ISSN 1881-6436

Discussion Paper SeriesNo. 11-03
Returns to scale effect in labour productivity growth
Hideyuki Mizobuchi
March 2012

Faculty of Economics,
Ryukoku University
67 Tsukamoto-cho, Fukakusa, Fushimi-ku,
Kyoto, Japan
612-8577

Returns to scale effect in labour productivity
growthHideyuki Mizobuchi
March 2012

Abstract
Labour productivity is defined as output per unit of labour input. Economists acknowledge thattechnical progress and growth in capital inputs increase labour productivity. However, less focus isgiven to the fact that changes in labour input alone could also affect labour productivity. Because
this effect disappears for the constant returns to scale short-run production frontier, we call it thereturns to scale effect. We decompose growth in labour productivity into two components: 1) the
joint effect of technical progress and capital input growth and 2) the returns to scale effect. We
propose theoretical measures for these two components and show that they coincide with the index
number formulae consisting of prices and quantities of labour inputs and outputs. We then apply
the results of our decomposition to US industry data for 1987–2009. Labour productivity in theservices sector is acknowledged to grow much more slowly than in the goods sector during the
productivity slowdown period. We conclude that the returns to scale effect can explain a large partof the gap in labour productivity growth between the two industry groups.

Keywords: Labour productivity, index numbers, Malmquist index, Törnqvist index,
output distance function, input distance function
JEL classification: C14, D24, O47, O51
Acknowledgement: A part of this paper was circulated under the title, ―New Indexes of Labour
Productivity Growth: Baumol‘s Disease Revisited‖. We are grateful to Bert Balk for his helpfulcomments and suggestions. We also wish to thank Erwin Diewert, Jiro Nemoto, Takanobu
Nakajima, Toshiyuki Matsuura, Mitsuru Sunada and seminar participants in the 12 th European
Workshop on Efficiency and Productivity Analysis, June 2011, Verona, Italy and the annual
meeting of the Japanese Economic Association, Kumamoto Gakuen University, May, 2011,
Tohoku University, Keio University, Chuo University, Tokyo University, Kyoto University. Allremaining errors are the author‘s responsibility.

Corresponding author: Department of Economics, Ryukoku University, 67
Fukakusa Tsukamoto-cho, Fushimi-ku, Kyoto 612-8577, Japan. +81-75-645-8439,
mizobuchi@econ.ryukoku.ac.jp

11. Introduction
Economists broadly think of productivity as measuring the current state of thetechnology used in producing the firm‘s goods and services. The production
frontier, consisting of inputs and the maximum output attainable from them,characterises the prevailing state of the technology. Productivity growth is often
identified by a shift in the production frontier, reflecting changes in production
technology. 1 However, movement along the production frontier also derives
productivity growth.2
Even in the absence of changes in the production frontier, changes in the
inputs used for production can lead to productivity growth moving along the
production frontier and making use of its curvature. Productivity growth inducedby movement along the production frontier is called the returns to scale effect.
This effect does not reflect changes in the production frontier. Thus, to properly
evaluate improvements in the underlying production technology reflecting a shift
in the production frontier, we must disentangle the returns to scale effect fromoverall productivity growth.
Productivity measures can be classified into two types: total factor
productivity (TFP) and partial factor productivity. The former index relates abundle of total inputs to outputs, whereas the latter index relates a portion of total
inputs to outputs. The present paper deals with labour productivity (LP) among
several measures of partial factor productivity. LP is defined as output per labour
input in the simple one-output, one-labour-input case. Economy-wide LP is thecritical determinant of a country‘s standard of living in the long run. For example,
US history reveals that increases in LP have translated to nearly one-for-one
increases in per capita income over a long period.3 The importance of LP as asource for the progress of economic well-being prompts many researchers toinvestigate the determinants of LP growth. Technical progress and capital input
growth have been emphasised as the main determinants of a country‘s enormous
LP growth over long periods (Jorgenson and Stiroh 2000, Jones 2002) as well as
the wide differences in LP across countries (Hall and Jones 1999). 4 The present
paper adds one more explanatory factor to LP growth.

1

See Griliches (1987). Moreover, the same interpretation is found in Chambers (1988).
In principle, productivity improvement can also occur through gains in technical efficiency.
Technical efficiency is the distance between the production plan and the production frontier. The
present paper assumes a firm‘s profit-maximising behaviour, and in our model, the current
production plan is always on the current production frontier. The assumption of profit
maximisation is common in economic approaches to index numbers. See Caves, Christensen andDiewert (1982) and Diewert and Morrison (1986).
3
See the 2010 Economic Report of the President.
4
In addition, these authors found that improvements in the quality of labour inputs (in otherwords, human capital accumulation) play an important role for explaining changes in LP. If we
adopt the method of these authors; that is the number of workers or the number of hours worked
are adopted as the measure of labour input, changes in the quality of labour input raise the amountof output attainable from a given number of workers or a given hours worked, leading to an
outward shift in the short-run production frontier. However, because we differentiate qualities ofdifferent labour inputs, allowing wages to vary among them, our measure of changes in the total
labour input reflects changes in labour qualities among varieties of labour inputs. Thus,
improvements in labour qualities do not affect the short-run production frontier itself and we canignore the role of the labour quality growth for explaining LP growth throughout this paper. See
Footnote 5 for the unmeasured improvement in labour quality.
2

LP relates labour inputs to outputs, holding technology and capital inputs
fixed. The short-run production frontier, which consists of labour inputs and the
maximum output attainable from them, represents the capacity of current
technology to translate labour inputs into outputs. Both technical progress andcapital input growth, which have been identified as the sources of LP growth,
induce LP growth throughout the shift in the short-run production frontier.
However, the returns to scale effect, which is the extent of LP growth induced by
movement along the short-run production frontier, has never been exposed.
We decompose LP growth into two components: 1) the joint effect oftechnical progress and capital input growth and 2) the returns to scale effect.5
First, we propose theoretical measures representing the two effects by using theshort-run distance functions. Second, we derive the index number formulae
consisting of prices and quantities and show that they coincide with theoreticalmeasures, assuming the translog functional form for the short-run distancefunctions and the firm‘s profit-maximising behaviour.
Our approach to implementing theoretical measures is drawn from Caves,
Christensen and Diewert (1982) (CCD). Using the distance functions, CCD
formulated the (theoretical) Malmquist productivity index that measures the shift
in the production frontier, and show that the Malmquist productivity and the
Törnqvist productivity indexes coincide, assuming the translog functional form
for the distance functions and the firm‘s profit-maximising behaviour.6
The Törnqvist productivity index is a measure of the TFP growth calculatedby the Törnqvist quantity indexes. It is an index number formula consisting ofprices and quantities of inputs and outputs. Equivalence between the two indexes
breaks down if the underlying technology does not exhibit constant returns toscale. CCD showed that its difference depends on the degree of returns to scale in
the underlying technology, which captures the curvature of the production
frontier. Thus, following Diewert and Nakamura (2007) and Diewert and Fox(2010), we can interpret that CCD decomposed the TFP growth calculated by the
Törnqvist quantity indexes into the Malmquist productivity index and the returns
to scale effect.7 The former component captures TFP growth induced by the shift
in the production frontier. The latter component, which is the difference between
the Malmquist productivity and the Törnqvist productivity indexes, captures TFP
growth induced by the movement along the production frontier exploiting its
curvature.
CCD‘s formula for the returns to scale effect appeared as the residual of two
indexes and CCD did not explicitly model the returns to scale effect using theunderlying production frontier. 8 On the other hand, other studies model the5

In case when our measure of labour inputs fails to capture the improvement in labour quality, theunmeasured improvement in labour quality shifts the short-run production frontier. Thus, its effecton LP growth is captured by the joint effect of technical progress and capital input growth.
6
Since CCD are concerned with measurement of TFP, they deal with the underlying production
frontier that consists of total inputs (capital and labour inputs) and the maximum output attainablefrom them, indicating the capacity of current technology to translate total inputs into outputs. From
this point forward, ‗the underlying production frontier‘ or simply ‗the production frontier‘ means
this type of the underlying production frontier, in distinction from the short-run production frontier.
7
CCD used the word of ‗scale factor‘ for the returns to scale effect.
8
In the present paper, we show that our index number formula for the returns to scale effectcoincides with the growth in LP induced by movement along the short-run production frontier. As

growth in TFP induced by the movement along the underlying production frontier
but adopt different approaches to estimating the modelled returns to scale effectrather than relying on index number formulae. Lovell (2003) modelled the returns
to scale effect by using input and output distance functions and calls it the scaleeffect or activity effect. In Balk‘s (2001) decomposition of TFP growth, theproduct of scale efficiency change and input mix effect or that of scale efficiencychange and output mix effect summarised the TFP growth induced by movement
along the production frontier, and it can be interpreted as the returns to scaleeffect.9
Although scholars have recognised the significance of the returns to scaleeffect for TFP growth, its effect on LP growth has never been addressed eventhough it plays a more important role in explaining LP growth than in explaining
TFP growth. When the underlying technology exhibits constant returns to scale,the returns to scale effect disappears from TFP growth. However, it still plays arole in LP growth because even if the underlying technology exhibits constantreturns to scale, the short-run production frontier is likely not to exhibit constantreturns to scale.
Triplett and Bosworth (2004, 2006) and Bosworth and Triplett (2007)
observed that LP growth in the services sector of the US economy has been muchslower than in the goods sector since the early 1970s. As we discussed above,
there are two underlying factors to LP growth. Thus, different explanations are
possible for its stagnated LP growth depending on the factor emphasised. We
apply our decomposition result to US industry data to compare the relative
contributions of the two effects.
Section 2 graphically illustrates the two effects underlying LP growth. Section
3 discusses the measure of the joint effect of technical progress and capital input
growth in the multiple-inputs multiple-outputs case. Section 4 includes the main
result. It discusses the measure of the returns to scale effect in the multiple-inputs
multiple-outputs case. We show that the product of the joint effect of technicalprogress and capital input growth and the returns to scale effect coincides with LP
growth. Section 5 includes the application to US industry data. Section 6 presents
the conclusions.
2. Two Sources of Labour Productivity Growth
We graphically display the drivers of LP growth using a simple model of one
output y and two inputs, labour input
and capital input. Suppose that afirm produces outputs
andusing inputs
and
. The
period production technology is described by the period production frontierforand . Let us begin by considering how this joint
effect of technical progress and capital input growth raises LP. Figure 1 illustrates
the case in which the joint effect of technical progress and capital input growth
will be emphasised further on, our result implies that CCD‘s formula for the returns to scale effectcoincides with the TFP growth induced by the movement along the underlying production frontier.
9
For the decomposition of Nemoto and Goto (2005), we interpret the product of ‗scale change‘
and ‗input and output mix effects‘ as the returns to scale effect. Their result identified the
combined effect of changes in the composition of inputs and of outputs.

positively affects the productive capacity of labour. The lower curve represents
the period
short-run production frontier and indicates how much output can beproduced using a specified quantity of labour given capital input and technology
available in period . Similarly, the higher curve represents the period
shortrun production frontier and indicates how much output can be produced using a
specified quantity of labour given capital input and technology available in period.
Since the short-run production frontier shifts upward, the output attainablefrom a given labour input
increases between the two periods such thatfor all . Moreover, the corresponding LP grows such⁄
⁄ . Thus, the ratio⁄
that
captures the joint effect on LP growth oftechnical progress and capital input growth. Additionally, note that the ratio is ameasure of the distance between the short-run production frontiers of periods
andin the direction of theaxis, evaluated at . The ratio increases as the
distance between the period
and the period
short-run production frontiers
increases. Therefore, the joint effect of technical progress and capital input growth
can be captured throughout by measuring the shift in the short-run production
frontier.
[Place Figure 1 appropriately here]
Any quantity of labour input can produce more output in period
than inperiod , reflecting the positive joint effect of technical progress and capital input
growth. The firm increases its demand for labour input fromto, exploiting
the increased productive capacity of labour input. Suppose that production takes
place atfor period
and atfor period . The slope of the ray from theorigin toandindicates the LP of each period. Sinceis smaller than
, LP declines between the two periods. That LP can decline despite theoutward shift in the short-run production frontier suggests that another factor
contributes to LP growth.10 The path fromtocan be divided into two parts:
the vertical jump fromtoand the movement along the period 1 short-run
production frontier fromto . Along the vertical jump fromto , the⁄⁄changes fromto
. Its ratio
isconsidered to be the growth in LP induced by the shift in the short-run production
frontier, which is the joint effect of technical progress and capital input growth.
However, LP growth is offset by the change in labour input fromto. The
movement along the period
short-run production frontier fromto⁄reduces LP fromto
. We call the LP growth induced by⁄⁄
movement along the short-run production frontier
thereturns to scale effect.
However, the division of the path fromtointo two steps fromtoand fromtois only an example. Decomposing the path fromtointo
the movement along the period 0 short-run production frontier fromtoandthe vertical jump fromtois also possible. In this case, the former

10This is just an example of the fact that the shift in the short-run production frontier is not the
only contribution factor to LP growth. We do not exclude the case in which LP increases under theoutward shift in the short-run production frontier.

movement reflects the returns to scale effect, and the latter jump reflects the joint
effect of technical progress and capital input growth.
For measuring the joint effect of technical progress and capital input growth,
the important consideration is the quantity of labour input at which the distance
between two short-run production frontiers is evaluated. For measuring the returns
to scale effect, whether we consider the movement along the period 0 or 1, shortrun production frontier is significant. Hereafter, we generalise our discussion to
the more general multiple-inputs multiple-outputs case and propose measures forthe two effects that are immune from the selection of the arbitrary benchmark.
3. Joint Effect of Technical Progress and Capital Input Growth
A firm is considered as a productive entity transforming inputs into outputs. We
assume there are(net) outputs,
and
inputsconsisting oftypes of capital inputs,(); andtypes oflabour inputs,(). Outputs include intermediate inputs. If output
m is an intermediate input, then
. If output is not an intermediate input
but a (gross) output,. We also assume that any outputs and labour inputs
are non-zero such thatfor allandfor all .11 The period
production possibility set
consists of all feasible combinations of inputs andoutputs, and it is defined as
{

}.

(1)

We assume
satisfies convexity and Färe and Primont‘s (1995) axioms thatguarantee the existence of distance functions.12 The period production frontier,which is the boundary of , is represented by the period input requirement
functionand it is defined as follows:
{

(

)

}.

(2)

represents the minimum amount of the first labour input that a firm can use atperiod , producing output quantities y and holding capital inputs
and otherlabour inputs() fixed. This function, originally formulated
for characterising the period
production frontier, can also be used forcharacterising the period short-run production frontier. Given period capital
input, the set of labour inputs
and outputs
satisfying() forms the period short-run production frontier.

11

Excluding zero quantities for outputs and labour inputs is a crucial condition for deriving index
number formulae by aggregating the growth rates of input and output. Since capital input
quantities do not appear in the index number formula of the present paper, we do not impose thecondition of non-zero quantities for the capital inputs. There is an alternative approach in index
number theory called the ‗difference approach‘. As emphasised by Diewert and Mizobuchi (2009),
the difference approach can apply to the situation that there are certain inputs or outputs whose
quantities are zero.
12
Originally, Färe and Primont‘s (1995) axioms are for input and output distance function.
Moreover, they guarantee the existence of the labour input distance function that we introduce inthis paper.

We assume that() is differentiable atwithrespect to
and
for
and
and satisfies the following conditions.13
These conditions are necessary for discussing the relationship between the input
requirement function and the distance functions, which we introduce later.
(

)

,

(

(3)

)

.

(4)

CCD measured the shift in the production frontier using the output distancefunction. Adjusting their approach, we also use the output distance function tomeasure the shift in the short-run production frontier. Using the input requirement
function, the period output distance function forandis defined asfollows:
{

(

)

}.

(5)

Given capital inputs
and labour inputs,
is the minimum
contraction of outputs
enabling the contracted outputs,
capital inputs
and labour inputs
to fall on the period
production
frontier. If
is on the period
production frontier,equals . Note thatis linearly homogeneous in .
Furthermore, we can relate the period output distance function to the period
short-run production frontier. Given labour inputs,
is theminimum contraction of outputs
causing the contracted outputs
and labour inputs
to fall on the period
short-runproduction frontier. Thus,
provides a radial measure of the
distance ofto the period short-run production frontier. We measure the shift
in the short-run production frontier by comparing the radial distances fromtothe short-run production frontiers of the periods
and , which is defined asfollows:14
(

)

(

)

.

(6)

If technical progress and capital input growth have a positive effect on the
productive capacity of labour between periods
and , the short-run production
frontier shifts outward. Given labour inputs, more outputs can be produced.
Thus, the minimum contraction factor for given outputs
declines such that, leading to. Similarly, thenegative joint effect of technical progress and capital input growth leads to.
Each choice of reference vectors
might generate a different measureof the shift in the short-run production frontier from periods
to . We calculate
two measures using different reference vectors
and. Because
these reference outputs and labour inputs are, in fact, chosen in each period, they[
] is a column vector of the partial⁄⁄
Notation:∑
derivative ofwith respect to the vector
, and
.
14
CCD and Färe et al. (1994) introduced a measure of the shift in the production frontier using theratio of the output distance function. Given, Färe et al. (1994) measured the shift in the
production frontier by.
13are equally reasonable. Following Fisher (1922) and CCD, we use the geometricmean of these measures as a theoretical measure of the joint effect of technicalprogress and capital input growth,, as follows:15
√

(7)

.

The case of one output and one labour input offers a graphical interpretation of. In Figure 1, it is reduced to the following formula:
√

.

(8)

Given a quantity of labour input, the ratio of the output attainable from such alabour input at period
to the output attainable at period
represents the extent
to which the short-run production frontier expands.
is the geometricmean of those ratios conditional on
and.
is a theoretical measure defined by the unknown distance functions,and there are several methods of implementing it. We show that the theoreticalmeasure coincides with a formula of price and quantity observations under theassumption of a firm‘s short-run profit-maximising behaviour and a translog
functional form for the output distance function.16 Our approach is drawn from
CCD, which deal with the Malmquist productivity index, a theoretical measure of
the shift in the production frontier.
CCD showed that the first-order derivatives of the output distance function
with respect to quantities at the period actual production planarecomputable from price and quantity observations under the assumption of a firm‘sprofit-maximising behaviour and a translog functional form for the output
distance function. Then, using these relationships, CCD showed that the
Malmquist productivity index coincides with a different index number formula ofprice and quantity observations, the Törnqvist productivity index. 17 The
following equations (14) and (15), already derived by CCD, allow us to computethe first-order derivatives of the output distance function from price and quantity
observations. Moreover, they can be derived under our assumption in the same
way as CCD. For completeness of discussion, we outline below how to obtainthese equations.
The implicit function theorem is applied to the input requirement function( ⁄)to solve foraround.

15

Since the firm‘s short-run profit maximisation is assumed, it is possible to adopt a different
formulation for the measure of the shift in the short-run production frontier:
√(

(

)

(

)

)(

(

)

(

)

).

This above formulation is closer to the Malmquist productivity index introduced by CCD.
16
Alternative approaches involve estimating the underlying distance function by econometric orlinear programming approaches. Either approach requires sufficient empirical observations. Our
approach, originated by CCD, is applicable so long as price and quantity observations are available
for the current and the reference periods. See Nishimizu and Page (1982) for the application of theeconometric approach, and see Färe et al. (1994) for the application of the linear programming
approach.
17
CCD justified the use of the Törnqvist productivity index, which is the Törnqvist output
quantity index divided by the Törnqvist input quantity index.

(). We have the(),

Its derivatives are represented by the derivatives offollowing equations forand :18
(

)

(

)

[

(9)
)].

(

(10)

We assume the firm‘s short-run profit-maximising behaviour. Thus,
is asolution to the following period short-run profit- maximisation problem forand :19
{

(

)

}.

(11)

Outputs are sold at the positive producer prices, capital
inputs are purchased at the positive rental prices
andlabour inputs are purchased at the positive wages().
Note that() . The period
short-run profit-maximisation
problem yields the following first-order conditions for
and :
(

),

(12)

(

).

(13)

By substituting equations (12) and (13) into equations (9) and (10), we obtain thefollowing equations (14) and (15) for
and :20
⁄
[
[ ⁄

⁄

,

][
][

(14)
(

]

)]
(15)

Equations (14) and (15) allow us to compute derivatives of the distance function
without knowing the output distance function itself. Information concerning thederivatives is useful for calculating values of the output distance functions.
However, one disadvantage is that the derivatives of the period output distancefunction need to be evaluated at the period actual production planin equations (14) and (15) for
and . The output distance functions
evaluated at the production plan in different period such as
andalso constitute. Hence the above equations are
insufficient for implementing. In addition to a firm‘s short-run profit
maximisation, we further assume a translog functional form with time-invariant
18

Equation (3) implies that equations (9) and (10) are well defined.
We assume a firm‘s short-run profit-maximising behaviour, unlike CCD‘s assumption that evencapital inputs are optimally chosen. Thus, under our assumption, we cannot compute the first
derivatives of the output distance function with respect to capital inputs from price and quantity
observations. However, it is unnecessary to use the capital input counterpart to equations (14) and(15).
20
Equations (3) and (4) implies, meaning that equations (14) and (15) are well defined.
19second-order coefficients for the period
and , which is defined as following:
∑

( )∑

∑

( )∑

∑

∑

( )∑

∑

∑

∑

∑

∑

output distance function for

∑
(16)
∑

∑

where the parameters satisfy the following restrictions:for allandsuch as;(17)

for allandsuch as;(18)

for allandsuch as

;

(19)

∑
∑
∑
∑

(20)

;
for

;

(21)

for

; and

(22)

.

(23)

for

Restrictions (20)–(23) guarantee linear homogeneity in . The translog functional
form characterised in (16)–(23) is a flexible functional form, enabling it toapproximate an arbitrary output distance function to the second order at anarbitrary point. Thus, the assumption of this functional form does not harm any
generality of the output distance function. Note that the coefficients for the linear
terms and the constant term are allowed to vary across periods. Thus, technicalprogress under the translog distance function is by no means limited to Hicks
neutral, and various types of technical progress are allowed.
Under the assumptions of the short-run profit-maximising behaviour and thetranslog functional form, a theoretical measure
coincides with a formulaof price and quantity observations, as is shown in the following proposition. The
proof that CCD showed the equivalence between the Malmquist and the Törnqvist
productivity indices using equations (14) and (15) and the capital input
counterpart still goes through for. Thus, we can consider the following
proposition as a corollary of CCD.21
Proposition 1 (Christensen, Caves and Diewert, 1982)
Assume the following: output distance functions
andhave the translog
functional form with time-invariant second-order coefficients defined byequations (16)–(23); a firm follows short-run profit-maximising behaviour inperiods
and , as in equation (11). Then, the joint effect of technical21

Balk (1998) derived the same result as CCD under a more general condition, allowing technicalinefficiency. Balk‘s approach would allow us to generalise the result of the present paper so that it
still holds under the existence of a technical inefficiency type.

progress and capital input growth,and quantities as follows:, can be computed from observed prices
∏
∏

(
(

⁄

)

⁄

)

̅

,

where
and
are the average value-added shares of output
input , respectively, between periods
andsuch that
(

) and

(

(24)
and labour)

The index number formula in equation (24) can be interpreted as the ratio of aquantity index of output to a quantity index of labour input. Note that no data on
price and quantity of capital inputs appear in this formula. Although the shift inthe short-run production frontier reflects technical progress as well as the change
in capital input, we can measure its shift without explicitly resorting to capital
input data.
4. Returns to Scale EffectAs shown in Figure 1, the shift in the short-run production frontier is not the only
factor contributing to the growth in LP. Even when there is no change in the shortrun production frontier, the movement along the short-run frontier could raise LP,
exploiting the curvature of the short-run production frontier. We refer to LP
growth induced by the movement along the short-run production frontier as thereturns to scale effect. In the simple model consisting of one output and one
labour input, LP is defined as output per one unit of labour input. Therefore, LP
growth, which is the growth rate of LP from the previous period to the current
period, coincides with the ratio of the growth rate of output to the growth rate oflabour input. Since the returns to scale effect is the LP growth induced by the
movement along the short-run production frontier, it is computed by the growth
rates of output and labour input between the two endpoints of the movement.
Figure 2 shows how the movement along the period
short-run production
frontier from pointtoaffects. Comparing points
and , the growthrate of output isand the growth rate of labour input is. The growth rate of LP between the two points coincides with the growthrate of output divided by that of labour input in order that⁄
⁄ ⁄.
[Place Figure 2 appropriately here]
We generalise the growth rates of labour input and output between two points
on the period short-run production frontier to measure the returns to scaleeffect in the multiple-inputs multiple-outputs case. First, we investigate thecounterpart of the growth rate of labour inputs in the multiple-inputs multipleoutputs case. CCD defined the input quantity index, which is the counterpart of
the growth rate of total inputs, by comparing the radial distances fromthe two input vectors to the period production frontier. The input distancefunction is used for the radial scaling of total inputs. Adapting the input
distance function used by CCD, we introduce the labour input distance functionthat measures the radial distance from labour inputs
to the period
production frontier. The period labour input distance function forandis defined as follows:

{

(

)

}.

(25)

Given outputs ,is the maximum contraction of labour inputs
enabling the contracted labour inputs
and capital inputs
with outputs
to be on the period production frontier. If
is on theperiod production frontier,equals . Note thatislinearly homogeneous in.
Furthermore, we can relate the period labour input distance function to theperiod short-run production frontier. Given outputs ,is themaximum contraction of labour inputs enabling the contracted labour inputs
and outputs
to be on the period
short-run production
frontier. Thus,
provides a radial measure of the distance of xL to
the period short-run production frontier conditional on y. We construct thecounterpart of the growth rate of labour input by comparing the radial distancesfrom two labour inputs
andto the period short-run production frontierconditional on . It is defined as follows:.

(26)

If labour inputs increase between two periods
and ,moves further
away from the origin than
, indicating that the labour input vector
is larger
than the labour input vector. The maximum contraction of labour inputs
for producing outputs y with the period capital inputs
and the period
technology increases such that
. It leads to. Similarly, if labour input shrinks between two periods, xL1
moves closer to the origin than does xL0, leading to.
Second, we generalise the growth rate of outputs between two points on the
period short-run production frontier. In the multiple-inputs multiple-outputs
case, outputs attainable from given labour inputs
are not uniquely determinedby the short-run production frontier. Let
be the portion of the period
short-run production frontier conditional on labour inputs, consisting of the setof maximum outputs
attainable fromusing capital inputs and technology
available at period . It is defined as follows:
{

}.

(27)

Since
provides a radial measure of the distance ofto theperiodshort-run production frontier conditional on, it can also beinterpreted as a radial measure of the distance ofto. We construct thecounterpart of the growth rate of outputs between two points on the period t shortrun production frontier by measuring the distance between
and.
We begin with the reference outputs vector . We measure the distance between
and, comparing the radial distances fromtoand
. It is defined as follows:.

(28)

If labour input growth makes it possible to produce more outputs while holding
capital input fixed and using the same technology, the set of outputs attainablefrom,shifts outward to that of outputs attainable from,.
Thus, the minimum contraction factor for given outputs y declines such that, leading to. Similarly, if thechange in labour inputs allows a firm to produce less outputs while holding capital
input fixed and using the same technology,, shifts inward to,leading to.
Using the counterparts of the growth rate of outputs and labour inputs between
two points on the period
short-run production frontier, we can propose ameasure for the LP growth between these two points. When we consider the
movement along the period
short-run production and use outputs y as
reference, the returns to scale effect is defined as follows:22
(

(

)

(

)

) (

(

)

(

)

).

(29)

Each choice of reference short-run production frontier and reference output vector
may generate a different measure of the returns to scale effect between two
periods
and . We calculate two measures by using short-run production
frontiers and output vectors available at the same period: period
short-runproduction frontier and period
output vector; period
short-runproduction frontier and period
output vector. Since these sets of short-run
production frontiers and output vectors are equally reasonable, we use thegeometric mean of these measures as a theoretical index of the returns to scaleeffect,, as follows:
√

.

(30)

The case of one output and one labour input offers us a graphical interpretation of. In Figure 1, equation (30) can be reduced to the following formula:
√(

(

)

)(

(

)

).

(31)

Given the period short-run production frontier, the ratio of the LP associatedwithto the LP associated withrepresents the LP growth induced by the
movement along the period
short-run production frontier.
is thegeometric mean of those ratios conditional on the period 0 and 1 short-run
production frontiers.
is a theoretical measure defined by the unknown short-run distancefunctions, and there are several methods of implementing it. We adopt the same
approach as we do for. In addition to a firm‘s short-run profit-maximising
behaviour and a translog functional form for the short-run output distancefunction, we also assume a translog functional form for the labour input distancefunction. Similar to the case of, we begin by showing that the first-orderderivatives of the distance functions with respect to labour input and output
quantities at the period actual production plan
are computablefrom price and quantity observations.23 We apply the implicit function theorem tothe input requirement function()
to solve for

22This formulation is a counterpart of the return to scale effect on TFP growth proposed by Lovell(2003; 450). Lovell‘s definition is based on the input distance function instead of the labour input
distance function. We return to this point in the last section.
23
is defined using the output and labour input distance functions. Since we have alreadyshown how to compute the first derivatives of the output distance functions in equations (14) and(15), we now focus on the short-run labour input distance functions.

.24 Its derivatives are represented by the). We have the following equations foraroundderivatives ofand :25

(

(

[
)

(

(

)

),

(33)

)].

(

(32)

We assume thatis a solution to the period
short-run profit
maximisation problem (11) for
and . By substituting equations (12) and(13) obtained from the profit maximisation into equations (32) and (33), we obtainthe following equations (34) and (35) for
and :
,
[ ⁄

][

(34)

]

).

(35)

Equations (34) and (35) allow us to compute the derivatives of the labour input
distance function without knowing the labour input distance function itself.
Information concerning the derivatives is useful for calculating the values of, which is defined by the distance functions. However, one disadvantage isthat the derivatives of the period
short-run distance function need to beevaluated at the period actual production planin equations (34)and (35) for
and . The distance functions evaluated at the production
plan in different periods such as
andalso
constitute. Hence, the above equations are insufficient for obtaining. In addition to a firm‘s short-run profit maximisation, we further assume atranslog functional form with time-invariant second-order coefficients for theperiod labour input distance function forand , which is defined asfollowing:
∑
∑

( )∑
( )∑

∑

( )∑

∑

∑

∑

∑

∑

∑
(36)

∑
∑

∑

where the parameters satisfy the following restrictions:24

for allandsuch as;(37)

for allandsuch as;(38)

for allandsuch as;(39)

It corresponds to CCD applying the implicit function theorem to the input requirement function
to solve the input distance function.
25
Equation (4) implies that equations (34) and (35) are well defined.

∑
∑
∑

for
for

∑

(40)

;

for

;

(41)

; and

(42)

.

(43)

Equation (36) is the same functional form defined by equation (16) that weassumed for the output distance function in the discussion of. However,parameters in both functional forms are independent and allowed to be varied.26
Moreover, the restrictions on parameters on the labour input distance function
differ from those on the output distance function. We replace restrictions (20)–(23) with that of (40)–(43). While restrictions (20)–(23) guarantee the linear
homogeneity in outputs
for the output distance function, restrictions (40)–(43)guarantee the linear homogeneity in labour inputs
for the labour input
distance function.
The translog functional form characterised by equations (36)–(43) is a flexible
functional form and it can approximate an arbitrary labour input distance function
to the second order at an arbitrary point. Thus, the assumption of this functional
form does not harm any generality of the labour input distance function. Note thatthe coefficients for the linear terms and the constant term are allowed to varyacross periods. Thus, technical progress under the translog distance function is by
no means limited to Hicks neutral, and various types of technical progress areallowed. Under the assumptions of short-run profit-maximising behaviour and thetranslog functional form, a theoretical index of the returns to scale,,coincides with a formula of price and quantity observations as is shown in thefollowing proposition.
Proposition 2
Assume the following: output distance functions
andhave the translog
functional form with time-invariant second-order coefficients defined byequations (16)–(23); labour input distance functions
andhave thetranslog functional form with time-invariant second-order coefficients defined byequations (36)–(43); a firm follows short-run profit-maximising behaviour in

26

Beginning from an output distance function that has the translog functional form defined byequations (16)–(23), one can derive the labour input distance function that corresponds to the
output distance function (the corresponding labour input distance function). This labour input
distance function will not have the translog functional form. However, we assume an independenttranslog functional form defined by equations (36)–(43) instead of the corresponding labour input
distance function. It is because we will encounter approximation errors with respect to the truelabour input distance functions in either case. The translog functional form is considered thesecond-order approximation to the true function. Thus, the approximation errors attributed to thethird and higher-order derivatives exist in output and labour input distance functions that has thetranslog functional form. Therefore, the corresponding labour input distance function does notdisentangle itself from the influence of the approximation error of the output distance function.
Both errors are very small and, to our knowledge, it is difficult to judge which is more serious.
Thus, there is no reason to adopt the corresponding labour input distance function rather than alabour input distance function that has the translog functional form. See Appendix B.

periods

and , as in equation (11). Then, the returns to scale effect,, can be computed from observed prices and quantities as follows:
∏

(

⁄

)

∏

(

⁄

)

.

̅

(44)

where
is the average value-added shares of labour input
and ̅
average labour-compensation share of labour input
between periods
such that() and ̅(is theand).

The index number formula on the right-hand side of equation (44) can beinterpreted as the ratio of the quantity indexes of labour inputs. Both terms are theweighted geometric average of the growth rates for labour inputs. The numerator
uses the ratio of labour compensation for a particular type of labour input to thetotal value added as weight, and the denominator uses the ratio of labour
compensation for a particular type of labour input to the total labour compensation
as weight. Thus, if labour income share, which is the ratio of the total labour
compensation to the value-added, is large, the difference between two terms
becomes small: hence, making the magnitude ofsmaller. Conversely, if
labour income share is small, the magnitude of
becomes larger.
Beginning from the understanding that the two contribution factors exist for
the LP growth, we independently reached the index number formula for thesefactors. However, our result does not deny the possibility that other unknown
factors explain LP growth. Fortunately, two factors ofandcanfully explain LP growth. The product ofandcoincides with theindex of LP growth, as follows.
Corollary 1
Assume the following: output distance functions
andhave the translog
functional form with time-invariant second-order coefficients defined byequations (16)–(23); labour input distance functions,andhave thetranslog functional form with time-invariant second-order coefficients defined byequations (36)–(43); a firm follows short-run profit-maximising behaviour inperiods
and , as in equation (11). Then, the product ofand
can be computed from observed prices and quantities as follows:
∏
∏

(
(

⁄

)

⁄

)

̅

,

(45)

where
is the average value-added shares of output
and ̅
average labour-compensation share of labour input
between periods
such that:() and ̅(is theand).

The right-hand side of equation (45) represents growth in LP, its numerator
coincides with the Törnqvist output quantity index, and the denominator is the
Törnqvist labour input quantity index. Thus, we simply call the right-hand side ofequation (45) the Törnqvist LP growth index. Equation (45) allows us tocompletely decompose LP growth into two components,
and,when multiple inputs and outputs are employed. This decomposition is justifiableas a generalisation of the one-input one-output case in which LP growth isinduced by the shift in the production frontier and the movement along the
production frontier in Figure 1.
Balk (2005) provided a general framework for decomposing productivity
indexes. Balk argued that, for meaningful decomposition, each factor indecomposition should be independent of other factors.27 Several decomposition
results dealing with the Malmquist TFP index are criticised from this point ofview. The difficulty in these decompositions of the Malmquist TFP index is
attributed to the fact that the Malmquist TFP index itself is not transitive in input
and output quantities.
On the other hand, our theoretical measures ofandaredefined, independent of each other. Thus, seeing at a glance whether a mere
multiplication of two indexes coincides with LP growth is difficult. They coincide
only when the underlying distance functions have translog functional forms.
Therefore, our decomposition result is immune from Balk‘s criticism. Moreover,
we emphasise that the Törnqvist LP index, the logarithm of which appears on theright-hand side of equation (45), satisfies transitivity in labour input and output
quantities for fixed shares of value added and labour compensation.
5. An Application to US Industry Data
Having discussed the theory of the decomposition, we now explore its empiricalsignificance with industry data. The industry data covering the period 1987–2009
is taken from the Bureau of Labour Statistics (BLS) multifactor productivity data.
We use a gross output, three intermediate inputs (energy, materials and purchased
services) and a labour input at current and constant prices by 59 industries, whichconstitute the non-farm private business sector. Labour input at constant prices
measures the number of hours worked.28 These industries are categorised either
as goods-producing industries (goods sector) or services-providing industries(services sector).
[Place Table 1 appropriately here]
Table 1 compares LP growth and its components across the non-farm private
business, the goods and the services sectors. For the entire sample period 1987–
2009, the returns to scale effect had a negative impact on LP growth of 2.19 percent per year in the non-farm private business sector. Whereas the joint effect oftechnical progress and capital input growth was an annual average of 2.38 per cent,it was largely offset by the returns to scale effect of –0.19 per cent on average per
year. During the same period, the returns to scale effect appeared differently in
two sectors. Whereas the positive returns to scale effect raised the services sector
LP by 0.36 per cent per year on average, the negative returns to scale effectlowered the goods sector LP by −0.39 per cent per year on average. During theperiod 1987–2009, the average growth rate of the goods sector LP was 2.43 percent, about 0.3 per cent higher than that of the services sector LP. However, once27

‗Now, of course, every mathematical expressioncan, given any other expression , bedecomposed as. However, not all such decompositions are meaningful‘. (Balk 2005).
28
Thus, this measure of labour input does not appropriately capture changes in labour quality. The
joint effect of technical progress and capital input growth includes LP growth induced by changes
in the characteristics of labour input.

the returns to scale effect is controlled, the order is reversed, resulting in an
average growth rate of the goods sector LP of 2.07 per cent, which is about 0.4
per cent lower than that of the services sector LP.
[Place Table 2 appropriately here]
Table 2 summarises the growth in labour input for the non-farm private business,the goods and the services sectors. According to both the weighted and theunweighted average of the detailed industries, labour inputs in the goods sector
decreased on average, whereas labour inputs in the services sector increased on
average. The different role played by the returns to scale effect in both sectors is
attributed to the difference in the growth of labour input between two sectors.
With reference to Table 1, dividing the entire sample period 1987–2009 into
three periods is useful: the ‗productivity slowdown‘ period 1987–1995; the‗productivity resurgence‘ period 1995–2007 and the ‗great recession‘ period
2007–2009. A productivity slowdown in the US economy began in the early
1970s with an average annual growth rate of 1.42 per cent for the non-farmprivate business sector during the period 1987–1995. Productivity growth surged
after 1995 with an average annual growth rate of 2.75 per cent during the period
1995–2007. During the global financial crisis, labour input used for production
sharply declined at a much faster pace than real value added, leading LP growthwith an average annual growth rate of 1.84 per cent in 2007–2009.
Triplett and Bosworth (2004, 2006) and Bosworth and Triplett (2007) found,in US industry data, that LP growth in the services sector was stagnant and lower
than LP growth in the goods sector.29 Our dataset also documented the difference
in LP growth between the goods and the services sectors. The services sector LP
grew at an average growth rate of 1.19 per cent during the period 1985–1995,
much lower than an average annual rate of 1.97 per cent for the goods sector.
However, once we control for the returns to scale effect and consider only thejoint effect of technical progress and capital input growth, the services sector withan average annual rate of 1.82 per cent comes close to the goods sector with anaverage annual rate of 2.04 per cent. Thus, although the services sector LP grewmuch slower during the period 1987–1995 than the goods sector LP, theproductive capacity of labour in the services sector, which is the output attainablefrom given labour inputs, grew at a comparable pace to the goods sector. The factthat the service sector LP grew slower than the goods sector LP reflects that thegreater increase in labour input in the services sector restrained LP fromincreasing significantly.
In reference to Table 2, labour input in the goods sector only slightly increased,leading to a modest returns to scale effect during the period 1987–1995, and evendecreased, leading a positive returns to scale effect during the period 1995–2007.
In contrast, labour input in the services sector steadily increased until 2007,
leading to the negative returns to scale effect in the periods 1987–1995 and 1995–
2007. LP growth in the goods sector is still larger than that in the services sector
LP during the period 1995–2007. The gap in LP growth between two sectors ismuch smaller than during the period 1987–1995. However, as shown in Table 1,
once we control the returns to scale effect, the order is reversed, resulting in LP
29

Triplett and Bosworth (2006) used the term Baumol’s disease to identify with the situation inwhich LP growth in the services sector is likely to stagnate. They argued that this disease wascured in the mid-1990s.

growth explained by technical progress and capital input growth at an averageannual rate of 3.18 per cent for the service sector, higher than the 2.81 per cent for
the goods sector. Thus, although LP growth during the period 1995–2007 waslower in the services sector than the goods sector, the productive capacity oflabour increased more in the services sector than the goods sector.
The pattern that Triplett and Bosworth (2004, 2006) and Bosworth andTriplett (2007) pointed out dissolved after 2008. The services sector LP grew at an
average annual growth rate of 2.09 per cent, even higher than the goods sector at
an average annual rate of 1.42 per cent during 2007–2009. During this period, thedeclining labour inputs lead to positive returns to scale effects in both sectors. The
goods sector shows a particularly large returns to scale effect with an averageannual rate of 3.69 per cent, which is more than a three-fold average annual rate of
1.02 per cent for the services sector. However, even this significantly large returns
to scale effect in the goods sector cannot compensates a negative effect oftechnical progress and capital input growth with an annual growth rate of –2.27
per cent, leading LP growth lower than the goods sector.30[Place Tables 3, 4 and 5 appropriately here]
Tables 3, 4 and 5 show LP growth and its components and growth in labour
input and labour income share by industry during the periods 1987–1995, 1995–
2007 and 2007–2009. The pattern found in the aggregate study based on the sector
data in Table 1 is also documented in the detailed industries. During the period
2007–2009, most industries in both sectors showed significantly positive returns
to scale effects, reflecting sharp declines in labour inputs. Most industries in theservices sector show negative returns to scale effects until 2007. They areparticularly significant during the period 1987–1995. However, the returns toscale effects for most industries in the goods sector are very modest before 2007.
They are negative during the period 1987–1995 and positive during the period
1995–2007.
There are exceptional industries in both the goods and the services sectors.
Two industries show significant and positive returns to scale effects with anaverage annual rate of more than 1 per cent during the period 1987–1995: 1.99 percent for oil and gas extraction industry; 1.16 per cent for petroleum and coalproducts industry. Support activities for mining industry shows a significant andnegative returns to scale effect with an average annual rate of more than −2.17 percent during the period 1995–2007. Conversely, three industries in the services
sector show positive returns to scale effects during both 1987–1995 and 1995–
2007: utilities industry, rail transportation industry and pipeline transportationindustry. In these industries, there is a trend of decrease in labour input throughout
the entire sample period, unlike other industries in the services sector. During theperiod 2007–2009, when decreasing labour inputs lead to positive returns to scaleeffects in most industries of the goods and services sector, three industries show
significant and negative returns to scale effects, reflecting greatly increasing
labour inputs: −3.55 per cent per year for oil and gas extraction industry; −5.7 percent per year for water transportation industry and −2.07 per cent per year for
education services industry.

30

This negative growth rate is mainly accounted for by the decline of TFP based on our own
calculation.

Equation (44) tells us that the returns to scale effect depends on labour income
share as well as growth in labour input. The returns to scale effect will apparently
get smaller under large labour income share. The detailed industry study reveals
cases when the returns to scale effect induced by labour input growth can begreatly mitigated by the large labour income share. The wood product industry
shows an extremely large decrease in labour input with an average annual rate of−19.15 during the period 2007–2009. However, its returns to scale effect is anannual average of 1.73 per cent, relatively small in magnitude during this period.
Its large labour income share of 91.85 per cent offset the impact of a large declineof labour input for this industry. The rental and leasing services and lessors ofintangible assets industry shows one of the largest returns to scale effect with anaverage annual rate of 5.45 per cent. Growth in its labour input is an averageannual rate of –6.57 per cent, which is a comparable scale. The returns to scaleeffect of this industry was amplified by its smallest labour income share of 16.91
per cent among all industries in both sectors.
6. Conclusion
This paper distinguished two effects on LP growth by examining the short-run
production frontier. The joint effect of technical progress and capital input growthappears as growth in LP induced by the shift in the short-run production frontier.
The returns to scale effect appears as the LP growth induced by movement along
the short-run production frontier. The LP growth calculated by Törnqvist quantity
indexes is fully decomposed into the product of these two effects. We applied thisdecomposition result to US industry data for the period 1987–2009. A large partof the difference in LP growth between the goods sector and the services sector
can be attributed to the difference in the returns to scale effect.
It is possible to give our decomposition result a different interpretation. Once
we make labour inputs include all the capital inputs,
becomes merely thetechnical progress effect, measuring the shift in the underlying production frontier.
Similarly, under the same setting, the returns to scale effectrepresents
the TFP induced by the movement along the underlying production frontier. Notethatcoincides with the geometric mean of Lovell‘s (2003) scale effect.31
Thus, Corollary 1 means that the TFP growth calculated by Törnqvist quantity
indexes is fully decomposed into the product of the two effects. Our result differsfrom the previous studies in that we give an exact interpretation of the index
number formula for the returns to scale effect in Proposition 2. Although CCD
derived the same index number formula that is a function of the degree of returns
to scale, they did not explain what this formula itself measures. On the other hand,
while Lovell (2003) proposed a measure of the returns to scale effect representing
the TFP growth induced by movement along the underlying production frontier,Lovell did not offer any index number formula that equals or approximates it. The
contribution of the present paper to the literature of TFP growth decomposition is31

In a strict sense,
coincides with the period activity effect in Lovell (2003),
which is the returns to scale effect evaluated on the period production frontier. Thus, when we
consider the returns to scale effect on the TFP growth between two periods, the geometric mean of
the period
andactivity effectis an appropriate√
measure.

in showing the equivalence between CCD‘s index number formula and Lovell‘stheoretical measure.
This paper assumes the firm‘s profit-maximising behaviour and ruled out
inefficient production processes. If we relax the firm‘s profit-maximising
behaviour, another factor—technical efficiency change—appears in thedecomposition of LP growth. Even with no change in the short-run production
frontier and no change in labour input, a firm can approach closer to the short-run
production frontier by improving technical efficiency. For example, a firm
improves technical efficiency by increasing output up to the maximum levelattainable from given labour inputs under current technology. For implementing
the decomposition of LP growth without assuming a firm‘s profit-maximising
behaviour, we can estimate the distance function using econometric and linear
programming techniques. However, we leave this exercise for future research.

Appendix A
Proof of Proposition 2

( )

((
( )

( )

)⁄(
((

((

)⁄(
)(

( )

))
))

))

((

)(

))

Since the firm‘s short-run profit maximisation is assumed, the period
production plan is on the period production frontier for
and .
( )∑(
)

(

)

)

(

)

( )∑(

using the translog identity in CCD
( )∑(

)

(

)

( )∑(

)

(

)

)

(

)

)

(

)

( )∑(

( )∑(

( )∑(

)

(

)

( )∑(

)

(

)

(

)

from equations (16) and (36)
( )∑(

)
( )∑(

from equations (14), (15), (34) and (35).

())

Appendix B
In this paper, we assume the translog functional form for the output and labour
input distance functions (translog output distance function and translog labour
input distance function), allowing parameters to be independent. However, sincedistance functions are defined with reference to the same production frontier,there should be some mathematical relationship in the functional form andparameters between two distance functions. Beginning with the period translog
output distance function, we can derive the corresponding period labour
input distance function
as follows:
{

(

)

}

(B.1)

Clearly, this labour input distance function corresponding to the translog output
distance function (the corresponding labour input distance function) does not havethe translog functional form.
However, we should not necessarily assume the corresponding labour input
distance function in addition to the translog output distance function. In thisappendix, we consider the problem of choosing the translog labour input distancefunction or the corresponding labour distance function when we assume thetranslog output distance function.
The translog labour input distance function approximates an arbitrary labour
input distance function to the second order at an arbitrary point of approximation.32 Thus, irrelevant of the type of functional form the true distancefunction has, the translog functional form is a good local approximation to it.
Needless to say, since the translog distance function does not consider the further
approximation based on the third- and higher-order derivatives, the values of thetranslog and the true labour input distance functions diverge as the point at which
the functions are evaluated moves from the approximation point(Type 1 error). From the same reasoning, the translog output distance function
suffers from the approximation error, due to neglecting third- and higher-orderderivatives. Thus, if we derive the labour input distance function from the translog
output distance function as in (B.1), the corresponding labour input distancefunction reflects such an approximation error of the translog output distancefunction. Thus, the values of the corresponding labour input distance functions
also differ from that of the true labour input distance function as the evaluation
point moves from the approximation point (Type 2 error).
Since we cannot analytically compare the magnitude of these two types ofapproximation errors, we use a numerical example to discuss how to choose thelabour input distance function. To implement, it is necessary to evaluate
the period
output and labour input distance functions atand, and the period
output and labour input distance functions at
and. Suppose that the period
andtranslog output
and labour input distance functions are the local approximations atand. 33 Then,
,
,
and
32

Strictly speaking, it provides the second-order log approximation.
The following argument can be applied to the case for which the period 0 and 1 translog shortrun labour input distance functions are the local approximations at
and
.
33

calculated from the translog as well as the corresponding distancefunctions differ from those calculated from the true distance functions.
We consider the simple case consisting of two outputs, one capital input and
two labour inputs() so that production took
place atandas follows:

The period
input requirement function
defined as follows:

for

and

is

(B.2)

(B.3)
The period output and labour input distance functions are constructed withreference to the period short-run production frontier described by the above
equations as follows:
(

(

(

)

(

(

(
(

)

(

(

Since the period

)

(

)

)
(

(B.4)

))

)

)

))

)

))

)

))

)

(
(B.5)

(

(

(

(
)

(

(

(

)

(B.6)

(
(B.7)

production takes place at the period production frontier,and.

The periodtranslog output and labour input distance functions are derived
by applying the second-order Taylor expansion to the period
output and labour
input distance functions in (B.4) and (B.6) at. Similarly, the periodtranslog output and labour input distance functions are derived by applying thesecond-order Taylor expansion to the period
output and labour input distancefunctions in (B.5) and (B.7) at. Once we derive the translog output
distance functions, we can also derive the corresponding distance function
following equation (B.1). Thus, we computeunder three difference
approaches in this example, as shown in Table B.1.
[Place Table B.1 appropriately here]
Approach 1 assumes the independent translog functional form for output andlabour input distance functions. This paper takes this approach. Approach 2 uses
the corresponding labour input distance function instead of the translog one.
Approach 3 uses the true distance functions of (B.3)–(B.6), and is a reference forcomparison of Approaches 1 and 2. The approach that gives us the value ofcloser to that of Approach 3 is the better one. Because
andare approximation points,,,
andequal 1 across Approaches 1, 2 and 3.
[Place Table B.2 appropriately here]
Table

B.2

compares the values of,
andunder three different approaches.34 First, the differences between
the true values based on Approach 3 and the estimates based on Approaches 1 and
2 are very small, at approximately a 0.2 per cent difference. This means that
Approaches 1 and 2 are capable of approximating the true value with great
accuracy. Second, if we compare Approaches 1 and 2 for estimating
andby constraint, Approach 2 is better. Although
the difference between them is extremely small, at less than 0.002 per cent, thecorresponding labour input distance function can approximate the true labour
input distance function above
andmore accurately
than the translog labour input distance function in this example. However, forestimating, Approach 1 is better than Approach 2. The estimate based on
Approach 1 is closer to the true value based on Approach 3 than that on Approach
2, by about 0.003 per cent. Thus, the translog labour input distance function canapproximate the true value ofmore accurately than the corresponding
labour input distance function in this example.
Thus, we conclude the following from our numerical example. First, bothType 1 and Type 2 errors are rather small. This means that assuming the translog
labour input distance function is as good as assuming the corresponding labour
input function. Second, even though Type 1 error (which comes from Approach 1)
might be more serious than Type 2 error (which comes from Approach 2) in somecases, this does not necessarily mean that the estimate ofbased onApproach 2 is more accurate than that on Approach 1. Therefore, it is hardly
possible to judge Approaches 1 and 2 from our numerical example. In our present
knowledge, there is no reason to adopt the corresponding labour input distancefunction instead of the translog labour input distance function.

34

We also need to calculateand. However, since the sametranslog output distance function is used in Approaches 1 and 2, we omit their values from Table
B.2.

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Table 1: Sources of Sectoral Labour Productivity Growth
1987-2009 1987-1995 1995-2007 2007-2009
Non-farm private business sector
Labour productivity growth
2.19
1.42
2.75
1.84
Technical progress and capital input growth
2.38
1.87
3.08
0.25
Returns to scale effect
-0.19
-0.45
-0.32
1.60
Goods sector
Labour productivity growth
2.43
1.97
2.90
1.42
Technical progress and capital input growth
2.07
2.04
2.81
-2.27
Returns to scale effect
0.36
-0.07
0.09
3.69
Services sector
Labour productivity growth
2.10
1.19
2.71
2.09
Technical progress and capital input growth
2.49
1.82
3.18
1.08
Returns to scale effect
-0.39
-0.63
-0.47
1.02
Note : All figures are average annual percentages.

Table 2: Labour Input Growth and Labour Income Share
1987-2009 1987-1995 1995-2007 2007-2009
Labour input growth (un-weighted average)
Private non-farm business
0.62
1.41
0.96
-4.63
Goods-producing industries
-1.01
0.21
-0.39
-9.65
Services-producing industries
1.24
1.97
1.47
-3.06
Labour input growth (weighted average)
Private non-farm business
0.15
1.22
0.24
-4.69
Goods-producing industries
-1.65
-0.14
-1.51
-8.59
Services-producing industries
1.30
2.09
1.36
-2.20
Labour income share
Private non-farm business
67.86
68.27
68.00
65.38
Goods-producing industries
66.87
68.79
66.44
61.73
Services-producing industries
68.20
68.02
68.59
66.59
Note : All figures are average annual percentages. The un-weighted average growth rate of labour productivity is thearithmetic mean of the growth rate of industry labour productivity. The weighted average growth rate of labour productivity
is calculated using labour income in each industry divided by the sum of industry labour.

Table 3: Sources of Industry Labour Productivity Growth, 1987–1995
Industry

Goods sector
Forestry, fishing, and related activities
Oil and gas extraction
Mining, except oil and gas
Support activities for mining
Construction
Food and beverage and tobacco products
Textile mills and textile product mills
Apparel and leather and applied products
Wood product
Paper products
Printing and related support activities
Petroleum and coal products
Chemical products
Plastics and rubber products
Nonmetallic mineral products
Primary metals
Fabricated metal products
MachineryComputer and electronic productsElectrical equipment, appliances, and components
Transportation equipment
Furniture and related products
Miscellaneous manufacturing

113-115
211
212
213
23
311,312313,314315,316
321
322
323
324
325
326
327
331
332
333
334
335
336
337
339

Services sector
Utilities
22
Wholesale trade
42
Retail trade
44,45
Air transportation
481
Rail transportation
482
Water transportation
483
Truck transportation
484
Transit and ground passenger transportation
485
Pipeline transportation
486
Other transportation and support activities
487,488,492
Warehousing and storage
493
Publishing industries
511,516
Motion picture and sounds recording industries
512
Broadcasting and telecommunications
515,517
Information and data processing services
518,519
Federal reserve banks, credit intermediation, and related activities 521,522
Securities, commodity contracts, and investments
523
Insurance carriers and related activities
524
Funds, trusts, and other financial vehicles
525
Real estate
531
Rental and leasing services and lessors of intangible assets
532,533
Legal services
5411
Computer systems design and related services
5415
Miscellaneous professional, scientific, and technical services
5412-5414,5416-5419
Management of companies and enterprises
55
Administrative and support services
561
Waste management and remediation services
562
Educational services
61
Ambulatory health care services621
Hospitals and nursing and residential care facilities
622,623
Social assistance624
Performing arts, spectator sports, museums, and related activities 711,712
Amusements, gambling, and recreation industries
713
Accommodation
721
Food services and drinking places
722
Other services, except government
81
Note: All figures are average annual percentages.

Labourproductivity
growthTechnical
progress andcapital input
growthReturns to scaleeffectLabour input
growthLabour income
ratio

-4.45
3.11
7.35
1.60
0.50
1.61
2.81
5.77
-1.57
0.49
0.76
5.45
-0.40
2.01
1.07
1.07
1.40
-1.17
19.98
-3.57
-1.85
0.61
2.41

-3.36
1.12
6.62
1.04
0.64
1.93
2.61
5.14
-1.47
0.59
0.95
4.29
-0.15
2.70
1.17
0.95
1.64
-0.85
19.68
-4.01
-1.92
0.62
2.84

-1.09
1.99
0.73
0.56
-0.14
-0.33
0.20
0.63
-0.10
-0.10
-0.20
1.16
-0.25
-0.69
-0.10
0.12
-0.24
-0.32
0.30
0.45
0.07
-0.01
-0.43

2.28
-2.93
-1.81
-1.96
0.85
0.72
-0.94
-2.67
0.34
0.26
1.41
-1.64
0.53
2.02
0.12
-0.35
0.79
1.33
-1.40
-1.09
-0.49
-0.08
1.38

53.22
30.33
58.58
69.54
85.42
52.66
75.97
76.43
72.25
58.33
86.15
28.92
48.66
66.22
69.21
73.11
72.60
73.21
76.69
59.65
78.37
82.06
67.67

3.47
2.50
3.38
2.90
5.14
8.53
3.84
-0.87
2.86
-3.38
4.68
2.42
-3.80
4.63
-2.60
0.18
5.90
2.29
0.12
0.29
1.61
-0.47
2.04
1.11
-0.77
0.32
1.14
0.76
-1.74
-2.75
2.27
2.49
-0.48
1.73
-0.64
-0.18

3.18
2.83
3.71
4.29
4.51
8.15
4.10
-0.20
2.22
-2.17
4.86
2.91
-2.36
4.80
-1.66
-0.12
6.19
2.50
1.55
1.10
4.02
-0.32
2.60
1.39
-0.80
0.98
1.90
1.52
-1.17
-2.08
2.42
2.72
1.08
2.30
-0.28
0.10

0.29
-0.33
-0.32
-1.39
0.63
0.39
-0.27
-0.67
0.63
-1.21
-0.18
-0.49
-1.43
-0.16
-0.95
0.30
-0.29
-0.21
-1.42
-0.81
-2.41
-0.15
-0.56
-0.28
0.03
-0.67
-0.76
-0.76
-0.57
-0.66
-0.15
-0.23
-1.57
-0.56
-0.36
-0.28

-0.39
1.15
1.41
5.11
-2.50
-0.80
1.17
3.22
-1.27
6.34
0.88
1.71
5.19
0.24
4.37
-0.69
1.69
1.21
1.87
0.96
3.19
1.46
5.43
2.50
0.08
4.74
2.01
2.55
3.69
3.33
2.12
1.57
5.47
1.87
1.94
2.25

25.90
72.75
77.65
75.51
73.19
53.05
77.49
79.29
50.13
81.10
81.20
71.38
72.44
38.12
78.22
57.02
78.21
82.09
22.12
15.58
24.35
89.86
89.29
88.73
94.27
85.81
62.58
65.30
84.48
79.69
92.95
85.77
70.81
71.12
81.68
87.48

Table 4: Sources of Industry Labour Productivity Growth, 1995–2007
Industry

Goods sector
Forestry, fishing, and related activities
Oil and gas extraction
Mining, except oil and gas
Support activities for mining
Construction
Food and beverage and tobacco products
Textile mills and textile product mills
Apparel and leather and applied products
Wood product
Paper products
Printing and related support activities
Petroleum and coal products
Chemical products
Plastics and rubber products
Nonmetallic mineral products
Primary metals
Fabricated metal products
MachineryComputer and electronic productsElectrical equipment, appliances, and components
Transportation equipment
Furniture and related products
Miscellaneous manufacturing

113-115
211
212
213
23
311,312313,314315,316
321
322
323
324
325
326
327
331
332
333
334
335
336
337
339

Services sector
Utilities
22
Wholesale trade
42
Retail trade
44,45
Air transportation
481
Rail transportation
482
Water transportation
483
Truck transportation
484
Transit and ground passenger transportation
485
Pipeline transportation
486
Other transportation and support activities
487,488,492
Warehousing and storage
493
Publishing industries
511,516
Motion picture and sounds recording industries
512
Broadcasting and telecommunications
515,517
Information and data processing services
518,519
Federal reserve banks, credit intermediation, and related activities 521,522
Securities, commodity contracts, and investments
523
Insurance carriers and related activities
524
Funds, trusts, and other financial vehicles
525
Real estate
531
Rental and leasing services and lessors of intangible assets
532,533
Legal services
5411
Computer systems design and related services
5415
Miscellaneous professional, scientific, and technical services
5412-5414,5416-5419
Management of companies and enterprises
55
Administrative and support services
561
Waste management and remediation services
562
Educational services
61
Ambulatory health care services621
Hospitals and nursing and residential care facilities
622,623
Social assistance624
Performing arts, spectator sports, museums, and related activities 711,712
Amusements, gambling, and recreation industries
713
Accommodation
721
Food services and drinking places
722
Other services, except government
81
Note: All figures are average annual percentages.

Labourproductivity
growthTechnical
progress andcapital input
growthReturns to scaleeffectLabour input
growthLabour income
ratio

2.35
-3.42
-0.23
-1.55
-2.10
-0.52
5.83
8.58
1.98
2.74
2.08
4.80
4.37
3.51
-0.10
0.14
1.18
2.52
27.30
0.88
4.67
3.13
5.48

2.52
-3.63
-0.39
0.62
-1.72
-0.57
4.26
6.31
1.77
1.52
1.76
3.79
3.55
3.02
-0.13
-0.63
1.14
2.06
27.15
-0.10
4.20
2.85
5.25

-0.18
0.22
0.16
-2.17
-0.37
0.05
1.56
2.28
0.21
1.21
0.33
1.01
0.82
0.48
0.03
0.77
0.04
0.46
0.14
0.98
0.47
0.28
0.23

0.49
-0.62
-0.55
4.83
2.46
-0.11
-5.95
-9.55
-1.43
-2.97
-2.53
-1.37
-1.52
-1.34
-0.18
-2.88
-0.28
-2.00
-2.79
-2.90
-1.56
-1.17
-0.69

63.48
21.10
50.28
62.80
85.45
50.98
72.98
76.02
78.57
58.08
86.73
19.48
44.88
62.47
63.74
67.29
69.43
72.82
79.05
65.63
72.19
76.31
66.56

2.53
5.74
3.37
10.35
2.17
3.58
2.37
2.08
5.60
2.07
1.38
6.07
1.32
6.89
4.75
1.09
9.26
1.40
5.05
0.93
3.27
1.05
3.53
1.57
1.13
2.05
0.07
-1.00
0.23
-2.62
2.88
2.03
0.59
1.06
2.03
-0.29

1.28
5.93
3.47
9.99
1.74
5.12
2.61
2.31
4.71
2.21
2.04
5.98
1.70
7.10
4.81
1.82
9.63
1.55
7.23
1.98
4.11
1.11
4.09
1.95
1.26
2.36
0.99
0.25
0.72
-2.11
3.09
2.19
0.92
1.38
2.40
-0.18

1.25
-0.19
-0.10
0.36
0.43
-1.54
-0.24
-0.23
0.89
-0.14
-0.66
0.09
-0.38
-0.20
-0.06
-0.72
-0.37
-0.15
-2.18
-1.05
-0.84
-0.06
-0.56
-0.37
-0.13
-0.31
-0.92
-1.25
-0.49
-0.52
-0.21
-0.16
-0.32
-0.32
-0.37
-0.11

-1.76
0.62
0.42
-2.35
-1.56
2.93
1.07
1.08
-1.70
0.64
3.12
-0.17
1.30
0.26
2.21
1.60
2.84
0.64
2.62
1.28
1.10
0.78
5.87
2.36
0.96
2.73
2.23
3.62
3.21
2.54
2.82
0.94
1.06
0.88
1.73
0.85

29.69
71.36
76.64
75.87
69.04
45.89
79.17
77.97
41.71
77.10
78.95
66.94
66.79
43.40
65.16
54.21
86.52
77.70
15.78
18.20
19.90
92.09
88.85
83.48
86.61
89.76
57.97
61.23
85.00
78.66
92.71
85.10
71.74
64.62
78.57
88.23

Table 5: Sources of Industry Labour Productivity Growth, 2007–2009
Industry

Goods sector
Forestry, fishing, and related activities
Oil and gas extraction
Mining, except oil and gas
Support activities for mining
Construction
Food and beverage and tobacco products
Textile mills and textile product mills
Apparel and leather and applied products
Wood product
Paper products
Printing and related support activities
Petroleum and coal products
Chemical products
Plastics and rubber products
Nonmetallic mineral products
Primary metals
Fabricated metal products
MachineryComputer and electronic productsElectrical equipment, appliances, and components
Transportation equipment
Furniture and related products
Miscellaneous manufacturing

113-115
211
212
213
23
311,312313,314315,316
321
322
323
324
325
326
327
331
332
333
334
335
336
337
339

Services sector
Utilities
22
Wholesale trade
42
Retail trade
44,45
Air transportation
481
Rail transportation
482
Water transportation
483
Truck transportation
484
Transit and ground passenger transportation
485
Pipeline transportation
486
Other transportation and support activities
487,488,492
Warehousing and storage
493
Publishing industries
511,516
Motion picture and sounds recording industries
512
Broadcasting and telecommunications
515,517
Information and data processing services
518,519
Federal reserve banks, credit intermediation, and related activities 521,522
Securities, commodity contracts, and investments
523
Insurance carriers and related activities
524
Funds, trusts, and other financial vehicles
525
Real estate
531
Rental and leasing services and lessors of intangible assets
532,533
Legal services
5411
Computer systems design and related services
5415
Miscellaneous professional, scientific, and technical services
5412-5414,5416-5419
Management of companies and enterprises
55
Administrative and support services
561
Waste management and remediation services
562
Educational services
61
Ambulatory health care services621
Hospitals and nursing and residential care facilities
622,623
Social assistance624
Performing arts, spectator sports, museums, and related activities 711,712
Amusements, gambling, and recreation industries
713
Accommodation
721
Food services and drinking places
722
Other services, except government
81
Note: All figures are average annual percentages.

Labourproductivity
growthTechnical
progress andcapital input
growthReturns to scaleeffectLabour input
growthLabour income
ratio

3.35
9.91
11.85
17.57
1.47
-7.67
-1.38
0.99
-20.18
-4.67
-2.42
15.88
-15.62
-9.36
-2.14
1.39
-17.35
0.53
11.94
6.26
-11.77-24.10
4.17

1.83
13.46
8.58
14.06
-0.06
-8.58
-6.34
-1.48
-21.91
-7.50
-3.71
14.56
-17.79-13.21-5.74-4.08-20.78
-2.53
10.00
2.77
-14.95-25.57
0.74

1.52
-3.55
3.27
3.51
1.53
0.91
4.95
2.47
1.73
2.83
1.29
1.31
2.17
3.84
3.60
5.47
3.42
3.06
1.94
3.48
3.18
1.47
3.43

-5.48
4.19
-5.36
-6.64
-12.90-1.91-14.46-12.96-19.15
-6.56
-9.52
-1.55
-3.79
-11.78-12.49-13.07-10.72
-9.55
-5.07
-8.43
-12.09
-9.17
-9.01

72.37
16.66
38.54
47.97
88.62
52.15
65.47
82.85
91.85
56.86
86.36
13.36
42.96
68.14
70.79
58.85
67.91
67.97
61.60
58.31
73.67
85.18
61.92

-0.71
7.50
-0.48
11.60
-4.01
1.03
-2.48
-4.59
-6.92
-0.98
-3.03
3.43
-0.72
4.84
6.81
6.76
-3.49
8.54
7.68
1.07
2.93
-1.81
2.09
8.30
-0.35
-1.15
-1.80
-1.62
1.31
2.95
1.88
-1.45
-3.29
-1.31
-6.41
-4.59

-0.16
6.30
-1.36
9.70
-5.98
6.73
-4.06
-4.75
-5.78
-2.18
-3.11
1.09
-1.50
3.28
4.86
4.34
-4.15
8.23
5.68
-2.14
-2.51
-1.97
2.34
6.74
-0.23
-1.68
-2.55
0.45
1.64
3.22
1.85
-1.89
-4.12
-2.80
-6.76
-4.82

-0.55
1.20
0.89
1.89
1.97
-5.70
1.58
0.16
-1.13
1.20
0.08
2.35
0.78
1.56
1.95
2.42
0.66
0.30
2.00
3.21
5.45
0.16
-0.25
1.55
-0.12
0.52
0.75
-2.07
-0.33
-0.27
0.03
0.43
0.84
1.49
0.35
0.23

0.76
-4.00
-3.98
-6.95
-4.62
9.57
-7.35
-1.16
1.72
-4.82
-0.42
-5.99
-2.42
-2.50
-3.63
-5.10
-4.41
-1.33
-2.24
-3.79
-6.57
-1.99
1.84
-7.46
0.83
-4.63
-1.80
4.83
2.06
1.09
-0.47
-2.68
-3.56
-4.14
-1.77
-2.24

28.11
70.02
77.69
71.34
57.30
40.57
78.81
85.07
35.04
75.06
78.76
60.73
68.42
37.34
46.10
52.11
85.13
77.02
10.76
14.64
16.91
91.70
86.33
79.09
87.49
88.60
58.10
57.31
83.68
72.48
92.90
82.98
77.04
63.92
80.16
89.86

Table B1: Different Approaches to Calculating SHIFTApproach 1
Approach 2
Approach 3

Output distance function

Labour input distance function

Translog
Translog
True function

Translog
Corresponding
True function

Table B2: Comparison of Accuracy of Different Approaches
Approach 1
Approach 2
Approach 3

0.816458
0.816440
0.814507

1.758342
1.758287
1.758306

0.558346
0.558252
0.558258

Figure 1: Labour Productivity Growth and Shift in the Short-run Production Frontier

Figure 2: Returns to Scale Effect and Movement along the Short-run Production
Frontier