Discussion Papers In Economics
And Business

Alteration in Skills and Career-Enhancing
in a Frictional Labor Market
Yosuke Oda
Discussion Paper 08-09

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANAlteration in Skills and Career-Enhancing
in a Frictional Labor Market
Yosuke Oda
Discussion Paper 08-09

March 2008

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANAlteration in Skills and Career-Enhancing
in a Frictional Labor Market∗
Yosuke Oda †March 2008

Abstract
This article constructs a job-search model in which worker’s ability varies over time; a highability unemployed might lose her skills due to prolonged unemployment whereas a low-ability
employed might acquire her skills due to (an implicit) on-the-job training. We numericallyshow that both pecuniary reward for short-term unemployed and reduction in unemployment
beneﬁts leads to lower unemployment rate, however, the former policy does stimulate careerenhancing of long-term unemployed whereas the latter does not. In addition, numericalanalysis suggests that mixture of the two policy can lead to higher aggregate welfare thanunder a sole policy.
Keywords: job-search model; cyclical change in skills; career-enhancing separation
JEL classiﬁcation: J64; J68∗

I am grateful to Masaru Sasaki, Jun-ichiro Ishida, Noritaka Kudoh and seminar participants at Kansai university (Search Theory Workshop) and at Osaka university, and JEA Annual Meeting Fall 2006 held at Osaka
City University for their helpful comments and discussions. I am particularly indebted to Ryo Horii for his helpful
suggestions and comments.
†

Graduate school of Economics, Osaka university, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, Japan. E-mailaddress: dg011oy@mail2.econ.osaka-u.ac.jp

1

Introduction

This paper constructs a general equilibrium job-search model in which individual’s their skills varyover time; high-skilled jobless workers lose skills due to prolonged unemployment whereas lowskilled employed workers acquire their skills due to (an implicit) on-the-job training. The aim of
this article is, by making use of the model, to consider a policy that would overcome the serious
issues below.
OECD (2002) documents that on average 30% of unemployed workers are long-term unemployed
who has been unemployed for one year or more in 2000, and in ten countries of them, the proportions
are over 40%. These proportions are historical high, and OECD (2002) concludes that “the longterm unemployed appear to be relatively more likely to go on to become very-long-term unemployedin some countries, and more likely to leave the labor force in others”. As frequently discussed, theprolonged unemployment duration makes workers discouraged and/or less skilled, which makes
again reemployment of the workers more diﬃcult. Notably, such a negative spiral aggravate social
welfare, in the sense of both that high unemployment rate worsens social welfare and that prolonged
unemployment deteriorates welfare for long-term jobless workers. So it is worthwhile to consider apolicy that overcomes the serious issues.
Heterogeneity in skills among individuals is treated in some earlier studies. For instance, Pissarides (1992) analyzes a worker’s loss of skills in an overlapping generations framework. In his
model, an old who was employed when young has higher productivity than an old who was unemployed when young. Thus individual’s skill is changed for once in her life and the skill is notaccumulated at all. As another example, Albrecht and Vroman (2002) studies a job-search modelin which distribution of worker’s skill is two-point (high or low). Since their paper focuses on ﬁrm’s
behavior, skill of each individual is assumed to be constant over time.
In contrast to these studies, this article assumes that individual’s skill level varies over time.
Following Albrecht and Vroman (2002), we assume that distribution of skills is of two-point (either
high or low). Under the assumption, high-skilled unemployed loses her skills if she cannot ﬁnd a
job within certain duration while a low-skilled unemployed acquires her skills if she works at a job
for certain duration. The rationale behind the assumption is that the former is due to prolonged
unemployment while the latter is due to (an implicit) on-the-job training. Given the circumstance,
we consider eﬀects of two labor policies which are stated below on unemployment rate and onsocial welfare.
The policies considered here are quite simple. One is reduction in unemployment beneﬁt and2the other is reemployment bonus. The former is straightforward. High unemployment beneﬁts
lead to high value of being unemployed, which results in high unemployment rate. Thus cutbackin the beneﬁts would decrease unemployment rate. The latter is originally planned as an economic
experiment (for summary, see Meyer, 1995), which is a reward for workers who are hired withincertain duration after ﬁred. It directly increases worker’s incentive to be employed, which resultsin lower unemployment rate. In summary, the former is a stick while the latter is a carrot as
employment-boosting policy.
Theoretically, it is obvious that the two policies have similar eﬀects on unemployment rate but
not on social welfare, since the stick policy decreases welfare for jobless workers while the carrot
one beneﬁts for reward-qualiﬁed workers. This paper, however, focuses on another eﬀect of the
carrot policy on a worker’s behavior. In this paper we do not treat on-the-job search, so if a worker
wants to change her job, she must separate from the current job and once become unemployed
to seek a new better job. In that context, reduction in unemployment beneﬁts would make her
discouraged to enhance her career since the value of unemployment is decreased.
Recalling that workers are either unemployed or employed and either high-skilled or low-skilled
in our setting, a policy that reduces only the rate of unemployment would be insuﬃcient from thepoint of view of social welfare. This is because the most socially desirable situation in the economy
is to increase the number of high-skilled employed. So the policy really needed is such that not
only the rate of unemployment is fallen oﬀ but low-skilled workers pursue a more productive job.
In that context, reduction in unemployment beneﬁts would make a worker discouraged to enhance
her career since the value of unemployment is decreased, so a worker who is hired by low-skilled
job does not intend to enhance her career. The pecuniary reward, however, can give a worker
who is employed by low-skilled job an incentive to enhance her career. In summary, reduction in
unemployment beneﬁts does operate as an employment-boosting policy but does not as careerenhancing policy while implement of pecuniary bonus program operate as both policies. Notably,
since the career-enhancement increases the number of individuals who are employed at a moreproductive job it has a positive eﬀect on social welfare.
Given above, this paper shows examples of policy eﬀects on unemployment rate and on social
welfare. As predicted above, we numerically show that, if the tax that ﬁnances unemployment
beneﬁts and/or pecuniary bonus is not too high, (i) both lower unemployment beneﬁts and higher
pecuniary rewards lead to lower unemployment rate, (ii) both higher unemployment beneﬁts andhigher pecuniary bonus result in higher welfare, and (iii) if the two policies are implemented simul-

3

taneously, moderate level of unemployment compensation and reemployment bonus can achieve
higher social welfare than when one of the two policies solely implemented.
The rest of the paper is composed as follows. Section 2 describes the model and deﬁnes theequilibrium, section 3 is devoted to numerical analysis, and section 4 concludes.

2

The Model

2.1

Basic Assumptions

Workers
This paper considers a continuous-time job-search model in which workers are inﬁnitely-lived andrisk-neutral. We focus only on a steady-state equilibrium. A measure of workers is ﬁxed andnormalized to one. Workers are either employed or unemployed and either high-skilled or lowskilled. Let u be a rate of unemployment. All jobless workers receive unemployment beneﬁt z.
Following Albrecht and Vroman (2002), our model assumes that a distribution of skills amongindividuals is a two-point distribution; a fraction γ of the unemployed workers are high-skilled, afraction 1 − γ of them are low-skilled, a fraction ϕ of employed workers are high-skilled, and afraction 1 − ϕ of them are low-skilled. Note that, unlike Albrecht and Vroman (2002), we assume
that workers’ skills level vary over time so that γ and ϕ are endogenously determined in equilibrium
as described in detail below.
This paper assumes that a high-skilled unemployed worker becomes low-skilled unemployed ata Poisson rate λ, which implies that a high-skilled unemployed worker might lose skills if she does
not work for certain duration (on average 1/λ).

1The paper also presumes that a low-skilledemployed acquires skills at a Poisson rate µ. For simplicity, the paper does not treat on-the-job
search. Thus, a low-skilled employed who acquired skills must become a high-skilled unemployedonce to improve her career. 2 Suppose that high-skilled unemployed workers (whose unemployment
1

Regarding the assumption, it would be appropriate to assume a time-varying unemployment beneﬁt rather thana constant unemployment beneﬁt, however, such an assumption requires a more complicate setup. See footnote 2.
2

Note that high-skilled unemployed workers are necessarily short-term unemployed worker but low-skilled unemployed workers are not necessarily long-term unemployed worker in the model. This is because low-skilled
unemployed workers include a worker who has just lost her job (that is, they are short-term unemployed).
Given the fact, if we assume a time-varying unemployment beneﬁts, the model needs three states of unemployment; high-skilled short-term unemployment, low-skilled short-term unemployment, and low-skilled long-term
unemployment. To avoid the complexity, the paper regards all low-skilled unemployed as substantively long-term
unemployed even if they are short-term unemployed. Hence, throughout the paper, the words high-(low -)skilled are4duration is necessarily short as in described in footnote 2) can receive reemployment bonus B if
they are hired.
Firms
Jobs are either ﬁlled or vacant. For simplicity, unlike Albrecht and Vroman (2002), assume that
there are ﬁrms which hire only high-skilled worker and only low-skilled worker. The former iscalled type h ﬁrm and the latter type l ﬁrm. Thus there exists practically two labor markets inthe model. For convenience they are called type h market and type l market, respectively.
When a job is ﬁlled, the job produces output yi and payes wage wi in the type i(= h, l) ﬁrm.
Assume that yh > yl . The wage is determined by bilateral Nash bargaining as described below.
Filled jobs break up at an exogenous Poisson rate δ. If a job is vacant, the type i ﬁrm incurs costci and the cost is assumed ch > cl . The markets are assumed to be free entry/exit so that ﬁrms
enter or exit the market so as to maximize their proﬁts, as described below.
Government
The role of government in this model is to collect tax to ﬁnance unemployment beneﬁt z andreemployment bonus B. The tax rate is endogenously determined to hold balanced ﬁnance at any
moment. The details are in subsection 2.4.
Matching Technology
Workers seeking a job and ﬁrms recruiting a worker meet randomly through a matching process.
The matching technology in each market are speciﬁed as follows:1

M (γu, vh ) = (γuvh ) 2 ,
1

M ((1 − γ)u, vl ) = [(1 − γ)uvl ] 2 ,where vi (i = h, l) denotes a measure of vacancies. Given the matching technology, rates of matching
1

1

1

1

for workers in each market are given by (γuvh ) 2 /γu = θh2 and [(1 − γ)uvl ] 2 /(1 − γ)u = θl2 , whereθh ≡ vh /γu and θl ≡ vl /(1 − γ)u are known as labor market tightness. Similarly, matching rates
−1

1

1

−1

for ﬁrms in each market are given by (γuvh ) 2 /vh = θh 2 and [(1 − γ)uvl ] 2 /vl = θl 2 , respectively.

used rather than short-(long-)term.

5

2.2

Asset Value Equations

Before describe asset value equations, recall the assumption that a low-skilled employed acquires
skills at a Poisson rate µ. Then, does the worker immediately separate from the current job? Inother words, does the worker necessarily become a high-skilled unemployed worker to improve her
career? The answer is no. If the worker behaves rationally, she would consider whether such a
career-enhancing separation is beneﬁcial or not. If so, the worker separates from the current job
to improve her career as soon as she has acquired skills. If not, she still works at the current job
until an exogenous job destruction occurs even if she has acquired skills.
Given the fact, we should consider both economy with and without career-enhancing separation(hereafter, abbr. CES). Note that CES arises or not is endogenously determined because value ofeach state is endogenously determined. To do the analysis, the paper proceeds in two steps, guessand verify: the following subsections in this section describe CES economy supposing that CES
condition holds (the guess), and the next section examines whether the condition meet or not byusing numerical calculus (the verify). The economy with no-CES is summarized in Appendix A.1.
Asset Value Equations for Workers
We use the following notations: Uh (Ul ) is the present-discounted value of high- (low-) skilled
unemployment, Wh is the value of high-skilled employment and Wlh (Wll ) is the value of being
employed at type l ﬁrm where a worker has acquired skills (where a worker is still low-skilled),
respectively. Assume that all individuals are levied capitation tax τ which is used to ﬁnance
unemployment beneﬁt z and bonus B. Suppose also that all jobless workers receive unemployment
beneﬁt z regardless of their skills. Recall that a high-skilled unemployed loses skills at the rate λwhereas a low-skilled employed acquires at the rate µ and that all jobs face to an exogenous job
destruction at the rate δ. Letting r be a discount factor which is common to all individuals, thevalue functions are given by the following equations:1

rUh = z − τ + θh2 (Wh + B − Uh ) + λ(Ul − Uh ),
1
2

(1)

rUl = z − τ + θl (Wl − Ul ),(2)

rWh = wh − τ + δ(Uh − Wh ),(3)

rWlh = wl − τ + δ(Uh − Wlh ),(4)rWll = wl − τ + δ(Ul − Wll ) + µ max{Uh − Wll , Wlh − Wll }.

(5)Note that the fourth term in (5) represents a low-skilled employed worker’s decision whether the6

worker separates from the current job if she acquired skills (Uh − Wll ) or the worker still works at
the current job even if she acquired skills (Wlh − Wll ). The former corresponds to CES whereas thelatter no-CES. Using these notations, CES condition is given by Uh ≥ Wlh . 3
Asset Value Equations for Firms
Firms discount the future at the rate r as well as workers. Let Vi (i = h, l) denotes a presentdiscounted value of vacancy for type i ﬁrms and let Ji (i = h, l) stands for a present-discounted
value of ﬁlled job for type i ﬁrms. The values of vacancy and ﬁlled job are recursively represented
as follows:−1

rVh = −ch + θh 2 (Jh − Vh ),
− 12

(6)

rVl = −cl + θl (Jl − Vl ),(7)

rJh = yh − wh + δ(Vh − Jh ),(8)

rJl = yl − wl + (δ + µ)(Vl − Jl ).

(9)Eq. (9) shows that type l ﬁrms face to an exogenous job destruction at the rate δ + µ because theworker-ﬁrm match breaks up not only when a shock occurs at the rate δ but when the employee
acquires skills at the rate µ. Supposing that the labor market is free entry, ﬁrms post a vacancy
until the expected value of job oﬀer equals to zero, which implies that Vh = Vl = 0 holds inequilibrium (the free entry/exit condition).

2.3

Equilibrium

In this subsection we characterize the equilibrium in CES economy. We begin with description of
ﬂow conditions which determines distribution of workers.
Flow Conditions
In the steady state, population in each state does not vary over time so that an inﬂow and outﬂow
in each state must be equal. Recall that u represents the rate of unemployment, γ denotes theratio of high-skilled in unemployed workers and ϕ indicates the ratio of high-skilled in employedworkers in CES economy.
More precisely, CES condition should be stated as Uh ≥ Wll since the expression Wlh is not needed in CES economy, however, to emphasize the fact that CES condition implies that the state value of high-skilled unemployment
is higher than (or equal to) the state value of low-skilled employment after skill acquisition, we use Wlh .
37

First, consider the ﬂow condition on high-skilled unemployment state. The inﬂow consists ofworkers from high-skilled employment due to an exogenous job destruction, ϕ(1 − u)δ, plus workers
from low-skilled employment due to the workers’ career-enhancing separation, (1−ϕ)(1−u)µ, while
the outﬂow consists of workers to low-skilled unemployment due to prolonged unemployment, γuλ,1plus high-skilled employed who ﬁnd a job, γuθh2 . The ﬂow condition on high-skilled unemployment
1is thus ϕ(1 − u)δ + (1 − ϕ)(1 − u)µ = γuθh2 + γuλ. Similarly, the condition on low-skilled unem1

ployment is given by γuλ + (1 − u)(1 − ϕ)δ = (1 − γ)uθl2 , which states that the inﬂow (workers
from high-skilled unemployment due to prolonged unemployment plus workers from low-skilled employment due to an exogenous job destruction) equals to the outﬂow (workers who are employed).
1

Finally, the condition on high-skilled employment state is γuθh2 = ϕ(1 − u)δ, which indicates thatthe inﬂow which is high-skilled workers who are employed equals to the outﬂow which is composedby high-skilled workers who lose a job due to a job destruction shock.
By using the three conditions, we obtain the rate of unemployment u, the ratio of high-skilled
in unemployed workers γ, and the fraction of high-skilled in employed workers ϕ in steady state,
which are arranged as follows:1u=δ(λδ + λµ + µθl2 )
1

1

1

,

(10)

θl2 (λδ + µθh2 ) + δ(λδ + λµ + µθl2 )
1

γ=

µθl2
1
2

,

(11)

λδ + λµ + µθl
1

ϕ=

µθh2
1

.

(12)

λδ + µθh2

Wage Determination
When a match is formed, the wage wi (i = h, l) is determined so as to maximize a matching surplus:
wh = arg max(Wh +B −Uh )β (Jh −Vh )1−β and wl = arg max(Wll −Ul )β (Jl −Vl )1−β , where β denotes
a bargaining power for workers. The sharing rules are given by (1 − β)(Wh + B − Uh ) = β(Jh − Vh )and (1 − β)(Wl − Ul ) = β(Jl − Vl ). Using these conditions, state values (1)-(9), and free entry/exit
conditions Vh = Vl = 0, we obtain the following expressions:β(rch θh + λcl θl ),r+λβ[(r + λ + µ)cl θl − µch θh ]wl = βyl + (1 − β)z +.
r+λwh = βyh + (1 − β)z − (1 − β)(r + δ)B +

8

(13)
(14)

Eq.(13) shows that wh is increasing in both θh and θl . Since higher θh implies larger vacancies
relative to high-skilled jobless workers, it is diﬃcult for ﬁrms to recruit a worker, which makes
workers more advantageous, which leads to higher wage. In addition, since higher θl implies larger
employment opportunity in type l labor market, high-skilled jobless workers do not care whether
the bargaining is approved or not, which also makes workers ascendant. In contrast, as (14) shows,
wl is increasing in θl but decreasing in θh . Since higher θh indicates that it is easy to ﬁnd a job
in type h market, which makes the state value of high-skilled unemployment higher. Low-skilledjobless workers want to move the state, however, since they have to be employed once in type lﬁrm, they would accept lower wage. Hence higher θl leads to lower wl . Note that, from (13), wh isdecreasing in bonus level B. This implies that since higher B makes workers more hunger to get
the bonus, they are willing to accept much lower wage.
Job Creation
As noted above, ﬁrms open vacancy until the expected proﬁt equals to zero in the steady state.
The fact is represented by the free entry/exit conditions, Vh = Vl = 0. Using them and eliminating
Jh and Jl from (6)-(9), we have the following expressions:θh = [(yh − wh )/ch (r + δ)]2 ,(15)

θl = [(yl − wl )/cl (r + δ + µ)]2 .

(16)

By making use of (13)-(16), we can obtain the equilibrium values of wh , wl , θh , and θl .
Government Budget
As described in Section 2.1, the expenditure for unemployment beneﬁts z and reemployment bonus
B is ﬁnanced by capitation tax and the tax is determined so as to balance the government budget
at any moment. The tax is determined by the following budget constraint:1

τ = uz + γuθh2 B,(17)where the ﬁrst term of the right hand side indicates the expenditure for unemployment beneﬁt z
and the second term represents the expenditure for reemployment bonus B.
Characterization of Equilibrium
Up to this point, we have obtained all expressions that characterize the equilibrium in our model.
The equilibrium consists of 8-tuple, {u, γ, ϕ, θh , θl , wh , wl , τ }. They are successively derived as
9follows. First, the wage wi and tightness of each market θi (i = h, l) is determined by wage
bargaining and ﬁrm’s optimal entry strategy (13)-(16). Second, the rate of unemployment u,
the fractions of high-skilled in jobless workers γ and in employed workers ϕ are derived by ﬂow
conditions (10)-(12). Finally, the capitation tax τ is determined so as to satisfy the government
budget (17). We can conﬁrm that the equilibrium is uniquely determined, which is shown inAppendix A.2.

3

Numerical Analysis

In the previous section we describe an economy with career-enhancing separation by assuming thatsuch a behavior arises: i.e., the CES condition Uh ≥ Wlh is assumed to hold. However, the guess
may not be true since the state values in the condition are endogenously determined in a generalequilibrium. In other words, the equilibrium stated in the previous section is just a candidate, but
not a certiﬁable equilibrium. Hence, before the analysis, we must rule out candidates of equilibrium
if the CES condition does not hold. We examine whether the CES condition holds or not withvarying policy variable, unemployment beneﬁts z, pecuniary rewards B or both since we focus on
the policies.
Before the examination, we can conﬁrm that two employment policies, one is increase in pecuniary bonus B and the other is decrease in unemployment beneﬁts z, have a diﬀerent eﬀect on thedecision about career-enhancing separation. Regarding the CES condition, Uh ≥ Wlh , the former
policy directly increases the left-hand side whereas the latter one directly decreases the left-handside (of course, both B and z indirectly aﬀect the state values). This suggests that the both
policies can operate as employment-boosting program, however, the policy that cuts unemployment beneﬁts is less apt to be career-enhancing policy. Since career-enhancing separation increases
the number of employed in high-skilled job (which is the most valuable state in our economy),consequently, pecuniary reward policy seems to be more desirable from a point of view of social
welfare.
Social Welfare Function and Parameters
As preparation for our analysis, deﬁne the measure of social welfare and set parameter values.
Following Cahuc and Lehmann (2000), we use an expected utility for each type of individual asa measure of welfare (for example, the measure of welfare for a high-skilled unemployed is rUh ).
In line with this manner, we deﬁne a measure of aggregate welfare by the weighted sum of all10

individual’s welfare:Ω = r{(1 − u)[ϕWh + (1 − ϕ)Wll ] + u[γUh + (1 − γ)Ul ]}.

(18)

It is diﬃcult to obtain some of parameter values we need, in particular, we do not have decisive
evidence on the rate of loss and acquirement of individual’s ability, λ and µ. In consideration ofplausibility, we set λ = 1.0 and µ = 0.4. This implies that the average duration of being highskilled when unemployed is 12 months (12 × (1/1.0) = 12.0) and that the average duration of being
low-skilled when she works at type l ﬁrm is 30 months (12 × (1/4.0) = 30.0). 4 In other words, ahigh-skilled worker loses her skills if she cannot ﬁnd a job within one year on average after ﬁred,
and a low-skilled worker acquires her skills if she works at a certain job for two and a half years
on average. Regarding the rest of parameter values, we set yh = 5.0, yl = 3.0, β = 0.5, r = 0.05,
ch = 1.0, cl = 0.5, δ = 0.2, z ∈ [0, 3.5], and B ∈ [0, 10]. 5
In the rest of the paper, we focus on the economy with CES and show results under (i) pecuniary
reward policy where B is policy variable given z, (ii) reduction in unemployment beneﬁt where z
is policy variable given B, and (iii) a mixture of the two policies where both B and z are policy
variables in order.
Pecuniary Reward
Here we examine eﬀects of reemployment bonus program on the unemployment rate and on social
welfare. To focus on this policy and since the level of unemployment beneﬁt is taken as given
here, we assume that z = 3.5. This implies the policy that reduces unemployment beneﬁt is not
implemented at all.
Numerical results are placed in Figure 1. (1-i) represents social welfare deﬁned in (18) withvarying bonus levels. It has hump-shaped, which implies that welfare improves as bonus increases
and after that worsens. This is because higher reward beneﬁts for bonus-qualiﬁed workers, however,it heavily burdens as the tax that ﬁnances increases. (1-ii) and (1-iii) represents unemployment
rate for high- and low-skilled, respectively. These are monotonically decreasing in bonus level as
predicted. Comparing (1-ii) to (1-iii), one can see that the number of low-skilled jobless workers
are more decreased that high-skilled. This causes through two channels. First, since higher bonus
4

These calculations are followed from Albrecht and Vroman (2002).

5

Note that since the elasticity of the matching technology with respect to vacancy is 0.5, β = 0.5 implies that
we focus on an eﬃcient economy in the sense that Hosios condition holds (see Hosios, 1990).

11makes bonus-qualiﬁed workers more apt to get a job, inﬂow to low-skilled unemployment (i.e.,
prolonged unemployment) reduces. Second, since low-skilled jobless workers are also induced toget a job as discussed before, the outﬂow from low-skilled unemployment increases and part ofthem move to high-skilled unemployment state due to career-enhancing separation, which results
in increase in high-skilled unemployed. Consequently, these two eﬀects extremely decrease thenumber of low-skilled unemployed and moderately decrease the number of high-skilled unemployed.
This result would suggest that, if tax burden is not too heavy, pecuniary bonus program seems toquite a policy as employment-boosting and career-enhancing.

H1-iL social welfare

H1-iiL unemployment rate of high-skilled H1-iiiL unemployment rate of low-skilled

3.15

0.06

0.2

3.1

0.058

0.175

3.05
2
2.95

4

6

8

10

B

0.056

0.15

0.054

0.125

0.052
2

2.9

4

6

8

10

B

2

4

6

8

10

B

0.075

Figure 1: reemployment bonus (z = 3.5)

Reduction in Unemployment Beneﬁt
We examine eﬀects of reduction in unemployment beneﬁt on social welfare and on unemployment
rate for high- and low-skilled here. To focus on this policy, we assume that there is no bonus
program (B = 0). Results are summarized in Figure 2, which are quite similar to the case ofreemployment bonus. Social welfare (2-i) has hump-shaped, which suggests that moderate level of
unemployment beneﬁt maximizes social welfare. If the level is too high, heavy tax burden worsens
individual’s welfare while if the level is too low, it directly reduces individual’s welfare. (2-ii) and(2-iii) show unemployment rate for high- and low-skilled unemployed, respectively. As discussed
above, lower beneﬁt leads to lower unemployment rate because jobless workers are more apt to
get a job to escape from the current state. 6 Note that generous beneﬁt decreases the numberof high-skilled unemployed, which results in drastic increase in low-skilled unemployed as (2-iii)
indicates.

6

In a model with search eﬀorts of workers, it can be easily conﬁrmed that lower unemployment beneﬁts leads tohigher job-search eﬀorts, which results in lower unemployment rate.

12

H2-iL social welfare

H2-iiL unemployment rate of high-skilled H2-iiiL unemployment rate of low-skilled
0.063

3.15

0.2

0.062

3.1

0.175

0.061

3.05

0.5

0.5

1

1.5

2

2.5

3

3.5

z

1

1.5

0.059

2

2.5

3

3.5

z

0.15
0.125
0.5

0.058

2.95
2.9

1

1.5

2

2.5

3

3.5

z

0.075

0.057

0.05

Figure 2: reduction in unemployment beneﬁts (B = 0)

Mixture of the Two Policies
Finally we analyze eﬀects of mixture of the two policies on social welfare and on unemployment
rate. To capture the shape of plane by numerical calculation, levels of pecuniary reward andunemployment beneﬁt are divided into 10 grids, which are shown in Figure 3. (3-i) shows welfare
level under arbitrary pairs of bonus B and unemployment beneﬁt z.
When z is low, welfare is monotonically decreasing in bonus level. Such a counterintuitive
situation occurs because bonus-qualiﬁed workers (high-skilled) can gain the bonus if they are hired
whereas no-bonus-qualiﬁed workers (low-skilled) are heavily imputed the tax burden though they
do not gain beneﬁts from pecuniary reward. In aggregate, reduction in welfare of low-skilled workers
dominates increase in welfare of high-skilled workers, which results in worsening of social welfare.
When z is high, however, welfare is hump-shaped with respect to bonus level. Consider an extreme
case, z = 3.5. Under such a generous unemployment compensation, there are many unemployedworkers. Given the situation, an increase in pecuniary reward has two opposite eﬀects on tax
rate. First, it straightforwardly increases the tax rate, which worsens welfare. Second, since there
are a lot of jobless workers with generous unemployment compensation, reduction in the numberof jobless workers by bonus program drastically decreases the tax burden due to unemployment
compensation, which improves welfare (even if the level of pecuniary reward is decent). When thebonus level is not too high, the second eﬀect dominates the ﬁrst one, which improves aggregate
welfare, and vice versa.
Note that, by comparing (3-i) with (1-i) and (2-i), the maximized level of welfare under mixtureof the two policies is higher than welfare level under a sole policy. The reason is straightforward.
Since the tax burden due to unemployment beneﬁt is reduced if the number of unemployed decreases, the two positive eﬀect on social welfare. Reduction in jobless worker’s welfare due tocurtailed unemployment beneﬁt is dominated by the positive eﬀect, so aggregate welfare improves.

13(3-ii) and (3-iii) show the unemployment rate of high- and low-skilled, respectively. Both rates
are nearly increasing in unemployment beneﬁt and decreasing in pecuniary reward. From (3-ii),
high unemployment beneﬁt and low bonus lead to reduction in the number of high-skilled unemployed, because it drastically increases the number of low-skilled jobless workers as (3-iii) indicates.
Note that, by comparing (3-iii) with (1-iii) and (2-iii), the number of low-skilled unemployed under
mixture of the policies is greatly lower than under a single policy, which suggests the usefulness of
mixture of the two policies.

H3-iL social welfare

H3-iiL unemployment rate of high-skilled

3.2
3.1
3
2.9
2.8

0.06

10
8
6

2
4
8

10

0.05
0.045

8
6

2

B

4

4

6
z

0.055

0.125
0.1
0.075
0.05

B

4

6
z

2

H3-iiiL unemployment rate of low-skilled

8

2
10

10

10
8
6

2
4
z

B

4

6
8

2
10

Figure 3: mixture of the two policies

4

Conclusion

This paper constructs a job-search model in which individual’s skills vary over time and examines
eﬀects of employment-boosting policy on unemployment rate and on social welfare. We show that,(i) lower unemployment beneﬁt and/or higher pecuniary bonus result in lower unemployment rate,(ii) higher unemployment beneﬁt and/or higher reward lead to higher social welfare, and (iii)
implement of the two policies can achieve higher social welfare than implement of single policy,
if the tax ﬁnances these transfers is not too high. These are because pecuniary bonus programinduces workers to seek a more productive job which is socially beneﬁcial . In summary, reductionin unemployment beneﬁt decreases unemployment rate at the cost of individual’s welfare, however,pecuniary bonus operates as both employment-boosting policy and career-enhancing policy, which
results in much higher social welfare.

14Appendix
A.1 . No-CES Economy
Here we describe the economy with no career-enhancing separation (no-CES). To distinguish theendogenous variables in this economy from CES economy, we denote the variables with tilde in1

no-CES economy. Matching technologies are given by M (γ̃, ũ) = (γ̃ ũṽh ) 2 and M ((1 − γ̃), ũ) =
1

1

−1

[(1 − γ̃)ũṽl ] 2 . So the rate of matching for type i (i = h, l) workers (ﬁrms) is θ̃i2 (θ̃i 2 ).
Asset value equations for workers and for ﬁrms are represented as follows:
1

(1′ )

rŨl = z − τ̃ + θ̃l2 (W̃ll − Ũl ),

1

(2′ )

rW̃h = w̃h − τ̃ + δ(Ũh − W̃h ),(3′ )rW̃lh = w̃l − τ̃ + δ(Ũh − W̃lh ),(4′ )rW̃ll = w̃l − τ̃ + δ(Ũl − W̃ll ) + µ(W̃lh − W̃ll ),(5′ )rŨh = z − τ̃ + θ̃h2 (W̃h + B − Ũh ) + λ(Ũl − Ũh ),−1

rṼh = −c + θ̃h 2 (J˜h − Ṽh ),

(6′ )

−
rṼl = −c + θ̃l 2 (J˜l − Ṽl ),(7′ )rJ˜h = yh − w̃h + δ(Ṽh − J˜h ),(8′ )rJ˜l = yl − w̃l + δ(Ṽl − J˜l ).

(9′ )

1

Flow conditions are given by:1ũ =δ(λδ + λµ + µθl2 )
1
2

1
2

1
2

,

(10′ )

θ̃l (λδ + λµ + µθ̃h ) + δ(λδ + λµ + µθ̃l )
1

γ̃ =

µθ̃l2
1
2

,

(11′ ).

(12′ )λδ + λµ + µθ̃l
1

ϕ̃ =

µθ̃h2
1
2

λδ + λµ + µθ̃hWage equations and job-creation conditions that determine equilibrium wage and market tightness
are derived as follows:β(rch θ̃h + λcl θ̃l ),r+λβ{[(r + λ)(δ + µ) + δµ]cl θ̃l − δµch θ̃h }w̃l = βyl + (1 − β)z +,(r + λ)(δ + µ)(13′ )

θ̃h = [(yh − w̃h )/ch (r + δ)]2 ,(15′ )

θ̃l = [(yl − w̃l )/cl (r + δ)]2 .

(16′ )w̃h = βyh + (1 − β)z − (1 − β)(r + δ)B +15(14′ )

The capitation tax is determined so as to balance the following government budget:(17′ )

τ̃ = ũz + γ̃ ũθ̃h B.
Welfare function in no-CES economy is given by:Ω̃ = r{(1 − u)[ϕWh + (1−ϕ)δ
Wll + (1−ϕ)µWlh ] + u[γUh + (1 − γ)Ul ]}.
δ+µδ+µ(18′ )

A.2 . Uniqueness of Equilibrium
As stated in section 2.3, the endogenous variables which construct the equilibrium are successively
determined. Recall that wage wi and market tightness θi (i = h, l) are ﬁrstly determined by(13)-(16). Eliminating wh and wl , we have the following expressions:1(r + δ)(r + λ)ch θh2 + β(rch θh + λcl θl ) − (1 − β)(r + λ)[yh − z + (r + δ)B] = 0,
1
2

(r + δ + µ)(r + λ)cl θl + β[(r + λ + µ)cl θl − µch θh ] − (1 − β)(r + λ)(yl − z) = 0,

(A.1)
(A.2)

which are simultaneous equations with respect to θh and θl . By implicit function theorem, we caneasily obtain the shape of these expressions:
∂θh
βλcl
=−
< 0,
−1
∂θl
1
(r + δ)(r + λ)c θ 2 + βrc
2

∂θh
=
∂θl

h h

(A.1′ )

h

− 12
1
(r
+
λ)(r
+
δ
+
µ)c
θ
+ βcl (r + λ + µ)
l
l
2

βµch

> 0.

(A.2′ )

Thus in θl -θh plane, (A.1) is monotonically decreasing while (A.2) is monotonically increasing,which guarantees uniqueness of the solution. Given the values of θh and θl , wages are determined11

by (15) wh = yh − (r + δ)ch θh2 and by (16) wl = yl − (r + δ + µ)cl θl2 , which is obvious that thesevalues are also uniquely determined. Since the rest of endogenous variables are evidently unique,uniqueness of the equilibrium is proved.

16

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