Labor Immobility in Japan: Its causes andconsequences∗
Kenn Ariga and Ryosuke Okazawa
Institute of Economic Research, and Graduate School of Economics(resp.), Kyoto University, Kyoto Japan

Abstract
This paper builds and calibrate a model of competitive search that can
reproduce a set of stylized facts concerning major impacts of the decade longstagnation and subsequent changes in the labor market in Japan. We highlightthe role played by varying degrees of relation specific investments in forming
employment relations. Depending upon the degree of specificity, a job slot canbe immediately destructed, or, only the filled jobs survive, or, jobs survive as
far as workers with specific training remain in the market. We obtain a richvariety of aggregate labor market responses to a negative productivity shock.
By embedding such a system of employment in an economy plagued with limited
capital mobility, we can generate upward drift in Beveridge curve, prolonged
periods of labor adjustments, decline in the share of jobs with costly investment
in training, and strong cohort effects. We also offer some preliminary resultsfrom numerical solutions of the model.

1

Introduction

In 1991, just before the onset of the decade long stagnation of the economy, 82%of the 44 million employees in Japan had regular1 and full time jobs. The rest of
employees, either temporary or part time or both, comprised the remaining 18%, or 7
and a half millions, out of which roughly 80% of them (5.5 million) were female. As oflate 2007, around 34 million had regular and full time jobs, which now account only
for 2/3 of the total employees, and the rest comprises the remaining third, or morethan 17 millions.2 The male workers in this category also rose steadily, comprising
now roughly one third, or more than 5 millions. Another way to highlight the change
is to compare net job losses and gains on regular, full time jobs and the rest. The
size of regular and full time employees peaked in 1994 at 38 millions. Since then,we lost 3.3 million such jobs while temporary and/or part time jobs increased by 7.5
millions. In a way, the only major thing Japanese labor market accomplished in thelast 15 years or so is to slowly but steadily slash permanent jobs and replace themby (more than one to one) temporary and part time jobs. This is shown in Figure 1.
∗ Very preliminary. Do not quote any part of the paper without permissions from the authors.
Of course, comments are most welcome.
1 ’Regular’ (Joyo) workers refers to those under the employment contracts without pre-specified
length of the employment.
2 11 millions of these temporary or part time workers - more than 60% of these, 25% of all theemployees in Japan - earn 2 million yen (roughly 20,000US$) per year or less.

1

The change hit hardest younger cohorts: in 1998, among the age 15-24 cohort, 83% of them had regular full time jobs. By 2006, this share dropped to 54%. Comparedto these younger cohorts, the impact is far more modest and limited among older age
groups. In 1998, 81% of workers in age 45-54 still had regular full time jobs. The
ratio declined some since then and stood at 75% in 2006. 3
As we can see from Figures 2 and 3, the stagnation of the macro economy during
the 1990s first appeared in substantial declines in both accessions and separations.
The general decline in labor mobility marked the start of the deep recession (around1993), as had been the case in all the post war recessions in Japan. In the latter
half of 1990s, we see the beginning of the second part, rapid increase in temporary
or part time jobs. This trend accelerated as the economy emerged out of the peril ofdeflation spiral. The upward trend of temporary / part time workers still continues
as of today.
In line with the rapid increase in temporary and part time jobs, the economy
witnessed long swing and sizable outward shift of the Beveridge curve. See Figure 4.
As of 2007, the registered vacancy rate stands at .037, somewhat higher than .034,
the figure for 1990, the peak year before the long stagnation. Yet the unemployment
rate is more than doubled compared to 2.1% in 1990. The sharp increase in unemployment from the last few years of the decade until 2002 mirrors the depressedlabor market and historical low rate of accessions. Even at the peak, the separation
rate is well below the level during the bubble years. In retrospect, the sharp increase
was an outcome not only of the increase in separations, but also of the historical low
rate of accessions. The declining mobility is masked, however, to some extent by thesteady increase in the share of part time and temporary jobs wherein the average
duration of single spell of jobs is much shorter. In short, it is the apparent sluggishness and declining share of the core labor force in Japan that shapes the landscapeof the labor market in Japan. According to an estimate in Miyagawa et al. (2004),
labor immobility alone contributed to .6% decline in labor productivity during thelatter half of 1990s. It was not the labor but the capital mobility which have far
more attention as the slowdown of the TFP growth and its cause has been one of
the research focus on the lost decade. And, of course, we all know and still vividly
remember the horror story of Zombie firms in Caballero et al. (2008).
All in all, Japanese economy in the 21st century seems to be stuck in a peculiar
and precarious pseudo equilibrium characterized by low capital and labor mobility
and shrinking primary sectors offering higher pays and job security.
This paper offers an interpretation of these key facts based on a model of competitive search. Our major innovation in the model analysis is to incorporate theimpact of a permanent negative technology shock in an economy populated withworkers with different degrees of trainability. If the cost of re-training is relatively
small, worker is highly mobile. They quit the job as soon as it is hit by the negative
productivity shock to search for a new job with higher productivity. Workers withsomewhat higher cost decide to retain the current job after the shock, whereas once
they lose the job, they move out of the depressed sector and try to search a job ina different sector. A group of workers with even higher cost of re-training may well
give up mobility and cling to the job and the skill they have: they remain in the
depressed sector even after they become unemployed. We show that the group of
workers with highest training cost actually never take up jobs that require training.
They spend their entire work life moving from one simple job to the other. Thus themodel equilibrium shows the worker allocation across types of jobs according to theirtrainability and demonstrate that the degree of specificity and cost of training can
3 The impact is far more sizable on the income side.

2

explain the heterogeneity and over all compositions of worker mobility. The model
also demonstrates that a macroscopic shock which permanently changes the magnitude and frequency of negative productivity shock can reproduce qualitatively the
major stylized facts we introduced in the beginning. Both long run unemployment
and vacancy rates increase, the share of the non core employees also increases. Moreover, using a calibrated example economy, we show that the time needed to absorb
macroeconomic shocks are highly heterogenous across types of jobs and workers.
In particular, the example will demonstrate that some facets of the labor market
take extremely long time to absorb the impact of the shock because they essentiallymoves at the speed of retirement and entry to the labor market. In particular, thisis consistent with slow but steady increase in the share of temporary and part time
workers.
In order to develop a model that can account for the major changes we witnessedin the last 15 years, our first task is to review and consolidate major findings takenfrom a fairly sizable body of empirical works on the Japanese economy during the lost
decade (and its aftermath), especially those on the labor market (section 2). We will
set out a simple and highly stylized ’model’ economy of the Japanese economies ca.
1990, and ca. 2006, and suggest how these two can be connected through injectionof ’shocks’ to the system. The ’model’ economies and ’shocks’ are given concise
representations in the model developed in section 3. We will focus our presentationsof the model on its core characteristics and its implications on the three key issues.
The formal presentation and the analysis of the model are relegated to Appendix.
Section44 put the model to work to explain the impact of the great recession and its
lasting impacts. We conduct a simple calibration exercise to pin down the parametervalues of the model. Our primary purpose is not to offer full fledged structural
estimate of the model. Instead, our focus is on the mutual consistency in the behaviorof key variables. We require them to be consistent with the historical record. [Section
4 is still incomplete.] Section 5 concludes.

2

The Lost Decade and its AftermathThis section reviews what we have learned about the lost decade and its aftermath in
the labor market by drawing upon the past empirical studies on the lost decade, particularly those on the labor market issues, and, by so doing, we propose three major
findings, against which we provide our own highly stylized model in the subsequent
sections.

2.1

Two major macroeconomic shocks

2.1.1

Macro shocks

There is no doubt that severe macroeconomic shocks in the early 1990s started thelong and arduous downward adjustments in the labor market. A somewhat lesswell known point is that there indeed were at least two, not one, shocks during thedecade: the first one hit the economy in 1992-3, and the next in 1997-8. From thevintage point of 1994 or 1995, the aftermath of the first macroeconomic downturn
seemed closer to the end by late 1996, at least as viewed in fig.2 and 3 wherein thehiring rate recovered to the level closer to the separation rate. It is also clear thatthe course of the (weak) recovery from 1994 seemed rather normal and similar to
the recoveries from the previous recessions. Turnover rates declined as the economy
entered the recession, as had been the case in the past. Unemployment rate climbed
4 This section is unfinished and the results reported are tentative and used for illustrative purpose
only.

3

up and by 1995, it rose by more than 1.5%, from the peak year of 1990-1991. Still
this increase was within the comparable to those in the past recession. For example,
in the recession after the first oil shock, unemployment rate increased roughly by
1% to around 3.5%. The impact of the second shock on the labor market differedin several dimensions. Unlike the past recessions, turnover rate gradually increased.
From 1997 onward, towards as late as 2003, the gap between hiring and separation
remained at the historical high level, plummeting unemployment rate upward. The
latter half of the lost decade witnessed layoffs of the core employees at major firms,
for the first time since the beginning of the high growth era. Along with waves oflayoffs at major listed firms and bankruptcy and mergers in the banking sector, somefirms dramatically altered the pay structure from seniority based to those individual
performance base salary system. The impact of the second shock on the labor market
was far larger and longer lasting. The sharp increase in unemployment during theperiod is marked by the increase of separation and the decline in accessions. It seems
fair to say that the 1997 macro shocks was the heaviest blow5 to the labor market, atleast for two reasons. First, because the shock occurred while labor market recovery
was at best weak and certainly not complete from the first shock. Second, because
the shock entailed near panic in the financial market and precipitous decline andone time near collapse of the lending market.
2.1.2

Reallocation and productivity shocks

The lost decade was also the period of an unprecedented change in information
technology. IT revolution directly reshaped the landscape of the competitivenessin the international markets of manufacturing, telecommunications, and software
industry. It also had important impacts as an input for virtually all segments of theeconomy. Given the rapid pace of internet technology diffusion, it is undeniable that
IT revolution did have important impacts on workplace in Japan. Internet mails(to some extent) substituted for lengthy meetings in smoke filled rooms, grocery
stores quickly adopted sophisticated POS system to record daily transactions. Manycase studies do find impacts of IT technology adoption on work organization, micro
productivity, time use, and other dimensions of daily work.
1990s was then also the period of increased turbulence which might have called
for rapid shifts in resources. Rapid technology changes and emergence of newly industrialized Asia induced de-coupling of production processes and it became common
for major Japanese manufacturers to shift major production facilities over seas. By
early this century, China became the largest trade partner for Japan.
Contrary to earlier anticipation of great waves of new IT firms and growth in
hiring, recent studies by Anton Braun et al. (2006) and Miyagawa et al. (2006)
both find that the short run response to (positive) TFP shock tends to reduce labor
inputs6 . None of these studies cover the period prior to 1990s. Thus it is difficult totell if the finding is due to the specific period of low labor immobility, or something
more structural and applicable to other periods. It is at least safe to say that theoverall labor market response to the IT revolution was weak and possibly perverse.
If the waves of globalization have left any visible impact, it was primarily throughthe process of ’restructuring’. With about the delay of a decade to the Americancounterpart, Japanese media now screams that Japanese jobs are destroyed by cheap
5 In terms of fixed investment, 1992-1993 decline was much larger than 1997-1998 decline. Private
fixed investment grew at annual rates of greater than 10% during the latter half of the 1980s. In
contrast, it declined by 8.9% in 1992, by 10.3% in 1993, and by 5.2% in 1994. Compared to thesesteep declines, the decline in investment were far modest in the late 1990s, ranging % to %.
6 Miyagawa et al. (2006) argue: ” The result indicates that real inflexibility in the labor market
contributes to the negative response of the labor input to the technological shock in the short run.”4

labor in China.7

2.2

Factor specificity and immobility

Employment ’rigidity’ can be defined in more than one ways. Rigidity can be definedin terms of the speed of adjustment. Given the economic function of labor or otherfactor of mobility to shift the productive resource away from lower, towards higher,
productivity sectors, employment rigidity refers to the sluggishness of labor mobility
in response to various measures of productivity/profitability/wage differentials. In
earlier empirical studies, the focus was distinctively on macroeconomic ones employing mostly macro time series data. They found adjustment of employment or totallabor inputs towards hypothetical steady state or ’equilibrium’ level is slow, at leastin comparison with the comparable estimate for the United States.
With the increased use of micro data starting in early 1990s, several types of rather
different studies tend to suggest that the economy during the lost decade suffered fromthe sluggishness of labor and other factors of immobility. By linking job creations(destructions) by new establishments (closures) to the financial performance of thefirm data, Fukao and Kwon (2006) finds that exiting firms have on average higher
TFP than those entering. Genda et al. (2008) on the other hand, finds continued
increase in job destructions during the lost decade, matched with mirroring declines
in job creations. They argue that depressed labor mobility, especially those withprevious work experience runs counter to what we would have expected if the lost
decade was also the decade of restructuring. Genda et al. (2008) thus concludes thatcyclical macroeconomic shocks shaped the pattern of job creations and destructions.
This view is to some extent shared by a study by Fukao et al. (2008) who findsthat as late as 2006, exiting firms have higher TFP than those surviving and it isattributed to the causal relation of recovery in TFP to restructuring. Namely, modest
recovery in firm level TFP growth in the current decade is due at least partially torestructuring. Although these studies disagree as to what extent the latter half ofthe lost decade was the time of restructuring, they all suggest that the labor mobility
induced by IT revolution was weak at best, and the response of the stock variables,such as employment and unemployment are highly sluggish.
Another strand of research focused upon ’cohort effects’ on employment, earnings
and other key variables, and they8 unanimously find strong and persistent effect ofcohort effects: i.e., macroeconomic situation in the year they enter the labor market
have long lasting and quantitatively large impacts. This is at least consistent with theview that labor mobility is severely limited after their first job. The strong cohort
effect can be explained in terms of relatively heavy investment in relation specifichuman capital in the beginning of the work career. If you miss this crucial timing
either because of the failure to land on a career job after school, or because of thequits in the mid-career, this will have lasting impact on the future prospect. Thosefinishing school in recession years have smaller chance to land on tenure track jobs,
or face severe competition for promotion within a firm. The aggregate outcome isthe visible and lasting impact of the labor market fluctuations in the year they finish
school on the rest of their lives. The impact of cohort effect is found especially large
for non-college graduates. Assuming that it is more costlier to train those without,than those with college educations, this finding is consistent with the view that theimportance of early investment (or the lack thereof) in training is responsible forlimited mobility of workers in mid-career. Available evidence suggests that net gains
7 See two papers by Fukao and Kwon (2006) and Fukao et al. (2008)
8 Recent studies include Kondo (2007) and Ohtake (2005).

5

from quits in mid-career was lower, not higher than before during the lost decade9 .
Related to these studies is the rapid increase in temporary and part time workers as
we noted in Introduction. Due at least partially to the deregulation of employment
matching business for profit, and partial lifting of prohibition of dispatched workers,
temporary workers dispatched from those new firms increased rapidly. The surveyof these workers indicate that they are less educated and they choose these jobs
primarily because they could not find regular full time jobs. Studies also indicatethat the probability of landing on regular, full time jobs rapidly declines with age
and continued experience of these marginal jobs.10 .
We make two general observations from these studies. First, factor specificity
may have played important roles in sluggish response (especially downward). Second, given the limited mobility of mid career workers, we observe persistent andquantitatively important impact from the experience during the first few years after
the entry to the labor market.

2.3

Structural Breaks and Constants

As the Japanese economy finally emerged out of the decade long stagnation around
2002, the labor market was still in the midst of absorbing the shocks. It was only in2003 to 2004 that the unemployment rate started to decline, and hiring of new school
graduates picked up. Then, a simple question which is not easy to answer is: do we see
the Japanese labor market after 15 turbulent years as the one essentially unchanged
from the prototype employment system of the post war era, or has anything changed
fundamentally? Let us start with what apparently has not changed.
2.3.1

Constants

Kato (2001) finds no evidence that the ten-year job retention rates of Japanese employees fell from the period prior to the bubble burst to the post-bubble stagnation
period. The use of this alternative data source turned out to produce similar results,
i.e., no evidence for the weakening lifetime employment (Chuma (1998)). On theother hand, using the most recent micro data, Kato and Kambayashi (this volume)
indeed does find some weakening of the employment security for the core workers
in the latter part of the lost decade . Investments in firm level training seems tohave declined, but we lack any hard evidence. The system and its crucial functionof the recruitment of new school graduates have not changed. It still is by far themost important means for hiring new workers. Significant gaps remain between theregular and full time versus the rest of employees in compensation, stability, andpromotions inside the firm. There is no evidence indicating a major change in theoverall sluggishness of employment adjustment. Together with the use of temporary
workers, work hour adjustment is still predominant in adjustments in total labor
inputs.
2.3.2

Changes

So, at the end of the day (decade!), it seems fairly clear that we see more of continuations, rather than any significant breaks in structural and organizational characteristics surrounding the Japanese labor markets. Having said this, some of the changes
are still noteworthy and even important for the subsequent analysis and the model
based characterizations. First of all, even if the core of employment system remainedlargely intact, it is undeniable that the size shrank, and continue to shrink, as can
9 See for example, Part II, Chapter 2, Section 3 in :Ministry of Welfare and Labor (formerly
Ministry of Labor),2004, White Paper on Labor (Rodo Hakusyo)
10 See Genda and Kurosawa (2001)

6

be seen most easily by the steady increase in non-core employment.
The lost decade was the period also of depressed metabolism of the economy
overall, and labor market is no exception. Of special importance is dwindling rateof start-ups and hence job creations by entry of new firms11 . There has been someweak recovery in the last few years, but, even that is way too small, even after taking
account of the weak over all recovery of the economy.
If there is anything genuinely new in the labor market, they are found on bothends of the quality spectrum of the Japanese labor. At one end, we find the emergenceof enclaves of specific jobs where the external labor market is the dominant venue
through which qualified people are matched to jobs. Such markets are now found insome of highly skilled professionals: examples include IT related jobs, such as SE,
web designer, CG specialists, etc, etc. As we have noted already, we see the rapid
growth of temporary and part time workers at the other end.
For different reasons, both ends of labor are substantially more mobile acrossfirms than those in the middle. This perhaps to some extent reflects the changes inworkplace that occurred during the last 20 years or so. Some of junior level clerical
workers are completely disembodied from the main body of the firm12 : The majorityof them are now outsourced and partially substituted away by PC and the use ofother IT technology. Having observed early failures of internalizing some professional
positions13 , many firms perhaps decided to rely on either contractual outsourcing
or specialized subsidiaries to handle IT related tasks. We also find some evidence
showing some professional positions in the finance sector are filled by the use of headhunters and rely almost exclusively on mid career quits.
To sum up, we do see some important changes. As far as the factors related to
the limited labor mobility are concerned, however, no compelling evidence exists that
the Japanese economy changed.

3

The ModelThis section presents the model of competitive search. We relegate formal presentation and the analysis to the Appendix. In this section, instead, we focus on thecharacterization of the equilibrium and demonstrate that the model captures the key
observations assembled in section 2.

3.1

Focus

In Section2, We highlighted three key characteristics of the Japanese labor market.
1. Intensive investment in relation specific (human and physical) capital. This isapparently responsible for the relative immunity of the core employees.
2. Limited worker mobility. Worker mobility is largely limited to at most thefirst ten years since their entry to the labor market. The first jobs are particularly important. This is partially a consequence of fact 1 above. As a result,many empirical studies find statistically significant and sizable cohort effects
on employment probability, job security, and earnings.

11 See Genda et al. (2008) for relevant data.
12 It is symbolic of the change that the Japanese word ’office lady’ ,– often abbreviated as OL,
referring to women at office working primarily as secretaries or lower level clerks –, almost completely
disappeared from the daily conversation.
13 Recall many troubles in integrating the computer systems after mergers of major city banks.
They revealed that the ”legacy” of the system is extremely costly to undo.

7

3. Competitive fringe of the labor market. Complementary to the market forthe core employees, there exists highly flexible and competitive market for
temporary / part time workers.
Against these constants of the labor markets, the lost decade have brought about
three important changes.
1. Shrinkage of the core employment.
2. Outward drift of the Beveridge curve.
3. Stagnant job creations and productivity growth
In what follows, we offer a model that captures these key characteristics andchanges in the Japanese labor market.

3.2

A model of competitive search

Against these background and stylized features of the Japanese labor market, we
employ a variant of competitive search as our basis of the analysis14 . Here, we
describe the major features of the model without technical details.
We consider unit sized mass of population. We assume that at the rate d per unitof period, a worker retires from the labor market. The labor force is held constantby the equal amount of net inflow into the market. Each worker is endowed with
distinct trait which determines the cost of training. We denote by z, the cost of
training incurred when each worker is trained for the first time at a job that requires
training. The variable z is a random draw from the cumulative distribution, F (·). We
assume away any change in individual worker’s z until the retirement. Job searchersand vacancies are matched through search initiated by workers. Only the unemployedworkers search. Specifically, each worker contacts a vacancy at a fixed probability
per unit of period. Workers can pinpoint the search in such a way he can choose theexact vacancy he will apply to. The essence of the directed search is that each worker
has full knowledge of offers posted by vacancies. The firm must post employment
offers in order to recruit the worker. We assume that the jobs post all the payoff
relevant details of the employment contract. Moreover, they are all endowed withunlimited ability to commit to the posted contract.
Crucially, we require each job slot to choose a specific point in the continuumof ”location”. Once installed, all the jobs are totally immobile. We assume that
this continuum to expand continuously over time. The idea is that a specific job
is based upon installed technology and jobs are totally immobile. The technology
is public goods so there is no limit to the use of any specific location. Arrivals of
new technology is represented by the ever expanding continuum. Each location faces
the probability of negative productivity shock. The shock permanently lowers thejob productivity. Hence the ’state’ of the location is either ’good’, or ’ depressed’,
14 We believe, for our objective, the model of competitive search is a better choice, than a random

matching model. For one thing, in a random matching model, mismatches occur as an inevitableoutcome of matching technology. The nature of random matching is also highly problematic if it isapplied to the market for the market of school leavers in Japan. The implication, that a new college
graduate ends up in a job that he never wanted but he could not choose where to apply, seems
difficult to justify.
Also for the analytical tractability, a random matching is a poor choice in our case. Such amodel will be practically unmanageable with up to six types of employment and unemployment(state variables). Under random matching models, vacancy shadow prices will be weighted averageof values associated with respective match combinations, which render the dynamics practicallyintractable.

8depending upon if the shock already occurred. With these assumptions, we model
the shock to be permanent and common to all the job slots in the same location.
Aside from ’good’ or ’depressed’ state, all the locations are homogenous so that at
equilibrium the new start ups of job slots are even spread in ’good’ locations15 .
Each job slot can choose the location and type of job slot, and make the wage
and employment contingent on the state of location. That is, the firm can offer acontract that pays the worker different wage according to the state, or make theemployment contingent upon the state. Firms can commit the type of workers which
they would employ. By this assumption, we can divide the market into the continuumof sub-markets indexed by worker type z. In each sub-market, there are homogeneous
workers and job slots. The worker can direct their search target to the job which
gives the highest expected return. If there is more than one jobs with the same
highest return from the search, workers use mix strategy to spread their application
probabilities over these jobs. The end outcome of competition among job slots isthat only those jobs offering the market determined (highest) expected returns canattract (any number of) job searchers. In other words, at such equilibrium, individual
firms and workers treat as given the expected returns from search and each job slot
believes (correctly at equilibrium) that it can attract as many expected number ofapplicants as it wants as far as their posted contract provides each applicant withthe market rate of expected return from application.
¡ ¢Consider a typical sub-market. Denote by ψ VU the probability that a vacant
job slot receives at least one application, assuming there are V equally attractive
vacancies and there are U equally qualified job searchers. We assume this probability
is strictly increasing and concave in its argument. Then the total number of matches
per unit of time is given byµ ¶Um(V, U ) = V ψVwhich is linear homogenous in V and U. The probability for a searcher to be matched
to a vacancy is given by
µ ¶
µ ¶
U
m(V, U )
V
U
φ
≡
= ψ
V
U
U
V
3.2.1

Two types of jobs

We assume an economy wherein production technology is embodied in each job slot
as it is created. With instant and free access to any technology for a newly createdjob, zero profit condition for job slots always hold.
Denote by ρ the probability of shock per each period. If a job slot is hit by ashock, its productivity is reduced to θ(< 1) and it will remain at that level forever16 .
Henceforth we call the collection of jobs with permanently lower productivity as d sector, in contrast to g-sector where jobs are yet to experience such shocks. Namely,d -sector collects all the jobs located in the depressed locations. Given the symmetryof locations which are all held implicit in the model, each location in g-sector has
15 As you might have guessed there is no new start ups of job slots in the depressed locations.
16 The idea is that installed jobs runs the risk of being left behind as newer technology arrives tothe industry they reside. Rather than modeling explicitly the productivity growth of new technology, we model such technology shock affecting negatively the relative position of the existing jobs.
This shortcut simplifies the model greatly, but, interested readers should consult Mortensen and
Pissarides (1998) and Hornstein et al. (2007) for explicit modeling of technology advancement insearch theoretic models.

9the same size and they are indistinguishable. The same holds true for locations in
d -sector.
To highlight the key heterogeneity among the types of employment, we consider
two types of jobs. Each job slot can be occupied by a single worker. We represent thecompetitive fringe of the labor market by type s jobs which requires no training andany worker can be employed to produce the same q s units of (homogenous) output
per period of time. Each type s job slot can be instantaneously and costlessly created
but it costs ps per period to maintain with or without a worker in place. Destructions
are automatic and instantaneous if they stop paying the cost. On the other hand,type c jobs require investment in training. For simplicity, we assume that it takes
once and for all investment by the amount z to train a worker who has never received
any kind of training, or, ²z if he has been trained at a different firm17 in the same
location, and (² + m)z if he received training in the past at a firm belonging to adifferent location18 . Naturally, we assume0≤²≤²+m≤1
We also denote by pc (> ps ) per period rental cost of type c job slot.
Finally, we assume that exogenous separations occur in either type of jobs anddenote by δi , the probability per period that such separations occur within eachperiod.
Notice that the specification above incorporate varying degree of transferabilityof skills across jobs and industries. Since workers are always free to apply for any
vacancy (provided that they meet the specified qualifications as detailed below),
the mobility of workers are made endogenous. In particular, workers without any
previous training is perfectly mobile because he has not sunk any investment in
training. Note that this mobility reflects not only the nature of training costs as
specified above, but it also reflects the nature of competitive search: the mobility isensured by the assumption that a worker can pinpoint the job offer he applies19 .
3.2.2

Constrained social efficiency of competitive search equilibrium

The market equilibrium mediated through competitive search is known to be constrained socially efficient: the market allocation maximize the net social surplus,
given the matching and production technology. Intuitively, the efficiency of the equilibrium can be understood by a simple analogy of the competitive search to a perfectcompetition. In the latter, all the participants take as given the equilibrium price andmaximizes her expected utility. The marginal condition ensures the optimality as theequilibrium price equates (locally non-increasing) marginal benefit for the buyer tothe (locally non-decreasing) marginal cost of output. In the competitive search equilibrium, job searchers and vacant job slots take the expected returns from job search
17 We ignore the measure zero probability of being matched to the same job slot where he received
the previous training.
18 Notice that the formulation accommodates various special cases. For example, if m = 0, thetraining contain some firm specific element but it is general across industries. ² = 0 on the other
hand corresponds to a case wherein training produces human capital shared by all the firms in anindustry. So a completely firm specific human capital implies² = 1 so that none of the previous
trainings are helpful outside the firm they are trained.
19 This is an extreme but useful property of the competitive search. Given the time and money
school leavers and the firms invest in the market for new school graduates, conventional model ofrandom matching seems to ill fit the nature of the friction in the market. After all, random matchingassume that workers cannot choose the type of jobs they apply. In this sense, immobility is due tomatching technology. Needless to say, ’churning’ do exists even in the markets for the new school
graduates, but they are not central to the issue of labor immobility.

10

as determined in the market and maximizes respective surplus. The constrained efficiency is ensured by the equality between marginal (and non-increasing) gains in
the probability of match (and production) to the opportunity cost (i.e., applying tothe other jobs). The joint outcome of these decisions generate allocation identical tothat of social planner because expected returns from search acts as the price at which
each job slot can attract as many expected number of applicants as possible. Search
friction transforms competitively priced input (the expected number of applicants)
into the probability of match and output (and hence profit). This transformation
is determined completely by matching technology and it is strictly increasing andconcave. The end outcome of this maximizations on the both sides of the market isthe first order optimality condition which ensures the efficiency.
In our current model, we added inter-temporal element with irreversible investment. This does not alter the fundamental property of the efficiency of the equilibrium in that we allow the job slot to commit to posted offer which can specify in full
all the contingent elements of pay and employment. 20
3.2.3

Market equilibrium

At the market equilibrium, each worker (and the corresponding search market) is
indexed by the cost of training and work experience. Consider a typical group of
workers whose training cost level is z. If a worker has never been trained before, his
choice is to apply to a type s job or a type c job without requirement for any priortraining. The market equilibrium value of such offers can be denoted as Ug (n, z)
wherein n as the first argument in Ug signifies he has never been trained at any firm
before. Since the type s job does not require any training, its equilibrium offer is
independent of z, so we may denote this by Us . Hence his best choice is simply
U (n, z) = max [Ug (n, z), Us ]As higher training cost reduces the joint surplus at type c jobs, Ug (n, z) is decreasing
in z. Thus there exists a threshold value of z such that untrained workers choose typec job if and only if her training cost is equal to or lower than z s . Given the stationarityof the economy as well as individual worker’s life, a worker’s optimal choice of job atequilibrium is also stationary and it depends only upon her training cost and work
experience. Specifically, it is evidently optimal, for a worker exogenously separated
from type s job, to resume searching for a type s job again because type s job does
not offer training opportunity nor is it required. Since a trained worker costs strictly
less to retrain, a trained worker never apply to a type s job. Hence the decision to
search for a type s (c) job is final and they will continue to do so after exogenous
separations. To be blunt, type s is a dead end job.
For a worker separated from type c job, his job search will be different from those
without prior training because his training cost at a new type c job is either ²z or(² + m)z, depending upon whether he remains in the location where he received thetraining in the past. Again, their choice is completely characterized by respective
value functions, Ui (g, z), Ui (d, z) (i = g, d) wherein subscript i signifies the sector in
20 Note that this is certainly not meant to deny the possibility that some types of competitive
search with training can generate inefficient allocation. For example, if a job slot cannot commit
to a contingent contract and chooses the action which is ex post optimal after the shock, such anequilibrium may entail inefficiency. See subsection 4 in Appendix. In short, the market equilibriumwith competitive equilibrium in our paper is an exact analogue to Arrow Debreu model of generalcompetitive equilibrium.
Another note on the wage schedule. Since employment continues more than one periods, competitive search can be supported by more than one types of wage offers provided that the expected
present value of the offer is the same. See appendix for the details.

11which a worker searches for a job. If the location he is trained in the past belongs to
g-sector, there is no need to move to a different location where a larger re-trainingis needed. On the other hand, if the location he received training is now in d-sector
and productivity of jobs are lower, it might make sense to move out of the locationand search for a job in g-sector. In Appendix, we prove the following∂Ug (d, z) ∂Ud (d, z)
<
<0
∂z
∂z
∃z u such thatUg (d, z) ≷ Ud (d, z)
asz ≶ zuNamely, the impact of training index z is smaller if she stays in the same sector thanif he moves out to a g-sector, i.e., Ud (d, z) is flatter than Ug (d, z). This makes sense
because if he moves out, re-training cost is (² + m)z, whereas if he stays, it costs only
²z. Thus the dependence of his value function on training cost is proportionately
smaller if he stays. If the training cost is sufficiently small, it is evidently optimal
to incur re-training cost and move to a location in g-sector where jobs are moreproductive. Thus there exists threshold z u such that those with training cost lower
than the threshold will move out of d-sector. The remaining question is what if theworker is employed when the job is hit by the productivity shock. The optimal choice
is again characterized by yet another threshold, z e . If the training cost is smaller than
the threshold, it is optimal to terminate the employment and move to g-sector andsearch for a job. Since his value of being employed is higher than being unemployed
and searching for a job in the d -sector, we haveze < zu
Consequently, three thresholds completely characterize the competitive equilibrium in each market indexed by z.
Figure 5 summarizes the discussion above, with slightly different formulation
which is used in the formal presentation of the equilibrium in Appendix. The figuretakes the cost of training z on the horizontal axis, and on the vertical axis we takethe value of the contracts at the equilibrium. Given the future probability of negative
productivity shock, the contract offer contains contingency in the event of the shock.
First, notice that the value of type s contract is a horizontal line since the job does
not require any training and the output is the same irrespective of z or previous
training. The first threshold, z s is the point where the value of type c job offer tountrained worker cuts the horizontal line. Thus workers with training cost smaller
than threshold search for a type c job. The decision on mobility after the shock isan outcome of competition among offers with different contingency. Note that thisdecision concerns only those with previous training. Untrained workers can movecostlessly across different sub-markets so they never stay in locations hit by thenegative productivity shock.
Then, suppose a worker has previous training and have been searching for a jobin the location when the industry is hit by the negative productivity shock. Whether
or not he should search in the same location or move out depends upon the twovalue functions, each representing the stay or move. At market equilibrium which issocially optimal, this decision is equivalent to whether or not an offer with conditional
employment contract that stipulates separation after the shock has higher value than
12the offer available for the trained worker in d-sector. Thus the second threshold, z u ,is given by the intersection of the two value functions. Finally, consider if a worker
employed should retain the job when the productivity shock occurs. In terms ofcompetition of offers, this is tantamount to the comparison between the offer withunconditional employment and the one that stipulates separation at the moment of
the shock. The intersection of these two schedule determines the last threshold, z e .
3.2.4

Worker mobility under three types of equilibrium

For workers with training cost above z s , they only apply for type s jobs. While
employed at a type s job, he may be separated for exogenous reasons (that occur
with probability δs ), retires (with probability d) or the job is hit by the negative
productivity shock. Depending upon the configuration of parameters, either the job
is destructed and employment is terminated, or, the employment continues until
separations occur for exogenous reasons21 .
Workers with training cost lower than z s search for a type c job. Their employments are terminated either by exogenous separations or by retirement of workers.
Upon the productivity shocks, we obtain three types of configurations. Those workers with training cost between z u and z s will remain in the same industry until theirretirement, even when the industry is hit by the negative productivity shock. Thus
workers are necessarily less mobile than type s workers. Among workers in the nexttier, i.e., those with training cost between z u and z e , their mobility depend upon thetiming of the productivity shock. If he is employed when the industry is hit by the
productivity shock, he retains the job until he is separated by exogenous reasons.
Once unemployed, however, he moves out of the location now in d -sector and seeksa job in g-sector. Therefore this tier of workers are more mobile than the first tier.
Finally, workers with the training cost lower than z e is the most mobile among workers who choose type c jobs. He moves out of the industry whenever it is hit by the
productivity shock, even if he is employed at the moment of shock.
In sum, the model predicts the worker mobility is U-shaped across index of training cost: those with highest training cost actually never receive training and mobile,
while workers with training cost below z e will always move after the shock.
3.2.5

Zero profit condition and optimal queue length

In appendix, we show that market equilibrium is characterized by the first ordercondition for the optimal queue length, i.e., the average number of applicants to ajob. For a population of workers with given training cost z, there exists potentiallythree types of offers for type c job. One type of offers is for the untrained workers.
The second is for those trained in a different job in the same location. Finally, an offer
to a worker who had been trained in a different location. Note that the last groupof workers must have been in a location hit by the productivity shock. Otherwise,
given the symmetry of the location continuum, there is no reason to move.
In sum, there are potentially four distinct sub-markets each with distinct offers,
depending upon whether or not and at where a worker is trained, as shown in Table
1. Let us start with job slots in d sector. As we explained above, no worker moves
to a job in d sector. If he is trained, there is no gain from moving into a location
with lower productivity. By the same reason, untrained worker has no reason to
search for a job in d-sector. Consequently, the market is potentially active only for
those who have been trained in the same location already. As we have shown above,
21 Because this type of jobs do not require training, the optimal choice is the same for all the types jobs. Hence only one of the two possible cases is observed at equilibrium. It should be noted alsothat the surplus from employment is due entirely to the search friction and we should expect thatjob destructions and separations are more likely outcome in comparison with type c jobs.

13Worker typenot trained
trained in the same location
trained elsewhere

Location typeg-sector
(n,g)
(g,g)
(d,g) if z ≤ z u

d -sector
not active(d,d ) if z ≥ z u
not active

Table 1: Table of sub-markets

this segment is inactive if z ≤ z u . For a job in g-sector, two types of sub-markets,(n,g), and (g,g) are always active, whereas (d,g) is inactive if z ≥ z u . The zero profit
condition for each active sub-market uniquely determines the optimal queue length.
By complementary slackness condition, there is no vacancy and the market is inactive
if the zero profit condition is not binding.

3.3

The Full Market Equilibrium and Impacts of Macroscopic
Technology Shocks

3.3.1

The Full Market Equilibrium

The full market equilibrium is determined by two sets of conditions. First set contains
equations which jointly determine optimal queue lengths and corresponding values of
employed and unemployed workers. The model equilibrium is such that these sets ofconditions do not contain state variables, i.e., employment and unemployment withor without training. The equilibrium values of queue lengths for respective offers areused to obtain the values of offers, or, equivalently,the shadow prices for unemployedworkers for given training cost and the location of the last training. They together
thus determine ”prices” that jointly support the efficient allocation of workers toquene and hence to jobs.
The ’price’ block of the model is determined completely within themselves, independently from the state variable dynamics. Since the dynamics of ’price’ block is
totally unstable, the solution, or the market equilibrium is such that the equilibrium
prices are always at respective steady state values. Consequently they jump immediately to the new steady state values whenever the model is disturbed by a change ina parameter. Employment and unemployment evolves over time towards the steadystate with the speed which partially depends upon ’prices’, the details of which are
shown in Appendix.
Worker mobility at the market equilibrium is depicted in the three diagrams
shown in Figure 6 through Figure 8. Figure 6 shows the possible moves for the least
mobile group, i.e., those with training cost between z u and z s . The arrows directingfrom g sector to d sector is caused by the shock and they never endogenously leave
the job they are matched for the first time. Even if they are separated from the firstjob, they remain in the same location. Figure 7 shows the moves for the middle
tier of the workers in type c jobs. Note the arrow from ud to eg indicating the movefrom the depressed location to search for a job in healthy location. Finally, the mostmobile group is shown in Figure 8. They never stay in d sector.
We now employ the model equilibrium to trace out the development of the labormarket in Japan from the onset of the lost decade until early years of the new century.

14

3.3.2

Two types of shocks: increased turbulence and deeper impact

Needless to say, conventional types of macro shocks must have hit the economy during
the lost decade. After all, no one doubts that the economy experienced severe decline
in the aggregate demand. To the extent that those shocks were cyclical, however,tracing out the impact of such shocks is not our main focus, thus we ignore them.
It seems fair to say that at least two type of technological shocks also occurred
during the period and both of them are likely to be permanent. The advent of
information technology (IT) most likely shifted comparative advantages of industries
and organization forms. Another important factor is rapid shifts in comparativeadvantages in the international markets. Lower trade barriers, emergence of East
Asian countries, etc all made the market more volatile. We model this increased
turbulence as an upward shift in ρ. At the same time, rapid shifts in technology
frontier made it more costly to retain and continue to use the current technology.
We model this as a deeper impact of negative productivity shock to the existing jobs,
i.e., lower θ.
Although both types of changes reduce the overall efficiency of type c jobs, theydiffer in their impacts on the labor market dynamics. To begin with, note that theendogenous labor flows in our model is driven by the mobility from d -sector to gsector. An increase in ρ reduces the net productivity gains from this mobility because
if the probability of permanent productivity shock increases, the expected duration ofa job in g-sector is reduced. Thus moving from d sector to g sector generates smaller
gains as the new location is more likely then before the macro shock to become d sector. Thus such a shift lowers the mobility after the productivity shock. Overall
impact of the shock on labor mobility is therefore negative.
On the other hand, an decrease in θ does the opposite: the impact of productivity
shock is larger so that the gains from the mobility is also larger. As a consequence,
workers are more likely to move out of d -sector to search for a job in g-sector. Thus
the labor mobility should increase as a result of the shock. Therefore, a combinationof changes in these two parameters tend to reinforce each other in terms of lowering
overall productivity of type c jobs (vis a vis type s sector), whereas the impacts on
short run labour mobility tend to canceled out each other. As a result, both types ofshocks shifts the overall labor allocation away from type c jobs and type s positions
will expand.
3.3.3

Short run and long run impacts

Suppose these types of shocks are totally unexpected and changes are also permanent.
The immediate impacts of these types of shocks differ depending upon whether or
not the impact is favorable to the d -sector employment or not.
An increase in ρ lowers the difference in values of g-sector and d -sector jobs as
g-sector jobs are more likely to move to d -sector. In this case, however, it still
remains that an increase in employment at d -sector can occur only through theincrease inflows out of gsector to d, and, also through increased matching at d sector unemployment pool. In short, employment level at d-sector cannot jump.
Consequently, shocks which favors the declining sector will move the economy only
gradually over time.
On the other hand, shocks which favor the g-sector jobs can bring about immediate job destructions at d -sector. Thus the change will produce immediate job
destructions and resulting move to unemployment pool for the trained (at d sector)
workers to g sector. We would expect to have a sudden increase in separation andaccession rate. In other words, the model predicts that the decline in turnover rates
should be gradual and long lived, whereas an increase tend to be more sporadic and15

rapid.
In the long run, both of these shocks reduce the overall surplus of type c employment, reduce the equilibrium quene length, and increase the steady state unemployment. The end result is outward shift of Beveridge curve and lower vacancy/unemployment rate.
3.3.4

Long and persistent impacts

Apparent sluggishness of the Japanese labor market in absorbing macro economic
shocks has been the research agenda for a long time. The development in the lost
decade and its aftermath only made this characteristics more puzzling, given theseverity and the length of the stagnation. The model developed in this paper represents the overall mobility as the aggregated outcome of individual worker mobilities
who differ in their trainability and their experience. We have shown above that agroup of workers in type c jobs with relatively high training costs are highly immobile.
They are totally immobile across ”locations”. They cling to the skills they acquiredin the previous job even after the location lost its productivity edge permanently.
For those jobs to disappear, the workers occupying these jobs need to retire. This
makes the process of adjustment extremely slow22 . Unlocking employment relations
with relation specific investment is costly for themselves as well as to the economy
as a whole. On the other hand, it is misleading to project the aggregate outcome
based solely on this group. By definition, this group is under represented in thelabor mobility data (because they do not move, at least not endogenously). The
over all labor mobility, hence the time needed to absorb macroeconomic shocks, arethe aggregated outcomes of heterogenous workers’ individual mobility decision, thus
it is sensitive to the share of workers in terms of trainability and work experience.
Needless to say, the impacts on individual workers mobility and incomes differ.
The other side of relative immobility of trained workers is that the joint surpluses
protect their jobs, while those jobless, especially those without training, take theblunt of the shock.
The available evidence suggests the lost decade was also the time when the aggregate mobility diminished. Population composition was changing rapidly and newentry to the labor market declined sharply. The distortions in the capital market
which manifested so vividly during the last years of the century only contributed todecelerate the labor mobility. This impact was felt on both creations and destructions of jobs. Decline in new start ups must have reduced the available new openings,
whereas the survival of ”zombie” firms contributed to the longevity of employment
in depressed sectors above and beyond warranted under unfettered competition.
3.3.5

Cohort Effects

Two macro technology shocks, represented by changes in θ or ρ, shift the relative
positions of value functions and hence positions of three thresholds. For example,
suppose that z s threshold moves left after the shock, say from z0s , to z1s (< z0s ) .
Workers with training cost between z0s and z1s will have dramatically different mobility
and income, depending upon the timing of the first employment. If a worker is lucky
to get a job before the shock, he must have been employed at type c job and he musthave received training, thus making him largely immune from the shock because hisvalue function reflects the investment which is already sunk. On the other hand,those who fail to be employed (strictly speaking experience employment) at type c
22 According to Census of Agriculture, 66% of farmers (those considered primary in the sense
farming is their main economic activity) are aged 60 or older. The share was 36% in 1985. Aged asthey are, there are still more than 6 million who at least partially work as farmers, and more than
3 million whose main economic activity is farming.

16

job before the shock now find themselves no longer needed in type c jobs. Only jobs
available now for them is type s which offer no training. This impact is permanentand cannot be undone, thus leaving the visible scar to the work history of thosecohorts.
Intuitions also suggest that this change occurs over an extended period of time.
A worker who was lucky enough to jump into the boat of secure, life time job in typec will live a life not very different from the one before the shock, and this situation
lasts as long as he remains active. The economy soaks in full impact only when thoselast lucky ones eventually retires from the labor market23 .

3.4

Summary and Limitations

This section presented without technical details the essence of the model of competitive search. Our model captures some of the key findings on developments of theJapanese labor market since 1990s. Sluggishness of the overall response to the shockis modeled as the aggregate mobility of heterogenous workers with different training
costs and work experiences. We highlighted the fact that for some of workers it isnever optimal to move to a different job even if the current job is hit by permanent
productivity loss. Our model equilibrium is consistent with the presence of strong
cohort effects. We also have demonstrated that the combinations of shocks may cancel out each other on their impacts on short run mobility, even though they bothincrease the long run unemployment rate in the long run.
Before closing this section, we should note the limitations of our model as employed for the narrative of the Japanese labor market in the last 15 years. To begin
with, our model cannot capture some of the key macroeconomic issues, which musthave occupied the minds of policy makers during the lost decade, such as the fearof deflation spiral, or for that matter, any macroeconomic dynamics associated withprice rigidity, market failures, etc. Needless to say, these concern shaped many of thepolicies which we analyze below. Admittedly, it is unfair to consider only the typeof efficiency issues in the framework of equilibrium analysis because by construction
any intervention that alters the equilibrium allocation reduces efficiency.
Far more important and serious limitation of the model is somewhat more subtle.
In spite the complexity, the model equilibrium is fairly straight forward in implications and its complexity of the model analysis remain within manageable range.
This relative clarity and sharpness are bought by the competitive search and zeroprofit condition on vacancies. If we acknowledge the time delay and irreversibility ofcapital investment, this condition is highly restrictive and evidently not applicable24 .
Historical records of job creations or vacancies show clearly the underlying sluggishness of the start ups in response to new opportunities. Banks forbearance, a varietyof government policies to subsidize interest rate payments for depressed industries
only aggravated the problem. For these reasons, the model analysis necessarily failto keep track of cyclical fluctuations in accessions and separations as we observed
during the lost decade. In the next section, we offer some calibrations results with
due precautions as required by these limitations.

23 The crucial point is that the training investment is sunk cost. Therefore, even if a second
technology shock completely washes out the change in either θ or ρ at the first shock, those training
investment still have permanent impacts on the economy as long as they remain in the labor market.
24 This trade off between the model transparency and poor prediction power ( at least in terms ofcyclical functions) is not uncommon among calibrated models of the labor market based upon costly
search. See, for example, Fujita and Ramey (2007) for their simulations results and the better fit
obtained by introducing sunk cost for job creations.

17

parameter
r
d
µz
σz
²
m
ρ
θ
δc
δs
qc
qs
pc
ps
A
η

value
0.005
0.0063
20.0
14.5
0.15
0.33
0.013
0.88
0.006
0.0188
2.0
1.0
0.575
0.165
0.46
0.3

description
discount rate
retirement probability
mean of training cost
standard deviation of training cost
generality of experiencemobility cost
the frequency of productivity shock
the scale of negative shock
separation rate of type j job
separation rate of type s job
productivity of complicated task
productivity of simple task
rental cost of type c capitalrental cost of type s capitalefficiency of matching function
parameter of matching function

Table 2: Benchmark Values

4

Model Calibration (incomplete, extremely tentative)

4.1

Steady State Equilibrium with benchmark specifications.

We use the model outline in section 3 to trace out the development of the labormarket for the period 1991-2006. The model has 16 free parameters as shown in
Table 2. Our strategy is simple. We pin down these parameter values by employing
various sources of the past empirical studies We show our tentative benchmark valuesof 16 parameters as shown in Table 2 . Below we have a brief description on how we
set the value for each parameter.
• The one period of the model corresponds to a quarter of a year.
• The annual discount rate is set at 2%, roughly the long run growth rate of theJapanese economy. Thus we set r = 0.005• Assuming the average work career to last 40 years, we set the quarterly retirement probability at d = 0.0063• q s , per period output for a type s job in the state g, is used as the benchmark
so we set q s =1
• q c , the output for a type c in the state g ise set at 2.
• Matching function. We assume that it is a simple Cobb Doughlas type.
M = AV η U 1−η
with the coefficient for vacancy, η = 0.3. This specification approximate
the coefficient estimates in Kano and Ohta (2005) and Kambayashi and Ueno
(2006). We also set A = 0.45 so that at the bench mark equilibrium, theaverage monthly job finding rate is around 15%• According to Fukao and Kwon (2006), the 7 year (between 1994-2001) survival
rate of firms in the upper 50% bracket is estimated as 68%. Thus we set
ρ = 0.013 so that exp [−28ρ] + .68.
18

• We use the results from Ito and Lechevalier (2008) for the across firm distribution of labor productivity of full time jobs. Specifically, we set θ = 0.88 so thatthe log productivity differential between upper 25% and lower 25% corresponds
to their results.
• On exogenous separation rate for type c jobs. We use the average (1991-2006)
separation rate for regular full time jobs, 11.3%. We subtract 51.7% of themas those separations with tenure less than 2 years. This adjustment is madeon the ground that our model assume away separations due to unknown match
specific productivity, which are likely to be concentrated among employees withshallow tenure. In net, we set the exogenous separation rate in such a way thatthe total separation rate ( including endogenous ones due to negative shocks)
be equal to 5.9%=.517×11.3%. The best choice turns out to be δc = 0.006.
• The average (1991-2006) ratio of separation rate for temporary/part time job tothat of the regular, full time job is 2.6. Thus we set δs such that the equilibrium
separation rate (including endogenous separations) is equal to 15.2% per year,
i.e.,2.6× 5.9%. δs = 0.025 satisfies this condition.
• We have almost no information directly relevant to this parameter for Japan,
but Koike (2005) asserts the firm specific components of human capital is at
most 10-20%. Thus we set ² = 0.15. We know even less on magnitude of m.
We chose m = .33 as the value (jointly with other parameter specifications)gives us .52, the target rate of job changes with industry changes to the total
accessions (see the explanation for this target ratio below).
• Distribution of the (first job) training cost. We assume F (·) to be a normal
distribution with mean µz and standard deviation, σz , truncated at x = 0 andre-normalized density such that the probability sums up to unity. The value ofσz is set at 14.5 so that at the mean, µz (= 20), the re-normalized density also
has the peak.
• µz = 20. This corresponds to 2.5 years’ worth of output for a type c job in thestate g. Ariga et al. (2006) reports average On the Job Training is 82.5 hours
per year among the surveyed sample firms in Manufacturing, which corresponds
to 4-5% of annual work hours, which in turn corresponds to 2 years of 40 year
work life. Ariga et al. (2006) also have been conducting by annual surveys at
two auto assembly plants and they also find the average monthly training hours
around 8-10 hours per month among assembly line employees. This suggests
annual training hours around 100. This translates into 5% of total work hours.
• We set pc =0.575, and ps =0.165, so that the equilibrium aggregate unemploymentvacancy rate is unity.
• We set the target share of type s employment at the bench mark equilibrium
to be at .085, the figure for male workers at 1991.
• We set the target ratio of job accessions involving location changes to the total
accessions at .52, which is the estimated ratio of job changes involving changes
in industry affiliation of jobs in Japan.

19

Unemployment rate
Vacancy/Unemployment
Thresholdsz̄z̃ẑJob Finding Rate
part-time job
full-time job
Annual worker turnover rate (full-time %)Annual worker turnover rate (part-time %)The share of accessions with location change
Employment share (%)
at part-time job
at full-time job in d
at full-time job in g
Gross output

benchmark
0.036
1.00

θ shock
0.042
1.05ρ shock
0.036
1.03

two shocks
0.044
1.10

40.3
29.6
9.9
0.45
0.56
0.43
11.8
30.5
0.52

37.0
16.6
0.46
0.56
0.43
14.0
30.5
0.64

39.5
24.7
7.1
0.46
0.56
0.43
11.6
34.5
0.51

35.4
32.9
12.8
0.46
0.56
0.43
13.9
34.5
0.68

8.6
41.3
50.1
1.75

12.9
27.0
60.1
1.71

9.4
52.0
38.7
1.72

15.3
36.9
47.8
1.65

Table 3: Steady States
4.1.1

Steady States of the model under base parameterization and macrotechnology shocks

We configured parameters in order for the steady state equilibrium of the model
to mimic closely the normalized actual data on unemployment rate, accessions andseparations rates for the regular-full time and part time/temporary workers. The
results are shown in the first column of Table 3. In the second column, we show
the steady state values when we lower the value of θ from .88 to .84. Similarly, thethird column is for the case wherein ρ is decreased from .018 to .013. Finally, thelast column show the steady state when two changes are combined.
The steady state values of the variables listed in Table meets the targets weoutlined above: the share of type s employment is 8.6%, corresponding to the target
figure, 8.5%, shown above; the mobility across locations is 52% of the total separation;
and, the vacancy size matches total unemployment rate. Job finding rate is also close
to the target. The unemployment rate at the bench mark is 3.6%.
About 50% of employment is at type c jobs which are yet to be hit by the technology shocks, whereas 40% have type c jobs which are already in depressed state.
Although we set exogenous separation rate of type s at 7.5% per year, 5% higher
than the corresponding rate for type c jobs, the annual turnover rate for type s is30.5 (thus separation rate is 15.2) compared to 11.8 (5.9%). The result indicates that
more than a half of the net difference in separation rate, 10.7%, is accounted for by
the difference in endogenous separation. This of course reflects the equilibrium effectof training investment. The share of aggregate training cost is 8.6% of gross output,
which is probably over estimate and this may account for some of the extremely
sluggish response in some of the state variables, as we will show below.
The long run effects by two types of macro technology shocks replicate qualitative characteristics we verbally explained in section 3.3. In terms of quantitative
effects, we lack any empircal counterparts to evaluate. We only warn you an obvious
but important feature of the numerical values of equilibria. Namely, the impact oncompositions of employment are highly sensitive to the specification of the distri-

20

Unemployment rate
Employment share (%)
at part-time job
at full-time job in g

Unemployment rate
Employment share (%)
at part-time job
at full-time job in g

θ shock
50% benchmark
1.5

75% benchmark
1.5

26
0.5
ρ shock
50% benchmark
5.5

52
1
75% benchmark
24

33
6.5

63
14

Table 4: Transition paths
bution function for training cost. With this due warning, we offer some tentative
assessments.
Compared to the modest impact on gross output by either type of shocks (2-3%),changes in employment shares are larger: roughly speaking, 1% GDP change due
to a decline in θ corresponds to : 3% increase in type s job, 4.3% increase in theshare of type c jobs in g sector, and 6% decrease in the share of type c jobs in the
depressed sector In the case of an increase in ρ, 1% decline in GDP is matched by.5 % increase in the share of type s job, 6.7% decrease in the share of type c in g,and 6.3% share increase of type c jobs ind. If we consider these ratios to be toolarge to be credible, an economy with lower training cost will show even more larger
response ratios of these shares relative to changes in GDP. The implication is thatthe alarming increase in type s jobs [or in terms of data, steady increase in temporary
or part time jobs] may not be so serious in terms of the accompanying loss in output.
The impact on the steady state unemployment is modest compared to thesechanges: .6% increase in θ shock, almost no change by ρ, and about .8% when two
shocks are combined. GDP declines by 5.8% with the combined shock so that theratio of GDP change to the change in unemployment rate is around 10. Setting aside
the exact number, the low sensitivity of unemployment to GDP change in Japan isa well known feature.

4.2

Dynamics

We traced the impact of two types of shock over time. We show two results. Figure9- 12 show the dynamic paths of major variables after shocks. In Table 4 we tabulate
the time needed for each of major variables to absorb the long run impacts. This ismeasured in term of time needed to move half way (50%) of the difference between
the two steady states, and the time needed for 75% benchmark. We notice almost
immediately that the dynamics differ sharply between the two shocks. The transition
to the new steady state is relatively fast in the case of θ shock, which is exactly what
we anticipated in section 3.3. In the case of the employment composition between g
sector and d sector, the transition from the benchmark steady state to the one withthe shock is 75% complete within 1 year. The only exception is the slow adjustmentin the share of type s relative to the combined share of type c. It takes 25 years toreach 50% benchmark, and 52 years to reach 75%25 . This is also exactly what we
25 You may know already why, but we note that the large numbers shown for the transition time
for the share of type s employment is due to the simplifying asumption: i.e., workers are assumed

21

benchmark
Unemployment rate
Vacancy/Unemployment
ThresholdszszuzeJob Finding Rate
Annual worker turnover rate (full-time %)The share of accessions with location change
Employment share (%)
at part-time job
at full-time job in d
at full-time job in g
Gross output

firing cost

training

0.044
1.10

capitalsubsidy-0.001-0.01-0.001±0-0.001
-0.04

35.4
32.9
12.8
0.46
13.9
0.68

+0.6
-1.0
-0.6
±0
-0.3
-0.1

±0
±0
-0.8
±0
-0.3
±0

+1.3
±0
±0
±0
-0.1
-0.1

15.3
36.9
47.8
1.65

-0.9
+1.7
-0.8
+0.01

±0
+1.1
-1.1
±0

-2.1
+1.5
+0.5
+0.02

Table 5: Policy effects
would have expected: the share moves roughly in line with the speed of retirement
and entry.
The shock in ρ generates far long adjustment process. Unemployment rate reaches
50% benchmark only after 5.5 years, and more than two decades to reach 75% level.
As expected, the adjustment in the share of type s takes longer than in the θ shock.
The difference is relatively smaller compared to other variables: 33 years to reach
50%, and 63 years to reach 75%. Again we note that the relatively smaller difference
is attributable to the fact that the pace of entry and exit from the labor market
determines the base speed of adjustment. With the risk of over simplification, thesluggishness of the s sector response is akin to the impact of a change in education
attainment on the average education of labor force. By the same token, a policy (say,a subsidy for training) intended to reversal the trend must also takes an excruciatingly
long time to generate visible impact. Perhaps, even the Liberal Democratic Partycannot wait that long!

4.3

Injecting various government policies

Table 5 show the long run impacts of three types of policies. In the extreme left
column, we show the bench mark steady state that corresponds to the steady state
shown in the last column of Table 3. Namely, each policy is introduced to the steadystate of the economy after the two shocks.
In the second column, we show the steady state associated with a program thatsubsidizes the rental cost for type c jobs in d -sector. In the third column, we show
the impact of legal restriction on firing which we represent by wasteful costs incurred
upon all types of separations in type c jobs. In the last column, we show the impactof a subsidy to training. The subsidy is given to all the unemployed workers when
he receives training for the first time. Re-training are excluded from the program.
We set the magnitude of each program in such a way that the expenditure (which
we presume to be financed by lump sum tax) is equal to .2% of GDP, which translates
to roughly 1 trillion yen per year in the current Japanese economy. Note that theageless and face a constant probability of retirement. But, remember, the counterpart, the rate of
new entry is not very far from the reality.

22impacts are computed under the assumption that the program is carried out withthe same magnitude forever.
Each program has more or less the same impact on ling run unemployment rate:
about .1% decline.
The program that subsidizes rental cost of job slot in d sector corresponds closely
to various government programs to assist the structurally depressed industries. Subsidies on interest payments are the most common. Our experiment reduces rental
cost of type c jobs by 1.5%. As a result, the share of d -sector job increases by 1.7%,
which is partially offset by .7% decline in g-sector job. A good news is that the overall profitability of type c jobs increases. Hence the share of type s jobs decline by.9%. Somewhat surprising result is the slight decline in vacancy/unemployment ratio
after subsidizing the cost of vacancy. This result obtains because of the composition
effect: the vacancy/unemployment ratio is higher in type s jobs, than in type c at
the benchmark. The decline in the share of type s jobs thus reduces the share oftype s vacancies. Not surprisingly, the labor mobility is somewhat (.3%) reduced bysuch a policy as the policy help survival of jobs in d sector.
In the second experiment, we considered the impact of penalizing separations.
Unlike other two, we gauge the total size of firing cost imposed to .2% of GDP. Thisfiring cost is considered social wastes. The results shown in the second column aremostly near zero except for the two variables. The biggest impact is found in thethreshold z e , i.e., those above the threshold optimally choose to be separated fromthe current employment once the negative productivity hits the location. Thus theimposition of firing cost reduces the mobility of the most mobile workers. The only
good news is that the suppressed separations reduce the unemployment. In turn, theshare of jobs in g sector declines by 1.1%, which is matched exactly by the rise in
the share of d sector jobs.
The last experiment is the impact of subsidy for training. The .2% of GDP
outlay in this experiment corresponds to a 3.3% proportional reduction of the firsttime training cost, z. Thus a simple way to visualize the experiment is to replace z by
.97z for all those who receive training for the first time. As a result, the share of typec employment increases by 2.1%, out of which 1.5% is due to the increase in type cjobs in d-sector. The reason is that the subsidy matters more to the workers closer tothe threshold between type c and type s. Those are the workers with highest training
cost among those in type c jobs. Consequently, the impact eventually translates into
the concentrated increase in the depressed sector jobs. This point can be confirmed
by comparing the impact on thresholds.

5

Conclusion (necessarily incomplete as Section 4)

We have shown that the delayed response of the labor market to a shock can beunderstood as the equilibrium outcome in an economy with search friction and (atleast partially) relation specific investment in training. That would be hardly a news,
though26 . Our modest contribution is to show that such an economy also have otherfeatures which we find in the Japanese labor market. We have shown that a relatively
modest change in a parameter governing the nature of productivity shock is enoughto cause long but steady increase in the share of jobs that do not require training.
This impact causes history dependence because of the sunk cost nature of training.
The same cohort of workers with the same innate ability can have a totally different
job history. The effect will last as long as this induced heterogeneity is eventually
washed away by the retirement of the cohort from the labor market. We also have
26 See for an early contribution by Fukao and Otaki (1993).

23

shown that a model size of negative macro technology shock is enough to explain asizable (say 1%) increase in the long run unemployment rate, without much visible
impact on the unemployment-vacancy ratio. I.e., Beveridge curve shifts out. When
we allow two technology parameters to change, the combined outcome shows all ofthese changes, yet the steady state values of employment turnovers differ little fromthe baseline. The big jump occurs, however in the share of employment in type s
jobs. Even after the deepening of the productivity shock, the new steady state has
roughly the same share of workers clinging to the jobs with lower productivity as theeffect of higher probability of permanent shock almost cancels out the first effect.
A large share of those will search and work in those jobs until retirement. The end
result is that the shrinking primary sector jobs in the g sector.
If these characterizations look too bleak, I only suggest to come to visit one ofJapanese villages with rice fields, or one of 2,921 mostly deserted fisherman’s ports27 ,or anyone tending the cashier at a convenience store. Those folks will tell you theirown versions of the stories why they do what they have been doing for the entire
working life.

A
A.1

Appendix: Formal Description of Market Equilibrium
Set upAs we explained in the main text, we assume the constant population of workers. All the participantsin the labor market are risk-neutral and use the common discount rate r.
We consider two types of jobs, type c and type s.The productivity of both jobs depend on thestate of the location. There is a continuum of the location and the state of each location evolves
according to the following simple stochastic process. There are two possible states, ‘g’ (good) and ‘d’(depressed). The collection of the locations in each state is called sector. Thus g-[d-] sector collects
all the g- [d-] locations. Each location in state g turns to the state d with probability ρ, representing
the permanent productivity shock. In order to prevent all the existing locations becoming d, we
assume the continuum of the location expands over time so that there always exists strictly positive
mass of the locations in state g. Let q c [q s ] denote the output per unit of time of a type c[s] job
in g-sector filled by a properly trained worker. As we explained in the main text, each worker isendowed with innate aptitudes for type c jobs and we represent by z the worker continuum in termsof the training cost needed. The value z represents the cost of the first training. We assume thatsuch a cost is payable in once and each training is complete once the cost is incurred. We denote
by F the distribution of z.
The basic building blocks of the model are the firms’ job offers and workers’ application strategies. The job offer must stipulate employment and compensation in all payoff relevant contingencies:
i.e., required type of the worker, the type of job slot, the location, and wage schedule which canbe contingent on the state of location. Such a job offer maximizes the profit given the application
strategies of each type of unemployed workers. Workers are distinguished by training cost z and thelast training she has taken. As we will see shortly, no worker receives training for the first time at a
type c job which is already in the state d. Still, an unemployed worker could have received training
at a job which is currently in g-sector, or, in d-sector, or, never been trained before. Although aworker can search for a job in any location, if the job where he received the training is currently ing sector, he has no reason to search in a different location because all the locations in the g sectors
are identical because of the imposed symmetry of the locations. A worker may move out, however,from the location where he received training if the location has moved to the d-sector and job
productivity is lower. Therefore a sub market in d - sector is populated only by those who received
the training in the same location, whereas sub markets for jobs in g sector can be populated byworkers with training in g as well as d.
Let e = g, d, denote the current state of the location where he received training and denoteby e = n for the untrained. Let T ≡ {g, d, n} × Z denote the set of types of unemployed worker.
Although a worker can be trained more than once, what matters is the most recent training28 . Thus
the notation of the type of training refers to the most recent one. Note that the training type of
27 The labor force statistics in 2005 reports 230 thousands listed as fishermen in Japan. Thus
there are only 78 fisherman per seaport. On the other hand, during the peak hours, each train (12cars) in Yamanote line accommodates more than 3,000 men and women at a time.
28 As no one choose to go back to the location now in the state d, all the past trainings before thelast must have been in the locations which are now all in d sector.

24the worker can change over time for two reasons. First the state of the location where he received
training can change. Second, once he is employed, he receives re-training.
The job offer can be identified by the pair of wage schedule w = (wg , wd ) and contract types,
wherein wg [wd ]denotes the wage offer contingent upon the location being in the state g [d]. For
each cohort of workers with given training cost index z, the following types of contract types are
possible. Let us start with contract types in type c jobs. First, the offer can be contingent upon
the state of the location. By contingent contract, we mean that the employment is terminatedonce the location is hit by the shock. For simplicity, we also assume that in this type of contract
the job slot itself is self-destroyed when hit by the productivity shock29 . Since we only consider
a permanent shock, the state d is permanent. A job in g-sector can also offer an unconditional
employment contract, except for the exogenous separations. Namely, an unconditional contract
guarantees employment after the shock (but a worker can of course walk away any time they want).
Since the state d is permanent, all the employment offer is unconditional guarantee. Hence we needto consider contingent employment offer only for jobs currently in g-sector. By the same token, nontrivial contingent wage schedule applies only to one type, i.e., the one that promises unconditional
employment.
For untrained workers, no offers from jobs in d sector can compete with those in g sector
because, for the untrained workers, the only difference between the two is lower productivity ind. Thus all the untrained workers will choose locations in g sector. We impose symmetry on theselection among the totally homogenous locations in the g sector so that each location is populatedwith the same density of the untrained workers.
We now consider contract types for type s jobs. It is easy to see that there can be only two
types of contracts, conditional and unconditional employment in g-sector because unfilled job slots
are immediately discarded as soon as the location is hit by the shock. This is the case because types jobs do not need any training. Thus there is no advantage of retaining the vacancies in d-sector.
Consequently there is no active job slot that will post an offer. Again we assume each location in g
sector has the same size of the workers searching for a job30 .
Consequently we need to consider the following five types of contracts: unconditional type c in
the g-sector(ḡ), conditional type c in the g-sector (ĝ), (unconditional) type c in the d-sector (d),unconditional type s (s̄), and conditional type s (ŝ). While unconditional employment contract isnever terminated by the employer, conditional employment contract is terminated when the sector
is hit by a permanent productivity shock without any compensations.
Since the firm can make an offer contingent on the type of worker, we can define the typecontingent employment contract by the pair {C, W}, where C : T → C and W : T → 2+ are
mapping from worker type to offer types and wage schedule, respectively. C ≡ {ḡ, ĝ, d, s̄, ŝ} is theset of type of contract. Thus each sub-market is fully specified by the job offer. In each sub market,
all the job offers are of the same type, and all the workers are of the same type. Denote by x theratio of job searchers to the vacancy, which we call queue length. As explained in the main text, ineach period, with probability A, a worker can send an application to a job that she chooses. Thenthe probability that a vacancy receives at least one application is given by ψ(x), which is increasing
and strictly concave in x.
Now that the description of each sub market is complete, an allocation of the economy can be
fully specified by a tuple {C, W, x} where x : T → + is the queue length in each sub market. Given
the allocation {C, W, x}, we can compute the expected present value of profit stream for a job slot
and the corresponding value of the expected income stream for a worker. Let Vc (w, x) be such value
for a vacancy that post the offer (c, w), where c ∈ C is type of offer. Then we haveRR

rVc (w, x) = −p(c) + ψ(x)[Jc (w) − Vc (w, x)].

(A1)

where p(c) = pc if c = ḡ, ĝ, d and p(c) = ps otherwise. The value of filled job depends upon thestate contingent wage schedule w = (wg , wd ) and the type of contract c and given byrJd (w) = θq c − wd − pc − (δc + d)Jd (w),rJḡ (w) = q c − wg − pc − (δc + d)Jḡ (w) + ρ[Jd (w) − Jḡ (w)],rJĝ (w) = q c − wg − pc − (δc + d + ρ)Jĝ (w),(A2)

rJs̄ (w) = q s − wg − ps − (δs + d)Js̄ (w) + ρ[Jds (w) − Js̄ (w)],rJŝ (w) = q s − wg − ps − (δs + d + ρ)Jŝ (w),
29 This simplifying assumption is used only to avoid further crowding of notations. As we impose
the zero profit condition, all the existing vacancy at equilibrium will have zero value anyway.
30 Notice that because of the free access to the underlying technology, single location can accommodate any size of job vacancies and workers. Although uneven distribution across locations are
immaterial as far as they remain in g sector, the impact of the technology shock on the aggregate
labor market obviously will depend upon the size of the location hit by the shock. We avoid thisunnecessary complications by imposing symmetry.

25

where Jds (w) is given by
rJds (w) = θq s − wd − ps − (δs + d)Jds (w)In words, the first is the value function for an active contract in state d. The first three terms sum
up to the net flow profit, and the last term corresponds to the capital loss upon worker retirement or
exogenous separations. The second and the third are the value function of type c jobs for contracts
in g state. The last two equations are for type s jobs in the state g. Note that a permanent shockto the sector generates capital loss in the value of filled job only if the job is currently in the gsector. The unconditional employment contract continues even after the sector is hit by permanent
shock but it incurs the capital loss due to the change of state. On the other hand, the conditional
employment contract terminates and the job slot is destroyed if the sector is hit by permanent shock.
Thus the value of job turns to be zero.
Next, we consider the value of unemployed worker given the allocation and the equilibrium
value of each type of unemployed worker. The rational unemployed workers choose their application
strategies taking their future change of training into account. Here, we incorporate the equilibrium
value into the value function instead of considering the future decision directly. Let Uc (t, w, x, U ∗ )be the value of type t unemployed worker given the wage schedule, contract type, queue length, andequilibrium value of unemployed worker, which we denote by U ∗ . In this formulation, we treat asgiven the value of different types of unemployed workers, even though in the future he may become
one of those types. We thus focus on the optimal choice of the current application strategy. We
have
rUc (t, w, x, U ∗ ) =φ(x)[Ec (z, w, U ∗ ) − Uc (t, w, x, U ∗ ) − κ(c, t)](A3)− dUc (t, w, x, U ∗ ) + I(e)ρ[U ∗ (d, z) − Uc (t, w, x, U ∗ )],where I(·) is indicator function that takes value one if e = g and zero otherwise. The last termin the square bracket is the capital loss associated with productivity shock which is applicable only
for a worker trained in g-sector.
The training cost κ(c, t) is given by31
80
if c = s̄, ŝ>><z
if e = n and c 6= s̄, ŝ(A4)κ(c, t) =²z
if (e, c) = (d, d), (g, ḡ), (g, ĝ)>>:(² + m)z otherwise
In words, the first case applies to type s jobs irrespective of contract types as no training is required
for this type of job. If a worker is never trained and he applies for a type c job, full training costmust be incurred, irrespective of the contract type. In the third line, we show the following: aworker have to pay ²z if he decides to search and apply for a type c job in the same location where
he received the last training. Finally the same worker has to incur (² + m)z if he decides to moveout of the current location and being matched to a type c job in the g-sector.
The value of employed worker depends on the training cost index z and wage. They are represented by the following value functions for employment.
rEd (z, w, U ∗ ) =wd − δc [Ed (z, w, U ∗ ) − U ∗ (d, z)] − dEd (z, w, U ∗ ),rEḡ (z, w, U ∗ ) =wg − δc [Eḡ (z, w, U ∗ ) − U ∗ (g, z)] − dEḡ (z, w, U ∗ )+ ρ [Ed (z, w, U ∗ ) − Eḡ (z, w, U ∗ )] ,rEĝ (z, w, U ∗ ) =wg − δc [Eĝ (z, w, U ∗ ) − U ∗ (g, z)] − dEĝ (z, w, U ∗ )+ ρ U ∗ (d, z) − Eĝ (z, w, U ∗ ) ,(A5)rEs̄ (z, w, U ∗ ) =wg − δs [Es̄ (z, w, U ∗ ) − U ∗ (n, z)] − dEs̄ (z, w, U ∗ )+ ρ [Eds (z, w, U ∗ ) − Es̄ (z, w, U ∗ )] ,∗rEŝ (z, w, U ) =wg − δs [Eŝ (z, w, U ∗ ) − U ∗ (n, z)] − dEŝ (z, w, U ∗ )+ ρ [U ∗ (n, z) − Eŝ (z, w, U ∗ )] ,where
rEds (z, w, U ∗ ) = wd − δs [Eds (z, w, U ∗ ) − U ∗ (n, z)] − dEds (z, w, U ∗ )This completes the specifications of all the value functions for potentially active contract and job
types.
31 Who actually pays the training cost is immaterial. If you so wish, we could add anotherdimension in the contract type, depending upon who pays the training cost. Since training cost is
not contingent upon the state (although its consequence does depend upon the state in the future),
adding this dimension is redundant. To put it differently, if a firm offers a contract in which they
pay the training cost, the equilibrium value of the offer will be the same as the current one in thatwage schedule will be adjusted accordingly. Needless to say, who pays the training cost does matterat least potentially, if we allow incompleteness of the contract. See section 5 of this appendix below.

26

A.2

Market equilibrium

We now complete the specification of the model by imposing the zero profit condition for all theactive vacancies. Note by the complementary slackness, expected net value of inactive vacancies
must be strictly negative.
We are ready to define the market equilibrium.
Definition 1. The market equilibrium is defined by the set {C ∗ ,W ∗ ,x∗ ,U ∗ ,V ∗ } that satisfies thefollowing conditions.
1. For any type t ∈ T , firms post vacancies so as to maximize their values under the constraintthat the offer must guarantee the equilibrium value U ∗ (t), i.e., given that queue lengths
are determined so as to be consistent with the equilibrium value of unemployment U ∗ (t).
Therefore, we have(C ∗ (t), W ∗ (t), x∗ (t)) ∈ arg max Vc (w, x∗ ) ≡ V ∗ (t)c,w,xsubject toU ∗ (t) ≥ Uc (t, w, x, U ∗ )and x ≥ 0 with complementary slackness, where U ∗ solves∀t ∈ TU ∗ (t) = UC ∗ (t) (t, W ∗ (t), x∗ (t), U ∗ )2. By the free entry condition, the maximized value of active vacancies, V ∗ (t), must be equal
to zero.
In equilibrium, a job slot computes the value of deviation based on the belief that the queue
length that corresponds to alternative job offer should be adjusted so as to guarantee to the unemployed workers the market determined present value of the expected income stream. In order toobtain the market equilibrium, we can solve the problem,above, or, equivalently, we can solve thedual. That is, we maximize the value of the unemployed worker conditional upon profit maximization of all the active job offers and the zero profit condition.
U ∗ (t) = max Uc (t, w, x, U ∗ )
c,w,x

s.t.

(A6)

Vc (w, x) = 0

Before we move on to analyze the market equilibrium defined above, we offer verbal explanations why the market equilibrium defined above coincides with that of a social planner. Given thefull array of contractual arrangements, the market equilibrium defined above evidently solves theresource allocation problem for a social planner endowed with the same search technology, technology evolutions underlying productivity shocks to job slots. In order to see through the logic behind,
let us assume for the time being that both job slots and workers are identical among themselves
and assume away also the productivity shocks, etc. None of these additional factors matters forthis explanation. Under this simplified setting, an individual offer is simple a wage rate at matchedworker receives.The competitive search equilibrium is such that the invididual agents take the valueof the unemployed worker, N ∗ given. From the viewpoint of each job slot, its own wage offer mustsatisfy the constraint
φ(x)M (w) = N ∗Namely the product of the probability that an application results in an offer ,φ(x), and the valueof the offer, M (w),should be equal to the market determined N ∗ . Sinceφ(x) ≡

ψ(x)xwe haveN∗ =ψ(x)M (w),xnamely, this is the trade off between more attractive (hence higher wage) offer and the probability
that a job is filled. Since the offer competition gurantees that N ∗ is the shadow price of theunemployed worker, the profit maximization condition ensures that the optimal choice of quene
length, x, coincides that of a social planner. 32 .
32 See Moen and Rosen (2004) for a more formal proof in a similar model. The formal proofs(omitted) for our case involves straight forward but lengthy derivations of optimal policy for asocial planner that solves the corresponding Hamiltonian defined upon the net social output. The
solutions of course coincide with those given here for market equilibrium.

27

We now solve the problem (A6) in two steps. First, we calculate the value of unemployment
worker given the type of employment contract, and then compare these contracts. To be specific,
we derive the following value at firstUc (t, U ∗ ) ≡ max Uc (t, w, x, U ∗ )
w,x

s.t.

Vc (w, x) = 0

then solve the functional equations
U ∗ (t) = max Uc (t, U ∗ ).
c

In order to derive market equilibrium, we must look for the optimal employment contract for eachtype of worker. The following points that we already made are helpful in deriving the desired
functional. First, recall that the untrained or those trained in the g sector should apply to the jobin the g-sector since they have no advantage to work in d-sector. In addition, the worker who search
for type s jobs should also apply to the job in the g-sector. In both cases, they have no proper
training or no trainings necessary so that they have no point choosing the d sector jobs which, after
all, have lower productivity. The crucial remaining problem is whether or not to move for those
who received training in the location which now belongs to d-sector. When the unemployed worker
applies to the job in the g-sector, should she choose the unconditional employment contract or not?
Next, note also that the trained workers never apply to type s job in the steady state equilibrium.
This is evident from the stationarity of the optimal policy: the fact that he received trained in thepast implies that it was optimal to apply for a type c job. Then, it should be optimal to do so now,
as well. Therefore the third remaining question is which type of untrained worker should apply totype s job.
The following proposition gives the answer to the second. For the last question, we have to wait
until section A.4.
Proposition 1. The comparison between the value of unconditional contract and conditional contract can be implemented by the checking the following inequality.
Uḡ (t, U ∗ )? Uĝ (t, U ∗ ) ⇔ (r + d)U ∗ (d, z) 7 θqc − pc

The intuition behind the Proposition 1 is as follows. From (A2) and (A5), the joint surplus thatis gained by a match in d-sector is given by
Ed (z, w, U ∗ ) + Jd (z, w, U ∗ ) − U ∗ (d, z) =θq c − pj − (r + d)U ∗ (d, z)
Rj

Therefore, Proposition 1 says that the unconditional employment contract is preferred to the conditional contract if and only if the joint surplus from declining job is positive. The proposition also
implies that the comparison between unconditional contract and conditional contract hinges only
upon innate trait z, it does not change according to career path of worker. The unemployed worker
trained in the [current] g-sector always should apply to the type c job in the g-sector. Hence theequilibrium value must satisfyUḡ ((g, z), U ∗ ) if (r + d)U ∗ (d, z) ≤ θq c − pjU ∗ (g, z) =Uĝ ((g, z), U ∗ ) if (r + d)U ∗ (d, z) > θq c − pj
Therefore, the optimal application of type (g, z) unemployed worker depends on the expected incomeof type (d, z) unemployed worker. We solve the optimal application problem of type (d, z) worker,
which is the most important decision in our model, in the following section.

A.3

The optimal policy for trained workers

In order to derive the optimal policy of type (d, z) [ those trained in the location which is currently ind-sector] unemployed worker, we consider the case wherein the worker always chooses the same typeof contract irrespective of his past choices. The option values that correspond to these strategies
can be defined recursivelyŨc (t) ≡ Uc (t, Ũc ) = max Uc (t, w, x, Ũc )
w,x

s.t.

Vc (w, x) = 0

Here we can find the equilibrium value of type (d, z) unemployed worker by comparing these values.
If the type (d, z) worker should apply to the job in the d-sector, then his re-training (costing ²z)
would not change his training type. Therefore stationarity implies that he should apply to the same
type of job in the future as well. On the other hand, by Proposition 1, if it is optimal to move out

28and search for a new job in g sector, he should choose the same policy in the future even though retraining could change the type of worker. Therefore, in equilibrium, type (d, z) unemployed worker
should apply to the same type of contract even after he receives re-training. That is, we haveU ∗ (d, z) =maxc∈{d,ḡ,ĝ}Ũc (d, z)

We can show that the optimal contract for type (d, z) worker is monotone in z.
Lemma 1. Let xc (t) be the queue length that maximizes the present value of the expected income
stream of type t unemployed given the type of contract c and the subsequent value U (t). Then thevalue can be written as the function of queue length,∆(xc (t))p(c) + I(e)ρU (d, z)
Uc (t, U ) =r + d + I(e)ρ
where ∆(x) = ψ 0 (x)/(ψ(x)−ψ 0 (x)x). Moreover, if U (t) = Uc (t, U ), then we have xc (g, z) ≤ xc (d, z)
for any c and z. That is, type (g, z) worker has larger probability to receive an offer than type (d, z)
worker if he seek the same type of job.
Proposition 2. Suppose that type (d, z u ) unemployed worker is indifferent between staying in the dsector and re-entering the g-sector, i.e., Ũd (d, z u ) = Ũg (d, z u ), where Ũg (d, z) = max{Ũḡ (d, z), Ũĝ (d, z)}.
Then, we haveŨd (d, z) if z ≥ z u
U ∗ (d, z) =Ũg (d, z) if z < z u
In addition U ∗ (d, z) is decreasing in z.
Proposition 2 can be stated in words as follows. The difference in z yields the difference invalue of unemployed worker which is proportional to the additional training cost in future. Thus
the impact of z is smaller if you stay at d-sector because re-training cost is proportionally larger if
you move out to g-sector, i.e., the negative slope of U is steeper if you move out, than at d-sector.
Therefore, only the high ability worker (z > z u ) should move out to a new location in the g-sector
after the productivity shock, whereas the low ability worker (z < z u ) should stay in the same
location after the shock.
We can also show that the choice between unconditional contract and conditional contract ismonotone in z because U ∗ (d, z) is decreasing in z. Let define z e by U ∗ (d, z e ) = (θq c − pc )/(r + d).
Then those with lowest training cost (z < z e ) should move every time after the shock , i.e., theychoose type ĝ job if z e < z u . (see Figure 5)

A.4

The optimal policy for untrained workers

Given the optimal choice of experienced workers, as summarized in the two thresholds, z u and z e ,we can finally solve the optimal strategies of untrained workers. The untrained workers must incur
the same amount of training cost z regardless of the state of sector as long as they apply to typec job, thus they should apply to the job in the g-sector if they apply to type c job. The choice
between unconditional contract and conditional contract is determined by z e as in the case of type(g, z) worker. That is, we haveUḡ ((n, z), U ∗ ) if z ≥ z e
Ug (n, z) ≡ max{Uḡ (n, z), Uĝ (n, z)} =Uĝ ((n, z), U ∗ ) if z < z e
The last remaining problem is who should apply for a type s job. Since this type of job does not
require training, the choice should be unanimous for those who choose type s. Let Us = max{Us̄ , Uŝ }denote the value of unemployed worker who searches for type s job. If Us > Ug (n, z), type z worker
should apply to type s job when he enters the labor market. Let z s be the threshold that satisfies
Us = U (u, z s ). Since we can show that Ug (n, z) is decreasing in z, the worker should apply to types job if z > z s .
Suppose that z e < z u < z s . Then, we can summarize the optimal strategy of each ability of
unemployed worker as follows. First, the most talented workers (z < z e ) always apply to type c
job in the g-sector and leave the job if the sector is hit by permanent shock. Second, z ∈ [z e , z u )workers also apply to the type c job in the g-sector but they stay in the d-sector as long as they are
employed. Third, type z ∈ [z u , z s ) workers apply to the type c job in the g-sector when they enter
the labor market and stay in the same sector even though they become unemployed. Finally, theleast adaptable z ≥ z s worker always apply to the type s job in the g-sector.
Now we have made the full circle and the market equilibrium is completely determined except
for the evolution of the state variables, which are shown in section A.6. Notice that we have thecomplete system of equations which jointly determine the equilibrium values of unemployment andemployment. See the last section of this appendix for the details of derivations for the optimal queue
lengths. These values are independent from the dynamics through which state variables converge
to the steady state.

29

A.5

Ex Post Optimality and Incomplete Contract

In the definition of market equilibrium, we assumed that job slots offer two types of contract:contingent and non-contingent employment contracts. The former stipulates that the employment
is terminated at the moment of the productivity shock. On the other hand, in unconditional
contract, wage is made contingent upon the state. If we deprive of full commitment ability, andassume instead agents are restricted to offer non-contingent wage and employment. In that case,time inconsistency problem may arise: namely, when they post the vacancy, their optimal choice ofoffer entails unconditional employment at a wage rate which is also unconditional. Ex post, when
the job slot filled by a worker is hit by the productivity shock, the job slot may well find it optimal
to renege on the promise as the expected return under the depressed state may well be negative.
Now let us return to the contingent wage schedule contract. Whether wage schedule is statecontingent or not matters only for the case in which a job slot posts an unconditional employment
contract. In the proof of Lemma 1, we have shown how the queue length for each type of worker isdetermined by the optimal (first order) condition, whereas the optimal wage schedule is derived bysubstituting the queue length into the zero profit conditions.
Since single zero profit condition can pin down only the discounted sum of the state contingent
wage, there are (infinitely) many wage schedules that satisfy this condition. That is, the optimal
wage schedule of the unconditional employment contract is not unique. The conditions, by which
the retention of employment in d-sector is made ex post optimal for both sides, are:
θq c − wd − pc
≥ 0,
Rc
∗
wd + δc U (d, z)≥ U ∗ (d, z).
Rc

(A7)
(A8)

which are equivalent (respectively to)
Jd (w) ≥ 0,
Ed (z, w, U ∗ ) ≥ U ∗ (d, z).
where Rc = r + d + δc . Therefore, if wd satisfies(r + d)U ∗ (d, z) ≤ wd ≤ θqc − pj ,then the unconditional contract is ex post optimal. We can show that this condition can be satisfied
when the joint surplus from the retention of job in d-sector is non-positive. This always holds true
if type ḡ contract is ex ante optimal.
Proposition 3. In equilibrium, the firm can make the type ḡ contract ex post optimal by theappropriate state contingent wage schedule if the type ḡ contract is ex ante optimal.

A.6

Steady State

Given the equilibrium allocation {C ∗ , W ∗ , x∗ }, we consider the flow and distribution of workers at
steady state. Let es (z) and u(e, z) denote the proportion of employment in the state s sector andthe proportion of unemployed with experience e among trait z workers, respectively. Let definefunction ιe (z) and ιu (z) as
1 if z ≥ z e
1 if z ≥ z u
ιe (z) =, ιu (z) =e
0 if z < z
0 if z < z uThen the flows of workers that apply to type c jobs (z ≤ z s ) are given by33 (see Figures 6-8)u̇(n, z) = d − (φn + d)u(n, z)u̇(g, z) = δj eg (z) − (φg + ρ + d)u(g, z)u̇(d, z) = ρ(u(g, z) + (1 − ιe (z))eg (z)) + δj ed (z) − (φd + d)u(d, z)ėd (z) = ιe (z)ρeg (z) + ιu (z)φd u(d, z) − (δj + d)edX
φe u(e, z) + (1 − ιu (z))φd u(d, z) − (ρ + δj + d)eg (z)ėg (z) =(A9)e∈{n,g}

33 Simple but extremely tedious computations will show that the state variable subsystem is locallystable. It involves confirming for each sub-case the linearized transition matrix to have non-positive
eigen values only. We have not encountered any (non-local) instability in numerical computations
we used for the analysis in Section 4 of the main text.

30

where φe = φ(x∗ (e, z)). From (A9), at steady state, we havedd + φndδφn ω2 (z)u(g, z) =(d + φn )Ω(z)dφu ρω1 (z)u(d, z) =(d + φn )Ω(z)dφu ρ[ιe (z)(d + ρ + φg )ω2 (z) + ιu (z)φd ω1 (z)]ed (z) =(d + δj )(d + φn )Ω(z)u(n, z) =eg (z) =d(d + ρ + φg )φn ω2 (z)(d + φn )Ω(z)whereΩ(z) = (d + ρ)(d + ρ + δj + φg )ω2 (z) − (1 − ιu (z))ρφd ω1 (z)ω1 (z) = (d + δj )[(1 − ιe (z))(d + ρ + φg ) + δ] + δj ιe (z)(d + ρ + φg )
ω2 (z) = d(d + δj + φd ) + (1 − ιu (z))δφd
On the other hand, low ability z > z s worker always apply type s job. The steady state
distribution for these worker, which is independent of trait of worker, is simply given by(d+δs
if Us̄ ≥ Uŝ
d+δs +φnu(n, z) =d+δs +ρ
if Us̄ < Uŝd+δs +ρ+φn(φnif Us̄ ≥ Uŝ
d+δs +φnes =φnif Us̄ < Uŝd+δ +ρ+φsnwhere es is the share of employment at type s job, which is independent of z, for z > z s .

A.7

Proofs

Proof of Proposition 1
From (A3), we havew,x

φ(x)[Ec (z, w, U ∗ ) − κ(c, t)] + I(e)ρU ∗ (d, z)
r + d + φ(x) + I(e)ρ

s.t.

Vc (w, x) = 0

Uc (t, U ∗ ) = max

Note that κ(ḡ, t) = κ(ĝ, t) for any t ∈ T . From (A4),Eḡ (z, w, U ∗ ) =Rc wg + ρwd + δ(Rc U ∗ (g, z) + ρU ∗ (d, z))
Rc (Rc + ρ)Eĝ (z, w, U ∗ ) =wg + δU ∗ (g, z) + ρU ∗ (d, z)
Rc + ρ

Use the zero profit condition to substitute for w, we get
Uc (t, U ∗ ) = maxx

φ(x)[Ẽc (z, x, U ∗ ) − κ(c, t)] + I(e)ρU ∗ (d, z)
r + d + φ(x) + I(e)ρ

whereq̃ − pc
δ(Rc U ∗ (g, z) + ρU ∗ (d, z))
pc
+
−
,
Rc
Rc (Rc + ρ)ψ(x)q c − pc + δU ∗ (g, z) + ρU ∗ (d, z)
pc
Ẽĝ (z, x, U ∗ ) =−Rc + ρ
ψ(x)Ẽḡ (z, x, U ∗ ) =R +ρθ

where q̃ ≡ Rj +ρ q c . Since we have the following, the proof is complete.
j∀xẼḡ (z, x, U ∗ )7 Ẽĝ (z, x, U ∗ ) ⇔ (r + d)U ∗ (d, z) ? θqc − pc

31

Proof of Lemma 1
The optimal queue length solves the following problem
Uc (t, U ) = maxx

φ(x)[Ẽc (z, x, U ) − κ(c, t)] + I(e)ρU (d, z)
r + d + φ(x) + I(e)ρ

where Ẽc (z, x, U ∗ ) is the value of employment that satisfies the zero profit condition. By the firstorder condition, we have(r + d + ψ 0 (x) + I(e)ρ)pc = γ(x)[(r + d + I(e)ρ)(Ẽc (z, x, U ) + p(c)/ψ(x) − κ(c, t)) − I(e)ρU (d, z)]where γ(x) = ψ(x) − ψ 0 (x)/x. By arranging terms, we get
Ẽc (z, x, U ) − κ(c, t) =∆(x)x
[ψ 0 (xc (t))/γ(xc (t))]p(c) + I(e)ρU (d, z)p(c) +
ψ(x)r + d + I(e)ρ

By substituting this into the objective function, we complete the proof of the first half of the
proposition.
Second, we will show the second half of the proposition. By using the results above, we write
Uc ((g, z), U ) =∆(xc (g, z))p(c) + ρUc ((d, z), U ),r+d+ρ

∆(xc (d, z))p(c).
r+dSince Uc ((g, z), U ) ≥ Uc ((d, z), U ) and ∆0 (x) < 0, we have xc (g, z) ≤ xc (d, z).
Uc ((d, z), U ) =Proof of Proposition 2
Let define z̄ u and ẑ u by Ũd (d, z̄ u ) = Ũḡ (d, z̄ u ) and Ũd (d, ẑ u ) = Ũĝ (d, ẑ u ), respectively. We will
prove that zz̄ u ⇔ Ũd (d, z)Ũḡ (d, z) and that zẑ u ⇔ Ũd (d, z)Ũĝ (d, z), which willcomplete the proof if we let z u = max{z̄ u , ẑ u }.
First, we show that ∂ Ũḡ (d, z̄ u )/∂z ≤ ∂ Ũd (d, z̄ u )/∂z < 0, which assures that zz̄ u ⇔Ũd (d, z)Ũḡ (d, z). Since I(d) = 0 and p(c) = pc for c = d, ḡ, ĝ, lemma 1 implies xd (d, z̄ u ) =xḡ (d, z̄ u ) and xḡ (g, z̄ u ) ≥ xḡ (d, z̄ u ). Let x0 = xd (d, z̄ u ) = xḡ (d, z̄ u ) and x00 = xḡ (g, z̄ u ).
From (A3) and (A5), Ũd (d, z) is written by

?

?

?

?

?

?

wd − Rc ²z

Ũd (d, z) = max

wd ,x (r + d)(Rc + φ(x))

s.t.

φ(x)

Vd (w, x) = 0

By the envelope theorem, we have
∂ Ũd (d, z̄ u )Rc ²φ(x0 )=−∂z(r + d)(Rc + φ(x0 ))Similarly, we haveŨḡ (d, z) = max

wg ,wd ,xs.t.

Rc wg + ρwd − Rc (Rc + ρ)(² + m)z + δc Rc Ũḡ (g, z)
φ(x)(r + d)(Rc + ρ)(Rc + φ(x)) + φ(x)δc Rc

Vḡ (w, x) = 0

andŨḡ (g, z) = max

wg ,wd ,xs.t.

φ(x)(Rc wg + ρwd − Rc (Rc + ρ)²z) + ρ[Rc (Rc + ρ) + φ(x)δc ]Ũḡ (d, z)
Rc (r + d + ρ)(Rc + ρ + φ(x))

Vḡ (w, x) = 0

Again by the envelope theorem, we have−(Rc + ρ)Rc (² + m) + δc Rc · ∂ Ũgu (g, z)/∂z∂ Ũḡ (d, z̄ u )=φ(x0 )∂z(r + d)(Rc + ρ)(Rc + φ(x0 )) + φ(x0 )δc Rc(A10)∂ Ũḡ (g, z̄ u )−Rc (Rc + ρ)²φ(x00 ) + ρ[Rc (Rc + ρ) + φ(x00 )δc ]∂ Ũgu (d, z)/∂z=∂z
Rc (r + d + ρ)(Rc + ρ + φ(x00 ))(A11)

By substituting (A11) into (A10), we have
∂ Ũḡ (d, z)
Rc φ(x0 )[(r + d + ρ)(Rc + ρ + φ(x00 ))(² + m) + δc φ(x00 )²]=−∂z(r + d)[(r + d + ρ)(Rc + φ(x0 ))(Rc + ρ + φ(x00 )) + δφ(x0 )(Rc + φ(x00 ))]32

Taking the ratio∂ Ũḡ (d, z̄ u )/∂z(r + d + ρ)(Rc + φ(x0 ))(Rc + ρ + φ(x00 ))(² + m)/² + δc φ(x00 )(Rc + φ(x0 ))=u )/∂z(r + d + ρ)(Rc + φ(x0 ))(Rc + ρ + φ(x00 )) + δc φ(x0 )(Rc + φ(x00 ))∂ Ũd (d, zu
Since m ≥ 0 and x00 ≥ x0 , the numerator is not less than the denominator. Therefore, we have
∂ Ũḡ (d, z̄ u )/∂z ≤ ∂ Ũd (d, z̄ u )/∂z < 0.
Since we can also show ∂ Ũĝ (d, ẑ u )/∂z ≤ ∂ Ũd (d, ẑ u )/∂z < 0 by the same procedure, the firstpart of the proposition is proved.
The second part of the proposition is obvious because ∂ Ũc (d, z)/∂z is non-positive for c = d, ḡ, ĝ.

Proof of Proposition 3
If type ḡ contract is ex ante optimal, we have (r + d)U ∗ (d, z) ≤ θq c − pc from Proposition 1.
Therefore by setting wd so as to satisfy (8), the firm can make the contract ex post optimal.

A.8

The Optimal Queue Lengths

We briefly indicate how to solve for the optimal queue lengths. As in the proof of Lemma 1, theoptimal queue length satisfies(r + d + ψ 0 (x) + I(e)ρ)pc= γ(x)[(r + d + I(e)ρ)(Ẽc (z, x, U ) + p(c)/ψ(x) − κ(c, t)) − I(e)ρU (d, z)]
As an example, we consider the case in which the unemployed trained workers choose to stay in thesame location, i,e, z ∈ (z s , z e ). The other cases can be solved in a similar fashion.
Let x∗d be the optimal queue length for type d worker. Since the unemployed worker chooses
the type d contract, c = d and κ(c, t) = ²z, then xd solves(r + d + ψ 0 (x∗d ))pc
= γ(x∗d )(r + d)[(Ẽd (z, x, U ) + p(c)/ψ(x∗d ) − ²z] cθq − pc + δc U ∗ (d, z)= γ(x∗d )(r + d)
− ²z
Rc
From Lemma 1, we have U ∗ (t) =∆(x∗ (t))p(c)+I(e)ρU ∗ (d,z), define, thus
r+d+I(e)ρ

(r + d + ψ 0 (x∗d ))pc c
θq − pcδc ∆(x∗d )pc− ²z= γ(x∗d )(r + d)+
Rc
Rc r + d
Since ∆(x) = ψ 0 (x)/γ(x), we havepc (Rc + ψ 0 (x∗d )) = γ(x∗d )(θq c − pc )On the other hand, the optimal queue length for type g worker, x∗g , satisfies(r + d + ψ 0 (x∗g ) + ρ)pc
= γ(x∗g )[(r + d + ρ)(Ẽgu (z, x, U ) + p(c)/ψ(x∗g ) − ²z) − ρU ∗ (d, z)]q̃ − pcδc (Rc U ∗ (g, z) + ρU ∗ (d, z))= γ(x∗g ) (r + d + ρ)+
− ²z − ρU ∗ (d, z)
Rc
Rc (Rc + ρ)




δ
Rc ∆(x∗g )pc − ρ(r + d)(r + d + ρ)U ∗ (d, z)
q̃
−
p
c
c
= γ(x∗g ) (r + d + ρ)
− ²z +
Rc
Rc (Rc + ρ)
By rearranging terms, we have
Rc (Rc + ρ + ψ 0 (x∗g ))pc = γ(x∗g ) [(Rc + ρ)(q̃ − pc − Rc ²z) − ρ∆(x∗d )pc ]Similarly, the optimal queue length for type n worker, x∗n , satisfies(r + d + ψ 0 (x∗n ))pc
= γ(x∗n )(r + d)[Ẽgu (z, x, U ) + p(c)/ψ(x∗n ) − z]q̃ − pcδc (Rc U ∗ (g, z) + ρU ∗ (d, z))= γ(x∗n )(r + d)+−z
Rc
Rc (Rc + ρ)(r + d)Rc ∆(x∗g )pc + ρ(r + d + ρ + Rc )∆(x∗d )pcq̃−pc= γ(x∗n )(r + d)
− z + δc
Rc
Rc (Rc + ρ)(r + d + ρ)(r + d)
Thus, we have(r + d + ψ 0 (x∗n ))pc(r + d)Rc ∆(x∗g )pc + ρ(r + d + ρ + Rc )∆(x∗d )pcq̃ − pc− z + δc
= γ(x∗n ) (r + d)
Rc
Rc (Rc + ρ)(r + d + ρ)(The end of the Appendix)

33

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Figure 1: Share of Non-Regular Staff

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Figure 3: Job Accession and Separations: Part time employees
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Figure 4: Beveridge Curve

38

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~
U d (d , z )~U g (d , z )U g (n, z )~U gˆ (d , z )

Us

ze

zu




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
 

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

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

z

zs


   Figure 5: Optimal Application

39

 

dug

d

dud

ρu g

un

ug

ud
δ c egdun

φnun

δ c ed

φgug

φd ud

eg

ed
ρeg

de g

de d

Figure 6: z u < z < z s

40

dug

d

dud

ρu g

un

ug

ud
δ c eg

dun

φnun

φgug

δ c ed

φd ud

eg

ed
ρeg

de g

de d

Figure 7: z e < z < z u

41

dug

d

dud

ρu g

un

ug

ud
δ c eg

dun

φnun

φgug

δ c ed

φd ud
ρeg

eg

de g

ed

de d

Figure 8: z < z e

42

0.12
unemployvacancy

0.11
0.1

u rate, vrate

0.09
0.08
0.07
0.06
0.05
0.04
0.03
0

10

20

30

40

50

60

periodFigure 9: Simulated Dynamics of Unemployed and Vacancy: θ shock

0.65
employ-demployment-g

0.6

0.55

employment

0.5

0.45

0.4

0.35

0.3

0.25
0

10

20

30

40

50

60

periodFigure 10: Simulated Dynamics of Employment Composition: θ shock43

0.0366

0.0364u rate, vrate

0.03620.036

0.03580.0356

unemploy

0.0354

vacancy
0.0352
0

10

20

30

40

50

60

periodFigure 11: Simulated Dynamics of Unemployed and Vacancy: ρ shock

0.49
0.48
0.47
employ-d0.46

employment-g
employment

0.45
0.44
0.43
0.42
0.41
0.4
0.39
0

10

20

30

40

50

60

periodFigure 12: Simulated Dynamics of Employment Composition: ρ shock44