labor (β> 1), a decline in cotton labor’s relative share
would require an elasticity of factor substitution
greater than one for the U.S. cotton sector (Table 1).
An elasticity of factor substitution greater than one
would reflect relative ease of substitution of capital
for labor in response to a secular increase in the
wage-rental ratio.
Time series data on the stock of capital invested
in machinery used in cotton production are not
available. Thus, it is not possible to estimate the
elasticity of factor substitution (σ) based on a CES
production function where the capital-labor ratio and
the wage-rental ratio are used as explanatory vari-
ables. However, it is feasible with available data, to
use two other alternative approaches to estimate σ.
The CES production function may be expressed
as:
Y = (αo t^'k K)~p + (βo L)^~p ~^1'p (8)
where:
Y = output
K = capital
L = labor
p = substitution parameters
αo, β0 = distribution parameters, and
tγk, p½' — ra∣-e 0f factor augmentation for capital
and labor respectively [11].
Differentiating (8) with respect to labor (L)
yields:
9Y∕3L = (Y∕L)1+p (βo Pβ)-P (9)
Assuming the real wage rate (w) is equal to the
marginal physical product of labor (9Y∕9L), re-
arranging and substituting terms in (9), and convert-
ing to logarithms gives:
log Sl = (σ-l) log ∕3o + (l-σ) log w
+ γβ(σ-1) log t (10)
Based on data in Table 2, the following estimates
were obtained by ordinary least squares.3
log Sl = -0.473 - 0.509 log w - 0.336 log t(ll)
(0.478) (0.062)
R2 = .90
d' = 1.056
Although the Durbin-Watson (d') is in the
inconclusive range and the coefficient of the real
wage variable is significant only at the 0.15 level,
statistical results are consistent with a priori expecta-
tions. Given the estimated coefficient for the wage
variable, elasticity of factor substitution is 1.5.
An alternative method of estimating elasticity of
factor substitution is suggested by R. G. D. Allen [1,
p. 373].
El = -(1-Sb)(σ) + (Sl)(T7) (12)
where
El = price elasticity of demand for labor
Sl = labor’s relative share
η = price elasticity of product demand, and
σ = elasticity of factor substitution.
Tyrchniewicz and Schuh [15] report a long-run
price elasticity of demand for hired farm labor in the
United States of —0.49 and for unpaid family labor
of —3.0. Wallace and Hoover [19] estimated a price
elasticity of demand for hired and family farm labor
of —1.433. Unpaid family labor and operator labor
represent a major portion of the traditional share-
cropper cotton labor force which has been replaced
with the modernization of cotton production.4
Hence, long-run price elasticities of demand for
cotton labor of —1.0 and —1.5 appear to be reason-
able estimates.
Blakeley’s [2] and Martin’s [13] estimates of the
price elasticity of demand for cotton are —0.86 and
—0.89, respectively. Cotton labor’s average relative
share for the period 1952-1969 is 0.23.
Using these parameter estimates, the Allen
formula gives values between 1.0 and 1.7 for the
elasticity of factor substitution. These estimates are
consistent with the previous estimate of σ.
SUMMARY AND CONCLUSIONS
Although knowledge of bias of the technological
change occurring in a given economic sector may be
indicative of how labor’s relative share of output
value may change, knowledge of elasticity of factor
substitution of capital for labor is required before any
conclusive statement can be made about how labor-
saving technology may be affecting labor’s relative
share. If a capital input can be easily substituted for
labor, then labor’s relative share will tend to decline.
If, however, the ease of substitution of capital for
ɜ Standard errors are contained in parentheses under their respective regression coefficients.
4In 1959 about 15 percent of the U.S. Cotton crop was grown by 65 percent of the cotton producers. These farms relied
heavily on family and operator labor [3] .
140