On-the-job Training and On-the-job Search:
Wage-Training Contracts in a Frictional Labor Market∗
Seung-Gyu (Andrew) Sim†
December 22, 2014

Abstract
This paper analyzes the coexistence of on-the-job training and on-the-job searchin a frictional labor market in which both skilled and unskilled workers engage inon-the-job search behavior and firms post skill-dependent wage-training contracts topreemptively back-load compensation and extract more surplus in earlier periods. The
paper demonstrates that because back-loaded compensation after training eﬀectively
enlarges the expected duration, and joint value, of the match as if it were job-specifictraining, general training is over-intensified overall. Comparative static analysis
predicts aggregate training intensity to be reinforced, but net output to improvetoward the Beckerian outcome, as search friction is mitigated.
Keywords: On-the-job Training, On-the-job Search, Wage-Training Contracts
JEL Classification: J24, J31, J64∗

This is a revised version of the third chapter of my Ph.D dissertation. I am deeply indebted to John
Kennan, Rasmus Lentz, and Chris Taber for their insightful guidance and invaluable support. I also thankJunichi Fujimoto, Tim Huegerich, Hidehiko Ichimura, Michihiro Kandori, Young Sik Kim, Chul-In Lee,Dan Sasaki, Yasuyuki Sawada, Yongs Shin, Serene Tan, Kenichi Ueda, and all seminar participants at
Hitotsubashi University, Seoul National University, University of Tokyo, and Yokohama National University
for their helpful comments and warm encouragement. I also acknowledge financial support by the Grantsin-Aid for Scientific Research (Kakenhi No. 26780170) from the Japan Society for the Promotion of Science.
All remaining errors are mine.
†
Address: University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan, 113-0033. Phone: +81-3-58415620. Email: sgsim@e.u-tokyo.ac.jp1

1

Introduction

Most economists believe human capital accumulation on the job to be the major contributor to both individual wages and broader economic growth.1 Rapid technologicalprogress and knowledge spillover dictate that workers continuously upgrade their skills
post schooling, and firms pay the cost of on-the-job training to encourage workers tocatch up and keep up with new technologies.2 Absent commitment by workers to staywith their firms,3 it is unclear why, and how much, firms will pay for general training. This paper investigates how firm productivity, wage payments, and investment
in human capital interact in response to workers’ on-the-job search behavior. The focus is on eﬃciency analysis of the market equilibrium and comparative static analysis
associated with the mitigation of search friction.
Becker (1964) documents that under perfect competition firms provide the sociallyeﬃcient level of general training to retain workers, such training paid for by workers
through lower wages during training. This result is shown by Acemoglu and Pischke(1999), however, to be subject to a wide range of deviation from perfect informationand competition in the labor market. Firms provide ineﬃciently low levels of training
and share the cost of training in a labor market with high job turnover, according to
Acemoglu (1997), because training firms cannot raise a claim on the expected benefit
of next employers (the free-rider problem). Sanders and Taber (2012) acknowledge
the widely accepted notion that potential job turnover causes under-investment ingeneral, and over-investment in job-specific, training, especially in a wage bargaining
environment.4 Fu (2011) investigates on-the-job training and wage dispersion in africtional labor market by incorporating general training and the piece-rate sharing
rule into the framework proposed by Burdett and Mortensen (1998) and Burdett et al.
(2011), who, consistent with Sanders and Taber (2012), shows general job training to
be under-provided in the market equilibrium relative to the constrained social planner’s
problem.
This paper extends the Burdett and Mortensen (1998) framework by diﬀerentiating
workers according to skill level. Workers enter the labor market as unskilled andacquire general skills through firm-sponsored training on the job. Both employed andunemployed workers search for better employment opportunities. Deviating from Fu(2011) with respect to the piece-rate sharing rule, this paper posits that recruiting firms,
being heterogeneous in terms of productivity, post skill-dependent lifetime values thatreflect their optimal recruiting strategies. Committed values are delivered to employedworkers according to operating firms’ optimal training and retention strategies. When1

For example, Altonji and Shakotko (1987) argue that general human capital accumulation accounts forthe lion’s share of wage growth, and Topel and Wald (1992) report that the wage of a typical male worker
in the U.S. labor market doubles over the course of a 40-year career.
2
Lynch and Black (1998) find that more than half of (but not all) U.S. firms provide and pay for generaltraining, for example, in computer skills and teamwork.
3
Fallick and Fleischman (2004) find 60 percent of vacancies in the U.S. labor market to be taken by
employed, and only 40 percent by unemployed, searchers, Topel and Wald (1992)find that a typical worker
holds seven jobs during the first 10 years of employment, and Sim (2013) reports the average job durationof white male high school graduates in the United States to be slightly more than two years.
4
An interesting exception is Moen and Rosen (2004), who argue that an eﬃcient level of training canbe achieved in a frictional labor market if employers and employees are able to coordinate eﬃciently, as bymeans of long-term contracts on wages, training intensity, and search intensity. The authors do not account,however, for the possibility of over-investment.

2

unskilled as well as skilled workers engage in on-the-job search behavior, training firms
should, because they lose the entire rent when trainees leave for better paying jobs, careabout job turnover even before skills are acquired. This consideration leads training
firms to commit to paying trainees the entire surplus after training, choose the training
intensity that maximizes the sum of the workers’ and firms’ values, and extract surplus
by paying lower wages during training. In other words, firms exploit the training
opportunity to back-load compensation as much as possible in order to extract more
surplus in earlier periods. The underlying motivation for the back-loading wage scheme
coincides with that of the wage-tenure contract proposed by Burdett and Coles (2003),
Stevens (2004), and Shi (2009).5
It is useful to compare the market equilibrium with the problem of the constrainedsocial planner, who, taking labor market friction as given, maximizes the present valueof the expected net output flow throughout the life of a newly born worker. The constrained social planner, not caring from which firm a worker receives training, considers
only productivity improvement in determining training intensity. That training firms
exploit training for purposes of back-loading compensation as well as improving productivity eﬀectively increases the expected joint value and duration of matches post
training. Relative to the social planner, training firms thus provide over-intensified
general training as if it were job-specific training, which possibility has thus far not
been properly considered, the literature having paid little attention to job turnoveramong unskilled workers or training firms’ preemptive responses thereto. The theback-loading scheme, although it dilutes the problem of free riders raised by Acemoglu(1997), suggests other sources of ineﬃciency, namely, ‘training ineﬃciency’ and ‘allocation ineﬃciency.’ According to the quantitative analysis based on the calibrated model,
the market equilibrium outcome, relative to the constrained social planner’s outcome,retains a larger steady state mass of skilled workers (due to the former ineﬃciency),
but produces fewer units of the actual and net outputs (due to the latter ineﬃciency).
One possible take on Acemoglu (1997) is that, although he delivers a plausible
insight, as search friction is mitigated the market outcome converges to almost “notraining” in the opposite direction to the prediction of Becker (1964). The current
paper’s contracting solution predicts that accelerating the job turnover rate among
employed workers both enables them to find more eﬃcient matches quickly and discourages firms from providing general training, thereby improving, but not as muchas in the constrained planner’s problem, the market equilibrium’s ineﬃciency in job
turnover and training. It also predicts that firms will provide more intensified training
as the job finding rate of unemployed workers is accelerated because the value diﬀerential between skilled and unskilled workers becomes larger. When oﬀer arrival rates to
employed and unemployed searchers are accelerated in the same proportion, aggregate
training intensity is reinforced further from, but net output increased towards, theplanner’s outcome owing to the enhanced skill supply and eﬃciency improvement in
job turnover behavior in spite of higher training cost. The calibrated model demonstrates that as search friction is mitigated through labor market reforms or the arrival
5

Stevens (2004) shows firms, in the presence of on-the-job search, to take in early periods, and leave forthe workers in later periods, the entire surplus. Burdett and Coles (2003), assuming risk aversion, describea unique back-loading wage schedule such that diﬀerent firms choose diﬀerent starting wages and all firms
follow the wage schedule. Shi (2009) develops a directed search model with the wage-tenure contract andproposes ‘block recursivity.’ The current paper departs from the foregoing studies in proposing that training
firms improve the probability of retention by ceding the entire surplus to workers post training.

3

of an eﬃcient search technology, aggregate outcome in the planner’s problem moves
towards the Beckerian outcome and the outcome of the market equilibrium follows theplanner’s outcome.
The rest of the paper proceeds as follows. Section 2 develops the theoretical modeland characterizes the steady state equilibrium. The eﬃciency analysis is presented in
Section 3, the quantitative assessment in Section 4. Section 5 concludes.

2

Model

2.1

Environment

This paper incorporates on-the-job training into the Burdett and Mortensen (1998)
framework to analyze how firm productivity, wage payment, and provision of job
training interact in response to workers’ on-the-job search behavior. It defines unit
measures of risk neutral workers and firms, both of which discount the future at rate
r. A newly born worker enters the labor market as an unskilled unemployed worker.
The unskilled worker acquires general skills through on-the-job training and becomesa skilled worker.6 “Skilled” and “unskilled” are denoted throughout the paper by thesubscript i ∈ {s, u}. Firms are heterogeneous in terms of productivity, H(p) denoting
the proportion of firms with productivity no greater than p and H(·) continuously
diﬀerentiable with support [p, p]. Each firm has one vacant job at every instant. A
recruiting firm posts and commits skill-dependent lifetime values, and an operating
firm delivers the committed value through its wage-training schedule. The operating
job with productivity p occupied by an unskilled worker accrues revenue p, by a skilledworker, revenue p + s, at every instant. Assume that p + s < p.
To ensure the existence of an equilibrium, assume that there are ε-measure ofnoise firms such that they randomly draw and post skilled wages from the cumulative
distribution F̂N : [p + s, p + s] → [0, 1]. The overhead hat (b) indicates a distributionof wages, not values. It is assumed that the steady state equilibrium has a suﬃciently
small ε(> 0). Noise firms are introduced as a means of removing a mass point in the
distribution of skilled wage earnings (a detailed explanation of their role is provided infootnote 10), the only requirement being that the noise firms’ wage oﬀer distribution
be continuously diﬀerentiable and strictly increasing.7

Workers An unemployed worker collects unemployment benefit b and finds an employment opportunity at the Poisson arrival rate λ0 . An employed worker receives
a wage, finds another employment opportunity at rate λ1 , and is laid oﬀ at rate δ.
All workers retire (or die) at rate ρ, retirees being replaced by newly born workers
who enter the labor market as unskilled and unemployed. Denote by Ui and Ei thelifetime values of an i-type unemployed and employed worker, respectively, and let
Fi : R+ → [0, 1] be the steady state distribution function of the values oﬀered to i-type
workers. The latters’ support, denoted by [E i , E i ], as well as the distribution function
Fi , will be endogenously determined later. The lifetime value of an unemployed worker
6

Quercioli (2005) uses the Burdett and Mortensen (1998) framework to analyze the case in which firms
provide training opportunities for firm-specific skills.
7
The concept of ε-measure of noise firms is borrowed from Galenianos and Kircher (2009).

4

is described byrUi = b − ρUi + λ∫0max{z − Ui , 0}dFi (z), for each i ∈ {u, s}.

(1)

The left-hand side of asset equation (1) represents the opportunity cost of holding
asset “i-type unemployment,” the right-hand side, the dividend flow from the asset,
potential losses from retirement shock, and gains from job-finding.
Given that recruiting firms with diﬀerent productivity post diﬀerent values, thevalue of skilled employment oﬀered by a particular recruiting firm with productivity p
is denoted by Es (p). The committed value is delivered through skilled wages, potential
losses from exogenous retirement and separation shocks, and potential gains from jobto-job transition, described by∫1
rEs (p) = ws − ρEs (p) + δ(Us − Es (p)) + λmax{z − Es (p), 0}dFs (z).
(2)
The one-to-one relationship between Es (p) and ws , described in equation (2), allows
us to denote by ws (p) the skilled wage flow paid by the firm with productivity p.
The unskilled employed worker receives on-the-job training and unskilled wage wu .
Given training intensity x, chosen by the firm, the worker acquires skills at rate µx,
and upon acquiring skills is awarded a new labor contract with a higher value.8 In
the steady state equilibrium, skilled workers who remain at the firms at which they
were trained and promoted are termed “promoted (skilled) workers,” workers hired as
skilled who acquired the skills elsewhere “recruited (skilled) workers.” The value of
promotion, denoted by Est , is obtained by plugging skilled wages after promotion, wst ,into the value equation (2). Let Es (p) be the value of unskilled employment oﬀered
by the firm with productivity p. Given (wu , x, Est ), the expected value of unskilledemployment is given byrEu (p) = wu + xµ(Est − Eu (p)) − ρEu (p) + δ(Uu − Eu (p))∫
1
+λmax[z − Eu (p), 0] dFu (z).

(3)

The value of unskilled employment, like that of skilled employment, is delivered throughwage payments during training, expected gains from skill acquisition and subsequent
promotion, potential losses from exogenous retirement and separation shocks, andpotential gains from job-to-job transition. Equation (3) is used as the promise-keeping
constraint in the training firm’s optimization.

Operating Firms Let Js (Es (p), p) and Ju (Eu (p), p) be the values of the skilled
and unskilled matches having productivity p and committed values Es (p) and Eu (p),respectively. These can be rewritten as Js (p) and Ju (p) interchangeably. Given the
one-to-one relationship between Es (p) and ws (p) in equation (2), the asset value of theskilled match producing p + s at every instant is given byrJs (p) = p + s − ws (p) − [ρ + δ + λ1 (1 − Fs (Es (p)))]Js (p).
8(4)

Owan (2004), to shed light on the promotion strategy and hierarchical structure of firms, treats strategic
promotion and training provision separately. This paper, the organizational structure of firms being beyond
its scope, assumes that an unskilled worker who acquires skills will immediately be promoted to a skilled
position.

5

The firm having an operating unskilled match chooses a wage-training schedule todeliver the committed value. The cost of training associated with training intensity
x is given by c(x) = xγ , where γ > 1. Note that c(·) is (infinitely) continuously
diﬀerentiable with c(0) = 0, c′ (0) = 0, c′ > 0, and c′′ > 0. The value of an unskilled
match producing p at every instant is described byrJu (p) = max

wu ,x,Est

p−wu −c(x)−[ρ+δ+λ1 (1−Fu (Eu (p)))]Ju (p)+xµ(Js (p)−Ju (p)) (5)subject to constraints (3) and (4). Plugging (3) into (5) and taking the derivative withrespect to Est yields the first order condition (dFs (Est )/dEs )Js (Est , p) ≥ 0, where thestrict equality holds only when Js (Est , p) = 0, as long as Fs (·) is strictly increasing.9
The ε-measure of noise firms is introduced to avoid the case in which dFs /dEs reaches
zero quickly.10 Equation (4) implies that Js (Es , p) = 0 if and only if ws = p + s. Let
wst (p) be the wage payment after promotion by the training firm having productivity
p. After promotion, the firm thus oﬀers∫
wst (p) + δUs + λ1 E t (p) zdFs (z)ts, where wst (p) = p + s > ws (p).
(6)
Es (p) =r + δ + ρ + λ1 (1 − Fs (Est (p)))
Equation (6) implies that the training firm makes no claim after training, leaving theentire surplus for the trained worker. Apparently, no recruiting firms commit the entire
surplus to potential employees. The training firm’s back-loading strategy eﬀectively
deters potential job oﬀers from slightly better poaching firms, as if the training firm
and worker jointly accumulated job-specific human capital.
Given productivity p ∈ [p, p], the first order condition with respect to x yields
c′ (x) = µ(Est (p) − Eu (p) − Ju (p)).

(7)

The left-hand side represents the marginal cost, the right-hand side the marginal benefit, of providing training. As c(x) is convexly increasing with the Inada condition andthe right-hand side independent of x, a global maximum is uniquely obtained in theinterior. For each p ∈ [p, p], the interior solution is denoted by x(p). Once the valueof promotion Est (p) and training intensity x(p) are determined, the promise keeping
constraint (3) determines wage payment wu (p) at each productivity level p. Equation(7) implies that the training firm that commits to paying the worker the entire surplus
after promotion chooses the training intensity that seemingly maximizes the summation of the worker’s and firm’s net gains from training, as Moen and Rosen (2004)
suggest. Unlike Moen and Rosen (2004), however, the intensity that maximizes thesummation of the worker’s and firm’s gains is not jointly eﬃcient due to job turnoverby unskilled workers. This will be examined in the next section.
Note that the optimal decision on Est satisfies the second order suﬃcient condition as well.
Suppose there are no noise firms. For any p ∈ [p, p], the firm with productivity p never posts a skilled
wage more than or equal to (p + s), ws (p) < p + s. If ws (p) < p + s, the first order condition with respect
to Est implies that all training firms, regardless of productivity, choose wst (·) = w(p) to retain their trained
workers. If ws (p) ≥ p + s, there exists p̂ ∈ [p, p] such that wst (p̂) = ws (p) due to the continuity argument.
Firms with productivity p ∈ (p̂, p] set wst (·) = ws (p). There being, in both cases, a mass point at ws (p) in
the wage earning distribution, no equilibrium exists.
9

10

6

Recruiting Firms Let {Gu (Eu ), Gs (Es )} denote the proportions of unskilled andskilled workers that receive lifetime values no greater than {Eu , Es }, respectively, and{uu , us } the total mass of unskilled and skilled unemployed workers, respectively. The
total measure of workers being fixed at unity in the steady state equilibrium, a recruiting firm with productivity p posts the skill-dependent values (Eu , Es ) to maximize theexpected value, [λ0 us + λ1 Gs (Es )]Js (Es , p) + [λ0 uu + λ1 Gu (Eu )]Ju (Eu , p). Taking firstorder conditions with respect to Es and Eu yields[
λ0 ui + λ1 Gi (Ei )dGi ] 1dFi
λ0 ui + λ1 Gi (Ei ) =+λ Ji (Ei , p),(8)
r + ρ + δ + λ1 (1 − Fi (Ei )) dEi
dEi
for each i ∈ {u, s}. Equation (8) implicitly determines the optimal pair of(Eu (p), Es (p)) posted by the recruiting firm with productivity p, and equations(2) and (3) show how the committed values are delivered. [Figure 3] reveals bothEs (p) and Eu (p) to be strictly increasing in p under reasonable parameter values.
Let F̂s : [ws , ws ] → [0, 1] be the cumulative distribution function of skilled wages
oﬀered by profit-maximizing normal firms, and F̂sε : [p + s, p + s] → [0, 1] be thecumulative distribution function of the skilled wages oﬀered by both noise firms andprofit-maximizing normal firms, obtained, given ε(> 0), by F̂sε (w) = (1 − ε)F̂s (w) +εF̂N (w). As ε → 0, F̂sε (w(p)) converges to F̂s (w(p)) = Fs (Es (p)). Because the noise
firms randomly draw skilled wages from the wage oﬀer distribution F̂N : [p + s, p +s] → [0, 1], the supremum value oﬀered, E s , is strictly higher than Es (p), the actual
supremum value posted by the profit-maximizing normal firms. As ε goes to zero,
Fs (Es (p)) converges to one and Fs (E s ) − Fs (Es (p)) to zero.

Equilibrium Configuration

Given firms’ productivity distribution H(p), a
steady state equilibrium with on-the-job training and on-the-job search consists of value
equations {Ui , Ei , Ji }i∈{u,s} , compensation packages {(wu (·), x(·), Est (·)), (ws (·))}, andsteady state measures {Fi , Gi , ui }i∈{u,s} that jointly satisfy the following conditions.
(i) Given {Fi }i∈{u,s} , workers make optimal job turnover decisions, which determines{Ei , Ui }i∈{u,s} from (1), (2), and (3) together with the firms’ training decisions.
(ii) Given H(p) and {Fi , Ei , Ui }i∈{u,s} , each operating firm with a skilled worker delivers Es (p), which determines (4). The operating firm with an unskilled worker
optimally chooses (wu (p), x(p), Est (p)) following (3), (6) and (7), which determines (5).
(iii) Given {Gi , ui }i∈{u,s} and {Ji }i∈{u,s} , each recruiting firm posts a contract which
satisfies the first order condition in (8).
(iv) {Fi , Gi , ui }i∈{u,s} are stationary and consistent with workers’ job turnover decisions and firms’ provision of training.

[Figure 1] illustrates the dynamic worker flow in the steady state equilibrium. The
left, middle, and right vertical lines represent the equilibrium wage supports for unskilled workers, promoted workers, and recruited workers, respectively. Newly born
workers start their careers as unskilled and, once employed, are aﬀorded opportunities
for training and continue to search for better paying jobs. Arrow “a” in [Figure 1]
represents unskilled workers’ job-to-job transition, arrow “b,” promotion after acquiring skills. An unskilled worker receiving unskilled wage wu (p) is promoted to a skilled7unskilled wages

6
support of‘after training’ wages

support ofunskilled wages 6awu (p) s
3
support ofwst (p) s cskilled wages

>

s

ws (p) 6db
wst (p)skilled wages
6
DD‘promoted workers’D&‘recruited workers’
 by noise firms
DD‘promoted workers’D&‘recruited workers’LL‘recruited workers’
only

ws (p)wu (p)
unskilled workers

skilled workers

Figure 1: Equilibrium Support of Wages
position with wage wst (p), which is strictly larger than the skilled wage received byskilled workers recruited by the same firm ws (p). “Promoted workers” leave their jobs
voluntarily or involuntarily and become “recruited workers” or “unemployed workers,”
respectively. Arrow “c” represents “promoted workers’,” arrow “d” “recruited workers’,” job-to-job transition. [Figure 1] depicts two wage supports for skilled workers,
one exclusively for recruited, the other for both recruited and promoted, skilled workers. The dotted line on top of the support of skilled wages indicates that they are, in
the interval, oﬀered by noise firms.
Training firms give their workers the entire surplus after training for the same
reason firms oﬀer wage-tenure contracts in Burdett and Coles (2003), Stevens (2004),
and Shi (2009). Because unskilled workers engage in on-the-job search behavior during
training, training firms discount the future much more severely than their workers sothat they trade the surplus before and after training. Training firms take the surplus inearlier periods and leave the full surplus for workers after training. Because firms pay
much higher skilled wages to internally trained and promoted workers than to recruited
workers, the former are less likely to leave for slightly more productive poaching firms.
This strategic back-loading scheme leads to general training being exploited as a meansof enlarging the expected duration, and thus joint value of, the match as if it were jobspecific training, further reinforcing the investment in general training.

2.2

Characterization of Steady State Equilibrium

It is natural to think that the least productive firm having p, being able to attract
unemployed but not employed searchers, posts for each i-type worker the lifetime
value equivalent to Ui , which determines Ei (p) = E i , the infimum value oﬀered toi-type workers.

8

Lemma 1 Suppose that (Uu , Us ) are given. The optimal strategy by the least productive
firm then implies
]
λ1 [
(r
+
ρ)U
−
b
,
s
λ0
∫
[
p
p + s + δUs + λ1

ws (p) = (r + ρ)Us −Est (p) =1
r+δ+ρ

wu (p) = (r + ρ)Uu −(9)(1 − F̂s (w′ ))dw′

p+s r + δ + ρ + λ1 (1 − F̂s (w ′ ))

]
,

]
λ1 [
(r
+
ρ)U
−
b
− µx(p)(Est (p) − Uu ), andu
λ01(r + ρ + δ + λ1 )c′ (x(p)) + x(p)c′ (x(p)) − c(x(p))µλ1= (r + ρ + δ + λ1 )Est (p) − [p + δUu + 0 ((r + ρ + λ0 )Uu − b)].
λ

(10)
(11)
(12)

The above strategy by the least productive firm with p results in
Es (p) =ws (p) + δUs + λ1 ((r + ρ)Us − b)/λ0,
r + ρ + δ + µx(p)(13)

Eu (p) =wu (p) + δUu + λ1 ((r + ρ)Uu − b)/λ0 + µx(p)Est (p),r + ρ + δ + µx(p)(14)

Js (p) =Ju (p) =p + s − ws (p), andr + ρ + δ + λ1
p − c(x(p)) − wu (p).
r + ρ + δ + λ1 + µx(p)

(15)
(16)

Given that Us > Uu , Lemma 1 implies that as long as unemployed workers are more
eﬃcient than employed workers in job search, the infimum of skilled wages, λ0 ≥ λ1 ,is strictly greater than that of unskilled wages, ws (p) > wu (p). It is trivial that
Fu (Eu (p)) = Fs (Es (p)) = Gu (Eu (p)) = Gs (Es (p)) = 0, in which case the steadystate equilibrium is characterized by the system of diﬀerential equations (1)-(8) and(A4)-(A6) in Appendix A together with initial condition (9)-(16) in Lemma 1. The
diﬀerential equations (A4)-(A6) are derived from the following argument.
The unskilled, unemployed worker finds a job at rate λ0 ; the unskilled, employedworker is laid oﬀ at rate δ. A retiree is replaced with a newly born, unskilled, unemployed worker. Equating the outflow from, and inflow to, the steady state unskilled
unemployment yields(λ0 + ρ)uu = δGu (Eu (p)) + ρϕ.

(17)

The steady state unemployment rate being given by (ρ+ δ)/(ρ+δ + λ0 ), the proportionof skilled unemployed workers is given by
us = (ρ + δ)/(ρ + δ + λ0 ) − uu .

(18)

Given that Eu (p) is uniquely defined and strictly increasing in p, the unskilled, unemployed worker finds a job with productivity no greater than p at rate λ0 Fu (Eu (p)). The

9

unskilled worker working at a job with productivity less than p switches to a higher
valued job at rate λ1 (1 − Fu (Eu (p))), acquires skills at rate µx(p), is laid oﬀ at rate δ,and retires at rate ρ. The steady state measure Gu (Eu (p)) is characterized by∫ p
λ0 Fu (Eu (p))uu = (ρ+δ+λ1 (1−Fu (Eu (p))))Gu (Eu (p))+µx(p′ )gu (Eu (p′ ))dp′ , (19)p

where gu (Eu (p)) = dGu (Eu (p))/dp. There are two types of skilled employed workers, recruited and promoted workers. Denote by Grs (Es (p)) (Gts (Est (p))) the massof recruited (promoted) workers whose values are less than Es (p) (Est (p)). Then,
Gs (Es ) = Grs (Es ) + Gts (Es ). Equating the inflow to and outflow from the steadystate measure of recruited workers receiving less than value Es yields
∫ Es
1
r
0
1
[ρ + δ + λ (1 − Fs (Es ))]Gs (Es ) = λ Fs (Es )us + λ[Fs (Es ) − Fs (z)]dGts (z). (20)Est (p)

The left-hand side implies that recruited workers experience retirement, separation,and job-to-job transition; the right-hand side represents the flow of skilled workers
newly recruited at values no greater than Es . Promoted workers leave their training
firms due to retirement, separation, and job-to-job transition. Equating the inflow andoutflow yields
∫ p

′

′

′

µx(p )gu (Eu (p ))dp

= (ρ + δ)Gts (Est (p)) + λ1

p

∫ Est (p)(1 − Fs (Est (p)))dGts (z). (21)Es

Taking the derivatives of (19), (20), and (21) with respect to p yields a system ofdiﬀerential equations with initial conditions Gu (Eu (p)) = 0, Grs (Es (p)) = 0, andGts (Est (p)) = 0.
Lemma 2 Solving the unskilled workers’ optimal job turnover decision and firms’
optimal training decision yields∫ p
Iu (p′ )λ0 (dFu (Eu (p′ ))/dp)uu−1
Gu (Eu (p)) = Iu (p)dp′ , where (22)
1 (1 − F (E (p′ ))) + µx(p′ )
ρ
+
δ
+
λ
u
u
p
[∫ p
]
1
′
−λ (dFu (Eu (p ))/dp′ )′Iu (p) := exp
dp
.
(23)
1
′
′
p ρ + δ + λ (1 − Fu (Eu (p ))) + µx(p )The implied flows of skilled workers are thus described by∫ pµx(p′ )(dGu (Eu (p′ ))/dp)
Gts (Est (p)) =dp′ and(24)
1 (1 − F (E t (p′ )))
ρ
+
δ
+
λ
s
p
s
∫ p
Is (p′ )(λ0 us + λ1 Gts (Es (p′ )))(dFs (Es (p′ ))/dp′ ) ′r−1
Gs (Es (p)) = Is (p)dp , where (25)
ρ + δ + λ1 (1 − Fs (Es (p′ )))p[ ∫ p −λ1 (dF (E (p′ ))/dp′ )]ss′Is (p) := exp
dp
.
(26)
1
′
p ρ + δ + λ (1 − Fs (Es (p )))Plugging the relevant expressions in Lemma 2 into equation (8), solving for theskill-dependent values oﬀered by individual recruiting firms, and integrating those10values yields (Fs , Fu ). Taking (Fs , Fu ) as given, solving for the optimal strategies by
economic agents, and aggregating the worker flow constitutes a fixed point argument.
Given the complexity of the overall system, in section 4 the analytical proof of theexistence and uniqueness of the result is replaced with numerical experiments with
diﬀerent initial values. This section ends with the characterization of training intensity.
Proposition 1 The training firm with productivity p chooses a training intensity suchthatc′ (x(p))(r + ρ + δ)/µ + x(p)c′ (x(p)) − c(x(p)) = s + δ(Us − Uu )
∫ Es
∫ Es
t
1
1
[z − Es (p)]dFs (z) − λ+λ[z − Eu (p) − Ju (p)]dFu (z).
Est (p)(27)

Eu (p)

Moreover, x(p) < x(p) for any p ∈ [p, p) if and only if the summation of the workers’
and firms’ gains from job turnover after training is greater than the summation of theirgains before training, that is,∫ Es
Est (p)[z − Est (p)]dFs (z) −∫ Eu[z − Eu (p) − Ju (p)]dFu (z) > 0.

(28)

Eu (p)The left-hand side in equation (27) rises convexly from the origin with x(p). The
right-hand side is independent of x(p). Equation (27) implies that the training decision made by each firm is aﬀected by the sum of value diﬀerentials before and after
training, which consists of the value diﬀerentials of skilled and unskilled unemployment, (Us − Uu ), and diﬀerentials of joint gains from job-to-job transition after and∫E∫E
before completing training, E ts(p) [z −Est (p)]dFs (z)− Euu(p) [z −Eu (p)−Ju (p)]dFu (z), asswell as productivity improvement through training, s. Interestingly, training intensity
rises with the joint expected gains from job-to-job transition after, and falls with thejoint gains from job-to-job transition during, training. That the optimal back-loading
scheme permits trained workers only jointly eﬃcient job turnover keeps the expected
gains positive. Moreover, the sum of expected gains from job turnover by unskilledworkers is more likely to be negative due to the training firm’s loss, especially when thefirm’s values are suﬃciently high. The former being always positive under the optimal
back-loading scheme, the latter can be negative due to the firm’s loss, especially when
p is suﬃciently high or λ1 is suﬃciently small. This implies that condition (28) is not
so restrictive, and also explains why the current study, which considers job turnoverby unskilled workers, unlike previous studies that considered only job turnover after
training, finds the possibility of over-investment in general training.

3

Eﬃciency Analysis

Consider the problem of the constrained social planner who maximizes the present
value of the expected output flow throughout the life of a newly born worker. Variables
associated with the planner’s problem are designated by an asterisk. A typical worker
produces b when unemployed, p − c(x∗ (p)) when employed as unskilled, and p + s whenemployed as skilled, depending on productivity p ∈ [p, p]. Denote by Ss∗ (p) (Su∗ (p))

11the present value of the output flow of a currently employed, skilled (unskilled) worker
with productivity p ∈ [p, p].
(r + ρ)Ss∗ (p)=(r + ρ)Su∗ (p) =∫ p[Ss∗ (p′ ) − Ss∗ (p)]dH(p′ ),(29)p
max p − c(x∗ (p)) + δ(Uu∗ − Su∗ (p)) + µx∗ (p)(Ss∗ (p) − Su∗ (p))
x∗ (p)
p + s + δ(Us∗ − Ss∗ (p)) + λ1
∫ p

1

+λ

[Su∗ (p′ ) − Su∗ (p)]dH(p′ ), and

(30)

p

(r + ρ)Ui∗

∫ p

0

= b+λ

(Si∗ (p′ ) − Ui∗ )dH(p′ ), for i ∈ {u, s}.

(31)

p

The socially eﬃcient training intensity, x∗ (p), is determined byc′ (x∗ (p)) = µ(Ss∗ (p) − Su∗ (p)).

(32)

Once training intensity is obtained at each p ∈ [p, p], the steady state measures of{uu , us , Gu (·), Gs (·)} should be adjusted as follow:dG∗u (p)dpdG∗s (p)dp==

λ0 u∗u + λ1 G∗u (p)
dH(p)
, and
ρ + δ + λ1 (1 − H(p)) + µx∗ (p) dp
λ0 u∗s + λ1 G∗s (p) + µx∗ (p)(dG∗u (p)/dp) dH(p)ρ + δ + λ1 (1 − H(p))
dp

(33)
(34)

where G∗u (p) = G∗s (p) = 0, u∗u = ρϕ(ρ + δ + µx∗ (p))[(ρ + δ + µx∗ (p))(ρ + λ0 ) − δλ0 ]−1
and u∗s = (ρ + δ)/(ρ + δ + λ0 ) − u∗u .
Proposition 2 The constrained social planner chooses a training intensity such thatfor each p ∈ [p, p],c′ (x∗ (p))(r + ρ + δ)/µ + x∗ (p)c′ (x∗ (p)) − c(x∗ (p)) = s + δ(Us∗ − Uu∗ )(35)

In particular, dx∗ /dp = 0, dx∗ /dλ1 = 0, and dx∗ /dλ0 > 0.
Proposition 2 implies training intensity in the social planner’s problem to be flat
regardless of productivity, and aﬀected by acceleration of the job-finding rate of unemployed, but not by acceleration of the job turnover rate of employed, searchers.
Intuitively, there being no heterogeneity in training technology across productivity
levels, the training intensity in the social planner’s problem is aﬀected by neither productivity diﬀerentials nor job turnover. If the job-finding rate of unemployed searchers
is accelerated, the value of skilled unemployment rises further than that of unskilled
unemployment because the opportunity cost of being unemployed is larger in the former than in the latter case. This raises the marginal benefit of training in equation(35) and reinforces training intensity at all productivity levels.
Comparing x(p) in (27) and x∗ (p) in (35) results in interesting breakdowns. The
diﬀerence in training intensity between the market equilibrium and the planner’s
problem reflects the diﬀerence in unemployment value diﬀerentials (Us − Uu ) and(Us∗ − Uu∗ ) and the diﬀerence in the expected joint gains from job-finding. The
back-loading compensation scheme, in particular, by imposing a ‘proper price of the

12

6

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x(p)

.........................................................
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x̃(p)

x (p)

x̃(p) = x̃∗ (p)x̃ (p)p(a) Without Separation Shocks (δ = 0)
p

p

-

p
(b) With Separation Shocks (δ > 0)

Figure 2: Training Intensity
skill,’ enables training firms to internalize the positive externality for subsequentpoaching firms. That training firms are not able to raise a claim when their trained
workers switch to other firms through unemployment, however, still discourages
firms from providing general training (the free-rider problem as in Acemoglu (1997))
if (Us − Uu ) < (Us∗ − Uu∗ ). To isolate the eﬀect of the training firms’ strategic
back-loading scheme from the free rider problem raised in the skilled employmentunemployment-employment flow, consider the case without the separation shock,δ = 0, as in Moen and Rosen (2004).11 To keep a positive mass of skilled unemployedworkers, it is assumed that a newly born worker enters the labor market as unskilledwith probability ϕ ∈ (0, 1] and as skilled with probability (1 − ϕ). Variables associatedwith the benchmark setting (hereafter alternative benchmark) are designated by anoverhead tilde (e). (Us − Uu ) and (Us∗ − Uu∗ ) are dropped in equations (27) and (35),enabling a clear comparison by removing the general equilibrium eﬀect of sifting theoﬀer distributions.
Proposition 3 Consider the alternative benchmark setting with δ = 0 and ϕ < 1.
(i) x̃(p) = x̃∗ (p).
(ii) x̃(p) > x̃(p) = x̃∗ (p) = x̃∗ (p) for each p ∈ [p, p) if and only if the sum of theworker’s and firm’s gains from job turnover after training is greater than the summationof their gains before training, that is,∫ Es
Est (p)[z − Est (p)]dFs (z) −∫ Eu[z − Eu (p) − Ju (p)]dFu (z) > 0.

(36)

Eu (p)

11

Moen and Rosen (2004), to show that the eﬃcient level of training can be provided through coordination
between the trainee and training firm, assume only retirement, and not job separation, shock, with the resultthat all unemployed workers in their model are newly born workers. They further assume that unskilledemployees are unable to search for other jobs. These are not innocuous assumptions in deriving their mainargument.

13

Proposition 3 reveals that the joint surplus maximizing training intensity in themarket equilibrium outcome can be over-intensified. Because workers at the mostproductive firm have no gains from on-the-job search behavior, (27) is the same as (35),hence, x̃(p) = x̃∗ (p) in the alternative benchmark setting. As shown in Proposition 1,
the most productive firm provides the least training under condition (36). Training
intensity in the market equilibrium is thus over-intensified relative to the sociallyeﬃcient level under condition (36). If the second term in (36) has a negative value ora suﬃciently small positive value, firms provide over-intensified training. As shownin Panel (a) in [Figure 2], which summarizes Proposition 3, more productive firms
provide over-intensified general training relative to the social planner. Whether lessproductive firms, including the least productive firm, provide over- or under-intensified
training, however, remains ambiguous.
Corollary 1 Suppose that δ > 0.
(i) x(p) > x∗ (p) if and only if Us − Uu > Us∗ − Uu∗ .
(ii) x(p) > x∗ (p) for each p ∈ [p, p], if and only if
δ(Us − Uu ) + λ

1∫ Es
Est (p)[z − Est (p)]dFs (z) − λ1

∫ Eu[z − Eu (p) − Ju (p)]dFu (z)Eu (p)> δ(Us∗ − Uu∗ ).

(37)

Corollary 1 circles back to the general environment with the exogenous separation
shock. The proof of Corollary 1 is dropped provided that it is straightforward from
Proposition 3. Panel (b) in [Figure 2] summarizes Corollary 1. When δ > 0, it isunclear whether the most and least productive firms provide over- or under-intensified
training relative to the social planner’s training intensity. Because it is aﬀected by the
distribution assumption and other parameter values, Corollary 1 provides a suﬃcientcondition.
Proposition 3 and Corollary 1 examine, through a point-wise comparison at eachproductivity level p ∈ [p, p], whether individual firms provide over- or under-intensified
general training. To determine social eﬃciency in aggregation requires a careful choiceof the distribution assumption and parameter values. The next section performs aquantitative assessment with a focus on the aggregate market equilibrium.

4

Quantitative Analysis

This section, which calibrates the model and illustrates its implications for social eﬃciency, shows that (i) the most productive firm provides less intensified training, (ii)aggregate training intensity is over-intensified in the market equilibrium relative to theplanner’s problem under reasonable choices of the distribution function and parameter values, (iii) the back-loading scheme causes eﬃciency loss due to over-intensified
training and partial deterrence of (socially) eﬃcient job turnover, and (iv) as oﬀer
arrival rates to both employed and unemployed searchers are accelerated in the same
proportion, social eﬃciency, measured by net output, is improved.

14

Table 1: Parameter Values[p, p]
the productivity supports
productivity improvement through trainingηthe shape parameter of H(p)µhuman capital accumulation rateγ
cost function parameterrinterest rateρ
retirement rateδseparation rate0
λjob finding rate by unemployed workers
1
λjob finding rate by employed workers

4.1[0.75, 1.75]
0.25
1.0
0.03
2.0
0.012
0.008
0.064
1.35
0.45

Specification and Parameterization

The simulation experiments in this paper proceed with H(p) : [p, p] → [0, 1], defined as
H(p) =

1 − (p/p)η
, where η > 0.
1 − (p/p)η

(38)

Productivity support is normalized to be the unit length interval, [p, p] = [0.75, 1.75],
and evenly discretized with 1,001 grid points such that pj+1 − pj = 10−3 for eachj = 1, 2, · · · , 1000. Using data on the entire population of tax-paying firms in the
United States, Axtell (2001) shows the size distribution of U.S. firms to be characterized
by the Pareto distribution with shape parameter between 0.99 and 1.1. This paper
assumes η = 1.0 following his finding, which assumes a bounded Pareto distribution.
Further experiments indicate that a Pareto distribution with lower η results in more
over-intensified training. Hornstein et al. (2011) suggest that the mean-min wage ratio,
M m-ratio, should lie between 1.7 and 1.9. To accommodate this empirical finding, thispaper sets s = 0.25, which results in an M m-ratio of 1.77 in cooperation with otherparameter values. Given the firm’s training cost c(x), the worker acquires skills at
rate µx. Without any good reference on firm training cost, this paper proceeds with
γ = 2.0 and µ = 0.03, and checks robustness.
Setting the quarterly interest rate at 0.012, which roughly targets an annual interest
rate of 0.048 and quarterly retirement rate at 0.008, implies that the 76 percent ofworkers who enter the labor market around age 20 retire before age 65. Setting thejob finding rate for unemployed workers at 1.35, as in Shimer (2005), implies that the
average U.S. unemployment spell is about 10 weeks. The steady state unemployment
rate (ρ + δ)/(ρ + δ + λ0 ) is usually targeted around 5 or 6 percent, which fixes theseparation rate δ at 0.064. Sim (2013) documents average job duration in the U.S. labormarket to be slightly more than two years among white male high school graduates.
This paper reconciles his suggestion by choosing λ1 = 0.45. This, together with otherchoices, results in average job duration of eight quarters, which is consistent with thetargeted value.
In addition to the baseline simulation, this section analyzes the alternative benchmark setting with δ = 0. To maintain the unemployment rate, ρ is set to be 0.072(=

15

(a) Workers’ Values(b) Firms’ Values(c) Wages(d) The Value-earning Distribution

Figure 3: The Baseline Simulation Result
The horizontal axes of Panel (a), (b) and (c) represent firm productivity, and the vertical axes workers’ values, firms’ values, and wages, respectively. Panel (d) presents the
cumulative wage earning distribution.

0.064 + 0.008). It is necessary, when δ is set to be zero, to assume a certain fractionof newly born workers to be skilled in order to avoid a zero mass of the skilled unemployed in the general equilibrium environment. Panel (a) of [Figure 4] reports theimplied training intensity under ϕ = 0.4. But note that the training intensity in themarket equilibrium is robustly over-intensified regardless of the choice of ϕ.

4.2

Baseline Simulation[Figure 3] summarizes the result of the numerical experiments based on the parameter values chosen in the previous section. In [Figure 3], the solid lines represent, in
Panels (a), (b), (c), and (d), respectively, workers’ values of unskilled employment,
firms’ values of unskilled jobs, unskilled wages, and the proportion of unskilled workers
receiving a certain level of wage or less. The dotted lines represent, in Panels (a), (b),(c), and (d), respectively, workers’ values of skilled recruitment, firms’ values of jobs
with recruited workers, skilled wages received by recruited workers, and proportion ofskilled workers receiving a certain level of wage or less. The dashed line represents, inPanels (a), (c), and (d), respectively, the value of, and wage payment to, the promoted
worker and proportion of employed workers receiving a certain level of wage or less.

16

(a) In the Alternative Benchmark Setting(b) In the Baseline Simulation

Figure 4: Training Intensity
The horizontal axis in each panel represents firm productivity, and the vertical axis training
intensity. Note that each panel has a diﬀerent range on the vertical axis.

unskilled skilledworkers workers

training
cost

total
outputNet
outputME
PP

0.60010.60270.3493
0.34670.01590.01561.44551.44581.42971.4303ME/PP

0.99581.00741.01920.99980.9995

ME: the market equilibrium outcome PP: the planner’s solution

ME/PP: the ratio of ME to PP

Table 2: The Outcome of the Baseline Simulation
The value of a firm with a promoted worker, always being zero, is dropped in Panel(b).
Panel (a) shows all lifetime values to increase monotonically with firm productivity,
reflecting the high value a productive firm attaches to worker recruitment. Panel(b) shows firm values to also increase monotonically with productivity, but relatively
less productive firms to realize higher values from skilled, and more productive firms
from unskilled, matches, implying that productive firms more eﬀectively trade wage
payments before and after training through the back-loading wage scheme. This isalso captured in Panel (c). Although wage payment after internal promotion increases
monotonically with firm productivity, the unskilled wage is monotonically increasing
depending on the parameter values. That the gap between wages before and after
training apparently becomes larger as productivity rises implies that firms with highproductivity eﬀectively suppress the unskilled wage by committing to higher skilledwages. Comparing the dotted and dashed lines in Panel (c) reveals the distinct intervalsof skilled wages, as in [Figure 1] and also reflected in Panel (d). There being four
distinct intervals overlapped with each other, there exist three kink points in the wage
support. Shifting the dashed line by the proportion of unemployed workers pushes it
up to one at the right end.
In [Figure 4], the solid and dotted lines represent the training intensity in the market
equilibrium and in the planner’s problem, respectively. Panel (a) in [Figure 4] shows thetraining intensity of the market equilibrium in the alternative benchmark setting, the17solid line mostly above the dotted line, to exhibit a hump-shaped relationship between
productivity and training intensity. This implies that the condition in Proposition 3
is satisfied at all productivity levels except for a short interval at the bottom. The
training intensity is characterized bydEt′µλ1 [ [Fs (Es (p)) − Fu (Eu (p))] dp − Ju (p)H (p) ].
c′′ (x(p)) r + ρ + δ + λ1 (1 − Fu (Eu (p))) + µx(p)
t

dx(p)
dp

=

(39)

t

The first term in the square bracket in (39), [Fs (Est (p)) − Fu (Eu (p))] dE
dp , is interpreted
as an increase in the encouraging eﬀect that leads productive firms to train theirworkers more intensively in response to the enlarged retention probability and, hence,increase the joint value after training. The second term, −Ju (p)H ′ (p), captures adecrease in the back-loading pressure that leads productive firms to provide training lessintensively consequent to mitigation of the threat posed by potential poaching firms.
As p increases, the improvement in the retention probability, Fs (Est (p)) − Fu (Eu (p)),
goes to zero and dx(p)/dp < 0 for any p ∈ (p − s, p]. This implies that optimal training
intensity declines with firm productivity among highly productive firms. But because
Fs (Est (p)) − Fu (Eu (p)) has a large value among relatively less productive firms (under
the Pareto distribution), optimal training intensity increases with productivity p amongless productive firms. As predicted by Proposition 2, the dotted line is a flat straightline regardless of productivity, that is, dx∗ (p)/dp = 0. Panel (b) shows that whenδ > 0 and the skilled employment-unemployment-employment flow exists, the mostproductive firm in the market equilibrium oﬀers under-intensified training relative tothe planer’s decision.
[Table 2] presents clear evidence of over-investment in general training in the steadystate market equilibrium. The first and second columns show there to be more skilled(unskilled) workers in the market equilibrium (planner’s problem) as a result of the‘over-intensified training’ shown in the third column. Total employment is fixed at
0.9494(≈ λ0 /(ρ + δ + λ0 )) in both cases. Although the mass of skilled workers is larger
in the market equilibrium than in the planner’s solution, total output in the fourthcolumn is smaller in the former than in the latter, indicating that some (socially)
eﬃcient job turnover is deterred in the market equilibrium, and lowering the average
productivity of skilled matches. Because the negative eﬀect of the higher aggregate
cost and partial deterrence of eﬃcient job turnover dominates the positive eﬀect ofenhanced skill supply, the market equilibrium results in smaller net output.

4.3

Comparative Static Analysis[Figure 5] shows how training firms adjust training intensity in response to variationin the magnitude of search friction. The baseline parameterization dictates λ1 ≤ λ0 ,which indicates that constraints on search behavior, such as time, render job searchless eﬃcient among employed than among unemployed workers. Panel (a) shows theextent to which training firms must adjust training intensity if employed workers areto eﬀectively overcome such hurdles. Training intensity in the market equilibrium isrepresented by solid, in the planner’s problem by dotted, lines. The thin lines follow
from the baseline simulation; the thick lines represent training intensity in each setting with 20 percent reinforcement in the oﬀer arrival rate to employed workers. If
the job turnover rate of employed workers is accelerated, training firms train workers

18

(a) With 20 percent acceleration of λ1(b) With 20 percent acceleration of (λ0 , λ1 )Figure 5: Comparative Statics
The horizontal axis in each panel represents firm productivity, and the vertical axis training
intensity.

unskilled skilledworkers workers

training
cost

total
outputNet
outputWith 20 percent acceleration of λ1MEPP0.60070.60270.34870.34670.01580.0156

1.4672
1.46771.45141.4521

ME/PP

0.99681.00561.01440.99970.99950

1With 20 percent acceleration of (λ , λ )MEPP0.60130.60480.35610.35270.01620.01581.48081.48091.46471.4651

ME/PP

0.99441.00961.02511.00000.9997

ME: the market equilibrium outcome PP: the planner’s solution

ME/PP: the ratio of ME to PP

Table 3: With the Acceleration of the Oﬀer Arrival Rates
less intensively. Intuitively, as the job turnover rate is accelerated expected job duration becomes shorter, which, together with other general equilibrium eﬀects, reduces
training intensity in the market equilibrium.
Panel (b) shows how training firms adjust training intensity in response to theimprovement of search eﬃciency among unemployed as well as employed workers. As
before, the thin lines represent the baseline simulation, the thick lines training intensitywith 20 percent reinforcement. Interestingly, the results reported in Panel (b) are
opposite those reported in Panel (a), both thick lines lying above the thin lines. In otherwords, training firms provide more intensive training as search eﬃciency improves,whether for employed or unemployed workers. In particular, the accelerated oﬀer
arrival rate to unemployed searchers increases their reservation value, forcing recruiting
firms to raise the value of their oﬀers to potential employees, and, hence, the valuesof skilled and unskilled unemployment. But because the opportunity cost of unskilled
unemployment is not as great as that of skilled unemployment, the value of the latter

19

(a) The Ratio of the Skilled Proportions(b) The Ratio of the Aggregate Training Cost(c) The Ratio of the Aggregate Output(d) The Ratio of the Aggregate Net Output

Figure 6: Comparative Statics
The horizontal axis in each panel represents the ratio of λ1 /λ0 . The diﬀerent curves ineach panel represent the values associated with diﬀerent intensities of λ0 . As a result,unilateral improvement in only λ1 generates ‘movement along the curve’ and proportional
improvement in λ0 and λ1 ‘shift-up.’

improves further with λ0 , encouraging training firms to provide more intensive training.
[Table 3] shows the impact of the changes on the market equilibrium outcome.
When only λ1 increases, the market equilibrium gets a smaller fraction of skilled workers
but generates more output than before because the skilled workers are more likely towork at more productive firms. Aggregate training intensity decreases, but output
increases towards the planner’s outcome. When λ0 and λ1 increase together, training
intensity, the fraction and average productivity of skilled workers, and outputs all
increase and net output improves toward the planner’s outcome.
[Figure 6] further compares the market equilibrium with the planner’s problem. The
horizontal axis represents the relative search eﬃciency, λ1 /λ0 , the vertical axis the ratioof the market equilibrium outcome to the planner’s outcome, in all four panels. The
dotted, solid, and dashed lines represent the market equilibrium outcomes associatedwith λ0 = 1.08, 1.35, and 1.62, respectively. The thin horizontal line in the middle
represents the benchmark case in which the market equilibrium outcome coincides withthe planner’s outcome. In Panel (a), the skilled proportion ratio rises with λ0 , but falls
with λ1 /λ0 , confirming the implication of [Figure 5]. When λ1 /λ0 is small (large), themarket equilibrium outcome possesses a larger (smaller) mass of skilled workers than20the planner’s outcome, which indicates ‘over-investment’ in general training. When λ0
is suﬃciently large but λ1 /λ0 small, Panel (b) consistently indicates a higher training
cost (over-investment) in the market equilibrium. Comparing Panels (a) and (c) raises
an interesting point. When λ1 /λ0 is small, the market equilibrium outcome keeps
a larger mass of skilled workers than the planner’s outcome (in Panel (a)), but theformer seldom produces more than the latter (in Panel (c)). Total employment being
the same in both cases, the eﬃciency loss caused by deterring eﬃcient job turnover
can be inferred to be substantial. The contracting solution’s deterrence of eﬃcient job
turnover translates into an aggregate output loss in the market equilibrium relative tothe planner’s outcome. But as λ0 and λ1 increase in the same proportion maintaining
λ1 /λ0 constant, the aggregate output loss shrinks as the measure of skilled workers
increases and the distribution is also improved. Because aggregate training cost declines
with λ1 /λ0 , the net output ratios in Panel (d) are flatter than those in Panel (c).
Overall, improvement in λ1 generates ‘movement along the curve.’ Starting from thebaseline point (λ0 , λ1 /λ0 ) = (1.35, 0.333), the ratios of skilled proportion, aggregate
training cost, aggregate output, and net output decline along the solid line in eachpanel. The accelerated job turnover rate reduces both training intensity and thenet outputs of the market equilibrium. Improving λ0 and λ1 in the same proportion
generates a vertical ‘shift-up.’ Finally, as search friction is mitigated, the net output
in the planner’s outcome is expected to move toward the Bekerian outcome and thenet output in the market equilibrium to catch up with the planner’s net output.

5

Conclusion

This paper develops a job search model with on-the-job training and on-the-job search
to analyze how productivity, wage payment, and training intensity interact in response
to potential job turnover. It demonstrates that when unskilled, as well as skilled,workers engage on-the-job search behavior and firms are allowed to post skill-dependentwage-training schedule, firms exploit training as a means of back-loading, as much as
possible, compensation to extract more surplus in earlier periods. Hence, the backloaded compensation after training eﬀectively deters potential oﬀers from slightly moreproductive firms, which encourage firms to make over-investment on general training, as
if it were job-specific training. The implied training intensities chosen by heterogeneousfirms increase with productivity among relatively less productive, and decrease among
more productive, firms.
The market equilibrium suﬀers from two sources of ineﬃciency, ‘training ineﬃciency’ and ‘allocation ineﬃciency.’ Overall, the market equilibrium provides overintensified general training relative to the planner’s problem because the latter considers only productivity improvement in determining training intensity, whereas theformer exploits general training for purposes of intertemporal substitution as well as
productivity improvement. Back-loaded compensation after training eﬀectively deters
job oﬀers from slightly more productive firms, rendering general training ineﬃciently
over-intensified (ineﬃciency in training), as if it were job-specific training. Deterringsocially eﬃcient job turnover distorts the distribution of skilled workers (ineﬃciency in
job turnover), which degrades aggregate match quality in the market equilibrium.
The quantitative assessment based on the calibrated model demonstrates that (i)if only the job turnover rate of employed searchers is accelerated, the ratios of training

21

intensity and net output of the market equilibrium outcome to the planner’s outcome
fall together, and (ii) if oﬀer arrival rates to employed and unemployed searchers are
reinforced proportionally, both ratios rise together. This predicts that as search friction
is mitigated due to labor market reform or the arrival of an eﬃcient search technology,
aggregate training intensity may be reinforced, but net outputs improve toward theplanner’s outcome.

A

Mathematical Appendix

Proof of Lemma 1 Reordering and rewriting (1) yields that∫∫1(z − Ui )dFi (z) = 0 [(r + ρ)Ui − b] = (z − Ei (p))dFi (z).
λ(A1)

The last equality follows from Eu (p) = Uu and Es (p) = Us . Note that the least
productive firm with p should oﬀer (Eu (p), Es (p)) = (Uu , Us ) on any equilibrium.
Plugging (A1) into (2), replacing Es (p) with Us , and reordering yields (9). The detailed
derivation of (10) is presented in Christensen et al. (2005). (11) is obtained via thesame procedure. Given∫
1(A2)
zdFu = 0 [(r + ρ + λ0 )Uu − b],λcombining (3), (5), and (A2) yields
1

p + δUu + λλ0 [(r + ρ + λ0 )Uu − b] − c(x(p)) + µx(p)E t (p).
Eu (p) + Ju (p) =r + ρ + δ + λ1 + µx(p)(A3)

Plugging (A3) into (7) and reordering yields (12). Equations (13)-(16) immediately
follow from (9)-(12).
Proof of Lemma 2 Taking the derivative of (19), (20), and (21) with respect to p
and applying Leibniz rule results in
dGu (Eu (p)) dEu (p)dEudprdGs (Es ) dEs
dEs
dp
t
t
t
dGs (Es (p)) dEs (p)
dEst
dp

=
=
=

λ0 uu + λ1 Gu (Eu (p))
dFu (Eu (p)) dEu (p),(A4)
1
ρ + δ + λ (1 − Fu (Eu (p))) + µx(p)dEudp
λ0 us + λ1 Gts (Es ) + λ1 Grs (Es ) dFs (Es ) dEs
, and
(A5)
ρ + δ + λ1 (1 − Fs (Es ))dEs dpµx(p)dGu (Eu (p)) dEu (p).
(A6)ρ + δ + λ1 (1 − Fs (Est (p)))
dEu
dp

Solving the system of diﬀerential equations (A4)-(A6) yields Lemma 2.
Proof of Proposition 1 Plugging (3), (5), and (6) into (7) and reordering yields(27). Then, the last two terms in (27) constitutes condition (28).
Proof of Proposition 2 Plugging (29), (30), and (31) into (32) and reordering resultsin (35). Note that for any p ∈ [p, p],∫ p[](Ss∗ (p′ ) − Ss∗ (p)) − (Su∗ (p′ ) − Su∗ (p)) dH(p′ ) = 0.
(A7)
p

22

From (35), it is trivially true that dx∗ (p)/dp = 0 and dx∗ (p)/dλ1 = 0. Since
∫ p[
]
λ0
∗
∗ ′
∗
∗ ′
Us − Uu =
S
(p
)
−
S
(p
)
dH(p′ ),su
r + ρ + λ0 p

(A8)

it is obtained that dx∗ (p)/dλ0 > 0.
Proof of Proposition 3 (i) In the alternative benchmark setting, the right handside of (27) and (35) are same at p = p by construction.
(iii) It’s trivially obtained by comparing (27) and (35).

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