison. However,
no one component of operating costs displays this characteristic to any
noticeable degree.

(4) Opposite tendencies exist for the ratios of average returns to
capital to I /erage total cost (i^/A"*!) and for average rentals to average
total cost (Ep/ALL). The rental-to-cost ratio tends to rise as median
volume rises, while the return-to-cost ratio tends to fall.

Another method of displaying these results is graphical. On Figures 1
and 2 we have plotted the average total cost curves for all ten railroads
in the sample. The lengths of the curved segments indicate the range of
volume observed for each railroad in the sample time period.

'For a discussion of the uses and calculation of this statistic and for
tables of significance for small samples, see Sidney Siegel, Nonparametric
Statistics for the Behavioral Sciences (New York: McGraw-Hill Book Co.,
1956), pp. 202-213, 284.

o

:-^

-4 >J

M
0)
DU

CO
V

o

OB
O

o
o

P4

o
o

CM

\1'

M 2

w i3

0

60 O

to t-4 H
M (0

0> 4J VI

> O (U

< H O.

<
I

' __. >.

I

o

in

1
o

o
o

- o

tt s

O

-o
o

! I

i i ' '

i..J:i

i

! ■ i

■

--

I

■ 1 ■

i

-I i

— .

1

'"

'

1

o
o

ON

o
o
00

o
o

o
o

o

o

o
o

u

0)

p.

<a

i-i

CO

o
o

o

o
o

CO

O 1

00 p

CO -< H

w a

U 4J M

> O ffl

1

< H Ck.

i

. o

m

o
o

«M

o
o

_ o

-9-

These curves are plotted using the single variate models of equHtions
(4) and (4*) because we did not have a complete set of equations (3) and
(3') for all ten roads„ However, the role of distance is indirectly
observable in the following way: Take any volume level experienced by two
or more railroads. If X length of haul has any effect on costs, then the
road(s) with the longer median length of haul (from Table 1) should exhibit
lower average total costs per ton-mile on Figures 1 and 2o Indeed, with
but a few possible exceptions, this proves to be the case. As examples,
consider the following: The Arcade & Attica had the shortest median length
of haul in the sample and the highest average total cost curve; the Pecos Valley
Southern had a longer median distance and a lower average total cost curve per
ton-mile than did the Amador; the same is true for the Apache compared with
the City of Prineville or the Mississippi Exports The only exceptions to
this general relation are those cases where the average total cost curves
for two roads intersect, so that the conclusions to be drawn about X average
length of haul cannot be clearcut,

II

For each cost item of each railroad, the elasticity of total cost

with respect to ton-miles was calculated at the median volume levels. The

8
elasticities were derived from the single variable equations as follows:

For several reasons, only the single variate models were used for
calculating the elasticities. Not only was the distance variable
insignificant for many cost items, but it also caused some of the volume
variables to change signs. Also, for some of the ten railroads the distance
variable could not be used because there was little or no variation in that
variable.

-10-

For the semi- log model (using equation 4'), we have average cost

A .; - I£- = a=b InV = a + InV
T-M

where AC = average cost
TC = total cost
T-M » ton-miles

M = miles of the road.
Then TC - a (T-M) + b(T-M)ln(T-M/M)

MC = i-S£- = a + b rin(T-M/M) + h\

d T-M L ^ ' ^

= a + b ln(T-M/M) + b
= AC + b
Therefore for Model I,

®TC • T-M = ^ ^^ • ^••°'
d T-M TC

= (AC + b) « i^
AC

_ MC

AC

For the inverse model (using equation 4), we have average cost:

AC => I^ = a + b(l/V) = a + b (M/T-M).

T-M

Then TC = a (T-M) + bM

MC = i-^^ = a.
d T-M

Therefore for Itodel II,

®TC • T-M « i-I^ ° 1^
d T-M TC

= a • IzH
TC

= M£
AC *

-11-

Note that the elasticities are equal to marginal cost/average cost
for both mod'^ls. When marginal cost is very small compared to average cost,
an Increase in the traffic on that road should substantially reduce average
cost and therefore substantially improve the profitability of the road.
Therefore, the elasticities measure the extent to which roads can be made
more economically viable by a traffic increase.

Table 5 contains the calculated elasticities for each railroad and
each cost category. Observe that:

(1) The roads are ordered with respect to their median volume levels
to see if the elasticities of light density roads are different from the
elasticities of the heavy density roads. Examination of the elasticities
indicates that the elasticities are independent of the volume of traffic
of the road. The correlation between volume and the elasticities of the
"ALL" variables is -.166 (almost zero). Low elasticities appear for bokj
high volume roads as well as low volume roads.

(2) Some of the elasticities of certain cost items are greater than
unity, imply, ig that as volume incre; jes, total cost increases more than
proportionately. This occurred most frequently for the maintenance of way

costs (C, ) (including both subcategories and roadway maintenance (C^ , )»

9
"other" maintenance of way costs (C^a-)), and costs of loss, damage,

casualties and personal injuries (C. ).

(3) Some of the elasticities are negative. For model II, this occilrs
when the sign of the coefficient a is not of the expected sign. Most of

This effect would likely vanish if a period of years was used instead
of a single year. When volume Increases noticeably, roads try to catch up
on badly deferred maintenance.

TABLE 5
ELASTICITIES, EACH COST CATEGORY, EACH RAILROAD

Group 1

Group 2

Group 2

Group 1

Group 2

Type of

Cost
^1

Mp^el

A&A

HNE

CADIZ

0.637

PVS

BC

1

0.297

0.723

-0.154

0.678

2

0.170 a

0.809

0.780

-0.193

a

0.718

3

^la

0.213

1.352

1.007

-0.294

0.770

4

*a

0.120 a

1.426

1.046

-0.339

0.755

5

«Ui

0.095

2.296

1.113

-0.754

0.769

6

-0.051 a

2.212

1.125

-0.876

0.750

7

^u.

0.399

0.745

0.032

0.196

0.769

8

0.385 a

0.891 a

0.384

0.227

a

0.764

9

<=lb

0.836

0.636

0.280

-0.215

0.453

10

0.901

0.668 a

0.530

-0.221

a

0.499

11

Cl.,

0.737

1.579

1.111

0.644

0.752

12

0.909

1.762 a

1.072 a

0.652

a

0.780 a

13

s

0.796

-0.628

0.435

-0.792

-0.037

14

0.828

-0.995 a

0.711 a

-0.830

0.039 a

15

=1.3

0.989

1.883

-0.182

0.149

1.225

16

1.069

1,806 a

0.144 a

0.158

a

1.189

17

•=10

0.117

0.795

-0.038

-0.053

0.903

18

-0,114

0.855

0.294

-0.104

a

0.923

19

c,

0.035

1.362

0.090

0,012

0.626

20

1=1

-0.239

1,495 a

0.378

-0.041

a

0.658

21

=lc,

0.089

4.260

1,248

0.027

0.881

22

-0.195 a

3.377

1.154

0.010

a

0.826

23

c,

0.601

3.770

-0.300

0.560

1.475

24

Ic3

0.497

3.428

-0.256 a

0,649

a

1.466

25

^1C4

0.200

0.627

-0.469

-0.270

1.143

26

0.035 a

0.675 a

-0.014 a

-0.352

a

1.150

27

^Id

0.270

0,358

1.431

-0.098

0.395

28

0.090 a

0.299 a

1.224

-0.139

a

0.455

29

B

-0.842

0.860

-0.471

-0.313

1.922

30

■ z

-1.356 a

0.771

-0.033 a

-0.362

a

1.789

31

E

0.296

-0.056

-0.200

0.062

a;205

32

c

0.131

-0.158 a

0.177

0.051

0.289

33

E^

0.691

1.099

-0.091

-0.059

0.219

34

t

0.765 a

1.097

0.294 a

-0.104

a

0.264 a

35

ALL

0.281

0.698

0.490

-0.134

0.709

36

0.156 a

0.752

0.681

-0.171

a

0.734

TABLE 5 continued

Group 1

Group 1

Group .

Group 2

Group 1

Type of

Cost
^1

Model
I

AMAD
-0.009

CoP

APACHE
-0.548

MISS
0.114

c&c

1

0.066

0.611

2

A

II

-0.031

a

0.163 a

-0.452 a

0.150 a

0,609

3

«u

I

0.401

-0.015

-0.740

-0,350

1.218

4

II

0.392

a

0.092 a

-0.648 a

-0.307 a

1.206

5

'^,

I

0.484

-0.024

-1.103

-0.246

1.453

6

II

0.475

a

-0.072 a

-1.003 a

-0,233 a

1.441

7

s

I

-0.992

0.713

0,589

-0.547

0.787

8

II

-0,978

a

0.719 a

0.637 a

-0.501

0.776

9

hh

I

-1.309

-0.118

-1.250

-0.258

1.887

10

^u

II

-1.533

a

0.008 a

-1.176 a

-0.228 a

1.792

11

<^lb,

I

-4.268

-0.322

-1.128

-0.397

0,221

12

i.Ol

II

-4.595

a

-0.149 a

-0.994 a

-0.478 a

0,193 a

13

<=!.,

I

0.203

0,326

0.550

-0.350

1,207

14

II

0.129

a

0,401 a

0.560 a

-0.409 a

1.160 a

15

^.3

I

2.489

0.774

-2.432

-0.011

2.339

16

II

2.281

a

0.756 a

-1.578 a

0.023 a

2.218

17

Ic

I

-0.196

0.366

-0.254

0.444

0.041

18

II

-0.158

a

0.429

-0.175 a

0.472

0.045 a

19

^ic,

I

-0.663

0.493

-0,505

0,385

-0.021

20

II

-0.590

a

0.539

-0.408 a

0.370 a

-0.021 a

21

^ic,

I

0.295

0.744

-1.777

0.350

-0.011

22

II

0.338

a

0.768

-1.664 a

0,414 a

0.022 a

23

^103

I

-3.906

1.818

0.311

-0.065

0.636

24

II

-3.806

1.692

0.437 a

0.028 a

0.672 a

25

^Ic

I

0.901

-0,238

0.656

0.573

-0.073

26

^''4

11

0.875

-0.104 a

0.724 a

0.592

-0.060 a

27

^Id

I

0.146

-0,708

0.437

0.447

-0.015

28

11

0.106

a

-0,542

0.457 a

0.476

-0.024 a

29

E

I

0.152

1.236

-0.498

1.829

-1.450

30

r

II

0.228

a

1.204 a

-0.449 a

1.742

-1.398 2

31

h

I

-0.137

0.361

0,305

0.173

0.302

32

II

-0.115

a

0.438

0.350

0,227

0.327 a

33

h

I

-0.609

0.488

0.942

-0.121

1.104

34

II

-0.508

0.532

0,980 a

-0.048 a

1.^074

35

ALL

I

0.036

0.200

-0.351

0.498

0.204

36

II

0.079

a

0.285 a

-0.269

0.517

0.216 a

•12-

those coefficients with the wrong sign, however, were insignificant at the
five percent level and were designated by an "a" on Table 3o For model I,
the negative elasticities occur vrtien the coefficient, b, of InV is of the
correct sign but large relative to ACo Usually these negative signs occur
for cost items that are very small, and may be a result of unsatisfactory
deflators,

(4) Six of the ten elasticities of "ALL" costs are very low (less
than .3). Since the elasticity is the ratio of marginal to average cost,
the small elasticities indicate that these firms vrere operating on the
low-output downward- sloping portions of their short run cost curves, and,
therefore, that substantial unused capacity existed for these firms during
the sample time period. These roads are: Arcade & Attica, Pecos Valley
Southern, Amador, City of Prineville, Apache, and Corinth and Counce.

(5) The other four roads have elasticities close to or abov .j.
Although an increase in traffic would reduce average cost ^^^r these roads,
it would not heve the substantial impact that it would for the other six
roads. These four roads include: HiHsboro & Northeastern, Cadiz,
Bellefonte Central, and Mississippi Export, These roads were likely
operating closer to minimum short run average cost than those in the previous
group--given their fixed inputs,

(6) Average costs, marginal costs, and the elasticities for the "ALL"
cost items are reported in Table 6 for each railroad. They illustrate more
clearly how the costs behave with a change in traffic of 1,000 ton-miles.

(7) To gat a better idea of the responsiveness of total cost to change
in volume, average elasticities for each of the two groups of railroads

TABLE 6

COSTS AND ELASTICITIES FOR THE SAMPLE RAILROADS
CALCULATED USING MODEL II AND THE MEDIAN VOLUME OF THE ROAD

Railroad

AC/IOQOT-M

MC/IOOOT-M

®TC°T-M

(1)

AScA

$542.40

$ 84.82®

0.156

(6)

AMAD

118.10

9.29^

.079

(8)

APAC

29.20

- 7.86^

- .269

(5)

BELC

188.80

138.57

.734

(3)

CAD

213.10

145.10

.681

(7)

CoP

51.80

14.75^

.285

(10)

C&C

34.30

a
7.39

.216

(2)

HNE

237.90

178.89

.7'^'^

(9)

MISS

31.90

16.48

.517

(4)

PVS

106.10

- 18.13*

- .171

Not signlfJ -antly different from zr -o at the five percent level.

■13-

described ebove. These crude average elasticities appear in Table ?,, For
most items, the elasticities for group I have substantially low3r figures
than group IIo Tha cost items which have higher elasticity for group I
than for group II are: Other maintenance of way (C, ), Maintenance of
equipment (C^, ), Equipment depreciation (C,. ), Other maintenance of equip-
ment costs (C., )5 and Tax payments (E ), Although C. and C. are
relatively small cost components, maintenance of equipment, equipment
depreciation, and tax payments represent a substantial proportion of this
group's cost.

(8) There are three cost items which have elasticities of apprcximutely

unity for group 11. These are roadway maintenance (Ci ), Locomotive

ia^

repairs (Civ^. ), and Equipment rentals (E )„ The average cost of these cost

items did not change with an increase in traffic. The high elasticities of

the first two presumably reflect "catch-up" maintenancG as traffic rises;

the third simply reflects proportionately greater car use. Two of the coct

Items for group II were much greater than unity: Trein fuel

costs (C, ) and Costs of Iocs, damnge, casualties, and personal injur-i-
lc2

(U, ), Ths average cost of these items actually increased with an increacc.
in traffic. The first must reflect running additional trains with less
volume per train; the second is likely accidental.

(9) The most interesting observations vhich can be aads about Table 7
concern those elasticities that are very low. The lowest two elasticitier

Wnen a negative elasticity appeared in Table 5, it was considerec
uo hs zero when these average elacticities were calculated.

TABLE 7
AVERAGE ELASTICITIES, VARIOUS COST CATEGORIES

Type of

Cost

Model

Group I

Group II

1

^

I

.162

.538

2

II

.157

.614

3

Cu

1

.305

.782

4

II

.302

.807

5

^lan

1

.339

1.045

6

**1

II

.319

1.022

7

^lao

I

.447

.387

8

2

II

.457

.510

9

^Ib

I

.454

.342

,10

II

.450

.424

11

^Ib,

I

.267

.861

12

ADJ

II

.292

.9039

13

^lb2

I

.514

,109

14

II

.513

.188

15

Clb3

I

1.123

.777

16

II

1,080

.791

17

he

I

.087

.536

18

II

«079

.636

19

'^H

I

.090

.616

20

II

.090

.725

21

^ICo

I

,193

1,685

22

i.C2

11

.190

1.443

23

^Ic

I

.663

1.311

24

3

II

.658

1.230

25

^Ic,

I

.293

.586

26

"■^4

II

.272

.604

27

^Id

I

.142

.658

28

xu

II

.109

.614

29

^r

I

.231

1.128

30

II

.239

1.0755

31

E

I

.221

.095

32

w

II

.296

.173

33

Et

I

.538

.330

34

la

II

.670

.414 '

35

ALL

I

.114

.599

36

II

.110

.661

TABLE 7
AVERAGE ELASTICITIES, VARIOUS COST CATEGORIES

Type of

Cost

Model

Group I

Group II

1

c,

I

.162

.538

2

1

II

.157

.614

3

<=i.

I

.305

.782

4

II

.302

.807

5

''^^

I

.339

1.045

6

II

.319

1.022

7

^Uo

I

.447

.387

8

*OQ

II

.457

.510

9

=ib

I

.454

.342

.10

II

,450

.424

11

=1H

I

.267

.861

12

II

.292

.9039

13

=u.

I

.514

,109

14

II

.513

.188

IS

Clb3

I

1.123

.777

16

II

1.080

.791

17

Clc

I

.087

.^36

18

A\»

II

,079

,636

19

^^=1

I

.090

.616

20

II

.090

.725

21

=u,

I

.193

1,685

22

II

.190

1,443

23

•^Ic

I

.663

1,311

24

"3

II

.658

1.230

25

%

I

.293

.586

26

II

.272

.604

27

^Id

I

.142

.658

28

II

.109

,514

29

h

I

.231

1.128

30

II

.239

1.0755

31

E

I

.221

.095

32

c

II

,296

.173

33

h

I

.538

.330

34

II

.670

.414

35

ALL

I

.114

.599

36

II

.110

.661

•14-

for group I are Transportation-Rail line costs (C, ) and the subcategory
of this, Emp''oyee Compensation (of t-ain crews) (C, ). The low elasticitlas
(almost zero) imply that average labor costs will decrease substantially
with an increase in traffic. Two other subcategories of Transportation-
Rail line costs (C, ) have low elasticities: Train fuel costs (C, )
Ic . ■I-C2

and Other transportation costs (C^ ), The very low elasticity for Train

fuel costs (C, ) indicates that an increase in volume will allow longer
ic2

trains and proportionately less switching and therefore, more efficient
fuel use. An inefficient use of labor for low levels of traffic is again
indicated by the low elasticities of Locomotive repairs (C,, ), a subcategory
of Maintenance of equipment costs (C,. ), and Traffic, administrative and
miscellaneous costs (C-,,). The low elasticity of the Rate of return on
railroad equipment (E ) indicates an inefficient use of equipment at low
volumes but the low elasticity of Equipment rentals (Ej.) is difficult to
explain. Total operating costs (C,) has a very low elasticity as well.
This is simply a reflection of the low elasticities of the cost components
of Total ope.ating costs.

Conclusion

The proposed abandonments of many light density lines and the reorganiza-
tion of the Northeastern quadrant's railroads indicate a need for a better
knowledge of railroad costs and revenues for evaluating the alternatives
to abandonment, such as subsidies or converting marginal branch lines into
either privately or municipally owned short lines.

We have attempted to shed some light on the responsiveness of the
various cost components of low density railroads to a change in traffic.

■15-

Our results show that for all roads in the sample, MC is well below AC,
and thus add:'*:ional traffic will redi-ce AC. For six of the ten roads in
the sample, MC is extremely low and additional traffic reduces AC dramatically;
for the second group, the reduction is less, though substantial. The two
major cost categories, train operating and wage costs and track maintenance
costs, show very low elasticities for the first group. The primary difference
between the two groups is in maintenance costs, suggesting that the higher
elasticity of the second group is due to catching up deferred maintenance
and, therefore, would be eliminated if a period of two or three years was
used instead of one year.

The basic conclusion therefore is that, in the short run, additional
traffic on light traffic lines will significantly lower train operating
and maintenance of way costs, and therefore improve the financial viability
of the lines. The roads are typically operating well below capacity. These
.results, however, are not conclusive about the ability of light traffic
lines to adjust over time to changed volume levels.

aouNoJS