GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.200

Efficiency of skill training for acquiring
Sector-specific skills with search frictions

Keisuke Kawata

May 2011

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, JapanE¢ ciency of skill training for acquiringsector-speci…c skills with search frictions
Keisuke Kawatay
Graduate School of Economics, Osaka University

Abstract
This paper develops a simple search model in which sector-speci…c trainings areendogenously determined with or without a negotiation between a worker and an
employer, and characterizes the allocation of two types of training. If a worker
and an employer can negotiate over the amount of skill training, the training hours
to acquire a skill speci…c to this employer’s sector may be longer or shorter inthe decentralized allocation than in the socially e¢ cient allocation. Meanwhile, ifthey cannot negotiate, the training hours are de…nitely longer in the decentralized
allocation than in the socially e¢ cient allocation.

JEL classi…cation: J24; J64
Keywords: sector-speci…c skills, job search, wage bargaining

I especially thank Masaru Sasaki, Ryosuke Okazawa, and the participants of the Summer Workshop
on Economic Theory for their useful comments and discussions. Financial aid from JSPS Research
Fellowships for Young Scientists is gratefully acknowledged.
yE-mail address: hge005kk@mail2.econ.osaka-u.ac.jp

1. Introduction
The e¢ ciency property of human capital investment has provoked a great deal of controversy. In search theory, it is well known that a worker’s training e¤ort to acquirea single-dimensional skill (general skill) is lower than what it would be in the sociallye¢ cient level1 . To the best of our knowledge, the e¢ ciency property of the investment
to acquire multi-dimensional skills2 has not been adequately discussed so far. Thus, inorder to …ll this gap, this paper constructs a search model with multi-dimensional skills
and focuses on the e¢ ciency of the allocation of (exogenous) training hours to acquire
each skill.
In this model, there are two sectors, and each job in a sector requires a sector-speci…cskill for production. A newborn worker is initially assigned to a sector (which is called an
initial sector) and conducts skill training to acquire necessary skills by spending exogenous
training hours. More precisely, workers decide to allocate exogenous training hours inorder to acquire these skills at birth. In the market equilibrium, workers acquire theskills of not only the initial sector but also another one, because if a worker allocates too
many hours to acquire a skill speci…c to the initial sector, she/he produces only a small
amount of output in her/his job when switching to another sector.
Previous papers have discussed the importance of considering the e¢ ciency of skill
investment in an environment where a worker and her/his employer can negotiate over theinvestment to acquire skills3 . In response to these discussions, I analyze the following twocases: The …rst case is that a worker and her/his employer can negotiate the allocationof training hours, and the second case is that they cannot negotiate.
There are potentially two sources of ine¢ ciency, and these sources have opposinge¤ects on the allocation of training hours. The …rst source is the hold-up problem, whichis due to a lack of complete contingent contracts of wages, which leads to over investment
in training to acquire a skill of an initial sector. The second source is the outside optione¤ect, by which workers determine the allocation to improve their outside option in wage
bargaining, which leads to under investment in training to this skill. If an employedworker and her/his employer can negotiate the allocation of training hours, both sources
arise, and the worker may allocate more or less hours to acquire the skill of an initial
sector than what would be socially e¢ cient. Meanwhile, if they can negotiate, only thehold-up problem arises, and then, the allocation of training hours for this skill must be
more than what would be socially e¢ cient.
The rest of this article is organized as follows. Section 2 presents the basic framework.
Section 3 characterizes the socially e¢ cient allocation. Section 4 de…nes the allocation
1

See Acemoglu (1997), Acemoglu and Shimer (1999), and Masters (forthcoming).
In recent years, Wasmer (2008) and Mukoyama and Sahin (2009) examined the skill investment
problem with multi-dimensional skills with search friction. However, they did not analyze the e¢ ciency
property.
3
See, for example, the survey by Acemoglu and Pischke (1999).
22

characterized in market equilibrium, and section 5 concludes.

2. Environment
I consider a partial equilibrium and discrete-time model, and the measure of each period
is one. Workers exit from the labor market with probability (death shock), and thesame number of newborn workers enter the labor market in each period. The number ofworkers in the steady state is then a constant and normalized unity. There are two sectors,
A and B, and two types of sector-speci…c skills, hA and hB , where hj2fA;Bg indicates theamount of a sector j speci…c skill. To simplify, an employed worker exogenously moves
to unemployment with probability s (job destruction shock), and an unemployed worker
meets a vacant job in sector j with exogenous probability pj 2 [0; 1] (and pA + pB 1).
Note that I assume that workers and …rms do not discount the future utility and pro…ts
due to the death rate.
An employed worker with a skill vector h = [hA ; hB ] produces yj (h) = y (hj ) in asector j 2 fA; Bg, where yj0 > 0 and yj (0) = 0. To guarantee an inner solution, yj
satis…es the following properties; yj00 < 0, yj0 (0) = 1, yj0 (1) = 0, and yj00 is small enough.
Moreover, I consider the equilibrium in which all workers acquire both hA and hB andaccept a job in both sectors.
Without loss of generality, I assume that newborn workers do not have any skills andare exogenously allocated to a job in sector A (called an initial job). They conduct skill
training to acquire skills before starting to produce in the initial job. Note that workers
cannot conduct additional skill training after their training period, in order to focus on
the allocation problem of training hours. The number of training hours to acquire hj
is hj , where is the parameter indicating the e¤ectiveness of skill training, andP thetraining hours constraints that are then faced by newborn workers are 1
j hj .
Moreover, they must decide the allocation of training hours for hA and hB :Timing in each period is as follows: (i) Newborn workers decide the allocation oftraining hours for hA and hB , (ii) employed workers produce outputs and conduct wage
bargaining, (iii) the labor market is open, and (iv) job destruction and death shocks
occur.
Flow conditions
Let ej and u denote the number of employed workers in sector j and the number of
unemployed workers respectively. The steady state condition of eA is+ (1) pA u = [1

(1

) (1

s)] eA :(1)

The LHS of equation (1) shows in‡ows into eA . The …rst term represents the number ofnewborn workers, and the second term, the number of workers who meet a job in sector
3

A. The RHS indicates out‡ows from eA , the number of workers who lose their own jobs
or die.
Similarly, the steady state condition of eB is(1
Because u = 1

eA

eB
=
eA
(1

) pB u = [1

(1

) (1

s)] eB .

(2)eB , equations (1) and (2) show the ratio of eB to eA :)2 spA + [1)2 spB) (1 s) + (1(1(1

) (pA + pB )](3):

Equation (3) implies that eA > eB if pA = pB , because newborn workers are initiallyallocated to jobs in sector A.

3. The social planner’s problem
First, I characterize a solution of the social planner’s problem. A social planner determines the allocation of training hours to maximize social surplus, within the training
hours constraint. The social planner’s problem is de…ned byXX
yj (h)ej s.t. 1 =hj :maxhA ;hAjj2fA;Bg

The …rst order conditions are given by y 0 (hj ) ej =, where is the Lagrangian
multiplier. Substituting the …rst order condition into equation (3) yields
yA0 (hA )=yB0 (hB )(1)2 spA + [1(1
(1

)2 spB) (1 s) + (1) (pA + pB )]:(4)

Given that yj00 < 0, if pA = pB , then hA > hB , because in the steady state, workers are
more likely to work in sector A than in sector B.

4. Market equilibrium
This section solves the problem of market equilibrium. To do so, I …rst de…ne the following
value functions.
According to Pissarides i(2000), the value of unemployment is, U (h) =
hP
P
(1
)
j pj Wj (h) + 1
j pj U (h) and the value of a vacant job is de…ned as

zero4 without loss of generality. The value of employment in sector j is Wj (h) = wj (h) +(1) [sU (h) + (1 s)Wj (h)], where wj (h) is wages, and the value of having a job …lled
is Jj (h) = yj (h) wj (h) + (1
) (1 s)Jj (h).
4

The results of this paper hold even when the value of the vacant job is positive.

4

The wage is determined through the basic Nash bargaining, and the …rst order condition of Nash bargaining yields Wj (h) U (h) = b(Wj (h) + Jj (h) U (h)), where b is
the parameter indicating the worker’s bargaining power. U (h) can then be rewritten asP(1) b j pj yj (h)U (h) =
:
(5)
1 (1
) (1
pb)
P
0
0
Because y 00 < 0 and 1
j hj , if pA y (hA ) < pB y (yB ), U (h) increases with training
hours to acquire hB and decreases with training hours to acquire hA .
Finally, I characterize the equilibrium allocation of the two types of training in thefollowing two cases: (i) a non-negotiation case in which a worker decides the allocation oftraining hours to maximize her/his expected lifetime utility, and (ii) a negotiation casein which the Coase theorem holds for the allocation of training hours.
4.1 Skill training: The non-negotiation caseIn this case, a newborn worker determines her/his allocation of training hours to maximize
WA (h). Then, using the …rst order condition of Nash bargaining, I rewrite the worker’s
problem as follows:X
max b (WA (h) + JA (h)) + (1 b) U (h) s:t:
hj = 1:hA ;hBj2fA;Bg

The above problem implies that a worker considers the e¤ect on the outside option in
Nash bargaining if b < 1, which is referred to as the outside option e¤ect. Meanwhile, thesocial planner does not consider this e¤ect because the level of the outside option onlya¤ects the share of output between a worker and an employer, but not the total output.
Furthermore, when pA y 0 (hA ) < pB y 0 (yB ) (this really happens in equilibrium), this e¤ect
leads to more training hours to acquire hB than there would be in the socially e¢ cient
allocation:
The …rst order condition implies that
b
1

(1

) (1

@yA (h)
+ (1
s)
@hj

)s

@U (h)
+ (1
@hj

b)

@U (h)
=
@hi

:

(6)

Given that yA (h) = y (hA ), this condition implies that the bene…t gained from an increase
in hB comes through the improvement of U (h). From (5), an increase in b increases thisbene…t, because the worker’s share of output determined by Nash bargaining increases.
In other words, the hold-up problem arises, and a worker undervalues an increase in
U (h), thereby leading to underinvestment to acquire hB when pA y 0 (hA ) < pB y 0 (yB ).
By (6) and (5) ; the equilibrium allocation in this case isy 0 (hA )=y 0 (hB )
(1

) (1

(1

(1 (1
) (1 s)) (1 s)b) pA + [1

5

b) (1
) pB
(1
) (1 s) + (1

) b (pA + pB )](7):

Comparing (7) with (4), the equilibrium allocation is the same as the socially e¢ cient
allocation if b = 1. The RHS of (7) is a decreasing function of b. Thus, I can state thefollowing proposition.
Proposition 1 If 0 b < 1, the market allocation of training hours is ine¢ cient, andworkers allocate excess training hours to acquire hB .
Intuitively, there are two sources of ine¢ ciency in the allocation, the outside optione¤ect and the hold-up problem. These two sources have opposing e¤ects on the training
allocation. If b < 1, the market allocation is ine¢ cient because both sources arise, andthe outside option e¤ect dominates the hold-up problem.
4.2 Skill training: The negotiation case
Following the Coase theorem, the allocation of training hours is determined to maximize
to the sum of WA (h) and JA (h). The allocation of training hours is then determined by
the solution to the following problem:X
max WA (h) + JA (h) ; s.t.
hj = 1:hj2fA;Bg

According to the …rst order condition and (5), the optimal condition isy 0 (hA )=y 0 (hB )b (1)2 spA + [1b (1)2 spB(1) (1 s) + b (1) (pA + pB )]:(8)

A comparison of (7) and (8) shows that a worker allocates more training hours toacquire hA through negotiation, because the outside option e¤ect is eliminated. The RHSof (8) is an increase function of b, and similar to the allocation in the non-negotiationcase, the equilibrium allocation is the same as the socially e¢ cient allocation if b = 1.
Thus, I can state the following proposition.
Proposition 2 If b < 1, hB is less in the equilibrium allocation than in the socially
optimal allocation.
Intuitively, while the outside option e¤ect is eliminated because of the Coase theorem,
the hold-up problem remains. Then, workers underestimate the bene…t from improving
outputs in sector B. Moreover, workers allocate all training hours to acquire hA if b = 0,
and the skill training is socially optimal if b = 1 since the hold-up problem is also
eliminated.

65. Conclusion
This paper investigates the allocation problem of training hours. There are two sourcesof ine¢ ciency in this allocation: the …rst is that workers consider an e¤ect on the outside
option in the wage bargaining, and the second is the hold-up problem. When newborn
workers are initially assigned to sector A, the …rst source induces them to acquire thesector B speci…c skill, and the second source discourages them from acquiring this skill.
If a newborn worker and her/his employer can negotiate, only the hold-up problem arises,
and training hours to acquire the sector A speci…c skill are longer than the socially optimal
hours. Meanwhile, if they cannot negotiate, the hold-up problem and the outside optione¤ect occur, and the acquisition level of the sector A speci…c skill could be either higher
or lower in the decentralized solution than in the socially optimal level.

7

References[1] Acemoglue, D. 1997. "Training and innovation in an imperfect labour market". Review of Economic Studies, Vol 64, No 3, 445-464.
[2] Acemoglue, D, and Pischke, J. 1999. "Beyond Becker: training in imperfect labour
markets". Economic Journal, Vol 109, No. 453, 112-142.
[3] Acemoglue, D., and Shimer, R. 1999. "Holdups and e¢ ciency with search frictions".
International Economic Review, Vol 40, No 4, 827-849.
[4] Masters, A. "Commitment, advertising and e¢ ciency of two-sided investment in competitive search equilibrium". Journal of Economic Dynamics and Control, forthcoming.
[5] Mukoyama, T., Sahin, A. 2009. "Specialization and human capital in search equilibrium". Journal of the European Economic Association, Vol 4, Issue 2-3, 503-512.
[6] Pissarides, C. 2000. Equilibrium Unemployment Theory. Cambridge, MA: Basil Blackwell.
[7] Wasmer, E. 2008. "General versus speci…c human capitals in labor markets with search
frictions and …ring costs". American Economic Review, Vol 96, Issue 3, 811-831.

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