Agricultural Labor in the American South, 1860-1870:
An Analysis of the Elasticity of Substitution and
Change in Functional Shares
THOMAS MILLARD ZEPP
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF TIE REQUIF:E'i-X'TS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
The author gratefully acknowledges the invaluable assistance
he received from Dr. Ralph H. Blodgett, Dr. Joseph M. Perry, Dr. Max
R. Langham, Dr. Milton Z. Kafoglis, and Dr. Gavin Wright. Their
knowledge, enthusiasm, and patience contributed materially to the
preparation of this study.
TABLE OF CONTENTS
ACKNOWLEDGMENTS . .
LIST OF TABLES . .
ABSTRACT . .
I INTRODUCTION AND THE SPECIFICATION OF THE PROBLEM
The Input and Institutional Matrix .
The Antebellum Period .
The Postwar Period .
The Theoretical Basis for the Analysis .
The Plan for the Analysis .
II ESTIMATED SUBSTITUTABILITY BEFIEEN LABOR AND
OTHER AGRICULTURAL INPUTS, 1850-1860:
THE MODEL . 13
The Assumptions of the Model .. 15
Constant Returns to Scale .. 15
Constant Elasticity of Substitution 16
Derivation of the Theoretical Model .. 18
III ESTIMATED SUBSTITUTABILITY BETWEEN LABOR AND
OTHER AGRICULTURAL INPUTS, 1850-1860:
THE ESTIMATE. . 25
The Implicit Slave-Hand Wage: "x" 25
The Value-Added per Slave-Hand: "y" .. 28
The Statistical Estimate of the ES ... 36
Preliminary Inferences from the Estimates .... 39
TABLE OF CONTENTS (Continued)
IV AN APPROXIMATION OF THE CHANGE IN THE LABOR
SHARE OF INCOME IN SOUTHERN AGRICULTURE, 1860-1870 46
Background . 47
The Proposed Resolution .. 51
The Estimated Change in Relative Shares .. 52
The Percentage Change in w .. 54
The Percentage Change in r 55
The Estimate .. 58
The Southern Agricultural Labor Share in 1860 59
The Southern Agricultural Labor Share in 1870 62
The Primary Estimate .. 62
The Secondary Estimate .. 63
Comparisons and Conclusions 65
V SUMMARY AND CONCLUSIONS .. 74
Summary of the Analysis ... 75
Conclusions . 78
Implications of the Analysis .. 78
APPENDIX A: THE ELASTICITY OF SUBSTITUTION .. 83
The Concept . .. 84
The Substitution Curve .. 86
The Economic Implications of the ES .. 90
Four Production Functions and Their Implied ES 92
The Cobb-Douglas Function .. 92
The CES Function .. 93
The Transcendental or VES Function .. 94
The Homothetic Production Function .. 95
TABLE OF COIt;NT.TS (Continued)
APPENDIX B: AN ALGEBRAIC AND GEOMETRIC COMPARISON OF
INTRA-MARGINAL SLAVE RENTS AND FREE
FARM LABORERS' WAGES ... 100
A Simple Model of Intra-Marginal Slave Rents .... 102
Free Farm Workers' Wages Related to the Level of
Intra-Marginal Slave Rents ... 108
SELECTED BIBLIOGRAPHY . ... 111
BIOGRAPHICAL SKETCH .... ... 117
LIST OF TABLES
3-1 Production and Wage Data for 1850 & 1860 .. 29
3-2 Ratio of Non-cotton-value-added to Cotton-value-
added on Average Farms in Selected Southern
States and Sub-regions, 1850 & 1860 .... 34
3-3 Summary of Regression Results .. 37
3-4 Variant II Estimated Rates of Return in Percent
for 9.5 Cent Cotton and No Breeding Returns,
1850 & 1860 .. 41
4-1 Average Monthly Farm Earnings with Board,
Relative Size of the Labor Force, and
Weighted Change in State and Regional Earnings
for Seven Southern States, 1860 & 1870 55
4-2 Estimated Total Wages, Income, Labor Share of
Income, and Labor Force in the Agricultural
Sector of Seven Southern States, 1860 & 1870 66
Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy
AGRICULTURAL LABOR IN THE AMERICAN SOUTH, 1860-1870:
AN ANALYSIS OF THE ELASTICITY OF SUBSTITUTION AND
CHANGE IN FUNCTIONAL SHARES
Thomas Millard Zepp
Chairman: Ralph H, Blodgett
Co-Chairman: Joseph M. Perry
Major Department: Economics
This study considered two inter-related problems concerning the
agricultural labor force in the antebellum and post-Civil War American
South. The primary problem was the change in the functional share of
income which could be attributed to southern agricultural labor in 1860
and 1870. Two alternative ways of approaching the problem were
examined. One procedure required data for the size of the 1870
southern agricultural labor force. Since the available labor force
data were known to be incomplete for 1870, this procedure would have
required a somewhat arbitrary assumption about the number of omitted
farm laborers. In an attempt to be less arbitrary, a second procedure
was adopted. Choice of this procedure required consideration of the
second problem. It required a determination of the elasticity of
substitution between farm labor and other agricultural inputs in seven
cotton states of the 1860 South.
The elasticity of substitution was estimated as a constant elas-
ticity of substitution by regressing the natural logarithm of value-
added per slave-hand (and free farm labor in slave-hand equivalent
units) on the natural logarithm of the implicit slave-hand wage.
Value-added per slave-hand was synthesized from census data on various
southern crops and population and from Foust and Swan's estimates of
bales of cotton per slave. The implicit slave-hand wage was shown to
be related, from the industry point of view, to free farm labor earnings
by an exposition of intra-marginal slave rents. Lebergott's free farm
labor earnings plus board were then taken as surrogates for implicit
labor (slave and free) wages. The tentative estimate of the elasticity
of substitution indicated that slaves and free farm labor were
relatively good substitutes for other agricultural inputs in 1860.
The point estimate of the elasticity of substitution was greater than
The change in the functional share of income which was received
or imputed to farm labor in 1860 and 1870 was calculated by using an
estimate of the 1860 ratio of functional shares (labor income to the
residual farm income), and the (average or arc) percentage change in
the ratio of shares. The percentage change in the ratio of shares was
found from the percentage change in the level of implicit wages, the
percentage change in the rate of return on other agricultural inputs,
the constant elasticity of substitution estimate for 1860, and an
assumption that the elasticity of substitution in 1860 was the same
as the elasticity of substitution in 1870. It was found that the
functional share of income received by agricultural labor in 1870 had
increased relative to the share imputed to other agricultural inputs,
but that the absolute share had fallen from the dollar amount received
and imputed to farm labor in 1860. Since the prices of agricultural
products were higher in 1870 than in 1860, and since the labor force
was larger in the seven states considered, this result indicated that
the Civil War had dealt a severe blow to the non-labor inputs employed
in agriculture, a blow that was still very much in evidence five
years after the end of the conflict.
INTRODUCTION AND THE SPECIFICATION
OF THE PROBLEM
The full effect of the Civil War on the economy of the American
South is still a matter of debate. Even after one hundred years of
examination by historians and economic historians, the impact is not
completely understood. Historians have been concerned primarily with
the changes brought about by the War in social and political institu-
tions. Few economic inferences can be made from these types of detailed
studies. Economic historians have asked questions which are much
more relevant to the economic re-adjustment of the postwar South. They
have studies long trends, examined the Civil War's effect on industrial-
ization, and have devoted entire volumes to the task of examining
different economic problems which occurred during and after the
hostilities.1 But with few exceptions, economic historians have avoided
explicit accounts of southern re-adjustment. They have focused their
narratives on output changes which occurred in the "national" economy.2
To assess fully the effects the Civil War had on the rate of recovery in
the South, additional information is needed. Changes in incomes received
by productive inputs as well as changes in the composition and level of
output must be included in the analysis.3 As Robert Gallman pointed
out in 1964, "the Civil War may have promoted [or retarded] growth
S. through changes in the distribution of income channeling
income into [or out of] the hands of those who would be prepared to
invest ." Studies of changes in the distribution of income are
conspicuously absent from the literature on southern recovery. This
gap must be eliminated before the postwar adjustment can be fully
This study examines two inter-related problems which concern
change in the distribution of income in the agricultural South.
The primary goal is to provide an analysis of the change in the
functional share of income which may be attributed to agricultural labor
in the South before and after the Civil War. An approximation of the
elasticity of substitution between farm labor and other agricultural
inputs is the secondary consideration. An estimate of this elasticity
will not only indicate the relative substitutability between agricul-
tural inputs but also will supplement the existing data and will enable
a more precise estimate of the change in the share of income which was
imputed or paid to agricultural labor. Though these two considera-
tions will not explain the adjustments in the southern economy after
the War, they will provide a sound basis for further inference about
The Input and Institutional Matrix
The array of institutional and input relationships in the South
had an effect upon the size of the share of income received by agricul-
tural labor. The Civil War brought about a dramatic change in this
The Antebellum Period. Slavery was the common denominator of all
economic activity in the antebellum South. As William N.-Parker notes,
surrounding all studies of the American antebellum
South, the subject of slavery lies like a great
fetid swamp from which historians emerge like alligators
to snap at one another. Though it may not be the
explicit .. it is the implicit subject of them all,
and its presence is felt in every corner.5
But the domination of agricultural production by the institution of
slavery does not mean that such production may be explained solely in
terms of slaves. The availability of other productive inputs and free
labor played just as important a role as the slaves. Slaves and free
labor were both labor inputs in southern agriculture; from an industry
point of view, most farmers and planters, some members of their families,
overseers, hired farm laborers, and slaves were all members of the
antebellum agricultural labor force. Though slaves and free (hired)
laborers were not customarily used on the same farm, they were both a
part of the labor force which was used to produce staple crops. From
an industry point of view, all labor, be it slave or free, was a part
of the same labor pool. The manner in which this labor was combined
with other inputs was a critical problem.
The labor force was the effective constraint upon industry output
throughout the antebellum period. No other agricultural input was so
inelastically supplied. There were still large tracts of land
available in Texas and the settled parts of the South, even as late as
1860. Given the necessary labor to clear the land, more acreage could
have been brought into production with no appreciable effect upon land
price. Building materials were readily available for new construction.
Only for lack of labor would the price of buildings have increased.
The reproduction of farm implements was also limited solely by labor.
In the production of most southern crops, the implements which were
used were simple tools, requiring a minimum of skill and materials.
Labor placed the limit on antebellum agricultural production in the
South. The basic materials for all other factors of production were
in elastic supply over any realistic range of output. This relation-
ship between productive inputs was to change dramatically after the
destruction wrought by the Civil War.
The Postwar Period. After Appomattox, the South faced the severe
economic consequences of a war fought on its own soil. More than
the "peculiar institution" had been destroyed. The ravages of war had
left buildings and farms in shambles, the railroads in a critical state
of disrepair, and few of the antebellum financial institutions in a
position to provide credit for a recovery. Each of these damaged
parts of the southern economy affected the recovery in agriculture by
a different magnitude.
It appears that the damage received by the railroads was the least
important hindrance to long-run recovery in southern agriculture. Even
though the roads had suffered severe physical and financial damage from
the hostilities, they were back to their antebellum levels of operation
as early as 1867.7 Some farmers were hurt during the readjustment of
the railroads as rails were taken from less traveled roads to replace
twisted and damaged rails on the main lines. But, from an industry point
of view, after the initial railroad readjustment, net differences in--
output can not be attributed to a poorer system of internal transportation.
Recovery in the financial sector did not occur so quickly. The
banking system had first seen its specie reserves dwindle, then
disappear, as more and more of the universally acceptable medium of
exchange found its way into the hands of those who supplied the South
with war provisions. What little specie did remain after the War was,
in many cases, confiscated by the Union. At the end of the struggle,
the South was left with little of the necessary backing for credit.
Only cotton remained, and what was not hidden was either confiscated
or severely taxed by the victor. On top of these problems, the South
did not have the means to generate or to obtain the desired money
During the War the Congress had passed the National Banking Act
and several associated laws which did not take into account the needs
of a region which was primarily agricultural. A tax had been placed on
notes issued by local state banks. Though the tax was initially avoided,
eventually its imposition was enforced and the local bank notes were
driven out of circulation.8 A circulating medium was conspicuously absent
except in those areas where large numbers of troops had been stationed.
In many areas, the local communities had to resort to a type of scrip,
or even barter, to carry on normal economic transactions.9 The new law
did not prohibit National Banks from being established in the South, but
a provision in the charters of such banks which prohibited loans
secured by real estate made a charter attractive to individuals in only
a few of the larger commercial centers of the region. The National Bank
was hardly the answer for the rural South.10
These financial problems unavoidably retarded the recovery of
southern agriculture. A large number of farms had been burned or
damaged during the hostilities. Those which had not felt the weight
of war directly had suffered as they had fallen into disrepair for
lack of materials and manpower.11 Without effective financial
institutions, the war-damaged and time-worn farm capital could not be
replaced very quickly. In addition, the lack of short-term capital --
or availability of short-term capital at only very high rates -- was
one of the primary reasons that the crop-lien and share-cropping
relationships were established. Credit between farmers who owned land
and tenants who farmed it replaced cash as a medium of exchange.
Productivity of the postbellum labor force in agriculture was thus
a net result of many interrelated effects of the war. Many of those
laborers who would have been effective postbellum farmers were either
dead or seriously maimed as a result of the hostilities.12 Those who
were alive were relatively less productive since there was a smaller
amount of farm capital to be combined with each. Crumbling buildings,
rusting farm implements and stolen or dead plow horses could not be
replaced with non-existent financing. Some land had been taken out of
production as the railroads had re-located their rails. Other land
had never been brought back into production; weeds and rubble had
replaced the stately stalks of corn and bolls of cotton. Labor relative
to the other agricultural inputs had increased drastically from its
The Theoretical Basis for the Analysis
John R. Hicks has shown that the change in the distribution of
income which occurs as various factors of production are used in
different proportions may be treated as a question of the change in
the distribution of income as only one input varies while all others
are held constant.13 Holding these other inputs at a constant level,
the change in the share of income received by.the variable input may
be established by reference to the following three rules or propositions:14
1. An increase in the supply of any factor of
production will increase the absolute share .
accruing to that factor if the elasticity of demand
for that factor is greater than unity.
2. An increase in the supply of any factor will
always increase the absolute share of all other
factors taken together.
3. An increase in the supply of any factor will
increase its relative share .. if its "elasticity
of substitution" is greater than unity.
The analysis of shares in this study will be based upon these
three fundamental propositions. The first and second propositions, while
they do provide some indication of the change in the.absolute share
received by southern agricultural labor, are not conclusive. There is
a chance that the effects would have been offsetting, or at least that
they moved in opposite directions. Cetetis paribus, if the elasticity
of demand for labor were greater than unity, the share received by labor
would have increased as the size of the labor force used in agriculture
increased. But the amount of the other productive inputs was not
constant; it was decreasing. By the second proposition (when output
price was constant), the share received by labor would have fallen.
Without further information, it is not possible to establish even the
direction of change -- much less the magnitude of change -- from these
two propositions. Thus the key to the analysis is the third proposition.
If the elasticity of substitution (hereafter, ES) between agricultural
labor and other agricultural inputs were known, and if either the
change in the input intensity ratio or the change in the ratio of input
marginal physical products (under competitive equilibrium, the ratio of
input prices) were established, it would be possible to determine the
change in labor's share relative to the share received by other agricul-
tural inputs.15 The direction and the magnitude of the change in the
ratio of relative shares may be established. In addition, if the
absolute share for either year could be established, the absolute
change in shares could be determined.
In a more general situation, it would be simpler to estimate the
difference in shares of income directly from the shares in each year.
The share for each year could be found by proper multiplication of the
number of laborers, their average wage level, and total income for the
sector. But in the case of the southern agricultural sector, the data
available for an estimate of the postwar labor share are not reliable.
The most comprehensive data on the size of the southern labor force,
those reported in the 1870 Census, are subject to an indeterminate
undercount.16 To estimate the change in the share of income received
by labor with this source of data, a rather arbitrary assumption about
those who had not been included in the census enumeration of occupations
would be needed.
Given the data available and the industry in question, it seems
more reasonable to make the less arbitrary assumption that the ES between
productive inputs was the same in 1870 as it was in 1860. Given this
assumption and an estimate of the ES for 1860, the absolute share of
income received by labor in 1870 may still be considered. The estimated
ES may be used to establish the change in relative income shares. This
change in relative shares together with an estimate of the absolute share
of labor for 1860 is sufficient to establish the absolute share going to
labor in 1870. The change in shares is then found by subtraction.
The Plan for the Analysis
A consideration of the theoretical relations between shares and
the extant data for the period in question has shown that two questions,
not one, need to be answered. An answer to the primary question of the
change in the labor share of income will provide an indication of the
change in the distribution of income in southern agriculture. But to
provide a reasonable estimate of this change in labor's share, a second
question about the elasticity of substitution between farm labor and
other agricultural inputs in the 1860 South must be answered initially.
The answer to this second question will permit estimation of the
change in relative shares which will provide, in turn, the basis for
the estimate of the change in the share received by labor.
Chapters II and III will be concerned with the establishment of
the elasticity of substitution. Chapter II will present the model which
was used to estimate the ES and a discussion of the applicability of
the model's underlying assumptions to the period and industry considered.
Chapter III will present the technique which was used to estimate the
ES. The data which were used for this estimate will be discussed at
length. Appendices A and B will supplement the crucial analytical
points of the two chapters with theoretical and mathematical extensions
of the text.
Chapter IV will present the various steps which were necessary for
an estimation of the change in the share received by agricultural
labor. It will present the change"in relative shares which was estimated
from the 1860 ES established in Chapter III and the change in the ratio
of input prices. The estimated absolute shares for 1860 and 1870 will
then be presented. At the end of the chapter, the dollar estimates of
shares for each year and the percentage of agricultural income that
the dollar shares represent will be summarized.
In the final chapter, the analysis will be summarized and some
implications of the study will be placed in their proper context with
regard to generally accepted conclusions about the period.
See Trends in the American Economy in the Nineteenth Century, National
Bureau of Economic Research, Studies in Income and Wealth, XXIV
(Princeton: Princeton University Press, 1960), passim; Output,
Employment, and Productivity in the United States after 1800, National
Bureau of Economic Research, Studies in Income and Wealth, XXX
(New York: Columbia University Press, 1966), passim; Ralph Andreano,
Ed., The Economic Impact of the American Civil War (Cambridge:
Schenkman Publishing Co., 1962), passim; Thomas C. Cochran, "Did
the Civil War Retard Industrialization?" Issues in American Economic
History, Ed. Gerald D. Nash (Boston: D. C. Heath & Co., 1964) pp.
287-294; Stephen Salsbury, "The Effect of the Civil War on American
Industrial Development," ibid., pp. 295-301; and David T. Gilchrist
and W. David Lewis, Eds., Economic Change in the Civil War Era (Green-
ville, Delaware: Eleutherian Mills-Hagley Foundation, 1965), passim,
for examples of studies which are indirectly related to the problem
of recovery in the South.
2 "National" is set off to emphasize the fact that the United States
was not really one economy at this point in time, but was a set of
partially inter-related regional economies. This distinction is
discussed further in Chapter IV.
3In the March, 1971, issue of the Journal of Economic History, James
Baughman pointed out that economic historians are changing their
emphasis from "productivity studies" to studies of the distribution
of income. One fundamental reason for this shift in emphasis is
that interesting questions -- such as the recovery in the South --
cannot be answered without knowledge of the input and output
markets. "A Note from the Convenors, Summaries of Doctoral
Dissertations," Journal of Economic History, XXXI, No. l(March, 1971),
4 Robert E. Gallman, "Discussion," Economic Change in the Civil War
Era, Eds. David T. Gilchrist and W. David Lewis (Greenville,
Delaware: Eleutherian Mills-IIagley Foundation, 1965), p. 169.
5 William N. Parker, "Introduction: The Cotton Economy of the Ante-
bellum South," Agricultural History, XLIV, No. l(January, 1970), 1.
6 The economic relationship between these two institutionally different
types of labor is treated at length in Appendix B.
7 John F. Stover, The Railroads of the South 1865-1900 (Chapel Hill:
The University of North Carolina Press, 1955), pp. 3, 58, 59; and
Reports of Committees, 2nd Session, 39th Congress, Affairs of Southern
Railroads, IV (Washington: Government Printing Press, 1867), passim.
This second source provides a great deal of background on the financial
problems facing railroads in the postwar period.
8 Theodore Saloutos, "Southern Agriculture and the Problems of
Readjustment, 1865-1877" Agricultural History, XXX, No. 2
(April, 1956), p. 63.
10 Ibid., p. 64.
11 See Emory Q. Hawk, Economic History of the South (New York:
Prentice-Hall, Inc., 1934), pp. 425-442, for one of the better
historical accounts of the aftermath.
12 The fall in the effective labor force was more a long-run than a
short-run consequence. See Allen C. Kelly, "Demographic Cycles
and Economic Growth: The Long Swing Reconsidered," Journal of
Economic History, XXIX, No. 4 (December, 1969), pp. 653, 654.
13 John R. Hicks, The Theory of Wages (2d Edition, New York: St.
Martin's Press, 1963), p. 115.
14 Ibid.,pp. 115-120.
15 See Appendix A.
16 U. S. Department of the Interior, Census Office, Statistics of
Agriculture (June 1, 1870) (Washington: Government Printing
Office, 1872), p. 72, and U. S. Department of the Interior, Census
Office, The Statistics of the Population of the United States, Ninth
Census, Vol. I (Washington: Government Printing Office, 1872),
p. 660. The latter discusses the undercount in general; the former
discusses the difficulties in collecting data for the South.
ESTIMATED SUBSTITUTABILITY BETWEEN LABOR AND OTHER
AGRICULTURAL INPUTS, 1850 1860: THE MODEL
In several recent studies of the profitability and viability of
slavery, the techniques which have been used to estimate "profits"
have precluded considerations of substitutability between slaves and
other productive inputs.1
In all of these studies, the profitability of slavery in the
production of cotton is based upon the comparison of an estimated
internal rate-of-return with an appropriate interest rate. The
internal rates-of-return are calculated in several different ways, but
all of the estimates appeal to some modification of the basic present
value of capital formula.2 The latest and most refined estimate
of the internal rate-of-return is the calculation made by Foust and
Swan (hereafter F&S). They estimate the internal rate-of-return by
dividing the total capital cost of a slave into the value of output per
slave (adjusted for average maintenance cost). Capital cost includes
not only the price of a slave, but also the value of land, farm
equipment, and animal stock on a per-slave basis.3
In general, the described procedure is only valid for one point
on an isoquant, the point established by a purely competitive industry
in equilibrium. It is only at that point that input marginal revenue
products equal their respective input prices; an additional dollar's
worth of any one input would provide the same increment to the total
output, as would an additional dollar's worth of any other input. Total
cost is equal to the sum of the partial costs and does exhaust total
revenue. But, if equilibrium is not assumed or can not be assumed,
lumping all inputs into one composite input is, in general, only
consistent with inputs which are perfect substitutes or perfect
complements. Perfect substitutability is obviously not what F&S had
meant to imply. If they had, they certainly would not have assumed
that labor (slaves) and farm-capital were complements in another
section of their study.4
As these profitability-of-slavery studies are formulated, the
estimated internal rate-of-return is consistent with a whole family
of isoquants. Since purely competitive equilibrium is implicitly
assumed, any isoquant with a slope equal to the ratio of input prices
might be the underlying relation between productive inputs. More
information is needed to determine the shape of the isoquant and the
substitutability between the inputs. This additional information can
be provided by the elasticity of substitution.
This chapter presents the model which will be used in Chapter III
to estimate the elasticity of substitution (hereafter ES). It is the
preliminary model outlined by Arrow, Chenery, Minhaus, and Solow
(hereafter ACMS) in their classic article on the constant elasticity of
substitution (CES) production function.5 Its derivation and applicability
to the agricultural sector of the mid-Nineteenth Century South is
The Assumptions of the Model
The ACMS model is suitable only if the necessary assumptions of
the model are appropriate for the period and sector being considered.
The two crucial assumptions are constant returns to scale and a constant
elasticity of substitution. How well each of these assumptions fits
the situation in the slave South is considered in turn.
Constant Returns to Scale. A chapter from Gavin Wright's
forthcoming book on antebellum southern agriculture is the primary
evidence for assuming that the returns to scale in slave agriculture
were constant. Using data from the University of North Carolina Yale
manuscript census sample for 1860, Wright determines the input to
input-productivity relationships for several agricultural inputs by
estimating a series of "reduced-form-like" equations (leaving the
structural equations unspecified). His estimates, though only tentative
at this time, indicate that if there were non-linear returns to scale,
economies existed for only the smallest of firms and diseconomies were
present in only the very largest.6
An inference favoring the assumption of constant returns to scale
may also be made from the historical observations of Gray and Russell.7
Both historians imply that the larger plantation had a competitive
advantage (economies of scale) in the production of rice and sugar, but
the growing of cotton and tobacco however -- and
these staples employed about eight slaves to every
one in sugar and rice -- it is very doubtful that
the plantation was superior to the small farm as a
unit of agricultural production.. The planter
was able to effect a division of labor among his
hands that was not possible on a small farm but
the operations and the machinery required in
farming in those days were too simple to permit
any considerable advantage to be gained from
that. In fact the division of labor on a large
plantation tended to become fixed. .
and this inflexibility may have led to certain comparative inef-
ficiencies since the hands on the smaller farms would be used where
they were needed during the peak periods in the agricultural cycle,
whereas the hands on the larger plantations would continue to perform
the same tasks. On the really large plantations, for example, it
would be completely unacceptable to use a coachman in the fields,
even though the plantation owner needed every worker he could get.8
The final evidence of constant returns must await the empirical
results of the next chapter.
Constant Elasticity of Substitution. The particular constant
elasticity of substitution (CES) function used in this study was
chosen after consideration of the relative predictive power of the
class of CES functions and the data available for the historical
If constant returns to scale is an acceptable assumption, and
if an estimate of the elasticity of substitution (ES) is desired, only
two classes of production functions, the variable elasticity of sub-
stitution (VES) and the CES, need to be considered.9 This consideration
was based primarily upon a paper presented by C. A. Knox Lovell at the
1970 Southern Economic Association meetings. Lovell used aggregate
U. S. manufacturing data for 1947-1963 and estimated and compared a
CES function with three different VES functions to determine their
consistency and predictive accuracy.10 He established that the
significant variation of [the capital-labor ES]
found in the three VES functions does not appear
to enhance their performance vis-a-vis the CES
function in tests of predictive accuracy. Nor
does it imply any resulting inconsistency on the
part of the CES function with the data used in its
estimation. It thus appears that the variability
of the elasticity of substitution brings no
discernable increase in performance for the data
we use and for the tests we conduct.11
Moreover, he concluded that the "more restrictive CES function out-
performs the VES functions as a group and individually, in both
consistency and prediction tests. While this edge is not large, our
preference for the CES function is well documented."12
Since cross-sectional data from a previous century and for a
different industry are used in this analysis, Lovell's conclusions may
not apply. But lacking evidence to the contrary, it seems more
reasonable to assume that his results apply to this period and sector
than to assume they do not. Since the predictive power of a function
is important to a part of this analysis, the class of CES functions
The particular form of the CES function used in this study was
chosen after considering the limited data on productive inputs. Of those
data that are available, the estimates of slave productivity and cost
have received the greatest attention. The problems of obtaining good
estimates of farm-capital (e.g., values of farms, farm implements,
cleared and uncleared land, animal stocks) are not much different from
the problems associated with slave estimates, but they have not been
the subject of debate and scrutiny. In many cases no estimate exists
and in others only an incomplete estimate is available. Since these
farm capital estimates are relatively inferior, greater confidence
should be placed in an estimate based solely upon the slave data. The
formulation of the CES function which is used, therefore, permits an
estimation of the ES which is based upon these data.
Derivation of the Theoretical Model
The model used to estimate the elasticity of substitution, the
(2-1) In(y) = A + B ln(x)
where y is value-added per slave-hand, x is the implicit slave wage, A
is a constant and B is the estimated constant elasticity of substitu-
tion (CES) between labor (slave-hands and free labor in slave-hand
equivalent units) and farm-capital. It is based upon the linearly
homogeneous production function
(2-2) V = F(T,H)
where V is value-added (annual), H is slave-hands and free farm workers
engaged in the production of industry value-added, and T is other
productive inputs, here called farm-capital.
When F is linearly homogeneous, (2-2) may also be written as
(2-2a) V = H[F(T/H, 1)] ;
(2-2b) V/H = F(T/H, 1) .
Defining y as V/H and k as T/H, (2-2b) may be written more simply as
y = f(k) .
The marginal products of the inputs may be written in terms of
y and k from the following:
aV 3(Hy) H[f(k)] HL[f'(k)]
(2-3) aT ~ T T = H
= f(k) = marginal product of T = ay
(2-4) V fH[f(k)] = fT(k) T
H a f (k)
= f(k) k[f'(k)] = marginal product of H.
Under the assumption of pure competition, the wage of H (call it
x) is equal to the marginal product of H when output is the numeraire,
(2-4a) x = f(k) k[f'(k)]
(2-4a) can be inverted to give a functional relationship between
k and x because the relationship is a one-to-one mapping. y = f(k)
is also a one-to-one mapping as y is a monotonically increasing function
of k. Considering y, then, as some function of x, call it y = ((x), it
is possible to establish
(2-5) y = (y k)
which is a differential equation with solution y = f(k, A), where A is
a constant of integration. It is then possible to reconsider equation
V = H [f(T/H; A)] ,
which to be representative of a reasonable production function must
have f'(k) > 0, and f"(k) < 0 for some value of the parameter A.
The elasticity of substitution between T and H is defined as
the elasticity of the input intensity ratio (T/H) with respect to the
marginal rate of technical substitution, 8V/3H In terms of k and
y, this elasticity becomes
(2-6) ES = [f'(k)] [f(k) kf'(k)]
k [f(k)] [f"(k)]
Reconsidering the relationships which have been established for
y and for x, it is possible to show that under constant returns to scale
that the elasticity of substitution may be expressed in terms of
productivity of H and the price of H, i.e., x. Consider the following
y = f(k)
x = f(k) kf'(k)
which come from (2-2c) and (2-4a). Differentiating (2-4a) with respect
to x, it is found that
(2-7) 1 dk dyL k[f"(k)] dy [f'(k)]. d
dy dx dy dx dy dx
and noting that dk/dy = l/f'(k) because the function can be inverted
and, further, that
dx/dy = f'(k)(y) k[f"l(k)]( ) f'(k)d)
= k[f"(k)][l/f'(k)] ;
it is possible to show that dy/dx = f'(k)/(k[f"(k)]).
(2-8 Ld l- f(k) k[f' (k)L Ff(k) 1
(2-8) dx/x = x F(k) (1)j [f" (k)
-= [f'(k)][f(k) k[f'(k)]] = ES
That is, the elasticity of the value-added per slave-hand with respect
to the implicit wage is, under competitive conditions and constant
returns to scale, equal to the elasticity of substitution between labor
From (2-8) it is possible to consider the particular formula used
in this study. (2-8) specifies that
(2-9) dx/x = B
where B denotes the CES. (2-9) may be re-written as a homogeneous
differential equation with variables separable,
B = 0 ,
which has integral curves given by
(2-10) Injly = B[lnlxl] + A
where A is a constant. The derivation is complete; regressing ln(y)
on In(x) does provide an estimate of the constant elasticity of
substitution. This estimate is B.
This chapter has presented the model which will be used to
estimate the elasticity of substitution in Chapter III. The
assumptions for the model were considered and found to be appropriate
for the available data and the agricultural sector of the southern
economy in 1850-1860. A derivation of the model was also presented.
It was shown that the elasticity of value-added per slave-hand with
respect to the implicit slave-hand wage is equal to the elasticity of
substitution between farm labor and other agricultural inputs (farm-
1Examples of such studies are Alfred H. Conrad and John R. Meyer,
"The Economics of Slavery in the Antebellum South," Journal of
Political Economy, LXVI, No. 2(April, 1958), pp. 95-130, which
is reprinted in Alfred -1. Conrad and John R. Meyer, The Economics
of Slavery and Other Studies in Econometric History (Chicago:
Aldine Publishing Co., 1964), Chapter II; Edward Saraydar, "A
Note on the Profitability of Antebellum Slavery," Southern Economic
Journal, XXXI,No. 4(April, 1964), pp. 325-332; Richard Sutch,
"The Profitability of Antebellum Slavery -- Revisited," Southern
Economic Journal, XXXI, No. 4(April, 1965), pp. 365-383; and
James D. Foust and Dale E. Swan, "Productivity and Profitability
of Antebellum Slave Labor: A Micro-Approach," Agricultural
History, XLIV, No. l(January, 1970), pp. 39-62.
2 Conrad and Meyer, 1964, p. 48.
3 Foust and Swan, p. 49. The exact equation they used was
r = a + K
where p = price of cotton at farm; a = rate of increase in slaves;
Y = output/slave; C = slave maintenance cost; K = total capital
cost per slave.
4 When F&S assume that the decennial rate of growth of value-per-acre
of land, buildings and equipment approximated the rate of growth of
non-slave investments-per-slave, they imply that inputs were
combined in fixed proportions. Foust and Swan, p. 52.
5 "Capital-Labor Substitution and Economic Efficiency," Review of
Economics and Statistics, XLIII, No. 3(August, 1961). It is in
this article that ACMS formally presented the CES function.
6 Letter from Gavin Wright, January 19, 1971. The tentative chapter
for his book was enclosed.
7 Lewis Cecil Gray, "Competitive Advantages of Negro Slavery Under
the Plantation System, an excerpt from History of Agriculture
in the Southern United States to 1860," Slavery and the Southern
Economy Sources and Readings, Ed. Harold D. Woodman (New York:
Harcourt, Brace & World, Inc., 1966), p. 33. And Robert R.
Russel, "Opportunities Available for the Ambitious and Efficient
Farmer, [an excerpt from] 'The Effects of Slavery Upon Nonslave-
Holders in the Antebellum South'," ibid., pp. 118-119.
8Russel, p. 119. But even if there were no productivity advantages
for the larger plantations, there were certainly advantages when
financing was needed. The larger plantation owner could finance
physical improvements and the purchase of land at a lower cost
than could his middle-size neighbor; moreover, the very small
farmers had extreme difficulty obtaining financing at any price.
When slaves are considered from an industry point of view, the
input prices should reflect the capitalized values in pure
competition. If some element of monopoly power was present, the
assumption about the industry structure should be modified, not
the assumption about the physical productivity relationship.
9 See Appendix A for the mathematical specification of these
functions as well as the Cobb-Douglas and more general homothetic
function. The Cobb-Douglas function was not considered here
because it pre-specifies a unitary ES.
10 C. A. Knox Lovell, "Estimation and Prediction with CES and VES
Production Functions" (paper presented at the Southern Economic
Association M1eetings, November, 1970).
11 Ibid., p. 33.
13 In Chapter IV of this analysis, the 1860 estimated-ES will be
combined with an estimate of the ratio of shares in 1860 to
"predict" the 1870 labor share of income.
14 Arrow, et al., pp. 227-229.
ESTIMATED SUBSTITUTABILITY BETWEEN LABOR AND
OTHER AGRICULTURAL INPUTS, 1850-1860: THE ESTIMATE
The model developed in Chapter II will be used in this chapter
to estimate the elasticity of substitution (ES) between slaves and
farm-capital. To make this estimate, data must be found for value-
added per slave-hand (y) and the implicit slave-hand wage (x). These
data are not readily available. The extant data must therefore be
partially synthesized to obtain the needed variables. Whenever
possible, these source data are combined in accordance with assumptions
which have already borne the test of academic criticism. Where new
assumptions are required, they will be discussed at length. After the
data are partially synthesized, the resulting size of the sample used
for the estimate of the ES will be small. The small sample precludes
placing too much confidence in the point estimate of the ES, but still
permits a test of the fixed-factor proportionality assumption which has
been made in some of the studies of the profitability of slavery.1
The Implicit Slave-Hand Wage: "x"2
One way in which the implicit slave-hand wage could be established
is to treat the slave as a capital investment and transform slave-hand
prices into implicit wages (the properly discounted cost-of-capital).3
This alternative was considered and rejected because of the large
number of assumptions which had to be made for unknown relationships
between slave-hand wages and slave-hand prices. At a minimum, the
price of slave-hands of equal productivity in the different regions,
the average slave owner's expected change in the price of a slave-
hand, the average slave owner's expected change in the stock of his
slaves as they reproduced, the average slave owner's expectation of how
many years the slave could work, the productivity differences for
different ages and sexes, maintenance cost, and the stream of returns
on his investment would need to be "known" before the slave-hand
price could be transformed into an implicit slave-hand wage. Even if
all of these average slave owner's expectations were equal to what
materialized, the compound effect of all of the different assumptions
which must be made would make it difficult to determine whether the
estimated slave-hand wage is more a result of the assumptions or of
In lieu of such a procedure, the estimated values of "x" were
established deductively. Economic theory shows that a rational firm
using slaves would not continue to use the marginal input if the
implicit price of a slave-hand, i.e., out-of-pocket maintenance cost,
subsistence cost (output which could have been sold), and capital
costs associated with the slave-hand price exceeded the value marginal
product of the slave. In like manner, the rational firm, finding that
the value marginal product of a slave-hand is greater than the implicit
cost of an additional hand, would bid against its neighbors until
slave-hand prices had increased. After the adjustment process is
completed, the capital costs will have changed to reflect the point
where value marginal product is just equal to the implicit wage of
slave-hands in the surviving firms; the interactions of all firms in
the purely competitive industry would have "forced" each firm to
pay (explicitly or implicitly) value marginal products. Thus, if a
firm had survived to become one of the many firms in purely competitive
equilibrium, it could not have paid less or could not have afforded to
pay more than the value marginal products of its inputs.
Slaves as economic factors were fixed to the industry and variable
to the firm.4 If the slave's value marginal product had exceeded his
maintenance cost, an intra-marginal factor payment would have been
generated in a competitive market which would have equated the marginal
value product of the slave-hand to the industry opportunity (or
transfer) cost. If one assumes, as is assumed in this study, that
the marginal product of a free (average) agricultural worker was the
same as the marginal product of an (average) male field hand, the
industry.opportunity cost is the wage which would have had to have
been paid to the marginal free worker.5 If it can also be shown that
free agricultural workers did compete as inputs in the "slave"
industries, then the wage paid these free workers should have been
equal to the capitalized slave wage.
It is possible to establish that the free workers did compete in
at least one slave industry; Foust and Swan have shown that the ratio
of slaves to free workers in three cotton-producing regions were
50:1, 20:1 and 17:1 in 1850 and 51:1, 21:1 and 28:1 in 1860.6 The
relative proportion of free workers indicated by these ratios is
certainly large enough to sustain the hypothesis that free workers
participated in sufficient numbers for their wages to be considered
as the industry opportunity cost of slaves.
Wage data for this period are extremely limited. After
examination of several sources, Stanley Lebergott's estimates were
chosen because of their applicability and completeness.7 Because
the implicit slave wage did contain an element of "board" as well as
capital cost, Lebergott's free-farm-labor-plus-board estimates appear
to be the appropriate wages to use as a surrogate for "x". Lebergott's
state and regional estimates for average monthly wages were multiplied
by twelve to make them annual estimates; they are included in
The Value-Added per Slave-Hand: "y"
This data series is based upon the set of bales-per-slave estimates
made by Foust and Swan (hereafter F&S).8 Using the University of North
Carolina-Yale manuscript census sample for 1860 and their own sample
for 1850, F&S estimated the bales-per-slave for various counties and
areas in seven deep-South states.
The Lebergott wage series was for regions and states while the
data available to establish value-added were for counties and areas.
Two alternative procedures were considered to make the two sets of data
conformable. One procedure was to assume that the different levels of
output in the different counties were random variations with respect to
the average wage in the state. If the hypothesis of the model is
correct, this procedure would artificially increase the variance of
the estimate because county wage data were not available and each of
the county output estimates (transformed into value-added) would have
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to be regressed against the respective average state wage. The second
procedure and the one adopted was to calculate the appropriate
weighted-average "y" estimate for each available wage data observation.
Each of the state weighted-average estimates was calculated by
weighting the F&S point-estimates of bales-per-slave within the state
by the relative number of farms which had been used to get the sub-
state estimates. These weighted average bales-per-slave estimates
were then used as the basic data from which the "y" estimates would
be calculated. Weighting the data by this procedure not only gives a good
estimate of average state bales-per-slave, but it also emphasizes
the role of the farmer or plantation owner as the input-mix decision
Converting the bales-per-slave estimates into weighted average
estimates solved only one of three problems which had to be resolved
before these data could be transformed into value-added per slave-
hand. The F&S data neglected crops other than cotton which slaves
were used to produce. Therefore, modification had to be made to
obtain total-value-added. Additionally, the estimates were for
"slaves", and the model which was used required estimates in terms of
"slave-hands". The data were converted by assuming a participation rate
between slaves and slave-hands.
Bales-per-slave were converted into bales-per-slave-hand by
accepting Foust and Swan's estimated participation rate of 50% and
multiplying each of their estimates of bales-per-slave by two. This
participation rate is a composite multiplying factor which takes into
account the fact that slaves working in the fields were different ages
and sexes and provides an adjustment for slaves who did not work in
the fields. The resulting bales-per-slave-hand estimates indicate the
average cotton output of an average male field hand.9
These physical estimates of bales-per-slave-hand were changed
into terms of cotton-value-added per slave-hand by assuming an at-the-
farm cotton price of 9.5 cents per pound and noting that the census
400-pound bale applies to these estimates.10 It was assumed that the
farms were self-sufficient; thus, the maintenance cost of slaves is
a part of the implicit wage (with board) and the cotton-value-added is
equal to the value of that cotton.11
But cotton was not the only surplus agricultural crop being
produced by the slaves. Depending upon the particular southern sub-
region, various amounts of several other crops were being produced with
slave labor as well. The value-added in cotton, while certainly the
major part of total value-added, did not fully account for the
productivity of slaves. Fogel and Engerman have suggested that an
addition of one-half bale-per-slave to F&S's estimates would compensate
for the amount of other crops being produced.12 The variant II
estimates below are based upon such an assumption; one-half bale-per-
slave was added to each of the F&S estimates to calculate total-value-
added per slave-hand. These estimates are tabulated in Table 3-1.
Gavin Wright considers such a procedure to be unacceptable. His
research has indicated that the amounts of corn -- by far the next most
important southern crop -- produced in different states varied a great
deal. He suggested that a better estimate of total value-added would
be found by adding an appropriate "corn-equivalent" to each of the
bales-per-slave-hand estimates.13 Extending Wright's suggestion to
encompass all other southern agricultural outputs, a procedure was
developed which generated the relative amount of non-cotton-value-added
per slave-hand in each state from census data on the total corn, cotton,
and tobacco production and population.14 These estimates were added
to the cotton-value-added per slave-hand estimates and were designated
as the variant I estimates of "y".
Corn was an extremely important crop for the southern farmer and
his slaves. Not only was the corn ground into meal and eaten, but it
was also used as feed for hogs. Thus, to estimate the surplus of corn
and other grains, the grain requirements for rural self-sufficiency must
be subtracted from the aggregrate grain production figures for each
state. This self-sufficiency requirement was based upon three
assumptions. It was assumed (1) that the population on farms in
1850-60 was 80% of the total population;15 (2) that on an annual basis,
13 bushels of corn were used for corn meal and 17 bushels of corn were
needed to feed the animals which were slaughtered for food; and (3) that
corn represented 80% of the grain nutrient produced in each sub-region
for feed.16 These assumptions provided a'multiplying factor of 21,
which was used to convert state population figures into self-sufficiency
corn (and corn-equivalent grains) for average state farms. These
estimates were subtracted from total corn production to find an
estimate of corn which could have been sold by the farmers of the
The respective corn total surpluses were then added to a tobacco
value-equivalent to find total non-cotton agricultural surplus. The
price of tobacco was taken to be 10 cents per pound at the farm and
the price of corn was taken to be 50 cents per bushel at the farm in
1850 and 60 cents per bushel in 1860 to transform tobacco in pounds
to bushel-value-equivalents.18 Thus, for every five pounds of tobacco
(or six pounds in 1860) produced in the state, one more bushel was
added to the corn surplus estimate.
Given the assumed farm prices of cotton and corn, the non-cotton
surplus in terms of bushels of corn at 50 cents or 60 cents, was
related to the value of cotton in terms of $38.00 bales. Using these
assumed prices, each 1850 bale was 76 times as valuable as a bushel
of corn and each 1860 bale was 63 times as valuable as a bushel of
corn. The resulting estimates of corn-equivalent units of value
divided by cotton values were then established. These estimates are
presented in Table 3-2.
RATIO OF NON-COTTON-VALUE-ADDED TO COTTON-VALUE-ADDED
ON AVERAGE FARMS IN SELECTED SOUTHERN STATES AND
SUB-REGIONS, 1850 & 1860
STATE LA ARK MISS ALA TEX GA SC WSCent
1850 .033 (.912) .264 .293 --- .294 .098 .316
1860 .041 .382 .165 .209 .141 .197 .014 .164
Source: See Text.
The ratios of non-cotton-value-added per cotton-value-added were
then multiplied by the cotton-value-added per slave-hand estimates to
find the non-cotton-value-added per slave-hand for an average farm in
each of the sub-regions. Adding this figure to the (much larger)
cotton-value-added per slave-hand estimates provided the total-value-added
per slave-hand for the variant I estimates of "y".19 These estimates
have also been included in Table 3-1.
In summary, total value-added per slave-hand was estimated for
each of the sub-regions in two different ways. The following algebraic
representation of the data variants clearly points out the differences
variant I: y. = (1 + Z.)[(B/S).(2)(400)(9.5)]
variant II: y. = [(B/S). + ](2)(400)(9.5) ,
where Z. is the ith ratio from Table 3-2, (B/S). is the weighted-average
bales per slave estimate for the ith state and year, "2" transforms
bales per slave into bales per slave-hand, "400" transforms bales
(and bale-equivalents) into pounds, and "9.5" is the price per pound in
terms of cotton. Both data variants assume a 50% participation rate,
a $38.00 cotton bale, and a self-sufficient average farm for each state.
The difference in the estimates arises from the way non-cotton-value-
added is treated. In the variant I case an approximation for each
sub-region was calculated; in the variant II series, an approximation
of the regional average was used for each of the sub-regions.
One final consideration of the data for 1850 was made. It would
have been possible to use two more 1850 observations than were used
to calculate the 1850 ES, the observations for Texas and Arkansas.
But because both of these states were on the edge of the population
frontier in 1850, the estimates of wages and output were expected to
be exceptionally poor and unrepresentative of the economic relationship
in the states and they were dropped from consideration. This adjustment
decreased the number of observations for the 1850 regression to six.
The Statistical Estimate of the ES
Given the following definitions:
Y = the observed (partially synthesized) value for y
(variant I or variant II)
Y* = the value-added per slave-hand: the desired value of y
X = the implicit slave-hand wage: .the observed value for x
X* = the value marginal product of a slave-hand: the
desired value of x,
the stochastic log-linear specification of the Arrow, Chenery, Minhaus
and Solow (ACMS) equation previously discussed is based upon the
Y* = aX*B*V
and on the form
ln(Y*) = A + B* ln(X*) + ln(V) ,
where V follows a lognormal distribution and a is the antilog of A.
But with historical data in general, and the procedure used in
this study to generate Y in particular, Y* may not be equal to Y. The
assumption that the farm-gate prices of cotton were the same for all
regions, the assumption that Foust and Swan bales-per-slave estimates
were appropriate, the implicit assumption that the value-added by slaves
in each state for the clearing of new land was a constant percentage
of y for all regions, or any of the less troublesome assumptions which
were made to estimate y could have led to measurement error. Specifying
the measurement error as In(Y**) = ln(Y) ln(Y*), upon substitution for
ln(Y*) it is found that
ln(Y) = A + B* ln(X*) + ln(U) ,
where In(U) = ln(V) + ln(Y**), i.e., where the disturbance term
accounts for both measurement and stochastic error. Again assume that
U follows a lognormal distribution. If it is reasonable to assume
that slave owners imputed value marginal products to slaves, i.e.,
X = X*, then
In(Y) = A + B* ln(X) + ln(U)
may be estimated by least squares and the estimated value of B* (ES)
will be a consistent unbiased estimator of B*. .The ordinary least
squares estimates of this equation are presented in Table 3-3.
Data were available for two different years, 1850 and 1860. It
was felt that the ES's may have changed between 1850 and 1860; therefore,
estimates were derived for each year with each data variant.
SUMMARY OF REGRESSION RESULTS
I 6 1850 .7989 .4655 .4691
II 6 1850 .6609 .3108 .5756
The above specification of the equation appealed to economic
theory and the assumption that the value-added per input was a function
of an externally generated wage. This assumption appears reasonable
when it is remembered that the wage used as the implicit slave wage
was the transfer cost to the industry for additional labor.
It was also assumed that all inputs were being paid their value
marginal products at that point in history when the data were observed.
Thus, if firms faced a purely competitive demand for final product
and final product price was taken as the nume'raire, all firms would have
had average product = marginal product = wage. The wage (X) -- if it
were correctly measured -- would have been equal to marginal product
(X*), the variable required by the definition of the ES.20 But if
X were not equal to X*, and the observed ln(X) values were larger or
smaller than the In(X*) by some random amount, In(X**), the estimates
presented in Table 3-3 would be biased and inconsistent, because there
would have been errors in both variables.
Specifying the error between In(X) and ln(X*) as
ln(X**) = In(X) ln(X*) ,
where the E[ln(X**)] = 0, the stochastic specification: Y* = aX*B*V
ln(Y) = A + B* ln(X) + ln(W)
after the substitution of Y and X for Y* and X*.21 The disturbance
term, ln(W), is a function of the logs of Y**, X** and V and takes on
ln(W) = ln(U) B* ln(X**) .
The non-vanishing covariance between ln(X) and In(W) implies that
ordinary least squares leads to an inconsistent estimator of the ES.
Furthermore, such an inconsistent estimator of B*, here noted as b,
underestimates B* as can be seen by22
plim b = -
1 + (o **2/ X*2)
where the subscripted a2's are the variances of the respective variables.
Because these variances are positive, as the sample becomes very large,
the estimate of B* will be less than B*. Thus, if there were errors in
both variables, the least squares estimates in Table 3-3 would be
inconsistent and would probably underestimate the true ES's.
Preliminary Inferences from the Estimates
The primary task of this chapter has been to establish a reasonable
estimate of the ES in 1860. Lacking the necessary data for such an
analysis, the data which were available were transformed and combined
in accordance with two different assumptions to generate the variant I
and variant II sets of data. From these data and several reasonable
assumptions about the industry, two constant elasticities of substitution
were estimated; 1.67 from the variant I data and 1.34 from the variant
II. No attempt to discriminate between the estimates of the ES was
made. Though more confidence is placed in the procedure used to
generate the variant I data, the relative merit of the procedure is not
conclusive. Both data variants will be considered in the next chapter.
Incidental to the establishment of the ES estimates was the
finding that both data variants led to estimates of the 1860 ES which
were significantly different from zero. The t-tests cast doubt upon
the assumption that slaves and other inputs were used in fixed
proportions in 1860. Those profitability-of-slavery studies which
have implicitly assumed such a relationship between inputs should be
carefully reconsidered to see if such an assumption is crucial to the
It was also found that the estimated ES's for 1850 and 1860 were
quite different. To attempt an explanation of these differences, the
data, plotted on scatter diagrams, and the coefficients of determina-
tion were examined. Both considerations indicated that the 1850
variance was relatively larger. From this relative difference in
variance it may be inferred that the sample error -- for some reason --
was larger in 1850 than in 1860.
Two alternative hypotheses about the way the wage (X) was related
to the value marginal product (X*) were considered above. Being able
to accept either one of these as a suitable assumption about the data
would permit a tentative inference which could explain the difference
in error. On the one hand, if inputs had been paid their marginal
value products, X would have been identical to X* and classical least
squares would be the appropriate estimating technique. The relative
difference in error would then be explained by the relative applicability
of the variant I and variant II procedures for generating the data in
the different years. On the other hand, inputs may not have been paid
their value marginal products in 1850. Thus X / X*, and the presence
of errors in variables could also justify the relative difference in
Foust and Swan's estimates of the internal rates-of-return
attributable to slaves suggest which of the two assumptions is the
more appropriate for each of the observational years. Their estimates
have been modified and tabulated in Table 3-4.23 The estimates for
1850 indicate that the three sub-regions considered were all receiving
a rate-of-return over and above the rate of interest necessary to
induce an investment in slaves, while in 1860, the average internal
rate-of-return was right in the middle of the range of interest rates
(6% 7%) usually given as those facing the slave owner.24
VARIANT II ESTIMATED RATES OF RETURN IN PERCENT FOR
9.5-CENT COTTON AND NO BREEDING RETURNS, 1850a AND 1860a
Area 1850a 1860a
Old South 8.6 5.8
Other New South 8.4 7.3
Alluvial 10.6 7.0
All Regions 8.9 6.6
Source: Foust and Swan, p. 55. The percentages above
are averages of the 9 and 10 cotton-price per-
centages given by F&S.
Notes: (a) F&S have called these "years" 1849 and 1859.
The 1850 and 1860 are used here as "census
years". See fn. 8, this chapter.
Everything considered, the 1860 estimates of the internal
rates-of-return reinforce the hypothesis that X = X*. Since the
rates-of-return were all very close to the range of interest rates,
it appears that if inputs were not being paid their marginal value
products that at least they were receiving payments which were close
enough to assume equality. The 1860 estimates of the ES in Table 3-3
would appear to be unbiased estimates of the true-ES in 1860.
The same conclusions can not be drawn for the 1850 estimates
of the ES or payment of inputs. The established two percentage point
differential between rates is too large to ignore. Given the technique
Foust and Swan used to estimate the internal rates-of-return, without
further information, one can not infer the amounts the various inputs
were underpaid.25 It is only known that from an industry point of
view, total revenue was not exhausted by input total cost and that the
average slave owner would have been receiving a quasi-rent (quasi-
profit). Value marginal products and input prices would have been in
adjustment, X would not necessarily have been equal to X*, and errors
in variables are present in the data. Thus, if the 1850 sample is
sufficiently large, the estimates of the ES presented in Table 3-3
are underestimates of the true-1850-ES. Though this result does not
mean that the ES's were the same in both years, it does suggest that
the change in the parameter over the ten-year interval was smaller than
is indicated by the ordinary least squares estimates.
1 See Foust and Swan, p. 52.
2 An algebraic and geometric presentation of the intra-marginal factor
payments to slaves is reserved for an appendix; see Appendix B.
3 Conrad and Meyer, 1964. Their procedure is used to establish the
profitability of slavery, but the same data problems arise.
4Stanley Lebergott, Manpower in Economic Growth: The American Record
Since 1800 (New York: McGraw-Hill Book Co., 1964), p. 19.
5 In other words, if either of the marginal inputs had been combined
with the same amounts of other inputs, they would each have contri-
buted the same increment to total product. If shares are competitively
imputed to the factors of production, then, each should receive the
same wage. That the average slave-hand may have been more or less
productive than the average free worker is not crucial to the
analysis if the productivity difference was a constant percentage
difference in all regions. If from either an institutional preference
for non-blacks or a productivity differential, such a constant per-
centage difference did occur, only A (the intercept of the estimated
function) would be different, not the estimate of the ES.
6 Foust and Swan, p. 42, fn. 9.
7 Lebergott, 1964, pp. 263, 539.
8 Foust and Swan, pp. 44-45. Strictly speaking, Foust and Swan's
output estimates are neither 1859 (1849) nor 1860 (1850) figures.
The census-year was not the agricultural-year and as the marshall
took the census, sometimes they would get output figures for 18-9
(i.e., for the agricultural year prior to the year in which the
census was taken), sometimes they would get estimates of the 18-0
crop and sometimes -- later on in the census year -- they would
get actual 18-0 figures. The year actually ran from September of
the previous year to September of the census year. Since there is
no way to weight the figures for one year or the other, they have
here been called 1850 and 1860 output figures. See discussion of
the problem in U. S. Department of the Interior, Census Office,
A Compendium of the Ninth Census (June 1, 1870) (Washington, D. C.:
Government Printing Office, 1872), pp. 690-692. Hereafter output
is given in census years unless otherwise stated. These output
figures are conformable with Lebergott's 1850 and 1860 wages.
9Foust and Swan, pp. 42-43. The resulting estimate is an estimate
for a male slave with average productivity.
10 Ninth Census, Compendium, pp. 695-696 for statement on the 400
pound bale. The 9.5 cent cotton price per pound is an average of
the prices suggested by Richard Sutch; he states, "if we use
9-10 cents a pound as the price of cotton, we should not be over-
stating the farm-gate price .", Sutch, p. 372.
11 Gallman presents a very convincing argument for self-sufficiency
in Robert E. Gallman, "Self-Sufficiency in the Cotton Economy of
the Antebellum South," Agricultural History, XLIV, No. 1(January,
1970), pp. 5-23.
12 Foust and Swan, pp. 54-55.
13 Wright, 1971.
14 Ninth Census, Compendium, pp. 8, 12, 14, 695, 696, 701.
15 The 80% population proportion was found by a comparison of series A
206 with series A 20 in Historical Statistics. U. S. Department of
Commerce, Bureau of the Census, Historical Statistics of the
United States, Colonial Times to 1957 (Washington, D. C.: Government
Printing Office, 1960), pp. 8, 14.
16 The estimates of corn consumption were based upon studies of self-
sufficiency in southern agriculture. Since these studies have
biased the corn requirement against the hypothesis that the farms
were self-sufficient, the estimates presented in the text have been
adjusted slightly downward. See Gallman, 1970; and Raymond C.
Battalio and John Kagel, "The Structure of Antebellum Southern
Agriculture: South Carolina, A Case Study," Agricultural History,
XLIV, No. 1(January, 1970), 25-37.
17 "Corn" represents corn and other grains in corn-equivalents. All
grain production is stated in terms of corn-equivalents since it was
by far the most important non-cotton crop. See Battalio and Kagel,
p. 28 and p. 28, fn. 8.
18 For the estimate of tobacco price, note the trend in tobacco prices
in Historical Statistics, p. 302. The corn prices used in the
text are adjusted from the port corn prices found in DouglassC. North,
The Economic Growth of the United States, 1790-1860 (Englewood Cliffs:
Prentice-Hall, Inc., 1961), pp. 263, 260. It was assumed that corn
port prices would be different from farm prices by a slightly larger
percentage than the differences in cotton.
19 Including the Arkansas estimate for 1850, the mean value of these
ratios is .234. Multiplying this mean ratio by the 2.22 mean bales
per slave estimated for 1860 by FES (Foust and Swan, p. 44) gives an
average regional corn-equivalent in bales which is slightly smaller
than the bale suggested by Engerman and Fogel.
20 The fundamental definition of the ES relates percentage changes
in the factor intensity ratio to percentage change in the ratio of
factor marginal products. It is only when pure competition is
assumed that one may say, in general, that the input intensity
ratio is related to the price ratio. See Appendix A.
21 The rationale for assuming that the E[ln(X**)] = 0 is as follows.
During the adjustment period, some firms would find that the
implicit wage > value marginal product (X*) and would be selling
slaves, other firms would find X < X* and would be buying slaves.
If, though the entire industry were in long run adjustment toward
the implicit wage such that in the majority of cases, to different
degrees, X < X*, then zero-expectation does not seem appropriate.
In this case, it would appear more appropriate to postulate that
E[ln(X**)] = k, where k < 0. But even in this case, the analysis
would not change. A constant plus a random variable has the same
variance as that random variable.
22 J. Johnston, Econometric Methods (New York: McGraw-Hill Book Co.,
1963), pp. 148-150, 156. See Also, Arthur S. Goldberger, Econometric
Theory (New York: John Wiley & Sons, Inc., 1964), pp. 282-284.
23 Foust and Swan, pp. 54-58. Their variant II is the same concept
(with respect to corn value-added) as the variant II used here. A
more detailed break-down of the internal rates-of-return are also
presented. Foust and Swan, p. 57.
24 The 6-7% range came from Conrad and Meyer. (Conrad and Meyer, 1964,
pp. 53-55.) Sutch and F&S assumed a 6% rate on alternative invest-
ments. (Sutch, p. 369; Foust and Swan, p. 53.) The rate con-
sidered should be a long term rate; therefore, the relatively high
short term rate in 1850 (Conrad and Meyer, 1964, p. 54) was not
25 Mishan has shown that, with a case of Marshallian short run quasi-rent,
resolution of the mis-payment of inputs demands fixed quantity
of one factor and fixed prices for each of the others. E. J. Mishan,
"What is Producer's Surplus?", American Economic Review, LVIII, No. 5,
Part 1 (December, 1968), 1269-1282.
AN APPROXIMATION OF THE CHANGE IN THE LABOR
SHARE OF INCOME IN SOUTHERN AGRICULTURE, 1860-1870
The most dramatic economic problem which confronted the South
after Appomattox was the mass of freedmen who had to be integrated into
southern agriculture as farm laborers instead of slaves. The scene is
set by Phillips as
the survivors of the cataclysm had to resolve the
problem of economic life afresh under conditions
of general derangement and almost universal poverty.
The landlords possessed land and managerial
experience -- and worthless Confederate currency.
The freedmen had liberty, and little else but a
residual acquiescence in the necessity of working
for a living.1
But resolution of the problem of institutional readjustment is only the
first episode of an incomplete epilogue on the aftermath of the Civil
War. The institutional problems with labor were solved noisily but
quickly. Within three or four years after the end of hostilities, the
crop-lien and share-cropping systems had been instituted.2 This chapter
is concerned with a more enduring consequence of the War, a part of the
story which is normally overshadowed by political reconstruction and
the spectacular economic upheaval caused by the abolition of slavery,
the destruction of the South's economic factors of production. Both
labor and farm-capital had been affected by the hostilities. About 280,000
men had died and rubble lay where many farms and buildings had
previously stood. The War had destroyed more than the peculiar insti-
tution. It is to this physical destruction and the resulting change in
the share of income received by southern agricultural labor that this
chapter is directed.
Two studies have been found with a more than cursory treatment of
the 1860-1870 changes in the relative amounts of productive inputs or
factor shares which resulted after the Civil War. These two studies
are Edward C. Budd's analysis of factor shares from 1850 to 1910 and
Stanley Lebergott's studies of wages and manpower in 'the Nineteenth
Budd's study neither asks nor answers the regional question. It
was concerned with changes in the size distribution of income of inputs
for the entire United States from 1850 to 1910.4 This study, in contrast,
is concerned with changes in the functional shares of labor within one
industrial sector of a certain region of the country over a relatively
short time span.
To make his analysis complete, Budd had to find some way to impute
wages to slaves. His treatment was based upon an average annual slave
wage of $268.00.5 When this figure is compared to the averages of the
annual wages in Table 3-1 ($170.88 for 1860 and $120.41 for 1850), his
estimate seems excessively high. But even if his aggregate estimates
based upon inflated wages could be accepted, there is a more fundamental
problem with using his study as a basis for this one.
Davis and Legler found that only after the 1870's could regions
within the country be considered as a unit with respect to government
activity in the economy. They state, "in a nation that was not a
national economy, attempts to analyze that nation in terms of a
national economy may be more misleading than revealing. For most of the
period, it appears that analysis on a regional basis is almost a
necessity."6 It is here suggested that the same may be true of
agriculture in the United States between 1860 and 1870. Though Budd
found that the aggregate share going to labor engaged in agricultural
pursuit fell between 1860 and 1870, his study says nothing about what
was happening in the South.7 His treatment of aggregate shares
glosses over the subtle changes which were occurring within the different
regions of the economy. The aggregate net change in shares misses the
point of historical importance to the South. A change in shares in the
South which was different or more intense than the aggregate change in
shares could provide a clue to the reason that the South moved so
slowly out of agriculture and into manufacturing in the late Nineteenth
Lebergott's study of Nineteenth Century labor was considered at
greater length.8 The quality of his series of wage estimates for the
different states has already been discussed; they are the most
complete and consistent series available. His wage data indicate that
throughout the South the postwar wages had fallen from their antebellum
levels. He explains the decline by stating that
such wage weakness, since cotton production was
up to prewar levels, could not well have reflected
any decline in the demands for labor. We infer
that the decline in wage rates paid to Southern
farm labor from 1860 to 1870 reflects in signi-
ficant degree the inferior bargaining position
of the individual farm laborer as compared with
that of the slave owner.9
His conclusion is.based on the hypothesis that, had "the slaves sold
their own labor as individuals in a completely free market, their own
reservation prices would have been somewhat above subsistence levels
but below that achieved by their masters." The master (supposedly)
set the hiring rate so that the slave maintenance cost and the cost of
an investment in a slave were covered.10
Lebergott's explanation of the fall in the level of wages must be
reconsidered because of its implications for relative factor shares.
While he does note that the rental price received for a slave should
cover the capital cost of that slave, he fails to note that the capital
cost is simultaneously determined with the intra-marginal factor payment
to slaves established by the wage which had to be paid to free workers.
Slave rental rates did not "support" free labor wages, rather the two
supported and determined each other.11 It has previously been argued
that given the nature of the slave industry, that one should expect the
wage of the free worker to have had a stronger effect upon the implicit
slave wage and rental rate than vise versa. Firms in the industry would
have continued to bid up the price of the slave until the intra-
marginal factor payment had wiped out the difference between maintenance
cost and industry transfer cost, the free worker wage.
In addition, new evidence which was not available to Lebergott
casts doubt upon the significance of his explanation of the fall in
wages. Robert S. Starobin's study of industrial slavery in the Old South
indicates that only about 5% of the slaves were ever involved in
industrial activity and that most of these slaves were either owned
by the firms or rented on an occasional basis from plantation owners
in the immediate area.12
A plantation owner who rented slaves occasionally would have
considered cost differently from an owner who rented his slaves to
firms for an entire year. A slave, as a field hand, was a constraint on
output during only one season of the year, the picking season. Slaves
could plant and cultivate far more cotton than they could pick. Thus,
during those non-picking seasons, the slave was -- to a greater or
lesser extent -- not fully occupied, i.e., underemployed. It is during
these slack periods that the rational part-time slave lessor would have
leased his slaves. In such periods, the opportunity cost of renting
some of his slaves was very low. The slave owner had only to consider
this low opportunity cost and the risk involved with such a short-term
lease to arrive at an acceptable hiring rate. He certainly did not
have to consider the longer run costs as Lebergott suggests. The
maintenance and capital costs were sunk costs in this short period; they
had to be covered by the picking of cotton, not by infrequent and
The third objection to Lebergott's explanation of the fall in
the wage is more important than the two which have already been presented.
He rejects the possibility that the demand for labor had changed after
the War when there are two good reasons to infer that it did change.
Lebergott had stated that cotton had recovered by 1870. A closer
examination of the data reveals that neither cotton production nor
industry revenues from cotton were back to prewar levels until 1874, and
that a complete recovery did not occur until the late '70's.13
Lebergott also implies that the destruction left by the Civil War did
not change the relative demands for productive inputs. As will be shown
below, this last implication cannot be accepted; both labor and farm-
capital had felt the sting of conflict. The absolute and relative
changes in productive inputs did have a significant effect upon the
level of wages and the change in the share of income received by labor.
The Proposed Resolution
A direct method of estimating the absolute labor shares of income
would involve calculating the respective labor incomes as a percentage
of total income for 1860 and 1870. The change in the share of income
could then be noted by a simple comparison. This method could be used
if it were possible to treat the residual (total income minus labor
income) as the share going to other productive inputs -- here lumped
together as "farm-capital" -- and if the data for wages, the labor
force, agricultural output, and the prices received by farmers for
that output were available for both years.
Acceptable data are available to estimate the share in 1860,
but the data available for an estimate of the 1870 share are inadequate.
Even the 1870 Census data for the 1870 labor force are admittedly
incomplete.14 If the 1870 Census data were used to estimate the
(explicit and implicit) share of income received by southern agricultural
labor, the estimate would be no better than the accuracy of the
assumption chosen to account for the omitted portion of the labor
An alternative approximation of the share in 1870 can be found
by an indirect analysis based on the assumption that the elasticity
of substitution (ES) between productive inputs was the same in 1870
as it was in 1860. If this assumption is appropriate, it is possible
to estimate the change in the ratio of relative shares from knowledge
of the change in input prices and an estimate of the ES. The change in
the ratio of relative shares indicates the change in labor's share
relative to the share received by other productive inputs but it does
not provide sufficient information to establish the change in absolute
share. The labor share in 1870 can be calculated, though, by using the
estimate in combination with the calculated ratio of shares for 1860.
The advantage of this indirect method of approximating shares is
'that when and if more comprehensive data become available, it may be
incorporated into the model and the estimate revised. The direct
procedure which uses only census data will always involve an unknown
magnitude of error. It is doubtful that a more comprehensive estimate
of the 1870 labor force can be found.
The Estinated Change in Relative Shares
To estimate the change in relative shares, three assumptions were
made. (1) It was assumed that, though the relative amounts of the
productive inputs did change dramatically after the destruction of the
Civil War, the estimated elasticity of substitution (ES) for 1860 was
sufficiently close to the 1870 ES so that they may be considered the
same. (2) It was assumed that the productive inputs were paid amounts
which were sufficiently close to their value marginal products to state
that value marginal products were equal to respective input prices.
And (3) it was assumed that Lebergott's wage data and the interpolation
made from the rates of interest and return data are correct values for
these input prices.
Given these assumptions, it was possible to estimate the relative
change in shares of income from two series of prices and an estimate
of the elasticity of substitution for 1860.
Defining L as labor to include free and slave 1860-labor and
all 1870 farm labor, w as the implicit and explicit wages (with board)
received by L, K as all other inputs (primarily land and physical
capital), and r as the rate of return which is also the price paid for
each marginal unit of K, it was possible to define the ratio of relative
shares (R) as15
(4-1) R = K (r) .[
Taking natural logs,
In(R) = In(K/L) + In(r) ln(w) ,
and then the total differential, (4-1) becomes
(4-2) d[ln(R)] = d[ln(K/L)] + d[ln(r)] d[ln(w)]
Further noting that from the definition of the ES16
d[ln(K/L)] = (ES) (d[ln(w/r)]) ,
it is possible to substitute into (4-2) to find an expression of the
percentage change in ratio of shares which is a function of only
w, r, and the (ES), i.e.,
d[ln(R)] = (ES) (d[ln(w/r)]) + d[ln(r)] d[ln(w)] .
This expression may be written more compactly because
d[ln(w/r)] = dw dr
may be substituted, such that
dR (ES)dw dr + dr dw
R Lw r w
(4-3) dR [(ES) -1] ;
which, for the discrete case, becomes
(4-4) %AR = [(ES) 1] [%Aw %Ar]
Equation (4-4) was used to approximate the change in relative
shares in southern agriculture between 1860 and 1870. To make such an
estimate, appropriate values for the three variables had to be found.
The elasticity of substitution used for the estimate was the estimated
ES for 1860 from Chapter III. The wage data which were used came
directly from Lebergott's average farm labor earnings plus board
estimates.17 The change in the price of other inputs was interpolated
from several different series of interest rates and rates-of-return and
will be discussed fully below.
The Percentage Change in w. The wages which were used to calculate
the percentage change in the average level of wages in the seven
cotton states which made up the sample are reproduced in Table 4-1.
These wages were weighted by the relative size of the labor force in
each state for each respective year to obtain an over-all weighted
average arc percentage change in the level of wages. It was found
to be 14.2% for the seven-state sample.18 For further comparison,
the arc percentage changes in wages for each of the states are included
in Table 4-1 also.
AVERAGE MONTHLY FARM EARNINGS WITH BOARD, RELATIVE SIZE OF THE
LABOR FORCE, AND WEIGHTED CHANGE IN STATE AND REGIONAL EARNINGS
FOR SEVEN SOUTHERN STATES, 1860 & 1870
Relative Size of Average Monthly
Farm Earnings 1860 to 1870
the Labor Force Plus Board Arc % Change
State 1860 1870 1860 1870 in Earnings
Alabama .18784 .19291 $12.41 $10.82 -13.68%
Arkansas .07123 .07079 14.25 13.52 5.26%
Ge(rgia .20172 .22256 11.95 10.83 9.83%
Louisiana .12034 .09646 17.00 14.34 -16.98%
Mississippi .17244 .17239 16.66 13.38 -21.84%
South Car. .15243 .13890 11.37 10.33 9.59%
Texas .09400 .10604 16.02 14.05 -13.10%
Regional 13.91 12.07 -14.2%
Sources: Lebergott, 1964, Table A-23, p. 539
Ninth Census, Compendium.
The Percentage Change in r. Establishing the change in the rate
of interest, i.e., the change in the price paid for other agricultural
inputs, required data from more than one source. Two internal rates-
of-return on other agricultural inputs were required. The rate for
1860 has been estimated to be 6.6%, but there is no comparable estimate
for 1870.19 Several different rates were considered in the attempt to
provide an approximation of the 1870 mortgage rate to use as a surrogate
for the internal rate of return on other agricultural inputs.
Short-term rates of return to reserve and non-reserve city banks were
initially considered because they were the only regional series which
extended back as far as 1870.20 These rates could not be considered
as surrogates for the mortgage rate, though-. The mortgage rate
surrogate had to be an approximation of the rate-of-return on land,
buildings, and other farm-capital. Such a rate is not a short-term
rate-of-return.21 In addition, the bank rates-of-return were not very
close to the short-term cost of capital to tenant farmers and landlords.
Two historians have estimated that the effective (short-term) rate paid
by these in agriculture probably fell within a range of from 18 to
24 percent, which is a far cry from the 5.3-6.5% rates-of-return
to banks.22 Thus, lacking a means to establish the desired rate
directly, the 1870 mortgage rate was inferred from associated data and
then checked for consistency with rate of return series which were
The mortgage rate was inferred from the History of Usury Laws
presented in the Twelfth Volume of the Eleventh Census.23 The super-
intendent noted that the trend in the maximum level of interest permitted
by law, the usurious rate, had fallen during the period being considered
and after.24 This fall in the general level of usurious rates occurred
notwithstanding the fact that the maximum rate in the South was being
relaxed and was actually increasing throughout the period 1865-1890.
Arkansas had removed the previously legislated 10% maximum as early as
1868. Georgia had felt the squeeze and had relaxed the 8% maximum rate
to 10% by 1871. Mississippi had removed the 10% maximum rate by
further .legislation in 1873. And while South Carolina had placed a 7%
ceiling on the rate in 1877, the rate had been changed within five
years back up to a level more in line with her sister states, 10%.25
Though it is not possible to assess how responsive the various
state legislators were to the economic pressures on their constituents,
it is possible to infer that a number of these constituents must have
expected a rate of return in excess of the old usury law rate or there
would have been no pressure for enactment of new legislation.
Assuming that those who had the influence to push for the enactment
of new laws with higher usurious rates faced credit conditions which
were not significantly different from the conditions faced by the
community at large, it is possible to infer that the rate of interest
on longer term debt was in the neighborhood of 10%.26
To test the plausibility of this hypothetical mortgage rate,
three comparisons were made. Initially, the hypothetical rate was
compared to the mortgage rates in the Mid-west. Given that both areas
had similar problems with inducing capital out of the Northeast, the
rates in the Mid-west should have been about the same as the rates
which had to be paid in the South. The comparison was consistent with
the 10% southern rate. Indiana, Iowa, and Illinois all had mortgage
rates of 10% for the period 1868-1876.27 The second comparison was
made to the profit rates of banks within the South. There were very few
banks in the South during this period and those which did have their
doors open had varying degrees of monopoly power.28 Given this monopoly
power in the banking sector, the rate of return on bank capital and
surplus should have exceeded the longer run returns to firms in the
non-monopoly sectors since an element of monopoly profit would be
included in the estimated average profits for banks. Sylla's estimates
of average profit for banks are also consistent with a 10% interest
rate. He estimated that the average rate of profit for reserve city
and non-reserve city banks were 13.3% and 16%.29 The third consistency
comparison which was made was with the yields in gold of American rail-
road bonds. Taking the average of the twelve-month period being
considered, the gold yield average for September, 1969 to September, 1870,
was 9.46%.30 Though this estimate is not directly comparable -- since
mortgages were in terms of greenbacks and there was a greenback premium
on gold -- it may be taken as an over-estimate of the rate in terms of
greenbacks.31 Since an interest rate differential was necessary to
induce capital to flow inter-regionally, this estimate is also
consistent with the assumed 10% rate.32
The three comparisons which have been made have all reinforced
the acceptability of the 10% mortgage rate. Such an estimate appears
to be reasonable for the analysis of the change in relative shares.
Thus, assuming a 10% rate for 1870 and the 6.6% rate for 1860, the
arc percentage change in the price of farm-capital was +40.96%.33
The Estimate. To find the percentage change in the ratio of
relative shares from equation (4-4), an estimate of the ES, the %Aw,
and the %Ar was required. Each of these variables has been estimated.
The ES was estimated in Chapter III from two different derivations of
output data, variant I and variant II, as 1.67 and 1.34, respectively.
The percentage changes in the level of wages and the rate of interest
were estimated immediately above; they were found to be -14.2% and
+40.96%. Replacing the variables in equation (4-4),
%AR = [(ES) 1] [%Aw %Ar] ,
with the estimated values, the change in the ratio of shares for the
two data variants is 34
variant I, %AR = -36.96%
variant II, %AR = 18.75% .
The percentage change in the ratio of relative shares is the relative
decrease in the share going to farm-capital or the relative increase
in the share going to farm labor; i.e., (%AKr)/(%ALw) = -37% (or -19%),
or (%ALw)/(%AKr) = +37% (or +19%). The estimated percentage change in
the ratio of relative shares indicated that the share received by farm
labor relative to the share received by other inputs increased, but it
did not indicate the magnitude of either share. To infer the size of
the functional shares in either year, additional information was
included in the analysis.
The Southern Agricultural Labor Share in 1860
In comparison with some of the data sources used by historians,
the 1860 Census is a fairly complete and accurate document. By making
a minimum number of assumptions about the way the data were generated
and by including some additional information on wages and agricultural
prices, it was possible to use census data to estimate the percentage of
southern agricultural income which was paid or imputed to labor in 1860.
Two estimates were needed to approximate the labor share, an
estimate of income received or imputed to labor and an estimate of total
income generated in the seven states which made up the sample.35 Total
wages paid or imputed to agricultural labor was computed from state
average wage and labor force estimates. The labor force was estimated
from a computation of slave and non-slave labor. The size of the non-
slave labor force was found by adding together the number of farmers,
planters, overseers and farm laborers which.are tabulated in the 1860
Census for each state.36 The size of the field-hand-equivalent labor
force for each state was found by assuming that 5% of the
slaves in each state were in non-rural activities and that Foust and
Swan's 50% participation rate applied to each of the seven states.37
This adjustment of slave population into field-hand equivalents permitted
addition of the number of non-slave laborers to the number of field-hand
equivalent units of slaves to find a total labor force which could be
multiplied by Lebergott's wage estimates. The total number of non-
slave farm laborers was 435,000 and the number of field-hand equivalent
slave laborers was 1,122,000. Combining the two figures gave an 1860
farm labor force of 1,559,000 in the seven states considered.
Each of the respective state sub-totals was multiplied by the
annual state average wages presented in Table 3-1 to find an 1860
labor income of $260,261,700 in the seven states.38
The estimate of agricultural income for the seven states in the
1860 Census agricultural year was based upon Eugene Lerner's agricul-
tural income estimates for the eleven states which seceded from the
Union.39 His estimate of $575.5 million was based upon farm output
times unit prices for selected crops. Among those individual crop
revenues calculated is an estimated $277.6 million for cotton. The
cotton estimate was based upon a cotton output figure of 2,373,000,000
pounds (5,932,500 400-pound bales) at a cotton price in excess of
$ .11 per pound.40 Both the output and the price figures are excessively
high; his output figure is 600,000 bales above the total number of
bales reported for the United States at the 1860 Census and his
price per pound is at least one cent higher than has been accepted as
a reasonable 1860 farm price for cotton.41 The cotton revenue was
re-calculated using the output data in the 1860 Census and a 9.5 cent
per pound cotton price. This adjustment provided a lower estimate of
the agricultural income in the eleven states of $500,972,000.
The agricultural income for the seven states was found from this
eleven-state income estimate and the percent of income which had been
generated in the seven states. In Census year 1870, 66.4% of the
eleven-state income came from the seven states in the sample.42 Noting
that 90.27% of the cotton was produced by the seven states in 1860 and
only 87.84% was produced in 1870 and that the amount of corn produced
in the sample states remained approximately 56% of the eleven-state
total for both years, it was assumed that all crops other than cotton
added the same relative amount to the eleven-state income in both
years.43 Based on this assumption, the relatively greater amount of
cotton produced in the seven states before the War indicated that
67.7% of the eleven-state revenue was generated in the seven states
Sixty-seven and seven-tenths percent of $400,972,000 is $339,158,044,
the estimate of agricultural income for the seven states in the sample.
Labor's share of that income was $260,261,700, which is 76.7% of total
agricultural income in the seven southern states.
The Southern Agricultural Labor Share in 1870
Two estimates of the share in 1870 were made. The primary estimate
avoided the use of the incomplete 1870 Census; it is based upon the
estimated change in the ratio of relative shares and the 1860 ratio of
shares. For comparison, a secondary estimate of the relative labor
share based upon the 1870 Census was also made.
The Primary Estimate. By combining the estimated change in the
ratio of relative shares with the ratio of shares in 1860, it was possible
to establish the ratio of relative shares for 1870 and then the labor
Defining the ratio of shares in 1870 as R70, the ratio of shares
in 1860 as R60 and the change in the ratio of shares between 1860 and 1870
as LR = (R70 R60), the ratio of shares for 1870 is
(4-5) R70 = R60 + AR.
Recalling that the %AR = AR/R = (R70 R60)/[(R70 + R60)/2], (4-5) may
also be written as
(4-6) R = R + (AR/R)(R) = R + (%AR)(R) ,
70 60 60
where R = (R70 + R60)/2.
Given that the share going to labor in 1860 was 76.7%, the ratio
of shares in 1860 is R60 = (1 .767)/.767 = .233/.767 = .3038.
Replacing the variables in (4-6) with the estimated values of
R60 = .3038 and the variant II estimate of %AR = -.1875, the ratio of
shares in 1870 was estimated as
R = [.3038 + (-.1875/2)(.3038)]/[1+ (.1875/2)]
R70 = .27532/1.09375
R = .25172
Furthermore, noting that R7 = (1 labor's share)/(labor's share),
the share of income received or imputed to labor (call it s) was
estimated as follows:
R70 = .25.72 = (1 s)/s ;
s = .799
Based on the variant II data, the labor share of income increased to
79.9% in 1870.
The same procedure was used to estimate the 1870 labor share from
variant I data. Based upon the variant I %AR, the 1870 labor share
was 82.7% and the ratio of shares in 1870 was .20903.
The Secondary Estimate. A secondary estimate of the labor share
of income in 1870 was made from census data for 1870 after adjusting
the size of the labor force to account for some of the omitted farm laborers.
The total agricultural income for the seven sample states was found
from another adjustment of Eugene Lerner's data. Lerner estimated that,
during calendar-year 1869, $527 million was generated in the agricul-
tural sector of the eleven states which had seceded from the Union.4
This estimate was adjusted to the census-year by recalculating the value
of cotton and assuming that the revenue generated by all other crops
was the same in the census and calendar years.45 The value of cotton
was found from a weighted average price of cotton of 20.4 cents per pound
and the output level reported in the eleven states at the Ninth Census,
3,009,000 bales.46 Lerner's calendar-year estimate of agricultural
income for all other crops, $216.4 million, was then added to the value
of cotton at the census, $245,534,000, to get an adjusted agricultural
income in the eleven southern states of $461,934,000. Sixty-six and
four-tenths percent of this figure is $306,724,176, the estimated
agricultural income for the seven states in census-year 1870.47
Total 1870 wages in the agricultural sector of the seven sample
states was derived from Lebergott's wage data and an approximation of
the 1870 labor force. It was assumed that those workers who were
paid in-kind received a share which equalled the cash payment received
by those whose wages had been used to establish the average wage for
the states.48 Under this assumption, average farm labor earnings plus
board are the average agricultural wages in each state.
The labor force estimate came from data in the 1870 Census and
an approximation of the number of farm laborers who were omitted from
the census enumeration. The bulk of the 1870 agricultural labor force
was found in the Ninth Census tabulation of occupations.49 But even
though the superintendent had given the marshals detailed instructions
for gathering the information for this section, he still found that one
out of every ten U. S. males of working age had not been included in
the list and an even higher percentage of working women had not been
listed with an occupation.50 Notwithstanding the known understatement,
1,501,000 farmers planters, overseers, and farm laborers were placed in
the agricultural labor force of the seven states.51 To estimate the
total labor force, this figure was added to an approximation of the
number of omitted farm laborers and farmers. The omissions were
calculated by assuming that 10% of the white males and 10% of the rural
blacks had not been included in the census count and that one-half
of those individuals were engaged in agricultural pursuits. This
technique provided an underestimate of the true number of men, women
and children who were in agriculture but were not counted by the
marshals.52 based on these two percentages, the number of omitted farm
workers for all seven states was estimated at 201,000.53 And the
total labor force was set at 1,702,000 in 1870.
After multiplying each state wage by the estimated labor force
for that state, total wages for the seven states considered was found
to have been $246,877,526. Using this estimate as the numerator of
the ratio'of total wages/total income, the secondary estimate of the
labor share in 1870 was found to be 80.5%.54
Comparisons and Conclusions
The results of this chapter are summarized in Table 4-2. Based
upon these estimates, farm income and agricultural wages received by
the 8.77% larger labor force had not recovered to their prewar levels
when the 1870 Census was taken.55 The share received by labor relative
to the .share received by other productive inputs had increased, but the
dollar amount of that share had fallen. The decline in the size of the
share was not due to the larger labor force. Ceteris paribus, the
dollar share would have increased since the demand for labor was
elastic.56 The fall in the labor share rather rests upon the fact that
the labor force was being combined with only about one-half of the
pre-Civil War stock of farm-capital.57 Though the prices of southern
agricultural cash crops were considerably higher after the conflict,
they had not increased enough to offset the decline in production.
ESTIMATED TOTAL WAGES, INCOME, LABOR SHARE OF INCOME,
AND LABOR FORCE IN THE AGRICULTURAL SECTOR OF
SEVEN SOUTHERN STATESa, 1860 & 1870
1. Total Wages
2. Total Sector Income
3. Percentage of Income Received
or Imputed to Labor:
Crucial (b) Variant I
Source (c) Variant II
4. Labor Force in Equivalent
NOTES: (a) Alabama, Arkansas, Georgia, Louisiana, Mississippi, South
Carolina, and Texas
(b) census agricultural year
(c) the labor force is the total number of average farm
workers and is not in a strict sense the "number-of-
workers". The two totals would not necessarily be the
same because there were many individual slaves who were
working at less than average productivity. If anything,
the number of farm workers would be larger than this
figure. The two estimates do reflect the relative
amount of labor at each census.
To avoid placing undue emphasis upon the decline in farm-capital,
it should be remembered that though the labor force was slightly larger
in 1870, it had increased less than it would have increased in the
absence of war. If the counterfactual conditional that the War did not
interfere with the normal rate of increase in population and the labor
force is posed, the impact of the War on labor is placed in a better
perspective. Between 1850 and 1860, the rates of white and slave
population growth had been 20.9% and 23.38%, respectively.58
Assuming that the lower rate of growth in population had continued to
1870 and that the percentage of the population in the labor force in
1870 was the same as the percentage in 1860, the realized rate of
growth in the labor force was only 42% of what it would have been if
the counterfactual conditional were obtained.59 The 1870 labor force
was 9.7% smaller than it would have been in the alternative situation
specified for the absence of War.60
The Civil War did have an effect upon the productive inputs of
the agricultural South which was still present five years after the
end of the conflict. The destruction of the farm-capital stock was
still very much in evidence and the shares of income received by the
inputs reflected the relative and absolute changes in factor proportions.
1 U. B. Phillips, "Plantations with Slave Labor and Free," Agricul-
tural History, XII, No. l(January, 1938), p. 90.
2 Saloutos, pp. 66-72. Saloutos presents a thorough discussion of
the alternative means of combining productive inputs and of finding
a suitable labor force which were considered during the postwar
3 Edward C. Budd, "Factor Shares, 1850-1910," Trends in the American
Economy in the Nineteenth Century, National Bureau of Economic
Research, Studies in Income and Wealth, XXIV (Princeton: Princeton
University Press, 1960), pp. 365-406; Stanley Lebergott, "Wage
Trends, 1800-1900," ibid., pp. 449-499; and Lebergott, 1964. The
second Lebergott study is an extension and partial revision of the
4 Since one should expect little difference in the size shares and
functional shares in the Nineteenth Century, the results found by
Budd should be compatible with the results herein (where they can
be compared). A good exposition of the difference between size shares
and the underlying functional shares can be found in Melvin W.
Reder, "A Partial Survey of the Theory of Income Size Distribution,"
Six Papers on the Size Distribution of Wealth and Income, Ed. Lee C.
Soltow, National Bureau of Economic Research, Studies in Income and
Wealth, XXXIII (New York: Columbia University Press, 1969), pp. 205-
253, see especially p. 207.
5 Budd, p. 384, fn. 32.
6 Lance E. Davis and John Legler, "The Government in the American
Economy, 1815-1902: A Quantitative Study," Journal of Economic
History, XXVI, No. 4(December, 1966), p. 526
7 Budd, p. 373.
8 Lebergott, 1964.
9 Ibid., p. 160.
10 Ibid., p. 159.
11 This is not to say that the implicit slave-hand wage (explicit
hiring rate) would have been equal to the free laborer's wage. There
could have been a continual differential between them.
12 Robert S. Starobin, "The Economics of Industrial Slavery in the Old
South," Business History Review, XLIV, No. 2(Summer, 1970), pp. 131-
13 Historical Statistics, p. 302. Series K-302 shows that though
1870 was a good year for cotton, that the average for a two-year
period was not up to the 1859-60 or 1860-61 level until 1874-75.
Industry revenue recovery may be found in Eugene M. Lerner,
"Southern Output and Agricultural Income, 1860-1880," Agricultural
History, XXXIII, No. 3(July, 1959), p. 124.
14 The Superintendent of the Ninth Census noted the difficulties of
collecting data in the South by stating "the Superintendent was
fain to accept whatever could be obtained in regard to the agricul-
ture of that region to have insisted upon a logical treatment
of the subject would have been equivalent to giving up the
agricultural statistics of the year." Statistics of Agriculture
(June 1, 1970) p. 72.
15 See Appendix A.
16 See Appendix A.
17 Lebergott, 1964, Table A-23, p. 539.
18 Lebergott has estimated the percentage fall in the level of wages
in the South for three different regions (Lebergott, 1964, p. 160).
His estimates included southern states in which cotton was not as
as important as it was in the seven states in the sample and Border
and Union States. Also, his estimates are percentage fall from the
1860 level of wages, not the arc percentage change considered here.
For both these reasons, his estimates should and do indicate a
smaller change. Lebergott did weight his estimates but he apparently
used a different criteria than was used here (Letter from Stanley
Lebergott, January 18, 1971).
19 The 6.6% internal-rate-of-return estimate is from Foust and Swan,
p. 55. See also the comments in Chapter III.
20 Lance E. Davis, "The Investment Market, 1870-1914: The Evolution
of a National Market," Journal of Economic History, XXV, No. 3
(September, 1965), pp. 355-399. See also Richard Sylla, "Federal
Policy, Banking Market Structure, and Capital Mobilization in the
United States, 1863-1913," ibid., XXIX, No. 4(December, 1969), pp.
21. If anything, the short term rate had more of an effect upon the cash
wage received by farm labor than it had on prices of other inputs.
Since the National Banking Act made direct loans to agriculture out
of the question, circulating capital received by the tenant farmer
was based upon the rate of interest that the landlord had to pay
his cotton factor who was able to borrow such funds for "commercial
purposes". Given that the landlord would have been able to have
used such capital in other ways, he had to consider the opportunity
cost of paying cash wages. This cost would be reflected in the
22 The short term bank rate is from Davis, pp. 362, 364. The range on
the effective rate paid by those in agriculture came from Saloutos,
p. 75, who states that the rate was 15%-25% and Coulter who states
that the 18%-24% range was common in the South. Coulter illustrates
the presence of high rates with an epitaph which was published in an
August, 1869 newspaper
Here lies thirty-six per cent.,
The more he got, the more he lent;
The more he got, the more he craved --
Good God! Can such a soul be saved!"
E. Merton Coulter, The South During Reconstruction, 1865-1877, Volume
VII of A History of the South, Eds. W. H. Stephenson and E. M.
Coulter (Baton Rouge: Louisiana State University Press, 1947), p. 194.
23 Department of the Interior, Census Office, Report on Real Estate
Mortgages in the United States at the Eleventh Census: 1890, Volume
XII of the Eleventh Census (Washington, D. C.: Government Printing
24 Ibid., p. 169.
25 Ibid., pp. 168-174.
26 This estimate is not a mathematical average but is an approximation
based on the information about the difference in rates of interest
on lots and acres, Diagram 9, ibid., p. 246; the periods in which
the usury laws were changed; the various penalties for usury in the
sample states, ibid., pp. 168-174; and the number of mortgages
written at usurious rates as a percent of total mortgages compared
to the usury law of the particular state, ibid., p. 174. This is
obviously a crude first approximation of the true rate in 1869-1870.
27 Davis, p. 377.
28 Sylla, passim.
29 Ibid., p. 679.
30 Frederick R. Macaulay, Some Theoretical Problems Suggested by the
Movements of Interest Rates, Bond Yields and Stock Prices in the
United States Since 1856 (New York: The National Bureau of Economic
Research, 1938), p. A217.
31 Ibid., pp. A215, 331-332.
32 Davis, p. 358.
33 To give some idea of the magnitude involved with an error in the
estimate of the 1870-"r", the arc changes for 9.5% and 10.5% were
calculated. The % difference in the assumed rate leads to
approximately a 5% difference in the estimated 1860-70 arc change.
The sensitivity of the estimate of the change in relative shares to
the choice of an 1870 interest rate can be noted in the table below
which presents the calculated change in shares for 9% and 10%
rates of interest as well as the assumed 10% rate.
Estimated Change in the
(with ES = 1.67)
Relative Shares for:
(with ES = 1.34)
35 The seven states are the same ones which have been considered before,
Alabama, Arkansas, Georgia, Louisiana, Mississippi, South Carolina,
36 Joseph C. G. Kennedy, Preliminary Report on the Eight Census, 1860
(Washington, D. C.: Government Printing Office, 1862), p. 134; and
U. S. Department of the Interior, Census Office, Population of the
United States in 1860; the Eight Census, Vol. I (Washington, D. C.:
Government Printing Office, 1864), pp. 11, 21, 77, 197, 273, 455, 491.
37 Slave population figures were taken from Preliminary Report on the
Eight Census, p. 134. The 5% slaves-in-cities estimate is from
Starobin, p. 132. The 50% participation rate is a complex
multiplying factor which converts slaves into field hand equivalents,
see Foust and Swan, pp. 42-43. The formula used to calculate field
hand equivalents is (total slaves -(.05)(total slaves))/(2), for
38 The estimated state agricultural
wages for 1860 are
39 Eugene Lerner, p. 124.
40 Ibid., p. 122.
41 Foust and Swan, Sutch, and Conrad and Meyer all use cotton prices
between 7 and 10 a pound in their profitability of slavery studies.
42 Ninth Census, Compendium, p. 692. Seven-state income was estimated
as $404,853,000 and eleven-state income put at $609,857,000. These
are both overestimates; it was assumed that the percentage
difference was the same.
43 Ibid., pp. 698-699, and 694-695.
44 E. Lerner, p. 124.
45 The value of cotton and corn was approximately 83% of the total value
of agricultural output in 1870 in the 11 southern states. The value
of cotton was recalculated because the price of cotton fluxuated a
great deal between 1869 and 1870. The value of corn for the.census
year is only two million dollars different than the Lerner estimate
for the calendar year. The values of the other agricultural crops
in calendar year 1869 and calendar year 1870 stayed about the same
since some increased in value and others fell. It appears that this
assumption is reasonable, given the quality of the data.
46 The cotton price is from Frederick Strauss and Louis H. Bean, Farm
Income and Indices of Farm Production and Prices in the United States
1869-1937, USDA Technical Bulletin No. 703 (Washington, D. C.:
Government Printing Office, 1940), p. 64. The price is a weighted
average of 25.8 and 18.0 cents per pound. The cotton output figure
is in 400-pound bales and is from the Ninth Census, Compendium,
47 The 66.4% was discussed above. It is calculated from the Ninth
Census, Compendium, p. 692.
48 Lebergott, 1964,-pp. 539, 263-264.
49 Statistics of Population, Ninth Census, pp. 659-804.
50 The directions to the marshall were found in ibid., p. xxxiii.
The incomplete count is discussed in ibid., p. 660.
51 Ibid., pp. 674-675.
52 There are several reasons to expect that this procedure will lead to
an underestimate of the true number of uncounted agricultural farm
workers. It should be expected that if 10% is the percent of
uncounted U. S. males and that the percent of uncounted women is even
higher, that a higher percent of uncounted laborers would be
realistic for the South. It should be noted that the 50% figure of
those in agriculture is also a-low estimate. Of those individuals
who did report their occupation, 74% of them listed occupations which
were directly related to agriculture ibidd.) Women and children
worked on farms as well as men; no attempt has been made to include
them in the estimate of omissions. The estimate of omissions has been
biased to a rather large extent. This bias should account for
unemployment in the South, though it is not possible to estimate the
amount of this unemployment. It is felt that the amount of this bias
is sufficient to preclude an over-estimate of the uncounted agricul-
53 Approximately 122,925 Negroes and 78,400 whites.
54 If no attempt had been made to account for omissions this procedure
would have led to an 1870 labor share estimate of 71% -- which from
the previous analysis of the change in relative shares is incon-
55 8.77% is the arc percentage change between 1,559,000 and 1,702,000.
56 Hicks, p. 115.
57 Ibid. The decline in farm-capital was calculated by assuming that
the share received by farm-capital exhausted the residual of farm
income (total farm income minus labor's share of the income) with
the following two-step procedure:
(1) The arc percentage change in farm-capital was calculated.
R = Kr/Lw, and
%AK = %AR (%Ar %AL %Aw).
After the substitution of previously calculated values for the terms
inside the parentheses, the expression reduced to
%AK = %AR 46.39%,
which for the respective variant estimates of %AR was
variant I %AK = -83.35%
variant II %AK = -65.14%.
(2) The arc percentage changes were converted into the 1870 pro-
portion of the 1860 capital stock (call it X) by defining the 1860
stock as "1" and calculating the following formula for each data
%AK = (X 1)/[(X + 1)/2]
From this formula the variant I estimate of the stock of capital (X)
in 1870 as a percentage of the stock in 1860 was found to be 41%.
The variant II estimate indicated that the 1870 stock of capital was
51% of the 1860 stock. These estimates provide the basis for the
approximation of one-half in the text.
58 Preliminary Report on the Eight Census, p. 7.
59 8.77 is 41.96% of 20.9.
60 20.9% of the 1860 labor force, 1,559,000 = 325,831. Thus, the
predicted 1870 labor force is 1,884,831. The difference between
this predicted labor force and the established 1870 labor force as
a percent of the predicted labor force: (1,884,831 1,702,000)/
1,884,831 = 9.7%.
SUMMARY AND CONCLUSIONS
This study considered two inter-related problems about the
agricultural labor force in the South before and after the Civil War.
The primary concern was the change in the functional share of income
which could be attributed to southern agricultural labor in 1860 and
1870. Two alternative ways to approach this problem were examined.
One procedure required data for the size of the 1870 southern agricul-
tural labor force. Since the 1870 Census data on the labor force, which
are the most complete data available, were known to be incomplete,
this procedure would have required a somewhat arbitrary assumption about
the number of farm laborers who were omitted from the census enumeration.
In an attempt to be less arbitrary, a second procedure was adopted.
The chosen procedure required an estimate of the elasticity of substi-
tution between farm labor and other agricultural inputs in the 1860
South. The analysis in Chapters II and III provided this estimate.
Combining the estimated 1860 ES with an estimate of the labor share in
1860 and input prices for 1860 and 1870 permitted the calculation of the
change in the labor share of income presented in Chapter IV.
Summary of the Analysis
The assumptions of constant returns to scale and the relevance
of a constant elasticity of substitution (CES) production function
were evaluated for the questions considered and the preliminary
specification of the CES function made by Arrow, Chenery, Minhaus,
and Solow was chosen for estimation of the ES. The model required data
on total value added per unit of labor and the wage for that unit of
labor. These data were not readily available for agriculture in the
antebellum South. The ES could be estimated only after those productivity
data which were available were partially synthesized to obtain total
value-added per slave-hand (and free labor in slave-hand equivalent
units) and after a surrogate for the implicit slave-hand wage was
Lebergott's data on free farm labor earnings plus board were
taken as surrogates for the implicit slave-hand wages. The theoretical"
relationship between payments to these two different types of antebellum
labor were considered at length. It was found that if all farmers paid
the same rate of interest to borrow funds, these wages would have been
equal when the industry was in purely competitive equilibrium. But if
there had been an imperfect capital market in which the small, marginal
farmer had to pay a higher price for funds, the free hired laborer would
have been paid a wage which was higher than the imputed slave-hand wage.
There is a possibility that the capital market was imperfect and that
the free wage was a somewhat greater than the implicit slave-hand wage.
But even in this case, Lebergott's series may still be used if there
were a constant percentage difference between the free and imputed
slave-hand wage in each of the sub-regions. Only the intercept value of
the estimated logarithmic equation would be different. The slope which
is the estimated ES would be the same.
Total value-added per slave-hand (and free labor in slave-hand
equivalent units) was found by establishing cotton-value-added per
slave-hand and non-cotton-value-added per slave-hand. Cotton-value-
added per slave-hand was based upon the most refined set of productivity
data available, Foust and Swan's estimates of bales per slave. Accepting
their participation rate of 50%, the bales per slave estimates were
converted into bales per slave-hand. Noting that a $38 bale was a
reasonable farm price for cotton in 1850 and 1860, the bales per slave-
hand were converted into cotton-value per slave-hand. In addition, since
southern farms have been shown to have been self-sufficient, these
estimates gave cotton-value-added per slave-hand.
Two different adjustments were made to obtain non-cotton-value-
added per slave-hand and thus total value-added per slave-hand. The
variant I procedure was based upon a suggestion made by Gavin Wright.
Census data on corn, cotton, tobacco, and population were used to
calculate non-cotton-value-added for each of the sub-regions in the
sample. The variant II procedure followed a suggestion that Fogel and
Engerman had made to Foust and Swan. One-half bale per slave was added
to each of the cotton productivity estimates to find total output in
terms of cotton and cotton-equivalent units and then total value-
added per slave-hand. No attempt to place a preference ordering on the
data variants was made. Both total value-added per slave-hand variants
were regressed on the implicit slave-hand wages using ordinary least
squares. A reconsideration of the underlying assumptions for such an
estimator reinforced the acceptability of the estimated 1860-ES but
cast doubt upon the estimate for 1850.
The estimate of the 1860-ES together with additional information
on input prices in 1860 and 1870 permitted calculation of the change
in the relative share of income received by labor before and after
the Civil War. It was shown that if inputs were paid their value
marginal products, the percentage change in the ratio of shares could
be expressed in terms of the ES, the percentage change in the wage
level and the percentage change in the appropriate rate of interest.
The two percentage changes in input prices were calculated and combined
with the estimated ES to find the percentage change in the ratio of
shares. Labor's share increased relative to the share received by
other agricultural inputs.
To find the percentage of income received by agricultural labor
in 1870, an additional relationship was noted. It was shown that the
ratio of shares in 1870 could be found by adding the ratio of shares in
1860 to the change in the ratio of shares. After calculating the ratio
of shares for 1860 from census data and after transforming the per-
centage change in the ratio of shares into the change in the ratio of
shares, both elements were available to calculate the ratio of shares
in 1870. The 1870 ratio of shares was calculated from these two
estimates. After simple algebraic manipulation, the competitively
imputed percentage of agricultural income received by labor in 1870
was established. These percentages and the dollars represented by
these percentage shares of income were presented in Table 4-2. The
dollar shares were found by a comparison of the competitively imputed
dollar share received by labor with the total agricultural income
generated in the seven-state sample.
The tentative estimates of the elasticity of substitution between
agricultural labor and other productive inputs indicated that these
inputs were relatively good substitutes in 1860. The point estimate of
the ES was greater than one. Ceteris paribus, a fall in the price of
labor would have led to a proportionately greater increase in the
quantity of labor demanded by the industry and the share going to labor
would have increased. A one-tailed t-test indicated that the hypothesis
that these inputs were perfect complements should be rejected at a 5%
level of significance.
The share of income received by agricultural labor in the 1870
South had increased relative to the share received by other agricultural
inputs, but had fallen from the dollar amount received and imputed to
labor in 1860. Since the prices for agricultural products were higher
in 1870 than in 1860, and since the labor force was larger in the
seven states considered, this result indicated that the Civil War had
dealt a severe blow to the non-labor inputs employed in agriculture, a
blow that was still very much in evidence five years after the end of
the conflict. Based upon the estimated change in the share of income
received by these inputs, only about one-half of the farm-capital being
used by the seven states in 1860 was in use in 1870.
Implications of the Analysis
This study has considered an aspect of the economics of slavery
which has not been questioned traditionally -- the relationship between
free and slave labor. Treating slaves as part of the industry labor
force implied a new important role that the capitalization of a large
part of the labor force played in the antebellum South. If there were
an imperfect capital market, and if the larger plantation owners had a
financial advantage in the purchase of slaves, slaves would have been
used more intensively than their free labor counterparts. If these
relationships were present in the antebellum South -- as it appears they
were -- slavery did have a negative effect upon economic growth since
the region was suited for extensive use of labor.
The analysis has also provided greater insight into the immediate
problems of adjustment in the postwar period which had impacts on the
rate of recovery and the slow movement out of agriculture and into
manufacturing. Conrad and Meyer have noted that "a great part of the
increase of wage share in agriculture took place in the period from
1860 to 1870 as a result of the emancipation of the slave-labor force
and the consequent swelling of the hired-labor ranks."I This study
has implied a more fundamental reason for the increase in wage share:
the destruction of large amounts of farm-capital during the War. This
destruction,coupled with an elasticity of substitution greater than one,
implied that in one region, at least, the relative share received by
farm labor increased notwithstanding the institutional change. It
further implies a tentative explanation for the slow movement out of
agriculture. Traditionally, the slow change in the composition of
southern industry has been explained in terms of immobilities. It now
appears that there was an economic rationale for this occurrence. The
destruction led to a relatively higher farm-capital value marginal
product; the rate of return on re-investment in southern agriculture
was attractive. An hypothesis that the slow movement out of agriculture
was prolonged by a relatively high rate of return on farm capital
seems justified. It would appear that immobilities alone do not
completely explain the slow changeover.
But, before the southern recovery is completely understood, a
number of other studies must be accomplished. The postwar financial
institutions are of first importance. The vague qualitative historical
studies of these institutions are far from sufficient for an under-
standing of the total picture. It appears that additional work must
be done on the changes in income shares, too. This study was of the
functional shares in agriculture. To answer questions about the change
in the stream of investment, the change in the distribution of income
by size must be established.2 Further study of free and slave wages
is needed, also.3 The size of the sample used in this study to
estimate the ES was limited to the detail provided by Lebergott's
data. Given the availability of the UNC-Yale manuscript census sample,
productivity data are not the constraint on estimation of the ES. If a
more refined estimate is desired, it must be based on a more detailed
source of antebellum or postwar wages.
A final comment should be made concerning the usefulness of the
procedure which was derived to estimate the change in income shares.
It could prove to be a useful tool for others who may decide to
examine factor shares for different periods. The assumptions are
explicit and are reasonable for a number of problems which have not
been considered, and the estimated change in shares is not too sensitive
to the ES as long as the ES is close to one.4 Usingthis technique the
historian has a choice between the incorporation of price ratio data or
input intensity ratio data. This choice may prove to be a real asset
in some historical studies.
1 Alfred H. Conrad and John R. Meyer, "Income Growth and Structural
Change: The United States in the Nineteenth Century," The
Economics of Slavery and Other Studies in Econometric History,
Eds. the authors (Chicago: Aldine Publishing Company, 1964), p. 170.
2 This point refers back to the Gallman quote at the beginning of
Chapter I. Gallman, "Discussion", p. 169.
3 There is one recent study of changes in the level of Negro agricul-
tural laborers' wages which was not available when this thesis was
written. For a summary of this work see Charles E. Seagrave,
"Summaries of Doctoral Dissertations: The Southern Negro Agricultural
Worker: 1850-1870," Journal of Economic History, XXXI, No. l(March,
1971), pp. 279-280.
4 M. Bronfenbrenner, "A Note on Relative Shares and the Elasticity of
Substitution," Journal of Political Economy, LXVII, No. 3(June,
1960), pp. 284-287.
THE ELASTICITY OF SUBSTITUTION
THE ELASTICITY OF SUBSTITUTION
The concept of the elasticity of substitution (hereafter ES) was
first presented by John R. Hicks in 1932.1 Hicks included it as one
of the three rules governing change in the distribution of income which
occurs when the supply of one factor increases.2 Shortly thereafter,
Joan Robinson defined the concept in two alternative ways, pointing
out that she had made use of the same concept prior to the publication
of Hicks' formulation.3 It is her precise definition of the ES, as
Hicks later remarked, which stimulated further development of the
Mrs. Robinson's first definition was expressed in terms of the
input intensity ratio and the ratio of input prices.5 The second
and more fundamental definition was based on the input intensity ratio
and the ratio of input marginal products. In Mrs. Robinson's words, the
ES is fundamentally "the proportionate change in the ratio of the
amounts of the factors divided by the proportionate change in the
ratio of their marginal physical productivities."6
For the two factor case where output (Q) is a function of two
inputs (K, L), it is possible to define the production function as
Q = F(K, L) and to define the marginal rate of technical substitution
of K for L (SKL) as a ratio of marginal products (FL/FK) or as a
ratio of differentials (-dK/dL). For this general two input production
function the elasticity of substitution can be expressed in terms of
its partial derivatives as7
A- ES FLFK(LFL + KFK) dln(K/L)]
KL(FLLFK 2FKLFKFL + FKKFL2) d[ln FL/FK ]
when the function is linearly homogeneous, this expression reduces to8
(A-lb) ES KL
As an alternative formulation, since the ratio of prices will equal
the ratio of marginal products in equilibrium pure competition, the
ES may be defined as9
(A-2) ES = d[ln(K/L)] ,
where w is the price of L and r is the price of K.
Using either formulation (A-l) or A-lb) or the ES, it can be
seen that it will always take on a value between zero and infinity. The
two limits on the range of values are of special interest since an
ES = 00 implies that the inputs are perfect substitutes and an ES = 0
implies that the inputs are in no way substitutes, but must be used
in certain fixed proportions.
The Substitution Curve
Since the ES is an elasticity between ratios of inputs and not
the single variables, some of the characteristics are not explicit.
These characteristics may be made more obvious by the graphical
presentation of a "substitution curve". In Graph A-i, the various
amounts of K and L which could be used to produce a given level of
output (Q ) are shown by isoquant Q0. Graph A-2 shows two different
mappings of the ratio of inputs into the ratio of input prices.
Notice that the respective ratios are now the variables of interest.
The absolute price of K (r) and the absolute price of L (w) affect the
ratio of prices only to the extent that a change in either affects
the relative price of one factor in terms of the other. The relative
price of K could increase through either an increase in r or a
decrease in w.
Input K Input Price Ratio (w/r)
0 0 I nput
S L2 L1 Input L K 1/L1 K /L2 Intensity
Curve G in graph A-2 permits visual establishment of the ES very
quickly, since the tangent extensions of the curve are analogous to
the tangent extensions of a supply curve. Two such tangents have
been drawn. The tangent at A indicates that the ES is greater than
one for K1/L1. The tangent at B indicates that the elasticity of
K2/L2 with respect to w2/r2 is less than one.
In general, the elasticity of substitution need not be as well-
behaved as is indicated by functions C-D and G. It may be different
at each point along Q0, crossing over between inelastic and elastic
ranges many times, or it may even be discontinuous. But, for
expositional purposes, the two curves are the only ones presented.
Curve C-D is the mapping implicit in all Cobb-Douglas production
functions; it implies that for each K/L ratio the ES equals 1
(since it passes through the origin). The G curve is a more general
case where the elasticity of substitution is shown to be increasing at
first and then monotonically decreasing.10 Both of these curves can
be considered as substitution curves since each does indicate the
relative K/L ratio when output is constant or the production function
is homogeneous and the relative price of the inputs changes.
The seminal exposition of a substitution curve was made by
Abba P. Lerner to provide a diagram "so that the elasticity of
substitution could be read off in the same way that the elasticity of
demand is read off a demand curve."11 His diagram is partially repro-
duced in graph A-3. A comparison of his graph of the substitution
curve with the graph previously presented will point out three minor
advantages in graph A-2 and an interesting conclusion about the use of
the price ratio or its reciprocal.
Input Price Ratio (r/w)
Lerner's Substitution Curve
KK/L1 K2/L2 Input Intensity Ratio (K/L)
The advantages are as follows:
(1) If a simple mapping is desired to permit an easy "read-off"
of elasticity, graph A-2 is preferred. This is easy to see in the
case of the Cobb-Douglas function, since its plot on graph A-2 is a
straight line through the origin, whereas it would be a rectangular
hyperbola on Lerner's graph.
(2) The ES in the A-2 presentation is by definition a positive
number.(0 < ES < "); Lerner's ES is always negative, as is the
elasticity of demand.
(3) The elasticity of a curve in graph A-2 is the elasticity of
the K/L with respect to the marginal rate of technical substitution;
i.e., calling this elasticity ES2,
ES2 d(FL /FK)/(F/FK)
However, the elasticity of Lerner's mapping, here called ES3, is
3 d(FK/FL) /(FK/FL
That the denominators of the two elasticities have the same absolute
value is not immediately obvious. Demonstrating that they are of
equal absolute value will not only show that JES21 = JES31 (since
the numerators are the same value) but will provide justification for
using either the w/r or the r/w relative price ratio in calculations
of ES. To demonstrate that the denominators are of equal absolute
value the following short proof is offered:
Expansion of the ES2 denominator can be shown to equal
K LL2KL L
of the ES3 denominator can be shown to be
F 2 F 2
Each of these expansions will reduce to the same absolute value after
suitable algebraic manipulation has been performed, this value being
- 2F +
Since the numerators are identical, and since the denominators are
equal except for algebraic sign, the absolute values of the elasticities
are the same. This result is important since it implies that either
ratio of prices -- given pure competition in equilibrium -- may be
used to establish the elasticity of substitution.
The Economic Implications of the ES
The ES has two economic implications that are of interest to this
study. On the one hand, it provides a measure of the substitutability
between factors of production and changes in the factor intensities
and factor prices. On the other hand, it indicates the changes in
relative income shares going to the respective inputs.
It has been shown that if factor markets are competitive, and
if output is constant or the production function is a member of the
homothetic class, then the ES indicates relative changes in the input
intensity and input price ratios. Knowing the true value of this
structural parameter would thus permit the establishment of the
percentage change in either the K/L or the w/r ratio from a knowledge
of the percentage change in the other. Based upon these changes in
the respective ratios, it is possible to establish the percentage
change in the relative shares.
If the market and production conditions outlined above hold, it
is possible to define the ratio of relative shares (R) as:
Taking natural logs, the total differential is
d[ln(R)] = d[ln(K/L)] d[ln(w/r)],
which may also be expressed as:
%A(R) = %A(K/L) %A(w/r) .
If, for example, the ES equals 2 and it were known that the input
price ratio had changed by (+)l% (an increase of 1% in w/r or a decrease
of 1% in r/w) one could establish that the K/L ratio had increased
by 2%. From this result it follows that the percent change in the
ratio of relative shares is (+)1%,12 i.e.,
%A(R) = 2% 1% = 1%
The ES will thus indicate what would happen to the competitively
imputed share of output of the more rapidly growing input; whether
it will rise, fall, or stay the same. Though the precise change
depends upon the true value of the ES, the expected cases may be broken
down into the following three categories:
0 < ES < 1 the K/L ratio is inelastic with respect to the w/r
and the imputed share of the more rapidly growing
input will fall,
ES = 1 the K/L ratio is unitary elastic with respect to w/r
and the imputed share of the more rapidly growing factor
will stay the same,
1 < ES < the K/L ratio is elastic with respect to the w/r
and the imputed share of the more rapidly growing factor
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