Human capital aggregation and relative wages across countries

Journal of Macroeconomics 30 (2008) 1587–1601

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Journal of Macroeconomics

journal homepage: www.elsevier .com/locate/ jmacro

Human capital aggregation and relative wagesacross countries

Manish Pandey *

Department of Economics, The University of Winnipeg, 515 Portage Avenue, Winnipeg, Manitoba, Canada R3B 2E9

a r t i c l e i n f o

Article history:Received 10 May 2007Accepted 9 June 2008Available online 17 June 2008

JEL classification:O40O47

Keywords:Growth accountingAggregationHuman capitalRelative wages

0164-0704/$ - see front matter � 2008 Elsevier Indoi:10.1016/j.jmacro.2008.06.001

* Tel.: +1 204 786 9289; fax: +1 204 774 4134.E-mail address: [email protected]

a b s t r a c t

Most of the growth accounting literature relies on an aggregateproduction function to determine the contribution of factors ofproduction relative to that of total factor productivity (TFP) inexplaining differences in incomes across countries. I show thatthe importance of TFP in accounting for cross-country income dif-ferences depends crucially on how skilled and unskilled labor areaggregated. Further, cross-country evidence on the relationshipbetween relative wages and relative endowments of skilled andunskilled labor suggests that the two types of labor should notbe aggregated into a single factor of production. Growth account-ing decomposition using a commonly used nested-CES aggregateproduction function that allows skilled and unskilled labor to beused as separate factors of production results in a significantlygreater role for TFP in accounting for income differences acrosscountries than that found by past studies. The finding that differentaggregate production functions lead to significantly different con-clusions about the role of TFP in accounting for cross-countryincome differences calls for a more general approach to under-standing such differences.

� 2008 Elsevier Inc. All rights reserved.

1. Introduction

Most of the growth accounting literature relies on the use of an aggregate production function (APF)in order to evaluate the contributions of human capital, physical capital and total factor productivity

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1588 M. Pandey / Journal of Macroeconomics 30 (2008) 1587–1601

(TFP) to differences in incomes across countries.1 In this paper, I show that the relative importance ofTFP in accounting for cross-country income differences depends crucially depends on how skilled andunskilled labor are aggregated.

Previous studies, such as Klenow and Rodríguez–Clare (1997) and Hall and Jones (1999), aggregateskilled and unskilled labor into a single factor of production, human capital, when evaluating theimportance of TFP. To begin with, I show that the importance of TFP depends on the value of the sub-stitution parameter used for the aggregation by these studies.2 To do so, using data for 51 countries, Iconsider a growth accounting exercise similar to that of Hall and Jones (1999) under different values forthe substitution parameter and, for each country, compute TFP relative to that of the US.3 Relative TFP forall countries increases when the substitution parameter is decreased from 1 (skilled and unskilled laborare perfect substitutes) to 0.75 (imperfect substitutes), indicating that the degree of substitutability be-tween skilled and unskilled labor plays a crucial role for determining the importance of TFP in accountingfor cross-country income differences.4

Given the importance of the substitution parameter for the aggregate production function used inthe above exercise, it becomes imperative to determine the value of the parameter that should be usedfor the aggregation. Using cross-country data, I plot the logarithm of relative wages against the loga-rithm of the ratio of the two types of labor and find that the slope, which provides an estimate of thesubstitution parameter for this particular aggregation, varies systematically with GDP per worker: lowincome countries have a steeper slope than high income countries.5 In other words, the relation be-tween relative wages and relative endowments of skilled and unskilled labor varies systematically acrosscountries.6 This evidence suggests that aggregation of the two types of labor into a single factor of pro-duction using a common substitution parameter will not be able to match this systematic cross-countrydifference in the relationship between relative wages and relative endowments of skilled and unskilledlabor.

A natural question to ask, then, is whether using aggregate production functions that allow forskilled and unskilled labor to be used as separate inputs for growth accounting would lead to conclu-sions that differ from those obtained by Hall and Jones (1999). The two-level or nested-CES class ofaggregate production functions provide a generalization of the aggregation used by Hall and Jones.7

Based on estimates obtained using data for the US (Krusell et al., 2000) and cross-country panel data(Duffy et al., 2004), the literature on capital-skill complementarity has argued in favor of one nestingof this class of production functions. I employ this preferred APF for the growth accounting exerciseand investigate whether doing so results in significant changes in the importance of TFP in explainingcross-country income differences.

When aggregating the two types of labor into a single input, for the Hall and Jones growth account-ing exercise, the labor share of income is assumed to be constant across countries while country-spe-cific relative wages of skilled and unskilled labor are used to compute the relative efficiency of the twotypes of labor. To ensure comparability of results with the previous exercise, for the preferred variantof the nested-CES APF, I use estimates of the substitution parameters from Duffy et al. (2004) and

1 The debate whether factors of production or TFP contribute more to explaining differences in incomes across countries is notonly important for designing policies to reduce these large differences but also for examining whether exogenous (augmentedSolow model used by Mankiw et al., 1992) or endogenous growth models should be used for understanding such differences. For adetailed discussion see Klenow and Rodríguez–Clare (1997).

2 Given the lack of consensus for how skilled labor should be defined, all the analysis in the paper is based on two differentdefinitions of skilled labor. Definition 1 defines skilled labor as those who have completed primary level of schooling whiledefinition 2 defines skilled as those who have completed secondary level of schooling.

3 When aggregating skilled and unskilled labor to construct human capital, both Klenow and Rodríguez–Clare (1997) and Halland Jones (1999) assume perfect substitutability between the two types of labor.

4 A low value for the parameter (a low elasticity of substitution) implies large differences in human capital stocks acrosscountries, and this lowers the differences in TFP required to account for cross-country income differences.

5 This systematic cross-country difference holds for both the definitions of skilled labor that I consider.6 A similar systematic difference in the relation between relative wages and relative endowments between low income and high

income countries has been documented by Acemoglu (2003). Investigating the role of capital-skill complementarity, Chmelarovaand Papageorgiou (2005) and Henderson (2005) find similar cross-country systematic differences in the substitutability betweencapital and the two types of labor.

7 Nested-CES production functions were first introduced by Sato (1967) and have recently been used to evaluate the role ofcapital-skilled complementarity (see, for example, Duffy et al., 2004, Krusell et al., 2000 among others).

M. Pandey / Journal of Macroeconomics 30 (2008) 1587–1601 1589

calibrate the two share parameters of the production function to match the country-specific relativewages of skilled and unskilled labor and the common labor share of income.8 These parameter valuesare then used to compute the TFP for each country relative to that for the US, and the results obtained arecompared with that for the aggregation used in the previous exercise.

Relative to the Hall and Jones aggregation, using the preferred APF for growth accounting increasesthe importance of TFP in explaining cross-country income differences. The contribution of TFP inexplaining differences in incomes increases from 54% (Hall and Jones aggregation) to 111% (preferredaggregation).9 To put it differently, a growth accounting exercise using the preferred aggregation con-cludes that differences in factors of production cannot explain any of the cross-country differences inGDP per worker. These results suggest that the choice of the aggregation plays a crucial role in determin-ing the relative importance of TFP in explaining cross-country income differences.

The rest of this paper is organized as follows: Section 2 provides a description of the cross-countrydata and illustrates the importance of the substitution parameter for the conclusions from a growthaccounting exercise when skilled and unskilled are aggregated into a single factor of production. Sec-tion 3 explores the relationship between relative wages and relative endowments of skilled and un-skilled labor to determine the substitution parameter for the aggregate production function usedfor growth accounting in the previous section. In Section 4, the implications for the relative impor-tance of TFP in explaining cross-country income differences using a commonly used nested-CES APFare presented and discussed. Section 5 concludes.

2. Aggregation matters for growth accounting

Although it has been recognized that it is important to distinguish between skilled and unskilledlabor, most studies assume that these inputs are perfect substitutes in production and aggregatethe two types of labor into a single factor of production. The example below shows that this assump-tion is far from innocuous when quantifying the relative importance of total factor productivity inaccounting for cross-country income differences.

Following Hall and Jones (1999), let the aggregate production function (APF) for country i in perworker terms be given by

8 Usisimilar

9 Thedetails.

10 It isfunction

11 Bilsunskille

yi ¼ Aikai h1�a

i ; ð1Þ

where yi is GDP per worker, ki is the physical capital stock per worker, hi is the human capital stock perworker, a is the capital share parameter and Ai is total factor productivity (TFP). The stock of humancapital is constructed by aggregating skilled ðsÞ and unskilled ðuÞ labor using the function:hi ¼ ðhq

u;i þ hs;ihqs;iÞ

1=q, where hj;i is the per worker aggregate human capital of skill type j in country i,q is the substitution parameter and hs;i is the country-specific efficiency of skilled labor relative to un-skilled labor.10 The elasticity of substitution between skilled and unskilled labor is r ¼ 1

1�q. With q ¼ 1,perfect substitutability between skilled and unskilled labor, we have the Hall and Jones specification.11

The TFP for country i relative to that for the US can be expressed as

bAi ¼yi=yUS

ki=yikUS=yUS

� � a1�a

hi=hUSð Þ

0B@

1CA

1�a

:

When factors are paid their marginal products, the relative wage of skilled and unskilled labor impliedby the production function for country i is

ng other estimates of the substitution parameters, for example those obtained by Krusell et al. (2000), lead to qualitativelyresults with regards to the importance of TFP in accounting for cross-country income differences.contribution of TFP is calculated using the decomposition suggested by Klenow and Rodríguez–Clare (1997). See below for

important to note that this aggregate production function is a special case of the nested-CES class of aggregate productions (specifically, Nesting 3 with / ¼ 0) to be introduced in Section 4.and Klenow (2000) and Klenow and Rodríguez–Clare (1997) also assume perfect substitutability between skilled and

d labor when constructing human capital stocks across countries.

12 The

where13 The

(unskilltypes oliteratuMincer

14 Thesome psome h

15 Thedifferenof schoacross cby leve


ws;i

wu;i¼ hs;i

hs;i

hu;i

� �q�1

: ð2Þ

Hence, given data on the relative wage and the ratio of skilled to unskilled labor for a country, the rel-ative efficiency of skilled labor ðhs;iÞ can be computed under different assumptions for the substitut-ability parameter, q. Note that with q ¼ 1; hs is equal to the relative wage of the two types of labor.However, with imperfect substitutability ðq < 1Þ; hs would depend not only on the relative wagebut also on the ratio of skilled to unskilled labor.

2.1. Data

Cross-country data for the ratio of skilled to unskilled labor and the relative wage for the two typesof labor, capital stock per worker and output per worker is obtained from Caselli and Coleman (2006),who obtained these data from the following sources:

1. Caselli and Coleman estimate relative wages for a country as expðbinÞ, where for country i, bi is thereturn to an additional year of schooling, and n is the difference in schooling years between skilledlabor and unskilled labor. The cross-country returns data are from Bils and Klenow (2000) (who inturn obtained the data from Psacharopoulos, 1994) and are for countries for which the Mincerregression included experience and experience squared as explanatory variables.12The year forwhich the estimated return to schooling is reported differs across countries, but all data are for the1970s and 1980s and hence provide a good approximation for the return in a country in 1985, theyear for which the data on schooling are used. Data for differences in schooling years are from Lee(2001).13

2. As is common in the growth accounting literature, schooling is used as a proxy for skill. To con-struct the ratio of skilled to unskilled labor Caselli and Coleman use three definitions of skilledlabor: primary-completed (skill definition 1), completed secondary schooling (skill definition 2)and completed college. For each definition, using cross-country data on schooling from Barroand Lee (2001) for the year 1985, the quantity of skilled and unskilled labor is constructed as aweighted sum of the subgroups that belong to each category with weights defined as the wage rel-ative to the lowest schooling group belonging to the category.14 With this procedure, for example,for the primary-completed definition of skilled labor, unskilled labor is measured in ‘‘no schoolingequivalents” while skilled labor is measured in ‘‘primary completed equivalents”.15

3. Data for capital stock per worker and output per worker are from Hall and Jones (1999). GDP perworker in international dollars while capital per worker is estimated using the perpetual inventorymethod. The data for both the variables are for 1988.

Mincer regression used for estimating the return to an additional year of schooling, b, for a country is given by

ln Wk ¼ constantþ bzk þ c1xk þ c2x2k þ �k;

for individual k, W is the wage, z is the years of schooling, x is the experience and � is an i.i.d. error term.data set provided by Freeman and Oostendorp (2000) allows for computing the skill premia between production

ed) and non-production (skilled) workers. However, the data set provides no information on relative quantities of the twof labor and hence cannot be used for estimating the substitution parameter. Moreover, to be consistent with previousre, skilled and unskilled labor are defined using schooling data and hence the estimates of return to schooling based onregressions are used to compute relative wages.Barro and Lee data set provides a breakdown of the population over 25 years of age into seven groups: (1) no schooling, (2)rimary schooling, (3) completed primary schooling, (4) some secondary schooling, (5) completed secondary schooling, (6)igher schooling and (7) completed higher schooling.

data does not correct for differences in quality of schooling across countries. Hanushek and Kimko (2000) argue thatces in quality of schooling are important for explaining differences in growth rates across countries. However, the quality

oling would matter for the estimate of the substitution parameter only if we suspect that there are systematic differencesountries in the relative quality between primary and higher levels of schooling. Due to lack of data on quality of schooling

l of schooling, I am unable to determine whether and how this would effect the results reported below.

ARG

AUS

BOLBWA

BRA

CAN

CHL

CHN

COLCRI

CYP

DOM

ECU

SLV

FRA

GHA

GRC

GTM

HND

HKGHUN

IND

IDN

ISR

ITA

JPN

KEN

MYS MEX

NLD

NICPAK

PAN

PRYPER

PHL

POL

PRT

KOR

SGPLKA

SWE

CHE

OAN

THA

TUN

GBR

USA

URY

VEN

DEU

1.0

1.5

2.0

2.5

Rel

ativ

e T

FP

, rat

io r

ho=

0.75

to r

ho=

1

0.0 0.2 0.4 0.6 0.8 1.0Relative GDP per Worker

Skill Def 1

Skill Def 2

Fig. 1. TFP relative to the US for different substitution parameters.


The cross-country data set contains information on measures of skilled labor ðhsÞ and unskilled ðhuÞlabor, and the relative wages of skilled and unskilled labor for 51 countries.16 The data set has infor-mation on 15 low income countries, 20 middle income countries and 16 high income countries.17 Thereis substantial disparity in incomes across countries in the sample. The GDP per worker of the richestcountry in the sample, the United States, is about 20 times that of the poorest country in the sample,Ghana. The average GDP per worker of the top 10% of the countries is about 13 times the averageGDP per worker for the bottom 10%.

2.2. Importance of the substitution parameter

I now illustrate the sensitivity of the conclusion of Hall and Jones’ (1999) accounting exercise todifferent values of the substitution parameter. As in Hall and Jones, I assume a ¼ 0:3.18 Using dataon relative wages and relative endowments of the two types of labor described above, h is computedfor q ¼ 0:75 (imperfect substitutability) and q ¼ 1 (perfect substitutability) and then used to aggre-gate the two types of labor into a measure of human capital for each country in the data set. The hu-man capital measure thus constructed is then used with data on output per worker and capital perworker to compute TFP for each country relative to the US for the two values of the substitutionparameter.

Fig. 1 plots the relative TFP computed for q ¼ 0:75 as a ratio of that for q ¼ 1 for skill definitions 1and 2. As is clearly evident from the Figure, the ratio is greater than 1 indicating that relative TFP of all

16 I exclude Jamaica from the group of countries for which Bils and Klenow (2000) report the Mincer coefficient on schooling asthe reported coefficient is unusually high and hence seems to be an outlier. Including it does not qualitatively affect the findingsthat follow.

17 Low income are countries with GDP per worker (in PPP terms) less than $ 7000, middle income are countries with GDP perworker between $ 7000 and $ 16,000 and high income are countries with GDP per worker greater than $ 16,000. This divides thesample of 51 countries into 15 low income, 20 middle income and 16 high income countries. This classification is consistent withthe World Bank definition of low, middle and high income countries based on GDP per capita measured using the purchasingpower parity (PPP) method. See World Development Report (1998).

18 This implies a labor share of income of 0.7. Gollin (2002) finds that labor share does not vary systematically across countriesand is bounded between 0.65 and 0.80 and argues in favor of using a common labor share of income for cross-country studies.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.4 0.5 0.6 0.7 0.8 0.9 1.0

Substitution Parameter

Rel

ativ

e TF

P

Fig. 2. Comparison of relative TFP for India for different substitution parameters.


countries increases when the substitution parameter decreases from 1 to 0.75 for both definitions ofskilled labor. A lower value for the substitution parameter (a lower elasticity of substitution) implieslarger differences in human capital stocks across countries, and this lowers the differences in TFP re-quired to account for cross-country income differences.

Fig. 1 also illustrates the difference for the two definitions of skilled labor from decreasing the sub-stitution parameter from 1 to 0.75. Using data for skill definition 2 for the computation, the line of fit(skill defenition 2) in the figure, shows that the increase in relative TFP is not as systematically relatedto GDP per worker as that for skill definition 1. Relative to skill definition 1, skill definition 2 has a low-er cross-country variation for both skilled and unskilled labor. While for q ¼ 1 the two definitions leadto similar cross-country variation in aggregate human capital, for q ¼ 0:75, the variation in aggregatehuman capital for the first definition is higher than that for the second definition of skilled labor.19 Thisresults in the observed larger difference in relative TFP for skill definition 1 than for skill definition 2.20

To illustrate the point further let us consider India relative to the US. With perfect substitutabilitybetween skilled and unskilled labor, human capital per worker in India is about 35% that of the US andthe TFP for India relative to the US is 0:46.21 However, with imperfect substitutability, q ¼ 0:75, hu-man capital per worker in India is only 10% of that in the US and the relative TFP for India is 1:07. Thatis, relative to the US, India goes from being 46% as efficient in the use of its resources to being 7% moreproductive than the US. As Fig. 2 illustrates, TFP for India relative to the US, as with all countries in thedata set, is inversely related to the substitution parameter q. However, for values of q greater than 0.8,the change in relative TFP from increasing q is negligible. Hence, if the substitution parameter wasgreater than about 0.8 (elasticity of substitution, r ¼ 5), then assuming perfect substitutability

19 Standard deviation of human capital across countries for skill definition 1 for q ¼ 1 is 48.81 and for q ¼ 0:75 it is 178.90; whilefor skill definition 2 the standard deviations for the two values of the substitution parameters are 48.84 and 82.70, respectively.

20 Given that there are significant differences in the conclusions for the two definitions of skilled labor, both definitions of skilledlabor are employed for the analysis that follows.

21 Using a different cross-country data set for relative wages and endowments of skilled labor and unskilled labor, Hall and Jones(1999) (p. 91 Table 1) report a lower TFP for India relative to the US of 0.27.

0.0

0.2

0.4

0.6

0.8

Log

Rel

ativ

e W

age

-1.0 0.0 1.0 2.0 3.0 4.0Log Ratio of Skilled to Unskilled

Fit for LICs

Fit for HICs

Fig. 3. Differences in the substitution parameter between low income and high income countries (skilled definition 1).


between the two types of labor would not matter for the conclusions about the relative importance ofTFP in accounting for cross-country income differences.22

The example suggests that, when aggregating skilled and unskilled labor into a single factor of pro-duction, assumptions about the substitution parameter have a very large impact on the relativeimportance of total factor productivity differences in accounting for cross-country income differences.Hence, it becomes imperative to determine the value of this parameter that should be used for aggre-gating the two types of labor into a single factor of production.

3. Relative wages and relative endowments

In this Section 1 use the data from Caselli and Coleman (2006) described above and investigate therelationship between relative wages and relative endowments of skilled to unskilled labor acrosscountries to gain insights into the value of the substitution parameter for the aggregate productionfunction, defined by (1), used for growth accounting in the previous section. The substitution param-eter for this APF can be estimated by regressing logarithm of relative wages on the logarithm of rel-ative endowments of skilled and unskilled labor and allowing for differences in the intercept termacross countries using Eq. (2).

To investigate whether the two types of labor are perfect substitutes in production, in Fig. 3, I plotthe logarithm of relative wages against the logarithm of the ratio of skilled to unskilled labor for skilldefinition 1 and examine the line of fit to determine whether the two types of labor are perfect sub-stitutes in production, that is, whether q should be 1 for the example above. The two types of laborwould be perfect substitutes if the slope of the line fit was 0 or close to it.

The scatter plot clearly shows that the line of fit (not shown in the figure) would be downwardsloping, suggesting that skilled and unskilled labor are imperfect substitutes in production. Further-more, as illustrated in the figure, the line of fit for low income countries (LICs) is steeper than thatfor high income countries (HICs).23 To determine whether the difference in the slope of the line of fit

22 See Klenow and Rodríguez–Clare (1997) for a similar argument.23 Low, middle and high income countries are as defined in footnote 17.

0.0

0.5

1.0

1.5

2.0

Log

Rel

ativ

e W

age

-4.0 -3.0 -2.0 -1.0 0.0 1.0Log Ratio of Skilled to Unskilled

Fit for LICs

Fit for HICs

Fig. 4. Differences in the substitution parameter between low income and high income countries (skilled definition 2).


is statistically significant between the low income and high income countries, I estimate the followingregression:

24 The

with an25 Usi

GDP peare deficountrirecordeagain, l

lnws

wu¼ c þ d1 ln

hs

huþ c1DM þ c2DH þ d2 DM � ln

hs

hu

� �þ d3 DH � ln

hs

hu

� �þ e; ð3Þ

where, using dummies DM and DH for middle income and high income countries, respectively, the con-stant and slope are allowed to differ across the three groups of countries. It is assumed that the errorterm, e, is i.i.d. with a normal distribution. Estimating Eq. (3) results in statistically significant esti-mates for d1 and d3.24 Hence, the difference in slopes between LICs and HICs illustrated in Fig. 3 is sta-tistically significant. In other words, the slope, and hence the substitution parameter, variessystematically with GDP per worker across countries.25

Given that there is no consensus on the definition of skilled and unskilled labor, a natural questionto ask is whether the difference in relationship between relative wages and relative endowments forLICs and HICs is robust to different definitions of the two types of labor. I do this, again using the Cas-elli and Coleman (2006) data and now define skilled labor as those with at least completed secondarylevel of schooling (skill definition 2). In Fig. 4, I plot the logarithm of relative wages against the loga-rithm relative wages for this definition of skilled and unskilled labor. The lines of fit for LICs and HICs

estimated regression is given by

lnws

wu¼ 0:37� 0:12 ln

hs

huþ 0:21DM � 0:10DH þ 0:02 DM � ln

hs

hu

� �þ 0:11 DH � ln

hs

hu

� �ð0:00Þ ð0:03Þ ð0:25Þ ð0:35Þ ð0:21Þ ð0:08Þ

adjusted R2 of 0:21 (p-values in brackets).ng a rolling averages regression leads to a similar conclusion that the substitution parameter varies systematically withr worker across countries. The advantage of the rolling averages approach is that it does not depend on how LICs and HICsned. The procedure is as follows: first, countries are sorted in ascending order by GDP per worker and then for the first 20es, ln ws

wuis regressed on ln ns

nuand the estimated slope, the estimated coefficient for ln ns

nu, and the average GDP per worker are

d for the group. Next, the 1st observation is dropped and the 31st is included forming a second sample of 20 countries, andn ws

wuis regressed on ln ns

nuand the estimated slope and the average GDP per worker are recorded for this group of countries.


for this figure show a very similar difference in the relationship between relative wages and relativesendowments as that in Fig. 3.26

This difference in slope between low income and high income countries implies that the relation-ship between relative wages and relative endowments of skilled and unskilled labor varies systemat-ically with GDP per worker across countries: LICs have a steeper slope (lower substitution parameter)than HICs.27 Hence, the aggregation considered in Section 2, which uses a common substitution param-eter to aggregate skilled and unskilled labor to into a single factor production, human capital, cannotmatch the relationship between relative wages and relative endowments of the two types of labor. Thisleads me to consider other commonly used aggregations that allow the two types of labor to be used asseparate factors of production. The next section evaluates the consequences for the importance of TFP inaccounting for differences in incomes across countries when such an aggregation is used for the growthaccounting exercise.

4. Aggregation of skilled and unskilled labor

In this Section 1 investigate whether using aggregate production functions that allow for skilledand unskilled labor to be used as separate inputs for growth accounting lead to conclusions that differfrom those obtained in Section 2. For this purpose, I consider variants of the two-level or nested-CESclass of aggregate production functions that allow skilled and unskilled labor to be used as separatefactors rather than be aggregated into a single factor of production.

A two-level CES production function for country i in per worker terms given by

26 Estisignifica

27 Aceendowm

28 Kruequipm

where kof subscussedbetweeable to

yi ¼ Ai aX/1;i þ ð1� aÞ bXl

2;i þ ð1� bÞXl3;i

h i/=l� �1=/

; ð4Þ

where X1;i;X2;i, and X3;i are any permutation of the factors of production ki;hs;i, and hu;i; Ai is the coun-try-specific total factor productivity parameter, and /;l 2 ½�1;1� and a; b 2 ð0;1Þ are substitutionand share parameters, respectively.

There are three possible permutations of k, hs, and hu and all of these nestings have been used in theliterature. Krusell et al. (2000) and Duffy et al. (2004) use a nesting similar to X1 ¼ hu; X2 ¼ k andX3 ¼ hs (Nesting 1) whereas Stokey (1996) uses a variant of the nesting X1 ¼ hs; X2 ¼ k and X3 ¼ hu

(Nesting 2).28 The APF used by Caselli and Coleman (2006) and Heckman et al. (1998) is a variant ofthe third permutation, X1 ¼ k; X2 ¼ hs and X3 ¼ hu (Nesting 3). The Hall and Jones aggregation with per-fect substitutability between skilled and unskilled labor used for growth accounting in Section 2 is a spe-cial case of Nesting 3 with substitution parameters / ¼ 0 and l ¼ 1.

The three nestings differ in their implied direct partial elasticities of substitution between the threepairs of inputs. The direct partial elasticities of substitution between any two pairs of inputs for theAPF specified as (4) are

mating regression (3) with data for skill definition 2 shows that the difference in the slope for LICs and HICs is statisticallynt at 5% level of significance.

moglu (2003) documents a similar systematic difference in the relationship between relative wages and relativeents across low income and high income countries.

sell et al. (2000) use a four input aggregate production function, dividing physical capital into capital structure and capitalent. The nesting they use is given by

y ¼ Akas ahq

u þ ð1� aÞ bkhe þ ð1� bÞhh

s

h iq=h� �ð1�aÞ=q

;

s is capital structure and ke is capital equipment. The implied relative wage of skilled to unskilled labor and the elasticitytitution between skilled and unskilled labor for this four input production function are the same as for Nesting 1 (dis-later). The division of the capital stock into structures and equipment may be important if there are complementaritiesn the latter and skilled labor. However, due to lack of good cross-country data on the break-up of capital stock, I am un-pursue the course suggested by Krusell et al. (2000).

29 Seeproduct

30 Foret al. (2in capitstudies

31 Thicountri


rX2 ;X3 ¼1

1� l

rX1 ;X2 ¼ rX1 ;X3 ¼1fr þ 1

fsj

ð1� lÞ 1fs

j� 1

fs

� �þ ð1� /Þ 1

fr þ 1fs

� � ; ð5Þ

where fr is the relative expenditure share of X1; fs is the relative expenditure share of X2 and X3; whilefs

j is the relative share of either X2 or X3 in total expenditure.29

The partial elasticities of substitution in (5) imply that Nestings 1 and 2 allow for the elasticity ofsubstitution between capital and skilled labor to differ from the elasticity of substitution between cap-ital and unskilled labor, while Nesting 3 implies that these elasticities are the same between the twopairs of inputs. In other words, Nestings 1 and 2, with appropriate restrictions on the substitutionparameters, allow for capital-skill complementarity while Nesting 3 does not. A large body of empir-ical studies, since the seminal work by Griliches (1969), find that capital better substitutes unskilledlabor than skilled labor and hence support the use of Nestings 1 and 2.30

However, between Nestings 1 and 2, the former has been the preferred APF for a number of studies.When examining the role of capital-skill complementarity, Duffy et al. (2004), using cross-countrypanel data, and Krusell et al. (2000), using panel data for the US, argue in favor of Nesting 1 and againstNesting 2 based on the results they obtain when estimating the two variants of the nested-CES aggre-gate production function. While Duffy et al. (2004) find implausible estimates for parameters usingNesting 2, Krusell et al. (2000) argue that the restriction that rk;hs ¼ rhu ;hs imposed by Nesting 2 isat odds with the finding that the elasticity of substitution between skilled labor and unskilled laboris higher than the substitution elasticity between skilled labor and capital reported in Hamermesh(1993).

Since Nesting 1 has been the preferred variant of the nested-CES APF in a number of studies on cap-ital-skill complementarity, in the next Section, I use this aggregate production function for undertak-ing a growth accounting exercise to determine the sensitivity of the conclusions of the exercise to thechoice of aggregation production functions.

4.1. Growth accounting with preferred aggregation

Using the preferred aggregate production function, Nesting 1, I undertake a growth accountingexercise to evaluate whether the choice of aggregation matters for the importance of TFP in explainingcross-country income differences. For the growth accounting exercise in Section 2, the parameters forthe aggregate production function were calibrated to match a common labor share of income of 0.7and the relative wages of skilled and unskilled labor across countries. To ensure comparability of re-sults with the previous exercise, I calibrate the share parameters for Nesting 1 to match the sametargets.31

The aggregate production function for country i in per worker terms used for the growth account-ing exercise is given by

yi ¼ Ai aih/u;i þ ð1� aiÞ bik

li þ ð1� biÞh

ls;i

� �/=l� �1=/

; ð6Þ

where the share parameters a and b are country-specific while the substitution parameters l and /are common to all countries. The labor share of income ðlabshÞ and the relative wage of skilled to un-skilled labor for country i implied by (6) are

Sato (1967) and Duffy et al. (2004) for further details on partial elasticities of substitution for the nested-CES aggregateion functions.instance, support for capital-skill complementarity is provided by Krusell et al. (2000), using US time-series data, and Duffy004), using cross-country panel data. In addition, Flug and Hercowitz (2000) using cross-country data find that investmental equipment raises the relative demand for skilled labor. Similar results are reported by a number of microeconomicsurveyed in Hamermesh (1993).

s requires that the share parameters for the nested-CES production function given by (4) be allowed to vary acrosses.

Table 1Growth accounting with different aggregations

Country Relative TFP

Y/L HJ-APF (q = 1) HJ-APF (q = 0.75) Nesting 1

USA 1.000 1.000 1.000 1.000Canada 0.941 1.559 1.901 1.252Switzerland 0.874 1.136 1.715 0.986Australia 0.843 1.341 1.998 1.036Italy 0.834 1.824 3.728 0.858

Malaysia 0.267 0.628 1.247 0.253Colombia 0.264 0.616 1.269 0.223Chile 0.263 0.498 0.964 0.226Costa Rica 0.257 0.614 1.283 0.208Poland 0.238 0.630 1.038 0.320

Botswana 0.094 0.261 0.587 0.068India 0.086 0.461 1.074 0.067China 0.060 0.302 0.667 0.054Kenya 0.056 0.256 0.591 0.042Ghana 0.052 0.367 0.849 0.044

For all countries:Correlation Y/L (logs) 1.000 0.913 0.770 0.990Contribution of TFP 0.544 0.398 1.109

Note: See Section 2.1 for data source. The contribution of TFP for each aggregation is computed using the methodologydeveloped by Klenow and Rodríguez–Clare (1997).

32 TheChmelasensitivsubstitu

33 Estiet al., 20data anparameobtainethe subthe resu


labshi ¼aih

/u;i þ ð1� aiÞ bik

li þ ð1� biÞh

ls;i

� �/=l�1ð1� biÞh

ls;i

aih/u;i þ ð1� aiÞ bik

li þ ð1� biÞh

ls;i

� �/=l� �

ws;i

wu;i¼ 1� aið Þ 1� bið Þ

aibi

ki

hs;i

� �l

þ ð1� biÞ� �/�l

l hs;i

hu;i

� �/�1

:

Given l and /, the share parameters for each country a and b are calibrated to match a labor share of0.7 and the data on relative wage of skilled to unskilled labor for each country obtained from Caselliand Coleman (2006).32

Duffy et al. (2004), using cross-country panel data, present estimates of the substitution parameterfor Nesting 1 for both the definitions of skilled labor used in this paper. I use the estimates of thesubstitution parameters obtained by them for skill definition 1 (l ¼ 0:51848 and / ¼ 0:64170).33 Withthese values for the substitution parameters, Nesting 1 implies that capital and skilled labor are morecomplementary as inputs than are capital and unskilled labor.

Using the substitution parameters, the calibrated share parameters, and data on output per worker,capital per worker and endowments of skilled labor and unskilled labor from Caselli and Coleman(2006), I compute TFP for each country in the data set relative to that of the US. For skill definition1, Table 1 compares relative TFP computed using the two APFs considered in Section 2 where skilled

calibration strategy employed here assumes that the substitution parameters do not vary across countries. However,rova and Papageorgiou (2005) find evidence of systematic cross-country differences in / and l. While I check for theity of the results for different substitution parameters, calibration using more targets to recover country-specifiction parameters is left as an endeavor for future research.mates of the substitution parameters for this production function using time-series data for a country (for example, Krusell00) differ from those provided by Duffy et al. (2004). Given that the Duffy et al. estimates are obtained using cross-country

d are provided for the definitions of skilled and unskilled labor used in this paper, I use the values for the substitutionters provided by them for skill definition 1 in Table 1 (p. 333). I check for the sensitivity of the results using the estimatesd for skill definition 2 (l ¼ 0:43718 and / ¼ 0:54344) and find that the results are very similar for the two sets of values forstitution parameters. Further, using other values for the substitution parameters (see footnote 37) did not qualitatively alterlts of the growth accounting exercise presented below.


and unskilled labor were aggregated into a single factor of production, the Hall and Jones (1999)aggregation (HJ-APF (q = 1)) and the Hall and Jones aggregation with q ¼ 0:75 (HJ-APF (q = 0.75)),and the preferred aggregation, Nesting 1, for a select group of countries.34

The Table illustrates the importance of the choice of the aggregate production function for conclu-sions from a growth accounting exercise. As discussed earlier, for the Hall and Jones aggregation whenthe substitution parameter is decreased from 1 to 0.75, relative TFP of all countries increases while thecorrelation between the logarithm of relative TFP and relative GDP per worker decreases (bottom pa-nel of Table 1). The preferred aggregation, Nesting 1, however, results in a lower relative TFP comparedto the Hall and Jones aggregation (HJ-APF (q = 1)), suggesting that differences in TFP are even moreimportant in explaining differences in output per worker across countries than the conclusions ob-tained by Hall and Jones (1999).

The Table also reports the contribution of TFP in explaining differences in GDP per worker acrosscountries using the decomposition developed by Klenow and Rodríguez–Clare (1997).35 The contribu-tion of TFP in explaining differences in GDP per worker decreases with the decrease in the value of thesubstitution parameter from 54% to 40%. The contribution of TFP is the highest for the nested-CES aggre-gation. In fact TFP differences required to explain differences in GDP per worker are greater than a100%.36 In other words, if the preferred APF were used for growth accounting we would conclude thatcross-country differences in factors of production cannot explain any of the differences in GDP per work-er.37 Hence, the choice of aggregate production function plays a crucial role for the conclusions obtainedfrom growth accounting exercises about the importance of TFP relative to that of availability of factors ofproduction in explaining cross-country differences in output per worker.

To evaluate whether the definition of skilled labor matters for the conclusions about the impor-tance of TFP in accounting for differences in incomes across countries, I use data on relative wagesand endowment of skilled and unskilled labor from Caselli and Coleman (2006) for skill definition 2and compute relative TFP for each country in the data set. In order to ensure comparability of the re-sults between the two definitions of skilled labor, when computing relative TFP with Nesting 1, I usethe same values of the substitution parameters for both the definitions. Fig. 5, which plots the ratio ofrelative TFP computed using the preferred aggregation, Nesting 1, to that for the Hall and Jones aggre-gation (q ¼ 1) for skill definition 1 and 2, shows a similar pattern for the two definitions of skilled la-bor. Relative TFP is significantly lower for all countries when computed with Nesting 1 than for theHall and Jones aggregation with q ¼ 1 for both the definitions of skilled labor.38 This suggests thatfor both definitions the importance of TFP in explaining differences in output per worker across coun-tries increases when Nesting 1 is used for the growth accounting exercise.

34 For brevity, Table 1 only presents relative TFP for top 5, middle 5 and bottom 5 countries in the data set ranked by output perworker.

35 For a production function given by YL ¼ AX, where X is some aggregation of factors of production (capital, skilled labor and

unskilled labor in per worker terms) and A is TFP, the decomposition suggested by Klenow and Rodríguez–Clare (1997) is given by:

1 ¼ COVðlnðY=LÞ; lnðXÞÞVARðY=LÞ þ COVðlnðY=LÞ; lnðAÞÞ

VARðY=LÞ ;

where COV is covariance while VAR is variance. The first term represents the contribution of factors of production while thesecond term is the contribution of TFP (reported in Table 1) in explaining cross-country differences in income. For further de-tails see Klenow and Rodríguez–Clare (1997).

36 The contribution of TFP is greater than 1 because the covariance between GDP per worker and aggregation of factors ofproduction (COV(ln(Y/L), ln (X))) is negative for this aggregation. Further, even though, between HJ-APF (q = 1) and Nesting 1, thereis a small increase in correlation between the logarithm of TFP and the logarithm of output per worker (from 0.913 to 0.990), thecontribution of TFP increases from 0.544 to 1.109. Contribution of TFP can be written in terms of correlation ascontribution ¼ correlation � rln A

rln y, where rln A and rln y are the standard deviations of log TFP and log output per worker,

respectively. The large difference in the contribution of TFP can be explained by the variance of relative TFP for Nesting 1 (0.84)being a lot higher than that for HJ-APF (q ¼ 1) (0.24).

37 To check for the sensitivity of the results obtained using Nesting 1 to the choice of the substitution parameters, I use theestimates obtained of the parameters obtained by Krusell et al. (2000) for the US (/ ¼ 0:401 and l ¼ �0:495) and undertake thegrowth accounting exercise. The contribution of TFP for this parametrization is 1.212 and does not qualitatively alter the findingthat for Nesting 1 TFP can account for more that 100% of the differences in incomes across countries.

38 The ratio of relative TFP computed using Nesting 1 and that computed using the Hall and Jones aggregation with q ¼ 1 is lessthan 1 for all countries.

0.0

0.2

0.4

0.6

0.8

1.0

Rel

ativ

e T

FP

, rat

io N

estin

g 1

to H

Jrho

=1


Skill Def 1

Skill Def 2

Fig. 5. Relative TFP for Nesting 1 for different definitions of skilled labor.


However, the upward sloping line of fit for both definitions of skilled labor suggests that, comparedto aggregation with q ¼ 1, Nesting 1 results in a systematically lower relative TFP for low incomecountries than for high income countries. This is due to differences in the calibrated share parameters(a and b) for the two groups of countries. Fig. 6 plots the share parameters in the aggregate productionfunction given by (6) for each country relative to that for the US for skill definition 1. While the

02

46

8C

alib

rate

d P

aram

eter

Rel

ativ

e to

US


beta

alpha

Fig. 6. Share parameters for Nesting 1.


calibrated value for a, the share parameter for unskilled labor, appears relatively constant across coun-tries, the calibrated value for b, the share parameter for capital relative to skilled labor, varies system-atically, with low income countries having a higher value for this parameter than high incomecountries. For a given endowment of factors of production, the higher weight on capital relative toskilled labor yields a higher prediction of output per worker for low income countries relative to thatfor high income countries. This results in the relative TFP for low income countries being lower thanthat for high income countries when Nesting 1 is used for growth accounting.

In sum, conclusions from growth accounting exercises are very sensitive to the choice of the aggre-gate production function used for the exercise. That is, the answer to the question of whether differ-ences in TFP or factors of production are more important in explaining differences in GDP per workeracross countries depends on which APF is used for the growth accounting exercise. While Nesting 1,the preferred APF, suggests that hardly any difference in GDP per worker can be explained by differ-ences in factors of production, the Hall and Jones aggregation with q ¼ 0:75 implies that about 60%can be explained by such differences.

5. Conclusion

This paper highlights the importance of the assumptions used for aggregating skilled and unskilledlabor for growth accounting exercises. It is shown that when aggregating the two types of labor to con-struct a single factor of production, human capital, the assumption about the substitution parameterused for the aggregation plays a crucial role for the conclusions about the importance of total factorproductivity relative to that of availability of factors of production in accounting for cross-country in-come differences. While differences in TFP are relatively more important for the Hall and Jones (1999)aggregation with perfect substitutability between skilled and unskilled labor, the relative importanceof differences in availability of factors of production increases when the value of the substitutionparameter for the aggregate production function used by them is decreased.

Given the importance of this parameter for the aggregate production function used by Hall andJones, it becomes imperative to determine its value for undertaking the aggregation. To do so, I inves-tigate the relationship between relative wages and relative endowments of skilled and unskilled laborusing data from Caselli and Coleman (2006), and find that the two types of labor are not perfect sub-stitutes in production. Moreover, and importantly, I find that there are systematic cross-country dif-ferences in the relationship between relative wages and relative endowments of the two types oflabor. This suggests that the substitution parameter for the aggregation used by Hall and Jones variessystematically with GDP per worker across countries. Hence, this aggregation, which uses a commonsubstitution parameter to aggregate skilled and unskilled labor to construct human capital, is found tobe inconsistent with the data.

The nested-CES class of aggregate production functions provide a generalization of the aggregationused by Hall and Jones (1999). These APFs allow skilled and unskilled labor to be used as separate fac-tors rather than be aggregated into a single factor of production. Given that past empirical studies,investigating the role of capital-skill complementarity, have argued in favor of one variant of thenested-CES aggregate production function, I use this APF to undertake a growth accounting exercise.

Comparing the results from growth accounting exercises using different aggregate productionfunctions, I find that the relative importance of TFP in explaining differences in incomes across coun-tries depends crucially on the APF used for the growth accounting exercise. While one APF implies thatabout 60% of the difference in GDP per worker can be explained by differences in factors of production,the preferred aggregation suggests that none of the differences can be attributed to such differences.Thus we require a consensus on the aggregate production function that should be used for evaluatingthe role of factors of production and TFP in explaining differences in GDP per worker acrosscountries.39

39 Recent literature has called into question the use of aggregate production functions for examining differences in rates ofgrowth across countries. For example, Temple and Woessmann (2006), when discussing the role of dualism for growth regressions,argue that results based on simple aggregations may be misleading since an aggregate production function is a goodapproximation for an economy only under very restrictive assumptions.


The findings also call for a more general approach to understanding differences in incomes as wellas use of technologies across countries. Since the aggregation of skilled and unskilled labor plays a cru-cial role in determining the sources of cross-country income differences, understanding the relation-ship between a country’s skill distribution and technology adoption becomes all the more important.Acemoglu and Zilibotti (1999) provide an example of such an approach where they explore the linkbetween availability of skilled and unskilled labor and technology use based on the idea that technol-ogies used by low income countries may be ‘inappropriate’ given their skill distribution.

Acemoglu (2003) extends this line of reasoning to argue that the higher relative price of skilled tounskilled labor in low income countries could be a result of these countries using technologies that arecomplementary to skilled labor. High income countries are constantly innovating, and hence the rel-ative wage of skilled to unskilled labor does not vary as much with the relative endowment of skilledand unskilled labor for these countries as it does for technology adopting, low income countries. Thismay provide a reason for the finding in this paper that skilled and unskilled labor are almost perfectsubstitutes in production for high income countries, but are imperfect substitutes for low incomecountries.

Acknowledgements

I am grateful to Jim Davies, Martin Gervais, Igor Livshits, Lance Lochner, Jim MacGee, and seminarparticipants at the Canadian Economic Association meetings in Toronto and at the University of Wes-tern Ontario for very useful discussions and suggestions. I thank Melanie O’Gorman and two anony-mous referees for detailed suggestions and comments that have significantly improved the paper.

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Human capital aggregation and relative wages across countries