The survivor technique and identification of optimal plant size using individual plant data

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Faculty Working Papers

THE SURVIVOR TECHNIQUE AND IDENTIFICATION OF
OPTIMAL PLANT SIZE USING INDIVIDUAL PLANT DATA

Jeremy Atack and Fred Bateman

#383

'

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

March 8, 1977

THE SURVIVOR TECHNIQUE AND IDENTIFICATION OF
OPTIMAL PLANT SIZE USING INDIVIDUAL PLANT DATA

Jeremy Atack and Fred Bateman

#383

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THE SURVIVOR TECHNIQUE AND IDENTIFICATION OF OPTIMAL
PLANT SIZE USING INDIVIDUAL PLANT DATA

Jeremy Atack
University of Illinois

Fred Bateman
Indiana University

Comments welcomed

ABSTRACT

The Survivor Technique and Indentif ication of Optimal Plant Size Using

Individual Plant Data

George Stigler's survivor technique has been gaining increasing
acceptance in recent years as a heuristic method for identifying minimum
long-run average costs or the range of plant sizes operating under
conditions of approximately constant returns to scale. The authors
investigate this hithertoo untested equivalence using micro-data and
conclude that the method has considerable merit but is not infallible.

Jeremy Atack
University of Illinois

Fred Bateman
Indiana University

The Survivor Technique and Identification of Optimal Plant Size

Using Individual Plant Data

George Stigler's "survivor technique" has never been applied
in its "ideal" form, using individual plant data. Nor have the results
been compared systematically with those applying alternative methods to

Che same figures, which as William Shepherd has stressed is crucial

2
for interpreting survivor technique results. Recently available samples

of plant-level data drawn from early federal census documents, and newly

completed work relying upon alternative methods, provide the first

3

opportunity to test the survivor technique under "ideal" conditions.

The Technique

The survivor technique seeks to identify that size class or those
classes of plant that not only survived the rigors of competition and

the test of time, but also succeeded in increasing their share of industry

4
value-added. That is, it seeks to identify plant sizes that grew in

relative importance through the long-run adjustment process. A number

of assumptions are implicit. The time span between observations must

be sufficiently long to permit long-run scale adjustments and for a

clear pattern to emerge. The long-run average cost curve is assumed to

retrain fixed, thus excluding major technological advances which may

shift the long-run average cost curve. Similarly, constant cost industry

conditions must be assumed so that the long-run adjustment process itself

doss not induce shifts in the long-run average cost curve through changing

factor prices. Finally, atomistic competition must be assumed so that

long-run demand shifts leave price unchanged due to compensating supply-
shifts through the adjustment process.

Collectively these assumptions ensure that optimally adjusted
plants will produce at minimum long-run average (private) cost. If there
are only internal economies of scale, then the optimally adjusted plant
produces under constant returns to scale as defined by the production
function and the long-run average cost curve. On the other hand, if
there exist substantial external economies which can only be achieved by
large scale operations, it is possible that a plant might be producing
under conditions of decreasing returns in production, but that the rise
in the value of long-run average costs is postponed by the realization
of these external economies. Thus, rising costs in the production process
say be offset by decreasing costs in the purchase of inputs or in marketing
and distribution ar i once again optimal plant size would appear to be
located under constant returns to scale identified with minimum long-run
average costs but not by production function estimates. The converse
of this argument would apply in cases where there is reason to suppose
that large scale operations encounter external diseconomies of scale
despite internal economies in the production process.

Results showing a single size class of plants gaining in relative
importance, therefore, suggest a U-shaped long-run average cost curve
and a determinate optimal plant size. The persistence of a wide range
of size categories with no clear gainers is suggestive of the presence
zi constant returns to scale ever a wide range of outputs and, hence,
of a relatively flat long-run average cost curve. Either result would
be consistent with cost function studies.

Application of the technique to monopolistic industries yields
imperfect results due to changes in optimal plant size over time with
shifts in demand since, under such a market structure, optimal plant
size is jointly determined by demand conditions and technology. In an
attempt to circumvent this problem, Weiss, rather than identifying the
optimal range (s) of plant size, identified only the "minimum efficient
plant size" defined to be the smallest plant size that increased its
relative contribution to total value-added.

The advantages, drawbacks and limitations inherent to the technique
have been detailed by previous users' but its most commonly-cited advantage —
its ability to analyze changes in optimal plant sizes over long time
spans rather than in a purely static context — is worth emphasizing.
Application of the survivor principle further "finesses the problem of

the capitalization of rents into costs, a process which drives disparate

g

measured average costs towards eauality." There are several recognized

limitations. Social costs are not captured by the method. Moreover,
it may embrace other effects, such as externalities or technological
change, which could lead to false impressions regarding internal scale
economies.

Application of the Survivor- Technique

The data used in this study wei e drawn from the 1850, 1860 and 1870
United States censuses of manufactures. Since the population from which
the random samples were selected was the state rather than the entire
nation, these data may not be representative of United States manufacturing
in general. However, the hypothesis that the pooled data were significantly

different from that expected from a random sampling of plants in the
entire nation was rejected at better than the five percent level. Neither
the industrial distribution of plants nor the characteristics of those

plants, particularly those with respect to size were significantly

9
different from those for the entire nation in any census year.

The availability of data from these census years permitted three
separate applications of the survivor technique to the periods 1850-1860,
1860-1870 and 1850-1860-1870. All other studies with the exception of
that made by Shepherd were limited to examining the possible existence
of optimally sized plants between only two years. In the absence of
externalities and technological change under conditions of atomistic
competition the availability of this additional observation should improve
the survivor technique results, but technological change, the accretion of
externalities or the existence of market power would lead to a clouding
of the results.

Twenty-five industries, or approximately one-fourth of the identifiable
industries existing during this time period were studied. Since the
results were not unequivocal in designating the presence or absence of
an optimal range of plant sizes across these industries the "quality"
of the results is shown in Table 1. Other practitioners of the technique
have not been consistent in thier choice of the appropriate index of
plant size, so the results in Table 1 offer four different measures of
size; capital assets, employment, gross output and value-added. These
results permit us to discriminate between the possible options. In

Table 1

THE QUALITY OF SURVIVOR TECHNIQUE RESULTS MEASURING PLANT SIZE
BY CAPITAL, EMPLOYMENT, GROSS OUTPUT AND VALUE-ADDED
1850-1860, 1860-1870 AND 1850-1860-1870

Percent of Survivor Technique Result of Each Type Measuring Plant Size By

Capital Employment Gross. Output Value-Added
Result 1850- 1860- 1850- 1850- 1860- 1850- 1850- I860-' 1850- 1850- 1860- 1850-
1860 1870 60-70 1860 1870 60-70 1860 1870 60-70 1860 1870 60-70

Clear

8%

28%

12%

20%

28%

24%

32%

4%

20%

32%

24%

28%

Clear After

Adjustment

24

36

44

32

24

20

36

32

20

56

40

43

Partially Clear

4

8

12

0

0

0

28

44

52

8

16

20

Partially Clear

36

72

68

52

52

44

96

80

9'2

96

80

96

40 15 24 40 40 56 4 4 4 0 4 4
Inconsistent 24 12 8 8 8 0 0 16 4 4 16 0

.ear or worse

64 28 32 48 48 56 4 20 8 4 20

Clear - One or more adjacent size classes increasing their share of industry value-added.

Clear After Adjustment - Two or more non-adjacent size classes increasing their share of

of industry value-added, but the apparent inconsistency is re-
solved by merging size classes.

partially Clear - Only the lower bound of the surviving size classes is observable.

f-rlear - Shifts in industry value-added between size classes erratic with no boundaries

:r.5i3tant - Incompatible with survivor technique because middle size classes were
shrinking relative to the smallest and largest size classes.

nerns of allowing us to draw inferences about the existence of an optimal
range of plant sizes. The results obtained using either capital assets
:r employment are clearly inferior to the alternatives. Gross output
cr value-added indices of plant size on the ether hand are approximately
the same unless we have a marked preference for more definitive statements
about the range of surviving plants in which case the value-added measure
of plant size is to be preferred. Since disclosure laws have prevented
use of the gross output measure for twentieth century data and because
of the relatively stronger results that we have obtained using value-added,
this represents our preferred measure of plant size.

In keeping with the heuristic nature of the survivor algorithm,
allocation of the results between the categories in Table 1 was not
particularly rigorous. A more mechanical procedure, such as designating
any result "inconsistent" where more than one intervening size class
was declining relative to those on either side, would have reduced the
number of results that were at least partially clear, particularly if
plant size were measured by capital assets or employment.

The results in Table 1 show little evidence that the use of two
census years is to be preferred to only three. Indeed in almost every
instance the use of 1850-60-70 data resolved some of the apparent incon-
sistencies in the sub-periods as might be expected if movements towards
an optimal range of plant sizes followed a random walk or if there were
no clear optimum.

The survivor technique implications for an optimal range of plant
sizes classified by value-added over the three census dates are shown

7

in Table 2. The results for flour milling (SIC industry code 2041) were

unclear and are therefore not shown in Table 2. In this industry six of
the twelve size groups showed marked increases in their share of industry
value-added but these were scattered across the entire spectrum of plant
sizes.

The results in Table 2 also reveal a wide range of optimal plant

sizes in each industry, suggestive of fairly broad flat long-run average
cost curves, with constant returns to scale prevailing across a wide
variety of different plant sizes. This result is consistent with the
conclusions of twentieth century cost function studies.

The Consistency of Survivor Technique Predictions

Although two alternative, and equivalent, methods of testing the
accuracy of the survivor technique predictions are possible — the estimation
of cost or production functions — analysis of these manufacturing data
is limited to production functions due to data inadequacies for cost
function estimation.

Ordinary least squares estimates of a production function, however,
imply linear cost functions and hence are inconsistent with the assumption
of a U-shaped long-run average cost curve underlying the survivor principle.
But in recent years a number of alternative production function forms,
estimated by the maximum likelihood technique, have been developed
that are consistent with a U-shaped long— run average cost curve within

certain parameter limits. A Cobb-Douglas variant of the Zellner and

12
Revankar function was used of the form:

ln(VA) = InA + u • InL + 3 ■ ln(K/L) [1]

where ln(V ) = InV + QV, a monotonic transformation of V.

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Returns tc scale, f- , are then:

e = y/(l + 6V)

and depend upon the paramater, G , Che estimate u and upon the level of
value-added, V. For G > 0, returns to scale are monotonically decreasing
and for 0 > 0 and u > 1, this production function implies that at low
levels of value-added, plants are subject to increasing returns to scale
eventually giving way to a range of approximately constant returns with
decreasing returns tc scale apparent for !»...-. ;;iough V.

The logarithm of the liklihood function corresponding to equation
[1] is:

lnit = constant - -| lna2 + In J(A;V)

- i I i In (VX), - InA - y • InL. - 0 • ln<^_

2.V- i=1 j 1 1 L± j

2

where a is the variance of the nomally and independently distributed

random error term with mean zero, n is the number of observations and
J(X;V) is the Jacobian of the monotonic transformation. Ordinary least
squares minimizes the last term of equation [2] for any predetermined
value of 0 , yielding a conditional maximum for the likelihood function.
3y varying 0 and evaluating (ln£ - constant) the global maximum of likeli-
hood function can be determined.

This non-linear maximum likelihood method was applied to the plant
data for the twenty-four industries shown in Table 2. The results are
given in Table 3 and are generally consistent with the survivor technique
oredictions.

10

Table 3

DECREASING SCALE ELASTICITY ESTIMATES OF RETURNS TO SCALE
IN MINIMUM EFFICIENT PLANTS, 1870

[ndustry Description

Returns
To Scale

SIC
Code

Industry Description

Returns
To Scale

2 Gil y.eac Packing
2C51 Bread and Bakery Products
£032 Halt Liquors
2GS5 Distilled Liquors

21 - Tobacco

2211 Co c tons

r

Voolens

Hen's Clothing
2351 Millinery
2-21 ? Savnills
2431 Milivork
2511 Vcod Household Furniture

1.55*
1.48*
1.21
1.25

-86
1.00
1.06

.93
1.04
1.13*
1.19

.91

27
2892

>799

i Printing and Publishing

i

• Explosives

; Leather Tanning

! 3131 j Boots and Shoes

3199 j Other Leather Products

!| 3251 | Brick and Tiles

f i

j 3312 I Blast Furnaces

!! ]

3321 1 Iron Foundaries

'•I I

|i 5444 ; Sheet Metal

j; 3511 Steam Engines

i 1

! :

3522 Agricultural Implements

Transportaion Equipment

_I

.96

1.41

1.06

.94

.76*

.92

.87

1.06

1.05

1.09

1.17

.73*

indicates returns to scale parameter significantly different from 1.00 at the
fivs percent level (2-tailed test).

11

As the results indicate, standard errors were large, but the
minimum efficient size plants in nineteen of the twenty-four were
operating in the range where returns to scale were not significantly
different from unity. Of the five industries that failed the test,
three — Meat Packing, Bread and Bakery Products and Sawmilling — exhibited
significantly increasing returns to scale in the minimum efficient size
plants but larger plants, still within the ranges identified as optimal,
were operating under constant returns to scale. However, the minimum
efficient size plants in both the manufacture of miscellaneous leather
products and transportation equipment were estimated to be operating
under conditions of decreasing returns to scale. These results directly
contradict the predictions of the survivor technique, particularly
since these results were classified as "clear after adjustment" and
"clear" respectively in Table 1 and applying more rigorous standards
would not affect these categorizations. At the same time the decreasing
scale elasticity estimates for these two industries lead us to suspect that
there may have been unidentified cost curve shifts in both industries as scale
elasticity was estimated to be increasing with increasing plant size.

The survivor technique predictions of the range of approximately
constant returns to scale were less consistent than those for the
minimum efficient scale of plant. According to the decreasing scale
elasticity production function estimates, the largest surviving plants

in nine industries were operating outside of the range of approximately

13

constant returns to scale/"1 which suggests that Weiss1 choice of the

criterion of minimum efficient plant size rather than an optimal range

of plant sizes might be correct albeit for the opposite reason to that

, . . 14
given by him.

Some Supplementary Evidence Upon the Consistency of Survivor Technique Predictions

In the process of sampling the manufacturing manuscript censuses,
the accounts of the operation of forty-six steamboats during the 1850
census year were uncovered. These data differed substantially from those
of the manufacturing plants and provided sufficient information, when
supplemented by other sources, to estimate a steamboat cost function.

During the antebellum period, the western river steamboat was
the principle supplier of transportation services to the agricultural
trans-Appalachian West. From an early date after its first successful
trial on the Ohio and Mississippi Rivers in 1811, the western river
steamboat took on certain structural characteristics which made it
unsuitable for coastal navigation, and this has allowed us to apply the
survivor technique to this mode of transportation. By examining a
listing of all steamboats constructed in the United States between 1804
and 1868, it was possible to isolate those constructed for use in the
Mississippi basin on the basis of where they were built and their first
home port. The size distribution of newly constructed western river
steamboats by decade is shown in Table 4. Due to the short average, life-
span of the western river steamboat of between five and six years, if
there existed an optimal range of vessel sizes, a rapid convergence
towards this optimal can be expected.

Table 4 shows such a convergence. Prior to 1840, steamboat operators
showed a clear preference for steamboats of 200 gross tons or less.
However, after 1830 an increasing number of vessels of over 300 gross
tons were constructed, coinciding with renewed efforts to improve

13

Table 4

DISTRIBUTION OF WESTERN .STEAMBOAT TONNAGE BY VESSEL TONNAGE
AND DATE OF CONSTRUCTION, 1810-1859

Decade

Percent of Total Tonnage of Western
; Constructed Each Decade by Vessel

River
L Size
501-6C

Tons

Steamboats
Class1

1-100

Tons

101-200

Tons

201-300
Tons

301-400
Tons

401-5G0
Tons

)0

bUi-ZUU
Tons

Over
700 Tons

1810-1819

2%

26%

29%

28%

10%

0%

0%

0%

1820-1329

9

47

27

12

6

0

0

0

1830-1839

12

49

17

14

4

2

1

1

1840-1849

8

34

28

17

7

4

1

2

1350-1359

4

23

22

17

13

8

7

6

Percentages nay not add to 100 due to rounding

Source: Reconstructed from William M. Lytle, Merchant Steam Vessels of the United
States, 1804-1868, Mystic, Conn.: The Steamship Historical Society of
America, Publication No. 6, 1952, pgs. 1-208. Otherwise known as the
"Lytle List".

•14

navigation of the western rivers and the percentage of total tonnage of
western river steamboats represented by these larger steamboats increased
each decade from 1830. Apparently the removal of naviational constraints
upon steamboat size, permitted steamboat operators to build more
optimally sized steamboats and these were of at least 300 gross tons.
However, remaining impediments to navigation particularly on certain
stretches of the river prevented steamboat operators from replacing the

entire fleet with optimally sized vessels from the standpoint of

■ - - 15

OTniiiumi long-run average cost.

The 1850 steamboat data from the manuscript census, supplemented
by other evidence on freight rates and technological characteristics
defining steamboat capacity were used to estimated an envelope long-run
average cost curve of the form:

AVC = a + b-T + c-T2 + d-T-C + e-C + f"C2 [3]

as suggested by Johns ton1 , where AVC = average variable costs per one
thousand ton-miles, T = steamboat gross tonnage and C = steamboat
capacity in thousands of ton-miles. The resultant estimate of equation
[3j for the forty-six steamboats is shown in Table 5 and the envelope
of short-run average variable cost curves for steamboats with a capacity
of between five and forty million ton-miles is shown in Figure 1. This
envelope curves declines sharply at first but begins tc flatten out between
400 and 500 gross tons, reaching a minimum for vessels of about 675
gross tons. This result is entirely consistent with the data presented
in Table 4.

15

Table 5

COST FUNCTION ESTIMATE FOR FORTY-SIX WESTERN RIVER STEAMBOATS, 1850

(Standard Error)

Constant T T2 T-C C C2 R2

24.767 0.0658803 0.000494b -0.000021b -0.002581b 0.0000002b .57376
(0.037719) (0.000214) (0.000008) (0.000837) (0.0000001)

3 Significant at the 10% level (2-tailed test)
Significant at the 5% level (2-tailed test)

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17

Conclusions

Although our test results of the consistency and accuracy of the
survivor technique predictions vis a vis those from alternative methods
of estimating scale economies confirm William Shepherd's caveat that
the survivor estimates "need to be screened against alternative evidence..,
even where its numerical results are perfectly deal," our conclusion
is much less pessimistic about the value of this method. The survivor
technique does indicate the range of plant sizes operating under con-
ditions of constant returns to scale and minimum long-run average cost.
Out of twenty-five opportunities to test the survivor principle using
different methods to estimate the numeric significance of scale economies,
only two industries "failed" the test of consistency. Given our use of
the ninty-five percent confidence interval we would have expected at
least one rejection due to random error even if the hypothesis that the
different methods yield identical results is true. Further, in both
instances, there is reason to suspect that the production function
estimates may be in error due to unidentified shift-parameters and hence
it is quite possible that the survivor technique gives superior results.

Not only might the survivor technique be superior to other methods
in identifying constant returns to scale, but the method also enjoys
s:-~e unique advantages. Although the method i^ not sLeg^nt and involves
considerable elements of judgment, the data which it uses are precisely
those which the census authorities currently make available albeit at a
:=cre aggregated level than might be desired. The researcher is therefore
not constrained by disclosure laws or data deficiencies.

18

Finally, use of the survivor principle to indicate the minimum
efficient scale of plant as suggested by Leonard Weiss proved to be
superior in terms of accuracy in identifying constant returns to scale
than the use of this method to specify an optimal range of plant sizes.
This latter conclusion must be something of a disappointment to those
who would urge use of the survivor technique for the resolution of
public policy issues in the anti-trust field. It is apparently much
easier to say what is optimal or efficient than to define what is sub-
optimal or inefficient.

19

Footnotes

"See George J. Stigler, "The Economies of Scale," Journal of Law and
Economics, 1, 3 (October 1958), 54-71,

•?

"William A. Shepherd, What Does the Survivor Technique Show About Economies

of Scale," Southern Economics Journal, 34, 1, (July 1967), 113-122.
Says Shepherd, "the survivor technique cannot safely be used on its
own. Its estimates need to be screened against other evidence...
(T)he survivor technique may, under favorable conditions, yield
preliminary or supplementary ic -.icstions of certain ranges in industry
cost function. But as an estimator of scale economies its applicability
is limited, even where its numerical results are perfectly dear."

3

The samples, drawn from the 1850, 1860 and 1870 censuses, contain data

on ownership, invested capital, labor force, wages, quantity and value
of inputs and outputs, location and motive power for 17091 firms.
Tests of the samples' representativeness are given in Jeremy Atack,
"Estimation of Economies of Scale in Nineteenth Century United States
Manufacturing and the Form of the Production Function," (unpublished
doctoral dissertation) Indiana University, 1976, chapter 3.

See George J. Stigler, op. cit.; T. R. Saving, "Estimation of Optimum
Size of Plant by the Survivor Technique," Quarterly Journal of
Economics, 75, 4, (November 1961), 569-607; Leonard W. Weiss, "The
Survival Technique and the Extent of Suboptimal Capacity," The Journal
of Political Economy, 72, 2 (June 1964), 246-261; and William G.
Shepherd, op. cit.

20

See the results summarized in A. A. Walters, "Production and Cost
Functions," Econometrica, 31, 1-2, (January-April 1963); 1-66
especially 39-52.

See Leonard W. Weiss, op. cit.

Especially by William G. Shepherd, op. cit.

8Ibid., 116.

9

Given our selection of the ninty-five percent confidence interval and

the null hypothesis that the sample statistics and industrial
distribution of plants were identical to those of the parent population,
the number of "rejections" was no more than would be expected
through random error. Copies of the sample tests are available
upon request.

See, A. A. Walters, op. cit.

See Marc Nerlove, "Returns to Scale in Electricity Supply," in C. F.

Christ (ed.), Measurement in Economics, Stanford: Stanford University
Press (1963), 167-198; A. Zellner and N. S. Revankar , "Generalized
Production Functions," Review of Economic Studies, 36, 2, (April 1969),
241-250; D. Soskice, "A Modification of the CES Production Function
to Allow for Changing Returns to Scale over the Function," Review of
Economics and Statistics, 50, 4, (November 1968), 446-448; V.
Ringstad, "Some Empirical Evidence on Decreasing Scale Elasticity,"
Econometrica, 42, 1, (January 1974), 87-102.

21

12

Zellner and Revankar, op. cit.

13

Tae largest surviving plants in the following nine industries were

operating outside of the range of approximately constant returns to
scale: Woolens (SIC 2231), Men's Clothing (SIC 2321), Household
Furniture (SIC 2511), Boots and Shoes (SIC 3131), Blast Furnaces
(SIC 3312), Iron Foundaries (SIC 3321), Sheet Metal (SIC 3444),
Steam Engines (SIC 3511) and Agricultural Implements (SIC 3522).

14

Leonard W. Weiss, op. cit. , 247.

The structural evolution and operation of western river steamboats
is discussed at length in L. C. Hunter, Steamboats on the Western
Rivers, Cambridge Mass: Harvard University Press, 1949.

John Johnston, Statistical Cost Analysis, New York: McGraw-Hill,
1960, especially 71-73.

William G. Shepherd, op. cit. , 116.

jOUND^