LEVY-GRANT-SCHEMES IN VOCATIONAL EDUCATION
STEFAN BORNEMANN*
Munich Graduate School of Economics
Abstract:
The German apprenticeship system is often considered a role model for vocational education. Its
influence on economic growth and technological progress through the provision of human capital to
the workforce is widely acknowledged. But recent declines in the number of apprenticeships has
lead to increasing unrest among policy makers. To counter this development, government is
considering to introduce a system that subsidizes apprenticeship training and collects training levies
from non-training firms. Such "levy-grant-schemes" already exist in several industrialized countries
and even in some sectors in Germany. Yet, economists differ greatly in opinion to this policy. More
surprisingly, however, a general economic analysis to this policy instrument is still lacking, as
recent contributions have relied on partial analyses only. This paper aims to close this gap.
Following the training literature, we use a simple oligopsonistic labor market model. Such a setting
allows to explain why firms provide and (at least partially) finance general vocational training. It
can also be shown, however, that a positive externality arises as other firms benefit from vocational
training through poaching. In principle the Pigouvian prescription of a subsidy scheme financed
from a non-distortionary tax could restore the social optimum. A levy-grant-scheme, by contrast, is
a particular scheme where subsidies and levies are interlinked. This paper unveils that it is basically
an uniform subsidy on apprenticeship training that is financed by a distortionary tax on labor. We
show that introducing such a levy-grant-scheme can be ambiguous to welfare. The paper compares
the proposed scheme with existing institutional settings and provides some proposals on the design
of such schemes if their introduction is indeed intended.
Keywords: Vocational Training, Poaching, Levy-Grant-Scheme
JEL classification: H23, I22, I28, J24
1 Introduction
The German apprenticeship system is often considered a role model for vocational education. It
provides basic vocational training to a large share of the workforce. 1 Training is well-structured and
leads to a certified qualification that is widely recognized among employers. Moreover, the system
is renowned for its dual nature of simultaneous schooling and training on-the-job. Several studies
have analyzed the German apprenticeship system and pointed to its role for the provision of human
capital investment to the workforce and thereby on economic growth and technological progress.2
For several years, however, the number of registered apprenticeships has been declining and youth
unemployment has risen simultaneously.3 With general unemployment already high, this
development is increasingly raising concerns and provoking doubts about the apprenticeship system
*
Version of 08/06/2005. I would like to thank Christian Holzner and Gerrit Roth for many helpful comments and
suggestions. Remaining errors are, of course, my own.
1
Roughly two thirds of the age-group from 15 to 24 attend vocational education through the apprenticeship system
(BMBF, 2004, 78).
2
See for instance Franz/Soskice (1995) or Harhoff/Kane (1997) and the references provided there.
3
The number of registered apprenticeships has declined from 1,629,312 in 1991 to 1,581,629 in 2003. Since the peak in
2000 at 1,702,017 this is a decline of 7%. Similarly, the number of new apprenticeship contracts decreased in the same
period from 571,206 to 564,493 peaking in 1999 with 635,559 (BMBF, 2004, 9).
1
in general. It is questioned whether this decline is of cyclical and thus only transitional nature or, to
the contrary, resulting from insufficient returns to training due to poaching of trained apprentices by
other firms. The fact that the proportion of firms that actually provide training is small in absolute
terms, is taken as an indication for the latter.4
As a remedy, government is considering to introduce a system that subsidizes apprenticeship
training and collects training levies from non-training firms. Similar “levy-grant-schemes” already
exist in several industrialized countries for vocational training and sometimes also for continuous
training. In Germany, such a transfer system has been introduced through collective agreements in
the construction industry, but attempts for legislation on the federal level have failed so far.
Economists differ greatly in opinion to this policy proposal. Given the importance of the question, it
seems surprising that a theoretical analysis of this policy instrument is still lacking. Some agree
with the idea that it would internalize positive training spillovers and thereby provide additional
incentives for training. Some accept that due to poaching, private and social returns from training
diverge, but consider the instrument to be overly bureaucratic and not conforming to the
institutional order of a market economy. Yet others question the existence of an externality from
training altogether and therefore see little need for this economic policy.5
This paper aims at closing this gap. We will ask at first whether there is indeed a positive
externality from apprenticeship training caused by poaching. We will then investigate whether the
introduction of a levy-grant-system for vocational education would be a welfare-improving policy.
Following the recent training literature, we study a simple oligopsonistic labor market model. Such
a setting allows to explain why firms provide and also (at least partially) finance general vocational
training. At the same time, however, a positive externality from training can be shown to exist as
other firms benefit from poaching. It will be shown that in principle the Pigouvian prescription of
an ideal subsidy scheme financed by a non-distortionary tax would restore the social optimum. A
levy-grant-scheme, by contrast, is a particular scheme where subsidies and levies are interlinked.
This paper unveils that it is equivalent to an uniform subsidy on apprenticeship training that is
financed by a distortionary tax on labor. We show that introducing such a levy-grant-scheme can be
ambiguous to welfare.
The paper proceeds as follows: The next section reviews the state of the literature on vocational
training and positive spillovers. In the third section we will introduce a simple model of
4
In 2002, only 31.3% of all firms were actually providing apprenticeship training (BMBF, 2004, 102).
For an overview of the policy debate see, for example, the differing contributions in the special issue of ifo
Schnelldienst, Vol. 57, No. 6. See Franz (1983) for a study on an earlier policy proposal for a levy-grant-scheme for
vocational training in Germany. His analysis is however only partial and does not address the question of an externality.
5
2
apprenticeship training conforming to present research. In the fourth section this structure will be
put to study incentives schemes for training. At the end the results are summarized and conclusions
for vocational training policy are drawn.
2 Vocational Education and Positive Spillovers
The question of positive spillovers in vocational education has attracted the interest of economists
long ago. Early contributions hinted at the problem of social returns exceeding private returns to
training and proposed policy measures to improve welfare. Later contributions, quite to the
contrary, questioned the existence of such positive spillovers and offered completely different
policy conclusions. A short review of the state of the literature on vocational education and positive
spillovers is therefore mandated.
The possible existence of educational spillovers was noted already by classical economics. Adam
Smith points out that returns to education accruing to society in addition to private returns justify
some role for partial public contributions to education. While Smith's analysis was concerned with
education in general, later writers dealt more specifically with on-the-job training. Pigou (1912) is
acknowledged to have first identified poaching as the source of the discrepancy between private and
social returns to training. He emphasized that a training firm may be inhibited from obtaining the
full returns to its training efforts and argued that with workers quitting for another firm, the new
employer participates in the returns to training. Pigou observed that this positive spillover from
training is lost to the investing firm, but not to society as a whole. From a social point of view,
therefore, training will be undersupplied because private investment is only carried out to the level
where the additional private returns equal the additional costs. With the extra social returns not
taken into account by the training firm, welfare is short of its optimal level (Pigou, 1912, 153).
Pigou identified a role for the state to improve overall welfare by attributing the social returns of
some beneficial action to the investing party and thereby internalizing the externality. He designates
fiscal incentives or disincentives as a suitable policy instrument in order to attribute social benefits
or detriments to an individual decision (Pigou, 1912, 164). Modern economics therefore commonly
refers to this policy instrument as Pigouvian subsidy or Pigouvian tax. Economists widely accepted
Pigou's conjecture of a poaching externality in vocational education and, not surprisingly,
subsequent public policy towards vocational education supported subsidies for worker training. 6
However, Pigou's assessment of training, and the role of public policy therein, did not remain
unchallenged.
6
The British Training Act of 1972 is exemplary for revisions of training policy to economic knowledge. Aiming to
overcome market failure in training, it introduced levy-grant schemes for apprenticeship training at the sector level.
These schemes operated until 1982 but were then abandoned in favor of more market-oriented solutions. See in
particular Stevens (1999) for an account of vocational training policy in Britain.
3
With the emergence of human capital theory, the economics of training experienced a major
revision. This literature explicitly accounts for the analogy between human capital and physical or
financial capital. Commensurate to the general concept of capital, human capital is perceived as the
stock of knowledge, skills, health, or abilities that is embodied in a person and that can be put to
productive work. This work potential can be increased, alike investments in physical or financial
capital, by training, schooling or health provisions. In his seminal work, Becker (1962)
distinguishes general from specific human capital. While general human capital enhances the
productivity of a worker in all firms, specific human capital is attached to a firm and, therefore, is
beneficial only while working there. Irrespective of the type of human capital, however, the theory
negates the existence of a poaching externality as envisaged by Pigou.
An investment into general human capital raises the marginal product of labor in all firms. If labor
markets are competitive, a worker will receive a wage according to the marginal product of labor. A
worker receiving a wage below the marginal product could otherwise quit for another firm and
obtain the prevailing market wage. It is therefore not possible for a training firm to recover any
training costs because general training leads to a congruent wage increase. The returns from the
general human capital investment are reflected in the higher wage and accrue fully to the worker.
Receiving the full benefits, the worker has proper incentives to bear the investment costs. Firms
might provide general training, but workers will pay for it either directly or indirectly via wage
reductions in the investment period. Contrary to Pigou there is no spillover from general training on
other firms (Becker, 1962, 17).
An investment into specific human capital, by contrast, carries an increase in the marginal product
of labor only in the respective firm. Because these skills are of no use to other firms, the
competitive market wage remains unaffected. All returns from this investment accrue to the firm
through the increase in worker productivity. But since this investment is rewarding only as long as
the worker stays with the firm, some of the returns from this investment will be shared between firm
and worker in order to prevent workers from changing firms. As all the returns are collected by the
private parties there is consequently no spillover on other firms by this type of investment as well
(Becker, 1962, 21).
In summary, standard human capital theory negates positive spillovers from vocational training. As
private and social returns do not diverge, human capital investment will be at its socially optimal
level. General training is paid for by the worker, specific training by both employer and worker.
Becker acknowledges that insufficient general training may indeed arise if workers are credit
constraint and/or risk averse (Becker, 1962, 41f.). In this case, however, training subsidies or public
training provision are not warranted. Educational policy should rather address the credit and
insurance market problems that cause training limitations.
4
Despite its theoretical appeal, standard human capital theory has severe shortcomings. Firstly, it is
at odds with the empirical evidence. Contrary to its central proposition, a large number of empirical
studies confirm the existence of firm-provided and firm-financed general training. In reality, firms
seem to incur substantial training costs to provide general skills to its workforce. Acemoglu and
Pischke (1998) survey empirical evidence for vocational and on-the-job training. Several studies
have documented this empirical fact also for apprenticeship training in Germany. Harhoff and Kane
(1997), display various cost estimates of training apprentices by sector and firm size. The gross
costs of apprenticeship training across sectors are estimated at $17,645 per apprentice and year with
on average higher costs in larger firms and industrial sectors. Even after taking the apprentice’s
productivity into account, net costs to firms remain high at $10,657 per apprentice and year or 60%
of the gross costs.7 Secondly, human capital theory is unable to explain some characteristic features
of apprenticeship training, such as the fixed contract duration or the restrictions on unilateral
termination. Under the assumption of competitive labor markets with full information and perfect
contracting, these particularities should be superfluous (Smits and Stromback, 2001, 32).
The contradictions between human capital theory and the empirical evidence have ignited new
interest in the economics of training and led to several publications investigating economic
rationales for firm-financed general training. The role of labor market conditions was already noted
by Becker. He regarded the dichotomy of general and specific skills a useful simplification and left
oligopsonistic labor markets unconsidered. Present contributions analyze training technology and
labor market conditions separately.8 In particular, these studies question the assumption of a perfect
labor market for general skills (Stevens, 1994a/b; Acemoglu and Pischke, 1998). Albeit their
differences, they share the common element of labor market imperfections that gives rise to a
compressed wage structure. In such a setting, firms can extract a rent from employing skilled
workers, as the market wage is below the marginal product of labor. General training increases the
marginal product of labor. Because of the compressed wage structure, however, wages increase to a
lesser degree. This turns technologically general skills into de facto firm-specific skills. Contrary to
standard human capital theory it is therefore profitable for a firm to finance general training. At the
optimum, a firm will provide a training level where the marginal increase in the rent equates the
marginal cost of training.
7
For detailed studies assessing the cost of apprenticeship training see in particular Bardeleben et al. (1995) or
Winkelmann (1997).
8
In effect, the notion of specific training is presently questioned. Lazear (2003) forwards that few skills are limited only
to a single firm and thereby doubts sizeable firm-specific skills to exist. Instead he argues that firms seek specific inputcombinations of general skills, stressing the importance of labor markets rather than skill-types.
5
The literature has identified several mechanisms causing a compressed wage structure. They relate
to production technology, market competition as well as informational and institutional conditions.
For instance, a compressed wage structure could result from technological complementarities. For
instance, capital and labor input may be complements. Such complementarities in the production
technology increase the marginal product of labor. The value of a trained worker is therefore higher
to the training firm than to outside firms (Acemoglu and Pischke, 1998, 559). Similarly,
complementarities may also arise in the training process when simultaneously instructing general
and firm-specific skills is complementary (Franz and Soskice, 1995, 219f.).
Competitive conditions on the output market could also account for wage compression.
Hentschirsch (1999) analyzes training investments when the product market is characterized by
Cournot or Bertrand competition. Similarly, Gersbach and Schmutzeler (2003) show that firms will
provide training if competition on the final market is sufficiently soft.
Another source of wage compression may be informational asymmetries. Katz and Ziderman
(1990), in a pioneering contribution, show that when training is not verifiable to outside firms,
workers will be unable to receive a wage reflecting their training level. The training firm has an
informational advantage regarding training with respect to other firms. Consequently, training will
be worthwhile, as productivity increases more than wage payments. Likewise, a training firm may
also possess superior knowledge about a worker's innate characteristics, i.e. abilities, talents and
personal qualities. They are difficult to assess by other firms so that outside wage offers will fall
short of a worker's marginal product (Acemoglu and Pischke, 1998, 556f.).
Other labor market frictions impeding workers to quit instantaneously for outside firms may result
from mobility restrictions and personal preferences. For instance, commuting or relocation costs
render job turnover costly to workers. Other frictions are in general related to job search, such as
search costs, search uncertainties and matching problems. Workers will abstain from perfect job
turnover in reflection of the costs associated with it. Again, the employing firm can appropriate a
rent and wages are compressed, rendering general training investments profitable.
Also labor market institutions have been identified to cause wage compression, e.g. minimum
wages, collective bargaining, dismissal protection and unions (Acemoglu and Pischke, 1998, 563f.;
Smits and Stromback, 2001, 56).
In conclusion, human capital theory with imperfect labor markets can explain why firms finance
some general training even though workers are mobile and able to quit. This literature specifies
labor market imperfections resulting into a compressed wage structure and making general training
profitable for firms. But will human capital investment be carried out to a socially optimal extent?
In effect, these models also point out that training in imperfect labor markets is necessarily
6
accompanied with an externality. Labor market frictions in fact cause workers not to react
instantaneously to wage differentials and thereby provide some training incentives. Frictions slow
turnover, however, they cannot inhibit quits entirely. Thus, positive spillovers arise because nontraining firms also benefit from training by employing workers trained in other firms. This is
commonly referred to as poaching.9 Due to an externality, however, training will not be provided to
an optimal extent (Stevens, 1994a).
In summary, labor market imperfections allow to explain firm-financed vocational education, but
they also assert the existence of an externality. This contrasts to conventional human capital theory
and therefore invites to reconsider appropriate public policy. Comprehensive studies analyzing
different policy instruments are lacking. The next section therefore introduces a simple model in
order to put different policy instruments into analysis.
3 Simple Model of Apprenticeship Training
This section presents a simple model of apprenticeship training. It allows to explain why firms
partially finance vocational training by considering an oligopsonistic labor market with frictions. At
the same time, an externality can be shown to exist.
3.1 Model Structure and Assumptions
Consider an economy where firms require skilled workers in order to produce goods and services.
Firms can obtain skilled workers either by training unskilled workers themselves or by recruiting
workers that have been trained by other firms through the labor market. To keep things simple, we
restrict ourselves to a two-period model with a present and a future period, t  1, 2 . In the present
period, firms compete for the currently existing skilled workforce. In addition, they may decide to
train unskilled workers that will become skilled workers in the future. We call this period the
training period. In the future period, firms compete anew for skilled workers. With training being
sufficiently general this allows for poaching of the newly trained workers by other firms. We
therefore refer to this period as the poaching period.
Firms compete for the services of skilled workers by posting wage offers. Workers arbitrage
between firms’ wage offers and choose to work for the firm with the higher wage. If labor markets
were perfect, workers would instantaneously move to the highest paying firm. In equilibrium, the
wage of a skilled worker would equal the marginal product. In such a situation, firms could not
obtain any rewards for general training expenses. They would therefore shift the costs of training
9
More precisely, poaching may be active or passive in nature, i.e. attempts to recruit skilled workers may be systematic
or resulting from general job turnover. However, we do not pursuit this distinction further.
7
onto workers or provide no general training at all. Essentially, this is the standard result of human
capital theory according to Becker (1962).
In reality, of course, labor markets are imperfect. Skilled individuals usually do not quit a firm
instantaneously although they could obtain a slightly higher wage elsewhere. The literature review
above has specified a number of labor market frictions that inhibit workers from perfect wage
arbitrage, such as search and switching costs, information asymmetries, and institutional conditions.
We abstain from modeling such frictions in full detail and use a reduced form instead. More
precisely, we simply represent such frictions by assuming that workers favor high wages but also
have some preferences over firms.10 We thereby bring about some attachment to firms even when
wage differences are present. Workers will consequently only quit for another firm if the wage
increase at least compensates for the lost attachment to the firm. As an example, firms could differ
in their geographical location and workers incur commuting costs from their residence to the
workplace. Therefore, when choosing among job offers, workers will trade off firms’ wage offers
against the costs to travel to work.
Formally, assume workers to differ in location. We represent this by the differentiation parameter
 . Let there be a mass of homogeneous skilled workers which are uniformly distributed along a
Hotelling-circle with length normalized to unity. In order to work for a firm, workers suffer
commuting costs that monotonously increase in the distance to work. For simplicity, let  be the
linear rate of commuting expenses and qi the location of the firm. Thus, the net wage of worker 
when working for firm i , wi , can be represented by (1).
(1)
wi (wi |  )  wi   qi 
Let there by n firms which are located evenly across the geographic space. Thus, the position of a
firm is given by qi  i n1 , i  1...n . Figure 1 depicts this situation for the case of 4 firms.
Figure 1: Hotelling Circle for
10
n4
Such preferences could be grounded, for instance, to the liking of particular sectors, products and services, working
conditions, workplaces, etc.
8
Firms post wage offers to attract skilled workers. The higher a firm’s wage offer, the greater will be
the share of skilled workers employed by the firm,  ti . Let firms’ production technology use skilled
labor as the only input and possess constant returns to skilled labor. We thereby assume skills being
technologically general.11 Equation (2) denotes the production function of firm i where vti is a
worker’s productivity and N ti is skilled labor employed by firm i in period t .
yti  vti Nti
(2)
Notice that two interpretations of labor productivity can be given. The parameter may indicate a
firm’s technological productivity. Alternatively, it may signify a firm’s competitiveness in the
output market.
In this simple setting, let labor supply be an increasing function of workers’ net wages, Nt ( wt ) .
With labor supply rising in work income,
N t ( wt )
wt
 0 , alternative choices of workers’ time outside of
the labor market are taken into account. Such alternatives could be, for instance, the value of home
production or unemployment benefits.
When firms offer apprenticeship training to unskilled workers they affect labor supply by enlarging
the skilled workforce that can potentially be recruited by firms. Skilled workers available for
employment in the future periods, Nt 1 , consist of the skilled workforce from the previous period
and the sum of newly trained apprentices.12
(3)
Nt 1 (.)  Nt (.)   Ati
i
Training apprentices brings about costs to firms. Training involves direct costs that are made up of
remuneration to trainers, material expenses and direct pay to apprentices. Moreover, there are
indirect costs that result from forgone production when staff or machines are used for training.
Apprenticeship training may also carry some direct returns to firms that must be taken into account
if apprentices are employed in production. However, empirical studies accounting for output
contributions emphasize that firms incur substantial training expenses.13 We therefore assume
positive net training costs. Let ci ( Ai ) denote the net training cost function of firm i where Ai is the
number of apprentices trained by the firm. We assume increasing marginal training costs in order to
11
Apprenticeships, at least in Germany, certainly are to be regarded as providing mainly general vocational education.
Contents and schedule are regulated and training leads to a widely accepted training certificate.
12
With this formulation we abstract from retirement and assume all apprentices to enter the labor market which is
clearly a simplification.
13
See supra for references.
9
display limitations to the capacity of training facilities and training staff. Equation (4) summarizes
these properties.
ci  Ai  with ci  0, ci  0, ci (0)  0
(4)
Also note that by presuming an explicit training cost function we tacitly assume that production and
training technology can be separated. This assumption is often made in the literature to simplify the
problem.14
The time structure of this two-period model can be summarized as follows:

Training period, t  1:
Firms make wage offers w1i and take Ai apprentices for training.
N1 (.) workers choose their employer and production takes place.

Poaching period, t  2 :
Firms make wage offers w2i
N 2 (.) workers choose their employer and production takes place.
3.2 Private Optimum
With this being the basic set-up let us now solve this two-stage game using backward induction:
Consider first the poaching period. Firms strive to attract skilled workers by making appropriate
wage offers w2i . Workers, on the other hand, aim for high wage income net of commuting costs.
They will choose to work for the firm yielding the highest net wage w2i . By consequence, the
workforce will be allocated depending on firms’ wage offers and workers’ residence location.
A worker will decide between the wage offers of his two neighboring firms. Let ˆi ,i 1 be the worker
in the interval  qi , qi 1  who is indifferent between working for firm i or firm i 1 in this period.
This worker fulfills the arbitrage condition (5) where net wages of both firms equal.
(5)



w2,i   ˆi ,i 1  qi  w2,i 1   qi 1  ˆi ,i 1

After simplifying and rearranging this allows to obtain the indifferent worker ˆi ,i 1 as a function of
wages w2,i and w2,i 1 .
ˆi ,i 1 
 qi 1  qi    w2,i 1  w2,i 
2
2
14
Clearly, this does not represent reality in small companies where older, experienced employees train apprentices and
thereby necessarily incur opportunity costs from foregone production. By contrast, large companies often have special
training departments and respective instructing staff so that this assumption would be in line with reality in large firms.
10
Workers in the interval ˆi 1,i ,ˆi ,i 1  will choose to work for firm i . The share of workers employed
by the firm in this period is therefore given by
(6)
ˆi ,i1
 2i   ˆ
i1,i
q  q  q  q  
f   d  ˆi ,i 1  ˆi 1,i  i 1 i i i 1 
2
2w2,i  w2,i 1  w2,i 1 
2
Recall that firms and workers are assumed to be evenly distributed across space. Consequently, the
distance between two firms is q  qi 1  qi 
 2i 
(7)
i 11 i 1 1

 . We may therefore write N i as
n
n
n
1 1

 2w2,i  w2,i1  w2,i1 
n 2
Firms post appropriate wage offers in order to maximize profits. On the one hand, wages are a
costly input that reduces profits. On the other hand, wages are a means to attract a larger fraction of
the skilled workforce and thereby increase profits. The firm’s optimization problem is given by
(8)
max  2i   v2i  w2i   2i N 2   v2i  w2i 
w2 i

1
n

 21  2w2i  w2,i 1  w2,i 1  N 2
and the first order condition is
1 1
1
 
2w2i  w2,i 1  w2,i 1    v2i  w2i   0

n 2

Proceeding likewise for all firms yields a system of n linear equations.
1
1
1
1
v2i  w2i  w2,i 1  w2,i 1 
 0 i  1...n
2
4
4
2n
For the purposes of this paper, consider only the case of identical firms.15 With homogeneous
productivity, v2i  v2 j  v2 , firms’ wage offers will be identical, w2i  w2 j  w2 . The prevailing
market wage in this period is thus
w2*  v2 
(9)
Proposition 1:

n
In equilibrium, firms’ wage offers fall below the marginal product of labor,
w2*  v2 , although workers do not differ in skills.
Frictions inhibit workers from perfect wage arbitrage. This lends some wage-setting power to firms
such that in equilibrium the wage is below the marginal product of labor.16 Consequently, firms
yield some employment rents and firms’ equilibrium profits in the symmetric case are then simply
15
The second order condition is fulfilled with 
 / w2i  0 .
2
16
In the case of heterogeneous productivity there will be no common uniform market wage. In equilibrium, wages
differ across firms with firm productivity. However, due to frictions, they nevertheless remain below productivity.
Essentially, the case of heterogeneous productivity replicates the results of Montgomery (1991).
11
 2i   v  w   2i N 2 
(10)

n2
N2
From comparative-static analysis follows that wages converge to the marginal product of labor with
a larger number of firms, lim w2*  v . Intuitively, the more firms strive for skilled workers, the more
n
competitive is the labor market, driving up workers’ wages to the marginal product of labor.
Moreover, the equilibrium wage declines in the commuting rate  . Higher frictions reduce worker
turnover. They increase the monopsony power of firms and yield larger profits to firms.
Now consider the training period. Firms will compete for a high share in present period’s skilled
workforce, N1 . Similar to (7), firm i ’s share of the skilled workforce depends on own wage and
other wages,  1i 
1 1

 2w1i  w1,i1  w1,i1  . Additionally, firms will decide whether to take
n 2
apprentices for training. Doing so, they will trade off the increase in future profits by training
apprentices against the training costs. The firm’s decision problem is therefore aiming to maximize
the sum of discounted future profits and present profits from employment reduced by the training
costs,  2i   1i  ci ( Ai ) .
(11)
 

arg max   2 N2 ()   v1  w1   1i () N1  ci ( Ai ) 
w1i , Ai
 n

The first order conditions are
(12)

1 1
1
:  
2w1i  w1,i 1  w1,i 1    v1i  w1i   0

w1i
n 2

(13)


:  2  ci( Ai* )  0
Ai
n
with second order conditions again fulfilled. Equation (12) states firm i ’s wage reaction function as
before. Consequently, for homogeneous firm productivity, the equilibrium wage for skilled labor in
the training period is given by
w1*  v1 

n
.
Equation (13) equates the firm’s return of an additional apprentice to the marginal cost wherefrom,
firm i ’s optimal training Ai*   i (  ,  , n) can be inferred.
Proposition 2:
Despite vocational training being general, firms rationally offer costly
apprenticeships. From 

n2
 0 and ci  0 follows Ai*  0 .
12
In contrast to conventional human capital theory claiming firms not to bear any cost of general
training this model allows to explain firms to provide and (at least partially) finance costly
apprenticeships. With labor market frictions slowing worker turnover, it is profitable for firms to
provide general training. The question is now, whether the privately optimal choice of training is
also socially optimal.
3.3 Social Optimum
The social return to apprenticeship training consists of the net wage a newly skilled worker receives
and the additional rents from employment to all firms,

*
i
 w* . The social welfare function can
thus be depicted by
W
      w ( )d  N   c ( A )
t 1
ti
t 1,2
t
t
i
i
Dissolving the integral and rearranging obtains (14). For derivation see appendix.


W    t 1 vt  4n Nt   ci ( Ai )
(14)
t 1,2
A benevolent social planner sets firms’ training levels such that the marginal social returns to
training are equal to the additional training costs.


arg max W   ci ( Ai ) 
Ai
i


Thus, socially optimal training is defined by the condition
  v  4n   ci( Aio )  0 i
(15)
Comparing condition (15) with the condition for the private optimum (13) allows the following
proposition.
Proposition 3:
With social returns exceeding private returns to training, the level of vocational
training is inefficient. Too little apprenticeships are provided, Aio  Ai* .
The reasoning for this result can also be illustrated as follows: The return from vocational training
for apprentices is the wage for skilled workers. However, due to frictions, the wage is below the
marginal product of labor.

Rw   w   v  n  4n

By contrast, frictions allow for some returns from training to firms. Because workers do not react to
small wage differentials, firms enjoy some monopsony power and can earn a rent from employing
skilled workers that can be used to finance training.
Rf  
13
1
nn
Private returns to vocational education that are collected by the parties in the training contract are
therefore

R priv  Rw  R f   v  nn1 n  4n

With social returns to vocational education given by

Rsoz   v  4n

it can be easily seen that there is a positive externality associated with vocational education.
X  Rsoz  R priv  
n 1 
n n
In conclusion, this section affirmed the existence of a poaching externality. Using a fairly simple
model it was shown that firms on a frictional labor market enjoy some monopsony power and can
thereby earn rents from employing skilled labor. This provides an incentive for general training. At
the same time, however, other firms competing for skilled workers will benefit from positive
spillovers of apprenticeship training. Thus, social returns exceed the private returns to training and
from a social point of view, too few apprenticeships are provided. In the following section,
emphasis will be put on the second question whether training policy possesses suitable instruments
to internalize the externality and thereby increase social welfare.
4 Incentive Schemes
This section investigates incentive schemes to internalize positive training spillovers. Although the
focus is on levy-grant-schemes for vocational education, an ideal Pigouvian subsidy will at first be
analyzed as a reference case.17
4.1 Pigouvian Subsidy Scheme
Consider an ideal Pigouvian subsidy scheme. In particular, this assumes a benevolent government
that can acquire all necessary information to determine the appropriate subsidy. Financial funds can
be obtained from the general budget using a lump-sum tax. Moreover, the mechanism itself is
costless to implement and operate. Such a perfect scheme grants firms with a subsidy per
apprentice, zi . By consequence, firms receive a payment equal to  i  zi Ai . The firm’s decision
problem in the training period is thus modified to:
(16)
arg max   2i   2i  ci ( Ai )  zi Ai 
w1i , Ai
This alters the optimality condition for training from (13) to (17).
(17)


n
2
 ci( Ai* )  zi  0
17
Other policy instruments could also address externalities from training. Another paper concentrates on the regulation
of training contracts.
14
With a positive subsidy rate the private training level will increase, Ai* / z  0 . By setting the
subsidy to the value of the positive training spillovers, z  
n 1 
, equation (17) can be set to
n n
coincide with the social condition.
Proposition 4:
A perfect Pigouvian subsidy scheme allows to restore efficiency. Private and
social incentives to training are aligned and first best training levels can be
achieved.
The intuition to this result is straightforward. By introducing a subsidy equal to the additional social
returns from apprenticeship training, incentives for training are provided and the social optimum
can be achieved. Comparative-static analysis shows that the optimal subsidy amount increases with
frictions. By contrast it decreases with the number of firms in the labor market.
Of course, the assumptions of an ideal Pigouvian subsidy scheme are highly unrealistic. Firstly,
there are costs to administer, monitor and verify the scheme. In particular, non-negligible costs are
associated with the collection and distribution of funds or with the gathering and processing of the
necessary information. Secondly, the instrument demands very precise information to calculate the
appropriate subsidy amounts, a requirement that faces severe difficulties in practice. In fact, the
interests of firms and government to disclose the information may diverge.18 With only limited or
aggregate information available, a uniform subsidy must be calculated that fits a sector or even the
whole economy. Thirdly, as lump-sum taxes are not feasible, financing the subsidy payments may
require a tax increase or the introduction of a new distortionary tax. Thus, practical problems will
require government to use an instrument that necessarily departs from the theoretical ideal.
4.2 Levy-Grant-Scheme
Levy-grant-schemes are often proposed to internalize the positive spillovers from training. This
policy proposal has recently received much attention in Germany. Moreover, it exists in several
countries such as Britain, Denmark, Australia or France. 19 This section now aims for a rigorous
formal analysis of this policy proposal.
The basic idea of levy-grant-schemes is to reward training firms while penalizing firms that train
insufficiently or not at all. If apprenticeship training is short of a certain standard, a firm has to pay
a training levy. By contrast, if training exceeds this standard, a firm receives additional training
18
In particular, an adverse selection problem may arise. Government faces asymmetric information with respect to
firms’ productivities. Firms have an interest to sex up productivity in order to obtain larger subsidies.
19
For an overview and international comparison on various policies towards financing training see in particular
Gasskov (1994). Greenhalgh (1999), Stevens (1999) and Bosch (2004) provide some account of levy-grant-schemes
throughout the world.
15
grants. To illustrate, consider the levy-grant-scheme that has been proposed for Germany.20 Let ti
again be the net payment in the training period to firm i resulting from the scheme. Moreover, let
 ti define the training quota of firm i , i.e. the firm’s ratio of apprentices to skilled employees.
 ti 
(18)
Ati
Nti
Let ̂ be the mandatory training quota specifying the desired training standard. If a firm exceeds
the mandatory level,  ti  ˆ , the firm receives a grant z per apprentice it trains in addition to the
mandatory training level. The training grant is thus
grant   Ati  ˆ Nti   z
By contrast, if firm’s training falls short of the mandatory level,  ti  ˆ , the firm has to pay a levy
p for every skilled worker it employs. However, the workforce equivalent to the actual number of
apprentices at the targeted training quota is deducted from the levy base. The penalty thus accounts
for differences in firm training and decreases for every apprentice the firm trains. Training levies
are thus calculated according to
1 

levies    N ti  Ati   p
ˆ 

Equation (19) summarizes the net payment resulting from the levy-grant-scheme to firm i .
(19)
 
1 
  Nti  Ati   p if  ti  ˆ
ˆ 
 ti   
  A  ˆ N   z if   ˆ
ti
ti
ti






Factoring out N ti , using (18) and rearranging then gives
p

 ti  ˆ   N ti if  ti  ˆ
 ti  
ˆ
  ti  ˆ   zN ti if  ti  ˆ






If the same financial incentive is set for an additional apprentice, regardless of whether the firm
exceeds or undershoots the training quota, then the penalty and subsidy must obey the relation
p  zˆ . The scheme thereby simplifies to
ti  ti  ˆ  zNti  zAti  zˆ Nti
(20)
The scheme can be split up in two parts, one depending on the number of apprentices, the other
depending on the number of employees, leading to the following conclusion:
Proposition 5:
A levy-grant-scheme subsidizes apprenticeship training through an additional
employment tax.
20
For the text of the proposed bill and the parliamentary proceedings see Deutscher Bundestag (2004).
16
Essentially, a levy-grant-scheme is a tax-subsidy-system. Setting   zˆ the payment term
simplifies to ti  zAti   Nti whereby it becomes evident that firms receive a subsidy z per
apprentice and pay a tax per employee  .21
A levy-grant-system is self-financing if the budget balances. This is the case of all levies raised
equal grants and administrative expenses. For expositional purposes, assume the scheme to work
without any cost. The budget constraint is thus given by
(21)

ti
  zNti   ti  ˆ    zAti   Nti  0
i
i
i
t
i
Now turn to firms’ training decision in the presence of the levy-grant-scheme. The optimization
problem in the training period is now given by arg max  1i   1i  ci ( Ai )  i  which reduces to
w1i , Ai
(22)
arg max  2i   v1i  w1i     1i N1  ci ( Ai )  zAi 
w1i , Ai
The first order conditions are


:  2  ci( Ai* )  z  0
Ai
n
(23)
(24)

:
w1i

1 1
1

2w1i  w1,i 1  w1,i 1    v1i  w1i     0

n 2

From condition (23), as under a Pigouvian subsidy scheme, firm i ’s optimal training
Ai*  i ( z ,  ,  , n) can be inferred which again increases in the subsidy. Condition (24) defines the
wage reaction function of firm i . Similar to above, by proceeding for all firms, we obtain a system
of n linear equations that differs to before only in the last term.
1
1
1
1 1
v1i  w1i  w1,i 1  w1,i 1 
   0 i  1...n
2
4
4
2n 2
We again consider the case of symmetric firms. With homogeneous productivity, v1i  v1 j  v1 ,
firms’ wage offers will be identical, w1i  w1 j  w1 . The prevailing market wage is thus
w1*  v1 
Proposition 6:

n

The employee tax to finance the training subsidies is fully shifted onto workers.
With the tax shifted onto workers, firms’ equilibrium profits remain unaffected at
21
Note the close similarity of the simplified payment equation (20) to incentives scheme for the private provision of
public goods that punishes or awards deviations from average (Falkinger, 1996).
17
 1i   v1  w   1i N1 

n2
N1
Effectively, a levy-grant-scheme subsidizes additional training through a tax on employment. It has
two immediate effects. Firstly, the training subsidy encourages additional apprenticeship training.
Secondly, the employment tax reduces the wage offers to the existing skilled workforce and thereby
discourages labor supply.
Let us turn to the decision problem of a benevolent government aiming to determine optimal
subsidy and tax rates. It must trade off the welfare increase from training against the welfare loss for
the present workforce under the constraint of a balanced budget. The government’s welfare problem
is thus
max W  W2 (.)  W1 (.)   i  ci ( Ai )  zAi 
z ,
s.t.
 zA (.)   N (.)  0
i
i
By solving a Lagrangian problem optimal values for z* and  * can be determined.22
We address the question whether the introduction of a levy-grant-system for vocational training
would be welfare-improving. Inserting the budget constraint, the welfare problem can be rewritten
as a function W  z, ( z)  . Take the total differential with respect to the subsidy rate,
dW W W 


, which is stated in (25).
dz
z
 z
(25)
A (.)
A (.) 
N (.) 
dW
  v  4n   i  v  4n  1
  ci  i
0
dz
z
 z
z 
Proposition 7:
The introduction of a levy-grant-scheme can be ambiguous to welfare.



If the total differential were positive for z  0 ,

dW
dz
z 0
 0 , the introduction of a levy-grant-scheme
would be welfare-improving. Equation (25) consists of three terms. The first term represents the
discounted welfare gain of training brought about by a subsidy. The second term is negative with
N1
 0

and
z
0

. It depicts present period’s welfare loss resulting from reduced labor supply. Lastly,
the third term states the marginal cost of training that is to be deducted from welfare when training
incentives are introduced. Without specific functional forms, however, the total differential cannot
be signed.
22
Of course, this assumes sufficient information on the functional forms to be obtainable.
18
5 Summary and Conclusion
The aim of this paper was twofold. Firstly, it sought to understand whether externalities arise from
firms providing costly apprenticeship training which would lead to socially inefficient training.
Secondly, it was asked whether incentive schemes, in particular levy-grant-schemes, would be a
welfare-improving policy.
Following recent developments in the training literature a simple frictional labor market model was
introduced. In this setting, firms can be shown to rationally provide costly apprenticeships although
training is mostly general and workers are subject to poaching by other firms. At the same time,
however, it can also be demonstrated that vocational training brings about positive spillovers on
other firms. With “poaching” therefore causing an externality, training will fall short of the socially
optimal level. This raises interest for public policies internalizing the externality and increasing
training incentives.
At first an ideal Pigouvian subsidy was analyzed as a reference case. In principle, this scheme could
restore the social optimum by providing the training firm with a subsidy equal to the value of the
positive spillover. Nonetheless, such an instrument is not feasible due to informational problems
and non-distortionary funding unavailable. The focus therefore turned to analyze levy-grantschemes which have recently been proposed for Germany and which are in fact already existing in
several industrialized countries. Such schemes could be identified to be a particular type of taxsubsidy-systems. Effectively, a tax is levied on employment and paid out per apprentice. The
analysis revealed further that the tax burden of this instrument will be fully shifted onto workers. A
levy-grant-scheme for vocational education can therefore be characterized as a “pay-as-you-go”financing of vocational education where present workers pay for the vocational education of future
workers.
Moreover, it could be pointed out that the introduction of a levy-grant-scheme can be ambiguous to
welfare. On the one hand, it provides additional incentives for apprenticeship training and thereby
increases welfare. On the other hand, it reduces wages of the skilled workforce. Thus, labor supply
will shrink and welfare be reduced. Ultimately, net welfare will depend on the size of both effects.
What can be concluded from this exercise for the policy debate on incentive schemes for vocational
training? Essentially, proponents and opponents of a levy-grant-scheme are somewhat right. Firms
offering costly apprenticeship training induce positive spillovers on other firms. Because not all
returns accrue to the training firm this may indeed lead to under-provision of training. There is thus
a case for training policy to internalize the externality and provide additional training incentives. A
levy-grant-scheme may nevertheless not be warranted as it brings along economic distortions that
may actually exceed the benefits from additional training.
19
From this analysis, what advise is to be given towards the proposal of introducing a levy-grantscheme for apprenticeship training in Germany? If such an incentive scheme is indeed intended,
less distortionary means of financing and subsidizing should be sought. In principle, funding from
general tax receipts would be desirable, but given the state of the public budget at present this seems
not a feasible choice. When additional sources of financing are therefore needed, a levy depending
on payroll, i.e. firm’s total wage bill, should be given preference over the present scheme which
refers only to the number of employees. Otherwise, firms may strive to substitute low-skilled
workers for fewer high-skilled workers bringing along an additional distortion from taxation.
Overall, however, the case for a levy-grant-scheme must be considered as rather weak. Apart from
its informational demands, high administrative costs are to be expected for government and firms as
well. Moreover, this scheme brings along significant distortions arising from uniform subsidization
as well as from raising funds. Allowing apprentices and training firms to sign reimbursement
clauses seems an attractive alternative to provide increased training incentives (Alewell and Richter,
2001, 162). In perfect labor markets, an agreement on reimbursing for training expenses essentially
equals a training loan. By contrast, an analysis of reimbursement clauses within frictional labor
markets with monopsony power is lacking at present. This is, however, the focus of another paper.
6 Appendix
Derivation of equation (14):
Recall that the net wage of a worker  at firm i is given by firm’s wage minus the commuting
costs, w* ( )  wi*   qi  . Firm i employs all workers in the interval ˆi*1,i ,ˆi*,i 1  . Commuting
qi

costs of all workers at firm i are thus given by the sum
ˆ*
i 1,i
 (qi   )d  
symmetric case, we can insert qi  i n1 , ˆi*1,i  22i n1  1n and ˆi*,i 1 
the integrals and obtain
ˆi*1,i
qi
2i 1
2n
 (  qi )d . In the
which then allows to dissolve


. The social costs of commuting in all n firms are therefore
.
2
4n
4n
Social welfare in period t consists of firms’ profits n

n
2
N t and net wages (wt*  4n ) Nt which is
simply Wt  (vt  4n ) Nt . Summing over both periods and including training costs

c (A )
i
i
then

yields W    t 1 vt  4n Nt   ci ( Ai ) which is (14).
t 1,2
Alternatively, the commuting costs can be calculated from Figure 2. In the symmetric case, the
distance for an indifferent worker to the firm is
1
2n
which is multiplied by the commuting rate  .
Total commuting costs are represented by the surface of both triangles.
20
Figure 2: Commuting Costs of Workers at firm i
7 References
1. Acemoglu, Daron & Pischke, Jörn-Steffen (1998). Why do firms train? Theory and evidence,
Quarterly Journal of Economics 113 (1): 79-119.
2. Alewell, Dorothea & Richter, Julia (2001). Die Ausbildungsabgabe als Instrument zur
Internalisierung externer Effekte, in: Weizsäcker, Robert K. v. (Ed.). Bildung und
Beschäftigung, Duncker & Humblot, Berlin, 140-177.
3. Bardeleben, Richard v., Beicht, Ursula & Fehér, Kálám (1995). Betriebliche Kosten und
Nutzen der Ausbildung: Repräsentative Ergebnisse aus Industrie, Handel und Handwerk,
Bertelsmann, Bielefeld.
4. Becker, Gary S. (1962). Investment in human capital. A theoretical analysis, Journal of
Political Economy 70 (5): 9-49.
5. Bosch, Gerhard (2004). Die Krise der Selbstverpflichtung, ifo Schnelldienst 57 (6): 16-20.
6. Bundesministerium für Bildung und Forschung (BMBF) (2004). Berufsbildungsbericht,
Bonn. [Report of the Federal Ministry of Education and Research on vocational education.]
7. Deutscher Bundestag (2004). Entwurf eines Gesetzes zur Sicherung und Förderung des
Fachkräftenachwuchses und der Berufsausbildungschancen der Jungen Generation
(Berufsausbildungssicherungsgesetz - BerASichG). BT-Drucksache, 15/2820.
[Parliamentary proceedings on the federal bill on training.]
8. Falkinger, Josef (1996). Efficient private provision of public goods by rewarding deviations
from average, Journal of Public Economics 62: 413-422.
9. Franz, Wolfgang (1983). Die Berufsausbildungsabgabe: Theoretische Analyse einer
"Schubladenabgabe", in Heinz König (ed.), Ausbildung und Arbeitsmarkt, Lang, Frankfurt am
Main, pp. 77-105.
10. Franz, Wolfgang & Soskice, David (1995). The German apprenticeship system, in Friedrich
Buttler, Wolgang Franz, Ronald Schettkat & David Soskice (eds.), Institutional Frameworks
and Labor Market Performance, Routledge, London, pp. 208-234.
21
11. Gasskov, Vladimir (1994). Alternative Schemes of Financing Training, International Labor
Office, Geneva.
12. Gersbach, Hans & Schmutzler, Armin (2003). A product market theory of worker training.
University of Zurich, Socioeconomic Institute, Working Paper, No. 0214.
13. Greenhalgh, Christine (1999). Adult vocational training policy in Britain and France, Oxford
Review of Economic Policy 15 (1): 97-113.
14. Harhoff, Dietmar & Kane, Thomas J. (1997). Is the German apprenticeship system a panacea
for the U.S. labor market?, Journal of Population Economics 10 (2): 171-196.
15. Hentschirsch, Eva (1999). Strategische Ausbildungsinvestitionen auf Oligopolmärkten mit
Spillovers, Metropolis, Marburg.
16. Katz, Eliakim & Ziderman, Adrian (1990). Investment in general training: The role of
information and labour mobility, Economic Journal 100 (403): 1147.1158.
17. Lazear, Edward P. (2003). Firm-specific human capital: A skill-weights approach, National
Bureau of Economic Research, WP 9679.
18. Montgomery, James (1991). Equilibrium wage dispersion and interindustry wage differentials,
Quarterly Journal of Economics 106 (1): 163-179.
19. Pigou, Arthur C. (1912). Wealth and Welfare, MacMillan, London.
20. Smits, Wendy & Stromback, Thorsten (2001). The Economics of the Apprenticeship System,
Edward Elgar, Cheltenham.
21. Stevens, Margaret (1994a). Labour contracts and efficiency in on-the-job training, Economic
Journal 104 (424): 408-419.
22. Stevens, Margaret (1994b). A theoretical model of on-the-job training with imperfect
competition, Oxford Economic Papers 46: 537-562.
23. Stevens, Margaret (1999). Human capital theory and UK vocational training policy, Oxford
Review of Economic Policy 15 (1): 16-32.
24. Winkelmann, Rainer (1997). How young workers get their training: A survey of Germany
versus the United States, Journal of Population Economics 10 (2): 159-170.
25. Wößmann, Ludger (2004). Entwicklung betrieblicher Kosten und Nutzen der
Berufsausbildung: Einige Anmerkungen zu den Ursachen des Ausbildungsplatzmangels, ifo
Schnelldienst 57 (6): 21-24.
22