I
Measurement of Productivity
Improvements: An Empirical Analysis
RAJIV D . BANKER*
SRIKANT M . DATAR*
MADHAV V. RAJ AN*
In this paper, we test for productivity gains resulting from the
introduction of a productivity-based incentive program in a large
manufacturing plant of a Fortune 500 corporation. We develop a
methodology based on a stochastic nonparametric frontier estimation technique to evaluate productivity in the postincentive plan
period relative to the pre-incentive plan period. We also test for
productivity gains using stochastic parametric frontier approaches.
The results of both the nonparametric and parametric stochastic
frontier analyses indicate that the incentive program has a positive
effect on indirect labor, manufacturing services, and materials productivity and relatively little effect on direct labor productivity.
1. Introduction
Productivity improvement and cost control have become key objectives
of U.S. corporations in recent years. As a result, many corporations have
introduced productivity improvement programs, especially productivitybased incentive payments to workers. The implementation and evaluation
of the impact of such incentive programs have placed demands on management accountants to develop reliable measures of productivity and manufacturing efficiency—an issue largely ignored in the management
accounting literature.
The agency theory literature studies the motivational effects of providing
incentives to woricers. The site at which a productivity-based scheme has
recently b ^ n introduced serves as a natural experiment for testing the effect
of an incentive contract intencted for improving labor productivity. Evaluation of its inqjact depends ontfjemettiod employed to measure productivity.
Univosity
319
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JOURNAL OF ACCOUNTING, AUDITING AND FINANCE
We use data from the site to compare conclusions drawn by altemative
methods of measuring productivity about the impact of the incentive plan
on productivity improvements.
In our model, productivity measures the efficiency with which each of
four inputs (direct labor, materials, indirect labor, and manufacturing support
services) is consumed in producing two outputs. Accounting systems in
practice are not geared to evaluating productivity. Simple input-output quantity ratios implicitly assume constant returns to scale (CRS) and the absence
of multiple inputs and outputs. The use of input prices further obfuscates
measures of productivity. Standard usage variances ignore indirect labor
productivity and additionally assume linear and separable technologies.
Advanced management accounting textbooks, such as Kaplan (1982),
discuss ordinary least squares (OLS) approaches for estimating cost functions. This can be adapted for estimating production functions and improvements in productivity by regressing each input on the outputs produced and
testing for decreases in input consumption in the' 'event'' period—the period
after the introduction of the gain-sharing program. The specification of
stochastic disturbance terms with zero means implies Ae estimation of an
average production function. The theoretical definition of a production function expresses the minimum amount of each input to produce given outputs
with a fixed technology. An ordinary least squares analysis is therefore
inconsistent with a frontier production function that forms the core of microeconomic theory.
This has led to the development and estimation of parametric frontier
cost and production functions in the economics literature (see, for instance,
Aigner and Chu [1%8] and Aigner, Lovell, and Schmidt [1977]). A weakness of such parametric fh)ntier estimation is its inability to theoretically
substantiate or statistically test the maintained hypottiesis about the paran^tric form for the production function and the postulated distribution for
the disturbance term. Furthennore, the restrictions imposed on the production correspondence by these hypottieses are not immediately apparent. We
adopt an altemative nonparametric stochastic frontier estimation technique
called Stochastic Data Envelopment Analysis that only imposes conditions
of monotonicity and concavity on the production function, llie technique
we employ is sufficiently general to allow for multiple inputs and ou^uts
and for some of the inputs to be fixed.
We examine these altemative n^thods of evaluating productivity using
data from a large manufacturing plant that has recently establisted a productivity-based incentive compensation plan for its workers. Tlie plant manufactures traditional engineering products. It has gained a leadership position
by providing quality products at low cost. Maintaining cost advantage
I
MEASUREMENT OF PRODUCTIVITY IMPROVEMENTS
321
through {M-oductivity improvements is critical in this mature, stable, competitive industry. The gain-sharing bonus scheme is an incentive to enhance
labor productivity by sharing the financial gains from improved productivity
with employees. Indirect plant labor and salaried staff are included in the
plan to provide incentives for savings in the shop floor-related portion of
indirect labor, including time spent on repairs and maintenance, equipment
handling, set-ups, and inspection of set-ups.
Strong links exist between methods discussed in this paper and those
of traditional capital markets research. Although we specify a production
economics model, as opposed to a financial economics one, we perform a
variation of residual analysis based on an estimated standard. Patel (1976),
for instance, estimates a referent market model for the relationship between
firm and market returns and uses deviations from this as a measure of
abnormal returns and thus the infonnation content of eamings forecasts.
Similarly, we estimate a referent production set for the input-output correspondence and compute deviations from it to measure improvements in
manufacturing productivity and thereby the impact of the incentive scheme.
Further, we use the 0-1 variables technique as in Schipper and Thompson
(1983) to identify productivity gains in different periods of interest relative
to a common referent correspondence. The objective of our analysis is to
describe a methodology to examine the productive impact of the gain-sharing
scheme in the setting described; additional refinements to the methodology
may be warranted based on the specific situation encountered by a researcher
and his or her observations on the production process being studied.
This paper has the following structure: In Section 2 we discuss the
empirical setting of the problem and issues involved in obtaining and handling the data. Section 3 examines the merits and demerits of various economic models used to identify stochastic input consumption frontiers in
order to measure deviations of actual input consumption from the frontier
in the event period. We consider both parametric and nonparametric estimation techniques and describe the structure imposed and the corresponding
estimation procedure employed. Section 4 discusses the results and interprets
our findings.
2. Empirical Setting
Our site is a manufacturing plant within a highly diversified Fortune
500 company. Productivity improvement is a key component of the division's comp^tive strategy. The division leads its industry in technological
advancement and market share. It has secured and retained its position by
iwoviding better quality and more reliable products at a lower cost than its
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competition in a mature, no-growth industry. Productivity improvements
are critical for long-run competitive advantage. Output prices are controlled
by competitive maiket forces and reductions in input prices are generally
available to competitors as well.
Historically, the company has made great strides in productivity improvements, by producing more outputs with the same or lesser quantity of
inputs, through technological innovation, and by efficient shop floor management rather than by substituting labor for capital. This fact is stressed
by its chairman, who notes in the company's annual report that the company's strategy was to put "increased emphasis on new technology and new
engineering capacity, training, product quality, productivity and cost reduction." Among management's stated "high-priority" areas were "applying technology to new and improved products and processes" and
"improving quality, productivity and employee motivation." To continue
this trend and to maintain its cost leadership, the company has embarked
on a IKW campaign to improve productivity.
The behavioral setting of our investigation is cost minimization. Production requirements are determined by the marketing department and taken
as given by the manufacturing plant.* The plant's focus is on minimizing
resource consumption while producing the outputs required. Productivity
gains are manifested via reduced quantities of inputs required to produce
specified quantities of outputs.
Ilie particular plant we focus on is labor intensive with relatively stable
capital. As a direct consequence of this nonemphasis on capital, depreciation
accounts for only 3% of total costs and is a relatively minor item in the
plant's monthly expense summary accounts. Direct labor, on the other hand,
constitutes 20-25 percent of total expenses, and indirect labor and supervisory costs 25-30 percent. As part of its campaign to increase productivity,
a gain-sharing program^ was instituted at the plant with benefits tied predominantly to improvements in labor efficiency. Hie gain-sharing plan includes indirect plant labor and salaried staff b(»:ause these elements are a
significant percentage of t ( ^ labor costs and offer consider£d)le potential
for {Hoductivity improvement. Including indirect labor in the gain-sharing
arrangement also facilitates union negotiations because the incentive arrangement encompasses all wOTkers in tiie plant.
We describe below the basic steps of the gain-sharing computation.
Labor "pnxluctivity" in successive periods is computed relative to a base
1. If inpats and outputs are simultaneously determined, a simuhimeous equMions model must be
estimated (see Zellaor. Kmetta, uid Droe {1966}).
2. The design <^ this gain-diaring program is discussed in detail in Baidta- and Datar (1987b).
MEASUREMENT OF PRODUCTIVITY IMPROVEMENTS
323
period benchmaii^. The first step entails a computation of the standard direct
labor hours in die base period (denoted by s^,) obtained by multiplying the
standard direct labor hours per unit for each product (based on industrial
engineering estimates) by the quantity of each product produced in the base
period. The actual total labor hours (denoted by a^) including direct labor,
indirect plant labor, and salaried staff hours are also computed for the
base period. The ratio of actual direct and indirect labor hours to standard
direct labor hours in the base period determines a base ratio (denoted by
Tb — Ot/st,). In each subsequent nranth t, the ratio r, of actual total labor
hours a, to the standard direct labor hours Sf (based on the direct labor content
of products produced in period t) is computed. The gain-sharing fraction g,
for period t is calculated as -^ = ~^.
r,
Values of ^, greater than one indicate
a,l Sy
"productivity" gains; values of g, less than one signal "productivity"
declines.
The gain-sharing agreement calls for woricers to be paid at base-period
wage rates if g, in a period is less than one. When g, is greater than one,
half of the "productivity" gains are paid to workers. For example if g, =
1.14, which signals a 14 percent increase in "productivity," each worker
receives a bonus of 7 percent over the base wage or salary.
Our objective is to identify increases in productivity in each of the four
inputs in the 15 months following the program's introduction relative to the
33 months preceding it. Monthly data were available for a 48-month period
labeled 1 through 48, with the gain-sharing program taking effect in month
34. Monthly data on the physical quantities of the two products produced
were obtained firom plant production reports. The summary of manufacturing
costs provided details of the actual number of direct and indirect labor hours
employed each period. Gains in labor productivity are measured via reductions in hours worked. This provides a good measure of productivity because
the mix of workers at various skill levels and pay levels has remained
constant over the entire period of study. Productivity gains as reflected in
a reduction of labor hours is achieved by employing fewer temporary
laborers.
Tht two [Hoducts manufacbired in the plant use a common metal which
accounts for 90 percent of total raw material cost (which constitutes about
15-20 percent of total cost). We deflate material cost of production by the
increase in material prices each period to obtain a constant-cost estimate of
material consunqition. For miscellaneous manufacturing oveiiieads, we
group several relatively minor costs such as power (about 3 pereent of t c ^
cost), gas (1-2 percent), perish^le tools and jigs (4 percent), and janitorial
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services (1-2 percent) that represent important support services for the operation of the plant, under the single category of manufacturing support
services (25-30 percent of full cost). We deflate the cost each period by
appropriate indices, based on plant records and suppliers' bills, to obtain a
constant-cost estimate of consumption of manufacturing services.
The financial reporting focus of the accounting system required significant assumptions to be made in our analysis. The only information available
with respect to material cost was the material cost of goods sold for each
product. The material cost of production for each product is calculated by
multiplying the (deflated) material cost of goods sold by the ratio of goods
produced to goods shipped for each individual product. The total material
cost of production is derived by aggregating material costs over all products.
The material consumption data are thus noisy and approximate and our
results with respect to material costs must be interpreted cautiously.
There are four cost components: direct labor, materials, indirect labor,
and manufacturing services. The stability and relative maturity of the manufacturing process limits the potential for improvement in materials and
direct labor productivity. The input-output relationships including the noise
and stochasticity in these relationships are well known, and management
can control these costs on the basis of inputs consumed and outputs produced.
Indirect labor and manufacturing services inputs, on the other hand, are
discretionary in nature with no identified direct relationship between inputs
and outputs. Consequently, these costs cannot be controlled by monitoring
outputs and inputs. Instead, incentives need to be provided to influence the
behavior and effort of workers. The labor-based gain-sharing program is an
example of such an incentive. Consequently, we expect the gain-sharing
program to result in improvements in indirect labor and possibly manufacturing services. The impact on manufacturing services is likely to be smaller
because the program does not directly provide incentives to improve manufacturing services productivity. Nevertheless, the general focus on improving labor productivity may positively influence manufacturing services
pnxluctivity as well.
3. Methodfrilogy for Testing the Impact of Productivity
Improvement Programs
In describing the methodology, we (tenote die' two out{Hits produced as
yi and yz written in vector form as y = (yttyz)- The {^ysical inputs Xi, X2,
X3, and X4 are denoted by the vector x = (jt,,X2,JC3,jC4) where JC, represents
direct laixx, x^ indirect hSofX, x-^ materials consumption, aiMi X4 consumption
of manufacturing wrvices. ITie [Htxluction technology at the pknt permits
MEASUREMENT OF PRODUCTIVITY IMPROVEMENTS
325
little substitution among inputs. Because the plant is labor intensive, the
capital employed is small and relatively constant over the period of our
analysis. Similarly, material consumption cannot be reduced by substituting
other inputs for materials.
Our analysis of the production process indicated that the consumption
of each input depends on only the quantity of outputs y, and )'2 produced
and in particular is independent of the level of consumption of the other
inputs. That is,
Xi = fi(y,,y2) for all i = 1,2,3,4.
Our objective is to evaluate if, after controlling for the outputs produced,
input consumption in the post-gain-sharing period is less than the input
consumption in the pre-gain-sharing period.
The usual approach for testing for efficiency gains in the post-gainsharing period relative to the pre-gain-sharing period is to use a least squares
regression by fitting prespecified functional forms for the correspondence
between outputs and each input. Following the methodology of Schipper
and Thompson (1983), dummy (0-1) variables representing the pre- and
post—gain-sharing periods could be introduced to capture shifts in the relationships across these periods. For instance, specifying a loglinear relationship between each input and output y, and ^2 yields the following
estimation model:
[A] log Xi, = aoi + a,i log yi, -t- a2i log y^, + b A + ^a
Vi=l,...4, t=l,...48
where D, = 0 for t = 1 , . . . 3 3
= 1 for t = 3 4 , . . . 48
The eleiments of Ci are assumed to be distributed i.i.d. N(O,CT^) and
uncorrelated withji andy2- Significantly negative values ofbi indicate lower
input costs Xi (and productivity gains with respect to input i) in the postgain-sharing period.
Hiere are two important limitations in applying least squares techniques
to estimate the input consumption relationship between each input Xi and
the output prcKluced y = (y^ya). First, tiie regression-based approach estimates ibe average amount of input consumed to produce given levels of
ouQmts, whereas the theoretical definition of a production function expresses
the minimum amount of input for given levels of outputs. Moreover, from
a managennnt ccmtrol perspo^ve, comparing future period input consunq)tion with theregressicm-basedestimated consumption indicates whether
input consumption in tiie post-gain-sharing period has been less tiian av-
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erage, rather than whether input consumption is lower than the best diat
was achieved in die pre-gain-sharing period. Furthermore, least squares
regression estimates assume that the disturbance term arises from an i.i.d.
stochastic process so that deviations of actual observations from the estimated function are a s s u n ^ to result fiom random deviations. In reality,
these deviations result from extemal random factors as well as inefficiencies
of plant workers. Indeed, the productivity-based incentive plan is aimed at
motivating workers to put in greater effort to reduce inefficiency and improve
productivity. Second, regression-based parametric methods assumed a particular and often arbitrary functional form on die input-output correspondence. This problem is partly mitigated by assuming a flexible parametric
relationship between inputs and outputs such as a translog or loglinear
functional form. In die next section we provide a methodology to test for
productivity gains assuming a loglinear stochastic production technology.
3.1. Parametric Stociiastic Frratier Estimatitm
Estimating a frontier production function involves the specification of
the error term as being made up of two components, one normal and die
other from a one-sided distribution. That is, the error structure is given by:
e,, = Vi, -f- Uu V i = 1 , . . . 4 and t = 1 , . . . 48.
TTie error component Ui, represents a symmetric disturbance, where for each
I, {UjJ are assumed to be independendy and identically distributed as
N(O,(ji.). The error component Vj, is assumed to be distributed independently
of «i, satisfying Vj, S: 0. In particular, {vJ are a s s u n ^ to be independently
and identically distributed from a half-normal distribution Ar^(0,o^) trunc a t e below at zero.
The logic untterlying this specification is that die production process is
subject to two disturbances. Hie irannegative disturbance Vj, refl«;ts the
condition that for each input the input consumption level must lie above the
fircmtier (a minimum omsumption level) over all time periods. These deviations are attributable to factcvs umler the worker's control such as iiwfficiencies, wastage, die effort provicted by employees, and the extent of
reworked, ctefective, and dmnaged products. For each i, die random disturbance term Ujt reixeseirts die stochastic nature of die frontier itself over
time, v ^ much like d n random disdirbance term in a least square regression model. Hie Uj, t^m is the result of favtHabie as well as unfavtnable
raiuknn extemal events not controllable at ihe plant level, such as rand(»n
p^cnmance, iiKMtel specificioicm, and cm»s of ob%rvati<»i and
MEASUREMENT OF PRODUCnVlTY IMPROVEMENTS
INPUT
OUTPUT
Figure 1
of Composed Error Specification
n:^asurement. Figure 1 distinguishes between average and frontier relationships and illustrates the notion of composed errors in the single-input and
single-output case.
The point M represents an ou^ut level of OL and input consumption
of LM. Under ttie composed error model, the input consumption frontier
level is estimated to be LN. The total deviation MN comprises the inefficiency
component NP and a random effect PM which exceeds the level of
inefficiency.
The stochastic input p(»sibility frontier expresses the Pareto efficient
input combinations necessary to {Hoduce specified vectors of outputs given
die existing technology. Input consumption in excess of frontier levels is a
reflectk»i of ii^fficiency in implementing production. Reducing the degree
of inefficiency in production is an inqxntant motivation for the introduction
of a gain-sharing {nt^ram. It could alternatively be argued that incentives
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{»x)vided by the gain-sharing agreement induce new ways of organizing
production and result in shifts in the input possibility frontier. We take the
position that the input possibility frontier is not shifting (note that capital
investment in technology is also relatively stable over the period of our
fuialysis) and test whether the probability distribution generating the inefficiency terms decreases with the introduction of the gain-sharing program.
Our objective is to examine if productivity in the 15 months following
the gain-sharing program increases relative to the 33 months preceding it.
Production inefficiencies are measured by the nonnegative disturbance term
Vj, and represent deviations firom the frontier attributable to factors under
the workers' control. Note that since for each i, Vj, is assumed to be independently and identically distributed from a half-normal distribution
N^{O,ail,) truncated below at zero, any increase in productivity will decrease
both the mean and the variance of the distribution of Vu (because the mean
and variance of a half-normal distribution are not independent). One way
to examine this is to test if v,, is distributed as half-normal A^^(O,CTVJ) for
the 33-nionth pre-gain-sharing period and as N^(O,(TI — 80 for the 15month post-gain-sharing period.
Assuming a loglinear relationship (or, alternatively, a translog function),
we could proceed by estimating the following model:
[B] log Xj, = aoi + au log y,, + azi log yz, + Cj,, i = 1 , . . . 4, t = 1 , . . . ,48
where €(, = Ui, -I- Vj,
and Ui, ~ N(O,a^.)
Vi, ~ N " ( 0 , a J . ) f o r t = l , . . . , 3 3
Vi, ~ N^(O,aJ. - 8i) for t = 3 4 , . . . ,48.
Henceforth, cFy. is used to denote (TI. for observations 1—33 andCTJ.~ Si for
observations 34—48.
An approach to estimating the stochastic frontier production function
models as in [B] discussed by Aigner, Lovell, and Schmidt (ALS) (1977)
and Olson, Schmidt, and Waldman (OSW) (1980) is a maximum likelihood
estimator (MLE). Following Weinstein (1964), the density function of €i
for each i = 1 , . . . 4 is given by:
wtere of = o^. + o^., Xj =CTv/o^Bjand f*() and F*(-) are tfie standard
nonnal (tensity and distdbutiiHi functicms, reflectively.
We tl^refore have:
MEASUREMENT OF PRODUCTIVITY IMPROVEMENTS
329
that is.
Ln fied = Ln - ^ -ITT
The relevant loglikelihood function for all 48 observations is given by:
Therefore,
L ( ) = 48Ln
IT
CT
48
Ln[F*
+ tE
= 34
33
V .
2
48
- 1
Ji - 8,
where €i, = ln Xjt — doi — «» In ^u ~ fl2i In )'2f The loglikelihood function
can then be maximized with respect to the unknown parameters Ooi, au, 021,
(TIJ (TI. and 8i using a nonlinear search algorithm (such as Fletcher-Powell).
A test of the null hypothesis of 8; = 0 would then provide evidence on
productivity gains and reduction in inefficiency with respect to input i in
the post-gain-sharing period. The maximum likelihood estimator of 6i is
consistent and asymptotically efficient, but its finite sample distribution is
not known.
An alternative approach maintains somewhat different assumptions and
it models the input-ou^t relationship as:
[C] log Xi, = aa + au log y,, + a2i log y2, + €;,
w t e r e €i, = Uj, -I- Vj,
and
uu ~ N(0,cr2.)
fOT t = 1 , . . . ,33 wtere Vj, = vi -I- b,
for t = 3 4 , . . . ,48.
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Note that in model [B], the inefficiency terms V;,, both pre- and post-gain
sharing, are drawn from a half nonnal distribution that ranges over [0,<»),
with the post-gain-sharing distribution hypothesized to have a lower mean
and, accordingly, a lower variance. In contrast, in model [C], pre- and
post-gain-sharing inefficiencies are drawn from distributions with the same
variance, but ranging over [8i,oo) and [0,«>),respectively,with 8; hypothesised (in the altemate hypothesis) to be positive. A positive value of Sj in
model [C] indeed implies tiiat mean inefficiency is greater pre-gain sharing,
but it also implies that, in every instance of the pre-gain-sharing period,
there is inefficiency in input consumption, relative to frontier levels, of at
least e^i. This is apparently a restrictive feature of this model.
Model [C] is also consistent with an altemative set of maintained assumptions, namely, a neutral shift in the frontier unaccompanied by any
shifts in the probability distribution from which the inefficiency terms arise.
Of cotirse, this does imply that this model cannot be employed to distinguish
between the two altemative sets of assumptions. Conversely, if tiiere is no
a priori evidence to maintain one set of assumptions rather than the other,
model [C] provides a robust formulation.
The model in [C] can be estimated as:
[C]
log X,, = aoj + a,i log y,, + a2i log y2, - B A + Uj, -I- \„
where Uj, ~
V, ~ N"(O,aJ.)
D, = O f o r t = l , . . . , 3 3
= 1 fort = 34, . . . , 4 8 .
Hie maximum likelihood estimation technique discussed earlier can be
employed to test the null hypothesis of Sj = 0 versus the altemative that 8^
>0.
Stochastic frontier production function models as in [C] can also be
estimated as discussed by ALS (1977) and OSW (1980) using a corrected
ordinary least squares (denoted by CDLS) estimator. Hie COLS coefficients
are obtained by estimating an ordinary least squares (OLS) regression for
the composed error model in [ C ] . Except for the constant term, the OLS
estimator is unbiased and consistent. Tlie bias of the constant term is the
mean of €i = + \/2hT (Tyj. Consistent estimates of the variances
l
^i can be obtained by:
4)il3i]^ and ^
= A-^i
IT
where |j4i and |i4i are the second and third monwnts of the OLS residuals.
MEASUREMENT OF reODUCnVITY IMPROVEMENTS
331
A consistent COLS estimate of the constant term is obtained by subtracting
\/5/iir dvi ftom the OLS estimate of the constant term. This COLS estimate,
however, is not asymjHotically efficient and its finite sample distribution is
unknown.
In a Monte Carlo experiment designed to compare the COLS and MLE
estimators mentioned above, OSW (1980) find diat die COLS estimator is
more n»an square error (MSE) efficient for sample sizes 200 and below.
At sample sizes of 400 and 800, die MLE is MSE efficient for estimating
al., (TI., and a? but COLS is stUl superior for d,i and dj,. OSW (1980) could
not reject the null hypothesis diat there is no difference in variance between
MLE and COLS parameter estimates for any parameter for sample sizes
greater dian 25. OSW (1980) conclude diat COLS and MLE techniques are
both ai^licable in estimating parameters ofthe equation in [B] in moderately
sized samples.
The above discussion also suggests that the computationally simple
COLS estimators are preferred to the MLE estimators in smaller samples.
There is, however, one important problem widi the COLS estimator in diat
the estimator may not exist (in a meaningful form) in some samples. This
may happen in one of two ways. A "Type I " failure occurs if dv. is negative.
The problem occurs when X; = a^./CTu^ is small. A "Type H" failure occurs
when d^ < 0 and corresponds to die situation when X is large. This problem
does not' exist in the case of MLE estimators because the MLE procedure
simply maximizes the loglikelihood function widi respect to X and as reported
by OSW (1980) provides unbiased estimates of a^, au, and Oii- Indeed, as
the variance of o^. of the one-sided efficiency term increases, die MLE
estimators dominate because die MLE mediodology takes die exact nature
of the asymmetry of the distribution of the disturbance into account.
Because we encounter situations in which the COLS estimators do not
exist, we report the results of both die COLS and MLE estimations of each
input on the ou^uts i»oduced.
3.2 N<Mq>anunrtric Stochietk Frontkr Estimation
Although die paranwtric stochastic estimation of production frontiers
described in Section 3.1 overcomes tte conceptual difficulties of estimating
an averse relationship between inputs and outputs inherent in the usual
regression analysis, it does assume a particular functional form for the
production cOTtespomience airi the error stmcture. TTw statistical distributitms of the i»rameters are based on ttese assumed functional forms. Inferences based on die statistical tests are consequently conditional on die
specification of tbe nKxlel ctwiectly reflecting the un(teriying
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JOURNAL OF ACCOUNTING, AUDITING AND FINANCE
production relation (see Hildenbrand [1981], Varian [1984], and Banker
and Maindiratta [1987]). But the choice of a particular functional form is
difficult to justify on a priori grounds. This problem can be partially mitigated
by using fiexible functional forms such as the translog that can be used to
approximate various production functions. Unfortunately, these forms require the estimation of a large number of parameters relative to the 48
available observations. Furthennore, the underlying regularity conditions of
monotonicity and strict quasi-concavity are violated at many points of most
data sets, thus biasing inference; for a theoretical analysis of regularity
conditions see Caves and Christensen (1980) and Bamett and Lee (1985).
The problems inherent in parametric estimation can be overcome by
estimating a nonparametric stochastic frontier using the approach of Stochastic Data Envelopment Analysis (SDEA) (see Banker [1986a]). This
technique is an extension of Data Envelopment Analysis (DEA), which was
introduced by Chames, Cooper, and Rhodes (1978). DEA is a nonparametric
method for evaluating productivity which assumes only the regularity conditions of monotonicity ofthe prodtiction function and convexity ofthe input
possibility frontier; it imposes no additional stmcttire on the specified functional form. Banker, Chames, and Cooper (1984) show its flexibility in
modeling production operations in the presence of multiple outputs.
The DEA approach has been used in a variety of empirical settings.
Examples include program evaluation (Chames, Cooper, and Rhodes
[1981]), evaluation of school district efficiencies (Bessent et al. [1983]),
productivity measurement for manufacturing operations (Banker [1985];
Banker and Maindiratta [1986]), and tiie estimation of hospital production
fimctions (Banker, Conrad, and Strauss [1986]). Some other settings in
which the DEA technique has been employed are steam-electric power
generation (Banker [1984]), coal mines (Bymes, Fare, and Grosskopf
[1984]), pharmacy stores (Banker and Morey [1986a]) and fast-food restaurants (Banker and Morey [1986b]).
DEA's limitation lies in the fact that it does not allow for the possibility
of extemal random errors impacting the production process. Any difference
between the actual input consumption and the estimated frontier level is
therefore attributed to inefficiency. TTie SDEA model, on the otiier hand,
allows for the possibility of random errors in model specification or measurement via a symmetric random error component, in addition to the onesicted deviations attributable to inefficiency in the use of input resources.
TMs formulfttion for the error term resembles tte composed error specifications of the mo(tels of Aigner, Lovell, and Schmidt (1977) ar^ Meeusen
and van det Broeck (1977) discussed in Secticm 3.1.
The nonlinear maximum likelilKxxl estimation models require an a pricm
MEASUREMENT OF PRODUCTIVITY IMPROVEMENTS
333
(and often arbitrary) specification of the parametric distributions of the two
error terms. On the other hand, the linear programming-based formulation
of SDEA requires the relative weights for the two types of deviations (or
error terms) to be specified in the objective function. By varying the relative
weights, we examine the sensitivity ofthe estimation results to the postulated
importance of deviations due to inefficiency or external random factors. In
fact, for specific extreme values of these weights, the model includes the
traditional nonstochastic DEA model (in which all variations of actual values
from the predicted frontier are attributed to inefficiency) and also the minimum absolute deviations (MAD) regression model (in which all variations
of actual values from the predicted values are attributed to external random
factors).
Since the consumption of each input is independent of the consumption
of other inputs, we employ the SDEA model to estimate a separate stochastic
production frontier for each input i, that is, x, = f{y) relating the input
consumed x, to the output vector y, with f:Y-^R
where Y is the convex
hull of y. We do not impose any parametric form on / and only assume
that/i is monotonic and convex.
We model the technological specification and the input possibility frontier for all inputs to be the same in the pre- and post-gain-sharing periods.
Our objective is to examine if productivity of input consumption is greater
in the post-gain-sharing period than in the pre-gain-sharing period. To do
so, we split the data comprising 33 observations in the pre-gain-sharing
period and 15 observations in the post-gain-sharing period into two sets of
24 observations each. The first set comprises all odd-numbered observations
and includes 17 observations from the pre-gain-sharing period and 7 observations from the post-gain-sharing period. We refer to this sample as the
estimation sample because this sample is used to actually estimate the stochastic nonparametric frontier for each input i.^ We then computed the
efficiency scores for all observations in the second sample, referred to as
the test sample, by comparing the estimated minimum input consumption
with the actual input consumption. These efficiency scores are used to test
whether productivity in the post-gain-sharing period is significantly greater
than pnxiuctivity in the pre-gain-sharing period. The frontier values ii, =
fi(yu, y^) are estimated by specifying the structure imposed on the deviations
of ;Ci, from 4 . As in Section 3.1, the deviation Xn - 4 is expressed as the
3. We also re-eaimated tf» fhwlier using a random sample of 17 observations from the 33 pregain-sharing observations and 7 (*servati<»is ftran the 15 pmt-gain-shahng observations. The results
were sinilm' to Ibose repcxted in detail hoe.
334
JOURNAL OF ACCOUNTING, AUDITING AND FINANCE
sum of two components; Vj, represents the excess of input i consumed in
period t due to inefficiency and u,, represents the effect of random factors
including specification and measurement errors. That is,
Xi, -
X,, =
Vi, -I- Ui,.
(1)
Because Vj, measures input inefficiency relative to the input consumption
frontier, Vi, is nonnegative and the symmetric term u,, is unconstrained in
sign. Unlike the parametric stochastic frontier estimation of Section 3.1; no
particular parametric form is assume for Uj and Vi. As in goal programming
formulations, the symmetric error Un is expressed as:
u., = Uit -
UiT widi Ui^; Ui7 > 0,
andSr=,Ui: = 2r.,Ui7
(2)
(3)
Therefore,
Vi Xi, = Xi, = Vi, + Urt -
Ui7 with Ui^. Ui7 > 0 ,
The stochastic, nonparametric input consumption frontier values Hi, =
fi(y,,,y2,) are estimated by minimizing a weighted sum of different components of deviations subject to die monotonicity and convexity constraints.
The monotonicity and convexity conditions for ii, = ^(y,) can be represented
by inequality (4) as follows:
For each t, i^i, — ii, S: Wi,(ys — yj for all s = 1,—n
(4)
where Wa is a nonnegative vector (see, for instance, Bazaraa and Shetty
[1979] and Banker [1986a]). The intuition for (4) follows fitom die fact diat
all points of a monotonic and convex function lie above die tangent hyperplane at any point t. Substituting (1) and (2) into (4) yields:
x« - Xi, 2: Wi,(y, - yd + (Vis - Vj,) -I- (Ui^ ~ u^ - \i^ + u^)
foralls=l,...,n.
(5)
For each input i, i= 1 , . . . ,4, the linear program to be solved is given
by:
[D]
Minimize 2 ^ , (u^ -I- u^ subject to
MEASUREMENT OF PRODUCTIVITY IMPROVEMENTS
[D.I]
[D.2]
[D.3]
335
for e a c h t = l , . . . , 2 4 .
x« - Xi. > Wi,(y, - y.) + (vu - v,,)
-I- (Uit - Ui7 - u.t + Ui7) for all s = 1 , . . . ,24, s 7^ t
2 , ^ , (u,r - Ui7) = 0
Wi, > 0 , Vi,, Ui:,Ui, > 0 .
TTie weight Cj > 0 in the objective function is a prespecified constant.
Varying the value of c, gives different estimates of the production frontier
values. Small values of Ci corresponds to greater weight being placed on
the inefficiency term Vi, and for c, < - leads to the conventional DEA
n
formulation in which all variation is attributable to inefficiency. Increasing
Ci increases the amount of variation attributed to the random factors reflected
in the «i, terms and for Ci > 2 corresponds to MAD regression. By estimating
the model for various values of c,, we are able to assess the sensitivity of
the estimation to assumptions about the relative weights assigned to the
different sources of deviations of actual values from estimated frontier
values.
In Figure 2, we illustrate the estimation of the production frontier corresponding to different values of c for the case of a single input and a single
output. Here, for small values of Cj (< 0.2) we obtain tiie DEA estimates,
which assume no random specification or measurement errors and the linear
program estimates the minimum amount of input consumption for a given
level of output assuming monotonicity and convexity. The frontier is computed based on available observations and without recourse to any a priori
assumptions about the specific underlying functional form ofthe input-output
correspondence. For each input, the input productivity measure in any period
t is the ratio of the minimum amount of input for the level of output produced
as determined by tiie estimated fr^ontier, and the actual input consumption
in that period. Thus, for period 4 the productivity equals ABIAC. This DEA
measure of productivity is a relative measure because it evaluates the productivity of any period relative to available observations subject to tiie
conditions of monotonicity, convexity, and minimum extrapolation.
For Ci = 0.8, the input consumption frontier is pulled upward because
some of tte deviation of actual input consumption from estimated values is
attributed to random stochastic factors rather than inefficiency alone. The
inefficiency scores for various observations is, in general, lower. For still
larger values of c,(Ci = 1.2), variations of actual from estimated values are
entirely ^tributed to random factc»s and yield the MAD regression equation.
TMs has tte effect of fiirtter pushing up tiie estimated input consumption
function and reducing tite iirefficiency scores. Note, however, that the MAD
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JOURNAL OF ACCOUNTING, AUDITING AND FINANCE
INPUT
OUTPUT
FIGURE 2
Estimation <tf Sto<^astic, Ntrnpanunetrk Input Consumption Function
regression is a flexible, nonparametric formulation for estimating monotonic
and convex functional correspondence of outputs and inputs and does not
impose any parametric form for the production relationship.
A stochastic, nonparan^tric frontier is e^imat^ for each input i and
for each value Cj basc^ on the 24 observations in die estimation sample.
Tliese yield estimates of d^ minimum amount of input consunq^cm ij, for
given levels of o u ^ t s {yu,y-hi assuming a one-sided (feviation due to inefficiency Vjt imd a symmetric two-sided emn* conqx)nent u^ attributable to
random factors including model specificaticm and nwasurement errors. E>ifferent values of Cj provide different weights on Vjt and u^.
MEASUREMENT OF PRODUCnVITY IMPROVEMENTS
337
Productivity (efficiency) scores are then computed for each of the 24
observations in the test sample for each input i and for all values c, as the
rado of the estimated consumption ii, and the actual consumption Xi,. The
Mann-Whitney (1947), Welch (1937), and Kolmogorov-Smimoff (Conover
[1980]) tests are used to examine if the average productivity of the 8 observations in the post-gain-sharing period is significandy greater than the
average productivity of the 16 observations in the pre-gain-sharing period.
The tests are run for all inputs i, / = 1 , . . . ,4 and all values of Ci. This
enables a determinadon of the sensitivity of our conclusions to changes in
the relative weights attributable to the random and inefficiency factors.
Comparing the results ofthe nonparametric and parametric stochastic frontier
analysis provides some insight into the robustness of our conclusions about
the impact of the gain-sharing program at the particular site.
In addition to estimating productivity measures for each of the inputs,
we compute an overall measure of productivity for each period using a
generalization ofthe Davis (1955) method. The overall productivity measure
aggregates the individual input productivities in each period using the actual
cost shares of the inputs in that period as weights."* The productivity of each
input may be analyzed in terms of productivity variances analogous to direct
material and direct labor usage variances in cost accounting.' The aggregation described above is equivalent to computing total variance for a period
as the weighted sum of the individual input variances. If inputs are not
separable, cost savings can be realized by subsdtuting one input for another
in die event of changes in the relative prices of inputs. As in Banker (1985),
we can compute two separate variances: an allocative variance that evaluates
die ratio at which inputs are employed relative to their prices and a technical
variance that examii^s the physical consumption of inputs reladve to the
estimated firontier consumption for the given mix of inputs. The product of
the two variances represents the aggregate variance. In the next section, we
discuss and interpret our findings based on both a nonparametric and a
parametric analysis of stochastic input consumption functions.
4. Results and Interpretatioiis
Given the methodological advantages of nonparametric stochasdc fronder estimadon, we start by discussing the results of Stochasdc Data En4. WeiiJiting each input productivity by the share of tfiat input in total cost emphasizes gains in
the most significimt ekments (tf total cost in the computation of total productivity.
5. Ualike vaHances, the n ^ measures described in this paper control for volume changes and
laoss periods.
I
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JOURNAL OF ACCOUNTING, AUDITING AND FINANCE
velopment Analysis. Tables 1 through 5 contain the results of tests for
differences in productivity scores in the pre- and post-gain-sharing periods
for direct labor, indirect labor, materials, manufacturing services, and aggregate inputs. The one-sided Welch two-means test makes inferences on
the relative magnitude of means in the two periods. The one-sided MannWhitney test measures die presence of significant differences in the locations
of the underlying distributions in the pre- and post-gain-shMing periods.
The Kolmogorov-Smimoff test evaluates a more general form of differences
among productivity scores in the two periods. The alternative hypothesis is
that productivity scores in periods after gain sharing "tend to be higher"
than those before its introduction.
Table 1 presents the results for direct labor productivity for various
estimates of c that increases the weight on random deviations relative to the
inefficiency component. The table indicates that direct labor productivity is
not significantly different in the post-gain-sharing period relative to tiie pregajn-sharing period for all values of c. A likely explanation is the limited
potential for improvement in direct labor productivity for a mature technology. The input-output relationship for direct labor is well documented
and can be effectively monitored via engineering standards. Besides, improvements in direct labor productivity may be constrained by machineoperating constraints and strict quality-control standards in place at the plant.
Results on the impact of the gain-sharing program on indirect labor
productivity are presented in Table 2. Welch's two-means test, the MannWhitney test, and the Kolmogorov-Smimoff test indicate tiiat indirect labor
productivity is significantly greater (at the 10 percent level) in the postgain-sharing period than in the pre-gain-sharing period. This basic conclusion is relatively st^le across all values of c. These results are consistent
with our hypothesis that providing gain-sharing incentives influences workers' behavior and motivates tiiem to determine ways to increase productivity
of indirect labor. The input-output relationship in Has. case of indirect labor
is not directly identified, so that unlike direct labor, monitoring indirect
labor productivity via engineering standards is considerably more difficult.
Table 3 describes tiie results for materials productivity and indicates
significant gains in materials productivity in the post-gain-sharing period.
Although these gains could be attributed to reduced scrap, wastage, and
reworic, the labor-tosed gain-sharing program does not directly motivate
efforts to improve materials productivity. Morraver, mataials {ooductivity
is more effectively controlled via evaluating materials consunoption requirements of the ou^uts produc»l. As indicate in Section 2, data on ti^ {riiysical
units of materials consumed each period were •aot available and our results
may be an artifact of the noise in our estimates (rf material
MEASUREMENT OF PRODUCTIVITY IMPROVEMENTS
339
TABLE lA
Welch's Two-Means Test on Direct Labor Productivity
c=0.04 (DEA)
0 = 0.2
c = 0.4
0 = 0.6
0 = 0.8
0=1.0 (MAD)
95% Conf. Interval
for Difference in
Average Productivity
r (Test
Statistic)
Degrees of
Freedom
Level of
Significance
(-0.109,0.088)
(-0.109,0.088)
(-0.109,0.088)
(-0.116,0.108)
(-0.112,0.109)
(-0.119,0.122)
-0.22
-0.22
-0.22
-0.08
-0.02
+ 0.02
15.2
15.2
15.2
14.6
14.4
14.0
0.59
0.59
0.59
0.53
0.51
0.49
TABLE IB
Mann-Whitney Test on Direct Labor Productivity
95.4%
Cor^idence
Interval
Point Estimate
for Difference in
Average Prodttctivity
-0.0093
-0.0093
-0.0093
0
0
-0.0017
o o o o o o
o o o o o o
1 1 1 1 1 1
c = 0.04 (DEA)
0=0.2
0 = 0.4
0=0.6
0 = 0.8
0=1.0 (MAD)
W (Test
Statistic)
Level of
Significance
96
100
100
99
0.5
0.5
TABLE IC
K<dmogorov-Snilm(^ Test tor Direct Labor Productivity
T,* (Test Statistic)
Level of Significance
c = 0.04
(DEA)
c=0.2
c = 0.4
c = 0.6
0.125
>0.10
0.125
>0.10
0.125
>0.10
0.187
>0.10
-0.8
0.187
>0.10
c=I.O
(MAD)
0.187
>0.10
The difference in manufacturing services productivity in the pre- and
post-gain-sharing period shown in Table 4 demonstrates some, though not
significant, improvement in productivity. Manufacturing services productivity is di£ficult to monitor based on tiie quantity of services consumed and
tHi^Hits produced, and providing incentives to influence behavior may be
useful in motivating incieased efficiency. Ontiieother hand, the gain-sharing
{Hognun focuses on labor productivity aloiK rather than total input productivity, and consequently provides no direct incentives for improving manufacturing services jwoductivity. Thus, although some improvement in
jHoductivity is realized, tiiese gains are not significant.
Tidde 5 describes changes in tiie overall jMtxiuctivity in the two periods
based on an aggregation of individual input productivities. The table signals
I
340
JOURNAL OF ACCOUNTING, AUDITING AND FINANCE
TABLE 2A
Welch's Two-Means Test on Indirect Labor Productivity
c = 0.04 (DEA)
c = 0.2
c = 0.4
c = 0.6
c = 0.8
c=I.O
c = 1.2 (MAD)
95% Conf. Interval
for Difference in
Average Productivity
T (Test
Statistic)
Degrees of
Freedom
Level of
Significance
(-0.1,0.438)
(-0.1,0.438)
(-0.11,0.444)
(-0.11, 0.451)
(-0.12,0.467)
(-0.12,0.48)
(-0.12,0.493)
1.48
1.48
1.46
1.44
1.38
1.40
1.42
7.5
7.5
7.5
7.5
7.6
7.6
7.8
0.091
0.091
0.094
0.097
0.11
0.10
0.099
TABLE 2B
Mann-Whitney Test on Indirect Labor Productivity
Point Estimate
for Difference in
Average Productivity
95.4%
Confidence
Interval
W (Test
Statistic)
Level of
Significance
0.0753
0.0753
0.0859
0.0849
0.086
0.0834
0.0932
(0, 0.21)
(0, 0.21)
(0, 0.21)
(-0.02,0.21)
(-0.03,0.22)
(-0.02,0.25)
(-0.02,0.26)
130.5
130.5
131.5
128.5
119.5
123.5
125.5
0.0331
0.0331
0.0288
0.0432
0.1223
0.0795
0.0629
c = 0.04 (DEA)
c = 0.2
c = 0.4
c = 0.6
c = 0.8
c=1.0
c = 1.2 (MAD)
TABLE 2C
Kolmogorov-SmimofF Test for Indirect Labor Productivity
c = 0.04 c=0.2
(DEA)
Tl (Test Statistic)
Level of Significance
0.5
0.05
0.5
0.05
c = 0.4
c = 0.6
c=0.8
c = 1.
(MAD)
0.5
0.05
0.437
0.10
0.375 0.437
>0.10
0.10
0.437
0.10
gains in aggregate productivity since introduction of the gain-sharing program. This reflects the weights assigned to individual input productivities
(based on the actual cost shares of various inputs) in computing aggregate
productivity as well as the productivity gains experienced by indirect labor,
materials, and to a lesser extent manufacturing services.
We next compare the results of the nonparametric analysis with the
conclusions based on a parametric estitimticm of the input consumption
function. Using mo<tel [C], tl^ latter requires a jKinuneiric specification of
the functional relationship between iapats and o u ^ t s as well as distiibutional assumptions about the random error term and inefficiency conqtoi^nts.
341
MEASUREMENT OF PRODUCTIVITY IMPROVEMENTS
TABLE 3A
Welch's Two-Means Test on Materials Productivity
c = 0.04(DEA)
c = 0.2
c = 0.4
c=0.6
c = 0.8
c=1.0
c= 1.2 (MAD)
95% Conf. Interval
for Difference in
Average Productivity
T (Test
Statistic)
Degrees of
Freedom
Level of
Significance
(0.032, 0.245)
(0.032. 0.245)
(0.022. 0.248)
(0.022.0.251)
(0.015. 0.264)
(0.011.0.301)
(0.010. 0.305)
2.71
2.71
2.49
2.48
2.34
2.29
2.27
22 0
22.0
21.7
21.3
19.5
16.2
16.0
0.0065
0.0065
0.011
0.011
0.015
0.018
0.019
TABLE 3B
Mann-Whitney Test on Materials Productivity
Point Estimate
for Difference in
Average Productivity
95.4%
Confidence
Interval
W (Test
Statistic)
Level of
Significance
0.163
0.163
0.1591
0.1566
0.1395
0.1694
0.1686
( 0.001.0.25)
( 0.001.0.25)
(-0.002. 0.276)
(-0.002,0.280)
(0. 0.3)
(0.002. 0.331)
(0. 0.322)
133
133
131
129
133
134
133
0.0233
0.0233
0.0309
0.0405
0.0233
0.0201
0.0233
c = 0.04(DEA)
c = 0.2
c = 0.4
c = 0.6
c = 0.8
c=1.0
c= 1.2 (MAD)
TABLE 3C
Kolmogorov-SmirnofF Test for Materials Productivity
c = 0.04 c = 0.2
(DEA)
T,* (Test Statistic)
Level of Significance
0.562
0.025
0.562
0.025
= 0.4
c = 0.6
c = 0.8
c=1.0 c=1.2
(MAD)
0.5
0.05
0.5
0.05
0.5
.05
0.562
0.025
0.437
0.10
In Tables 6A and 6B we describe the COLS and MLE estimates assuming
a translog production function that includes as special cases the CobbDouglas and CES functional forms.
Table 6A indicates productivity gains significant (at the 5% level) in
indirect lahor, materials, and manufacturing services (see estimates of 5,).
The asymptotic f-statistics should be interpreted cautiously because the sample size is 48 and small sample distributions of COLS are not known. Many
of the coefficients on ttie loglinear and logquadratic (translog) terms are not
significant, arguably due to multicollinearity among the independent variables. Furttermore, as discussed by Caves and Christensen (1980) and
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JOURNAL OF ACCOUNTING, AUDITING AND FINANCE
TABLE 4A
Wekh'!s Two-Means Test cta Maniifactoring Services ProdiKtivity
0 = 0.04 (DEA)
0 = 0.2
0 = 0.4
0 = 0.6
0 = 0.8
0 = 1 . 0 (MAD)
95% Conf. Interval
for Difference in
Average Productivity'
TfTest
Statistic)
Degrees of
Freedom
Level cf
Significance
(-0.15,0.376)
(-0.15,0.376)
(-0.15,0.380)
(-0.15,0.381)
(-0.16,0.387)
(-0.16,0.414)
0.%
0.96
0.96
0.95
0.96
0.97
10.1
10.1
9.8
9.7
9.6
lO.l
0.18
018
0.18
0.18
0.18
0.18
TABLE 4B
Mann-Whitney Test on 1Vfanufacturing Services Productivity
Point Estimate
for Difference in
Average Productivity
95.4%
Cor^idence
Interval
W (Test
Statistic)
Level of
Significance
0.0712
0.0172
0.0807
0.0766
0.0766
0.0957
(-0.06,0.21)
(-0.06,0.21)
(-0.05,0.23)
(-0.05,0.23)
(-0.05,0.22)
(-0.07,0.29)
119.5
119.5
117.5
116.5
115.5
116.5
0.1223
0.1223
0.1489
0.1636
0.1792
0.1636
0 = 0.04 (DEA)
0 = 0.2
0 = 0.4
0 = 0.6
0 = 0.8
0 = 1 . 0 (MAD)
TABLE 4C
Kolmogorov-Smimoff Test for Manufacturing Services Productivity
T,* (Test Statistio)
Level of Signifioanoe
0 = 0.04
(DEA)
0 = 0.2
0 = 0.4
0=0.6
0.313
>0.10
0.313
>0.10
0.313
>0.10
0.313
>0.10
0 = 0.8
0.313
>0.10
0=1.0
(MAD)
0.375
>0.10
Bamett and Lee (1985), the regularity conditions of monotonicity and strict
quasi-concavity are often violated at many points of data sets. Etesides, the
translog form requires the estimaticHi of a large number of parameters relative
to the 48 available observsrtions.
In addition, the COLS estimation suffers from "Type I" failure for
direct labor, indirect labor, and materials as discu^ed by Olson, Schmidt,
and Waidman (1980), because tl^ third mrni^ot of the OLS residuals is
positive, implying tiiat a , is negative. Hie bias of 0.073 in die i n t e n d
term in the case of manufacturing %rvices is cfHiected using tiie n^tiKxloIogy
described in Section 3.
In order to overconw tiie "Type I" failure in the COiJS estimatkm, we
also estimate the translog infmt consuroiHion function using Maximum Like-
NfEASUREMENT OF PRODUCnVlTY IMPROVEMENTS
343
TABLE SA
Welch's Two-Means Test on OveraO Productivity
95% Conf. Interval
for Difference in
c = 0.04 (DEA)
c=0.2
c=0.4
0=0.6
c = 0.8
c=1.0
c= 1.2 (MAD)
Average Productivity
T(Test
Statistic)
Degrees of
Freedom
Level of
Significance
(-0.064,0.293)
(-0.064,0.293)
(-0.068,0.297)
(-0.072,0.304)
(-0.077,0.316)
(-0.078,0.336)
(-0.078,0.341)
.45
.45
.42
.40
.37
.41
.42
9.5
9.5
9.3
9.3
9.1
9.2
9.2
0.091
0.091
.094
0.097
0.10
0.096
.095
TABLE SB
Mann-Whitney Test on Overall Productivity
Point Estimate
Average Productivity
95.4%
Confidence
Interval
W (Test
Statistic)
Level of
Significance
0.0904
0.0904
0.0928
0.0852
0.0799
0.0894
0.0914
(-0.012,0.191)
(-0.012, 0.191)
(-0.015,0.194)
(-0.022,0.204)
(-0.024,0.222)
(-0.012,0.242)
(-0.020 0.242)
126.5
126.5
129.5
128.5
126.5
129.5
129.5
0.0557
0.0557
0.0379
0.0432
0.0557
0.0379
0.0379
for Difference in
c = 0.04 (DEA)
c = 0.2
c=0.4
c = 0.6
c=0.8
c=1.0
c= 1.2 (MAD)
TABLE SC
for Overail Productivity
= 0.2
c = 0.4
c = 0.6
c = 0.8
c= I
(DEA)
T|* (Test Statistic)
Level of Significance
0.437
0.10
(MAD)
0.437
0.10
0.437
0.10
0.437
0.10
0.437
0.10
0.437
0.10
0.50
0.05
lihood Estimation (MLE). This is done using the program LIMDEP (see
Greene [1985]). The results reported in Table 6B are very similar to the
COLS estimation with consumption of indirect labor, materials, and manufacturing services significantly lower in the post-gain-sharing period (see
estinuttes of 5) resulting in greater productivity and efficiency. It is interesting to note that the MLE estimate of <Ty, turns out to equal zero in the
case of direct labor, indirect l^x>r, and materials. Consequently, for these
inqputs, tte COLS estimates are maximum likelihood resulting in identical
panaaster estimates in T^les 6A and 6B.
results of the translog model are difficult to interpret in a straight-
344
JOURNAL OF ACCOUNTING, AUDITING AND FINANCE
TABLE 6A
Estimates of the Translog COLS Model
log X, = ao, + a,i log y,, + a^, log y,, + a,, (log y,,)'
+ a* (log ya)' + %, (log y,,) (log y^) - 8,D, + e,
where i = l , . . . , 4 ; t = l
48,
and €„ = !!„ + v«. u. ~ N(0, o i ) , v, ~ N*(0, aid
Estimates
ao
a,
aj
a3
a,
as
8
Estimate of Bias
in
Intercept
Direct
Labor
Indirect
Labor
Materials
Cost
Manitfacturing
Services
16.697
(1.98)
-2.891
(-1.37)
-0.102
(-0.12)
0.305(2.07)
0.057
(1.77)
-0.067
(-0.63)
-0.013
(-0.61)
33.990~
(3.24)
-7.072~
(-2.69)
-0.529
(-0.50)
O.537~
(2.93)
0.031
(0.76)
0.037
(0.28)
-7.793
(3.02)
-13.980
(-0.72)
3.747
(0.77)
1.201
(0.61)
-0.271
(-0.79)
-0.139
(-1.86)
0.107
(0.43)
0.162~
(3.24)
1.182
(0.40)
1.900
(1.59)
0.170
(0.82)
0.081
(1.80)
-0.406~
(-2.73)
0.065~
( 2.16)
—
—
—
0.073
0.08r
Figures in parentheses indioate t-statistios.
The superscript ~ indicates a ooeffioient signifioant at the 5% level.
TABLE 6B
Estimates of the Translog MLE Model
log X, = aa + a,, log y,, + aj. log y^ + a,, (log y,,)'
+ a4, (log ya)' + a,, (log y,,) (log y j - 8,D, + e,,
where i = l , . . . , 4 ; t = l , . . . , 4 8 ,
and €, = u,, + V,. u, ~ N(0, ai), v, ~ N*(0, al,)
Estimates
Direct
Labor
Indirect
Labor
Materials
Cost
Mantrfacturing
Services
16.697
(1.98)
-2.891
(-1.37)
-0.102
(-0.12)
0.305(2.07)
0.057
(1.77)
-0.067
(-0.63)
-0.013
(-0.61)
33.t«6~
(3!24)
-7.072~
(-2.69)
-0.529
(-0.50)
O.537~
(2.93)
0.031
(0.76)
0.037
(0.28)
0.081(3.02)
-13.975
(-O.72)
3.747
(0.77)
1.201
(0.61)
-0.271
(-0.79)
-0.139
(-1.86)
O.tO7
(0.43).
0.162"
(3.24)
-5.S76
(-0.32)
1.075
(0.22)
1.552
(1.21)
0.137
(0.42)
0.067
(I.M)
-0.331(-2.06)
0.047~
(1.96)
Figures in {HuentlKses indioate t-statistios.
The superscript — indicates a ooefficwnt significant at the 5% tevel.
MEASUREMENT OF PRODUCTIVITY IMPROVEMENTS
345
TABLE 7A
Estimates of the LogUnear COLS Model
log X, = aa -I- a,, log y,, -I- aj, log y^ - 8 A + e,,
where i = l , . . . ,4; t = l , . . . ,48,
and e. = u, + v,. u, ~ N(0, a^), v« ~ N*(0, al,)
Estimates
Direct
Labor
Indirect
LtUmr
Materials
Cost
Manufacturing
Services
3.638"
(10.97)
0.725"
(16.34)
0.213"
(8.47)
-0.008
(-0.36)
6.988"
(16.76)
0.352"
(6.32)
0.135"
(4.28)
0.093"
(3.35)
-0.449
(-0.61)
0.842"
(8.56)
0.0514
(0.92)
0.164~
(3.36)
-0.024
Estimate of Bias
in Intercept
0.641"
(9.99)
0.237"
(6.54)
0.073"
(2.27)
0.105
Figures in parentheses indicate t-statisdcs.
The superscript ~ indicates a coefficient significant at the 5% level.
TABLE 7B
Estimates of the LogUnear MLE Modei
log x, = ao, + a,, log y,, + a,, log y^ - 8 A + e,
where i = l
4; t = l
48,
and e. = u. -H V. u. ~ N(0, ai), v» ~ N*(0, aid
Estimates
Direct
Labor
Indirect
Labor
Materials
Cost
Manitfacturing
Services
3.638"
(10.97)
0.725"
(16.34)
0.213~
(8.47)
6.988"
(16.76)
0.352"
(6.32)
0.135"
(4.28)
0.093"
(3.35)
-0.449
(-0.61)
0.842"
(8.56)
0.051
(0.92)
0.164"
(3.36)
0.412
(0.75)
0.644"
(10.27)
0.201"
(5.32)
0.062"
(2.12)
(-0.36)
Figures in {xirentheses indicate t-statisdcs.
The superscript ~ indicates a coefficient significant at the 5% level.
forward manner because of the presence of second-order terms. Further, as
noted earlier, the results are adversely affected by multicollinearity among
the indepen^nt variables and the large number of parameters estimated
relative to available oteervations. To provide a partial resolution to these
issues, we estim^e a restiicted form of the translog, the logline^ functional
form. TTie results, described in Tables 7A and 7B, sue very similar to those
obtaiiKd in tiw more general translog model. The gain-sharing agreement
346
JOURNAL OF ACCOUNTING, AUDITING AND FINANCE
has a positive impact on productivity of indirect labor, materials, and manufacturing services. Once again the COLS estimations presented in Table
7A suffer from "Type I " failure due to negative estimates for av. The
loglinear form is re-estimated using MLE (see Table 7b) and yields estimates
very similar to those obtained using COLS. In the cases of' 'Type I'' failures,
MLE estimates av to equal zero.
The results of the nonparametric and parametric stochastic frontier analyses are remaricably similar. They suggest that the gain-sharing progr^n
has a positive effect on indirect labor, manufacturing services, and materials
productivity and relatively little effect on direct labor. Our conclusions
appear to be robust to the assumed functional forms ofthe input consumption
functions although we are somewhat skeptical about the reliability of the
materials consumption data. Nevertheless, the consistency ofthe parametric
analysis with the stochastic nonparametric estimation increases the degree
of confidence in our analysis.
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