GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.98

Changing Worker’s States and Inefficient Decisions on Turnover

Keisuke Kawata

November 2009GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Changing Worker’s States and Ineﬃcient
Decisions on Turnover ∗Keisuke Kawata†

Abstract
This paper considers an on-the-job search model that includes wage bargaining
and employer-employee mismatch. There are two states of workers in relationshipto their ﬁt for a particular job, good match versus bad match (mismatch). Thesestates change in accordance with a stochastic process. There are two main results;
the ﬁrst is that the turnover level that workers ﬁnd optimal is lower than thesocially optimal level. The second is that the level of the ﬁrm’s entry is not optimaleven though the Hosios condition is hold. The ﬁrst result is clearly distinct fromprevious studies.
JEL classiﬁcation: J63; J81
Keywords: on-the-job search, wage bargaining, mismatch, turnover∗I especially thank Kenzo Abe, Koichi Futagami, Hiroshi Kitamura, Yusuke Oda, Fumio Ohtake,
Masaru Sasaki, Katsuya Takii, participants of Special Lectures in Economics (Macro Dynamics), Seminar: Entrepreneurship, Organization and Economic Development, Joint Seminar of Ohtake and Sasaki
at Osaka University, Osaka Workshop on Economics of Institutions and Organizations, Search Theory Workshop, and 2009 Japanese Economic Association Spring Meeting at Kyoto University, for theirhelpful comments and suggestions.
†
Graduate School of Economics, Osaka University E-mail address: hge005kk@mail2.econ.osakau.ac.jp

11. Introduction
Many authors have studied job-to-job transitions. Most of these studies are based on
on-the-job search models. On-the-job search models have welfare implications for theeﬃciency of turnover decisions. In the case of a wage bargaining model where ﬁrms
bargain a wage contract with an encountered worker (for example, [6], [11], and [13]), theturnover decisions by workers become socially optimal because high-productivity ﬁrms
oﬀer higher wages than low-productivity ﬁrms.
This paper’s purpose is to show that the turnover decision is not socially optimal,
even though a wage contract determined by the bargaining process, if it involves a mismatch between a worker and a ﬁrm in an on-the-job search model with wage bargaining.
This study also shows that the levels of job-to-job transitions are underrepresented in adecentralized economy. The central assumption of this paper is that the worker’s stateis changed by a shock, which implies that the productivity of a worker-ﬁrm pair is either
high or low. These results imply the necessity of a policy that induces workers to moveto a more productive ﬁrm, which is clearly distinct with the result of previous literatures.
Several examples support this assumption. Young-aged workers have a comparativeadvantage for a physical work while older workers have a comparative advantage forskilled work. When a young worker matches with a ﬁrm specializing in physical tasks, it
is a good match. However, if the worker grows old, this match is no longer good. Anotherexample is a change of preferences for working hours. The worker’s perceived disutility
from working hours can be triggered by some shocks, such as a childbirth and marriage.
If the disutility is low, she/he prefers to work long hours in a full-time job and earns
a high wage. On the other hand, if the disutility is high, she/he chooses to work shorthours in a part-time job earning low wages. In the latter case, childbirth switches her/his
preference to work, motivating her/him to move to a part-time job from a full-time job.
Some empirical studies ([3], [9], [7], and [12]) have suggested that, for working hours of
one’s choice, people switch between full-time jobs and part-time jobs.
I develop a tractable on-the-job search model with ﬁrms of type L or type S andworkers of state l or state s. Let (i, j) denote a pair of a state i worker and a type j2

worker. Pairs (l, S) and (s, L) suﬀer from mismatch and produce low output. Worker’s
states switch back and forth between type l and type s when a shock occurs, causing or
eliminating a mismatch. A mismatched worker, while on the job, is allowed to search fora new ﬁrm and decides whether to quit and move to a new ﬁrm or stay in the current
ﬁrm when ﬁnding the new ﬁrm. In this decision, the worker chooses either to eliminate
the current mismatch by moving or to wait for a good match in the future by staying
at the current ﬁrm. If the current pair is already a mismatch, the mismatch may beeliminated by a shock in the future. Alternatively, even through the current pair may
not be mismatched at present, they become mismatched by a shock later. Moreover, a
low-productivity pair may become a high-productivity pair after a shock, of vice versa.
In equilibrium, some workers stay in a low-productivity ﬁrm because they expect thatthe current pair will become a good-match in the near future. The worker’s behavior
regarding turnover is classiﬁed into three types; (i) they continue to stay type L ﬁrms
regardless of their worker type, (ii) they continue to stay type S ﬁrms regardless of theirworker type, and (iii) they continue to move no matter when the current pair becomesmismatched.
This paper ﬁnds two important implication from the present study. One regards aworker’s decision on turnover. Fewer workers choose the third type of behavior in adecentralized economy than in a socially eﬃcient economy. The reasoning behind thisphenomenon is that workers do not take into consideration the future poaching ﬁrm’s
capital gains from the future turnover. When a worker moves to a well-matched ﬁrm,
she/he will suﬀer from mismatch if her/his state changes after moving. She/he can theneliminate the new mismatch by moving to a new suitable ﬁrm again, but the value of thisturnover is underestimated because a part of the capital gains from this turnover goes tothe new poaching ﬁrm. Thus, the value of a well-matched pair is underestimated as well.
In contrast to this paper’s ﬁndings, other research found the worker’s decision on
turnover to be optimal in on-the-job search models with wage bargaining ([6], [11], and[13]). Similarly to this paper’s ﬁndings, however, they suggested that part of the capitalgains from turnover went to the poaching ﬁrm. The diﬀerence between this paper’s modeland the previous models is that productivity is changed in accordance with a stochastic
3

process in this model while productivity of a worker-ﬁrm pair was constant in the previous
models.
The second implication is that the number -of ﬁrm entering the labor market is notsocially optimal because both labor market tightness and the ratio of type L ﬁrms to type
S ﬁrms are distorted. In this paper, besides the well -known congestion externalities, a
new source of distortion occurs in that both ﬁrms and workers under estimate the ﬁrm’s
capital gain from poaching. This is the same intuition as the above proposition. This
proposition implies that the number of a ﬁrm’s entry is not socially optimal even if the
Hosios condition is held.
As another source of distortion, some studies showed that a worker’s turnover decision
was ineﬃcient. [11] showed that the level of turnover was ineﬃcient in Nash bargaining
in the double breach, and that they presented the eﬃciency wage bargaining model.
Regardless of the single breach case or their model of wage bargaining, this paper showsthat the level of turnover was ineﬃcient. [5] showed that the level of turnover exceeded
the optimal level because workers put excess eﬀort into the- on-the-job search. This
study’s model assumes that eﬀort for on-the-job search is exogenous. [2] showed that thelevel of turnover was below the optimal level when workers were risk-averse; this diﬀers
from the present study, in which workers are risk-neutral. In wage postings, where ﬁrms
post a wage contract and commit to it before meeting workers (For example, [4]), theturnover decision by workers is not socially optimal. Because a wage is dispersed, highproductivity ﬁrms may post lower wage than low-productivity ﬁrms and workers may
move to low-productivity ﬁrms from high-productivity ﬁrms.
This article is organized as follows. Section II presents the basic framework. Section
III deﬁnes the partial equilibrium in which both the rate of matches for workers andthe rate of matches for vacancies are exogenous, and then demonstrates the welfare
implications. In section IV, the paper expands its consideration to the general equilibrium
model in which, by the free entry condition, contract rates are endogenously determined.
This section shows both the existence of this equilibrium and the two welfare implications
for a ﬁrm’s entry.

42. The Model
A. Environment
This study uses a continuous time model with search. For simplicity, the study discusses
only a steady state. The number of workers is normalized to unity and they are inﬁnitely
lived. At any given point in time, workers are in one of two states: l state or s state.
A worker switches back and forth between the two states via exogenous shocks: πls
represents the rate of a shock that turns from state l to state s, while πsl is the rate of ashock that turns in the reverse direction. These rates vary among workers. Firms choose
either type L or type S at entry, given the two types of workers. The number of ﬁrms isdetermined by the free entry condition.
To focus attention on job-to-job transition patterns, no separation is assumed for
employed workers1 , implying that in the steady-state, no one is unemployed.
A pair’s output varies by the matched pattern. This study assumes that an L ﬁrmmatched with an l worker can gain higher proﬁt than one matched with an s worker, andin the same manner, an S ﬁrm matched with an s worker can gain higher proﬁt than one
matched with an l worker. Let an instantaneous surplus produced by an i state worker
and a j type ﬁrm denote yij . This study assumes that the order of instantaneous surplus
is deﬁned by ylL > ylS and ysS > ysL .
matching technology
To eliminate mismatches, workers are allowed to search while they are on the job. This
study, for simplicity, assumes that the cost of the search is zero, which encourages workers to constantly look for better jobs2 . Any vacancy and worker, regardless of being
unemployed or employed, come together via a matching technology µ(1, v) where 1 isthe number of workers and v is the number of vacancies. The function µ(1, v) is twice
diﬀerentiable and increasing in its arguments, and it exhibits constant returns to scale.
1

When separation occurs endogenously, the lower separation rate implies that the unemployment islower. Even though the exogenous job separation is incorporated into the model, the main implications
remain the same2[15] discusses the case of endogenous search intensity.

5The ﬂow rate of matches for a vacancy is then obtained µ(1, v)/v = q(θ), where q(θ)
is a diﬀerentiable decreasing function, and θ = v/1 is the tightness of the labor market. It also immediately follows from the constant returns to scale assumption that theﬂow rate of matches for a worker is µ(1, v)/1 = θq(θ) = p(θ). In addition, this study
also makes the standard Inada-type assumptions on µ(1, v), which ensures that θq(θ) is
increasing function and that limθ→∞ q(θ) = 0, limθ→0 q(θ) = ∞, limθ→∞ θq(θ) = 0, andlimθ→0 θq(θ) = ∞. Let γ denote a fraction of L type ﬁrms; accordingly, the eﬀective
arrival rate of a job oﬀer from a L type ﬁrm is γθq(θ). For convenience, I denote γθq(θ)
as pL , (1 − γ)θq(θ) as pS .
B. Wage Contract Determination
This study assumes that a wage contract is determined by the bargaining model oﬀeredby [6]. I discuss for other bargaining form in later. They constructed an explicit model
including an on-the-job search that allowed the incumbent ﬁrm to pose a wage counteroﬀer to its worker. This study uses the following notations: Ui is the value for an
unemployed worker of type i, Wij is the value for a worker of type i working with a typej ﬁrm, Vj is the value of a vacancy owned by a ﬁrm of type j, and Jij is the value of a
type j job ﬁlled in by a worker of type i. Additionally, this study deﬁnes Tij as the total
value of match, which implies Tij = Wij + Jij . Upon encountering a worker, a ﬁrm oﬀers
the worker a wage depending on the worker’s type, which is then written into a wage
contract.
The bargaining process for an employed worker is not as simple as the standard
process. When an employed worker contacts an outside ﬁrm, a trilateral wage bargaining
game among the poaching ﬁrm j, the employed worker i, and the incumbent ﬁrm j ′occurs.3 The game of bargaining has two steps described below.
The ﬁrst step is to play a second price auction game between the two ﬁrms. There
are two possible cases: the total surplus with the incumbent ﬁrm is higher than with thepoaching ﬁrm, Tij > Tij ′ , or a reverse case, Tij < Tij ′ . Because no employer will pay more3

In [6], the wage is determined by the Rubinstein-type inﬁnite-horizontal game of the strategic bargaining.

6

than match productivity, then the incumbent ﬁrm i can continue to employ the worker if
Tij > Tij ′ , and the poaching ﬁrm i′ can hire the worker if Tij < Tij ′ . Moreover, the ﬁrm
with larger T does not have an incentive to pay its worker over the maximum wage thatthe counterpart ﬁrm with smaller T can pay, which ensures W (w) = Tij ′ if Tij > Tij ′ , orW (w) = Tij if Tij < Tij ′ .
At the second step, the employed worker is allowed to renegotiate with a ﬁrm, so
the outcome of this renegotiation is determined through the standard Nash bargaining
process, in which the threat value of the worker is the value of employment under thewage which is oﬀered by another ﬁrm on ﬁrst step or the total value of a match. Then,if Tij > Tij ′ , the bargaining over the wage solves:
Wij (wij ) = Tij ′ + β(Tij − Tij ′ ),(1)

where β represents a worker’s bargaining power.
(1) shows that the value of the employed worker contracting the wage wij consists of
the outside option value, that is, the total value of matching between the i worker andthe j ′ ﬁrm, plus a share β of the social capital gain (Tij − Tij ′ ).
The result of the wage bargaining game is summarized as follows,(i) If Tij ≥ Tij ′ and Wij ≥ Tij ′ + β(Tij − Tij ′ ), the worker keeps the current wage
contract w with ﬁrm i, because the current wage is best for her/him. Then, after
the bargaining, the worker’s value W̄ij , the incumbent ﬁrm’s value J¯ij , and thepoaching ﬁrm’s value J¯ij ′ are,W̄ij = Wij , J¯ij = Jij , J¯ij ′ = 0.

(2)(ii) If Tij > Tij ′ and Wij < Tij ′ + β(Tij − Tij ′ ), the worker obtains a wage rise fromher/his current employer because the new wage determined by the wage bargaining
is more proﬁtable for the worker than the old wage.
Then, after the bargaining new values are,W̄ij = Tij ′ + β(Tij − Tij ′ ), J¯ij = (1 − β)(Tij − Tij ′ ), J¯ij ′ = 0.
7

(3)

(iii) If Tij < Tij ′ , the worker moves to the poaching ﬁrm. Then, after the bargaining
values are,W̄ij = Tij + β(Tij ′ − Tij ), J¯ij = 0, J¯ij ′ = (1 − β)(Tij ′ − Tij ).

(4)

The result (iii) means that the wage for the poaching ﬁrm is higher as the total value
with the incumbent ﬁrm is higher.

3. Partial Equilibrium
This section’s purpose is twofold; the ﬁrst purpose is to deﬁne and characterize partial
equilibrium of the decentralized economy in which the labor market tightness (θ) is
taken as given. The second purpose is to show the novel result, which is diﬀerent from[6], in that the level of turnover in market equilibrium is not eﬃcient. Analyzing thegeneral equilibrium framework with frictions when θ is determined endogenously is more
complex; therefore, this study ﬁrst reduces to the tractable partial equilibrium model,
later extending to the general equilibrium model.
A. Basic Bellman Equation
This study now develops expressions for various value functions. Workers and ﬁrms
discount the future at the common rate r. It is assumed, for convenience, that a worker’s
instantaneous utility is linear in the income ﬂow(the wage for an employed worker).
It begins with the value of a worker of type i employed in a type j ﬁrm as follows:
rWij = wij + πii′ (Wi′ j − Wij ) +∑pj ′ (W¯ij − Wij ), i ̸= i′ .

(5)j ′ =L,S

The second term on the right-hand side represents the expected capital change with rate
πii′ by switching to state i′ . The ﬁnal term is the expected capital gain with rate pj
by encountering a new ﬁrm through on-the-job search activities, and the values for eachtype of ﬁrms ﬁlling in a vacancy are given:rJij = yij − wij + πii′ (Ji′ j − Jij ) +∑j ′ ={L,S}

8pj ′ (J¯ij − Jij ).

(6)

These value functions are constructed in a similar manner to those for an employee. The
ﬁrst terms on the right-hand side of (6) represents instantaneous proﬁt, the second termis the expected capital change by switching to the worker’s state, and the ﬁnal term
indicates the expected capital loss with the rate pj throughout the wage competition
against another ﬁrm.
To focus attention on job-to-job transition patterns, this study assumes that both thevalue of unemployment and the value of holding an open vacancy are suﬃcient low; this
implies that no one chooses to be unemployed voluntarily.4 .
The next section incorporates the free-entry conditions into the model, which thereby
ensures that the value of holding an open vacancy is zero.
B. Equilibrium in Decentralized Economy
There are three patterns of the worker’s turnover behaviors, which is a policy function ofher/his states and types of the current and new ﬁrms, depending on the combination ofher/his parameters (πls , πsl ).
The ﬁrst behavior pattern is called ”Perfect Separation Behavior”(PSB) characterized
by a worker always trying to match with a suitable ﬁrm. If an s type worker originallyworking in an S type ﬁrm and turns out to be l type, she/he quits the S type ﬁrm andstarts to search for a job in L type ﬁrms on the job. Then, the condition for PSB is
TlL > TlS and TsL < TsS , which implies that the total value of a non-mismatched pair ishigher than that of a mismatched pair.
The second pattern is referred to as ”Stay at L-ﬁrm Behavior”(SLB) characterized by
any worker’s, regardless of worker type, preferring to stay in L ﬁrms. The s type worker
who frequently switches to l type (high πsl ) chooses to stay in the L type ﬁrm. If a l typeworker working in an L type ﬁrm turns out to be the s type, the worker remains in the
L type ﬁrm. Then, the condition of SLB is TlL > TlS and TsL > TsS , which implies thatthe total value of a worker-type L ﬁrm pair is higher than that of a worker-type S ﬁrm
pair.
4

In the next section, the value of holding an open vacancy (Vj ) is zero with the free-entry condition.
Meanwhile, the low value of unemployment is feasible if the beneﬁt of unemployment is low.

9The ﬁnal one is symmetric to the second one, referred to as ”Stay at S-ﬁrm Behavior”(SSB), which is characterized by any worker’s, regardless of the worker type, preferring to stay in S ﬁrms. Then, the condition of SSB is TlL < TlS and TsL < TsS , which
implies that the value of a worker-type S ﬁrm pair is higher than that of a worker-type
L ﬁrm pair.
According to the process of bargaining, the conditions determining these turnover
patterns are summarized as follows:• If TlL > TlS and TsL < TsS , the turnover pattern is PSB,
• If TlL > TlS and TsL > TsS , the turnover pattern is SLB,
• If TlL < TlS and TsL < TsS , the turnover pattern is SSB.
Workers change jobs if and only if the total value of a match with a newly encounteredﬁrm exceeds that of the current ﬁrm.
Next this study calculates the total values of a match and illustrate the ranges ofpoints (πls , πsl ) within which each turnover pattern occurs. The expression for the total
value TlL can be derived from the sum of (5) and (6),rTlL = ylL + πls (TsL − TlL ) + pS max{β(TlS − TlL ), 0}.

(7)

The ﬁnal term represents the expected capital gain by a type l worker’s movement to a
type S ﬁrm. The worker obtain the capital gain from her/his turnover if β(TlS − TlL ) is
positive. The new ﬁrm also obtains the capital gain, (1 − β)(TlS − TlL ). Then, the total
capital gain from her/his turnover is TlS − TlL .
Similarly, TsL , TlS and TsS are given:rTsL = ysL + πsl (TlL − TsL ) + pS max{β(TsS − TsL ), 0},(8)

rTlS = ylS + πls (TsS − TlS ) + pL max{β(TlL − TlS ), 0},(9)

rTsS = ysS + πsl (TlS − TsS ) + pL max{β(TsS − TsL ), 0}.

(10)

The turnover decision depends on the signs of the diﬀerences, TlL − TlS and TsS − TsL .
This study denote ylL − ylS as ∆l > 0 and ysS − ysL as ∆s > 0. The conditions for workers
10choosing the PSB are,∆l − πls (TsS − TsL ),r + πls + βpL∆s − πsl (TlL − TsL )TsS − TsL =.
r + πsl + βpS
TlL − TlS =

(11)
(12)

(11) and (12) imply the expected net gain by switching to a non-mismatched ﬁrmfrom a mismatched ﬁrm.
Using (7)-(10), a worker’s pattern of turnover is determined.
Proposition 1 The properties of the turnover patterns are illustrated as follows,∗∗∗(i) In the πls − πsl space, there exists a threshold (πls(πsl ), πls(πsl )) (formally deﬁned

TlL = TlS and TsL = TsP ) such that,(ylL − ylS )(πsl + βpS + r)(ylL − ylS )πsl + (βpL + r)(ysL − ysS )∗∗∗πls(πsl ) =and πls(πsl ) =.
ysS − ysL
ysS − ysL(ii) For any (πls , πsl ),∗∗∗• If πls(πsl ) > πls > πls(πsl ), then the behavior of the worker is PSB∗∗• If πls(πsl ) > πls , then the behavior of the worker is SLB∗(πsl ), then the behavior of the worker is SSB• If πls > πls

The proof of Proposition 1 appears in Appendix. This argument is illustrated in ﬁgure∗∗∗∗∗
1. πls(πsl ) and πls(πsl ) is increasing function, and πls> πls
for any πsl

Proposition 1 has two important implications. First, any worker who frequently
changes from type l to type s (which means πls is quite high) but seldom changes fromtype s to type l (which means πsl is quite low), chooses the SSB. Even through an ltype worker moves to an L ﬁrm to eliminate the mismatch, because the worker more
frequently changes to type s, the worker soon suﬀers from mismatch with the L ﬁrm bya change to type s. Therefore, this worker’s optimal behavior is not to move to the Ltype ﬁrm. Meanwhile, workers who are high πsl and low πls are more likely to choose the
SLB.

11

Secondly, the comparative statics exercise illustrates that the higher bargaining power∗∗of workers (higher β) or the higher arrival rate for workers (higher p) shifts locus πls(πsl )∗
upward but πls(πsl ) downward, causing the dominance of PSB. This implies that thediﬀerences, TlL − TlS and TsS − TsL , are increasing functions for β and p.
There are two eﬀects of the bargaining power on the turnover decisions for workers
whose optimal choice is PSB. The study is restricted to the turnover behavior of an lworker in an S ﬁrm. Currently, they are mismatched. According to the third term on theright-hand side of (9), an increase in the bargaining power raises the mismatched value
TlS . From (11), however, the total capital gain TlL − TlS decreases. This eﬀect is called
the direct eﬀect.
The second eﬀect of the bargaining power comes through a channel of the direct
eﬀect for an s worker. According to (8), an increase in the bargaining power raises themismatched value TsL , which leads to an increase in TsL − TlL from the second term on
the right-hand side of (7), thereby the non-mismatched value TlL . It implies a reductionof the extent of the capital loss. Then, the total capital gain TlL − TlS increases. Thiseﬀect is called the indirect eﬀect.
Similarly, there are the direct and indirect eﬀects for a s type worker; TsS − TsL
decreases from the direct eﬀect but increases from the indirect eﬀect. The sign of TlL −TlS
depends on the measure of TsS − TlS ; that is, TlL − TlS is positive if πls (TsS − TlS ) < ∆l .
The indirect eﬀect has a signiﬁcant role in eﬃciency (discussed in detail in subsection C).
The same argument can be applied in a case of an increase in p.
The next section deals with the solution for a social planner problem, and then the
study demonstrates that the turnover decision in the decentralized economy is diﬀerent
from the social optimal decision.
C. Social Planner Problem for the Turnover
The purpose of this sub-section is to illustrate the eﬃcient conditions of the turnover andto show that the level of the turnover in a decentralized economy is lower than the level
determined by the social planner.
The number of type l workers are denoted as El and type s workers similarly are12denoted as Es . For convenience, the numbers reach the level of steady state, then El
equals πsl /(πsl +πls ) and Es equals πls /(πsl +πls ) with steady state conditions πsl (1−El ) =πls Es . Let Eij denote the number of type i workers in type j ﬁrms.
The total surplus can be written as:∫ ∞{}e−rt ElL ylL + ElS ylS + EsS ysS + EsL ysL dt − c(vL + vS ).
TS =

(13)

t=0

The total surplus is equal to the total ﬂows of net output minus the vacancy cost. The
eﬃcient turnover is calculated to maximize the TS subject with the following equations:∫ 1∫ 1E˙ij =ėij dπls dπsl , i = {l, s}, j = {L, S}.
(14)
0

0

where ėij represents the law of motion of the number of type i workers working in a typej ﬁrm characterized by (πls , πsl ).
The law of motion of elL and esS are,elL
˙ = pL µL elS + πsl esL − (πls + (1 − µL )pS )elL

(15)

esS
˙ = pS µS esL + πls elS − (πsl + (1 − µS )pL )esS ,(16)

where µL ∈ [0, 1] is the probability that a type l worker who faces a choice between a L
ﬁrm or a S ﬁrm decides to move to the L ﬁrm, and µs ∈ [0, 1] is the probability that a
type s worker facing the same choice decides to move to the S ﬁrm.
TS is maximized with respect to µL and µS , subject to (13), (14), els = el − elL , andesL = es − esS . The optimal µL and µS are as follows,(πsl + pS + r)∆l


1
if
π
<
ls


∆s


(πsl + pS + r)∆lµL =(0, 1) if πls =

∆s



(πsl + pS + r)∆l
 0 if πls >
,
∆s
and


πsl ∆l + (pL + r)∆s
1 if πls >


∆s


πsl ∆l + (pL + r)∆s
µS =(0, 1) if πls =

∆s



π
∆
+
(p
L + r)∆s
 0 if πls < sl l
.
∆s
13

(17)

(18)

(Appendix B contains more detail.)
Proposition 2 abstracts these conditions.
∗
Proposition 2 There are the two loci of points such that π̂ls
(πsl ) = ∆l (πsl + pS + r)/∆s∗∗(πsl ) = (∆l πsl + ∆s (pL + r))/∆s . The eﬃcient turnover patterns are characterizedand π̂ls

below;∗∗∗• If π̂ls(πsl ) > πls > π̂ls
(πsl ), then the turnover pattern of the worker is PSB∗∗• If π̂ls(πsl ) > πls , then the turnover pattern of the worker is SLB∗(πsl ), then the turnover pattern of the worker is SSB• If πls > π̂lsThis argument is illustrated in ﬁgure 2. The social planner deals with the cost of mismatch
because that if a worker’s type changes, her/his productivity is lowered. If pj is higher,
the cost of the mismatch is lower because the worker’s transition to a new job is very
easy, and the expected period of mismatch is very short.
The study compares the social planner’s solution and the decentralized economy’ssolution.
Proposition 3 If ﬁrms have the bargaining power (β ̸= 1), the eﬃcient domain of PSB
is larger than the domain of PSB in the decentralized economy.
The shaded areas in ﬁgure 2 represent diﬀerences between the domain of PSB in thesocial planner and the domain of PSB in the decentralized economy.
Proposition 3 is clearly diﬀerent from the result of [6]. In their model, independentlyof the worker’s bargaining power, the social planner’s decision for the turnover is equal
to the worker’s decision. Proposition 3 implies that there exist workers who should have
moved to non-mismatch poaching ﬁrms.
From (7) to (10), the worker’s capital gains from turnover is part of the total capital
gain. Meanwhile, the social planner considers not only the worker’s capital gains but also
the poaching ﬁrm’s capital gains. Then, the value of turnover for the social planner is thetotal capital gain. This means that workers underestimate the value of the third term on
14

the right-hand side of (7) to (10). Thus, from based upon Proposition 1, the diﬀerences,
TlL − TlS and TsS − TsL , are smaller than those determined by the social planner.
In a case where workers have monopolistic bargaining power β = 1, there is nodiﬀerence between their decision and the social planner’s decision. This results from theworker’s gain being equal to the total capital gain from turnover (and the poaching ﬁrm’s
gain is zero), then the diﬀerences in the decentralized economy equal the one determinedby the social planner.
In [6], similarly to this study’s model, the worker’s value of turnover is diﬀerent
from the social planner’s value. However, their model showed only the direct eﬀect on
TlL −TlS and TsS −TsL through a change of β in the denominator of (11) and (12) because
a productivity change resulting from a shock is not incorporated into their model. The
direct eﬀect does not change the sign of the diﬀerences, TlL −TlS and TsS −TsL . Moreover,in their model, both workers and the social planner always look for a better match andmove to a high productivity ﬁrm because the matching quality dose not switch back andforth between mismatch and non-mismatch in a given ﬁrm. Therefore, the order of thetotal values of each ﬁrm for a worker coincides with that for the social planner’s.
This proposition implies that labor market policies that cause turnover may improvea social eﬃciency.

4. General Equilibrium
In this section, the study expands to consider the general equilibrium model in whichlabor market tightness is determined by the free entry condition and derives welfare
implications for the ﬁrm’s entry.
A. Free Entry Condition
The Inada condition for the matching function guarantees an existence of the ﬁrm’s entry.
This study demonstrates that there are both L type and S type ﬁrms.
A ﬁrm with an unﬁlled vacancy can gain a proﬁt when the ﬁrm meets an employedworker who is hired by a diﬀerent type of ﬁrm and it succeeds at poaching her/him. If

15

the ﬁrm poaches an employed worker from the same type of ﬁrms, it does not gain at all
because the ﬁrm has to compete with the incumbent ﬁrm, leading to zero proﬁt.
Then, there are two cases where a type L ﬁrm can gain a proﬁt. The ﬁrst case is thatthe ﬁrm meets a type l worker who chooses PSB and works in a type S ﬁrm. The secondcase is that the ﬁrm meets any type of worker who chooses SLB and works in a type S
ﬁrm.
Thus, the value of a type L ﬁrm with a vacancy, VL is as follows:[∫ ∫ πls∗rVL = − c + q(1 − β)(TlL − TlS )elS (πls , πsl )dπls dπsl
∫ ∫ πls∗
+

∗∗
πls

{

]
}
(TlL − TlS )elS (πls , πsl ) + (TsL − TsS )esS (πls , πsl ) dπls dπsl ,(19)where c is the ﬁrm’s instantaneous search cost. The ﬁrst double integral term is the
expected capital gain by meeting a type l employed worker who chooses PSB and works
in a type S ﬁrm, and the second double integral term is the expected capital gain bymeeting a worker who chooses SLB.
Similarly, the value of a type S ﬁrm with a vacancy, VS is as follows,[∫ ∫ πls∗(TsS − TsL )esL (πls , πsl )dπls dπslrVS = − c + q(1 − β)
∗∗
πls
∫ ∫
]
{
}
+
(TlS − TlL )elL (πls , πsl ) + (TsS − TsL )esL (πls , πsl ) dπls dπsl .

(20)

∗
πls

This study considers the steady-state equilibrium where both L type and S type ﬁrms
exist. There are two conditions that allow ﬁrms to enter freely. The ﬁrst condition isthat the expected value of a type L vacancy equals that of a type S vacancy, that is,
VL = VS . From (7) to (10), this condition is rewritten,∫ ∫ πls∗∗∫ ∫ πls∗
Adπls dπsl +

∗∗
πls

∫ ∫
Bdπls πsl −

16

∗
πls

Cdπls dπsl = 0,(21)where{(βpL + πsl + r)∆l − πls ∆s }elS − {(βpL + πls + r)∆s − πsl ∆l }esS,
(pL + πls + r)(βpL + πsl + r) − πls πsl{(βpS + πsl + r)∆l − πls ∆s }elS − {(βpL + πls + r)∆s − πsl ∆l }esL
B=,
(βpL + πls + r)(βpS + πsl + r) − πls πsl{(βpS + πls + r)∆s − πsl ∆l }esL − {(pS + πsl + r)∆l − πls ∆s }elL.
C=
(βpS + πls + r)(βpS + πsl + r) − πls πslA=

The calculation of the number of each type worker eij in the steady state appears inAppendix C. In the steady state, no worker who chooses SLB works in S ﬁrms, and in asimilar manner, no worker who chooses SSB works in L ﬁrms.
The proposition for existence of this equilibrium can be derived.
Proposition 4 When there are workers who chooses PSB, there exist both type L ﬁrms
and type S ﬁrms.
The logic behind proposition is very simple. A ﬁrm can gain proﬁt if and only if the
ﬁrm can poach a worker from another type ﬁrm. Then, if the number of type L ﬁrms
increases, the expected capital gain of a type S ﬁrm’s vacancy increases because it iseasier to ﬁnd an employed worker working in a type L ﬁrm. Meanwhile, that of a type L
ﬁrm’s vacancy decreases because it it more likely to meet an employed worker from thesame type ﬁrm. Moreover, if there are no type L ﬁrms, type S ﬁrms can not gain proﬁts
at all, which discourages them from entry. This mechanism guarantees that there exist
the two types of ﬁrms.
The second free entry condition is that both expected values of type L vacancies andtype S vacancies are zero. Then, from (19) and (20)∫ ∫ πls∗[∫ ∫ πls∗c =q(1 − β)Adπls dπsl +]{(βpS + πsl + r)∆l − πls ∆s }elSdπls dπsl ,∗∗ (βpL + πls + r)(βpS + πsl + r) − πls πsl
πls
(22)
∫
∫
∫
∫
∗
]
[
πls
{(βpL + πls + r)∆s − πsl ∆l }esL
c =q(1 − β)dπls dπsl .
Cdπls dπsl +∗∗∗ (βpL + πls + r)(βpS + πsl + r) − πls πsl
πls
πls
(23)
17

The Inada condition ensures the existence of the interior of solution according to (22)and (23).
B. Social Planner Problem for the Firm Entry
This study compares the level of ﬁrm’s entry in the decentralized economy to that of thesocial optimal level.
From pL = γp(θ) and pS = (1 − γ)p(θ), the social planner’s problem for the ratio of
the type L vacancies to type S vacancies (= γ) and the market tightness θ is deﬁned asfollows,

∫ ∞
e

max
γ,θ

−rt

{∫ ∫

}
{ylL elL + ylS (el − elL ) + ysS esS + ysL (es − esS )}dπls dπsl ) dt (24)0s.t. elL˙ = pL µL elS + πsl esL − (πls + (1 − µL )pS )elL(25)

and esS
˙ = pS µS esL + πls elS − (πsl + (1 − µS )pL )esS ,(26)

The solution for the above problem appears in Appendices D and E.
The optimal level of γ is,∫ ∫ π̂ls∗∗∫ ∫ π̂ls∗∫ ∫′′A dπls dπsl +B dπls πsl −
∗∗
π̂ls

∗
π̂ls

C ′ dπls dπsl = 0(27){(pL + πsl + r)∆l − πls ∆s }elS − {(pL + πls + r)∆s − πsl ∆l }esS,
(pL + πls + r)(pL + πsl + r) − πls πsl{(pS + πsl + r)∆l − πls ∆s }elS − {(pL + πls + r)∆s − πsl ∆l }esLB′ =,(pL + πls + r)(pS + πsl + r) − πls πsl{(pS + πls + r)∆s − πsl ∆l }esL − {(pS + πsl + r)∆l − πls ∆s }elLC′ =.
(pS + πls + r)(pS + πsl + r) − πls πsl

where, A′ =Comparing with (21), A = A′ , B = B ′ , and C = C ′ when workers have a monopolistic
bargaining power (β = 1). Then the study can obtain the following,
Proposition 5 When workers have monopolistic bargaining power(β = 1), the ratio of
the type L vacancies to type S vacancies is optimal, but when the ﬁrms have any bargaining
power(β < 1), the ratio is not optimal.
The logic is similar to that of proposition 2. In a decentralized economy, the values ofpoaching workers which are determined by, TlL − TlS and TsS − TsL , are underestimated18

because workers do not consider a poaching ﬁrm’s capital gain from the future turnover.
In addition, the extent of the underestimation is diﬀerent between TlL −TlS and TsS −TsL .
It implies that the diﬀerence between the optimal number of ﬁrms and the number ofﬁrms in a decentralized economy is diﬀerent between the type L vacancies and type S
vacancies. Then, the ratio of type L vacancies to type S vacancies in a decentralized
economy does not equal the social optimal ratio.
In a search model with two sectors, [1] showed a similar proposition, the ratio of aﬁrm’s entry in the two sectors was not optimal. Unlike this study’s model, he assumed
that the ﬁxed cost was diﬀerent in each sectors. Then, the holdup problem lowered the
ﬁrm’s entry in the high ﬁxed cost sector.
The optimal level of θ is,∫ ∫ π̂ls∗[∫ ∫ π̂ls∗′′
A dπls dπsl +c =µ]{(pS + πsl + r)∆l − πls ∆s }elSdπls dπsl , (28)∗∗ (pL + πls + r)(pS + πsl + r) − πls πslπ̂ls∫ ∫ π̂ls∗[∫ ∫]{(pL + πls + r)∆s − πsl ∆l }esL′′c =µC dπls dπsl +dπls dπsl . (29)∗∗ (pL + πls + r)(pS + πsl + r) − πls πsl
∗
π̂ls
π̂ls

Comparing these two equations with (22) and (23) is more complicate. For the terms inthe bracket, (28) and (29) equal (22) and (23) if β = 1. This follows a similar logic tothat of Proposition 5.
Meanwhile, for the term outside the brackets, (28) and (29) equal (22) and (23),respectively when β = −(q ′ (θ)θ)/q(θ), which called Hosios condition5 .
Proposition 6 In a decentralized economy, the level of market tightness cannot achieve
the optimal level without unless the elasticity of the matching function is one.
This proposition implies that the level of market tightness is not optimal even if the
Hosios condition is held. This model has two sources of distortion for a ﬁrm’s entry. The
ﬁrst is the congestion externality, which can be eliminated by the Hosios condition. The
second is the underestimation of the total capital gain from poaching that is the sum of
the worker’s capital gain and ﬁrm’s capital gain. According to proposition 2, the total
capital gain from turnover is eﬃcient only if workers have monopolistic bargaining power.
5

See Hosios [10] and Pissarides [14].

19Thus, if and only if workers have monopolistic bargain power and the elasticity of thematching function equals one, market tightness is the optimal level.
C. Alternative Bargaining Procedure
The main propositions, Proposition 3, 5, and 6, can be applicable to [8]’s, [11]’s, and [13]’sbargaining procedure. The wage is determined by standard general Nash bargaining in[8] and [13]. In [11]6 , either workers or ﬁrms have monopolistic bargaining power. Theirbargaining procedures have two properties similar to the procedure in this paper. (i)workers move to a ﬁrm of the higher total value. (ii) total value functions under [8]’sand [13]’s are the same form to this paper’s , and total value functions under [11]’s are1
β = case of this paper’s. Then, from the same logic in this paper, proposition 1 and2later propositions hold under [8]’s, [11]’s, and [13]’s.

5. Conclusion
This study has developed an on-the-job search model with wage bargaining that involves
a shock that changes the worker’s states. The study demonstrated the welfare implication
when the level of turnover is socially ineﬃcient. This ineﬃciency results from a part ofcapital gain from turnover going to a poaching ﬁrm, discouraging a worker from switchingﬁrms. Besides the welfare implication on turnover decisions, this study demonstrated thatthe levels of the ﬁrm’s entry were not optimal. Also, the ﬁrm’s capital gain from poaching
decreases because another poaching ﬁrm in the future may capture a part of future capitalgain from the original turnover.
Finally, this study brieﬂy discusses a policy implication regarding a hiring subsidy.
A hiring subsidy is deﬁned here as a temporary subsidy to ﬁrms in hiring a worker.
This subsidy causes workers to turn over because the total value of a worker increases.
According to proposition 2, the level of turnover is lower than the optimal level, making
this eﬀect positive for social welfare.
Beside the decision of turnover, the hiring subsidy aﬀects the level of a ﬁrm’s entry.
6

Their model analyzes more general case, involving the double breach.

20

The hiring subsidy encourages the ﬁrm’s entry, because the poaching ﬁrms take a part ofthe subsidy. It has an ambiguous eﬀect on the social surplus because there are two possible cases of a ﬁrm’s entry; one case is the over-entry and the other case is the under-entry,
depending on the worker’s bargaining power. In the case where the worker’s bargaining
power is greater than the Hosios condition, the hiring subsidy certainly improves social
welfare according to Proposition 6. This results from the level of the ﬁrm’s entry being
lower than the optimal level from the two sources, the congestion externality and underestimation of the capital gain from turnover. Then, the subsidy can correct both theturnover decision and the level of the ﬁrm’s entry.

21

References[1] D. Acemoglu, Good jobs versus bad jobs, J. Lab. Econ 19 (2001), 1-21.
[2] G. Bertola, A pure theory of job security and labour income risk, Rev. Econ. Stud71 (2004), 43-61.
[3] R. Blundell, M. Brewer, M. Francesconi, Job changes and hours changes: understanding the path of labor supply adjustment, J. Lab. Econ 26 (2008), 421-453.
[4] K. Burdett, D.T. Mortensen, Wage diﬀerentials, employer size, and unemployment,
Int. Econ. Rev 39 (1998), 257-273.
[5] K. Burdett, R. Imai, R. Wright, Unstable relationships, Frontiers of Macroeconomics,
1 (1) (2004), pp. 44.
[6] P. Cahuc, F. Postel-Vinay, J.M. Robin, Wage bargaining with on-the-job search:
theory and evidence, Econometrica 74 (2006), 323-364.
[7] S. Connolly, M. Gregory, Moving down: women’s part-time work and occupationalin Britain 1991-2001, Econ. J 118 (2008), F52-F76.
[8] J.J. Dolado, M. Jansen, F.J. Jimeno, On-the-job search in a matching model withheterogeneous jobs and workers, Econ. J 119 (2009), 200-228.
[9] R. Euwals, Female Labor Supply, Flexibility of working hours, and job mobility,
Econ. J 111 (2001), C120-C134.
[10] A. Hosios, On the eﬃciency of matching and related models of search and unemployment, Rev. Econ. Stud 57 (1990), 279-98.
[11] N. Kiyotaki, R. Lagos, A model of job and worker ﬂows, J. Polit. Economy 115(2007), 770-819.
[12] A. Manning, B. Petrongolo, The part-time pay penalty for women in Britain, Econ.
J 118 (2008), F28-F58.
22

[13] C, Pissarides, Search unemployment with on-the-job search, Rev. Econ. Stud 61(1994), 457-475.
[14] C. Pissarides, Equilibrium Unemployment Theory, Cambridge, MA: Basil Blackwell,2000.
[15] F. Postel-Vinay, J.M. Robin, To match or not to match? optimal wage policy withendogenous worker search intensity, Rev. Econ. Dynam 7 (2004), 297-331.

23

Appendix

A

Proof of Proposition 1

Conditions for PSB
Conditions for PSB are TlL ≥ TlS and TsL ≤ TsS . Total values of PSB are,rTlL = ylL + πls (TsL − TlL ),rTsL = ysL + πsl (TlL − TsL ) + pS β(TsS − TsL ),rTlS = ylS + πls (TsS − TlS ) + pL β(TlL − TlS ),rTsS = ysS + πsl (TlS − TsS ).

(A-1)
(A-2)
(A-3)
(A-4)

Conditions for PSB are rewritten by taking the following steps;
step 1 From (A-1) and (A-3), TlL (πls , πsl ) − TlS (πls , πsl ) is,
TlL (πls , πsl ) − TlS (πls , πsl ) =∆l + πls (TsL (πls , πsl ) − TsS (πls , πsl )),r + pL β + πls(A-5)

where ∆l = ylL −ylS and ∆s = ysS −ysL . From TsL ≤ TsS , TlL (πls , πsl )−TlS (πls , πsl )is a decreasing function for πls . TlL −TlS < 0 if πls → ∞, and TlL −TlS > 0 if πls = 0.
∗∗∗
Then there are unique πls(πsl ) as TlL (πls
, πsl ) = TlS (πls
, πsl ) and TlL (πls , πsl ) >∗TlS (πls , πsl ) if and only if πls (πsl ) > πls . Using (A-1) to (A-4), simple algebra gives,∗
πls(πsl ) =∆l (πsl + r + βpS )
.
∆s

(A-6)

step 2 The condition TsL ≤ TsS is rewritten. From (A-2) and (A-4), TsS (πls , πsl ) −TsL (πls , πsl ) is;
TsS (πls , πsl ) − TsL (πls , πsl ) =∆s + πsl (TlS (πls , πsl ) − TlL (πls , πsl ))r + pS β + πsl(A-7)

Similarly to step 1, TsS (πls , πsl ) − TsL (πls , πsl ) is the decreasing function for πslfrom TlS (πls , πsl ) < TlL (πls , πsl ). TsS (πls , πsl ) − TsL (πls , πsl ) < 0 if πsl → ∞, and∗∗TsS (πls , πsl ) − TsL (πls , πsl ) < 0 if πsl = 0. Then there are πsl(πls ); TsS (πls , πsl) =∗TsL (πls , πsl ) Using (A-1) to (A-4), simple algebra gives:πls =∗∆l πsl(πls ) + ∆s (r + βpL ).
∆s∗
TsS > TsL if the worker’s parameters (πls , πsl ) satisfy πsl(πls ) > πsl .

24

(A-8)

step 3 From step 1 and step 2, a worker’s behavior is PSB if and only if her/his pa∗∗
rameters (πls , πsl ) satisfy the conditions, πls < πls(πsl ) and πsl < πsl(πls ). Thus,
conditions for PSB are:∆l (πsl + r + βpS )∆l πsl + ∆s (r + βpL )> πls >
.
∆s
∆s

(A-9)

Conditions for SLB
Conditions of SLB are TlL ≥ TlS and TsL ≥ TsS . Total values of SLB are as follow:
rTlL = ylL + πls (TsL − TlL ),rTsL = ysL + πsl (TlL − TsL ),rTlS = ylS + πls (TsS − TlS ) + pL β(TlL − TlS ),rTsS = ysS + πsl (TlS − TsS ) + pL β(TsL − TsS ).

(A-10)(A-11)(A-12)(A-13)

TlL > TsL from ylL > ysL , and TlS − TsS is {pL β(TlL − TsL )}/{pL β + πls + πsl + r} from (A12) and (A-13). Then TlS − TsS < TlL − TsL , so this is rewritten as TsL − TsS < TlL − TlS .
∗Thus TlL > TlS must hold if TsL > TsS hold. From simple algebra, πls = {∆l πsl(πls ) +∗∆s (r + βpL )}/∆s and TsL ≥ TsS if πls > πls(πsl ). Thus, a worker’s behavior is SLB if andonly if the worker’s parameters (πls , πsl ) satisfy πls < {∆l πsl + ∆s (r + βpL )}/∆s .
Conditions for SSB
The conditions of SSB are TlL ≤ TlS and TsL ≤ TsS . Then, total values of SSB are as
follow,rTlL = ylL + πls (TsL − TlL ) + pS β(TlS − TlL ),rTsL = ysL + πsl (TlL − TsL ) + pS β(TsS − TsL ),rTlS = ylS + πls (TsS − TlS ),rTsS = ysS + πsl (TlS − TsS ).

(A-14)(A-15)(A-16)(A-17)

By similar calculation to the condition for SLB, a worker’s behavior is SSB if and only
if, the worker’s parameters (πls , πsl ) satisfy πls ≥ {∆l (πsl + r + βpS )}/∆s .

B

Proof of Proposition 2

The Hamiltonian function is as follows:
H =e−rt [el ylS + ∆l elL + es ysL + ∆s esS ]−λ1 [pL µL (el − elL ) + πsl (es − esS ) − (πls + (1 − µL )pS )elL ]−λ2 [pS µS (es − ess ) + πls (el − elL ) − (πsl + (1 − µS )pL )esS ].
25

(B-1)

The optimal conditions are:µL = 1 ⇐⇒ −λ1 (pL (el − elL ) + pS elL ) ≥ 0,
µS = 1 ⇐⇒ −λ2 (pS (es − esS ) + pL esS ) ≥ 0,

(B-2)
(B-3)

e−rt (ylL ylS ) + λ1 (pL µL + πls + (1 − µL )pS ) + λ2 πls − λ̇1 = 0,(B-4)

e−rt (ysS − ysL ) + λ1 πsl + λ2 (p2 µS + πsl + (1 − µS )pL ) − λ̇2 = 0.

(B-5)

According to (B-2) and (B-3), there are µL = 1 if and only if λ1 ≤ 0 and µS = 1 if
and only if λ2 ≤ 0.
From (B-4) and (B-5), λ˙1 = −rλ1 , and λ˙2 = −rλ2 , the sign of both λ1 and λ2 are:(r + πsl + µS pL + (1 − µS )pS )∆l,∆sπsl ∆l − (r + µL pL + (1 − µL )pS )∆s
λ2 < 0 ⇐⇒ πls >.
∆s
λ1 < 0 ⇐⇒ πls <

(B-6)
(B-7)

For any πsl , the right hand of (B-6) is higher than the right hand of (B-7), then the
optimal conditions for µL , µS are:(r + πsl + pS )∆l,∆sπsl ∆l − (r + pL )∆sµS = 1 ⇐⇒ πls >.
∆sµL = 1 ⇐⇒ πls <

C

(B-8)
(B-9)

Steady State Condition

The following calculation focuses on the ﬂow of workers who have any combination ofparameters (πls , πsl ). Let Nl be the fraction of state l workers and Ns be the fraction ofstate s workers. In the steady state, the fraction of changing type l equal the fraction ofchanging type s, then the steady state conditions are πsl Ns = πls Nl and Ns = 1 − Nl .
Then, in the steady state, the fraction of type l workers is Nl = πsl /(πls + πsl ) and thefraction of type s workers is Ns = πls /(πls + πsl ).
The fraction of state i workers working in type j ﬁrms is denoted as eij . According
to the assumption that both the endogenously and the exogenously job separations do
not occur, the proportion of unemployment also converge to 0. Then, elL + elS = Nland esL + esS = Ns . I present the steady-state condition for each worker’s behavior as
discussed below.
Perfect Separation Behavior
If the worker’s behavior is PSB, the steady state condition for the elL is,pL elS + πsl esL = πls elL .
26

(C-1)

The right hand of (C-1) is outﬂow to elL , which is the fraction of workers in type L ﬁrms
who change to s state from l state from a shock. The left hand is inﬂow to elL , the ﬁrst
term in the left hand is the fraction of l state workers who move to type L ﬁrms fromtype S ﬁrms, and the second term in the left hand is the fraction of workers in type L
ﬁrms who change to l state from s state from a shock.
Similarly, the condition for the esS is:pS esL + πls elS = πsl esS .

(C-2)

By elS = Nl − elL and esL = Ns − esS , the fraction elL and esS are:Ns πls πsl + Nl (πls (pL − πsl ) + pL pS ),πls pL + (πsl + pL )pS
Nl πls πsl + Ns (πsl (pS − πls ) + pL pS )esS =.
πls pL + (πsl + pL )pS
elL =

(C-3)
(C-4)

Thus, the steady state condition of workers choosing PSB is characterized by Nl , Ns ,(C-3), and (C-4).
Stay L Firm Behavior
In the steady state, the employed workers in type S ﬁrms converge to 0 if the workers’s
behaviors are SLB. This is because that workers in type S ﬁrms continue to move to L
ﬁrms. Then elL = Nl = πsl /(πls + πsl ), esL = Ns = πls /(πls + πsl ), and elS = esS = 0.
Stay S Firm Behavior
Similarly, the employed workers in type L ﬁrms converge to 0. Then elS = Nl = πsl /(πls +πsl ), esS = Ns = πls /(πls + πsl ), and elL = esL = 0.

D

The Social Planner Problem for the Type L Firms
to Type S Firms Ratio

Let λ1 (πls , πsl ) and λ2 (πls , πsl ) as shadow values of each parameters πls and πsl . The
Hamiltonian function is,∫ ∫−rt
H =e [{el ylS + ∆l elL + es ysL + ∆s esS }dπls πsl ]∫ ∫−λ1 (πls , πsl )[pL µL elS + πsl esL − (πls + (1 − µL )pS )elL ]dπls dπsl
∫ ∫−λ2 (πls , πsl )[pS µS esL + πls elS − (πsl + (1 − µS )pL )esS ]dπls dπsl .
(D-1)
27

Given the optimal condition for µL , µS , the ﬁst order condition for γ is,∫ ∫ πls∗∫ ∫ πls∗∗−λ1 els − λ2 esS dπls dπsl +−λ1 elS + λ2 esL dπls dπsl
∗∗
πls
∫ ∫
+
−λ1 elL + λ2 esL dπls dπsl = 0.

(D-2)

∗
πls

The functional forms of the shadow values are calculated for each parameter (πls , πsl ).
The functional forms are diﬀerent depending upon the behaviors of workers, and thenthe functional forms are characterize for each behaviors.
• For workers who take PSB, the laws of motion are:ėlL = γp(el − elL ) + πsl (es − esS ) − πls elL ,ėsS = (1 − γ)p(es − ess ) + πls (el − elL ) − πsl esS .

(D-3)
(D-4)

By λ˙1 = −rλ1 and λ˙2 = −rλ2 and the optimal conditions are:elL : e−rt ∆l + λ1 (pL + πls + r) + λ2 πls = 0,
esS : e−rt ∆s + λ1 πsl + λ2 (pS + πsl + r) = 0.

(D-5)
(D-6)

Then, the shadow values are:−e−rt ((pS + πsl + r)∆l − πls ∆s ),(pS + πls + r)(pL + πls + r) − πls πsl−e−rt ((pL + πls + r)∆s − πsl ∆l )λ2 =.
(pS + πls + r)(pL + πls + r) − πls πsl
λ1 =

(D-7)
(D-8)

• For workers who take SLB, the laws of motions are:ėlL = γp(el − elL ) + πsl (es − esS ) − (πls )elL ,ėsS = πls (el − elL ) − (πsl + γp)esS .

(D-9)(D-10)

By λ˙1 = −rλ1 and λ˙2 = −rλ2 , the optimal conditions are:elL : e−rt ∆l + λ1 (pL + πls + r) + λ2 πls = 0,
esS : e−rt ∆s + λ1 πsl + λ2 (pL + πsl + r) = 0.

(D-11)(D-12)

Then, the shadow values are:−e−rt ((pL + πsl + r)∆l − πls ∆s ),(pL + πls + r)(pL + πls + r) − πls πsl−e−rt ((pL + πls + r)∆s − πsl ∆l )λ2 =.
(pL + πls + r)(pL + πls + r) − πls πsl
λ1 =

28

(D-13)(D-14)

• For workers who take SSB, the low of motions are:ėlL = πsl (es − esS ) − ((1 − γ)p + πls )elL ],ėsS = (1 − γ)p(el − elL ) + πls (el − elL ) − (πsl + γp)esS .

(D-15)(D-16)

By λ˙1 = −rλ1 and λ˙2 = −rλ2 , the optimal conditions are:elL : e−rt ∆l + λ1 (pS + πls + r) + λ2 πls = 0,
esS : e−rt ∆s + λ1 πsl + λ2 (pS + πsl + r) = 0.

(D-17)(D-18)

Then, the shadow values are:−e−rt ((pS + πsl + r)∆l − πls ∆s ),(pS + πls + r)(pS + πls + r) − πls πsl−e−rt ((pS + πls + r)∆s − πsl ∆l )λ2 =.
(pS + πls + r)(pS + πls + r) − πls πsl
λ1 =(D-19)(D-20)

By substituting above shadow values into (D-2), the optimal condition of the social
planner problem is introduced.

E

The Social Planner Problem for Market Tightness

From (D-1), the ﬁrst order condition for θ is:∫ ∫ πls∗∫ ∫ πls∗∗′′[−λ1 γp′ elS − λ2 (1 − γ)p′ esL ]dπls dπsl[−λ1 γp elS + λ2 γp esS ]dπls dπsl +
k=
∗∗
πls
∫ ∫
−
[λ1 (1 − γ)p′ elL − λ2 (1 − γ)p′ esL ]dπls dπsl .
(E-1)
∗
πls

This is equivalence as:∫ ∫ πls∗[∫ ∫ πls∗∗
]
′
′
k =γ
[−λ1 p elS + λ2 p esS ]dπls dπsl +[−λ1 p′ elS ]dπls dπsl
+ (1 − γ)[∫ ∫ πls∗∗∗
πls[−λ2 p′ esL ]dπls dπsl +

∫ ∫

∗∗
πls

∗
πls

]
[λ1 p′ elL − λ2 p′ esL ]dπls dπsl .

(E-2)

From (D-2), the formula in ﬁrst brackets equals the formula in the second brackets. Then,the optimal condition for market tightness is:∫ ∫ πls∗∗∫ ∫ πls∗′′k=[−λ1 p elS + λ2 p esS ]dπls dπsl +[−λ1 p′ elS ]dπls dπsl
∫ ∫ πls∗=∗∗πls[−λ2 p′ esL ]dπls dπsl +

∫ ∫
∗
πls

29

∗∗
πls

[λ1 p′ elL − λ2 p′ esL ]dπls dπsl .

(E-3)

By substituting shadow values (D-19) and (D-20) into (E-3), the optimal condition of
the social planner problem is introduced.

30

∗
πls
(πsl )πls
SSB∗∗(πsl )
πls

PSB

SLB

0
Fig. 1. The domains of each worker’s behavior.

31

πsl

∗
π̂ls
(πsl )πls
SSB∗∗(πsl )
π̂ls

PSB

SLB

0
Fig. 2. The domains of social optimal behaviors.

32

πsl