the marginal physical products of capital and labor
unchanged is neutral. If the marginal physical product
of labor increases more (less) relative to the marginal
physical product of capital, it is a capital-saving
(labor-saving) technological change. The bias of the
technological changee (β) can be defined as:
β = 1 Hicksian neutral
β > 1 Labor-saving (capital-using)
β < 1 Capitabsaving(Iabor-Using) (2)
If each factor of production is paid its marginal
physical product such that total output is just
exhausted, Euler’s Theorem holds, Y = fL L + fκ K.
From Euler’s Theorem it follows that absolute shares
of capital and labor are Kfκ and Lfb, respectively,
and relative shares of capital and labor would be
Rκ = Kfκ∕Y and Rl = Lfb∕Y, respectively.
By differentiating labor’s relative share with
respect to time, and after some algebraic manipula-
tion, the change in labor’s relative share can be
expressed as a function of capital and labor’s absolute
shares, rate of technological change, bias of the
technological change, and elasticity of factor sub-
stitution. Differentiating labor’s relative share with
respect to time gives:
d(Rb∕Y)
dt = d(Lfb∕Y)∕dt
Y(Lfτ τ
4___Lj Li
+ fτ ) - Lfτ dY
----------b--- /dt
(3)
Substituting for Y and dY from Euler’s Theorem,
expanding, and rearranging terms yields:
Partially differentiating Euler’s Theorem with
respect to L yields:
<')Y
|£ = KfKL + LfLL + fL (6)
Substituting definitions (1) and (5) and the
derivations from (6) into (4), with some rearranging
of terms, yields: `
d(Rb∕Y) σ-l
-⅛- = Rl Rk λ(β- 1) (⅛⅛ (7)
Equation (7) expresses labor’s relative share as a
function of five parameters. By definition labor and
capital’s absolute shares (Rl and Rκ) are always
positive. Also, λ, the proportional increase in the
effective quantity of capital (K) per unit of time, is
positive. Hence, changes in labor’s relative share are
determined by two parameters: (a) bias of the tech-
nological change (β), and (b) elasticity of factor
substitution (σ)'. Once values of these two parameters
are known, changes in labor’s relative share can be
ascertained.
If either β or σ equals one, any change in
quantity of labor used will have no effect on labor’s
relative share. However, if β is greater than one
(labor-saving technological change), substitution of
capital for labor will decrease labor’s relative share
only if σ is greater than one. If β is greater than one
and σ is less than one, a decrease in use of labor will
increase labor’s relative share! The converse would be
true when β is less than one (capital-saving tech-
nological change). Table 1 summarizes the various
possible changes in labor’s relative share for different
values of β and σ, assuming a decline in use of labor in
a given economic sector or industry.
d(R1√Y) 1
dt Y2
Kfκ (fL dL∕dt + LfLL dL∕dt
+ LfLK dK∕dt) - Lfb (fκ dK∕dt
+ Kfκκ dK∕dt + KfKL dL∕dt) (4)
Johnson’s [7] definition of technical progress
over time is:
βλ= 1/L ∙dL∕dt = β ∙1∕K ∙dK∕dt (5)
where λ equals time derivative of technical change.
TABLEl. LABOR’S RELATIVE SHARE, ELAS-
TICITY OF SUBSTITUTION AND BIAS
OF TECHNOLOGICAL CHANGE
|
a |
β |
Change |
in Labor's Relacive Share Over Time |
|
- 1 |
No Change | ||
|
- 1 |
No Change | ||
|
> 1 |
> 1 |
Decrease | |
|
> 1 |
< 1 |
Increase | |
|
< 1 |
> 1 |
Increase | |
|
< 1 |
< 1 |
Decrease |
1 For more detail on algebra involved in this derivation see Johnson [7] and Martin [13].
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