Discussion Papers In Economics
And BusinessChanging demand for general skills, technologicaluncertainty, and economic growth
Masashi Tanaka
Discussion Paper 18-02

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANChanging demand for general skills, technologicaluncertainty, and economic growth
Masashi Tanaka
Discussion Paper 18-02

January 2018

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANChanging demand for general skills, technologicaluncertainty, and economic growth∗
Masashi Tanaka†
January 23, 2018

Abstract
We develop a simple endogenous growth model featuring individuals’ choices between general and ﬁrm-speciﬁc skills, endogenous technological innovation, and agovernment subsidy for education. General skills are less productive than are speciﬁc skills, but they enable workers to operate all technologies in the economy. We
show that demand for general skills increases as countries catch up to the worldtechnology frontier. Further, using aggregated data for 12 European OECD counties, we calibrate the model and compare the theoretical prediction with the data.
In cross-country comparisons, we ﬁnd that the returns on general skills and the impact of general education expenditure on GDP are higher in countries with higher
total factor productivity. These ﬁndings support our theoretical argument of thepositive relationship between ﬁrms’ demand for general skills and countries’ stagesof development.

Keywords: General and speciﬁc skills, Technological uncertainty, Education policy,
Distance to world technology frontier
JEL classification: J24, O33, O40, I22∗

I would especially like to thank Tatsuro Iwaisako for his invaluable comments and encouragement.
In addition, I am grateful to Masaru Inaba, Keigo Nishida, Ryosuke Okazawa, Kouki Sugawara, Katsuya
Takii, participants in the DSGE conference 2016 at Ehime University, participants in the workshop on
the economics of human resource allocation 2017 at Osaka University, participants in Nagoya Macroeconomics Workshop 2017 at Nagoya Gakuin University.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. E-mail address: masashi.tanaka.0203@gmail.com

1

1

Introduction

Although human capital theory suggests that higher education has a key role in sustained economic growth, countries diﬀer substantially in terms of their schooling andeducational structures (see Figure 1). For example, many European countries focus on
vocational education that provides highly specialized and job-related “speciﬁc skills,”
whereas the United States emphasizes general education that develops basically andbroadly usable “general skills.” As Gary Becker notes, there is a trade-oﬀ between the
two types of skills. Becker (1962) pointed out that speciﬁc skills have a stronger relationship to performance of the current job, but that they are diﬃcult to apply across jobs.
On the other hand, general skills are less productive than speciﬁc skills, but they canbe transferred to other jobs. This productivity–transferability trade-oﬀ between generaland speciﬁc skills creates mixed support for general and vocational education and, thus,
the diﬀerences in the educational structures across countries.
How do these diﬀerences in skill composition impact on growth performance of eachcountry? The literature on economic growth has examined this question, focusing on

Figure 1: Education type by country
Note: The ﬁgure shows the percentage of the tertiary general education graduates among all uppersecondary and tertiary education graduates. We calculate them using data from Education at a Glance(2014, OECD), which provides the general and vocational education population share for both secondary
and tertiary education. There are missing data for several countries, including the United States and,thus, we cannot use the data for these countries.

2the obsolescence of speciﬁc skills caused by the emergence of new technologies. The
common argument is that new production technologies replace outdated technologies ininnovation processes, and as a result, speciﬁc skills become obsolete more quickly than
general skills. Krueger and Kumar (2004a, 2004b), for example, argue that the European
emphasis on vocational education might have harmed European growth performance
relative to that of the United States because speciﬁc skills are less able to adapt totechnological development. Thus, the Schumpeterian concept of creative destruction
has developed a deep understanding of the importance of the skill obsolescence eﬀectunderlying long-term economic growth.
However, growth theories have little to say about the diﬀerence in skill transferability
between general and speciﬁc skills. A smooth transition to new or proﬁtable sectors is
helpful for the growth of the entire economy, but may be costly for individual workers
if their skills are less transferable across jobs. On the other hand, general skills enable
workers to deal eﬀectively with unexpected technological breakthroughs. Hence, in this
discussion, the key determinant of the skill composition of an economy is the extent of
uncertainty about future events, such as changes in industrial structures. Gervais etal. (2008) show that economies with lower uncertainty tend to have a larger share of
specialized labor, but that they are more vulnerable to economic turbulence, owing to
the inherent diﬃculty of reallocating workers. Our study also considers the productivity–transferability trade-oﬀ between the two skills, and examines the dynamic changes in
economic uncertainty and the relative importance of productive and transferable skills.
Our main interest is to study how the relative demand for general skills changes in
the catching up process of economic development, and how diﬀerent education systems
impact countries’ growth performance.
We develop a simple endogenous growth model featuring individuals’ choices between general and ﬁrm-speciﬁc skills, endogenous technological innovation by ﬁrms, andgovernment subsidies for general and speciﬁc types of education. We assume that ﬁrmspeciﬁc skills are more productive, but are useful only in the current ﬁrm, whereas
general skills are less but equally productive in all ﬁrms. The dynamic structure of thismodel is a country’s catching up process to the world technology frontier, based on theframework of Acemoglu et al. (2006). Under these assumptions, we analyze the economy’s composition of general and speciﬁc skills in terms of both transitional dynamics
3and long-run equilibrium. Three key features of this model are worth noting: (i) generaland ﬁrm-speciﬁc skills are perfectly substitutable; (ii) there is uncertainty in innovation
activities; and (iii) ﬁrms invest more in innovation activities as their technologies approach the world technology frontier. The assumption of perfect substitution between
the two types of skills helps us to identify the market values of productivity and thetransferability of skills. Further, innovation activity is a risker task (than imitating existing technologies), so the ex post realization of ﬁrms’ productivities diﬀer from the exante expected productivities. Hence, ﬁrms that succeed in innovation may demand additional labor for production. This creates demand for workers with general skills because
only general skills can contribute to the ex post labor reallocation across ﬁrms. Thatis, the uncertainty in innovation is the key factor determining the relative demand for,and the equilibrium composition of general skills. The third assumption, which is widely
accepted in Schumpeterian growth theory, and is a basic assumption in the distance-tofrontier model, implies that the intensity of ﬁrms’ innovation activity increases as they
approach to the world technology frontier. Therefore, the size of the demand for generalskills varies across countries, depending on their stages of development.
The analysis in this paper is divided into two parts. In the ﬁrst part, in sections
2 and 3, we begin with a simple model that brieﬂy illustrates households’ skill choices
and no governmental education subsidies. Here, we focus on the change in ﬁrms’ labordemand for general and speciﬁc skills in the catching up process of transition economies.
We show that ﬁrms’ demand for general skills increases as the country approaches theworld technology frontier. In relatively less developed countries, ﬁrms put less eﬀort into
innovation and, thus, place more importance on productivity than on the transferabilityof workers. Conversely, in countries that are closer to the technology frontier, moregeneral skills are demanded because ﬁrms extend their innovative activities and face great
uncertainty in production. Additionally, we ﬁnd that a follower country’s technology
monotonically converges toward a unique stable steady state, which is less than (orequal to) the world technology frontier. This indicates that, in follower countries withlower initial productivity levels, the demand for general skills increases over time in theprocess of development.
In the second part, we extend the model of the ﬁrst part by introducing an elastic labor supply and a governmental education subsidy in section 4. The comparative4

statistics in the long-run equilibrium show that a higher general education subsidy increases the supply of general skills, enhances ﬁrms’ innovation investment, and increases
the level of technology in the economy. Then, in section 5, we calibrate the model to 12
European countries1 and quantitatively analyze the eﬀect of a 1% increase in the general
education subsidy on the total output of each country. In the multi-country comparison,
there is no clear relationship between the population share with a general education andthe impact of the general education subsidy on total output. However, we ﬁnd that the
impact is larger in countries with higher total factor productivity (TFP), and that thereturn on the transferability of general skills is larger in higher TFP countries. Theseﬁndings support our theoretical argument of the positive relationship between ﬁrms’
demand for general skill and countries’ stages of development. That is, the theoreticaland quantitative results indicate that a country’s distance to the technology frontier is
an important factor in the appropriate composition of general and speciﬁc skills.
The present study contributes to the existing theoretical and empirical literatureon the trade-oﬀ between general and speciﬁc skills. The growing demand for general(non-specialized) skills and the resulting lack of workers with such skills, especially inhighly developed and industrialized economies, is well documented in the literature. As
argued by Goldin (2001), fundamental changes around the turn of the 20th century
made formal, general, school-based learning important to the emerging economic leaderof the world. Krueger and Kumar (2004a, 2004b) show that general education may
contribute to technology adoption and economic growth, especially during times of rapid
technological change. From a diﬀerent perspective, Gervais et al. (2008) construct amodel in which economic uncertainty is a key determinant of the skill composition ofan economy. The purpose of our study is similar to these studies, but we focus more on
the eﬀects of a change of the skill composition on the transitional dynamics. That is,
our primary argument is that structural changes in schooling institutions are required
as a country approaches the world technology frontier. This result complements theempirical ﬁnding of Hanushek (2013), who shows that high-performance skills are more
important for growth in developing counties, but that broad basic skills are more relevant
in industrialized countries.

1

We use data for Austria, Belgium, Denmark, France, Germany, Hungary, the Netherlands, Slovakia,
Slovenia, Spain, Switzerland, and the United Kingdom.

5

Outside of the literature on economic growth, numerous studies examine the relationship between skill composition and economic performance. Gould et al. (2001)
and Violante (2002) examine the eﬀect of technological changes on the depreciation of
speciﬁc skills, and show that an increase in the rate of technological progress increases
within- and between-group wage inequality, as occurred in the United States during thelast three decades of the 20th century.2 In terms of employment, Wasmer (2006) argues
that stringent job protection legislation induces skill specialization and, therefore, lessturnover.3 Hanushek et al. (2017) use microdata for 11 countries to show that the wage
and employment advantages of vocational as opposed to general education decrease withage, indicating that speciﬁc skills become obsolete too quickly.45
This study is also related to the rich and growing literature on the role of humancapital in technological progress. According to Nelson and Phelps (1966), a major roleof human capital is to enable workers to cope with innovation, technological progress,and diﬀusion. More recently, Galor and Tsiddon (1997), Caselli (1999), Galor and Moav
(2000), and Hassler and Rodriguez Mora (2002) examined the eﬀect of a technologicaltransition on the return on human capital. Acemoglu et al. (2006), on which theassumption of technological progress in the present study is based, stressed that as acountry catches up with the technology frontier, growth-maximizing governmental policy
should evolve in order to increase the intensity of ﬁrms’ innovation activities and theselection of highly skilled managers. Vandenbussche et al. (2006) also study on countries’
catching up processes, and argue the importance of a change in skill composition between
basic and higher human capital, according to the stage of development.

2

See also Kambourov and Manovskii (2009a, b), who emphasize the central importance of occupationspeciﬁc human capital to explain the increasing wage inequality in the United States, which occurred
simultaneously with an increase in occupational mobility. Dustmann and Meghir (2005) show the sourcesof wage growth for German workers by estimating returns on experience, sector-speciﬁc tenure, and ﬁrmspeciﬁc tenure. See Sanders and Taber (2012) for a review of the literature on ﬁrm-speciﬁc human capital,
industry- and occupation-speciﬁc human capital, and task-speciﬁc human capital.
3
Another interesting study along this line is that of Charlot et al. (2005), who assume that schooling
contributes not only to productivity, but also to the adaptability of skills. Then, they show that longer
schooling can reduce the unemployment rate, even frictional labor market. On the other hand, Decreuse
and Granier (2013), by treating investment in adaptability separately from investment in productivity,
model the educational trade-oﬀ between general and speciﬁc skills.
4
In addition to the above-mentioned papers, diﬀerences in education structures and skill compositions
have played a critical role as interrelated factors in sociopolitical development (Bertocchi and Spagat
2004; Malamud and Pop-Eleches 2010), and international trade and labor migration (Kim and Kim 2000;
Di Maria and Stryszowski 2009; Guren et al. 2015).
5
A new insight related to ﬁrm-speciﬁc human capital is oﬀered by Lazear (2009), who considers thateach job requires a slightly diﬀerent combination of a multiplicity of general skills. See also Silos andSmith (2015), who measure the speciﬁcity of human capital using data on college credits across subjects.

6

2

The Model

2.1

Economic Environment

The economy is composed of a continuum of a unit measure of workers, given exogenously
and constant over time. Workers live for one period and spend all of their income on
consumption in this period. All workers are equally endowed with one unit of eﬃcientlabor, which they supply to the intermediate goods sector inelastically. We assume thatlabor skills can be of two distinct types: general and sector-speciﬁc.
In every period, a unique ﬁnal good is produced competitively using a continuumof mass one of intermediate goods and any other ﬁxed factor, such as land or natural
resources, as inputs. The ﬁnal good is taken as the numeraire, with its price normalized
to 1, and is produced according to the following Cobb–Douglas production function:
1
Yt = Mt1−α
α

∫ 1
0

α
A1−α
i,t yi,t di,(1)

where Ai,t is the productivity of ﬁrm i, yi,t is the amount of the intermediate goodused in the ﬁnal good production, Mt is the amount of ﬁxed factors, and α ∈ (0, 1). A
representative ﬁnal good producer maximizes the proﬁtΠt = Yt −∫ 1

pi,t yi,t di − pMt Mt ,

0taking the price of the ﬁxed factors pMt and the intermediate good prices pi,t as given.
We normalize the total supply of the ﬁxed factors to one. Then, using the ﬁrst-ordercondition with respect to yi,t , together with Mt = 1, the inverse demand function forintermediate good i is given by:

α−1pi,t = A1−αi,t yi,t .

2.2

Education Choices

At the beginning of their life, workers choose between general and sector-speciﬁc skills.
Once a worker makes an education choice, it cannot be changed. General skills areless productive than speciﬁc skills, while they are equally productive in all sectors. In
contrast, skills speciﬁc to the i-th sector can only be used in production sector i. We
7assume that each intermediate good is produced by a single monopoly ﬁrm and, thus,
the sector-speciﬁc skills considered here can be rephrased as being ﬁrm speciﬁc. In thefollowing analysis, we focus on ﬁrms’ demand for general skills in economies at diﬀerent
stages of development. Thus, to simplify the argument, we assume that skill acquisition
requires no direct expenditure by workers (although we relax this assumption in section4). Henceforth, we refer to workers with general skills as “G-skill workers” and workers
with skills speciﬁc to ﬁrm i as “Si -skill workers.”

2.3

Firms and Technological Progress

Each intermediate good production is composed of a single monopolistic ﬁrm. Further,
we assume that each ﬁrm is owned by an entrepreneur who, like workers, lives for one
period and consumes all of her proﬁt.
Following Acemoglu, Aghion, and Zilibotti (2006) and Vandenbussche, Aghion, andMeghir (2006), henceforth AAZ and VAM, respectively, we assume that the world technology frontier, Āt , grows exogenously at rate λ, so thatĀt = (1 + λ)Āt−1 ,and characterize the technological innovation process of ﬁrm i at time t using the following linear function:
Ai,t = ϕĀt−1 + µi At−1 xi,t ,(2)where Āt−1 is the level of the world technology frontier at time t−1, At−1 is the country’s
local technology level at time t − 1, xi,t is the level of investment of ﬁrm i at time t, andϕ ∈ (0, 1). Here, µi represents an idiosyncratic shock to the innovation activity of eachﬁrm i, which takes a binary value: µi ∈ {µH , µL }, where µH > µL ≥ 0. We denote byπ ∈ (0, 1) the probability that each ﬁrm draws µH . We suppose that the local technology
frontier at period t − 1 in the country becomes common knowledge in period t. Thus,
the local technology At−1 in (2) is determined byAt−1 = max{Ai,t−1 }.

8(3)Further, the cost of investment in innovation takes the following form6 :c(x) =

Ā1−α
t
x2 .
2

In addition to the innovation activity, each intermediate ﬁrm i acts as a monopolist
and produces intermediate good i, according to the following production function:

yi,t = si,t + γgi,t ,(4)where si,t and gi,t respectively denote the numbers of Si -skill and G-skill workers employed by ﬁrm i at period t, and γ ∈ (0, 1) represents the relative productivity of generalskills. Note that γ < 1 indicates that the ﬁrm i speciﬁc skill, as compared to generalskills, has a productivity advantage in the production by ﬁrm i.
Furthermore, we suppose that, before the shock is realized, ﬁrms need to sign wage
contracts with workers, which restrict the ability of ﬁrms to ﬁre workers or to lower thewages after the realization of the productivity shock. This is relevant in most European
countries, where workers are protected against ﬁring by employment protection regulations. This assumption enables workers to invest in relationship-speciﬁc skills because
it contributes to avoiding the well-known hold-up problem: the returns on ﬁrm-speciﬁcskills are lost if a ﬁrm terminates the relationship.7 Note that employment protection
here plays a key role in guaranteeing the employment and wages of S-skill workers, but ismeaningless for G-skill workers because they can move freely across ﬁrms, even after theidiosyncratic shock. That is, there is no need for both ﬁrms and G-skill workers to limit
their ex post reallocation choices by using a contract. Thus, for expositional simplicity,
we assume that ﬁrms are required to make contracts only with S-skill workers.

2.4

Timing of Events

The timing of events in each period t is as follows. At the beginning of the period, eachintermediate ﬁrm i decides how many Si -skill workers to hire, and makes a contract
with these workers, that guarantees their payment and employment. We assume that
The assumption that investment cost is proportional to Ā1−αensures balanced growth.
t
A similar speciﬁcation is made by Gervais et al. (2008), where ﬁrms draw noisy signals about future
productivity. Then, after this but before the productivity shock, ﬁrms decide how many workers to hireand sign binding contracts with them.
6

7

9

the contract does not depend on the realization of idiosyncratic shocks to the ﬁrms. Thisimplies that ﬁrms bear the overall risk of holding speciﬁc skills. The number of Si -skill
S , are determined by the contracts. Because
workers, si,t , and the wage paid to them, Wi,t

the total number of S-skill workers is determined, the number of G-skill workers is also
determined at this time. That is, we suppose that worker’s education choices between
general and speciﬁc skills are made at the same time. Then, each ﬁrm i realizes the
productivity of innovation process µi ∈ {µH , µL }. Based on the realized productivity
level, the ﬁrm decides on the level of investment xi,t and whether to hire additional
G-skill workers. Then, the number of G-skill workers employed in ﬁrm i, gi,t , their wage,
G , and x
Wi,ti,t are determined.

The important assumption here is that before observing the idiosyncratic shock µi ,the payments and employment of S-skill workers are guaranteed by contracts. A similar
speciﬁcation of ﬁrm speciﬁc skills is employed by Gervais et al. (2008). Their speciﬁcation well describes the productivity–transferability trade-oﬀ between speciﬁc and generalskills. Equation (4) clearly shows that an Si -skill is more productive than a general skill
in the i-th good production and, thus, ﬁrms wish to secure a productive labor force.
On the other hand, G-skill workers play a complementary role in the ex post eﬃciency
of production by moving to high productive sectors. Note that S-skill workers are protected from outside shocks by long-term contracts. Therefore, it is only ﬁrms that face
the trade-oﬀ in their hiring decisions.
Before presenting the main analysis, several properties of the equilibrium wages are
now apparent. First, because intermediate goods ﬁrms are ex-ante homogeneous, all
S = WSﬁrms oﬀer the same wage and hire the same number of S-skill workers, that is, Wi,tj,tand si,t = sj,t hold for all i ̸= j ∈ [0, 1]. Second, G-skill workers are hired competitively
by ﬁrms who seek additional labor. Because G-skill workers are equally productive
G = W G must hold in equilibrium.
across ﬁrms, if ﬁrm i and j demand G-skill workers, Wi,tj,tFinally, because the aggregate state of the economy is not aﬀected by productivity shocks
to each ﬁrm, workers can perfectly predict the equilibrium wages, WtS and WtG , when
making their education choices. In order for workers to be indiﬀerent between acquiring
G-skill and S-skill, both skills must be paid the same wage. That is, we have WtS = WtG
in equilibrium, with positive numbers of G-skill and S-skill workers.

103Equilibrium

In this section, we study how the share of G-skill workers varies with the economy’s
distance to the world technology frontier. As discussed in the previous section, eventhough G-skill workers are less productive than S-skill workers are, and are perfectly
substitutable, Wts = Wtg must hold when ﬁrms employ G-skill workers in equilibrium.
The transferability of general skills is a key determinant of the value of general skills
relative to that of ﬁrm-speciﬁc skills. As we show next, there are two types of equilibriumoutcomes: a fraction of workers acquire general skills, or all workers acquire speciﬁc skills.
These two outcomes are distinguished by parameter values (equation (A1) below), but
we focus on the former case. That is, we analyze the equilibrium where both G-skill andS-skill workers emerge and, hence, Wts = Wtg (≡ Wt ) holds.
Further, we ﬁnd that the ex post realization of ﬁrms are either high productivity
ﬁrms with µH , or low productivity ﬁrms with µL . Hence, they can be distinguishedby i = H, L. For simplicity, we assume that µL = 0 for type-L ﬁrms, and denote theproductivity of type-H ﬁrms as µH = µ > 0.

3.1

Innovation and Demand for General Skills

The ﬁrms’ problem is divided into two stages: before and after observing the productivity
shock. In the second stage, based on the values of µi and st determined in the ﬁrst stage,ﬁrm i chooses gi,t and xi,t to maximize its proﬁt:
R(i, st ) ≡ pi,t yi,t − Wt gi,t − c(xi,t )1 1−α 2α= A1−αxi,t ,
i,t yi,t − Wt gi,t − Āt2
Substituting (4) into the above expression yields
[
]
1 2
α
R(i, st ) = a1−α
(s
+
γg
)
−
w
g
−
x
Ā1−α ,
t
i,t
t i,t
i,t
2 i,t t(5)where wt ≡ Wt /Ā1−αand ai,t ≡ Ai,t /Āt . Further, deﬁne at ≡ At /Āt ∈ (0, 1] as antinverse measure of the country’s distance to the world technology frontier. Therefore,

11the innovation process (2) can be rewritten as

ai,t =)1 (ϕ + µi at−1 xi,t .
1+λ

(6)

From (5) and (6), the second-stage problem of ﬁrm i can be expressed as follows:(max

gi,t ,xi,t

1
1+λ

)1−α

(

ϕ + µi at−1 xi,t)1−α1(st + γgi,t )α − wt gi,t − x2i,t .
2

Assuming interior solutions, the ﬁrst-order conditions with respect to gi,t and xi,t are,respectively,

[
]
( ) 1
1
γα 1−αgi,t =
−st +
ai,t
γ
wt

(7)

and
xi,t =(1 − α)µi(st + γgi,t )α a−αi,t at−1 .
1+λ

(8)

Substituting (7) into (8), we obtain(1 − α)µi
xi,t =
1+λ

(

γα
wt

) α

1−α

at−1 .

(9)

From (9), we have that xi,t increases in at−1 . That is, as in AAZ and VAM, the ﬁrms
enhance their innovation activities as the country approaches the world technology frontier. On the other hand, type-L ﬁrms choose xL,t = 0 because they gain no beneﬁt frominvesting in innovation activity. Therefore, the technology levels of type-H and type-L
ﬁrms are[]( ) α(1 − α)µ2 γα 1−α 2
1aH,t =
ϕ+at−1 ,
1+λ1+λwt
aL,t =

ϕ
.
1+λ

(10)

(11)

Clearly, we have that aH,t > aL,t . In other words, type-H ﬁrms can develop and employ
more advanced technology than type-L ﬁrms can. Further, from (3), we have that
aH,t = at holds in period t + 1 because the type-H ﬁrms are the local technology frontierin each period. Thus, the law of motion of the (inverse) measure of the country’s distanceto the frontier, at , can be derived using (10) (see section 3.4).

12

Even though we focus on the equilibrium in which both general and speciﬁc skills
are demanded, type-L ﬁrms never hire G-skill workers (see Appendix A for a formal
proof). Intuitively, because type-H ﬁrms enjoy higher productivity, they produce moregoods and demand more labor than type-L ﬁrms do. Because S-skill workers are moreproductive than G-skill workers, ﬁrms in the ﬁrst stage hire suﬃcient S-skill workers sothat they do not have to hire G-skill workers, even in the case of being type-L. At thesame time, type-H ﬁrms seek additional labor and, thus, hire G-skill workers to copewith the labor shortage in the second stage.8 Therefore, we have
]
[
( ) 1
1
γα 1−αgH,t =
−st +
aH,t
γ
wt

(12)

and gL,t = 0.

3.2

Demand for Specific Skills

Before the shock is realized, ﬁrms in the ﬁrst stage decide how many S-skill workers tohire so as to maximize their expected proﬁts. Substituting (12) and gL,t = 0 into (5),
we derive the revenues of type-H and type-L ﬁrms as functions of st , as follows:[(

R(H, st ) = (1 − α)

γα
wt

) α

1−α

]
1
1 2
aH,t + wt st − xH,t Ā1−α
,
t
γ
2

α 1−α
R(L, st ) = a1−α.
L,t st Āt

The ﬁrms’ problem in their ﬁrst stage can be written as
max πR(H, st ) + (1 − π)R(L, st ) − wt st Ā1−α
t
{
⇔

max π
st

st

(
(1 − α)

γα
wt

) α

1−α

1
1
aH,t + wt st − x2H,tγ2}
α+ (1 − π)a1−αL,t st − wt st .

We see from the above maximization problem that the existence of an interior solutionof st requires that π/γ < 1. Alternatively, gH,t = gL,t = 0 is realized. Intuitively, 1/γ
units of G-skill workers can be replaced with a single unit of S-skill workers. That is, byhiring an additional single unit of S-skill workers in the ﬁrst stage, ﬁrms can reduce, as
8Note that gL,t = 0 is not derived from the assumption that µL = 0. As long as µH > µL holds, theresult remains unchanged, even if we allow µL to be strictly positive.

13

expected, πγ wt of payments for G-skill workers. If this is always higher than the marginal
cost from hiring a single unit of S-skill workers, wt , ﬁrms never employ G-skill workers.
However, a world with no general skills seems unrealistic and, hence, we assume in thesubsequent analysis thatπ < γ.

(A1)

Under assumption (A1), the interior solution of st is given by:(st =(1 − π)γα(γ − π)wt

) 1

1−α

aL,t .

(13)

Finally, denote by Gt and St the aggregate demand for G-skill and S-skill workers,
respectively. From (12) and (13), we have
Gt = πgH,t
π
=
γ

[(

γα
wt

) 1

1−α

(
aH,t −(1 − π)γα(γ − π)wt

) 1

]

1−α

aL,t ,(14)

St = st(=(1 − π)γα(γ − π)wt

) 1

1−α

aL,t .

(15)

Further, from (14) and (15), the relative demand for G-skill workers can be expressed as
π
Gt
=
St
γ

[(

γ−π
1−π

) 1

1−α

]
e
at − 1 ,(16)

where eat = aH,t /aL,t represents the technology gap between type-H and type-L ﬁrms.
Here, (16) shows that the relative demand for G-skill workers increases in the technology
gap, and is the key factor in ﬁrms’ demands for general skills. The technology gap e
at canbe interpreted as the measure of uncertainty that ﬁrms face about future productivity.
That is, ﬁrms tend to place more importance on transferable skills than they do on
productive skills as the uncertainty about future technology increases. Therefore, ﬁrms
demand more general skills as the technology gap between the success and failure ofinnovation increases.
The feature of this model is that the uncertainty e
at is determined endogenously. Our

14next step is to examine how a change in the distance to the frontier aﬀects the technology
gap and the relative demand for general skills.

3.3

Labor Market Equilibrium

In this section, we analyze the equilibrium composition of the labor force between the
two types of skills. Because we suppose that the total labor supply is ﬁxed at 1, labormarket clearing requires Gt + St = 1. Substituting (14) and (15) into the labor market
clearing condition yields the market clearing wage:
(

γα
wt

) 1

1−α

[

π
(γ − π)=aH,t +
γ
γ

(

1−π
γ−π

]−1

) 1

1−α

aL,t

.

(17)

Combining (17) with (10) and (11), we ﬁnd that the equilibrium level of e
at satisﬁes thefollowing equation:
[

π
γ−π
(e
at − 1)
e
at +
γ
γ

(

1−π
γ−π

) 1 ]α1−α=(1 − α)µ2 a2t−1.
(1 + λ)1−α ϕ1+α(18)

The equilibrium composition of skills is determined by (16) and (18). We see from(18) that aet increases in at−1 . The positive relation between eat and at−1 stems from thesize of the technological spillover from the world technology frontier. As seen from (2),when a country is a long way from the world technology frontier, the major determinantof ﬁrms’ technologies is the spillover of the frontier technology. That is, ﬁrms’ innovation
does not contribute signiﬁcantly to their technologies in the early stages of development.
However, the closer the economy is to the frontier, the higher is the importance ofinnovation as a determinant of ﬁrms’ technologies and, thus, the technology gap between
type-H and type-L ﬁrms increases.
However, note that an increase in at−1 does not always expand the demand forgeneral skills. Equation (16) indicates that we have Gt = 0 for small values of e
at , thatis, for small values of at−1 . In other words, there exist b
a such that the number of G-skill
workers is zero when at−1 < ba. The next lemma shows the existence of the thresholdvalue, ba ∈ (0, 1).

15

Lemma 1.

Let[(1 + λ)1−α ϕ1+α
µ
b≡
1−α

(

1−π
γ−π

) α {(
1−α

1−π
γ−π

) 1

1−α

}] 21
−1

,

and suppose that (A1) and µ > µb hold. Then, there exists ba ∈ (0, 1), such that for all
at−1 ∈ (ba, 1], the number of workers with general skills is strictly positive.

Lemma 1 is proved in Appendix B. It implies that general skills are beneﬁcial for
economies that are closer to the frontier, that is, for more developed countries. As shownin Appendix B, µ > µb guarantees the existence of ba ∈ (0, 1). From the above discussion,
together with Lemma 1, we have the following proposition.

Proposition 1.

Suppose that (A1) and µ > µb hold. Further, assume that b
a < at−1 <1. Then, firms’ relative demand for workers with general skills, Gt /St , is higher whenat−1 is higher (i.e., when the economy is closer to the world technology frontier).

Proposition 1 is obtained directly from (16), (18), and Lemma 1. It argues thatthe relative importance of general skills increases as the country approaches the worldtechnology frontier. The driving force behind the growing demand for general skills
is the increase in e
at . As discussed above, when the country is closer to the frontier,ﬁrms gain relatively less from the frontier technology, but gain more from the local
technology. Hence, type-H ﬁrms increase their investment in innovation. This leads
to large technological uncertainty for ﬁrms and, thus, transferable skills become more
important than technical expertise.
The result shown in Proposition 1 complements an emerging body of literature thatemphasizes the importance of basic knowledge through general education, especiallyduring times of rapid technological progress.9 The heart of their critique is that speciﬁcskills become obsolete more quickly with the emergence of new technology and, thus, theneed for general skills grows. Furthermore, from (16) and (18) we have that an increase
in innovation eﬃciency (µ) increases the demand for general skills. However, we focus

9

See Krueger and Kumar (2004a, 2004b), Hanushek and Woessman (2012), and Hanushek et al.
(2017).

16

not on skill obsolescence, but on the change in ﬁrms’ demand for skill transferability. The
relative productivity of speciﬁc skills compared to that of general skills (1/γ) is ﬁxed andassumed to be greater than 1. In spite of this, accelerating innovative activities increases
the importance of labor ﬂexibility to unpredictable changes in industrial structure and,hence, increases the demand for workers with general skills.

3.4

Equilibrium Dynamics

In the comparative statics analysis of the previous subsection, we showed that the numberof G-skill workers increases as the country approaches to the technology frontier. We
now explore the dynamics of at . Substituting (17), (11), and aH,t = at into (10) yields[() 1 ]−αϕ(1 − α)µ2 π
ϕ(γ − π) 1 − π 1−α
at =
+
at +
a2t−1 .
1+λ
(1 + λ)2 γ
γ(1 + λ) γ − π(19)We assume that the follower countries do not overtake the frontier; that is, we supposethat the steady-state value a∗ satisﬁes a∗ ≤ 1. In the next proposition, we summarize
some of the properties of the transitional dynamics and the steady state of at .

Proposition 2.

Let[(1 + λ)(1 + λ − ϕ)
µ̄ ≡
1−α

{

π ϕ(γ − π)+γγ(1 + λ)

(

1−π
γ−π

) 1 }α ] 12
1−α,and suppose µ
b < µ ≤ µ̄ holds. Then, the economy with initial state ao ∈ (ba, 1] converges
monotonically to the unique stable steady state, where its distance to the world technology
frontier is weakly positive (i.e., a∗ ≤ 1).

As shown in Appendix C, a suﬃciently large value of γ ∈ (π, 1) guarantees theexistence of the non-empty interval [bµ, µ̄]. Here, µ ≤ µ̄ guarantees a unique and stablesteady state, where the value of a∗ is weakly less than 1. Figure 2 depicts the equilibrium
dynamics. It follows from Propositions 1 and 2 that if a0 ∈ (ba, a∗ ), then the economy
starts with a larger stock of speciﬁc skills, increases the ratio of workers with generalskills over time, and approaches the world technology frontier.

17

Figure 2: Equilibrium dynamics of at

4

Extensions and Policy Analyses

The analysis so far has established a simple theoretical framework for understanding theimportance of the transferability of labor skills in more developed countries. Contrary
to the theoretical prediction, countries have actually adopted very diﬀerent schoolingstructures, even in the most advanced nations. However, the previous model is notdirectly applicable to the analysis of educational policies in each country because we did
not pay much attention to households’ educational choices. To address this problem, we
extend the previous model to a more general setting in which workers with diﬀerent initialabilities choose between general and speciﬁc types of education. With this extension,
the equilibrium wages are diﬀerent between education groups, as is consistent with thewidely observed wage premium for general education graduates.10 The extension also
allows us to relax the theoretical assumption of γ < 1, which indicates that individuals
with a vocational education are more productive than those with a general education.
10

For example, OECD data show that, on average, individuals who have completed a general education
earn 34% more than those with a vocational education do within the group of tertiary graduates. See
Education at a Glance (2012, Table A6.1a).

18

Further, we examine the impact of a change in the education policy, which is capturedby the change in the relative subsidies for general and speciﬁc types of education. In thfollowing analysis, we focus on the steady-state equilibrium of the model economies.
We ﬁrst introduce workers’ preferences, following the work of Krueger and Kumar(2004b). There is a utility cost, C(θ) = 1/θ, of obtaining a general education, which
depends on a worker’s innate ability θ ∈ [0, 1]. We assume that θ is uniformly distributed
across the population, and that the the ability is irrelevant to workers’ productivity.
The heterogeneity of workers’ ability yields the wage diﬀerentials between educationgroups; that is, Wtg does not coincide with Wts in equilibrium. We suppose that workers’
preferences are represented by the utility
U g = log Wtg − log

1+ log νSg ,θ

when the worker chooses general skills. On the other hand,

U s = log Wts + log Ss ,when the worker chooses speciﬁc skills. Here, Sg and Ss represent the government
subsidies for general and speciﬁc types of education, respectively, and ν > 0 denotes therelative eﬃciency of the education subsidy for general education.11 Then, we identify
the worker, θ̄, who is indiﬀerent between acquiring general and speciﬁc skills, as follows

θ̄ =wts Ss.
νwtg Sg

All workers indexed by θ ∈ [0, θ̄] receive a speciﬁc education, while workers with (θ̄, 1]
receive a general education. Denoting as ḡ the total supply of G-skill workers, the above

11

We assume, for simplicity, that the subsidy yields utility directly. One justiﬁcation for this speciﬁcation is that it is the reduced form of the simple educational choice model. Suppose that a policy-makersets educational standards Eg and Es , and individuals choose their eﬀort in response to these standards.
Individuals are endowed with a unit of time and need to spend a fraction e of their time on education,and the rest on labor. Then, the budget constraint is given by C = Wt (1 − e). We assume that educational attainment is determined by the function (θSg )/(1 − e) for a general education and Ss /(1 − e) for a
speciﬁc education. Then, to meet the standards, individuals must exert eﬀort e = 1 − θSg /Eg to achievea general type education, and e = 1 − Ss /Es to achieve a speciﬁc education. Finally, the assumptionof logarithmic utility, log C, together with Eg = 1/ν and Es = 1, yields the above speciﬁcations of theutility function.

19

expression can be rewritten as follows:
1 − ḡ =wts Ss.
νwtg Sg(20)

We continue our analysis on the equilibrium, where there are strictly positive numbers of G-skill and S-skill workers, as in the previous section. However, note that theequilibrium wages in the present model must satisfy

wts <

wtg
,
γ

(21)

indicating that the wage per unit of labor is higher for G-skill workers than it is for
S-skill workers. Rewriting (21) yields γ < wp (≡ wtg /wts ). This means that the wage
premium for general skills, wp, is higher than the relative productivity of general skills,
γ, because the G-skill wage includes payment for their skill transferability. Because thetransferability wage premium is strictly positive, γ < wp must be satisﬁed in equilibriumwith a positive number of G-skill workers. Further, using this relation, we can estimate
the size of the transferability wage premium from the gap between wp and γ. We returnto this issue in section 5. A further important point is that γ in the present model does
not have to be lower than 1. Even if general skills are more productive than speciﬁcskills, S-skills remain valuable to ﬁrms as long as (21) holds.
We relegate the technical derivation of the ﬁrms’ proﬁt maximizing decisions toAppendix D, because they are much the same as the previous derivation. As shown inAppendix D, the total demand for G-skill workers in the steady state can be representedby[]( γα ) 1
π
1−α
∗
−s +
a ,
G=
γ
wg

(22)

and the total demand for S-skill workers is
ϕ
S=
1+λ

(

(1 − π)γαγws − πwg

) 1

1−α

.

(23)

Because we analyze a stationary equilibrium, we drop the time subscript from all variables. The labor market clearing conditions are ḡ = G for G-skill workers, and 1 − ḡ = S
for S-skill workers. Let us denote by g ∗ the steady-state equilibrium value of the number

20

of G-skill workers. Dividing both sides of (22) by (23), and using these market-clearing
conditions yields


( s
) 1
1−α
w
−
π
g∗
π
1 + λ γwg=
−1 +a∗  .
1 − g∗
γ
ϕ
1−π

(24)

Moreover, by substituting ws /wg of (20) into (24), we have


(
) 1
e − g ∗ ) − π 1−αg∗π1 + λ γν S(1=−1 +a∗  ,
1 − g∗
γ
ϕ
1−π

(25)

where Se = Sg /Ss . Finally, the dynamic equation (33) in Appendix D, evaluated at thesteady state, is represented by[]α1(1 − α)µ2 ( γα ) 1−α
∗ 2
a =
ϕ+
(a ) .
1+λ
1+λ
wg
∗

(26)

Further, substituting wg of (22), together with G = g ∗ and s = 1 − g ∗ , into (26), we
obtain(1 − α)µ2(1 + λ)a − ϕ =
1+λ
∗

(
)
γ − π ∗ α ∗ 2−α
1+
g
(a )
.
π

(27)

Here, g ∗ and a∗ in the steady-state equilibrium are determined from (25) and (27).
Appendix E shows that the dynamic structure of at is much the same as that in theprevious section. The results of the comparative statics with respect to the relative
e are summarized in the following proposition.
education subsidies, S,

Proposition 3.

The number of workers with general skills, g ∗ , and the level of tech-e Sg /Ss ).
nology, a∗ , increase in S(=

The formal proof of Proposition 3 is given in Appendix E. The ﬁrst key ﬁnding
in this proposition is that, although it may be obvious, the number of G-skill workers
increases as the subsidy for general education increases. This is because a higher general
education subsidy makes workers more willing to supply general skills for a lower wage
premium and, thus, makes the threshold ability θ̄ lower. On the other hand, the positive
relation between a∗ and Se is non-trivial. It indicates that an increase in the general21

education subsidy improves the steady-state level of technology. Therefore, the present
model shows that an increase in the supply of G-skill workers enhances the innovation
activities of ﬁrms, and thus, improves the technology level.
Although there is no fundamental change in the dynamic system from that describedin section 3, we cannot rule out the possibility of the existence of multiple steady states in
our analytical work. In the numerical study in the next section, however, we conﬁrm theuniqueness of the stable steady state using the data for the sample countries.12 Thus, inthe following analysis, we suppose that economies are located at the stable steady state.

5

Quantitative Analysis

In this section, we explore how the total output of the economies are aﬀected by theeducation subsidy on general education. Although Proposition 3 shows that a∗ increases
e we cannot conclude that a higher general education subsidy always increases thein S,gross domestic product of the economy. This is because if general skills are less productive than speciﬁc skill, an increase in G-skill workers decreases the total labor supplyof the economy. Thus, it is meaningful to explore whether and how an emphasis on a
general education subsidy contributes to countries’ GDP. Our ﬁnal goal is to estimate
the eﬀect of the marginal increase in Se on GDP.
The system of (20), (24), and (27) contains 11 variables. Using the available data
set for 12 European countries, namely, Austria, Belgium, Denmark, France, Germany,
Hungary, the Netherlands, Slovakia, Slovenia, Spain, Switzerland, and the United Kinge a∗ ) for each country. The paramdom13 , we ﬁrst determine the values of (g ∗ , ws /wg , S,eters describing the production technology, (λ, ϕ, α, π), are chosen to satisfy broadly
observable empirical evidence. The remaining three parameters, (γ, ν, µ), which aremodel-speciﬁc parameters, are derived from the three equations, (20), (24), and (27),12

Under the parameter values estimated in section 5, there are two steady-state points, where thatwith a lower value is stable and the other is unstable, as is the case of the dynamics of at in section 3(see Appendix C). Thus, we should expect economies to locate at the lower value. However, we cannot
rule out the possibility of more complex structures of the dynamic system of at in our analytical work.
13
We initially attempted to use all available EU countries from the OECD data set (22 countries),
but we exclude 10 countries for the following two reasons. First, we need data on attainment and per
student expenditure per education type (general/speciﬁc), but these data are missing for nine countries(Czech Republic, Estonia, Greece, Ireland, Italy, Luxembourg, Poland, Portugal, and Sweden). Second,
because we regard the United States as the frontier country, we cannot use data for countries whose
total factor productivity is higher than the United States. For this reason, we exclude Norway.

22

Table 1: Four key observations from the data
AUT

BEL

DNK

FRA

DEU

HUN

NLD

SVL

SVN

ESP

CHE

GBR

wp

1.29

1.23

1.11

1.34

1.25

1.64

1.08

1.39

1.32

1.47

1.17

1.26

Se

1.09

1.31

1.94

1.38

1.63

2.01

1.45

1.66

1.22

1.37

1.44

1.47

g∗

0.17

0.27

0.38

0.26

0.22

0.27

0.45

0.19

0.17

0.42

0.32

0.40

a∗

0.82

0.86

0.91

0.95

0.93

0.57

0.96

0.66

0.60

0.81

0.96

0.72

Note: wp = wg /ws and Se = Sg /Ss . The country codes are AUT=Austria, BEL=Belgium,
DNK=Denmark, FRA=France, DEU=Germany, HUN=Hungary, NLD=Netherlands,
SVK=Slovakia, SVN=Slovenia, ESP=Spain , CHE=Switzerland, GBR=United Kingdom.

and the eight predetermined variables. Then, we estimate the impact of a 1% increasein Se on the GDP of each country.

5.1

Data

Our primary data source is the Education at a Glance series: OECD indicators, which
provides a rich description of education indicators per education type (general/vocational).
Education at a Glance (2014) provides the relative earnings of workers aged 25–64 (normalizing the income of adults with upper secondary education to 100). These are given
for university and non-university tertiary categories14 and, thus, we regard the former as
the general type of education and the latter as the speciﬁc (vocational) type of education.
Although the OECD also classiﬁes secondary education into the same two categories forsome data, we cannot obtain the wage gap between secondary general and vocational
education graduates. Thus, we use the tertiary wage gap as a proxy for wg /ws . As
such, we obtain a general education wage premium wp ≡ wg /ws = 185/143 ≃ 1.29 forAustria, for example.
The data for the relative education subsidy Se for each country is also extracted from
Education at a Glance (2014), which provides the annual expenditure per student byeducational institution. Although it makes sense to use data on tertiary education to
14

The OECD discriminates university-level education from non-university tertiary education; theOECD calls the former “tertiary-type A” and the latter “tertiary type B.” The general pattern oftertiary type A education is to oﬀer programs that are broader and more general in orientation. Tertiary
type A institutions are known as universities in most countries. On the other hand, tertiary type B
education oﬀers shorter vocational programs and do not lead to the baccalaureate. Tertiary type B
institutions are known as further education colleges in the United Kingdom, community or two-yearcolleges in the United States and vocational education and training institutions in Australia.

23ensure consistency with the wage data, some countries have no data on tertiary education
expenditure by general/vocational category. Thus, following Kruger and Kumar (2004b),
we use governmental expenditure on all secondary education as a proxy for Ss , andexpenditure on all tertiary education as a proxy for Sg .15 Then, the relative subsidy forAustria, for example, is calculated as Se = 14895/13607 ≃ 1.09.
Education at a Glance (2014) also provides data on the educational attainment of thelabor force for OECD countries. It provides information on the share of the population
aged 25–64 having completed general or vocational education, at both the secondary
and the tertiary levels. However, in addition to tertiary type B graduates, we regard all
upper secondary graduates as S-skill workers. Then, for example, 60.0% of populationin Austria are regarded as having speciﬁc skills, and 12.7% have general skills. Thus,
we obtain the number of G-skill workers as g ∗ = 12.7/72.7 ≃ 0.17 for Austria, and g ∗ ofthe other countries are determined in the same way.
Finally, a proxy for the inverse measure of the distance to the frontier, a∗ , is deﬁned
as the total factor productivity (TFP) of each country divided by the TFP of the UnitedStates. We use the Penn World Table (PWT) 9.0 data set, which gives the TFP levels
in 2014 for several countries at current purchasing power parity relative to the Unitede a∗ ) for 12 countries are summarized in Table 1.
States. The values of (g ∗ , wp, S,

5.2

Predetermined Parameters

Next, we choose the values of the technological parameters, (λ, ϕ, α, π). The majority ofstudies employing the distance-to-frontier model take the United States as the frontiercountry and, thus, we regard λ as the US TFP growth rate. Let us suppose that one
model period corresponds to 15 years. The Penn World Table 9.0 data set shows thatthe average US TFP growth rate (at constant national prices, 2011 = 1) is 0.763% peryear for the period 2000–2014. Thus, the growth rate of the frontier economy is set to
λ = (1 + 0.00763)15 − 1 ≃ 0.1208.
The share parameter α in the production function is set to 0.7. Because α represents
the labor share of the ﬁnal product (the sum of workers’ income and entrepreneurs’
proﬁts) and, thus, a choice of α = 0.7 seems reasonable, as is commonly used. In the15

Seven countries, Austria, France, Germany Hungary, the Netherlands, Spain, and Switzerland, havedata on annual expenditure per student for both Tertiary type A (mainly university) and type B (nonuniversity); the other ﬁve countries do not.

24

Table 2: Parameters assumed to be common across countries
Parameter

λ

α

ϕ

π

value

0.12

0.70

0.64

0.78

latter part of this section, we examine the sensitivity of our results to diﬀerent values ofα.
The values of the spillover parameter ϕ and the share of type-H ﬁrms π are determined simultaneously from (2). We see from the expression that, in the frontier economy,
the TFP of the type-L sector, ĀL,t , grows at rate ϕ; that is,
ĀL,t
= ϕ.
Āt−1

(28)

On the other hand, the growth of the frontier economy is driven by innovation activitiesof the type-H sector in the frontier economy. Hence, we have
ĀH,t
Āt
=
= 1 + λ.
Āt−1
Āt−1

(29)

For information on the US industries’ TFP performance, we use the Bureau of Labor
Statistics data set (US Department of Labor), which provides US output per employee
data for 157 four-digit manufacturing and non-manufacturing industries. We calculate
the average TFP growth rate for the period 2000–2014 for each sector, and then rankthem based on the growth rate. Here, we regard the top π ratio of industries as the typeH sector, and deﬁne the lowest value as ĀH,t /Āt−1 . Similarly, we deﬁne the lowest TFP
growth rate among the bottom 1 − π industries as ĀL,t /Āt−1 . To ensure consistency
with (29), the lowest value of the top π ratio of industry groups must be consistent
with 1 + λ. The closest value is obtained by setting the top 122 industries as type-H.
Then, we have AH,t /At−1 = 1.1222 (the actual value of 1 + λ is 1.1208). Thus, we set
π = 122/157. Then, the lowest TFP growth rate of the bottom 35 industries is obtained
as 0.6384, which is used as the value of ϕ, according to (28).

25

Table 3: Estimation results
AUT

BEL

DNK

FRA

DEU

HUN

NLD

SVL

SVN

ESP

CHE

GBR

γ

1.24

1.18

1.07

1.29

1.20

1.63

1.06

1.35

1.29

1.45

1.13

1.25

ν

0.86

0.84

0.74

0.73

0.63

0.41

1.15

0.53

0.75

0.85

0.87

0.89

µ

3.06

3.05

3.06

3.05

3.10

2.65

3.06

2.90

2.84

2.84

3.08

2.88

wp − γ
wp

4.3%

3.7%

3.1%

4.2%

4.6%

0.5%

2.6%

2.8%

2.5%

1.0%

3.9%

1.0%

5.3

Calibration

Now, we calibrate the set of parameters (γ, ν, µ) by matching the above key statistics
from the data to the steady-state characteristic of the model in section 4. The estimatesof ν, γ, and µ are derived from (20), (24), and (27), respectively. The estimation results
are listed in Table 3.
As expected, the estimated values of γ are strictly greater than one for all countries.
They range from 1.06 for Netherlands to 1.63 for Hungary. That is, the estimatesshow that general education graduates are more productive than vocational education
graduates. The superiority of general skills is well discussed in the literature on theargument that speciﬁc skills become obsolete over time.16 However, the estimates of γ inTable 3 may be overestimated, because we set wp using the wage diﬀerentials of tertiary
graduates, which will higher than that of upper-secondary or secondary graduates.
Importantly, our estimates satisfy γ < wp; that is, the wage gap between G- andS-skill workers is greater than their productivity gap. In other words, a positive transferability premium on the wage of general skills is observed for all countries. The ﬁnal
row of Table 3 gives the transferability premium’s share of the wage diﬀerentials, which
is highest in Germany (4.6%) and lowest in Hungary (0.5%). Further, as shown in Figure 3, we ﬁnd a positive relation between (wp − γ)/wp and a∗ . This observation leads
to the hypothesis that in countries closer to the frontier, the supply of general skills is
insuﬃcient to meet ﬁrms’ needs for a transferable labor force. We test the hypothesisby examining the impact of an increase in the supply of general skills on GDP in the16
For instance, Hanushek et al. (2017) show using the German Microcensus data set that generaleducation individuals earn less at a younger age, but at an older age, earn more than vocational-educationindividuals. Although we use earning data on employed individuals aged 25–64, our data show that thegeneral/speciﬁc wage diﬀerential is lower in cohorts aged 25–34, but is higher in cohorts aged 55–64.
Thus, the skill obsolescence of vocational-education individuals is also observed in our data.

26

Figure 3: Distance to frontier and transferability premium

next subsection.

5.4

Impacts of an Education Subsidy on GDP

Now, we examine impact of an educational policy on the economy. Our primary interest
here is to examine how an increase in transferable skills contributes to GDP. Note,however, that the above estimates of γ > 1 mean that G-skill workers have a higher
productive skill than that of S-skill workers. That is, an increase in the number ofG-skill workers increases the GDP, not only by improving transferability, but also by
reinforcing the total labor input in the economy. Here, we focus on the former eﬀect and,hence, we use the output per eﬀective unit of labor Yt ≡ Yt /(s∗ + γg ∗ ) as an objective
measure in our analysis of the impact of a general education subsidy. Rewriting thedeﬁnition of ﬁnal good production in (1), together with s∗ = 1 − g ∗ , yields,
Yt =

1−α α
α
1 1−α 1−α πa1−α
H yH + (1 − π)aL yL
M
Āt
.
α
1 + (γ − 1)g ∗(30)

Four steady-state variables in (30), g ∗ , aH , yH , and yL , are aﬀected by the increase ine the former three variables increase in S,e while yL decreases in S.
eS:e The results are
We calculate the changes in g ∗ and Yt caused by a 1% increase in S.
summarized in Table 4. As shown theoretically in Proposition 3, the results in Table
4 show positive impacts of Se on g ∗ . Furthermore, we ﬁnd that the change rate of g ∗is negatively related to g ∗ in Table 1. The result in Table 4 also shows that a 1%
increase in Se increases Yt in all countries in our sample. Surprisingly, we cannot ﬁnd
27

Table 4: Change rates of g ∗ and Yt by 1% increase in Se
AUT

BEL

DNK

FRA

DEU

HUN

Change rate of g ∗

4.59%

2.67%

1.60%

2.87%

3.60%

2.55%

Change rate of Yt

0.68%

0.62%

0.52%

0.85%

0.80%

0.34%

NLD

SVL

SVN

ESP

CHE

GBR

Change rate of g ∗

1.19%

4.05%

4.46%

1.35%

2.09%

1.43%

Change rate of Yt

0.49%

0.50%

0.47%

0.43%

0.68%

0.37%

Figure 4: Transferability premium and change rate of Yt

clear relations between the values of g ∗ or the change rate of g ∗ and the change rate of
Yt . Hungary, Slovakia, and Slovenia have a relatively small share of general education
graduates. Thus, a marginal increase in a general education subsidy increases the general
education share in these countries signiﬁcantly. However, despite the higher change rates
of g ∗ , the general education subsidy has less impact on Yt in these countries. On theother hand, as shown in Figure 4, there is a strong and positive relation between thechange rate of Yt and the transferability premium ((wp − γ)/wp). These observations
indicate that a key determinant of the eﬀect of the general education subsidy on Yt is
not the absolute size of the general education share, but rather the size of ﬁrms’ relative
demand for general skills.
In addition, Figure 5 shows that change rate of Yt is higher in countries that are closer
to the frontier (i.e., countries with higher a∗ ). The reasoning behind this observation is28

Figure 5: Distance to frontier and change rate of Yt

given in Figure 3. The ﬁgure suggests the possibility of a short supply of general skills
in relatively high TFP countries, such as Germany, France, and Switzerland. In fact, a
relative increase in the general education subsidy has a signiﬁcant impact on the change
of Yt in these countries. Conversely, in relatively low TFP country, such as Hungary, thedemand for general skills is not as large and, hence, the subsidy for general education
has less impact on Yt .

5.5

Sensitivity Analysis

In section 5.2, we set α = 0.7 as a benchmark by considering that α represents the labor
share of the ﬁnal product. The rationale behind this assumption is that the revenue ofeach intermediate ﬁrm is divided between workers and the entrepreneurs and, thus, αis equal to the labor share of total output. However, the ﬁnal good production function
is deﬁned as the relation between the quantity of inputs of intermediate goods, yi , theﬁxed factor, M , and the quantity of outputs, Yt . Thus, the value of the share parameter
α will vary according to what M represents (however, note that the results in Table 4
are not aﬀected by the value of M ). Rather than deﬁning M , we check the sensitivityof the results to other values of α.
In Table 5 and Table 6, we report the results of the sensitivity analysis for α = 0.6
and α = 0.8, respectively. The tables show that our argument, thus far, does not dependon the value of α. That is, even at α = 0.6 and α = 0.8, we still observe positive impacts
of a general education subsidy on GDP for all countries, and ﬁnd especially large values

29for countries with a higher transferability premium and that are closer to the frontier.

Table 5: Results for α = 0.6
AUT

BEL

DNK

FRA

DEU

HUN

wp − γ
wp

5.51%

4.87%

4.06%

5.42%

5.96%

0.66%

Change rate of g ∗

4.68%

2.76%

1.66%

3.86%

4.14%

2.49%

Change rate of Yt

0.84%

0.85%

0.76%

3.01%

1.75%

0.29%

NLD

SVL

SVN

ESP

CHE

GBR

wp − γ
wp

3.42%

3.64%

3.27%

1.29%

5.07%

1.31%

Change rate of g ∗

1.29%

4.00%

4.39%

1.38%

2.60%

1.42%

Change rate of Yt

0.99%

0.49%

0.43%

0.57%

2.14%

0.38%

Table 6: Results for α = 0.8
AUT

BEL

DNK

FRA

DEU

HUN

wp − γ
wp

2.92%

2.55%

2.09%

2.86%

3.18%

0.32%

Change rate of g ∗

4.59%

2.66%

1.61%

2.79%

3.54%

2.60%

Change rate of Yt

0.63%

0.56%

0.47%

0.63%

0.65%

0.38%

NLD

SVL

SVN

ESP

CHE

GBR

wp − γ
wp

1.75%

1.88%

1.68%

0.64%

2.66%

0.65%

Change rate of g ∗

1.19%

4.10%

4.54%

1.36%

2.06%

1.45%

Change rate of Yt

0.42%

0.53%

0.52%

0.39%

0.55%

0.38%

30

6

Conclusion

In this paper, we have proposed a growth model where the composition of general andspeciﬁc skills changes during a country’s catching up process to the world technology
frontier. The key element underlying the growing importance of general skills in the development process is the increasing uncertainty about private sector productivity. Given
the assumption that ﬁrms’ innovation activities are riskier than copying and adopting
existing technology, more developed countries, which have to put greater eﬀort into innovation, face a higher productivity risk. This results in ﬁrms’ precautionary demand fortransferable workers who are adaptable to the unexpected sectoral changes. Thus, theproposed model predicts that ﬁrms place more importance on transferability than they
do on productivity for labor skills, and that their demand for general skills increases as
the country approaches the world technology frontier.
However, contrary to the models’ prediction, many of most developed and industrialized countries in Europe have a small share of general education. A number of studies
have examined the European educational composition and, in some cases, ﬁnd a need foran expansion of general education. To the best of our knowledge, this study serves ﬁrst
attempt to examine the relationship between the educational composition and growth
performance of the economy by focusing on the productivity–transferability trade-oﬀ
between general and speciﬁc types of education. Our numerical analysis shows that amarginal increase in general education expenditure increases the steady-state GDP for
all 12 European countries in our data. Interestingly, the cross-country comparison does
not show a clear relationship between countries’ initial share of the general education
population and the impact of general education expenditure on GDP. On the other
hand, the results show that the demand relative to the supply of general skills is larger
in higher TFP countries, and that the impact of general education expenditure is positively related to countries’ TFP levels. These ﬁndings suggest that the distance to theworld technology frontier is relevant to the debate on secondary and tertiary schoolingstructures. The results also indicate the possibility of an over-specialization in skills incountries at higher stages of development.
The arguments on cross-country comparisons in this study can be extended to crossindustry comparisons. Although the present model ignores sectoral diﬀerences in the31

distance to the frontier, the cross-industry variance of sectoral TFP levels within acountry might be not small. Thus, there may exist large diﬀerences in the demand forgeneral skills between industries, in which case, general skills will be more required inwell-developed and/or highly innovative industries.

Appendix
Appendix A: Proof of gL,t = 0
In this appendix, we prove that gL,t = 0 holds for any equilibrium. Conversely, we
assume that gH,t > 0 and gL,t > 0. Then, ﬁrms’ demand for general skilled workers is
obtained as follows:
]
[
( ) 1
1
γα 1−αgH,t =
aH,t ,
−st +
γ
wt
[
]
( ) 1
1
γα 1−αgL,t =−st +aL,t .
γ
wt
Using these expressions, the ﬁrms’ problem in the ﬁrst stage can be expressed as follows:
[ {
max
st

π

(
(1 − α)

γα
wt

) α

1−α

}
1
1 2
aH,t + wt st − xH,t
γ
2
]
}
{
( ) α
γα 1−α1+ (1 − π) (1 − α)aL,t + wt st − wt st .
wtγ

The maximization problem is linear in st and, hence, the non-negative constraint gL,t ≥ 0
has to be binding. Thus, we have gL,t = 0.

Appendix B: Proofs of Lemma 1 and Proposition 1
From (16), we have that Gt /St > 0 requires that
(
e
at >

1−π
γ−π

) 1

1−α

.

(31)

Combining (31) and (18), we have that at−1 must satisfy the following inequality:
(

1−π
γ−π

) α ((
1−α

1−π
γ−π

) 1

1−α

32

)
−1

<

(1 − α)µ2 a2t−1.
(1 + λ)1−α ϕ1+αSolving the above inequality with respect to at−1 yields[(1 + λ)1−α ϕ1+αat−1 >(1 − α)µ2

(

1−π
γ−π

) α {(
1−α

1−π
γ−π

}] 12

) 1

1−α

−1

≡b
a.

Further, in order to guarantee the existence of general skills, b
a < 1 must hold. Rearranging the condition yields[(1 + λ)1−α ϕ1+α
µ>
1−α

(

1−π
γ−π

) α {(
1−α

1−π
γ−π

}] 21

) 1

1−α

−1

≡µ
b.

Appendix C: Proof of Proposition 2
From (19), we have that the dynamics of at satisfy dat /dat−1 > 0, d2 at /da2t−1 > 0, andϕthat at = 1+λ> 0 when at−1 = 0. That is, at is increasing and convex in at−1 , and has

a positive intercept. Thus, the dynamics of at have at most two steady-state points. As
shown in the ﬁgure below, if two steady-state points exist, the one with a lower value
is stable and the larger one is unstable. A condition for the existence of a unique andstable steady state in the interval (0, 1] is that at−1 ≥ 1 at at = 1. Substituting thiscondition into (19), and rearranging, yields[(1 + λ)(1 + λ − ϕ)
µ≤
1−α

{

π ϕ(γ − π)+γγ(1 + λ)

33

(

1−π
γ−π

) 1 }α ] 12
1−α

≡ µ̄.

Finally, because µ must satisfy µ ∈ (bµ, µ̄), we have to check whether µ
b < µ̄ holds.
Substituting µb and µ̄ into µ
b < µ̄, and rewriting, yields
{(
ϕ1+α

1−π
γ−π

}

) 1

1−α

−1

{ (
}α
) 1
1−α
π
γ
−
π
ϕ(γ
−
π)
< (1 + λ)α (1 + λ − ϕ)
+
.
γ 1−π
γ(1 + λ)The left-hand side of the above inequality decreases in γ and is equal to 0 when γ → 1.
On the other hand, the right-hand side increases in γ and is equal to 0 when γ → π.
That is, γ has to be large enough to ensure that µ
b < µ̄. From the above discussion,
there exists γb ∈ (π, 1), such that µ
b < µ̄ holds for γ ∈ (bγ , 1).

Appendix DFirst, we return to the second-stage problem of ﬁrms, and resolve gH,t and xi,t . The
second-stage problem of type-H ﬁrms is as follows:(max

gH,t ,xi,t

1
1+λ

)1−α

(

ϕ + µat−1 xi,t)1−α1(st + γgH,t )α − wtg gH,t − x2i,t .
2

The ﬁrst-order conditions with respect to gH,t and xH,t are, respectively,
[
]
( ) 1
1
γα 1−αgH,t =
−st +
aH,t
γ
wtg

(32)

and
xi,t =(1 − α)µ(st + γgH,t )α a−αH,t at−1 .
1+λUsing the above two ﬁrst-order conditions, we obtain(1 − α)µxi,t =
1+λ

(

γα
wtg

) α

1−α

at−1 .

Then, we have the technology level of type-H ﬁrms as
[
]
( ) α
1
(1 − α)µ2 γα 1−α 2aH,t =
ϕ+at−1 .
1+λ
1+λ
wtg

34

(33)

Next, the ﬁrst-stage problem of ﬁrms can be written as
{
max π
st

(
(1 − α)

γα
wtg

) α

1−α

1
1
aH,t + wtg st − x2H,t
γ
2

}
α
s
+ (1 − π)a1−αL,t st − wt st .

Note that πwtg /γ < wts is required for the existence of an interior solution, that is, forthe existence of G-skill workers. Combining this condition with (21), we have

γ<

wtg
γ
< .
s
wt
π

(34)

We focus on the equilibrium that satisﬁes the condition given in (34), in which both
G-skill and S-skill workers exist. Section 5 showed that all estimates obtained from ourcalibration satisfy (34). Finally, the ﬁrst-order condition of this problem yields(
st =(1 − π)γαγwts − πwtg

) 1

1−α

aL,t .

(35)

Rearranging (32) and (35) by evaluating at the steady state and substituting aL,t =ϕ/(1 + λ) yields (22) and (23).

Appendix E: Proof of Proposition 3
In this appendix, we examine the steady-state characteristics using the system given ine Let(25) and (27). We ﬁrst show that the steady-state values, g ∗ and a∗ , increase in S.
e a∗ ) the solution of (25) with respect to g ∗ , as a function of a∗ and S.
e
us denote as g(S,e a∗ ) increases in both Se and a∗ . Substituting g ∗ = g(S,e a∗ )
From (25), we have that g(S,into (27) yields(1 − α)µ2(1 + λ)a − ϕ =
1+λ
∗

(
)
γ − π e ∗ α ∗ 2−α1+g(S, a ) (a ).
πHere, a∗ is determined from the above expression. The left-hand side (LHS) of the above
expression is increasing and is a linear function of a∗ with a negative intercept. At thesame time, the right-hand side (RHS) increases in a∗ , and passes through the origin.
However, we cannot examine the number of intersections between the LHS and RHS
curves because very little is known about the shape of the RHS function. However, wehave that if the LHS and the RHS curves have one or more intersections, the value of35

e This means that the smallest steady-state point
the smallest steady-state increases in S.
e a∗ ) increases in S.
e The
of a increases in Se and, thus, it becomes clear that g ∗ = g(S,steady-state point is described in the ﬁgure below.

Next, we investigate the stability of the steady state corresponding to the smallest
solution of a∗ . Let us denote as gt the total number of G-skill workers in equilibrium,
but not in the steady state. Then, substituting πgH,t = gt , st = 1 − gt , and aH,t = atinto (32) and (33) yields

and

[
]
( ) 1
π
γα 1−αgt =−(1 − gt ) +
at ,
γ
wtg

(32′ )[]( ) α
1(1 − α)µ2 γα 1−α 2at =ϕ+at−1 .
1+λ1+λwtg(33′ )

1[at−1 + ϕ(1 − at−1 )], (35)Moreover, using (20) together with st = 1 − gt and aL,t = 1+λcan be rewritten as((1 − gt ) =

) 1 (
1−α

1−π

e − gt ) − πγν S(1

γα
wtg

) 1

1−α

ϕ
.
1+λ

(35′ )The three endogenous variables, gt , at , and wtg , are determined by (32′ ), (33′ ), and (35′ ).
These equations show dat /dat−1 > 0 for at−1 ∈ [0, 1], and that at > 0 when at−1 = 0.17
By eliminating at and wtg from (32′ ), (33′ ), and (35′ ), we can see the positive relation between gt
and at−1 . From this observation, we have from (33′ ) and (35′ ) that at increases in at−1 .
1736

Therefore, the steady state determined by the ﬁrst intersection of the dynamic equationof at and the 45-degree line is stable, indicating that the dynamics are much the same as
those described in section 3 and in the ﬁgure in Appendix C. In the subsequent discussion,
we continued the analysis by supposing that economies are on this stable steady state,although we did not prove its uniqueness. Instead of deriving the parameter conditions
for the existence and uniqueness of the stable steady state, we conﬁrmed them in thequantitative analysis in section 5.

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