Chapter 10.
Productivity Measures: Business Sector and Major
Subsectors
Calculation
Procedures
Labor productivity Labor productivity, or output per hour, is computed
as:
Labor productivity = (Output index) /
(Hours of labor input)
or
P = O / H
The computation of labor compensation
per hour parallels the computation of output per hour.
Unit labor costs (ULC) are computed as labor compensation
(C) per unit of output, but are often represented as:
ULC = (C / H) / (O / H)
This form highlights the relationships
between unit labor costs, hourly compensation, and labor
productivity.
Real compensation per hour (RC) is
computed as hourly compensation deflated by the
seasonally adjusted Consumer Price Index for All Urban
Consumers (CPIU):
RC = (C / H) / CPIU
Unit nonlabor payments (UNLP) include
all nonlabor components of gross product originating in a
given sector — depreciation, rent, interest, and
indirect business taxes as well as profits and
profittype income — whereas unit nonlabor cost (UNLC)
excludes profit. These measures are computed as:
UNLP = (CU  C) / O
and
UNLC = (CU  C  PR) / O
where:
 CU is currentdollar output
 C is currentdollar compensation
 O is the output index
 PR is currentdollar profits.
Labor's share in current dollar
output in a given sector is simply the ratio of labor
compensation paid in that sector to current dollar
output:
LS = C / CU
and, analogously, the nonlabor or
capital share is defined as:
NLS = (CU  C) / CU = 1
 LS.
Most of the measures noted above are
presented quarterly in index form. Indexes are computed
from basic data or analytic ratios by dividing the series
by its own base year annual value (presently 1992) and
multiplying by 100. In addition, quarterly percent
changes at a compound annual rate and percent changes
from the same quarter in the previous year are computed:^{20}
Q_{t} = 100 (V_{t}
/ V_{t1}) ^{4}  100
Y_{t} = 100 (V_{t}
/ V_{t4})  100,
where:
t is a time subscript denoting the
quarter,
V is a series described above,
Q_{t} is the quarterly
percentage change in series V from quarter t1 to quarter
t, measured at a compound annual rate,
Y_{t} is the percentage change
in series V from quarter t4 (the same quarter 1 year
before) to quarter t.
Indexes and percent changes are
published to one decimal point. In order not to lose
precision, all computations are made from the underlying
measures themselves rather than from the published
indexes.
Multifactor productivity
BLS aggregates inputs for its multifactor productivity
measures using a Tornqvist chain index. Some of the basic
properties of this index are: It is calculated as a
weighted average of growth rates of the components; the
weights are allowed to vary for each time period; and the
weights are defined as the mean of the relative
compensation shares of the components in two adjacent
years. Hence, the growth rate of the index (I) for major
sectors is the proportional change over time (the
triangle (delta) refers to discrete change with respect
to time), such that:
% Δ I =
exp(Δ ln I) =
exp {1/2 * [^{s}k(t)
+ ^{s}k(t1)]
Δ ln K
+ 1/2 * [^{s}l(t) + ^{s}l(t1)] Δ ln L}
where ^{s}l(t) = labor costs(t)
/ total costs(t)
and ^{s}k(t) = capital costs(t) / total costs(t)
Similarly, both capital, K, and labor,
L are Tornqvist indexes. Each is a weighted average of
the growth rates of detailed types of capital, k_{i},
and labor inputs, l_{i}, respectively.
% Δ K = exp ( Δ ln K)
= exp{S_{i} 1/2 * [^{s}ki^{(t)} + ^{s}ki^{(t1)}] Δ ln k_{i}}
where ^{s}ki^{(t)} = ^{c}ki^{(t)}
* k_{i}^{(t)}/ total capital costs
and where^{ c}ki^{(t)} is the rental
price for capital asset k_{i}.
% Δ L = exp ( Δ ln L)
= exp{Si 1/2 * [^{s}li^{(t)} + ^{s}li^{(t1)}] Δ ln l_{i}}
where ^{s}li^{(t)} = ^{w}li^{(t)}
* l_{i}(t)/ total capital costs
and ^{w}li^{(t)} is the hourly
compensation for worker group l_{i}.
Changes in the index of labor
composition, LC, are defined as the difference between
changes in the aggregate labor input index, L, and the
simple sum of the hours of all persons, H.
% Δ LC = exp (Δ ln LC) = exp (Δ ln L  Δ ln H)
The Tornqvist index for major sector
multifactor productivity growth, A, is:
% Δ A = exp (Δ ln A) = exp(Δ ln Q  Δ ln I)
where Q is the FisherIdeal index of
sector output as measured by BLS.
For manufacturing and the 20 industries
which comprise manufacturing, aggregate input has a
conceptually similar definition except that there are 5
inputs rather than just the 2 used in the major sector
measures.
% Δ I = exp (Δ ln I) =
exp{1/2 * [sk(t) + sk(t1)) Δ ln K
+ 1/2 * [sl(t) + sl(t1)] Δ ln L
+ 1/2 * [^{s}e(t) + ^{s}e^{(t1)}]
Δ ln E
+ 1/2 * [^{s}m(t) + ^{s}m^{(t1)}]
Δ ln M
+ 1/2 * [^{s}s(t) + ^{s}s^{(t1)}]
Δ ln S}
where L = total hours at work
^{s}l(t) = labor
costs(t)/total costs(t)
^{s}k(t) = capital costs(t)
/ total costs(t)
^{s}e(t) = energy
costs(t)/total costs(t)
^{s}m(t) = materials
costs(t) / total costs(t)
^{s}s(t) = purchased
business services costs(t) / total costs(t)
and total costs are the current dollar
value of shipments adjusted for inventory change.
Using this definition for aggregate
input, multifactor productivity for manufacturing or any
of the 20 industries which comprise manufacturing is
identically defined as above.
% Δ A =
exp (Δ ln A) = exp(Δ ln Q  Δ ln I)
where Q is a Tornqvist output index
developed by BLS.
Footnotes ^{20} The estimation of quarterly (or subannual)
changes at compound annual rates as the differences
between movements in the underlying series involves
approximations. For changes in the neighborhood of 1 or 2
percent, these approximations are good; however, the
inexactness of these approximations is amplified by
relatively large changes in the economic measures such as
those experienced during periods of inflation, sharp
recession, and rapid recovery.
Since most of the productivity and
costs measures are reported as percentages to one decimal
place, e.g., 2.6 percent, questions sometimes arise
because the greater precision carried in the automated
computation results in differences in related measures in
the final decimal place.
Next: Uses and
Limitations
