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International Journal of Manpower
Is workforce diversity good for efficiency? An approach based on the degree of
concavity of the technology
Vincent Vandenberghe
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Vincent Vandenberghe , (2016),"Is workforce diversity good for efficiency? An approach based on the
degree of concavity of the technology", International Journal of Manpower, Vol. 37 Iss 2 pp. 253 - 267
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Is workforce diversity good
for efficiency? An approach
based on the degree of
concavity of the technology
Vincent Vandenberghe
Institut de recherches économiques et sociales,
Université catholique de Louvain, Louvain-la-Neuve, Belgium
Abstract
Purpose –The purpose of this paper is to answer the question of workforce diversity and efficiency.
It departs from the rather ad hoc approach used in most recent empirical papers exploiting
firm-level evidence, and suggests focusing on the estimation of the degree of concavity of the
production function.
Design/methodology/approach –Workforce diversity is optimal when the technology displays
concavity in the share of workers considered (e.g. decreasing marginal contribution of rising shares of
more productive/skilled workers). What is also shown in this paper is that a generalised version of the
production function à-la-Hellerstein-Neumark (HN) –where workforce diversity is captured via an
index of labour shares –is suitable for estimating the concavity of the technology, and thus for
assessing the case for/against workforce diversity.
Findings –The paper contains an application to two panels of Belgian firms covering the 1998-2012
period. The main empirical result is that of an absence of strong evidence that age, gender or
educational diversity is good or bad for efficiency.
Originality/value –The key idea of the paper is that the degree of convacity/convexity in the share of
workers considered of firm-level technology and the desirability/efficiency of workforce diversity are
intrinsically connected. It is also that a non-linear/CES version of the HN labour-quality index can be
used in empirical work to assess the degree of concavity/convexity of the technology and quantify the
efficiency gains/losses of workforce diversity.
Keywords Employment, Data analysis, Efficiency, Productivity rate, Concavity, Labour diversity
Paper type Research paper
1. Introduction
The popular press usually discusses workforce diversity as being beneficial for
efficiency. How do economists address this topical question?
A first stream of the economic literature adopts a rather micro and within-firm
perspective. It has its roots in personnel economics and human resources management
theory. Some authors active in that field argue that diversity can create negative effects
due to poor communication, lower social ties and trust, and also poor cooperation
among workers (Becker, 1957; Lazear, 1998, 1999). Others posit that diversity can be
beneficial to firm performance due to better decision making, improved problem
solving, enhanced creativity, or a better ability to interact with clients that are
themselves very diverse (Hong and Scott, 2001, 2004; Glaeser et al., 2000). Empirically,
International Journal of Manpower
Vol. 37 No. 2, 2016
pp. 253-267
© Emerald Group Publishing Limited
0143-7720
DOI 10.1108/IJM-01-2015-0010
Received 9 January 2015
Revised 28 April 2015
6 June 2015
Accepted 13 July 2015
The current issue and full text archive of this journal is available on Emerald Insight at:
www.emeraldinsight.com/0143-7720.htm
JEL Classification —J11, J14, J21
Funding for this research was provided by the Belgian Science Policy Office –BELSPO,
Research Grant No. TA/00/46. The author would like to thank anonymous referees for their
helpful comments and suggestions on previous versions of this paper.
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economists try to assess which of these two antagonist forces prevail by examining
how (within firm) workforce diversity translates into firm-level efficiency gains/losses.
The most recent contributions exploit the potential of firm-level longitudinal (i.e. panel)
data to explore how within firm changes of the degree of diversity of the workforce
affect output. Recent examples are Kurtulus (2011), Ilmakunnas and Ilmakunnas (2011),
Garnero et al. (2014) and Parrotta et al. (2012). Compared to studies based on cross-
sectional material, these provide evidence and results that are much more robust and
trustworthy. Findings generally show that educational diversity is beneficial for firm
productivity. In contrast, age and gender (i.e. demographic) diversity are found to
hamper firm-level added value per worker ceteris paribus.
We would argue that one of weaknesses of the above empirical papers resides in the
rather ad hoc specification of the underlying technology. The authors basically regress
productivity[1] on labour, capital[2] and descriptive indicators of labour diversity
(i.e. standard deviation, dissimilarity or Herfindhal/Simpson indices). The reduced-form
equations that are estimated do not explicitly derive from the standard textbook
production functions (Cobb-Douglas, CES, etc.). What is more, they do no connect with
another stream of the economic literature assessing the benefits/losses of diversity.
That literature is more structural. It has developed concepts like super(sub)modularity
of production (Milgrom and Roberts, 1990; Iranzo et al., 2008)[3], the O-ring theory
(Kremer, 1993), that of assortative matching (Becker, 1981; Durlauf and Seshadriand,
2003), or has examined the relationship between local stratification and growth
(Bénabou, 1994, 1996a, b). Also, it takes a more macro stance. Diversity/homogeneity is
discussed in terms of its impact on aggregate output (i.e. that of the different
neighbourhoods/regions forming a city/country, etc.), and results carry very specific
implications in terms of how diverse/heterogeneous individuals[4] should be allocated
across entities[5]. This said, we would argue here that both literatures ultimately
address the same key question, which is to determining –using Grossman and Maggi
(2000) terminology –whether cross-matching (all entities comprise a diversified set of
individuals) is preferable than self-matching (each type of individuals is concentrated in
one distinct entity).
In this paper, we suggest exploring the diversity/efficiency nexus, in the context of
private-economy firms, using a branch of that second literature; more specifically, the
framework of authors who have studied stratification/diversity and growth in the
context of cities (Bénabou, 1994) and/or educational systems (Vandenberghe, 1999).
Referring to the discussion above, that literature presents the advantage that it has
developed a structural and encompassing view on efficiency, and it deals explicitly
with the issue of optimal allocation of diverse individuals. What it essentially shows is
that cross-matching (i.e. diversity) is effective when the “local”technology (i.e. the one
characterising neighbourhoods, schools or firms) displays concavity; in other words,
decreasing marginal contribution to total output of rising shares of individuals of the
most productive type (e.g. highly educated).
The second methodological contribution of this paper is to show that a slightly
“augmented”version of the Hellestein-Neumark (HN) framework (Hellerstein and
Neumark, 1995) can be used to assess the degree concavity/convexity of the technology
in the share of a particular type of worker. The key idea of HN is to estimate a
production function where an heterogeneous/diverse labour input appears as a sum of
shares; and where different worker types (e.g. educated/uneducated; men/women,
young/old, etc.) potentially differ in terms of marginal product. Most authors have used
the HN framework to measure productivity/skills difference across different types of
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workers; with the aim of comparing them to wage differences (and assess the degree of
alignment of wage and productivity/skills). Our objective here is rather to show that an
HN framework, that allows for imperfect substitutability across labour types, is
suitable to address the question of concavity/convexity in the share of types of
workers, and thus that of the relationship between diversity and efficiency.
The rest of the paper is organised as follows. Section 2 exposes our analytical
framework in details. Section 3 presents our data as well as the econometric strategy. It
contains the results of its application to the analysis of Belgian firm-level data where
workers differ in terms of educational attainment, age and gender. Section 4 concludes.
2. Framework
2.1 Concavity/convexity and overall efficiency
Imagine an economy that consists of i¼1, …,Nfirms, each of them potentially
employing two (unequally productive) types of workers. The economy-wide output
(W) is the sum of output of the Nfirms. The proportion of (high/low) productive
workers in firm iis x
i
; while the corresponding proportion of the same type of workers
in firm Nis x
N
:
W¼Yx
1
ðÞþYx
2
ð Þþ þ Yx
N
ðÞ (1)
Starting from a situation synonymous with cross-matching x
1
¼x
2
¼ ¼ x
N
¼θ–
where θis the share of the workers of the type considered in the whole population –
consider the effect of raising their share in firm 1, at the expense of, say, firm N:
dW=dx1¼dY:ðÞ=dx1þdY:ðÞ=dxNdxN=dx1
(2)
By assumption the rise of the type’s share in firm 1 translates (leaving aside the
question of size differences across firms) into a reduction of their share in firm N.
Logically thus [δx
N
/δx
1
]¼−1:
dW=dx1¼dY:ðÞ=dx1dY:ðÞ=dxN(3)
In x
1
¼x
N
¼θthe two derivatives are equal, and expression (3) is equal to 0, meaning
that that point corresponds to an extremum. Whether it defines a maximum or a
minimum depends on the second-order condition:
d2W=dx1dx1¼d2Y:ðÞ=dx1dx1d2Y:ðÞ=dxNdxN:dxN=dx1
(4)
Or equivalently as, again, [δx
N
/δx
1
]¼1:
d2W=dx1dx1¼d2Y:ðÞ=dxidx1þd2Y:ðÞ=dxNdxN(5)
Thus if δ
2
Y(.)/δx
i
δx
i
W0 (i.e. the firm-level technology is convex in the share of the
high-productive type) optimality requires adopting corner solutions (i.e. self-matching/
minimal diversity). By contrast, if δ
2
Y(.)/δx
i
δx
i
o0 (the firm-level technology is
concave), the optimum is interior and symmetric (x
1
¼x
N
¼θ). Maximising output
requires cross-matching/maximal diversity.
Figure 1 illustrates the idea of concavity in x(i.e. the share of high(low) productive
workers) being good for efficiency. Of course, Figure 1(a) shows that a higher share of the
high-productive type (say in firm a) translates into a higher firm-level output. But, if we
assume that such a move translates into a reduction of the equivalent share elsewhere in
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the economy (say in firm b), the question of the net impact amounts to verifying that
output in cis higher than the aand baverage. The point to bear in mind is that intra-firm
diversity is higher if the economy consists of firms in crather than aor b.
Finally for this section, we would like to talk about the apparent contrast between
the framework of this paper and the one underpinning most existing works by
empirical economists on diversity. This paper focuses on workforce diversity and its
impact on aggregate efficiency, while the latter works generally care about firm-level
efficiency. Our view is that there is fundamentally no opposition between what matters
for a representative firm (and its managers) and what holds for the whole economy.
Assume for a moment that we exclusively consider the point of view of the firm and
its managers. They decide to increase the proportion of presumably more productive
workers[6] (xgoes up in Figure 2). That move (say from ato c) has two consequences.
First, it mechanically (i.e. linearly) increases the average of the individual productivities
characterizing the workers. The second consequence is that the firm becomes more
diverse. In order to determine whether diversity matters, managers need to determine
whether output Yis affected beyond what mechanically derives from the change of the
average of individual productivities. In Figure 2, that mechanical/linear effect
corresponds to segment (C1). And what comes on top to the segment (C2) to the
contribution of diversity[7]. That decomposition can be done using a traditional HN log-
linear model –where the labour shares appear as a simple sum –to which one adds an
Y
a
a
b
b
c
c
Y
xx
(a) (b)
Notes: (a) High productive type; (b) low productive type
Figure 1.
Concavity of
production
technology in a
given worker type
and overall efficiency
Y
X
c
a
b
(C2 )
(C1)
Figure 2.
Concavity and the
point of view of the
firm’s managers
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Herfindahl index[8]. The HN (productivity weighted) sum of labour shares will capture
the mechanical/linear output consequence of a higher x(in other words (C1)), and the
coefficient of the Herfindahl index will reflect (C2). What we propose in this paper is to
detect (C2) simply by allowing for nonlinearity in the labour-quality index, in other
words by replacing the traditional HN linear expression by a CES index [x
ρ
+λ(1x)
ρ
]
1/ρ
where ρ≠1 informs the managers (or the social planner) whether diversity matters for
efficiency (more on this in the next subsection).
2.2 Concavity and the HN framework
The next step is to specify a realistic (and econometrically tractable) firm-level
production function that is function of x
i
. The one we retain here owes a lot to
Hellerstein and Neumark (1995), but also to the literature on productivity and skill
diversity (Duffy et al., 2004; Iranzo et al., 2008), or the one studying the relationship
between age and productivity (Vandenberghe et al., 2013)[9]. In these works, the
production function of a representative firm (from now on, for simplicity of exposure,
we drop index i) writes as a Cobb-Douglas:
Y¼AKaQLb(6)
where Yis output (or productivity), Kis the stock of capital. The key variable is
what is called the quality of labour aggregate QL. Total labour is L. But what matters is
its decomposition into different types. Without loss of generality, we consider a
situation with two types (h,l) where L
h
is the number of (presumably) high-productive
workers in the firm. Parameters µ
h
, represents the types’contribution to output
(or actual skills)[10]:
QL ¼mhLh
r
þmlLLl
r
hi
1=r
(7)
We suggest specifying the quality aggregate as a CES index, where labour types are
not perfectly substituable and contribute to output non-linearly. The latter assumption
is essential for assessing concavity/convexity of the technology in a worker’s type, and
answering the question of the desirability of diversity in terms of overall efficiency.
By contrast, HN assume perfect substitutability ( ρ¼1) meaning the CES collapses to a
simple sum, and also, (as will become clearer after) that diversity does not matter for the
economy’s efficiency, as the firm-level technology is neither concave nor convex in a
worker’s type.
Expression (7) can be easily be rewritten in terms of labour shares, with x≡L
h
/Lthe
proportion of workers with contribution µ
h
. By definition, in a two-type setting, (1−x)is
the share of the other type of workers present in the firm:
QL ¼Lmhxrþml1xðÞ
r
1=r(8)
or equivalently, picking type hworkers as reference category:
QL ¼Lmh1=rxrþl1xðÞ
r
½
1=r(9)
with λ≡µ
l
/µ
h
reflecting the relative contribution of type lworkers to output.
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The point is that expression (9) now appears as function of the share of type
hworkers (x). This means that the key question raised in this paper (i.e. is diversity
good/bad for efficiency), amounts to determining whether QL is concave or convex in x.
Back to the full production function, one needs to inject (9) into (6):
Y¼
~
AKaLbxrþl1xðÞ
r
½
b=r(10)
where
~
A¼Amhb=r
Noting f(x)≡[x
ρ
+λ(1x)
ρ
]
1/ρ
the part of the labour quality aggregate that consists of
a CES index, the firm-level output’s second-order derivative with respect to xis:
@2YxðÞ
@x@x¼Amhb=rKaLbbfxðÞ
b1b1ðÞ
@fxðÞ
@x
2
=fxðÞ
"!
þ@2fxðÞ
@x@x#(11)
where:
@fxðÞ
@x¼xrþl1xðÞ
r
½
1=r1xr1l1xðÞ
r1
¼xrþl1xðÞ
r
½
1=r1xr11l1x
x
r1
!
(12)
and, most importantly, the second order derivative of the CES index is:
@2fxðÞ
@x@x¼r1ðÞxrþl1xðÞ
r
½
1=r2l1xðÞ
r2xr2(13)
The sign of (11) is entirely determined by those of parameters βand ρ. The first
parameter is nothing but the output elasticity with respect to total labour of the
Cobb-Douglas part of the production function. And, presumably, in the presence of
capital, it is inferior to 1. This means a diminishing marginal productivity for
total labour (L)[11]. And, by extension, that law also applies to any quality-adjusted
labour aggregate à-la HN. Assuming that labour types’marginal productivity differ
significantly (i.e. in expression (12) l1x=x
r1a1 or just that λ≠1 in case of perfect
substitutability ( ρ¼1)), then changes in the value of xamounts to changing the overall
level of (quality-adjusted) labour. That logically translates into a fall of marginal
productivity, captured in expression (11) by the term premultiplied by β−1.
The more interesting question is what happens with parameter ρconditional on a
certain value of β; or, said differently, to determine whether the law of diminishing
marginal productivity is positively (or negatively) affected by the diversity of the
labour force. And that amounts to determining if ρ≠1. If ρo1 we would conclude that
diversity is good for efficiency ceteris paribus.IfρW1 then diversity is a bad thing for
efficiency. And if ρ¼1 diversity becomes irrelevant.
3. Econometric analysis
3.1 Data
The empirical results of this paper derive from the analysis of two panels. The first one
contains around 8,000+firms with more than 20 employees. These firms are largely
representative of the Belgian private economy in terms of sector/industry, and are well
documented as to the capital they used and, their productivity performance[12]. Using
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firm identifiers, we have been able to add social security information[13] on the age and
gender of (all) workers employed by these firms, for a period running from 1998 to 2006.
Table I presents the descriptive statistics. Of particular importance are the ones
describing age and gender. Note in that age and gender diversity (as captured by the
Herfindahl index) seems to have risen between 1998 and 2006.
The second panel contains information about the educational attainment of the
workforce. It comprises a slightly smaller number of firms (4,000+); also from all
sectors forming the Belgian private economy. It runs from 2008 to 2012. Firms are also
well documented in terms of sector, overall size of the labour force, capital used, and
productivity (value added). But there is no information on the age and gender of the
workforce that would allow a more refined breakdown of educational categories.
Descriptive statistics, are reported in Table II. Of prime interest in this paper is the
breakdown by educational attainment. Table II shows that, during the observed period
(2008-2012), more than 75 per cent of the workforce of private for-profit firms located
in Belgium have, at most, an upper secondary school degree. This means less than
25 per cent of workers are in possessing of a tertiary education background; clearly
less than the percentage among the current generation of school leavers[14].
This discrepancy logically reflects the lower propensity of older generations to stay on
beyond secondary education, and complete a tertiary degree. But given the focus of this
Value added
(log)
No. of empl.
(log)
Capital
(log)
Secondary or
less
More than
secon.
Herf.
educ.
2008 9.515 5.313 10.248 0.784 0.215 0.181
2009 9.262 5.080 10.035 0.761 0.239 0.200
2010 9.340 5.133 10.095 0.762 0.238 0.199
2011 9.373 5.170 10.111 0.755 0.245 0.199
2012 9.391 5.171 10.119 0.751 0.249 0.203
n227,838
Notes: Main variables (weighted(£))/2008-2012. £: weights are equal to the firm’s number of workers
Source: Belfirst-Carrefour
Table II.
Descriptive
statistics: education
Value added (log) k
EUR
No. of empl.
(log)
Capital
(log)
Share 50
+
Herf
age
Share
female
Herf
gender
1998 10.072 6.146 8.111 0.132 0.213 0.249 0.277
1999 10.095 6.088 8.146 0.136 0.217 0.256 0.284
2000 10.140 6.056 8.198 0.139 0.223 0.262 0.289
2001 10.122 6.148 8.130 0.143 0.226 0.271 0.298
2002 10.353 6.356 8.428 0.154 0.240 0.280 0.306
2003 10.356 6.268 8.503 0.167 0.256 0.281 0.306
2004 10.424 6.270 8.522 0.174 0.264 0.284 0.308
2005 10.435 6.280 8.486 0.179 0.269 0.289 0.310
2006 10.510 6.263 8.665 0.188 0.278 0.294 0.314
n75,393
Notes: Main variables (weighted (£))/1998-2006. £: weights are equal to the firm’s number of workers
Source: Belfirst-Carrefour
Table I.
Descriptive statistics:
age, gender
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paper, perhaps the most important point worth observing is that educational diversity
inside firms located in Belgium (as reflected by the Herfindahl index reported in the last
column) has seemingly increased between 2008 and 2012.
3.2 Identification strategy
The econometric version of our model, that we apply to a panel of firms writes:
Yit ¼Ai0KitaQLit
ðÞ
bet:tþoit (14)
And its log equivalent is:
ln Yit ¼Bi0þaln Kit
ðÞþbln Lit
ðÞþbln f xit
ðÞþt:tþoit (15)
with B
i0
≡ln(A
i0
)+β/ρln(µ
h
), f(x
it
)¼(x
it
ρ
+λ(1−x
it
)
ρ
)
1/ρ
the CES index as a function of labour
shares (two-types case), and τthe constant rate of TFP growth, common to all firms.
We assume a three-component error term:
oit ¼yiþgit þdit (16)
meaning that the linear (or non-linear) least squares sample-error term potentially
consists of: first, an unobservable firm fixed effect ϴ
i
; second, a short-term shock γ
it
(whose evolution may correspond to a first-order Markov chain, causing a simultaneity
bias) observed by the firm (but not by the econometrician) and (partially) anticipated by
managers: and third, a purely random shock δ
it
.
The panel structure of our data allows for the estimation of models that eliminate
the fixed effects (ϴ
i
). For instance, resorting to the growth-equivalent of (15) (i.e. lag T
differences of logs, or log of ratio of Y
it
to its lagged Tvalues) leads to:
ln Yit =YitT
¼tTþaln Kit=KitT
þbln Lit=LitT
þb=rln f xit
ðÞ=fx
itT
ðÞ
þoitoit T(17)
where ω
it
−ω
it−T
¼γ
it
−γ
iT
+δ
it
−δ
it−T.
This said, another challenge is to go around the simultaneity bias caused by
short-term shock γ
it
. Equation (17) suggests estimating a model where the dependent
variable is the (estimated) TFP, following a two-step strategy[15]. The first step
consists of estimating the log of TFP as the residual of the regression of output on
capital and total labour:
ln d
TFPit
TFPitT
¼ln Yit =YitT
^
aln Kit=Kit T
^
bln Lit=LitT
(18)
It is when estimating that first-step equation that we control for the presence of γ
it
,
using the strategy developed by Levinsohn and Petrin (2003) (LP henceforth) and, more
recently by Ackerberg et al. (2006) (ACF henceforth). Both LP and ACF estimations
involve assumptions about the time of the choice of inputs. Capital is assumed quasi-
fixed (in the short- to medium run), whereas labour is more flexible and partially chosen
after the (unilaterally) observed productivity chock γ
it
. This makes least square
estimates for labour inputs endogenous. To go around this problem, LP assumes
that γ
it
can be proxied by a third order polynomial in the use of intermediate inputs
(i.e. purchases of raw materials, services, electricity, etc.) and also in capital[16].
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The sole presence of this proxy/polynomial at step 1 makes it possible to consistently
estimate alpha (greek letter) and beta (greek letter) using OLS or non-linear least
squares (NLLSQ). By extension, the residuals of that first-step LP-ACF equation are
also clear of γ
it
and can be used at step 2 to consistently estimate λand ρ(i.e. the
parameters of the CES index f(x
it
)) using NLLSQ:
ln d
TFPit=TFPitT
¼tTþb=rln fit xit
ðÞ=fitTxitT
ðÞ
þditdit T(19)
3.3 Econometric results
We report the key results of our analysis in Table III (age), Table IV (gender) and
Table V (education). In each of them we report the results for the level (1) and the
growth specification (2), (3), (4). The advantage of the growth specification is that it
accounts for firm-level fixed effects, known for being very important across firms
(Syverson, 2011). Among our growth specifications, we distinguish one-step (2) and
two-step models (3) (4). The first step of the latter implements the LP (3) or the ACF (4)
strategy to control for endogeneity/simultaneity and delivers unbiased estimates of (total
factor) productivity which can then (in the second step) be regressed on labour shares.
Alongside each of these four specifications, we also report the results obtained with the
traditional model used by empirical economists that consists of regressing output on total
capital and labour, the HN sum of shares for the different types of labour (bar the reference
one)[17] plus the firm-level Herfindahl index capturing workforce diversity.
A first result is that we find evidence of (marginal) productivity differences across
all the estimated models (i.e. l1x=x
r1a1). Younger workers appear more
productive than older workers, educated workers more than less educated ones, and in
all cases except one (Table V, model (4)), men seem more productive than women.
Second, as to the degree of concavity/convexity of the production function, our main
result is that of an absence of strong evidence that age, gender or educational diversity
is good or bad for efficiency. In Table III (age) and Table IV (gender), the probability
that ρo1 (i.e. concavity/diversity being good for efficiency) seems reasonably high
when estimating models (1) (2), but no longer when turning to the models that account
for endogeneity/simultaneity (3) (4); in particular ACF where for both age and gender
ρ’s appear very close to 1. This is also what we find for education, but this time for all
the econometric models estimated.
Third, our results match up with those delivered by using the traditional HN
+Herfindahl index approach. In Table III (age) and in Table IV (gender), the coefficient
of the Herfindhal index in models (1) (2) –akin parameter ρ–hints at diversity-related
efficiency gains. But these gains invariably vanish in the models (3) (4) that account for
endogeneity/simultaneity. As to education (Table V), we also conclude that the
coefficient of the Herfindahl index, like parameter ρ, and whatever the model used (1) (2)
(3) (4), points at an absence of any significant impact of diversity on efficiency.
4. Final comments
The key message of the paper is that looking at the degree of concavity/convexity of the
production function is useful to assess the efficiency costs/benefits of labour diversity.
The inspiration comes from the economic literature on (social) heterogeneity, stratification
and growth (Bénabou, 1994, 1996a, b; Vandenberghe, 1999). By focusing on concavity,
this paper departs from the approach used by most recent empirical economics papers
that consists of regressing output on descriptive indices of workforce diversity. We think
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Level Growth specification (FE)
(1) CES (1′) HN-Herf (2) CES (2′) HN-Herf (3) CES LP (3′) HN-Herf LP (4) CES ACF (4′) HN-Herf ACF
α(K) 0.0856 (0.00117)*** 0.0853 (0.004)*** 0.0237 (0.00271)*** 0.0237 (0.003)***
β(L) 0.923 (0.00227)*** 0.921 (0.007)*** 0.651 (0.00477)*** 0.651 (0.005)***
ρ0.594 (0.0405)*** 0.855 (0.0804)*** 0.960 (0.0947)*** 1.095 (0.232)***
λ1.450 (0.0844)*** 1.478 (0.102)*** 1.225 (0.0866)*** 1.233 (0.168)***
η(1−x) 0.627 (0.140)*** 0.333 (0.054)*** 0.194 (0.0543)*** 0.270 (0.0950)**
δ(Herf) 0.567 (0.113)*** 0.149 (0.044)*** 0.0612 (0.0446) 0.0818 (0.0821)
Controls Share part-time work, share blue-collar workers
No. of obs. 73,738 73,738 63,792 63,792 63,792 63,792 21,671 21,671
λ1.450 1.478 1.225 1.233
Pr(λ¼1) 0.000 0.000 0.009 0.166
RMP
a
1.176 1.680 1.368 1.512 1.198 1.248 1.297 1.353
Pr(RMP ¼1) 0.000 0.000 0.000 0.000
ρ0.594*** 0.855 0.960 1.095
Pr(ρ¼1) 0.000 0.072 0.670 0.683
Notes:
a
Implied relative marginal productivity of younger workers (ref: workers aged 50+)¼l1x=x
r1or η/β+1 in the case of the HN-Herfindhal model. Standard errors in
parentheses. *po0.05, **po0.01, ***po0.001
Source: Bel-first; Carrefour
Table III.
Age diversity
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Level Growth specification (FE)
(1) CES (1′) HN-Herf (2) CES (2′) HN-Herf (3) CES LP (3′) NH-Herf LP (4) CES ACF (4′) HN-Herf ACF
α(K) 0.0825 (0.00116)*** 0.0806 (0.00403)*** 0.0235 (0.00271)*** 0.0235 (0.00271)***
β(L) 0.924 (0.00225)*** 0.922 (0.00738)*** 0.652 (0.00478)*** 0.652 (0.00478)***
ρ0.644 (0.0104)*** 0.840 (0.0401)*** 0.966 (0.0487)*** 1.050 (0.168)***
λ1.529 (0.0153)*** 1.245 (0.0426)*** 1.147 (0.0372)*** 0.959 (0.0974)***
η(1−x) 0.462 (0.0292)*** 0.151 (0.0257)*** 0.112 (0.0258)*** −0.00218 (0.0599)
δ(Herf) 0.526 (0.0385)*** 0.0982 (0.0272)*** 0.0363 (0.0273) −0.0300 (0.0617)
Controls Share part-time work, share blue-collar workers
Nobs 73,736 73,736 63,788 63,788 63,788 63,788 21,671 21,671
λ1.529 1.245 1.147 0.959
Pr(λ¼1) 0.000 0.000 0.000 0.675
RMP
a
1.758 1.501 1.304 1.232 1.156 1.144 0.951 0.997
Pr(RMP ¼1) 0.000 0.000 0.000 0.000
ρ0.644*** 0.840*** 0.966 1.050
Pr(ρ¼1) 0.000 0.000 0.483 0.765
Notes:
a
Implied relative marginal productivity of male workers (ref: female workers) ¼l1x=x
r1or η/β+1 in the case of the HN-Herfindhal model Standard errors in
parentheses. *po0.05, **po0.01, ***po0.001
Source: Bel-first; Carrefour
Table IV.
Gender diversity
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Level Growth specification (FE)
(1) CES (1′) HN-Herf (2) CES (2′) Herf (3) CES LP (3′) Herf LP (4) CES ACF (4′) Herf ACF
α(K) 0.310 (0.000832)*** 0.310 (0.00279)*** 0.265 (0.00292)*** 0.265 (0.00292)***
β(L) 0.712 (0.00120)*** 0.711 (0.00310)*** 0.559 (0.00316)*** 0.559 (0.00316)***
ρ0.996 (0.00862)*** 0.968 (0.0153)*** 0.999 (0.0151)*** 0.942 (0.0485)***
λ1.386 (0.00933)*** 1.009 (0.00855)*** 1.012 (0.00800)*** 1.051 (0.0277)***
η(1−x) 0.224 (0.00802)*** 0.00491 (0.00496) 0.00732 (0.00500) 0.0466* (0.0220)
δ(Herf) 0.0571 (0.0120)*** 0.0143 (0.00817) 0.00260 (0.00822) 0.0286 (0.0374)
Controls Share females, share blue-collar workers
Nobs 227,564 227,564 172,816 172,816 172,816 172,816 7,536 7,536
λ1.386 1.009 1.012 1.051
Pr(λ¼1) 0.000 0.288 0.130 0.0658
RMP
a
1.395 1.315 1.057 1.009 1.014 1.012 1.145 1.060
Pr(RMP ¼1) 0.000 0.000 0.000 0.000
ρ0.996 0.968*0.999 0.942
Pr(ρ¼1) 0.616 0.036 0.943 0.236
Notes:
a
Implied relative marginal productivity of highly educated workers (ref: workers with an upper secondary degree or less) ¼l1x=x
r1or η/β+1 in the case of the
HN-Herfindhal model. Standard errors in parentheses. *po0.05, **po0.01, ***po0.001
Source: Bel-first; Carrefour
Table V.
Educational diversity
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that our approach is more structural. It explicitly addresses the question underpinning
most of the empirical works done by economists about workforce diversity and
efficiency; namely, whether cross-matching (all entities have a diversified set of
individuals) is more/less effective than self-matching (one type prevails in each entity).
And although it takes a more macro stance, the key issue remains the one that matters for
firm-level efficiency.
We show mathematically that if the technology used by individual firms is concave
in the share of a given worker’s type (e.g. old, female or educated), cross-matching/
diversity of the types is synonymous with efficiency. We then show that a generalised
version of HN labour-quality index –that has been extensively used by empirical
economists to analyse productivity-related issues –is suitable to assess the degree of
concavity of the technology. What HN have shown is that labour heterogeneity/
diversity can be represented, within a Cobb-Douglas function, as a sum of labour
shares. To all those interested in analysing the diversity-efficiency nexus, we simply
propose to aggregate these shares non-linearly as a CES index.
In the second part of the paper, we implement our innovate framework using two
panels of firms located in Belgium for which we have information on age/gender
(Panel 1) and educational attainment (Panel 2). We apply various treatments that are
aimed at controlling for the two main (potential) sources of bias: firm unobserved
heterogeneity and simultaneity. We address the first problem by resorting to a growth/
fixed-effect specification of our HN-with-CES-index production function. And we cope
with simultaneity by implementing both the LP and the Ackerberg et al. (2006) idea of
using observed intermediate input decisions (i.e. purchases of raw materials, services,
electricity, etc.) to control for/proxy unobserved short-term productivity shocks
causing simultaneity.
The main results of the paper is an absence of strong and systematic evidence that
age or gender or educational diversity is good/bad for efficiency.
Notes
1. Usually the log of value added per worker.
2. And not all of them have information on capital stock.
3. The latter narrowly corresponds to what is commonly considered as the cost/benefit of skill
diversity (Grossman and Maggi, 2000). Super(sub) modularity carries very specific
implications for the optimal organisation of production. If a technology is supermodular,
efficiency requires self-matching. An example is the O-ring technology imagined by Kremer
(1993), where output critically depends on each individual’s correct execution of his/her
task. In that case, workers should be sorted so that those with similar skills work together.
In contrast, when a technology is submodular, cross-matching (diversity) is indicated.
4. Mainly in terms of their skills.
5. The results of the more empirical and firm-centric literature implicitly carry similar
implications about optimal allocation. If for instance a representative firm is less effective
when age heterogeneity (as captured by the standard deviation of age) rises, the inevitable
implication is that maximising overall productivity requires age self-matching.
6. The reasoning is the same with a move synonymous with a rising share of the less
productive type.
7. We could have considered a case where [C2] is negative. The important thing is to detect
any deviation from the dashed line.
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8. When the information about firms’workforce takes the form of a vector of j¼1, …,n
shares, economists generally resort to the Herfindahl/Simpson index to quantify diversity at
firm level and assess its impact on output. The index writes Hi¼1Pn
jxij
2, and in the
case of n¼2 types Hi¼1xi21xi
ðÞ
2.
9. Which is relatively more developed than the literature on workforce diversity, and better
connected to the standard economic theory of production.
10. Note that here, contrary to Iranzo et al. (2008), workers’skills are not available or measured
ex ante. But the relative productivity (i.e. skills) by type can be estimated econometrically.
11. With the standard HN-Cobb-Douglas Y¼AK
α
QL
β
, one has that: @2YxðÞ
@L@L¼
AKabb1ðÞLb2o0ifbo1.
12. These observations come from the Bel-first database. Most for-profit firms located in
Belgium must feed that database to comply with legal prescriptions.
13. Compiled in the so-called Carrefour database.
14. Statistics Belgium estimates that they now represent between 35 and 40 per cent of a cohort.
15. Not to be confounded with the two-stage estimation characterizing the method of
Levinsohn and Petrin (2003) and Ackerberg et al. (2006), to estimate the parameters of a
production function.
16. The actual assumption made by LP is that the use of intermediate inputs is a monotonic
function of γ
it
and k
it
that can be inverted. And the inverse function can be approximated by
a third-order polynomial in intermediates and capital.
17. Meaning here, given our two-type setting, that the equation only contains one share.
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Corresponding author
Vincent Vandenberghe can be contacted at: vincent.vandenberghe@uclouvain.be
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