Discussion Paper No. 13
Long-run Consequences of Ranking Job Applicantsby Unemployment Duration: Theoretical andNumerical Analyses

Akiomi KITAGAWA

August 28, 2013

Data Science and Service Research
Discussion Paper

Center for Data Science and Service Research
Graduate School of Economic and Management
Tohoku University27-1 Kawauchi, Aobaku
Sendai 980-8576, JAPANLong-run Consequences of Ranking Job Applicantsby Unemployment Duration: Theoretical andNumerical Analyses
Akiomi KITAGAWA∗
August 28, 2013

Abstract
This paper considers the long-run consequences of ranking job applicants on the basis of their unemployment durations by using ageneral equilibrium model in which statistical discrimination by firms
against jobless workers may yield multiple stationary equilibria. Because the most ineﬃcient equilibrium is supported by the belief thatjobless workers have lost their employability, the government should
dissuade firms from holding this extreme belief, thereby creating second chances for jobless workers. Moreover, by reducing the incomesof jobless workers through taxation, the government can create a new
equilibrium in which job seekers can find new jobs without experiencing
long-term unemployment.
Keywords: ranking, unemployment duration, eﬃciency wages,
statistical discrimination.
JEL Classifications: E24, J64, J71.

∗

Graduate School of Economics and Management, Tohoku University, 27-1, Kawauchi,
Aoba-ku, Sendai 980-8576, JAPAN. E-mail: kitagawa@econ.tohoku.ac.jp1

1

Introduction

Long-term unemployment has become one of the biggest problems facing
developed economies. While details diﬀer from one economy to another,
there is one common feature found in all economies: the longer a worker isunemployed, the more diﬃcult it becomes for him or her to find a job. There
has been controversy over the cause of this phenomenon for more than three
decades.1 One group of researchers argue that the long-term unemployedcannot find a job simply because they are no longer employable. This argument is based on two presumptions. First, jobless workers gradually lose
their employability as their unemployment duration becomes longer. Second, at job interviews, employers detect and reject job applicants who havelost their employability. It is self-evident that these presumptions jointly
produce the above-mentioned observation. A second group of researchers
does not accept this explanation, arguing that the long-term unemployedcannot find a job because employers “rank” job applicants by their unemployment durations and tend to hire those with shorter unemployment
durations without checking whether they are really employable or not. Following their explanation, such a discriminatory practice, which Blanchard
and Diamond (1994) call “ranking,” makes it diﬃcult for the long-term unemployed to find jobs, which produces the observed duration dependence.
Tautological as it may sound, a worker is long-term unemployed because heor she is long-term unemployed.
Recently, a number of empirical and experimental studies have presented
evidence that supports the second view. For example, Eriksson and Lagerström (2006, 2012) use an Internet-based CV database in Sweden to investigate empirically whether being unemployed reduces the probability ofgetting contacted by firms. Because all workers looking for new jobs areinvited to submit their personal details to this database, the authors haveaccess to exactly the same information as the firms, and they find that an
unemployed applicant faces a lower contact probability than an otherwise
identical employed applicant. Eriksson and Rooth (2011) use unique datafrom a field experiment in the Swedish labor market to investigate how pastand contemporary unemployment aﬀect a young worker’s probability of being invited to a job interview; they find evidence that recruiting employers
use not past but contemporary unemployment to sort workers. While thesestudies emphasize that being unemployed reduces the probability of getting
contacted by firms, Oberholzer-Gee (2008), in a field experiment in Switzerland, documents that job market opportunities for unemployed workers diminish rapidly over time. According to this experiment, a person who has
been without a job for 2.5 years is 51 percent less likely to be invited to aninterview than an employed person. After 30 months, it makes little sense1

See Machin and Manning (1999) for a survey.

2

for an individual to keep applying for jobs because few firms will express aninterest in hiring this person. The results of these studies strongly suggestthat ranking is a common practice in the actual labor market.2
This paper considers the long-run consequences of ranking job applicantsby their unemployment durations in the framework of a simple general equilibrium model with the following features. First, unemployment arises fromthe problem of nonverifiability rooted in the employer–employee relationship. Specifically, the model assumes not only that employers can only
imperfectly observe the work eﬀorts of their employees, as assumed in themodel of Shapiro and Stiglitz (1984), but also that the work eﬀort of any
worker is unverifiable to third parties, including the law court. As arguedby MacLeod and Malcomson (1989), when the work eﬀort is unverifiable,
backloading wage payment systems, such as performance pay, are ineﬀective in eliciting workers’ eﬀort because such a system gives employers an
incentive to evade wage payment by asserting that the employee has not
expended work eﬀorts. This, in turn, induces the workers hired by an employer with that system to shirk in response to the prospective nonpaymentof wages. To avoid such an unproductive situation, employers choose to
adopt a non-backloading wage payment system as considered by Shapiro andStiglitz, which leaves some workers in the jobless state. Second, statistical
discrimination against the long-term unemployed arises from informational
asymmetry between employers and jobless workers. The model assumes not
only that employers can only imperfectly observe the employability of job
candidates, but also that the employability of any worker is unverifiable tothird parties. Under these assumptions, employers’ beliefs about the employability of jobless workers can influence labor market performance basedon the theory of statistical discrimination pioneered by Phelps (1972) andArrow (1973). Specifically, employers believe that jobless workers who havebeen unemployed for at least a given length of time are not employable, andthey set an admissible length of unemployment duration to qualify them for
employment. Jobless workers, on the other hand, decide whether or not topreserve their employability, given such a ranking behavior and the wages
they will receive after being rehired. This interaction between employers
and jobless workers potentially gives rise to multiple stationary equilibria,
2

In addition, it is worth mentioning the study by Ghayad and Dickens (2012) and afollow-up experiment performed by Ghayad. Their study shows that since June 2009, theBeveridge curve in the United States, which normally exhibits a downward relationshipbetween job openings and unemployment, has broken down for the long-term unemployedin such a manner that a rising number of job openings does not reduce the number ofthe long-term unemployed. This implies the possibility that employers rank the longterm unemployed lower than other potential candidates. To test this possibility, Ghayad
performs an experiment in which he sends out resumes describing the qualifications andemployment history of 4,800 fictitious workers and finds that those who reported having
been unemployed for six months or more received very few callbacks even when their
qualifications were better than those of the workers who attracted employers’ interest.

3each of which supports a distinct belief of employers. Third, the numberof employers is endogenously determined by free entry, which makes it possible to explore the implications of increased entry costs for labor market
performance and allocative eﬃciency. As the entry cost takes a larger value,
employers are forced to pay lower wages to their employees to keep investments in their firms profitable. When the entry cost is larger than a criticallevel, wages become too low to motivate employed workers, unless unemployment duration before finding a new job is expected to be suﬃciently
long. This necessitates a large scale of unemployment, which creates roomfor employers’ belief to play a crucial role in selecting job applicants.
Theoretical and numerical analyses show that the number of stationary
equilibria varies with the entry cost. When the entry cost is smaller than
the above-mentioned critical level, there are two stationary equilibria in thismodel. In one of them, which we call the “no-second-chance equilibrium,”or NSCE, employers believe that all jobless workers have lost their employability, and they fill their vacancies exclusively with new entrants into thelabor force, who are employable by assumption. This hiring policy discourages jobless workers from maintaining their employability since it completely
destroys their chances of being rehired. In the other equilibrium, which wecall the “second-chance equilibrium,” or SCE, employers believe that somejobless workers, whose unemployment durations do not exceed a given positive length of time, are still employable, and they fill their vacancies not
only with new entrants, but also with the jobless workers who are qualified
for employment. This eﬀectively assures all jobless workers of finding a newjob after experiencing one period of unemployment as long as they are employable, and thus, they choose to maintain their employability. Because
the SCE facilitates rapid access by the unemployed to jobs, and because its
allocation is far more eﬃcient than that of the NSCE, the government inthis case should guide the economy to the SCE by persuading employers toabandon the extreme belief that all jobless workers are unemployable. Whenthe entry cost is larger than the above-mentioned critical level but smaller
than another level, there are multiple SCEs and one NSCE in this model,
although the number of SCEs decreases as the entry cost takes a larger
value. Again, in this case, the NSCE is vastly inferior to the SCEs in employment and allocative eﬃciency. Thus, the government should persuade
employers not to hold the belief that all jobless workers are unemployable,
thereby preventing the realization of the NSCE. In addition, none of theexisting SCEs attains the maximal level of employment, which implies thatin those equilibria, jobless workers must experience more than one period of
unemployment to find a new job. This malfunction of the labor market canbe improved if the government can reduce the incomes of jobless workers
by taxation. This policy coalesces the existing SCEs into a single SCE thatattains the maximal level of employment, which works in favor of not only
workers but also investors with stakes in firms operated by employers. When4the entry cost is larger than the second critical level, there is no stationary
equilibrium but a single NSCE in this model. In that equilibrium, no production occurs because wages that can motivate employed workers are toohigh for investors in a new firm to make a profit or break even, and thus,
there is no labor demand in the economy. Again, in this case, the policyof reducing the incomes of jobless workers is eﬀective in restoring production activities, as it creates one NSCE and one SCE, both of which attaina positive level of employment; the latter, in fact, attains its maximal level.
However, this does not lead to an improvement in economic welfare since
it brings a substantial welfare loss to workers (although it always makes
investors better oﬀ). In this case, taxing jobless workers eﬀectively curtails
their income-earning opportunities other than employment, thereby forcing
them to work for low wages.
The idea that the prolonged unemployment of some workers results from
employers’ discriminatory treatment of job applicants based on their observable record of unemployment has been formalized by Lockwood (1991),
Blanchard and Diamond (1994), Acemoglu (1995), Kübler and Weizsäcker(2003), Kugler and Saint-Paul (2004), Eriksson and Gottfries (2005), andEriksson (2006). Among these theoretical works, that of Acemoglu is mostclosely related to this paper, as he also uses statistical discrimination to
explain long-term unemployment. Unlike in the model of this paper, unemployment occurs as a result of a mismatch in his model, but the models
have two common assumptions. First, jobless workers have to incur a costto maintain their skills (or employability). Second, in the current model,
employers can only imperfectly observe whether the jobless workers havedone so, whereas in Acemoglu’s model, once employment begins, employers
can perfectly observe whether their newly-recruited employees have skills.
These assumptions jointly produce multiple equilibria, in one of which employers discriminate against the long-term unemployed, who, in response,
allow their skills to atrophy. When the cost to maintain skills is suﬃciently
small and employers can fairly precisely observe whether the unemployed
have incurred that cost, Acemoglu’s model also has an equilibrium in which
employers do not discriminate against the long-term unemployed, who, in
response, decide to maintain their skills. As stated above, similar results are
obtained in this paper, which is a natural result based on the assumptions
that both models have in common.
Pioneering as it is, Acemoglu’s analysis has several issues. First of all, it
is not suﬃciently detailed to explore the full implications of his model. For
analytical simplicity, he has assumed that matching probabilities are constant, which markedly limits the scope of his model and possibly conceals
some implications that would be evident if these probabilities were endogenized, especially those of entrepreneurial environments for employment. A
more detailed analysis is needed, as the importance of entrepreneurship forjob creation is now widely recognized. In addition, a well-known criticism
5

against the theory of statistical discrimination is true of his model. Since,in his model, employers can perfectly observe whether their employees haveskills after employment has begun, discrimination against the long-term unemployed is eliminated if employers adopt an employment system in whichhigher wages and tenure are given to the workers who have skills at the stageof non-tenured employment. In other words, the occurrence of discrimination in his model depends crucially on the assumption that labor contracts
must be written before the employability of a job applicant becomes clear.
To advance the theory of long-term unemployment one step further inthe direction indicated by Acemoglu, this paper extends the eﬃciency wage
model of Shapiro and Stiglitz (1984) into a general equilibrium frameworkin which the number of employers is endogenously determined by free entry,
thereby exploring the implications of entry costs for labor market performance and allocative eﬃciency. To avoid the criticism against the theory ofstatistical discrimination, the model assumes that employers can only imperfectly observe the employability of job candidates and that the employabilityof any worker is unverifiable to third parties, including the law court, as already stated. Under these assumptions, employers are reluctant to adopt
the above-mentioned tenure-track system because the job openings of those
who have adopted it are likely to be avoided by job applicants, who deeply
suspect that those employers have an incentive to exploit that nonverifiability, firing their employees at the end of the non-tenured employment stage
under the pretense that they have turned out to be unemployable, thereby
avoiding the payment of higher wages. To dispel this suspicion, employers
need to enter into contracts with workers before their employability becomes
clear by screening job applicants based on their employment records before
interviews.
The rest of this paper is organized as follows. Section 2 presents themodel. Section 3 considers the labor contract between the employer andthe worker under the assumed informational frictions, and then derives the
profile of the aggregate labor demand. Section 4 derives profiles of the“aggregate incentive constraints,” which provide the wage level employers
would choose given the aggregate employment, although they are defined
only when those wages are high enough to motivate the jobless workers
whom employers believe to be employable to preserve their employability.
Section 5 derives the stationary equilibria numerically and examines theirwelfare properties to obtain some policy implications. Section 6 concludes.

2

The Model

The analysis is based on a simple dynamic general equilibrium model with
eﬃciency wages à la Shapiro and Stiglitz (1984). Time is discrete, extending
from negative infinity to positive infinity.

6

2.1

Workers

At the beginning of each period, a continuum of workers of measure [θ/(1 −θ)]N is born, where θ and N are constants satisfying θ ∈ (0, 1) and N > 0.
Workers are mortal: they die in a given period with a probability θ, andtherefore, their probability of surviving into the next period is 1 − θ. This
shock is idiosyncratic, and thus, the population of workers born s periodprior decreases to θ(1 − θ)s−1 N at the beginning of the current period. In
the first period of their lives, workers neither work nor consume, but devote
the period to job search activities. Because they do not work until the second
period of their lives, the population of the labor force in a given period, thatis, the total number of workers who can work in that period, is∞∑θ(1 − θ)s−1 N = N.

s=1

The constant population of the labor force is attained in such a manner thatat the beginning of each period, the old workers of measure θN die and exit
the labor force and, as a substitute, the same measure of new workers, whowere born in the previous period, enter it.3
The lifetime utility of a worker born in a generic period t is given by
Et

)
∞ (
∑
1−θ s
s=1

1+r

(It+s − et+s )where Et , r, It+s , and et+s represent, respectively, the conditional expectation evaluated at the beginning of period t, the discount rate that is positiveand constant over time, the real income earned in period t + s, and theeﬀort level chosen in that period. After entry into the labor force, a worker
is either employed by a firm or unemployed in any period. She is paid wages
by her firm when employed and endowed with some fixed amount (w units)of the consumption good when unemployed.4 Thus, It+s can be written as{
wt+s if employed in period t + s
It+s =wotherwise,
where wt+s denotes the real wage received in period t + s.
In each period, the worker sets the level of eﬀort at either zero or a fixed
positive, denoted by e, which aﬀects her employer’s production when employed and her own employability when unemployed. An employed worker
contributes to her employer’s production if and only if she is employable
For any generation of workers, by assumption, those of measure [θ2 /(1 − θ)]N die
before entering the labor force.
4
This unemployment allows for workers that are self-employed or in low-paid jobs inthe secondary labor market.
37and expends e units of work eﬀort. Every worker is employable when entering the labor force and costlessly maintains this ability during periods of
employment. During periods of unemployment, in contrast, a worker must
expend e units of eﬀort in every period to maintain her employability. If she
neglects this maintenance even in one period, her employability is lost andnever restored.

2.2

Firms and Investors

In this economy, firms are also mortal: in every period, new firms are established, while some of the existing ones disappear.
Establishing a new firm requires a lump sum payment of F , which isfunded by a continuum of investors who are assumed to live from the infinite
past to the infinite future. In each period, they attempt to maximize a
discounted sum of net income flows in which future incomes are discounted
by the rate of r, as in the utility function of workers. They can borrow andlend as much as they like at a market interest rate, the level of which equals
r in equilibrium and, thus, can establish as many firms as they want.
An established firm does not start production until its second period of
existence, but in the first period, it recruits workers for production in thenext period. In addition, at the end of each period except the first one, after
having paid wages and dividends in that period, some firms experience withprobability b such an idiosyncratic shock that they suddenly become unproductive. The firms that experience this shock are never productive again.
To formalize this assumption, the technology that firms use to produce theconsumption good is specified as
Y = z L̃α ,where Y and L̃ denote, respectively, the output of the consumption good
and the number of employees who are employable and have expended workeﬀorts. The value of L̃ does not necessarily correspond to the total numberof employees since some of them might be either “unemployable” or “employable but expending zero eﬀort.” Firms take α and z as given: the former
is a common constant satisfying α ∈ (0, 1) and the latter is a firm-specificproductivity that initially takes a unitary value but irreversibly switches toa zero value with probability b at the end of each period.
idenote, respectively, the number of employees
Let L̃t+s , Lit+s , and wt+s
who are employable and expending eﬀorts in period t + s, the number of
employees who complete the ith period of service in that period, and thelevel of the wage paid to those employees in that period. The assumption
on the production technology suggests that one of the tasks assigned to the
management of a firm ismake the value of L̃t+s correspond to the total∑tosinumber of employees,
i=1 Lt+s , for any value of s (> 0) by motivating
employable workers to expend work eﬀorts while preventing unemployable8ones from being added to the firm. This is not an easy task because of some
informational frictions built into this economy; these will be discussed in thenext subsection. In this subsection, we simply suppose that firms are able
to do this task. Then, the net gain from establishing a new firm in period t
is evaluated by investors as
∞ (

1 ∑
−F +
1+r
s=1

1−b
1+r

)s−1
[(1s · Lt+s )α − wt+s · Lt+s ] ,(1)where 1s , wt+s , and Lt+s denote, respectively, an s-dimensional all-ones
vector,1s
wt+s = (wt+s, · · · , wt+s),

andLt+s = (L1t+s , · · · , Lst+s ).
The firm maximizes (1) by optimally selecting a path of wages and employment, {(wt+s , Lt+s )}∞
s=1 , since it is in the best interest of their investors.

2.3

Informational Frictions

In a given period t, the state of a worker who has already entered thelabor force can be summarized by a pair (nt , Qt ). The first variable, nt ,represents the worker’s recent experience of (un)employment. Specifically,
if nt is a positive integer, this worker enters the nt th period of service withthe current firm in period t. If nt is a negative integer, she enters the −nt th
period of continuing unemployment in period t. If nt = 0, she is born inperiod t, and has not yet entered the labor force. Every job seeker has a
non-positive value for nt because workers are not allowed to hop directlyfrom one firm to another. If a worker separates from the current firm, thenshe must experience at least one period of unemployment before being hiredby the next firm. The second variable, Qt , indicates her employability, andtakes a unitary value if she is employable at the beginning of period t andzero otherwise. The following properties of this variable are easy to verify.
First, Qt = 1 for those who are born in period t − 1 since every worker is
employable when entering the labor force. Second, if Qt = 0 for some t,then Qt+s = 0 for all s > 0 since lost employability is never restored. Before
applying for a job opening in period t, jobless workers decide whether or not
to expend an employability-preserving eﬀort, and thus, at the point of a job
interview, their employability in the next period (i.e., the value of Qt+1 foreach job applicant) is already determined.
Firms can observe an applicant’s recent experience of employment (i.e.,the value of nt ) perfectly, but evaluate her employability in the next period9(i.e., the value of Qt+1 ) only imperfectly. In job interviews, they can detect an unemployable applicant only with a fixed probability q (∈ (0, 1)),although they never mistake an employable applicant for an unemployable
one. Even after having started an employment relationship with a worker,
the firm can observe this employee’s performance only imperfectly. Specifically, when the employee does not expend work eﬀorts, the firm can catch her
shirking only with probability q, although it never mistakes a non-shirking
employee for a shirking one. Neither the firm nor the employee herself canverify the employee’s employability or eﬀort expenditure to a third party
such as the law court. All they can verify is whether or not the employee
has shown up at the oﬃce and whether or not the firm has paid the promised
wage.

2.4

Sequence of Events within a Period

Events within a given period proceed as follows.
After the birth of new workers and the establishment of new firms, allfirms present in that period announce both their plan for current and future
hiring and wage levels for newly recruited and already-employed workers.
Firms may also announce the admissible duration of unemployment for job
applicants, which is used as a criterion to select the applicants to be interviewed. Given these announcements, jobless workers decide whether or not
to continue maintaining their employability, and employed workers decide
whether or not to expend work eﬀort in the current period. Then, firms
other than the newly established ones start production with the employees
recruited in or before the previous period. After production activities are
finished, wages are paid to employed workers. Then, a fraction b of operating firms become unproductive, and their employees lose their jobs. In
addition, labor contracts with workers who have been caught shirking are
terminated. Labor contracts with the other employed workers are renewed,
although some of them die at the end of the current period. To replace them,
the firms that survive into the next period post some job openings. Newly
born and jobless workers apply to these openings, and the firms interview
those who have passed the criteria, selecting new employees from among
them.

3

Labor Contracts and Aggregate Labor Demandin Stationary Equilibria

The model constructed in the previous section potentially has a huge varietyof equilibria that yield qualitatively diﬀerent outcomes. Instead of fully
characterizing them, we restrict our focus to a class of stationary equilibria
in which the wage paid to a worker depends solely on her length of service

10nwith her current employer. That is, if wt+s
denotes the level of wages paid
to the workers who are currently employed by a firm established in periodt and who complete their nth period of service in period t + s, then, innsuch a stationary equilibrium, wt+s
takes a common value of wn for any
ncombination of t and s, although w may vary with n. In this section,
based on the assumption that all employed workers are employable, we firstcharacterize the labor contract written in such a stationary equilibrium,then use the free-entry condition to examine the determination of wages
and employment at an individual firm, and finally profile the aggregate
labor demand.

3.1

Labor Contracts

Under the informational frictions assumed in the previous section, a labor
contract that is acceptable to workers must be such that in any period, wage
payment is made if an employee shows up at the oﬃce, independent of her
eﬀort expenditure. Wage payment cannot be conditioned on the employee’s
eﬀort expenditure since it is unverifiable. On the other hand, unconditional
payment of wages gives employees an incentive to shirk. To prevent thismoral hazard, the labor contract also stipulates that the wage level that thefirm should pay is significantly higher than the employee’s reservation, w;
that the term of the contract is one period; and that the contract terminates
unless both of the contracting parties agree to renew it. These clauses jointly
make termination of the contract highly damaging to the employee, thereby
enabling the firm to punish a shirking employee through that termination.5
Such an “at-will” employment relationship renders it diﬃcult for a pairof a firm and a worker to make the promise that the wage will increase
whenever the labor contract is renewed. If the starting wage is lower thanthose supposed to be paid in and after the second period of service, thefirm naturally has an incentive to save labor costs by terminating the contracts of current employees at the end of each period and recruiting newworkers at a low starting wage with the empty promise of a future raise.
Given the possibility of this moral hazard, newly recruited workers choose
to expend zero eﬀort if their current wages are lower than those they aresupposed to receive in and after the next period because they regard such an
upward-sloping wage profile as an informal notice that their contracts will
be terminated at the end of the current period. To alleviate their suspicions
and, thus, to elicit their work eﬀorts, the firm needs to make the starting
wage no lower than those paid in and after the second period of service.6
5Because an employee’s eﬀort expenditure is unverifiable, it is impossible for a firm todiscipline its employees by use of such a bonding system in which, prior to starting work,
the employees post bond to a third party, such as the law court, which would be forfeitedby that party in the case of being caught shirking.
6
This moral hazard incentive is introduced to make this model immune to the criticism

11

The next proposition characterizes a profile of wages paid to an employee.
It is “almost” flat in the sense that the employee receives a constant level ofwages in and after the second period of service with her employer, although
she is possibly better paid in the first period.
Proposition 1 (Wage Profile). Suppose that the economy is settled in astationary equilibrium, and let V (n, Q) and wn denote the lifetime utility ofa worker in the labor force whose current state is (n, Q) and the wage levelan employed worker receives in the nth period of service with her employer,
respectively. Moreover, define w2+ as
w2+ ≡

r+θ
1 + r − (1 − θ)(1 − b)(1 − q)
V (−1, 1) +e.
1+r
(1 − θ)(1 − b)q(2)Then, wages are paid as w1 ≥ w2+ and wn = w2+ for ∀n ≥ 2.
Proof. See Appendix A.
As long as wages are set as in Proposition 1, employed workers have no
incentive to shirk. In the rest of this paper, w1 and w2+ denote, respectively,
the wages paid to newly recruited workers and those paid to other employedworkers.

3.2

Aggregate Labor Demand

Our next task is to profile the aggregate labor demand, which is a relation
between the aggregate employment and a weighted average of w1 and w2+that is implied by firms’ optimizing behavior and the free-entry condition.
To this end, we first need to show that in any stationary equilibrium,
all operating firms keep their number of employees constant over time at acommon level.
Lemma 1. Suppose that the economy is settled in a stationary equilibrium,and let Lt+s represent the total number of employees a firm established in

by Carmichael (1985) of Shapiro-Stiglitz eﬃciency wage model. The point of his criticism
is that even if firms are restricted to eliciting work eﬀorts by paying eﬃciency wages,
they can still sell their jobs and require newly recruited workers to accept a low starting
wage level, or pay an entrance fee. He asserts that, contrary to the argument of Shapiro
and Stiglitz, unemployment cannot be involuntary since the starting wage or entrance fee
clears the job market in each period. In the present model, however, the starting wage
may fail to clear the job market when its market-clearing level is suﬃciently low. Firms
are reluctant to set the starting wage at such a low level, being apprehensive that this may
induce newly recruited workers to expend zero eﬀort. To make the labor contract withthem “credible,” firms rather choose to set the starting wage at a suﬃciently high level,
which causes job rationing and involuntary unemployment in the job market. Similararguments are made by Arvan and Esfahani (1993) and Ritter and Taylor (1994).

12

period t puts into production activities in period t + s. Then, for ∀t and∀s (≥ 1),
Lt+s = L,
where the value of L is given by
αLα−1 = w1 −(1 − b)(1 − θ)(w1 − w2+ )(≡ ŵ).
1+r(3)

Proof. See Appendix A.
This lemma might give a paradoxical impression since it asserts thatnew firms, which have just started production (thus having no option but touse only newly recruited workers) and old firms, which started production
one or more periods before (thus being not so dependent on newly recruited
workers) would choose the same number of employees even when w1 > w2+ .
This result is accounted for by the fact that in choosing the number of
employees, firms only consider the expected sum of wages that will be paid
to a marginally hired worker, which is given by
]
∞ [
∑
1+r
(1 − b)(1 − θ) s 2+
1
ŵ.
w =
w +
1+r
1 + r − (1 − θ)(1 − b)s=1

As suggested by this equation, the expected wage payment can be averagedout to ŵ per period. Regardless of their histories, all of the operating firms
view ŵ as the marginal cost of labor, and they thus choose the same numberof employees.
In a stationary equilibrium in which a positive number of firms are operating, the net gain from investing in a new firm is reduced to zero by freeentry. This uniquely determines the values of ŵ and L, given α, b, r and F .
Lemma 2. Suppose that the economy is settled in a stationary equilibrium,
where the net gain from investing in a new firm is reduced to zero by freeentry. Then, in that equilibrium, every operating firm sets ŵ and L as
]1/α
[
]
[ α
(r + b)F 1/αα (1 − α)1−α∗(≡ w ), L =(≡ L∗ ).
(4)ŵ =(r + b)1−α F 1−α1−αProof. See Appendix A.
Now we can state the following.
Proposition 2 (Aggregate Labor Demand). Suppose that the economy issettled in a stationary equilibrium, and let E denote the aggregate employment, whereas ŵ and w∗ are defined by (3) and (4), respectively. Then, inthat equilibrium, E and ŵ satisfy(ŵ − w∗ )E = 0,

E≥0
13

and

ŵ − w∗ ≥ 0.

(5)

ŵ

w∗

E

O

Figure 1: Aggregate Labor Demand
Proof. See Appendix A.
Figure 1 depicts a locus of pairs of E and ŵ satisfying (5). To see why
the locus is L-shaped, we need to recall the definitions of ŵ and w∗ . As
implied by (3), ŵ is the marginal labor cost incurred by a firm when it hires
an additional worker. To motivate this worker, the firm must, on average,
pay ŵ units of wage per period. On the other hand, w∗ is the break-evenlevel of ŵ, at which the net gain from establishing a new firm becomes zero.
When ŵ > w∗ , operating firms, if they exist, cannot discipline their employees without paying wages that are higher than their break-even levels. Thisdeprives investors of the incentive to establish a new firm, which ultimately
reduces the number of operating firms and, thus, the aggregate labor demand, to zero. When ŵ < w∗ , in contrast, operating firms can discipline
their employees by paying wages that are lower than their break-even levels.
This gives investors a strong incentive to establish a new firm, which ultimately makes the number of operating firms and, thus, the aggregate labordemand, infinitely large. However, such a large labor demand cannot be
met by any means, meaning that there is no such equilibrium with ŵ < w∗
in this model. Finally, when ŵ = w∗ , the net gain from establishing a new
firm becomes zero, which makes the number of operating firms and, thus,
the aggregate labor demand, indeterminate.

14

Table 1: Model Parameters
Workers

Firms

4

r
θ
w
e
N
α
b
q

Discount rate
Death rate
Income when unemployedWork eﬀort
Population of the labor force
Labor share in production
Business closing rate
Rate of detecting unemployable and shirking workers

0.01
0.03
1.00
0.50
1.00
0.70
0.04
0.90

Firms’ Belief, Ranking, and Aggregate Incentive
Constraints

To determine the equilibrium values of E and ŵ, we also have to derive
the aggregate incentive constraints (AIC). A profile of an AIC provides thelevel of ŵ firms would choose given the level of E, although it is defined
only when that level of ŵ is high enough to motivate the jobless workers
whom firms believe to be employable to actually preserve their employability.
As in other models of eﬃciency wages, firms choose a wage level with thegoal of preventing their employees from shirking. In this model, moreover,these wages then unintentionally keep some jobless workers from becoming
unemployable. Because our focus is on a stationary equilibrium in whichfirms’ belief is fully supported, the AIC must be consistent with their belief
about jobless workers’ employability.
We index firms’ belief about jobless workers’ employability by a nonpositive integer n, which means that they believe that job applicants withn ∈ [n, 0], who have been unemployed for no more than −n periods, arestill employable, while other applicants are unemployable. When firms havea belief indexed by n, they naturally “rank” job applicants by their unemployment duration or, more specifically, set the admissible level of unemployment duration for job applicants at n, thereby disqualifying applicantswith n < n. Obviously, this hiring policy discourages those who have been
unemployed for more than −n periods from preserving their employability,
which partially fulfills firms’ belief. The question is whether this really motivates other jobless workers to preserve their employability, as firms believe.
Checking whether or not this condition is met for qualified workers is such a
detailed process that we cannot but resort to a numerical method to derivean AIC profile for each value of n.
Figures 2 and 3 depict the profiles of the AICs that are obtained withthis numerical method under the parameter configuration of Table 1.7 For
7For the details of the derivation, see Appendix B. Since the expected length of aworker’s stay in the labor force is given by 1/θ, we have eﬀectively assumed that workers

15

1.64

w∗∗∗
1.62

n = −∞
w∗∗∗∗
1.6

ŵ

n=0
1.58

n = −31.56

w ∗∗n = −1

1.54
0.4

n = −2
0.5

E ∗∗

0.6

0.7

0.8

0.9

1

E ∗∗∗∗ E ∗∗∗E
Figure 2: AICs for Various Values of n

1.64

w∗∗∗
1.62

n = −∞
w

∗∗∗∗
1.6

ŵ
1.58

1.56

n = −2 n = −3n = −1
1.54

E ∗∗∗∗
0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96 E

E
Figure 3: AICs for Various Values of n (Enlarged)

16

∗∗∗

Table 2: New Notations
NotationE ∗∗E ∗∗∗E ∗∗∗∗
w∗∗
w∗∗∗
w∗∗∗∗

Definition[θ/(θ + b − θb)]N[1/(1 + b − θb)]N(1 − b)N
w + {[1 + r − (1 − θ)(1 − b)(1 − q)]/(1 − θ)(1 − b)q}ew + {[1 + r + (1 − θ)b]/(1 − θ)(1 − b)q}ew + [(1 + r)/(1 − θ)(1 − b)q]e

Implied Value
0.436
0.963
0.960
1.547
1.626
1.603

conciseness in expression, these figures use some new notations: E ∗∗ , E ∗∗∗ ,E ∗∗∗∗ , w∗∗ , w∗∗∗ , and w∗∗∗∗ . Their definitions and implied values under theabove parameter configuration are summarized in Table 2.
As shown in Figure 2, the profile of the AIC for n = 0 is derived for
relatively small values of E, while those of the AICs for n < 0 are derived for
relatively large values of E. This contrast is mostly attributable to whether
or not jobless workers are given “second chances,” that is, chances to be
rehired. When n = 0, firms fill their vacancies exclusively with new entrants
into the labor force, and thus there is no second chance for jobless workers.
Because of this exclusive hiring policy, even if all the new entrants are hiredin every period, the aggregate employment cannot exceed E ∗∗ , which, under
our parameter configuration, means that more than half of the labor force isleft in the jobless state. When n < 0, firms fill their vacancies not only withnew entrants but also with those who have been unemployed for −n periods
or less, and thus, there are some second chances for jobless workers. Suchan inclusive hiring policy makes it possible for the aggregate employment tobe as large as E ∗∗∗ . When E = E ∗∗∗ , in particular, all workers but those
who have just lost their jobs are employed; equivalently, every job seeker
can find a job by experiencing only one period of unemployment. These
imply that there are no workers who have been unemployed for two periods
or more, and thus that, as long as n < 0, the diﬀerence in the value of n has
no eﬀect on equilibrium outcomes. This is why profiles of AICs for n < 0have a common vertical part.
The reason profiles of the AICs for n < 0 cannot be derived for relatively
small values of E is explained as follows. As E takes a smaller value, thejob finding rate for jobless workers decreases its value, which enables firms
to discipline their employees by paying lower wages. At the same time, thelower rate of job finding and lower wages after being rehired gradually reduce
spend an average of 33.3 periods in the labor force by setting the value of θ at 0.03. The
life expectancy of a firm is 1/b = 25 periods, and the expected duration of an employment relationship is 1/(θ + b − θb) ≈ 14.5 periods. Moreover, the probability of “lifelong
employment,” that is, the probability that a newly recruited worker will not experience
unemployment until she dies, is θ/(θ + b − θb) ≈ 0.43, which implies that more than 56%
of newly recruited workers experience unemployment at least once during their lives.

17the incentive of jobless workers to preserve their employability, eventually
obliterating it. This runs counter to firms’ belief of n < 0, therefore, AICs
cannot be defined for such small values of E.
This argument helps us understand why profiles of the AICs for n < 0have an upward-sloping part, while that of the AIC for n = 0 has a horizontal
part instead. As already explained, when n < 0, a smaller value of E enables
firms to discipline their employees by paying lower wages, which explains
why the AICs for n < 0 have an upward-sloping part. When n = 0, on theother hand, there is no second chance for jobless workers, and thus, their job
finding rate always takes the zero value. As a result, neither the incentiveof employed workers to expend eﬀorts nor firms’ wage-setting behavior isaﬀected by any change in E, which is the intuition behind the horizontal
part of the AIC for n = 0.
Figures 2 and 3 also show that a decrease in n rotates the upwardsloping part of the AIC for n < 0 counterclockwise around its upper end,
gradually shortening its length.8 Such a decrease in n qualifies a larger
set of jobless workers for the given vacant positions, thereby reducing theprobability that a qualified worker will be rehired. From the viewpoint of
currently employed workers, this eﬀectively stiﬀens the penalty for shirking
since, if they lose their current jobs under the smaller value of n, they arelikely to experience a longer duration of unemployment before finding a newjob. This, in turn, enables firms to discipline their employees by paying lower
wages in accordance with the level of E, which leads to the counterclockwise
rotation of AIC. From the viewpoint of currently jobless workers, the samedecrease in n eﬀectively reduces the probability that they will be rehiredsince it enlarges the set of jobless workers who are qualified for employment.
This discourages them from preserving their employability, which cannot be
undone by anything but the prospect that they will receive suﬃciently highwages after reemployment. This is why the upward-sloping part of AIC isshortened as n takes a smaller value.

5

Stationary Equilibria with and without Second
Chances

We now are in a position to examine the stationary equilibria of this model.
A stationary equilibrium in which firms embrace the belief of n is an intersection of the loci of the aggregate labor demand and the AIC for n.
Considering the qualitative diﬀerence between the AIC for n = 0 and those

In Figures 2 and 3, the upward-sloping parts of the AICs for n ≤ −4 are not depictedexcept for n = −∞ because they are hidden by that of the AIC for n = −3. The only
diﬀerence among the upward-sloping parts of the AICs for n ≤ −3 is their lengths: as ntakes a smaller value, the length of the upward-sloping part becomes shorter.
8

18

Table 3: Critical Values of F
Notation
F1
F2
F3

Definition(1 − α)(α/w∗∗∗ )α/(1−α) /(r + b)
(1 − α)(α/w∗∗∗∗ )α/(1−α) /(r + b)
(1 − α)(α/w∗∗ )α/(1−α) /(r + b)

Implied Value
0.840
0.868
0.943

for n < 0, we call the equilibrium with n = 0 the “no-second-chance equilibrium” (NSCE), and those with n < 0 “second-chance equilibria” (SCE).

5.1

Diagrammatic Expositions

In examining the NSCE and SCE, we should pay attention to the entry costfor new firms, F , since, as shown in (2), it is the primary determinant of theaggregate labor demand, w∗ , defined by (4).
For analytical convenience, we define F1 , F2 , and F3 as in Table 3, thethird column of which reports their implied values under the parameter
configuration of Table 1. As is easily verified, w∗ is a decreasing function of
F with ∗∗∗if F = F1 ,
 w
∗
∗∗∗∗
w
if F = F2 ,w =w∗∗ if F = F3 ,where w∗∗ , w∗∗∗ , and w∗∗∗∗ are as defined in Table 2. Dependent on theamount of the entry cost, four patterns are conceivable for the existence of
the SCE and NSCE.
Figure 4 depicts a case in which the entry cost is suﬃciently small,F ∈ [0, F1 ], by configuring the parameters as in Table 1. In such a case, themodel has a unique SCE and a unique NSCE, which suggests that whether
firms believe that those who have experienced unemployment are no longer
employable is critical in determining the equilibrium of this economy; otheraspects of their belief have no eﬀect on this determination. This can be
explained as follows. Small costs of entry, which characterize the present
cases, promote the establishment of new firms, thereby making the labormarket so tight that all job seekers in each period can find jobs. In the SCE,
in particular, every jobless worker can find a new job if she experiences only
one period of unemployment. As a result, there are no workers who havebeen unemployed for more than one period, which makes all details of firms’
belief except for whether n = 0 is satisfied insignificant.
Figure 5 depicts a case in which the entry cost is not so small, F ∈(F1 , F3 ), using the same configuration of parameters as in Figure 4. Again,
there is a unique NSCE in this model. In addition, there are multiple SCEs,

19

1.68

1.66

NSCE

SCE

w∗
1.64

w∗∗∗
1.62

ŵ

1.6

1.58

1.56

w∗∗
1.54

1.52

0

0.1

0.2

0.3

0.4

E ∗∗ 0.5

0.6

0.7

0.8

0.9

E ∗∗∗ 1

E
Figure 4: SCE and NSCE when F ∈ [0, F1 ]

1.68

1.66

1.64

w∗∗∗
1.62

ŵ

1.6

1.58

w

SCE

NSCE

∗

1.56

w∗∗
1.54

1.52

0

0.1

0.2

0.3

0.4

E ∗∗ 0.5

0.6

0.7

0.8

0.9

E
Figure 5: SCE and NSCE when F ∈ (F1 , F3 )20

E ∗∗∗ 1

1.68

1.66

1.64

w∗∗∗
1.62

ŵ

1.6

1.58

NSCE
1.56

w∗ = w∗∗
1.54

1.52

0

0.1

0.2

0.3

0.4

E ∗∗ 0.5

0.6

0.7

0.8

0.9

E ∗∗∗ 1

E

Figure 6: NSCE when F = F3
as depicted in Figure 5. The reason SCEs are so diversified in this case will
be considered in the next subsection.
Figures 6 and 7 depict cases in which the entry cost is large, F ∈[F3 , +∞). In these cases, the SCE does not exist since any level of ŵ at whichinvestors can break even or profit is too low to motivate jobless workers topreserve their employability. In contrast, the NSCE still exists for thesecases, but its status changes according to the value of F . When F = F3 ,there is a continuum of NSCEs, which are diﬀerentiated from one anotherby E. Across them, the job finding rate increases its value from zero tounity as E increases from zero to [θ/(θ + b − θb)]N .9 When F ∈ (F3 , +∞),
there is a unique NSCE, the aggregate employment of which equals zero.
There, w∗∗ , which is the minimum level of ŵ that can motivate employedworkers, is too high for investors to break even or profit by investing in a
new firm, and thus, no firm is established. As a result, there is no job oﬀer
to newly born workers, who then have no choice but to live by consuming
their endowments in each period. Put another way, not only second chances
but also first ones are completely destroyed in this case.

9

Using welfare measures defined in the next subsection, we can even show that theseequilibria can be Pareto-ranked by E; that is, an equilibrium indexed by a larger value ofE dominates that by a smaller value in the Paretian sense. Intuitively, a larger value of E
improves economic welfare by raising the job finding rate, as well as by increasing the total
amount of the net profit payment, which is composed of the profits earned by currently
operating firms minus the entry costs paid by investors for establishing new firms.

21

1.68

1.66

1.64

w∗∗∗
1.62

ŵ

1.6

1.58

1.56

NSCE

w∗∗
1.54

w∗
1.52

0

0.1

0.2

0.3

0.4

E ∗∗ 0.5

0.6

0.7

0.8

0.9

E ∗∗∗ 1

E
Figure 7: NSCE when F ∈ (F3 , +∞)5.2

Numerical Derivation of Stationary Equilibria

Despite its comprehensibility, diagrammatic expositions can only provide asketch of the stationary equilibria that exist in this model. To take a close
look at these equilibria, we will numerically derive the equilibrium values ofŵ, E, UO , a, L∗ , E/L∗ , w1 , and w2+ for such values of n and F as
n = 0, −1, −2, −3, −4, −5, −10, −20, −30, −∞,
{
F =

0.1i
F11−0.1j F30.1j

for i = 1, 2, · · · , 8
for j = 0, 1, 2, · · · , 10

and F > F3 ,where UO and a, respectively, represent the population of unemployable
workers and the per-period job finding rate for employable workers.10
Tables 4–9 report the results of this numerical analysis. As suggestedby these tables, when the value of F is increased from below to above F1 ,the number of SCE suddenly explodes from one to positive infinity. To
understand why, we need to recall that the equilibrium value of ŵ, w∗ ,decreases as F takes a larger value. As long as F is smaller than F1 , thelevel of w∗ is high enough to motivate employed workers, even when finding
a new job takes only one period of unemployment experience. If F takesa larger value than F1 , in contrast, the level of w∗ becomes so low that it
cannot motivate employed workers unless finding a new job takes more than
one period of unemployment experience. As a consequence, only a subset of
10

For the details of the derivation, see Appendix C.

22

Table 4: Aggregate Employment (E)
n
F
ŵ
0
−1
−2
−3
−4
−5
0.1
4.047
0.436 0.963 0.963 0.963 0.963 0.963
0.2
3.007
0.436 0.963 0.963 0.963 0.963 0.963
0.3
2.527
0.436 0.963 0.963 0.963 0.963 0.963
0.4
2.234
0.436 0.963 0.963 0.963 0.963 0.963
0.5
2.031
0.436 0.963 0.963 0.963 0.963 0.963
0.6
1.878
0.436 0.963 0.963 0.963 0.963 0.963
0.7
1.758
0.436 0.963 0.963 0.963 0.963 0.963
0.8
1.660
0.436 0.963 0.963 0.963 0.963 0.963
0.840 (F1 ) 1.626 (w∗∗∗ ) 0.436 0.963 0.963 0.963 0.963 0.963
0.850
1.618
0.436 0.958 0.962 0.962 0.962 0.962
0.860
1.610
0.436 0.954 0.960 0.961 0.961 0.961
0.870
1.602
0.436 0.948 0.958 0.960 0.960 0.960
0.880
1.594
0.436 0.941 0.956 0.959 0.959 0.959
0.890
1.586
0.436 0.932 0.954 0.958 0.958 0.958
0.900
1.578
0.436 0.921 0.951 0.956 0.957 0.957
0.911
1.570
0.436 0.906 0.947 0.955 0.956 0.956
0.922
1.562
0.436 0.886 0.943 0.953 0.955 0.955
0.932
1.555
0.436 0.856 0.938 0.952 NANA∗∗
0.943 (F3 ) 1.547 (w ) 0.436 NA
NA
NA
NA
NA
F > F3
1.547 (w∗∗ )
0
NA
NA
NA
NA
NA
Note: The largest possible value is reported when (F, n) = (F3 , 0).

−10
0.963
0.963
0.963
0.963
0.963
0.963
0.963
0.963
0.963
0.962
0.961
0.960
0.959
0.958
0.957
NA
NA
NA
NA
NA

−20
0.963
0.963
0.963
0.963
0.963
0.963
0.963
0.963
0.963
0.962
0.961
0.960
0.959
NA
NA
NA
NA
NA
NA
NA

−30
0.963
0.963
0.963
0.963
0.963
0.963
0.963
0.963
0.963
0.962
0.961
0.960
NA
NA
NA
NA
NA
NA
NA
NA

Table 5: Population of Unemployable Workers (UO )
n
F
0
−1
−2
−3
−4
−5
−10
−20
0.1
0.564
0
0
0
0
0
0
0
0.2
0.564
0
0
0
0
0
0
0
0.3
0.564
0
0
0
0
0
0
0
0.4
0.564
0
0
0
0
0
0
0
0.5
0.564
0
0
0
0
0
0
0
0.6
0.564
0
0
0
0
0
0
0
0.7
0.564
0
0
0
0
0
0
0
0.8
0.564
0
0
0
0
0
0
0
0.840 (F1 ) 0.564
0
0
0
0
0
0
0
0.850
0.564 0.004 0.000 0.000 0.000 0.000 0.000 0.000
0.860
0.564 0.009 0.001 0.000 0.000 0.000 0.000 0.000
0.870
0.564 0.015 0.002 0.000 0.000 0.000 0.000 0.000
0.880
0.564 0.022 0.003 0.000 0.000 0.000 0.000 0.000
0.890
0.564 0.031 0.005 0.000 0.000 0.000 0.000 NA
0.900
0.564 0.043 0.007 0.001 0.000 0.000 0.000 NA
0.911
0.564 0.057 0.010 0.001 0.000 0.000 NA
NA
0.922
0.564 0.078 0.013 0.002 0.000 0.000 NA
NA
0.932
0.564 0.108 0.018 0.002 NA
NA
NA
NA
0.943 (F3 ) 0.564 NA
NA
NA
NA
NA
NA
NA
F > F3
1
NA
NA
NA
NA
NA
NA
NA
Note: The smallest possible value is reported when (F, n) = (F3 , 0).

23

−30
0
0
0
0
0
0
0
0
0
0.000
0.000
0.000
NA
NA
NA
NA
NA
NA
NA
NA

−∞
0
0
0
0
0
0
0
0
0
0
0
NA
NA
NA
NA
NA
NA
NA
NA
NA

−∞
0.963
0.963
0.963
0.963
0.963
0.963
0.963
0.963
0.963
0.962
0.961
NA
NA
NA
NA
NA
NA
NA
NA
NA

Table 6: Per-Period Job Finding Rate for Employable Workers (a)
n
F
0
−1
−2
−3
−4
−5
−10
−20
0.1
1
1
1
1
1
1
1
1
0.2
1
1
1
1
1
1
1
1
0.3
1
1
1
1
1
1
1
1
0.4
1
1
1
1
1
1
1
1
0.5
1
1
1
1
1
1
1
1
0.6
1
1
1
1
1
1
1
1
0.7
1
1
1
1
1
1
1
1
0.8
1
1
1
1
1
1
1
1
0.840 (F1 ) 1
1
1
1
1
1
1
1
0.850
1 0.997 0.987 0.986 0.986 0.986 0.986 0.986
0.860
1 0.992 0.974 0.972 0.972 0.972 0.972 0.972
0.870
1 0.987 0.962 0.959 0.959 0.959 0.959 0.959
0.880
1 0.981 0.949 0.945 0.945 0.945 0.945 0.945
0.890
1 0.974 0.937 0.932 0.931 0.931 0.931 NA0.900
1 0.964 0.924 0.918 0.918 0.918 0.918 NA0.911
1 0.951 0.911 0.905 0.904 0.904 NANA0.922
1 0.934 0.897 0.891 0.891 0.891 NANA0.932
1 0.907 0.881 0.878 NA
NANA
NA
0.943 (F3 ) 1 NA
NA
NA
NA
NA
NA
NA
F > F3
0 NA
NA
NA
NA
NA
NA
NA
Note: The largest possible value is reported when (F, n) = (F3 , 0).

−30
1
1
1
1
1
1
1
1
1
0.986
0.972
0.959
NA
NA
NA
NA
NA
NA
NA
NA

−∞
1
1
1
1
1
1
1
1
1
0.986
0.972
NA
NA
NA
NA
NA
NA
NA
NA
NA

Table 7: Number of Operating Firms (E/L∗ )
n
F
L∗
0
−1
−2
−3
−4
−5
0.1
0.003 151.27 333.95 333.95 333.95 333.95 333.95
0.2
0.008 56.20 124.06 124.06 124.06 124.06 124.06
0.3
0.014 31.49
69.52
69.52
69.52
69.52
69.52
0.4
0.021 20.88
46.09
46.09
46.09
46.09
46.09
0.5
0.029 15.18
33.51
33.51
33.51
33.51
33.51
0.6
0.037 11.70
25.82
25.82
25.82
25.82
25.82
0.7
0.046
9.39
20.72
20.72
20.72
20.72
20.72
0.8
0.056
7.76
17.12
17.12
17.12
17.12
17.12
0.840 (F1 ) 0.060
7.23
15.97
15.97
15.97
15.97
15.97
0.850
0.061
7.11
15.64
15.69
15.69
15.69
15.69
0.860
0.062
7.00
15.30
15.41
15.42
15.42
15.42
0.870
0.063
6.88
14.96
15.13
15.15
15.15
15.15
0.880
0.064
6.77
14.60
14.85
14.89
14.89
14.89
0.890
0.065
6.66
14.23
14.57
14.63
14.63
14.63
0.900
0.067
6.55
13.83
14.28
14.37
14.37
14.38
0.911
0.068
6.44
13.38
14.00
14.11
14.12
14.12
0.922
0.069
6.34
12.87
13.71
13.86
13.88
13.88
0.932
0.070
6.23
12.24
13.40
13.61
NA
NA
0.943 (F3 ) 0.071
6.13
NA
NA
NA
NA
NA
F > F3
0.071
0
NA
NA
NA
NA
NA
Note: The largest possible value is reported when (F, n) = (F3 , 0).

24−10333.95124.06
69.52
46.09
33.51
25.82
20.72
17.12
15.97
15.69
15.42
15.15
14.89
14.63
14.38
NA
NA
NA
NA
NA

−20
333.95124.06
69.52
46.09
33.51
25.82
20.72
17.12
15.97
15.69
15.42
15.15
14.89
NA
NA
NA
NA
NA
NA
NA

−30
333.95124.06
69.52
46.09
33.51
25.82
20.72
17.12
15.97
15.69
15.42
15.15
NA
NA
NA
NA
NA
NA
NA
NA

−∞
333.95124.06
69.52
46.09
33.51
25.82
20.72
17.12
15.97
15.69
15.42
NA
NA
NA
NA
NA
NA
NA
NA
NA

Table 8: Starting Wage (w1 )
F
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.840 (F1 )
0.850
0.860
0.870
0.880
0.890
0.900
0.911
0.922
0.932
0.943 (F3 )F > F3

ŵ
4.047
3.007
2.527
2.234
2.031
1.878
1.758
1.660
1.626 (w∗∗∗ )
1.618
1.610
1.602
1.594
1.586
1.578
1.570
1.562
1.555
1.547 (w∗∗ )1.547 (w∗∗ )033.59420.262
14.11410.356
7.744
5.788
4.249
2.996
2.556
2.453
2.350
2.248
2.146
2.045
1.945
1.844
1.745
1.646
1.547
1.547

−1
6.197
4.234
3.328
2.775
2.390
2.102
1.875
1.691
1.626
1.618
1.610
1.602
1.594
1.586
1.578
1.570
1.562
1.555
NA
NA

−2
6.197
4.234
3.328
2.775
2.390
2.102
1.875
1.691
1.626
1.618
1.610
1.602
1.594
1.586
1.578
1.570
1.562
1.555
NA
NA

−3
6.197
4.234
3.328
2.775
2.390
2.102
1.875
1.691
1.626
1.618
1.610
1.602
1.594
1.586
1.578
1.570
1.562
1.555
NA
NA

n
−4
−5
6.197 6.197
4.234 4.234
3.328 3.328
2.775 2.775
2.390 2.390
2.102 2.102
1.875 1.875
1.691 1.691
1.626 1.626
1.618 1.618
1.610 1.610
1.602 1.602
1.594 1.594
1.586 1.586
1.578 1.578
1.570 1.570
1.562 1.562
NA
NA
NA
NA
NA
NA

−10
6.197
4.234
3.328
2.775
2.390
2.102
1.875
1.691
1.626
1.618
1.618
1.602
1.594
1.586
1.578
NA
NA
NA
NA
NA

−20
6.197
4.234
3.328
2.775
2.390
2.102
1.875
1.691
1.626
1.618
1.610
1.602
1.594
NA
NA
NA
NA
NA
NA
NA

−30
6.197
4.234
3.328
2.775
2.390
2.102
1.875
1.691
1.626
1.618
1.610
1.602
NA
NA
NA
NA
NA
NA
NA
NA

−∞
6.197
4.234
3.328
2.775
2.390
2.102
1.875
1.691
1.626
1.618
1.610
NA
NA
NA
NA
NA
NA
NA
NA
NA

Table 9: Wage Paid in and after the Second Period of Service (w2+ )
n
F
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.840 (F1 )
0.850
0.860
0.870
0.880
0.890
0.900
0.911
0.922
0.932
0.943 (F3 )F > F3

ŵ
4.047
3.007
2.527
2.234
2.031
1.878
1.758
1.660
1.626 (w∗∗∗ )
1.618
1.610
1.602
1.594
1.586
1.578
1.570
1.562
1.555
1.547 (w∗∗ )1.547 (w∗∗ )

0
1.547
1.547
1.547
1.547
1.547
1.547
1.547
1.547
1.547
1.547
1.547
1.547
1.547
1.547
1.547
1.547
1.547
1.547
1.547
1.547

−1
3.865
2.903
2.460
2.189
2.000
1.859
1.748
1.657
1.626
1.618
1.610
1.602
1.594
1.586
1.578
1.570
1.562
1.555
NA
NA

−2
3.865
2.903
2.460
2.189
2.000
1.859
1.748
1.657
1.626
1.618
1.610
1.602
1.594
1.586
1.578
1.570
1.562
1.555
NA
NA

−3
3.865
2.903
2.460
2.189
2.000
1.859
1.748
1.657
1.626
1.618
1.610
1.602
1.594
1.586
1.578
1.570
1.562
1.555
NA
NA

25

−4
3.865
2.903
2.460
2.189
2.000
1.859
1.748
1.657
1.626
1.618
1.610
1.602
1.594
1.586
1.578
1.570
1.562
NA
NA
NA

−5
3.865
2.903
2.460
2.189
2.000
1.859
1.748
1.657
1.626
1.618
1.610
1.602
1.594
1.586
1.578
1.570
1.562
NA
NA
NA

−10
3.865
2.903
2.460
2.189
2.000
1.859
1.748
1.657
1.626
1.618
1.610
1.602
1.594
1.586
1.578
NA
NA
NA
NA
NA

−20
3.865
2.903
2.460
2.189
2.000
1.859
1.748
1.657
1.626
1.618
1.610
1.602
1.594
NA
NA
NA
NA
NA
NA
NA

−30
3.865
2.903
2.460
2.189
2.000
1.859
1.748
1.657
1.626
1.618
1.610
1.602
NA
NA
NA
NA
NA
NA
NA
NA

−∞
3.865
2.903
2.460
2.189
2.000
1.859
1.748
1.657
1.626
1.618
1.610
NA
NA
NA
NA
NA
NA
NA
NA
NA

job seekers are hired in equilibrium, which creates room for firms’ belief toplay a crucial role in selecting job applicants, thereby diversifying the SCEs.
All of the diversified SCEs, each of which corresponds to a single belief
such that n < 0, remain in existence as long as F ∈ (F1 , F2 ]. However, if
F exceeds F2 , they start to decrease in number, and cease to exist before
F reaches F3 . As shown in the tables, an SCE characterized by a smallern ceases to exist earlier than those by larger ns. A larger value of F and
its resulting lower value of w∗ weaken the incentive of jobless workers topreserve their employability, thereby making it more diﬃcult for an SCE tosatisfy all of its EPCs, the number of which is equal to the absolute valueof n. When F is suﬃciently close to F3 , even the SCE with n = −1 cannot
satisfy its single EPC. Thus, the last SCE vanishes from the economy.

5.3

Welfare Comparison among the Existing Equilibria

To evaluate the economic welfare of the existing equilibria, we constructthree measures of economic welfare: workers’ surplus (WS), investors’ surplus (IS), and their sum, which we call total surplus (TS), as follows.
)t
∞ (
∑
1
WS ≡
[w̃E + w(N − E) − e(N − UO )] ,
1+r
t=0

)t
∞ (
∑
1
E
IS ≡
· ∗ · [(L∗ )α − w̃L∗ − bF ],
1+r
L
t=0

TS ≡ WS+IS
)t [
]
∞ (
∑
1
bF E
∗ α−1
E(L )
+ w(N − E) − ∗ − e(N − UO ) ,
=
1+r
L
t=0

where w̃ ≡ w1 − (1 − θ)(1 − b)(w1 − w2+ ).11 As defined above, the WS is thediscounted sum of workers’ incomes minus their eﬀort expenditures. In eachperiod, employed workers receive, on average, w̃ units of labor income, andjobless ones receive w units of endowment; these add up to w̃E + w(N − E).
Moreover, employable workers, the population of which is equal to N − UO ,expend e units of eﬀort. Thus, the workers constituting the labor force
receive, on aggregate, w̃E + w(N − E) − e(N − UO ) units of surplus, which11
We can also express w̃ as the average of average labor costs. While marginal laborcosts are common across all firms and equal to ŵ defined by (3), the average labor costs of
new firms, which have just started production, may diﬀer from those of old firms, whichstarted production one or more periods before. On the one hand, the average labor costsof new firms, the number of which is b, are given by w1 since all of their employees are
newly recruited workers. On the other hand, the average labor costs of old firms, thenumber of which is 1 − b, is given by θw1 + (1 − θ)w2+ since newly recruited workers only
account for a fraction θ of their employees. By averaging out these average costs over all
operating firms, we can obtain the definition of w̃.

26

Table 10: Workers’ Surplus
n
F
0
−1
−2
−3
−4
−5
−10
0.1
200.17 344.69 344.69 344.69 344.69 344.69 344.69
0.2
159.78 244.45 244.45 244.45 244.45 244.45 244.45
0.3
141.15 198.23 198.23 198.23 198.23 198.23 198.23
0.4
129.76 169.98 169.98 169.98 169.98 169.98 169.98
0.5
121.85 150.34 150.34 150.34 150.34 150.34 150.34
0.6
115.92 135.64 135.64 135.64 135.64 135.64 135.64
0.7
111.26 124.07 124.07 124.07 124.07 124.07 124.07
0.8
107.46 114.65 114.65 114.65 114.65 114.65 114.65
0.840 (F1 ) 106.13 111.34 111.34 111.34 111.34 111.34 111.34
0.850
105.81 110.51 110.50 110.50 110.50 110.50 110.50
0.860
105.50 109.68 109.67 109.66 109.66 109.66 109.66
0.870
105.19 108.86 108.84 108.83 108.83 108.83 108.83
0.880
104.89 108.04 108.01 108.01 108.01 108.01 108.01
0.890
104.58 107.22 107.19 107.19 107.19 107.19 107.19
0.900
104.27 106.40 106.37 106.37 106.37 106.37 106.37
0.911
103.97 105.58 105.55 105.55 105.55 105.55
NA
0.922
103.67 104.75 104.74 104.74 104.74 104.74
NA
0.932
103.37 103.91 103.93 103.94
NANA
NA
0.943 (F3 ) 103.07
NA
NA
NA
NA
NA
NA
F > F3
101
NA
NA
NA
NA
NA
NA
Note: The largest possible value is reported when (F, n) = (F3 , 0).

−20344.69244.45198.23169.98150.34135.64124.07114.65111.34110.50109.66108.83108.01
NA
NA
NA
NA
NA
NA
NA

−30
344.69244.45198.23169.98150.34135.64124.07114.65111.34110.50109.66108.83
NA
NA
NA
NA
NA
NA
NA
NA

−∞
344.69244.45198.23169.98150.34135.64124.07114.65111.34110.50109.66
NA
NA
NA
NA
NA
NA
NA
NA
NA

is the summand without the discount factor in the defining equation of the
WS. The IS is a discounted sum of “net profit flows” from the firm sector to
the investors, that is, profits earned by currently operating firms minus theentry costs needed to establish new firms. In each period, operating firms
distribute their profits to investors, which add up to (E/L∗ )[(L∗ )α − w̃L∗ ].
On the other hand, the investors pay an aggregate of b(E/L∗∗ )F to fund
new firms. Thus, on balance, (E/L∗∗ )[(L∗∗ )α − w̃L∗∗ −bF ] units of net profit
flow from the firm sector to the investors. This is the summand without thediscount factor in the defining equation of the IS. The TS is the discounted
sum of the net output of this economy, that is, the sum of the produced
and endowed consumption goods minus investment in new firms and eﬀort
expenditures.12
Tables 10–12 report, respectively, the values of the WS, IS, and TS, which
are computed under the parameter configuration of Table 1. As shown inthese tables, for a given entry cost level, the decrease in n from 0 to −1
produces significant welfare gains, whereas other decreases in n produce
only small ones. To understand why, we must note that a decrease in n has
two conflicting eﬀects on TS. On the one hand, it increases the aggregate
employment, thereby enhancing the total amounts of labor incomes and net
profit flows, which contributes to an enhancement of the TS. On the other12

In fact, WS and IS can be interpreted as, respectively, the discounted sum of the
lifetime utilities of the workers constituting the labor force in the present and future periods
and the discounted sum of expected gains from past, current, and future investments. For
the details, see Appendix D.

27

Table 11: Investors’ Surplus
n
F
0
−1
−2
−3
−4
−5
−10
−20
0.1
28.29 35.82 35.82 35.82 35.82 35.82 35.82 35.82
0.2
18.95 26.25 26.25 26.25 26.25 26.25 26.25 26.25
0.3
14.64 21.84 21.84 21.84 21.84 21.84 21.84 21.84
0.4
12.01 19.15 19.15 19.15 19.15 19.15 19.15 19.15
0.5
10.18 17.27 17.27 17.27 17.27 17.27 17.27 17.27
0.6
8.81 15.87 15.87 15.87 15.87 15.87 15.87 15.87
0.7
7.73 14.76 14.76 14.76 14.76 14.76 14.76 14.76
0.8
6.85 13.86 13.86 13.86 13.86 13.86 13.86 13.86
0.840 (F1 ) 6.55 13.55 13.55 13.55 13.55 13.55 13.55 13.55
0.850
6.47 13.42 13.47 13.47 13.47 13.47 13.47 13.47
0.860
6.40 13.29 13.38 13.39 13.39 13.39 13.39 13.39
0.870
6.33 13.14 13.29 13.31 13.31 13.31 13.31 13.31
0.880
6.26 12.98 13.20 13.23 13.23 13.23 13.23 13.23
0.890
6.19 12.79 13.10 13.15 13.15 13.15 13.15 NA
0.900
6.12 12.58 12.99 13.07 13.07 13.07 13.07 NA
0.911
6.05 12.31 12.88 12.98 12.99 12.99 NA
NA
0.922
5.98 11.98 12.76 12.90 12.91 12.92 NANA
0.932
5.91 11.52 12.62 12.81 NA
NA
NA
NA
0.943 (F3 ) 5.84
NA
NA
NA
NA
NA
NA
NA
F > F3
0
NA
NA
NA
NA
NA
NA
NA
Note: The largest possible value is reported when (F, n) = (F3 , 0).

−30
35.82
26.25
21.84
19.15
17.27
15.87
14.76
13.86
13.55
13.47
13.39
13.31
NA
NA
NA
NA
NA
NA
NA
NA

−∞
35.82
26.25
21.84
19.15
17.27
15.87
14.76
13.86
13.55
13.47
13.39
NA
NA
NA
NA
NA
NA
NA
NA
NA

Table 12: Total Surplus
n
F
0
−1
−2
−3
−4
−5
−10
0.1
228.46 380.51 380.51 380.51 380.51 380.51 380.51
0.2
178.73 270.71 270.71 270.71 270.71 270.71 270.71
0.3
155.79 220.07 220.07 220.07 220.07 220.07 220.07
0.4
141.77 189.13 189.13 189.13 189.13 189.13 189.13
0.5
132.03 167.62 167.62 167.62 167.62 167.62 167.62
0.6
124.73 151.50 151.50 151.50 151.50 151.50 151.50
0.7
118.99 138.83 138.83 138.83 138.83 138.83 138.83
0.8
114.32 128.51 128.51 128.51 128.51 128.51 128.51
0.840 (F1 ) 112.67 124.89 124.89 124.89 124.89 124.89 124.89
0.850
112.29 123.93 123.97 123.97 123.97 123.97 123.97
0.860
111.91 122.96 123.04 123.05 123.05 123.05 123.05
0.870
111.52 122.00 122.13 122.14 122.14 122.14 122.14
0.880
111.15 121.01 121.21 121.24 121.24 121.24 121.24
0.890
110.77 120.01 120.28 120.33 120.34 120.34 120.34
0.900
110.39 118.98 119.36 119.43 119.44 119.44 119.44
0.911
110.02 117.89 118.43 118.54 118.55 118.55
NA
0.922
109.65 116.73 117.50 117.64 117.66 117.66
NA
0.932
109.28 115.43 116.55 116.75
NANA
NA
0.943 (F3 ) 108.91
NA
NA
NA
NA
NA
NA
F > F3
101
NA
NA
NA
NA
NA
NA
Note: The largest possible value is reported when (F, n) = (F3 , 0).

28−20380.51
270.71220.07189.13167.62151.50138.83128.51124.89123.97123.05122.14121.24
NA
NA
NA
NA
NA
NA
NA

−30
380.51
270.71220.07189.13167.62151.50138.83128.51124.89123.97123.05122.14
NA
NA
NA
NA
NA
NA
NA
NA

−∞
380.51
270.71220.07189.13167.62151.50138.83128.51124.89123.97123.05
NA
NA
NA
NA
NA
NA
NA
NA
NA

hand, it increases the number of employable workers, thereby increasing
the total expenditure of eﬀorts, which contributes to a reduction in the
TS. Therefore, the eﬀects of a decrease in n on the TS are theoreticallyinconclusive since it always increases IS and has mixed eﬀects on the WS.
When n is decreased from 0 to −1, both IS and WS increase their values,
and therefore, TS also exhibits a substantial enhancement. This decrease inn substantially increases the populations of employed and employable workers. As stated above, the increase in employed workers enhances IS and WS,
while the increase in employable workers reduces WS. Despite this friction,
WS increases its value in this case since the positive contribution throughthe increased labor income, (1 + r−1 )(w̃ − w)∆E, dominates the negative
contribution through the increased eﬀort expenditure, (1 + r−1 )e∆UO .
In contrast, when n is decreased from −1 or a smaller value to a further
smaller one, none of WS, IS, or TS exhibit a substantial change for any
level of the entry cost. When F is no larger than F1 , the existing SCEs arecoalesced into one equilibrium, and thus, any decrease in n has no marginal
eﬀect on WS, IS, or TS. When F takes a value between F1 and F3 but is
not so close to F3 , the existing SCEs are so close to one another that any
decrease in n can hardly change the values of E and UO , thus producing only
small changes in WS, IS, and TS at best.13 When F takes a value between
F1 and F3 that is suﬃciently close to F3 , a decrease in n may lead to a
substantial increase in E and a substantial decrease in UO . For example,
when F = 0.932, the decrease in n from −1 to −2 increases E by 0.082 andreduces UO by 0.090 (See Tables 4 and 5). However, these changes do notlead to a substantial enhancement of WS because the positive contribution
through the increased labor income, (1 + r−1 )(w̃ − w)∆E ≈ 4.58, is mostly
oﬀset by the negative contribution through the increased eﬀort expenditure,(1 + r−1 )e∆UO ≈ −4.56. On the other hand, the increased employment alsoenhances IS by 1.10, which explains the greater part of the enhancement of
TS observed in this case.

5.4

Policy Implications

The theoretical and numerical analyses made thus far yield the following
policy implications. First, second chances should be given to jobless workers.
It is clear from the second and third columns of Table 12, which summarize
the total surpluses for the cases of n = 0 and n = −1, respectively, that theabsence of second chances causes a substantial loss of economic welfare bydiscouraging all jobless workers from preserving their employability. Unless
13
Nevertheless, we should note that the decrease in n may reduce WS for some valuesof F . As shown in Table 10, the decrease in n from −1 to −2 reduces WS if F ranges
from 0.850 to 0.922, and the decrease in n from −2 to −3 also reduces WS if F equals
either 0.860 or 0.870. In these cases, the increment of labor income is dominated by thatof eﬀort expenditure, which causes the observed reduction in WS.

29

Table 13: Changes in Workers’ Surplus
n
0
F
w
NSCE
SCE
−1
−2
−3
−4
−5 −100.840 (F1 )
1
0
5.21
0
0
0
0
0
0
0.850
0.992 −0.50
4.71
0.02 0.03 0.03 0.03 0.03 0.03
0.860
0.984 −1.00
4.21
0.03 0.05 0.05 0.05 0.05 0.05
0.870
0.976 −1.49
3.72
0.05 0.07 0.08 0.08 0.08 0.08
0.880
0.968 −1.99
3.22
0.07 0.10 0.10 0.10 0.10 0.10
0.890
0.960 −2.48
2.73
0.10 0.12 0.13 0.13 0.13 0.13
0.900
0.952 −2.96
2.25
0.12 0.15 0.15 0.15 0.15 0.15
0.911
0.944 −3.45
1.76
0.16 0.18 0.18 0.18 0.18 NA
0.922
0.937 −3.93
1.28
0.20 0.20 0.20 0.20 0.20 NA
0.932
0.929 −4.41
0.80
0.25 0.23 0.23 NA NA NA
0.943 (F3 ) 0.921 −4.89
0.32
NA NA NA NA NA NA
1
0.883 −6.70−1.49 NA NA NA NA NA NA
2
0.495 −45.86 −40.65 NA NA NA NA NA NA
Note: The smallest possible changes are reported when (F, n) = (F3 , 0).

−20
0
0.03
0.05
0.08
0.10
NA
NA
NA
NA
NA
NA
NA
NA

−30
0
0.03
0.05
0.08
NA
NA
NA
NA
NA
NA
NA
NA
NA

−∞
0
0.03
0.05
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA

−20
0
0.01
0.03
0.04
0.05
NA
NA
NA
NA
NA
NA
NA
NA

−30
0
0.01
0.03
0.04
NA
NA
NA
NA
NA
NA
NA
NA
NA

−∞
0
0.01
0.03
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA

Table 14: Changes in Investors’ Surplus
n0Fw
NSCE SCE
−1
−2
−3
−4
−5 −100.840 (F1 )
1
0
7.00
0
0
0
0
0
0
0.850
0.992
0.04
7.01 0.06 0.01 0.01 0.01 0.01 0.01
0.860
0.984
0.08
7.01 0.13 0.03 0.03 0.03 0.03 0.03
0.870
0.976
0.13
7.02 0.21 0.06 0.04 0.04 0.04 0.04
0.880
0.968
0.17
7.02 0.31 0.09 0.05 0.05 0.05 0.05
0.890
0.960
0.21
7.03 0.42 0.12 0.07 0.06 0.06 0.06
0.900
0.952
0.25
7.03 0.57 0.16 0.09 0.08 0.08 0.08
0.911
0.944
0.29
7.04 0.77 0.21 0.10 0.09 0.09 NA
0.922
0.937
0.33
7.04 1.04 0.27 0.12 0.11 0.11 NA
0.932
0.929
0.37
7.05 1.44 0.33 0.15 NA NA NA
0.943 (F3 ) 0.921
0.41
7.05 NA NA NA NA NA NA
1
0.883
6.10
12.57 NA NA NA NA NA NA
2
0.495
4.64
9.34 NA NA NA NA NA NA
Note: The smallest possible changes are reported when (F, n) = (F3 , 0).

the entry cost is so high that no stationary equilibrium but the NSCE canexist in this economy, the government should make every eﬀort to persuade
firms not to hold the extreme view that all jobless workers have already losttheir employability. This would create second chances for those who havebeen unemployed for a relatively short span of time and would lead to an
extensive improvement in the aggregate employment. As shown in Figures 2
and 3, the aggregate employment cannot be larger than 0.436 when second
chances are absent, that is, n = 0, but it never falls below 0.820 when thesechances are present, that is, n < 0.
Second, if the entry cost takes such a value that multiple SCEs can exist
in this economy, taxing jobless workers may lead to an improvement in economic welfare in the sense that it enhances both WS and IS. When the value

30

Table 15: Changes in Total Surplus
n
0
F
w
NSCE
SCE
−1
−2
−3
−4
−5 −100.840 (F1 )
1
0
12.21
0
0
0
0
0
0
0.850
0.992 −0.46
11.72 0.07 0.04 0.04 0.04 0.04 0.04
0.860
0.984 −0.91
11.22 0.16 0.08 0.08 0.08 0.08 0.08
0.870
0.976 −1.37
10.73 0.26 0.13 0.12 0.11 0.11 0.11
0.880
0.968 −1.82
10.25 0.38 0.19 0.16 0.15 0.15 0.15
0.890
0.960 −2.27
9.76
0.52 0.24 0.20 0.19 0.19 0.19
0.900
0.952 −2.72
9.28
0.70 0.31 0.24 0.23 0.23 0.23
0.911
0.944 −3.16
8.80
0.93 0.38 0.28 0.27 0.27 NA
0.922
0.937 −3.60
8.32
1.24 0.47 0.33 0.31 0.31 NA
0.932
0.929 −4.04
7.84
1.69 0.57 0.37 NA NA NA
0.943 (F3 ) 0.921 −4.48
7.37
NA NA NA NA NA NA
1
0.883 −0.59
11.09
NA NA NA NA NA NA
2
0.495 −41.22 −31.30 NA NA NA NA NA NA
Note: The smallest possible changes are reported when (F, n) = (F3 , 0).

−20
0
0.04
0.08
0.11
0.15
NA
NA
NA
NA
NA
NA
NA
NA

−30
0
0.04
0.08
0.11
NA
NA
NA
NA
NA
NA
NA
NA
NA

−∞
0
0.04
0.08
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA

of w is reduced by taxation, workers are more drawn to the status of employment, which enables firms to motivate their employees by paying lower
wages than before. This, in turn, shifts the profiles of the AICs downward.
If w is reduced to such a level that the profiles of the AICs for n < 0 can
intersect with that of the aggregate labor demand at the upper end of their
upward-sloping parts, then, possible stationary equilibria are restricted to
one SCE and one NSCE, which we call the “new SCE” and the “new NSCE,”
respectively. In the new SCE, E and ŵ are determined as E = E ∗∗∗ andŵ = w∗ = w∗∗∗ , and, regardless of firms’ belief, jobless workers can find newjobs by experiencing only one period of unemployment. In the new NSCE,
E and ŵ are determined as E = E ∗∗ and ŵ = w∗ = w∗∗∗ . Since the new
SCE attains the maximum employment, it is quite natural to guess that the
government can enhance both WS and IS by guiding the economy from thecurrent equilibrium to the new SCE.
Tables 13–15 report the results of a numerical experiment conducted tojudge the validity of this guess, with all eﬀects evaluated under the parameter
configuration of Table 1.14 There are thirteen columns in each table; the firstlists the values of F , all of which, except for F1 , give rise to either multiple
or no SCEs before w is reduced; the second reports the maximal value of w
that can support the new SCE for a given value of F ; the third reports thesurplus gains from moving the economy from the old NSCE, which exists
before w is reduced, to the new NSCE; the fourth reports the surplus gains
from moving the economy from the old NSCE to the new SCE; and the fifth
to thirteenth report the surplus gains from moving the economy from an old
SCE, which exists before w is reduced, to the new one. The case in which
F = F1 serves as a benchmark, since in this case, the profile of the aggregate
14

For the details of the derivation, see Appendix E.

31

labor demand runs through the upper end of the upward-sloping parts of
the AICs for n < 0, although w is not reduced. When F ∈ (F1 , F3 ), thereduction in w coalesces the multiple old SCEs into the new one, in which E
is increased to E ∗∗∗ , UO is decreased to zero, but the levels of w1 and w2+ ,and thus, that of w̃ remain unchanged. It also creates the new NSCE, inwhich E, UO , and ŵ take the same values as in the old one, but w1 and w2+
have changed their values in such a manner that w̃ is reduced. As shownin Tables 13–15, the resulting changes in WS, IS, and TS are positive if
the economy moves to the new SCE, implying that in these cases, guiding
the economy from the current equilibrium to the new SCE surely improves
economic welfare. When F = F3 , the reduction in w coalesces the multiple
NSCEs into the one with E = E ∗∗ and w2+ < w̃ < ŵ = w∗ = w∗∗∗ < w1
while creating from scratch the new SCE with E = E ∗∗∗ and w2+ = w̃ =ŵ = w∗ = w∗∗∗ = w1 . Again, in this case, guiding the economy from an old
NSCE to the new SCE improves economic welfare, as implied by Tables 13–
15. When F > F3 , the situation changes drastically. Before w is reduced,
there is a single NSCE, in which no production occurs, and the reduction
in w restarts production activities in the economy by creating the new SCE
and the new NSCE, in both of which a positive population of workers ishired. However, this does not lead to an improvement in economic welfare
since it decreases WS even if the economy is guided to the new SCE, asshown in Table 13. In these cases, firms need to pay low wages to theiremployees to generate profit flows that are large enough to compensate fora large entry cost. Such low wages significantly erode the attraction of
employment and completely eliminate the incentive of workers to expend
eﬀorts unless w is substantially reduced. Put another way, when the entry
cost is substantially large, the reduction in w eﬀectively curtails workers’
income-earning opportunities other than employment, thereby forcing them
to work for low wages. As shown in Table 13, when F is as large as two, it
decreases TS even if the economy is guided to the new SCE, which implies
that production activities are no longer socially desirable. To sum up thisargument, taxing jobless workers can improve economic welfare only whenthe entry cost takes such an intermediate value that multiple SCEs can exist
in this economy. When the entry cost is substantially large, such taxation
makes only investors better oﬀ at the expense of workers’ welfare.

6

Conclusion

This paper has examined the long-run consequences of ranking job applicants by their unemployment durations by using a simple general equilibrium
model with eﬃciency wages, in which the statistical discrimination of firms
against jobless workers has a distorting eﬀect on labor market performance.
When the cost of establishing a new firm takes an intermediate value, the32model exhibits multiple stationary equilibria, which diﬀer from one anotherin allocative eﬃciency, and the most ineﬃcient one is realized by the beliefof firms that jobless workers have already lost their employability. Thus
the government should persuade firms not to hold such an extreme belief,
thereby creating second chances for jobless workers. Moreover, if the government can reduce the incomes of jobless workers through taxation, themultiple equilibria, in which a subset of jobless workers are given second
chances, are coalesced into a more eﬃcient one, in which every job seeker
can find a new job by experiencing one period of unemployment. These
policies eﬀectively nullify the distorting eﬀect of statistical discriminationconsidered here.

Appendix A: Proofs (Not for Publication)
Proof of Proposition 1
Proof. Let V (n, Q) denote the lifetime utility of a worker in the labor force
whose current state is (n, Q). If the path of wages paid to an employedworker is optimally chosen, then she finds it optimal to expend work eﬀort
in every period, and thus, for ∀n ≥ 1, the following relations hold true:1−θ
[(1 − b)V (n + 1, 1) + bV (−1, 1)]
1+r[]
1 − θ (1 − b)(1 − q)V (n + 1, 1)
n
≥ w +
.
1 + r +(b + q − bq)V (−1, 1)

V (n, 1) = wn − e +(6)The larger side of (6) measures the lifetime utility when this worker expends
e units of work eﬀort in the nth period. If she makes this choice, then, inthe next period, she will either keep this job with probability 1 − b, inwhich case her lifetime utility changes from V (n, 1) to V (n + 1, 1), or lose
it with probability b, in which case her lifetime utility changes from V (n, 1)
to V (−1, 1). The smaller side, on the other hand, measures the lifetime
utility when this agent expends zero eﬀort in the nth period. If she makes
this choice, then, in the next period, she will either keep this job withprobability (1 − b)(1 − q), in which case her lifetime utility changes from
V (n, 1) to V (n + 1, 1), or lose it with probability b + q − bq, in which caseher lifetime utility changes from V (n, 1) to V (−1, 1). By rearranging bothsides of (6), we can obtain∀n ≥ 1,V (n + 1, 1) ≥ V (−1, 1) +1+re.
(1 − θ)(1 − b)q(7)

The firm sets wages paid in and after the second period of service at sucha level that the equalities of (7) hold true, thereby minimizing labor costs.
Because workers are not allowed to hop directly from one firm to another,
33

the firm has no incentive to pay employees recruited in or before the previous
period higher wages than those necessary to elicit their work eﬀorts. Using
the equalities of (7) to eliminate V (n, 1) and V (n + 1, 1) from (6), we canobtain∀n ≥ 2, wn =r+θ
1 + r − (1 − θ)(1 − b)(1 − q)
V (−1, 1) +e = w2+ . (8)1+r(1 − θ)(1 − b)q

To elicit work eﬀorts from newly recruited workers or, equivalently, to convince them that their labor contracts will be renewed at the end of the first
period of service, the starting wage, w1 , cannot be lower than those paid inand after the second period of service, w2+ . When w1 ≥ w2+ , the workers
are convinced that unless they shirk, their contracts will be renewed since
their wages decrease from w1 to w2+ in the next period, which gives theirfirm a strong incentive to continue the employment relationship with them.
When w1 < w2+ , in contrast, the workers are so skeptical about the renewalof their contracts that they choose to expend zero eﬀort. If new workers
can be recruited at a starting wage that is lower than w2+ , the firm finds it
profitable to terminate its contracts with the current employees, replacing
them with new ones and, thereby, saving labor costs. From the viewpointof newly recruited workers, however, the starting wage lower than w2+ is an
informal notice that their contracts will terminate at the end of the first period of service. Thus, if such a low starting wage is oﬀered, they will choose
to expend zero eﬀorts while receiving that wage, which is paid independentof their eﬀort expenditure. We must also note that the cost minimizationof the firm cannot preclude the possibility of w1 > w2+ . Unlike employedworkers, job applicants can apply to multiple job openings, which makes
it possible for starting wages to take a larger value than w2+ . When theaggregate demand for labor is suﬃciently large, firms may not hire a desired
number of new workers without making the level of w1 higher than w2+ .
Proof of Lemma 1
Consider a firm established in period t, and let Lnt+s be the number of its
employees who complete the nth period of service in period t+s. Obviously,
Lt+s and {Lnt+s }sn=1 satisfy
Lt+s =s∑

Lnt+s .

(9)n=1

As shown in Proposition 1, if the firm sets the wage levels for the newly
recruited and other employed workers at w1 and w2+ , respectively, then, allemployees expend work eﬀorts. Hence, the expected profits of this firm can34

be written as
]
)α
[( s
)
s
∞ (
∑
∑
∑
1 − b s−1
wn Lnt+s
−
Lnt+s
1+r
s=1

n=1

n=1

= Lαt+1 − w1 Lt+1
+

)
∞ (
∑
1 − b s−1
s=2

1+r

[( s
∑

)α
− w1 L1t+s − w2+

Lnt+s

n=1

s
∑

]
Lnt+s

(10)

n=2

= Lαt+1 − w1 Lt+1
+

)
{
∞ (
∑
1 − b s−1 Lα
s=2

1+r

2+
t+s − w (1 − θ)Lt+s−1
1−w [Lt+s − (1 − θ)Lt+s−1 ]},the first equality of which is obtained from the fact that wn = w2+ for∀n ≥ 2, and the second from (9) and the fact that only a fraction (1 − θ) ofcurrent employees survive into the next period:∀t,∀s ≥ 1,∀n = 1, . . . , s,n
Ln+1
t+s+1 = (1 − θ)Lt+s .

If {Lt+s }∞s=1 is chosen to maximize (10), then∀t,∀s ≥ 1,1αLα−1t+s = w −(1 − b)(1 − θ)(w1 − w2+ ),1+rwhich implies our desired result.
Proof of Lemma 2
Proposition 1 and Lemma 1 jointly imply that the net gain can be rewritten
as
)
∞ (
)
1 ∑ 1 − b s−1 ( αL̃t+s − wt+s · Lt+s
−F +
1+r
1+r
s=1

∞ (

zLα − w1 L ∑= −F +
+
1+r
s=2

1−b
1+r

)s−1

Lα − [θw1 + (1 − θ)w2+ ]L
1+rzLα − w1 L 1 − b Lα − [θw1 + (1 − θ)w2+ ]L
+
·
1+r
r+b
1+r
[
]
1
(1 − θ)(1 − b)(w1 − w2+ )α1= −F +
L −w L+L
r+b
1+r= −F += −F +

Lα − ŵL,
r+b
35

the second line of which is obtained from the assertions of Proposition 1 andLemma 1 that every firm sets the wage for the employees in the nth periodof service at a constant level as in (8); that such a wage profile induces
employees to expend work eﬀorts, thereby making each firm’s eﬀective labor
input coincide with its total number of employees, L̃t+s = Lt+s ; and thateach firm keeps its total number of employees constant over time, Lt+s = L.
Also note that the last line of this equation is obtained from the definitionof ŵ, (3). Thus, if the net gain equals zero, the following must hold:−F + (Lα − ŵL)/(r + b) = 0.

(11)

By solving (3) and (11) with respect to ŵ and L, we can obtain our desired
results.
Proof of Proposition 2
Let m and L denote, respectively, the number of firms operating in such anequilibrium and the number of employees each firm hires. Then, m, L, E,
and ŵ satisfy (3),mL = E,and−F + (Lα − ŵL)/(r + b) ≤ 0.

(12)

Condition (12) means that the net gain from setting up a new firm cannot
be positive in equilibrium. To see why, suppose that the net gain is positive,−F + (Lα − ŵL)/(r + b) > 0,
which, combined with (3), implies thatŵ − w∗ < 0.

(13)

Because the number of firms is increasing as long as the net gain is positive,
there are infinitely many firms operating in equilibrium, m = +∞. We
must also note that the aggregate employment (i.e., E) takes a finite value
since the number of workers in the labor force equals N in any period.
These jointly suggest that the number of employees chosen by each operating
firm must be infinitesimally small, L = 0. According to (3), firms find it
optimal to select such a small employment size if and only if ŵ = +∞,which obviously contradicts (13). Thus, we can safely say that (12), andthus, ŵ − w∗ ≥ 0, are true in equilibrium. When the equality of (12) isvalid, operating firms set their levels of ŵ and L as in (4), as already shownin Lemma 2. On the other hand, their number cannot be determined since
any non-negative value of m is consistent with investors’ rational decisions.
36Thus, in this case, E can take any non-negative value. When the inequalityof (12) is valid, conditions (3) and (12) jointly imply that firms would set
the level of ŵ as ŵ − w∗ > 0 if they were operating in equilibrium. However,in equilibrium, no firm is established because m = 0, and thus, E must be
equal to zero.

Appendix B: Derivation of Aggregate Incentive Constraints (Not for Publication)
When n = 0 and n = −∞, equilibrium conditions are so simple that we
can analytically derive the profiles of the AIC for these cases. When n ∈(−∞, 0), equilibrium conditions are so complicated that we must resort toa numerical method to derive the profiles of the AIC for these cases.

AIC for n = 0
Let us begin by deriving the AIC for the case of n = 0. This case is special inthat firms’ belief is unconditionally self-fulfilling. Specifically, firms strongly
believe that jobless workers have already lost their employability, disqualifying currently jobless workers for employment and thereby nullifying the riskof hiring an unemployable worker. In response to this hiring policy, joblessworkers stop maintaining their employability, which completely fulfills firms’
belief.
Lemma 3. Suppose that there is a stationary equilibrium with n = 0. Then,in that equilibrium, (a) jobless workers, who have negative values of n, expend zero eﬀort, and their lifetime utilities are determined as∀n ≤ −1,∀Q = 0, 1,

V (n, Q) = [(1 + r)/(r + θ)]w,

and (b) the level of w2+ is determined as
w2+ = w +1 + r − (1 − θ)(1 − b)(1 − q)e (≡ w∗∗ ).
(1 − θ)(1 − b)q(14)

Proof. When n = 0, job openings are limited to newly born workers in eachperiod, and thus the lifetime utility of a worker unemployed at the beginningof period t can be written as
)
∞ (
∑
1−θ s
(w − et+s ).
1+r
s=0

Obviously, this utility attains its maximum, [(1 + r)/(r + θ)]w, at et+s = 0
for ∀s ≥ 0, which establishes part (a). Part (a) implies that in this case,V (−1, 1) = [(1 + r)/(r + θ)]w.
Substituting (15) into (2) produces (14).
37

(15)

Because every worker dies with probability θ, and because every job isdestroyed and re-created with probability b, there are [(θ + b − θb)/(1 − θ)]E
job openings in each period.15 When firms fill their vacancies exclusively
with new entrants into the labor force, then, the following must hold:a[θ/(1 − θ)]N = [(θ + b − θb)/(1 − θ)]E,(16)

where a ∈ [0, 1] denotes the job finding rate for workers believed to be
employable, who are the new entrants in this case. The value of a is notexogenously given, but endogenously determined in equilibrium.
Proposition 3 (AIC for n = 0). Suppose that there is a stationary equilibrium with n = 0, and let E, ŵ, and w∗∗ represent the aggregate employment,
the average wage defined by (3), and the constant defined by (14), respectively. Moreover, define E ∗∗ asE ∗∗ ≡ [θ/(θ + b − θb)]N.
Then, in that equilibrium, E and ŵ satisfy(E ∗∗ − E)(ŵ − w∗∗ ) = 0,

(17)

E ∗∗ − E ≥ 0(18)ŵ − w∗∗ ≥ 0.

(19)

and

Proof. When the aggregate labor demand is so large that all newly born
workers can find jobs, a = 1, their starting wages w1 and the aggregate
employment E must satisfy
E = [θ/(θ + b − θb)]N (≡ E ∗∗ ),w1 ≥ w∗∗ ,(20)

where w∗∗ is as defined in (14). When the aggregate labor demand is notlarge enough to ensure jobs for all newly born workers, a < 1, the starting
wages and the aggregate employment must satisfy
E < E ∗∗ ,w1 = w∗∗ .

(21)

Conditions (20) and (21) can be merged as(E ∗∗ − E)(w1 − w∗∗ ) = 0;
E ∗∗ − E ≥ 0;
w1 − w∗∗ ≥ 0,
which are equivalent to (17)-(19) since the sign of w1 − w∗∗ always coincides
with that of ŵ − w∗∗ .
38

ŵ

w∗∗

O

E ∗∗

E

Figure 8: AIC for n = 0
Figure 8 depicts the AIC for n = 0. As shown in that figure, when firms
embrace the belief of n = 0, it is impossible to attain a level of aggregate
employment that is larger than E ∗∗ . Under this belief, all vacant positions
must be filled with new entrants, which necessitates that the aggregate employment be no larger than E ∗∗ . When E = E ∗∗ , the labor market is sotight that all newly born workers can find a job, a = 1, and thus, wages
satisfy
w1 ≥ ŵ ≥ w2+ = w∗∗ .
When E < E ∗∗ , the labor market is so loose that some of the newly born
workers cannot find a job, a < 1, and thus, wages satisfy
w1 = w2+ = ŵ = w∗∗ .

AIC for n = −∞When n = −∞, firms do not care about an applicant’s record of unemployment because they believe that unemployed workers retain their employability. However, this belief is not suﬃcient to ensure the employabilityof unemployed workers. Even when firms share such a belief, unemployedworkers may find it optimal to stop preserving their employability if they15

Note that the total number of job openings is larger than that of the vacancies that are
expected to appear, (θ + b − θb)E. This is because firms recruit extra workers, anticipatingthat a fraction θ of their recruited workers die at the end of the current period.

39

expect to experience a suﬃciently long duration of unemployment before
being rehired or if they expect to receive suﬃciently low wages after being
rehired. Put another way, some additional conditions must be met for the
self-fulfillment of firms’ belief.
Lemma 4. Suppose that there is a stationary equilibrium with n = −∞.
Then, in that equilibrium, (a) the following conditions must hold:∀n ≤ −1,w−e+

1−θ[aV (1, 1) + (1 − a)V (n − 1, 1)]
1+r

1−θ
≥w+
[a(1 − q)V (1, 0) + (1 − a + aq)V (n − 1, 0)],
1+r

(22)

and (b) the lifetime utilities of employed and unemployed workers are determined as
VDU if n ≤ −1 and Q = 0



1
 VDE
if n = 1 and Q = 0VDE if n ≥ 2 and Q = 0,
V (n, Q) =VU if n ≤ −1 and Q = 1





V 1 if n = 1 and Q = 1 E
VE if n ≥ 2 and Q = 1
1 , V
1where the values of VDU , VDE
DE , VU , VE , and VE are given by

]
1−θ [
1
a(1 − q)VDE+ (1 − a + aq)VDU ,
1+r

(23)

1
VDE
= w1 +1−θ
[(1 − b)(1 − q)VDE + (b + q − bq)VDU ] ,
1+r

(24)

VDE = w2+ +1−θ
[(1 − b)(1 − q)VDE + (b + q − bq)VDU ] ,
1+r

(25)

]
1−θ [ 1aVE + (1 − a)VU ,
1+r

(26)

VE1 = w1 − e +1−θ
[(1 − b)VE + bVU ] ,
1+r

(27)

VE = w2+ − e +1−θ
[(1 − b)VE + bVU ] .
1+r

(28)

VDU = w +VU = w − e +

and

40

Proof. Part (a): Consider an employable worker who is currently experiencing her −nth period of unemployment. If this worker preserves her employability in the current period, then at the end of that period, she will
either be hired by a firm with probability a, in which case her lifetime utility
evaluated at the beginning of the next period is given by V (1, 1), or remain
unemployed with probability 1 − a, in which case her lifetime utility evaluated at the beginning of the next period is given by V (n − 1, 1). Thus,in this case, her lifetime utility evaluated at the beginning of the current
period can be expressed as
w−e+

1−θ
[aV (1, 1) + (1 − a)V (n − 1, 1)] .
1+r

(29)

If this worker stops preserving her employability in the current period, thenat the end of that period, she will either be hired by a firm with probabilitya(1 − q), in which case her lifetime utility evaluated at the beginning of
the next period is given by V (1, 0), or remain unemployed with probability
1 − a + aq, in which case her lifetime utility evaluated at the beginning of thenext period is given by V (n − 1, 0). Thus, in this case, her lifetime utility
evaluated at the beginning of the current period can be expressed asw+

1−θ[a(1 − q)V (1, 0) + (1 − a + aq)V (n − 1, 0)] .
1+r

(30)

The worker finds it optimal to preserve her employability if and only if thevalue of (29) is no smaller than that of (30). Therefore, (22) is necessary forthe existence of a stationary equilibrium with n = −∞.
Part (b): When n = −∞, the lifetime utilities of workers who havealready lost their employability satisfy[]
1 − θ a(1 − q)V (1, 0)∀n ≤ −1, V (n, 0) = w +,(31)
1 + r +(1 − a + aq)V (n − 1, 0)
1−θV (1, 0) = w +
1+r
1

[

(1 − b)(1 − q)V (2, 0)+(b + q − bq)V (−1, 0)

and∀n ≥ 2,V (n, 0) = w2+ +

1−θ
1+r

[

]
,

(1 − b)(1 − q)V (n + 1, 0)+(b + q − bq)V (−1, 0)

(32)

]
.

(33)

To understand why these conditions hold, consider an unemployable worker
who is currently experiencing her −nth period of unemployment. At theend of the current period, she will either be hired by a firm with probabilitya(1 − q), in which case her lifetime utility changes from V (n, 0) to V (1, 0),or remain unemployed with probability 1 − a + aq, in which case her lifetime utility changes from V (n, 0) to V (n − 1, 0). Alternatively, consider an41

unemployable worker who is experiencing her nth period of service with hercurrent firm. Her employment will either continue into the next period withprobability (1 − b)(1 − q), in which case her lifetime utility changes from
V (n, 0) to V (n + 1, 0), or terminate at the end of the current period withprobability b + q − bq, in which case her lifetime utility changes from V (n, 0)
to V (−1, 0). Condition (31) implies that V (n, 0) takes a constant value,∀n ≤ −1,

V (n, 0) = V (n − 1, 0).

Otherwise, it diverges toward either positive or negative infinity as n approaches negative infinity. Likewise, condition (33) implies that V (n, 0)
takes a constant value,∀n ≥ 2,V (n, 0) = V (n + 1, 0).

Otherwise, it diverges toward either positive or negative infinity as n approaches positive infinity. These results allow us to rewrite (31)–(33) as(23)–(25). Next, consider the workers who are still employable. Their lifetime utilities satisfy[]
1 − θ aV (1, 1),(34)∀n ≤ −1, V (n, 1) = w − e +1 + r +(1 − a)V (n − 1, 1)

V (1, 1) = w1 − e +

1−θ
1+r

[

(1 − b)V (2, 1)+bV (−1, 1)

]
,

(35)

and
∀n ≥ 2,V (n, 1) = w

2+

1−θ
−e+
1+r

[

(1 − b)V (n + 1, 1)+bV (−1, 1)

]
.

(36)

To understand why these conditions hold, consider an employable worker
who is currently experiencing her −nth period of unemployment. Since (22)is true, this worker finds it optimal to expend e units of eﬀort to preserve
her employability in the current period. At the end of that period, she will
either be hired by a firm with probability a, in which case her lifetime utilitychanges from V (n, 1) to V (1, 1), or remain unemployed with probability
1 − a, in which case her lifetime utility changes from V (n, 1) to V (n − 1, 1).
Alternatively, consider an employable worker who is experiencing her nth
period of service with her current firm. As shown in Lemma 1, the wage
profile is optimally designed such that this worker chooses to expend e units
of work eﬀort in the current period. Her employment will either continue
into the next period with probability 1 − b, in which case her lifetime utilitychanges from V (n, 1) to V (n + 1, 1), or terminate at the end of the current
period with probability b, in which case her lifetime utility changes from42V (n, 1) to V (−1, 1). Condition (34) implies that V (n, 1) takes a constantvalue,∀n ≤ −1,

V (n, 1) = V (n − 1, 1).

Otherwise, it diverges toward either positive or negative infinity as n approaches negative infinity. Likewise, condition (36) implies that V (n, 1)
takes a constant value,∀n ≥ 2,V (n, 1) = V (n + 1, 1).

Otherwise, it diverges toward either positive or negative infinity as n approaches positive infinity. These results allow us to rewrite (34)–(36) as(26)–(28).
As shown in this lemma, all of the unemployed workers choose to preserve their employability if and only if condition (22) is true. Otherwise,some unemployed workers stop preserving their employability, which makes
it diﬃcult for firms to embrace the belief that all of the unemployed workers
are still employable. We call (22) the employability preserving condition(EPC) for n = −∞.
Because both new entrants and currently unemployed workers are believed to be employable in this case, the total number of job seekers is givenby [θ/(1 − θ)]N + N − E, whereas [(θ + b − θb)/(1 − θ)]E jobs become open
in each period, as in the case of n = 0. Thus, in equilibrium, the following
must hold:a{[θ/(1 − θ)]N + N − E} = [(θ + b − θb)/(1 − θ)]E.

(37)

Then we can state the following.
Proposition 4 (AIC for n = −∞). Suppose that there is a stationary
equilibrium with n = −∞. Then, in that equilibrium, E and ŵ must satisfy{[] }r + θ + (1 − θ)b θ + b − θbEŵ − w −+·e(1 − θ)(1 − b)q(1 − b)q N − (1 − θ)E
(38)
(
)
1
×
N − E = 0,
1 + b − θb[]r + θ + (1 − θ)b θ + b − θbEŵ − w −+·e ≥ 0,(1 − θ)(1 − b)q(1 − b)q N − (1 − θ)E(39)

1
N − E ≥ 0,
1 + b − θb

(40)

E − (1 − b)N ≥ 0.

(41)

and

43

Proof. From (23)–(28), we can derive VDU and VU as
1+r
VDU =
(r + θ)[1 + r − (1 − θ)(1 − q)(1 − a − b)][1 + r − (1 − θ)(1 − q)(1 − b)]w
















 (1 − θ)(1 − q)a[1 + r − (1 − θ)(1 − q)(1 − b)] +ŵ
×
1 + r − (1 − θ)(1 − b)










2


(1 − θ) (1 − q)qa(1 − b) 2+




 −

w
1 + r − (1 − θ)(1 − b)

(42)

and
VU

=

(1 + r)[1 + r − (1 − θ)(1 − b)](w − e)(r + θ)[1 + r − (1 − θ)(1 − a − b)]+(1 + r)(1 − θ)a(ŵ − e).
(r + θ)[1 + r − (1 − θ)(1 − a − b)](43)

Lemma 4 implies that (22) can be rewritten as VU ≥ VDU . Thus, by substituting the values of VDU and VU into this inequality, we can obtain anotherexpression of the EPC:
[1 + r − (1 − θ)(1 − b)]w + (1 − θ)aŵ
−e
1 + r − (1 − θ)(1 − a − b)
≥ [1 + r − (1 − θ)(1 − q)(1 − a − b)]−1


[1
+
r
−
(1
−
θ)(1
−
q)(1
−
b)]w













 (1 − θ)(1 − q)a[1 + r − (1 − θ)(1 − q)(1 − b)] 



ŵ
+
.
×
1 + r − (1 − θ)(1 − b)












(1 − θ)2 (1 − q)qa(1 − b) 2+




 −

w
1 + r − (1 − θ)(1 − b)

(44)

Next, we will derive the relation between ŵ and a implied by firms’ optimal
wage setting behavior. According to Lemma 1, firms set the wages paid inand after the second period of service as
w2+ =

=

r+θ
1 + r − (1 − θ)(1 − b)(1 − q)
VU +
e
1+r
(1 − θ)(1 − b)q[1 + r − (1 − θ)(1 − b)]w + (1 − θ)aŵ
1 + r − (1 − θ)(1 − a − b)+

1 + r − (1 − θ)(1 − b)e,
(1 − θ)(1 − b)q
44

(45)

the second equality of which is obtained from (43). As is easily verified,condition (37) can be reduced toa=(θ + b − θb)E.
N − (1 − θ)E(46)

When a = 1, this condition can be further reduced to
E = [1/(1 + b − θb)]N.
In this case, the labor market is so tight that firms may not hire their desired
number of new employees without making starting wages higher than those
paid in and after the second period of service,w2+ ≤ ŵ ≤ w1 ,which, combined with (45), implies thatŵ ≥ w +1 + r + (1 − θ)be.
(1 − θ)(1 − b)q(47)

When a < 1, condition (46) implies thatE < [1/(1 + b − θb)]N.
In this case, the labor market is so loose that firms can hire their desired
number of new employees by setting starting wages at the same level as
those paid in and after the second period of service,
w1 = w2+ = ŵ,(48)

which, combined with (45), implies thatŵ = w +1 + r − (1 − θ)(1 − a − b)e.
(1 − θ)(1 − b)q

These results are summarized by[]
1 + r − (1 − θ)(1 − a − b)ŵ − w −e × (1 − a) = 0,
(1 − θ)(1 − b)qŵ − w −1 + r − (1 − θ)(1 − a − b)e ≥ 0,(1 − θ)(1 − b)q

(49)

(50)

(51)

and
1 − a ≥ 0.

45

(52)

We also need to examine whether optimal wages satisfy the EPC. Whena = 1, the EPC, that is, (44), is reduced to[1 + r − (1 − θ)(1 − b)]w + (1 − θ)ŵ−e
1 + r + (1 − θ)b≥ [1 + r + (1 − θ)(1 − q)b]−1
[1 + r − (1 − θ)(1 − q)(1 − b)]w
















(1
−
θ)(1
−
q)[1
+
r
−
(1
−
θ)(1
−
q)(1
−
b)]


+
ŵ
×
.
1 + r − (1 − θ)(1 − b)












(1 − θ)2 (1 − q)q(1 − b) 2+




 −

w
1 + r − (1 − θ)(1 − b)(53)

Using (45) to eliminate w2+ from (53), we can obtain a simplified version of
the EPC,ŵ − w ≥1 + r + (1 − θ)be.
(1 − θ)q(54)

This condition is true for any value of ŵ implied by optimal wage setting
behavior since such a value satisfies (47). When a < 1, condition (48) istrue, and thus, the EPC, that is, (44), is reduced toŵ − w ≥[1 + r − (1 − θ)(1 − a − b)][1 + r − (1 − q)(1 − θ)(1 − a − b)]e.
(1 + r)(1 − θ)aq

(55)

In addition, ŵ satisfies (49) in this case. Using (49) to eliminate ŵ − w from(55) and rearranging it, we can obtaina ≥ 1 − b,(56)which is the necessary and suﬃcient condition for a value of ŵ implied by
optimal wage setting behavior to satisfy the EPC. Finally, using (46), we
can rewrite (50)–(52) and (56) as (38)–(41).
Figure 9 depicts the AIC for n = −∞ as well as that for n = 0.16
16This figure is depicted under the condition that θ < (1 − θ)(1 − b). If this condition is
not met, the AICs for n = 0 and n = −∞ have one intersection, creating the possibility
that the aggregate employment attained under the belief that all unemployed workers are
unemployable dominates that attained under the belief that they are employable. Thispossibility need not be seriously considered because its realization requires either b or θ
to take an unrealistically high value. Because labor contracts are renewed every period in
our model, the length of one period should be considered to be one year or shorter, andthus, it is reasonable to assume that both b and θ are smaller than 0.1, which suﬃces tomake the above condition true.

46

ŵ
n̄ = 0w+

1 + r + (1 − θ)be(1 − θ)(1 − b)q

w+

1+r
e
(1 − ǰ)(1 − b)qw+n̄ = −∞w∗∗

1 + r − (1 −ǰ)(1 − b)e
(1 − ǰ)(1 − b)q
1N
1 + b − ǰbOǰNǰ+ b − ǰb(1 − b)NE

Figure 9: AICs for n = 0 and n = −∞As shown in that figure, the AIC for n = −∞ has vertical and upwardsloping parts. On the vertical part, the labor market is so tight that all
job seekers can find a job, a = 1, and thus wages satisfy w1 ≥ ŵ ≥ w2 .
On the upward-sloping part, the labor market is so loose that some job
seekers cannot find a job, a < 1, and thus, wages satisfy w1 = w2 = ŵ.
The AIC for this case has an upward-sloping part rather than a horizontal
one because it provides unemployed workers with second chances. In thepresence of second chances, the level of wages required to elicit work eﬀorts
from employed workers increases with the aggregate employment. As theaggregate employment increases, the probability that an unemployed worker
will be rehired improves, which eﬀectively eases the penalty for shirking
behavior by currently employed workers. To keep their employees fromshirking, firms need to pay higher wages, thereby rendering the loss of theircurrent job suﬃciently costly to them. Also note that the wages that are
set so as to discipline employees can also motivate unemployed workers if
and only if the aggregate employment is no smaller than (1 − b)N . In othercases, some unemployed workers stop preserving their employability, either
because their prospects of being rehired are slim or because the wages they
will receive after reemployment are too low. This is why the AIC for thiscase cannot be obtained when the aggregate employment is smaller than
(1 − b)N .

47

AIC for n ∈ (−∞, 0)
When n < 0, the jobless workers whose unemployment durations are nperiods or shorter are given “second chances,” that is, chances of being
rehired. The AIC for such a negative value of n requires not only that all of
the employed workers spend work eﬀorts, but also that the jobless workers
who are given second chances preserve their employability (whereas otherjobless ones stop preserving their employability).
Lemma 5. Suppose that there is a stationary equilibrium with n < 0, andlet a be the job-finding rate for the jobless workers who are believed by firms
to be employable. Then, in that equilibrium, (a) the following conditions
must hold:∀n ∈ [n, −1],

w−e+

1−θ
[aV (1, 1) + (1 − a)V (n − 1, 1)]
1+r

1−θ
≥w+
[a(1 − q)V (1, 0) + (1 − a + aq)V (n − 1, 0)],
1+r

(57)

and (b) the lifetime utilities of employed and jobless workers are determinedas
VO if n ≤ n − 1




Vn
if n ∈ [n, −1] and Q = 0


 DU
1

if n = 1 and Q = 0
 VDE
VDE if n ≥ 2 and Q = 0
V (n, Q) =
,

n


 VU if n ∈ [n, −1] and Q = 1V 1 if n = 1 and Q = 1 E
VE if n ≥ 2 and Q = 1
n , V1 , Vn1where the values of VO , VDU
DE , VU , VE , and VE are given by
DE

1−θ
VO ,
1+r

(58)

]
1−θ [
1
a(1 − q)VDE+ (1 − a + aq)VO ,
1+r

(59)

1
VDE
= w1 +]1−θ [−1(1 − b)(1 − q)VDE + (b + q − bq)VDU
,
1+r

(60)

VDE = w2+ +

]
1−θ [
−1
(1 − b)(1 − q)VDE + (b + q − bq)VDU
,
1+r

(61)

]
1−θ [ 1aVE + (1 − a)VO ,
1+r

(62)

VO = w +

n
VDU
=w+

VUn = w − e +48

VE1 = w1 − e +]1−θ [(1 − b)VE + bVU−1 ,
1+r

(63)

VE = w2+ − e +]1−θ [(1 − b)VE + bVU−1 ,
1+r

(64)

and if n ≤ −2, then, for n = n + 1, · · · , −1,
n
VDU
=w+

]
1−θ [
n−1
1
a(1 − q)VDE+ (1 − a + aq)VDU
,
1+r

(65)

]
1−θ [ 1aVE + (1 − a)VUn−1 .
1+r

(66)

VUn = w − e +Proof. Part (a): Note that condition (57) is the same as condition (22),except that the former must be satisfied for ∀n ∈ [n, −1], whereas the latterfor ∀n ∈ (−∞, −1]. As already explained in the proof of Lemma 4, thelarger side of (57) measures the lifetime utility of an employable worker who
is currently experiencing her −nth period of unemployment in the case that
she chooses to preserve her employability in the current period, whereas
the smaller side measures her lifetime utility in the case that she chooses
to stop preserving her employability. If condition (57) is not met for somen ∈ [n, −1], then, those who have been unemployed for −n periods findit optimal to stop preserving their employability, which contradicts firms’
belief. Therefore, condition (57) is necessary for the existence of a stationary
equilibrium with n ∈ (−∞, 0).
Part (b): When n ∈ (−∞, 0), those who have been unemployed for morethan −n periods will never be rehired, and thus, their lifetime utilities are
determined as∀n ≤ n − 1,

∀Q = 0, 1,

V (n, Q) = [(1 + r)/(r + θ)]w (≡ VO ).

(67)

The lifetime utilities of unemployable workers with n ≥ n (n ̸= 0) satisfy[]
1 − θ a(1 − q)V (1, 0)∀n ∈ [n, −1], V (n, 0) = w +, (68)
1 + r +(1 − a + aq)V (n − 1, 0)
1−θV (1, 0) = w +
1+r
1

[

(1 − b)(1 − q)V (2, 0)+(b + q − bq)V (−1, 0)

and∀n ≥ 2,V (n, 0) = w

2+

1−θ
+
1+r

[

]

(1 − b)(1 − q)V (n + 1, 0)+(b + q − bq)V (−1, 0)

Condition (70) implies that V (n, 0) takes a constant value,∀n ≥ 2,V (n, 0) = V (n + 1, 0).
49

,

(69)
]
.

(70)

Otherwise, it diverges toward either positive or negative infinity as n approaches positive infinity. These results allow us to rewrite (68)–(70) as(59)–(61) and (65). Next, consider the workers who are still employable.
When unemployed, such a worker finds it optimal to expend e units of effort to preserve her employability since (57) is true. When employed, such aworker finds it optimal to expend e units of work eﬀort since the wage profile
is designed as shown in Lemma 1. Thus, their lifetime utilities satisfy[]
1 − θ aV (1, 1)∀n ∈ [n, −1], V (n, 1) = w − e +, (71)
1 + r +(1 − a)V (n − 1, 1)
1−θ
V (1, 1) = w − e +
1+r

[

1

(1 − b)V (2, 1)+bV (−1, 1)

]
,

(72)

and
∀n ≥ 2,V (n, 1) = w

2+

1−θ
−e+
1+r

[

(1 − b)V (n + 1, 1)+bV (−1, 1)

]
.

(73)

Condition (73) implies that V (n, 1) takes a constant value,∀n ≥ 2,V (n, 1) = V (n + 1, 1).

Otherwise, it diverges toward either positive or negative infinity as n approaches positive infinity. These results allow us to rewrite (71)–(73) as(62)–(64) and (66).
Note that the EPC is modified as (57) since the equilibrium consideredhere requires that only unemployed workers with n ∈ [n, −1] choose topreserve their employability. In this equilibrium, moreover, those who havebeen unemployed for more than −n periods will be never rehired, regardlessof their employability. They have no choice but to live by consuming theirendowments in each period, which makes their lifetime utilities as low as[(1 + r)/(r + θ)]w.
We can also state the following.
Lemma 6. Suppose that there is a stationary equilibrium with n < 0, andlet E, Un and UO denote, respectively, the populations of employed workers,
that of those who have been unemployed for −n (≤ −n) periods, and thatof those who have been unemployed for more than −n periods. Further,
the job-finding rate for the jobless workers who are believed by firms to be
employable is a. Then, in this equilibrium, the values of E, Un , UO , and amust satisfyE=θa{1 − [(1 − θ)(1 − a)]−n+1 }(1 − θ)2 ab[(1 − θ)(1 − a)]−n + θ[1 − (1 − θ)(1 − a − b)]
50

N,

(74)

UO =

[(1 − a)θ + b(1 − θ)][1 − (1 − θ)(1 − a)][(1 − θ)(1 − a)]−nN,(1 − θ)2 ab[(1 − θ)(1 − a)]−n + θ[1 − (1 − θ)(1 − a − b)](75)

and for n = n, · · · , −1,Un =θ[(1 − a)θ + b(1 − θ)][1 − (1 − θ)(1 − a)][(1 − θ)(1 − a)]−n−1
N. (76)(1 − θ)2 ab[(1 − θ)(1 − a)]−n + θ[1 − (1 − θ)(1 − a − b)]

Proof. When n < 0, the population measures E, Un , and UO satisfy thefollowing conditions:(1 − a)θN + b(1 − θ)E = U−1 ,∀n = n + 1, · · · , −1,(1 − θ)(1 − a)Un = Un−1 ,(1 − θ)(1 − a)Un = θUO ,

(77)
(78)
(79)

and
N =E+

−1
∑

Un + UO .

(80)

n=n

Condition (77) is required by the fact that those who are currently experiencing their first period of unemployment were either born in or working forsome firm in the previous period. Condition (78) is required by the fact thatthose who are currently experiencing their −nth period of unemployment
were experiencing their (−n − 1)th period of unemployment in the previous
period. Condition (79) is required by the fact that the population of those
who have been unemployed for more than −n periods is constant over time.
Condition (80) is required by the fact that any worker in the labor force
is either employed or unemployed. By solving (77)–(80) with respect to E,
Un , and UO , we can obtain (74)–(76).
Conditions (74)–(76) are comparable to (37) in the case of n = 0, as
they keep the populations of employed and jobless workers constant in astationary equilibrium. Using Lemmas 5 and 6, we can show that the AICs
for n < 0 have a common vertical part.
Proposition 5. When a = 1, the AICs for n < 0 are identical and derived
asE=

1
N (≡ E ∗∗∗ ),1 + b − θbŵ ≥ w +51

1 + r + (1 − θ)be (≡ w∗∗∗ ). (81)(1 − θ)(1 − b)q

Proof. When a = 1, conditions (74)–(76) are reduced to
E = [1/(1 + b − θb)]N,(82)

U−1 = [(1 − θ)b/(1 + b − θb)]N,(83)

U−2 = · · · = Un = UO = 0.

(84)

Moreover, conditions (62)–(64) and (66) are reduced to
1−θ 1
V ,
1+r E
]
1−θ [
= w1 − e +(1 − b)VE + bVU−1 ,
1+r
]
1−θ [
2+
(1 − b)VE + bVU−1 .
= w −e+1+r

VU−1 = w − e +

(85)

VE1

(86)

VE

(87)

By solving (85)–(87) with respect to VU−1 , VE1 , and VE , we can obtain)[](
1 + r − (1 − θ)(1 − b)
1−θ
1+r
−1
w+
ŵ − e . (88)VU =r+θ1 + r + (1 − θ)b
1 + r + (1 − θ)b
Proposition 1 implies that when a = 1, the following are true:ŵ ≥ w2+r + θ −1 1 + r − (1 − θ)(1 − b)(1 − q)
=
V +
e
1+r U
(1 − θ)(1 − b)q
1 + r − (1 − θ)(1 − b)
1−θ
=
w+
ŵ
1 + r + (1 − θ)b
1 + r + (1 − θ)b
1 + r − (1 − θ)(1 − b)+e,
(1 − θ)(1 − b)q
the last equality of which is obtained from (88). By rearranging it, we canobtainŵ ≥ w +1 + r + (1 − θ)be = w∗∗∗ .
(1 − θ)(1 − b)q(89)

Clearly, (82) and (89) are our desired results.
This result is not surprising. When a = 1, the labor market is so tightthat every job seeker can find a job with one period of unemployment experience, and thus, the beliefs satisfying n < 0 have no eﬀect on labor market
performance. In other words, if the diﬀerences among such beliefs aﬀect
labor market performance, the job finding rate must satisfy a < 1.
When n ∈ (+∞, 0), however, it is diﬃcult to analytically derive the
AICs for the case of a < 1. Hence, we will derive the AICs for n ∈ (−∞, 0),
resorting to the following numerical method. First, we will increase the52value of a from 0 to 1 in steps of 0.001 and, for each value of a, computen , V1 , Vn1the values of VO , VDU
DE , VU , VE , VE , ŵ, E, Un , and UO satisfyingDE(58)–(76) and
w1 = w2+ = ŵ =r + θ −1 1 + r − (1 − θ)(1 − b)(1 − q)
V +
e.
1+r U
(1 − θ)(1 − b)q

The last condition is the firm’s optimal wage-setting policy, that is, (2).
Since a < 1, we can assume that w1 = w2+ = ŵ. In computing these values,
we take one period as one year, and we configure the values of r, α, θ, b, w,e, q, and N as in Table 1. Combined with these parameter values, a given
n , V1 , Vvalue of a uniquely determines a set of the values of VO , VDU
DE ,
DE
n
1
VU , VE , VE , ŵ, E, Un , and UO . Thus, this process yields 1,001 sets of thesevalues. For each of the obtained sets, we will examine whether it satisfies
the EPC,∀n = n, · · · , −1,nVUn ≥ VDU.

If a set satisfies the EPC, it will be kept. Otherwise, it will be abandoned.
In doing so, we eﬀectively pick up pairs of E and ŵ that can constitute an
AIC. By depicting the locus of such pairs of E and ŵ on the (E, ŵ) plane,
we obtain Figures 2 and 3.

Appendix C: Derivation of NSCE and SCEs (Notfor Publication)
As shown in Appendix B, the NSCE is always derived analytically. Moreover, when ŵ ≥ w∗∗∗ , the SCE is also derived analytically. Thus, in thesecases, all we have to do to derive the SCE and NSCE is substitute parameter values into the closed form of E, UO , a, E/L∗ , w1 , and w2+ . Whenŵ < w∗∗∗ , we fix the value of n and then compare the values of ŵ, which
are obtained in deriving the profiles of the AICs, with w∗∗ , choosing the one
nearest to w∗ . We regard the value of a that gives this nearest value of w∗
as the equilibrium value of a. Once the equilibrium value of a is obtained,
it gives the equilibrium values of E, UO , and E/L∗ , as in Appendix B.

53

Appendix D: Alternative Interpretations of Workers’ and Investors’ Surpluses (Not for Publication)
Workers’ Surplus
We can show that the WS equals the discounted sum of the lifetime utilitiesof the workers constituting the labor force in the present and future periods,
WS =E[(θ + b − θb)VE1 + (1 − θ)(1 − b)VE ] +−1∑

Un VUn + UO VO

n=n∞

θN ∑ V (0, 1)+,
1−θ
(1 + r)t

(90)

t=0

where V (0, 1) is the lifetime utility of newly born workers and VE1 , VE , VUn ,VO , E, Un , and UO are defined as in Appendix B.
Proposition 6. The RHS of (90) can be expressed as
)t
∞ (
∑
1
RHS =
[w̃E + w(N − E) − e(N − UO )] ,
1+r

(91)

w̃ ≡ w1 − (1 − θ)(1 − b)(w1 − w2+ ).

(92)

t=0

where

Proof. Since VO , VUn , VE1 , VE , E, UO and Un satisfy (58),(62)–(64) and (66),and since V (0, 1) is given by
V (0, 1) =1−θ[aVE1 + (1 − a)VU−1 ],1+r54

(90) can be rewritten as
RHS = E[(θ + b − θb)VE1 + (1 − θ)(1 − b)VE ] +−1∑

Un VUn + UO VO

n=n
∞
∑

V (0, 1),
(1 + r)t
t=0
{
}
1−θ
−1
1
= (θ + b − θb)E w − e +[(1 − b)VE + bVU ]
1+r
}
{
1−θ
−1
2+
+(1 − θ)(1 − b)E w − e +[(1 − b)VE + bVU ]
1+r
{
}
1−θ
1
+Un w − e +[aVE + (1 − a)VO ]
1+r
{
}
−1
∑
1−θ
Un w − e +[aVE1 + (1 − a)VUn−1 ]
+
1+r
n=n+1
(
)
1−θ
θN
+UO w +
VO +[aVE1 + (1 − a)VU−1 ]
1+r
1+r
∞
θN ∑ V (0, 1)+.
1−θ
(1 + r)t
+

θN
1−θ

t=1

Using w̃ to rearrange the above equation, we can obtain
RHS = w̃E + w(N − E) − e(N − UO ) + (1 + r)−1
]
 [

∑−1
1


a
(1
−
θ)
U
+
θN
V
n


n=n
E








+(1
−
θ)(1
−
b)EV
E




−1
+
[(1
−
θ)bE
+
θ(1
−
a)N
]
V
U
×
∑
n


+ −2


n=n (1 − θ)(1 − a)Un+1 VU








+
[(1
−
θ)(1
−
a)U
+
(1
−
θ)U
]
V
n
O
O


∑∞


t
+[θN/(1 − θ)] t=0 V (0, 1)/(1 + r)
= w̃E + w(N − E) − e(N − UO ) + (1 + r)−1
+ b − θb)VE1 + (1 − θ)(1 − b)VE ]  E[(θ
∑
n+U V
×
+ −1
O O
n=n Un VU ∑t +[θN/(1 − θ)] ∞t=0 V (0, 1)/(1 + r)
= w̃E + w(N − E) − e(N − UO ) + (1 + r)−1 RHS,
the second equality of which is obtained from (77)–(80). The obtained result
implies that WS must satisfy
WS =

1+r
[w̃E + w(N − E) − e(N − UO )] ,rwhich is equivalent to (91).
55

Investors’ Surplus
We can also show that the IS equals the discounted sum of investors’ expected gains from past, current, and future investments,
)t
[
]
∞ (
∑
1
bE
(L∗ )α − w∗ L∗E
IS =· ∗ · −F ++ ∗ [(L∗ )α − w̃L∗ ]
1+r
L
r+b
L
t=0
(93)
)
∞ (
∑
}
1−b t E { ∗ α12+ ∗+· ∗ (L ) − [θw + (1 − θ)w ]L ,
1+r
L
t=1

where w∗ , L∗ , and w̃ are defined by (4) and (92). The first term on the
RHS of (93) is the expected sum of net gains from establishing new firms
in and after the current period. As already seen in the proof of Lemma 2,
this term equals zero under free entry. The second and third terms are theexpected sum of the profits that currently operating firms will distribute totheir investors in future periods.
Proposition 7. The RHS of (93) can be expressed as
)t
∞ (
∑
1
E
RHS =
· ∗ · [(L∗ )α − w̃L∗ − bF ].
1+r
L

(94)

t=0

Proof. The second and third terms on the RHS of (93) can be reduced to
1+r E·· [(L∗ )α − w̃L∗ + b(w1 − w∗ )L∗ ] (≡ X).
r + b L∗
Since the first term equals zero, we can safely say that X gives the value of
the RHS. Note that][(L∗ )α − w∗ L∗
0 = b −F +r+b−r[(L∗ )α − w̃L∗ ] + b(w̃ − w∗ )L∗= (L∗ )α − w̃L∗ − bF +
r+b
∗
α
∗
r[(L ) − w̃L + b(w1 − w∗ )L∗ ]= (L∗ )α − w̃L∗ − bF −
r+b
∗
r
L
= (L∗ )α − w̃L∗ − bF −
·
X,
1+r E
the third equality of which is obtained from the fact thatw̃ − w∗ = w̃ − ŵ
= −[r/(1 + r)](1 − θ)(1 − b)(w1 − w2+ )= −r(w1 − ŵ)= −r(w1 − w∗ ),56

where ŵ is as defined in (3). The obtained result implies that the RHS mustsatisfy
RHS =

1+r E
· ∗ · [(L∗ )α − w̃L∗ − bF ],r
Lwhich is equivalent to (94).

Appendix E: Computation of Tables 13–15 (Not for
Publication)
Let w(F ) be a reduced level of w, at which the profile of the aggregate labordemand runs through the upper end of the upward-sloping parts of the AICs
for n < 0, given the value of F . When w is reduced to w(F ), the following
must be true:w∗ = w∗∗∗ = w(F ) +1 + r + (1 − θ)be,(1 − θ)(1 − b)q

which can be rearranged asw(F ) = w∗ −[=1 + r + (1 − θ)be(1 − θ)(1 − b)qαα (1 − α)1−α(r + b)1−α F 1−α

]1/α

(95)
1 + r + (1 − θ)b−e,(1 − θ)(1 − b)q

the last line of which is obtained from (4). By substituting thirteen valuesof F from 0.840 to 2 and other parameter values configured as in Table 1
into (95), we obtain the second column of Tables 13–15.
The reduction in w newly produces one NSCE and one SCE. In both
equilibrium, WS, IS, and TS can be expressed as
(
)
1+r
WS =
[w̃E + w(F )(N − E) − e(N − UO )] ,
(96)
r
)
(
E
1+r
· ∗ · [(L∗ )α − w̃L∗ − bF ],
(97)
IS =
r
L
(
)[
]
1+r
bF E
∗ α−1
TS =
E(L )
+ w(F )(N − E) − ∗ − e(N − UO ) (98),rLwhere L∗ = [(r + b)F/(1 − α)]1/α . Thus, to evaluate WS, IS, and TS in the
new NSCE or the new SCE, we need to compute the values of E, UO , andw̃ for each value of F at respective equilibria.

57

In the new NSCE, E, UO , w2+ , w1 , and w̃ are determined asθ
N,(θ + b − θb)
UO = N − E ∗∗ ,1 + r − (1 − θ)(1 − b)(1 − q)w2+ = w(F ) +e,(1 − θ)(1 − b)q(1 + r)w∗ − (1 − b)(1 − θ)w2+,
w1 =1 + r − (1 − b)(1 − θ)w̃ = w1 − (1 − θ)(1 − b)(w1 − w2+ ).
E = E ∗∗ =

(99)
(100)
(101)
(102)
(103)

By substituting the thirteen values of F and other parameter values into(99)–(103), we compute the equilibrium values of E, UO , w2+ , w1 , and w̃
for each value of F . Then, substituting the equilibrium values of E, UO ,and w̃ and other parameter values into (96)–(98), we evaluate the valuesof WS, IS, and TS in the NSCE for each value of F . Finally, taking thediﬀerences between the values of WS (resp. IS, TS) in the new NSCE andtheir counterparts in the old one, which are reported in the second columnof Table 10 (resp. Table 11, Table 12), we obtain the third column of Table
13 (resp. Table 14, Table 15).
In the new SCE, E, UO , w2+ , w1 , and w̃ are determined as
E = E ∗∗∗ =1N,(1 + b − θb)

UO = 0,
w

2+

(104)
(105)

∗

1

= w = w̃ = w .

(106)

By substituting the thirteen values of F and other parameter values into
(104)–(106), we compute the equilibrium values of E, UO , w2+ , w1 , and w̃
for each value of F . Then, substituting the equilibrium values of E, UO ,and w̃ and other parameter values into (96)–(98), we evaluate the valuesof WS, IS, and TS in the new SCE for each value of F . Finally, taking
the diﬀerences between the values of WS (resp. IS, TS) in the new SCE
and their counterparts in the current equilibrium, which are reported in thesecond to eleventh columns of Table 10 (resp. Table 11, Table 12), we obtainthe fourth to thirteenth columns of Table 13 (resp. Table 14, Table 15).

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60