The Hazards of Unemployment: A Macroeconomic Model of Job
Search and Résumé Dynamics
∗
Ross Doppelt
November 20, 2015
Abstract
I introduce a dynamic general-equilibrium model to investigate the relationship between the duration
of unemployment and the probability of nding a job. In particular, I analyze the hypothesis that a
long unemployment spell sends a negative signal about a worker's quality, which in turn aects the
worker's probability of being hired. Each worker has a permanent skill level, either high or low. At
every stage of an employment relationship, high-skill workers draw from a more favorable productivity
distribution. Consequently, high-skill workers are more likely to be hired, conditional on being seen by
a rm. Skills are unobservable, but a worker's history of unemployment and on-the-job productivity
provide observable signals of her underlying type. I refer to a worker's posterior probability of being
highly skilled, conditional on these observables, as the worker's résumé. Spending time in unemployment
damages a worker's résumé: Because high-skill workers are more likely to form matches, prospective
employers infer that unmatched workers are less likely to be highly skilled. Consequently, a worker's
job-nding rate changes over the course of an unemployment spell as her résumé declines. The match
surplus incorporates the fact that hiring a worker makes her résumé look better, and not hiring her
makes her résumé look worse. Wages therefore include a compensating dierential for the résumé value
of employment. I calibrate model to match data on job-nding rates as a function of duration. The
average worker experiences signicant negative duration dependence, and incomplete information also
generates considerable heterogeneity in job-nding rates. I extend the model to discuss how informational
concerns interact with human capital decay as a source of duration dependence. Finally, I discuss the
theory's empirical predictions and econometric implications.
∗ Department of Economics, Penn State University.
Contact:
lated under the shorter title, The Hazards of Unemployment.
ross.doppelt@psu.edu.
This paper previously circu-
Please download the most recent version on my website:
https://sites.google.com/site/rossdoppelt/. This work is based on the third chapter of my doctoral thesis at New York University; I am grateful to my thesis advisor, Tom Sargent, and committee members, Ricardo Lagos and Gianluca Violante, for their
feedback and criticisms. I also beneted from discussions with Gadi Barlevy, Dhruva Bhaskar, Saki Bigio, Katka Borovi£ková,
Joe Briggs, Je Campbell, Chris Flinn, Dan Greenwald, Victoria Gregory, Gregor Jarosch, Boyan Jovanovic, Nic Kozeniauskas,
Elliot Lipnowski, Lars Ljungqvist, Simon Mongey, Jo Mullins, Keith O'Hara, Laura Pilossoph, João Ramos, Edouard Schaal,
Shouyong Shi, and numerous seminar participants. Any imperfections are purely my own.
1
1
Introduction
In this paper, I build a theoretical macroeconomic model to investigate the relationship between the duration
of unemployment and the probability of nding a job. Figure A.1, constructed with data from the Current
Population Survey, shows that the probability of nding a job in the coming month is about 30 percentage
points lower for someone who has been looking for a year, relative to a newly unemployed worker.
1 Recent
evidence from the empirical microeconomic literature, which I will discuss below, suggests that some of
this correlation is causal: Fewer rms are interested in hiring applicants who have been unemployed for
a long time. Economists have oered several hypotheses to explain this relationship; my focus will be on
information stigma, the notion that the length of an unemployment spell sends a signal about worker quality.
After searching for a certain amount of time, a job seeker may fail to be hired because no rms saw her
application, or she was a poor t with all of the rms that did see her application. Prospective employers
will never know exactly why an applicant has struggled to nd work, but rms will draw inferences. If good
workers are more likely to be successful in making matches, then there is reason to think that someone who
fails to make a match is less likely to be a good worker. In turn, an individual worker's job-nding rate will
change as she spends time in unemployment because rms' demand for her services will change as the length
of her unemployment spell reveals information about her. Jobless spells therefore inict a kind of résumé
damage, which aects a worker's future employment and earnings prospects.
I explore this hypothesis by introducing a dynamic general-equilibrium model with heterogeneous agents,
incomplete information, and frictional labor markets. Workers in the model can have one of two skill levels,
either high or low, and this true type is not observed by anyone in the economy. Whereas skill is a permanent
feature of workers, productivity is stochastic and match-specic. The highly skilled are not necessarily more
productive, but they do have a more advantageous productivity distribution. Consequently, people's labormarket outcomes reveal information about their skills at each stage of their working lives. For the unemployed
to be hired, they not only need to be seen by a rm; they need to realize a productivity draw good enough
to justify forming a match. After being hired, a worker's productivity continues to evolve, with the highly
skilled being more likely to enjoy productivity increases, less likely to suer productivity losses, and less
likely to experience separations.
Of central importance is a worker's probability of being highly skilled, conditional on her history of
unemployment and on-the-job performance.
I will refer to this probability as a worker's résumé:
Each
period, beliefs about a worker's skill level get updated according to Bayes's rule, so the résumé summarizes
a worker's background and experience, as they pertain to her future productivity.
1 Data
details are in Appendix D.
2
Search markets are
segmented by expected skill level. When rms post vacancies, they solicit applications from workers with
a particular résumé, and the unemployed apply to the openings commensurate with their résumés. In the
market for résumé-r workers, a Pissarides-style matching function determines how many applications land
in the hands of rms. Prospective employers have access to a screening technology, which allows them to see
the applicant's productivity in her rst period on the job. If the productivity draw is good, then the rm
contacts the worker, and they begin production in the following period. If the draw is bad, the rm never
contacts the worker, who remains unemployed with a résumé updated to reect her failure to nd a job.
The job-nding hazard therefore depends on two things: (1) the relative demand for workers with dierent
résumés, and (2) the speed with which a worker's résumé deteriorates during unemployment. Moreover, the
amount of stigma associated with joblessness depends on how easy it is to be hired, which underscores the
importance of analyzing this problem in general equilibrium.
1.1 Main Results
The model allows me to make progress on three fronts. I begin by developing a novel theoretical framework
for analyzing informational dynamics in frictional labor markets.
Then, I use a calibrated version of the
model to provide a quantitative exploration of the economic forces at work. Finally, I ush out the theory's
empirical predictions and econometric implications. This approach allows me to address several questions.
Informational concerns play an important role in determining match surpluses and, by extension, wages
and job-nding rates. The total surplus from a match equals the expected discounted value of production,
minus the worker's value of search. Both of these values depend on the worker's résumé. By construction, a
worker's expected skill level provides a sucient statistic for forecasting her output. Moreover, the worker's
résumé will determine her employment prospects the next time she has to search for a new job. The match
surplus incorporates the fact that hiring a worker makes her résumé look better, and not hiring her makes her
résumé look worse. An employee's pay is determined by a linear surplus-splitting condition. Consequently,
wages include a compensating dierential for the résumé value of employment. In a sense, rms pay workers
for their time and output, while workers pay for the résumé improvement they get from being employed.
Using the model, I derive a wage decomposition that quanties the contribution of information to a worker's
take-home compensation.
Understanding the theory of how information is priced into wages is necessary for understanding the
association between résumés and job-nding rates. The model suggests that a worker's probability of being
hired is a hump-shaped function of her résumé. With full information, the most productive workers are in
the highest demand: The market for workers who denitely have high skills is tighter than the market for
3
workers who denitely have low skills. With incomplete information, however, the tightest markets are for
workers with good, but not pristine, résumés. Workers of uncertain quality are willing to take substantial
pay cuts to improve their résumés, whereas workers who are known to have high skills need to be paid high
wages. For rms posting vacancies, recruiting workers with mid-level résumés may result in lower expected
productivity, but higher expected prots. Thus, market tightness is a non-monotonic function of résumé.
After expositing the theoretical framework, I use the calibrated model to analyze the relationship between job-nding rates and unemployment durations, on both the aggregate and individual levels. Despite
the clear pattern in Figure A.1, the drop in average job-nding rates does not directly imply that an individual's probability of being hired actually changes over the course of an unemployment spell. An alternative
explanation is that people have heterogeneous but time-invariant job-nding rates: In that case, fast job
nders would be hired after a short amount of time, so the long-term unemployed would comprise the people
who had the lowest job-nding rates all along. Disentangling the relative importance of dynamic selection,
due to unobserved heterogeneity, and true duration dependence, due to time-varying job-nding rates, has
been a contentious issue in the literature. My model features both of these mechanisms. Not only is duration
dependence endogenous, so is the distribution of résumés across workers, which determines the amount of
heterogeneity in job-nding rates. For the key parameters governing stigma eects and skill heterogeneity,
I calibrate the model to match the moments of the aggregate data that are of primary interest, namely,
average job-nding rates, as a function of duration. Because I do not target the hazard curves of individual
workers, this approach is fairly agnostic about what causes the pattern in Figure A.1.
On the individual level, the results paint a rich picture of true duration dependence. A worker's résumé
is a decreasing function of unemployment duration, but the job-nding rate is a hump-shaped function of the
worker's résumé. Individuals will have heterogeneously shaped hazards, depending on the résumés they had
when they entered unemployment. The baseline calibration suggests that the tightest market is for workers
with probability .80 of being highly skilled. The vast majority of unemployed workers have résumés below
this level, and for them, duration dependence is negative. That is, their job-nding rates decline as their
unemployment spells progress. For a worker entering unemployment with the average résumé, the job-nding
probability drops by nearly half after 52 weeks. As I discuss below, some of the most recent and persuasive
empirical studies nd negative duration dependence; in this respect, the model is broadly consistent with
the microeconomic evidence. But for the workers with the highest résumés, duration dependence is positive:
Their job-nding rates actually go up from one period of search to the next, even though expectations
about their skills are revised downward. This result makes sense in light of how résumé dynamics inuence
wages. As high-résumé workers spend more time searching, they are eectively willing to take larger pay
cuts in order to stanch the stigma from joblessness. Nevertheless, after a suciently long unemployment
4
spell, beliefs about a worker's productivity will eventually deteriorate to the point that her job-nding rate
declines. In that sense, all workers will eventually experience negative duration dependence.
On the aggregate level, information stigma is necessary for the calibrated model's ability to account for
the observed correlation between unemployment durations and job-nding rates. Partly, this is a consequence
of the fact that most workers have downward-sloping hazards. The model allows us to look separately at
the eects of true duration dependence and dynamic selection.
Mechanically, the model features a lot of
dispersion in job nding rates, and this heterogeneity can generate much of the drop o in average job-nding
rates, even in the absence of true duration dependence. Economically, though, stigma and heterogeneity are
not orthogonal phenomena. Instead, I argue that information eects are responsible for creating much of the
dispersion in job-nding rates. The dierence in job-nding rates between workers with known skill levels
is much smaller than the range of job-nding rates across workers with unknown skill levels. When workers
have heterogeneous skills, but there is full information about those skills, the aggregate correlation between
job-nding rates and durations is much less pronounced.
Human capital decay is the leading alternative to information stigma for explaining negative duration
dependence, and the model can be extended to incorporate skill loss during unemployment and skill gain
during employment.
job-nding rates.
Relative to the baseline model, this modication can actually
attenuate
the drop in
When skills change over time, information about an unemployed worker's current skill
level is less valuable.
Employees are therefore less willing to accept low wages in order to improve their
résumés. Ultimately, the value of forming a match depends less on the résumé of the worker when she is
hired. In turn, job-nding rates become more uniform across the markets for workers with dierent expected
skill levels, so individual hazards become atter.
Finally, the model has useful implications for empirical research. Because information is revealed about
a worker's skill level every period, a worker's type will eventually be revealed after a suciently long time
in the marketplace. In the model, the résumé distribution will never degenerate because workers die and
are replaced by new-born agents, whose skills are unknown.
However, the model does suggest that, if
duration dependence were a byproduct of learning about worker quality, then older workers would have less
information revealed about them during their most recent unemployment spell. That being said, workers of
dierent ages will be subject to dierent amounts of true duration dependence, as well as dierent amounts
of heterogeneity in job-nding rates.
Thus, it's still necessary to have a careful econometric strategy for
disentangling true duration dependence from unobserved heterogeneity. Fortunately, the theory also provides
guidance for the specication and identication of econometric models, in particular the mixed proportional
hazard model. This framework provides a theoretical justication for tting a mixed proportional hazard
model, but only when looking at workers who enter unemployment with comparable résumés. Moreover, to
5
obtain identication of the proportional hazard model, the theory places restrictions on the information sets
of econometricians, relative to agents in the economy.
1.2 Advancement of the Theoretical Literature
Other papers have looked at information as a source of duration dependence from a theoretical macroeconomic perspective. Examples include Jarosch and Pilossoph [2014], Fernandez-Blanco and Preugschat [2014],
Acemo§lu [1995], and Lockwood [1991].
2 Those models unanimously conclude that information stigma leads
to negative duration dependence, although the calibration exercises by Jarosch and Pilossoph [2014] and
Fernandez-Blanco and Preugschat [2014] suggest that dynamic selection eects are quantitatively more important.
A shared limitation of these theories is that they break the connection between a worker's current job
search and her future career outcomes
value of information.
and this modeling choice is crucial for the way agents assess the
In each of the above papers, a worker's type is perfectly revealed after a match is
made and production commences. This feature is coupled with one of two assumptions: Either a worker
remains employed with the same job until she dies, or all information about a worker is reset once she
3 Such simplications allow authors to make progress on other fronts. However,
reenters unemployment.
these assumptions restrict the value of information: If a worker's planning horizon does not extend beyond
her next job, then her résumé becomes irrelevant for her subsequent employment and wage prospects. In
contrast, my results suggest that the résumé value of employment can be large when workers face the specter
of repeat unemployment spells, so preventing the stigma of unemployment contributes signicantly to the
surplus generated by forming a match. This fact is responsible for the non-monotonic relationship between
expected skills and job-nding rates, which is not found in other models.
To the best of my knowledge, Gonzalez and Shi [2010] provide the only other model in which learning
takes place across multiple unemployment spells, but those authors nd that individuals have unambiguously
upward-sloping hazards. Gonzalez and Shi construct a directed-search model in which workers learn about
their search ability, which determines how good they are at nding jobs when unemployed, rather than their
productive ability, which determines how good they are at creating output when employed. Qualitatively,
2 The
earliest general-equilibrium treatment of this problem appears in Lockwood [1991], who proposes a random-matching
model in which rms can test candidates' abilities; because these tests are imperfect, search duration provides an additional
signal of worker quality. Acemo§lu [1995] combines information asymmetry with human capital depreciation by positing that
workers have to exert eort, which is unobservable, to keep their skills from decaying during unemployment. More recently,
Jarosch and Pilossoph [2014] propose a model with two-sided heterogeneity and positive assortative matching between workers
and rms. All workers are assumed to meet rms at a constant and exogenous rate, but as a worker's search duration increases,
fewer rms are willing to pay the cost of screening her. Fernandez-Blanco and Preugschat [2014] construct a contract-posting
model in which rms oer wages contingent on unemployment duration.
3 Mathematically,
these are almost identical. Lockwood [1991] and Acemo§lu [1995] make the former assumption; Jarosch
and Pilossoph [2014] and Fernandez-Blanco and Preugschat [2014] make the latter.
6
those authors tell a story about discouragement during unemployment: As spells drag on, workers infer
that their search ability is poor, so they target lower-wage jobs in order to increase the probability of being
hired. However, as I discuss below, the most convincing microeconomic evidence suggests that true duration
dependence is largely negative, which is dicult to reconcile with Gonzalez and Shi's prediction of upward-
4 Nevertheless, on a theoretical level, Gonzalez and Shi's result demonstrates
sloping individual hazards.
that negative duration dependence is not a trivial outcome in models where information about workers is
revealed during unemployment spells. I also nd that some workers will have increasing job-nding rates
because they are willing to accept lower wages; however, this outcome manifests in only a minority of job
seekers.
Relative to existing theories, my model incorporates richer dynamics for individuals' skills, as well information about those skills.
One novelty of my model is the analysis of how skill change interacts with
information stigma. Of the above papers, only Acemo§lu [1995] looks at both skill loss and incomplete information. In his model, skill change is necessary for the existence of duration dependence; in mine, skill change
can make duration dependence less pronounced. Finally, all of the above models assume that learning takes
place exclusively in unemployment. The framework that I propose allows beliefs to evolve following hirings,
separations, and changes in productivity. These features of the model respect the large body of evidence
suggesting that information about a worker is revealed throughout her career.
5 It also becomes easier to
interpret the posterior beliefs about a worker's skill as a résumé when those beliefs reect information from
previous jobs.
6
1.3 Relation to the Empirical Literature
This model speaks to the large empirical literature that tries to measure duration dependence using microeconomic data. The complete set of observational studies is too large to catalogue here, but the results are, in
the words of Ljungqvist and Sargent [1998], mixed and controversial. See Machin and Manning [1999] and
Van den Berg [2001] for reviews of the labor and econometrics literatures. One reason for this discord is that
true duration dependence cannot be separated from dynamic selection eects in observational data without
imposing identifying assumptions, which are usually driven more by statistical practicality than economic
theory. Nevertheless, there exists good evidence for negative duration dependence.
Recently, authors have used eld experiments to circumvent the identication issues inherent in using
4 To
be fair, much of this evidence came out after Gonzalez and Shi [2010] was published. Also, because of dynamic selection
eects, Gonzalez and Shi's model can generate a negative correlation between job-nding rates and unemployment durations
in the aggregate, even though true duration dependence is positive.
5 Seminal papers along these lines include Altonji and Pierret [2001] and Gibbons and Katz [1991].
6 Morchio [2015], who cites an earlier version of my paper, adopts the rhetorical device of referring
skill level as her résumé. However, his model only has learning on the job, not during unemployment.
7
to a worker's expected
observational data, and the results are compelling. Oberholzer-Gee [2008], Kroft et al. [2013], and Eriksson
and Rooth [2014] sent out fake applications to real help-wanted ads, and the applications with the longest
gap since the last job were the least likely to elicit responses from rms. Moreover, some of these authors'
results point to information stigma as one of the economic mechanisms for duration dependence. In addition
to conducting a eld experiment, Oberholzer-Gee [2008] conducted a survey of managers in charge of hiring
decisions at Swiss rms. For managers who prefer not to hire the long-term unemployed, the most important
reason is the belief that a productive worker would have already found a job; skill decay is cited as the
second most important reason. Exploiting the variation in labor-market conditions across localities, Kroft
et al. [2013] nd that duration dependence is more strongly negative in places that have low unemployment.
7
Those authors interpret their results through the lens of the information-stigma hypothesis: Failing to nd
a job in a tight market sends a stronger signal about an individual's inability to get hired.
There is also direct evidence to support the search protocol in my model: Unemployment duration is one
hiring criterion that rms mention in help-wanted ads, and workers systematically apply to dierent openings
as their spells become longer.
The New York Times, The Wall Street Journal, and other media outlets have
documented that some rms advertise vacancies while stating an explicit preference for applicants who are
8 These stories conjecture that businesses expect the long-term unemployed
currently or recently employed.
to be of a lower average quality.
However, not all rms that discriminate on the basis of unemployment
duration are exclusionary. Although they draw less attention from the popular press, there are also helpwanted ads that specically encourage applications from the unemployed and underemployed.
9 By welcoming
applicants who have had little success nding jobs elsewhere, some rms are likely trying to nd people who
can be recruited easily or paid cheaply. I am unaware of any systematic evidence of which hiring practice is
more common, but it's clear that dierent rms target dierent types of workers, with dierent histories of
joblessness. Of course, current unemployment status is not the only thing that employers consider; most job
postings also specify desired levels of education and experience. Analogously, a rm in the theoretical model
solicits applications from people with a specic résumé, which is a composite of all past experiences of both
employment and unemployment. Furthermore, Kudlyak et al. [2013] provide evidence that workers apply
for lower-skill jobs as their unemployment spells grow longer. Their study exploits longitudinal data from
a job-posting website to see what kinds of people apply to what kinds of jobs. They nd that after several
weeks of search, well-educated workers eventually apply to the same jobs to which less-educated workers
apply at the outset. As those authors point out, their results are consistent with the hypothesis that agents
7 Using
U.S. data, Imbens and Lynch [2006] also nd that hazard curves are steeper in tight labor markets, although Biewen
and Stees [2010] nd no such correlation in German data. Both of these studies use observational data, so they are subject to
the same qualications mentioned above.
8 See, e.g., Rampell [2011] and Banjo [2012].
9 Searching for the words unemployed or unemployment
on a job-search website will turn up many such postings.
8
use unemployment duration to learn about worker quality.
10
I will proceed as follows. Section 2 describes the model environment. Section 3 denes an equilibrium
and discusses aggregation. In Section 4, I calibrate the model and choose parameters to match the aggregate
data on job-nding rates and search durations. Section 5 contains the results, including decompositions of
hazards and wages to quantify the importance of information stigma. Sensitivity analysis is in Section 6.
Empirical implications are discussed in Section 7. Figures are in Appendix A, computations are discussed
in Appendix B, proofs are in Appendix C, and data details are in Appendix D.
2
Model Environment
2.1 Preferences and Demographics
There is a unit mass of risk-neutral workers. During periods of unemployment, workers get leisure value
A worker also has a constant probability
µ
of dying from one period to the next. Each period, a measure
of workers die and are replaced by a measure
the eective discount factor is
β≡
µ
of new workers. Agents discount the future at rate
ρ;
λ.
µ
hence,
1−µ
1+ρ . Firms seek to maximize the present discounted value of prots, and
the measure of rms is determined by free entry.
2.2 Skills and Productivity
There are two types of workers, those with high skills and those with low skills.
Being skilled does not
necessarily confer a higher level of productivity, but skills do put workers in a more advantageous position
to make high-quality matches and master jobs more quickly. A worker's output is given by
aggregate productivity, and
x
is idiosyncratic. Match-specic productivity
process that depends on the worker's type. In particular,
states of the world, the process governing
workers.
x
x
has support
x
zx,
where
z
evolves according to a Markov
X ≡ {0, x1 , . . . , xnx },
and in all
for high-skill workers stochastically dominates that for low-skill
Upon encountering a rm, a high-skill worker realizes a value of match-specic productivity
with probability
Ω̄ (x | 1);
for low-skill workers, this probability is
probability of a high-skill worker transitioning from
probability is given by
10 Kudlyak
Ω (x, x0 | 0).
is
x to x0
Ω̄ (x | 0).
is given by
x
For the duration of a job, the
Ω (x, x0 | 1);
for low-skill workers, this
The distinction between high- and low-skill workers is captured by the
et al. [2013] cite Gonzalez and Shi [2010] as one theory consistent with their results; my model is also consistent
with these ndings.
9
stochastic-dominance assumptions:
X
Ω̄ (x | 1)
X
<
x≤X
X
Ω̄ (x | 0) .
Ω (x, x0 | 1)
X
<
x0 ≤X
Ω (x, x0 | 0) .
(2.2)
x = 0,
which is an absorbing state. The
x0 ≤X
I will assume that there is always positive probability of realizing
fact that
0 < Ω̄ (0 | 1) < Ω̄ (0 | 0)
(2.1)
x≤X
means that no one can perform all possible jobs, so even highly skilled
workers may fail to be hired. Similarly, the fact that
0 < Ω (x, 0 | 1) < Ω (x, 0 | 0)
means that everyone has
some risk of entering unemployment.
It will be convenient to dene for
r ∈ [0, 1]:
Ω̄ (x | r) ≡
Ω (x, x0 | r) ≡
rΩ̄ (x | 1) + (1 − r) Ω̄ (x | 0) .
(2.3)
rΩ (x, x0 | 1) + (1 − r) Ω (x, x0 | 0) .
(2.4)
Notice that each of the above quantities is a conditional probability. For instance, suppose that a rm encounters an unemployed worker who has probability
r of being highly skilled; then, Ω̄ (x | r) is the probability
of realizing an initial value of match-specic productivity equal to
of transitioning from
x
to
x0
for an employee who has probability
x.
r
Likewise,
Ω (x, x0 | r) is the probability
of being highly skilled.
2.3 Information and Filtering
In reality, rms cannot perfectly predict how fruitful an employment relationship will be; instead, prospective
employers draw inferences based on a candidate's history. Likewise, in the model, a worker's skill level will
not be directly observable, but rms will engage in a ltering problem to discern her ability using Bayes's
rule. To this end, I will adopt the following denitions.
Denition.
For each worker,
public information is the complete history of when the worker was unem-
ployed, when the worker was employed, and all realizations of
x
in jobs where the worker was employed.
Public information is available to workers and all prospective employers. Although information is incomplete, it is symmetric: Workers are also learning about their skills through the progression of their careers.
Notice that public information includes realizations of
realizations of
11 It's
x
x
in jobs where a worker has been employed
that result in an unemployed worker's application being rejected.
not
11 Also, if a worker holds
fair to ask whether an employee's on-the-job productivity is publicly observable. The evidence from empirical micro
studies is somewhat mixed: Schönberg [2007] and Kim and Usui [2013] suggest that this assumption is realistic, and Kahn
[2013] suggests that it is not. Nevertheless, the employer-learning literature commonly assumes that someone's idiosyncratic
10
a job with productivity
x,
and a transition to
x0
induces her to enter unemployment, then the realization
x0
is included in public information. That is, a prospective employer can see why an applicant left her last job.
For each newborn worker, there is a prior probability of that worker's being highly skilled. This probability
is drawn from an exogenous distribution
born with prior probability
Denition.
A worker's
r
F (·).
I will maintain the assumption that, amongst the workers
of being highly skilled, a fraction
r
is, in fact, highly skilled.
résumé, denoted r ∈ [0, 1], is the posterior probability of that worker's being highly
skilled, conditional on public information.
A worker's initial résumé is just the prior probability of her being highly skilled. This value corresponds to
12
someone's qualications upon entering the labor market, based on features such as academic performance.
Each subsequent period, a worker's résumé gets updated, depending on her labor-market experience. From
one period to the next, there are three things that can happen to a worker:
1. An unemployed worker can remain unemployed.
2. An unemployed worker can be hired by a rm with match-specic productivity
3. An employed worker's productivity can transition from
x
to
x0 ,
x.
possibly resulting in separation.
Each of these transitions sends a dierent signal. Suppose a worker is unemployed with résumé
worker is hired with productivity
will be denoted
résumé
B u (r).
x, then her posterior probability of being highly skilled
B h (r | x);
x
to
x0 ,
If the
i.e., her updated
if the worker remains unemployed, then this value will be denoted
Now, suppose a worker is employed with match-specic productivity
productivity transitions from
r.
then her résumé is updated to
B e (r | x, x0 ).13
B u (r) ≡
P [highly
skilled
|
unemployed
→
B h (r | x) ≡
P [highly
skilled
|
unemployed
→ x,
B e (r | x, x0 ) ≡
P [highly
skilled
| x → x0 ,
résumé
x and résumé r.
Concisely:
unemployed, résumé
résumé
r]
r] .
If the worker's
r]
(2.5)
(2.6)
(2.7)
The updating of beliefs will depend on the endogenous probability of making an employment transition.
Consequently, I will defer presenting the exact expressions for these functions until later, in Section 2.9.
productivity is observable to the entire marketplace, so current and prospective employers are using the same noisy signals to
form inferences about a worker's skill. For example, Altonji and Pierret [2001], Lange [2007], and Kahn and Lange [2014] take
this approach.
12 If
employers make inferences about worker quality on the basis of time-invariant characteristics, such as race or sex, these
features would also be subsumed into the prior.
13 Recall
that a transition from
updated according to
B e (r | x, x0 )
x
to
x0
may or may not result in the continuation of a match.
because the transition from
x
11
to
x0
is assumed to be observable.
Regardless, beliefs will be
When appropriate, I will denote the inverses of these functions by lowercase letters; e.g., the function
bh (r | x)
satises
r = B h bh (r | x) | x
14
.
2.4 Matching
The economy features a continuum of labor markets, indexed by
r ∈ [0, 1].
There is one search market for
each résumé value. When a rm posts a vacancy, it solicits applications from workers with a specic résumé
r;
in turn, workers with résumé
r
respond to the rms seeking to hire employees like them.
With some
probability, a prospective rm sees a worker's application; below, I will detail the matching technology for
determining this probability. Firms are assumed to have some screening ability, which allows them to discern
whether a particular worker will be a good match. Specically, the prospective employer can observe the
worker's initial value of
x.
If the employer decides the match is worthwhile, they contact the worker, and
they begin production in the following period. The timing of the matching process ensures that information
between employers and employees will be symmetric:
realization of
bad
x
x
A worker will only be contacted by a rm if her
makes for a viable match, so a worker who remains unemployed never nds out if she had a
draw, or if her application was simply never seen.
This search protocol has both conceptual and technical justications.
Segmenting the search market
along the dimension of expected skill is realistic: Help-wanted ads typically specify that a job is intended for
people with a certain level of experience, set of credentials, or history of unemployment, as I have already
documented. The setup also implies that workers with dierent résumés do not congest one another's search.
In a number of industries, rms have separate recruitment processes for junior and senior positions, so this
assumption provides a reasonable approximation of the hiring process. From a model-making perspective,
I am assuming that search markets segmented.
15 In addition to delivering tractability, the search protocol
in the present model is clearly a normal-form Nash equilibrium: Workers with résumé
of searching in market
r,
and rms posting vacancies in market
applicants who have résumé
r.
r
r
play the strategy
play a strategy of considering only those
Given the recruitment strategies of rms, there's no reason for workers to
deviate from their search strategies, and given the search strategies of workers, there's no reason for rms to
deviate from their recruitment strategies.
In each market
r ∈ [0, 1],
the number of meetings is determined by a constant-returns-to-scale function
of vacancies and searchers. Denote by
workers, in market
r.
θ (r)
the market tightness, or the ratio of vacancies to unemployed
If a worker is looking for a job in a market with tightness
θ,
then her application will
14 It will become apparent in Section 2.9 that B h (· | x) and B e (· | x, x0 ) are guaranteed to be invertible in their rst arguments.
In the numerical exercises,
15 Elsewhere,
B u (·)
is invertible as well.
in the directed-search literature, authors focus on the endogenous segmentation of markets. Examples include
Gonzalez and Shi [2010] and Menzio and Shi [2010, 2011].
12
min ζθ1− , 1 .
be seen by a rm with probability
If a worker is searching in market
r,
then the probability
of her being seen by a rm is:
n
o
1−
p (r) ≡ min ζθ (r)
,1 .
The probability of a rm encountering a worker in market
r
(2.8)
is:
n
o
−
q (r) ≡ min ζθ (r) , 1 .
(2.9)
2.5 The Firm's Problem
As will become clear in Section 2.7, the surplus of a match will depend not only on the worker's résumé,
but also on the résumé the worker would have if she were not employed. This counterfactual résumé, and
therefore wages, will depend on whether the worker was employed in the previous period. Let
denote a worker's tenure status:
s ∈ {e, h}
s = h signies that a worker is newly hired, and s = e signies that a worker
is in a continuing employment relationship.
Let
Gs (r, x)
be the value associated with owning a rm matched with a worker in state
(r, x, s).
Let
ws (r, x) be the wage paid to a worker in state (r, x, s); I will detail the wage determination process in Section
2.7. Firm owners have an outside option of zero. An active rm's Bellman equation is:
Gs (r, x)
= zx − ws (r, x) + β
s.t.:
Let
X
Ω (x, x0 | r) max {Ge (r0 , x0 ) , 0}
r0 = B e (r | x, x ) .
V (r) be the value associated with posting a vacancy in market r.
cost
κ
(2.10)
x0
0
For a potential rm, there is a constant
to maintaining a vacancy in any market in a given period. With probability
q (r)
the rm encounters
a worker. The potential rm's Bellman equation is:
!
V (r)
= −κ + β
[1 − q (r)] V (r) + q (r)
X
Ω̄ (x | r) max {Gh (r0 , x) , V (r)}
(2.11)
x
s.t.:
r0 = B h (r | x) .
Free entry requires that:
V (r) = 0, ∀r.
13
(2.12)
Hence, the value of posting a vacancy in any market must satisfy:
κ = βq (r)
X
Ω̄ (x | r) max Gh B h (r | x) , x , 0 .
(2.13)
x
2.6 The Worker's Problem
Let
Hs (r, x)
be the value associated with being an employed worker in state
associated with being an unemployed worker with résumé
Hs (r, x)
= ws (r, x) + β
X
r.
(r, x, s);
let
U (r)
be the value
The employed worker's Bellman equation is:
Ω (x, x0 | r) max {He (r0 , x0 ) , U (r0 )}
(2.14)
x0
s.t.:
r0 = B e (r | x, x0 ) .
The unemployed worker's Bellman equation is:
!
U (r)
=
λ+β
s.t.:
[1 − p (r)] U
rh0 = B h (r | x)
(ru0 )
+ p (r)
and
X
Ω̄ (x |
r) max {Hh (rh0 , x) , U
(ru0 )}
(2.15)
x
u
ru0 = B (r) .
2.7 Wage Determination
I will assume that wages are determined each period by a linear surplus-splitting rule, with a fraction
η
of
the total match surplus going to the worker. This value will depend on the counterfactual résumé that the
worker would have if a new match were not formed or if an extant match were not maintained. Consequently,
there are separate surplus-spitting conditions for new hires
(s = h)
First, consider an unemployed worker searching in market
updated to
updated to
market
r
B u (r),
and she receives value
B h (r | x),
gets value
U (B u (r)).
and she receives value
Gh
B h (r | x) , x ;
Hh
r.
and continuing employees
If she is not hired, then her résumé is
If she is hired with productivity
B h (r | x) , x .
(s = e).
x,
then her résumé is
A rm that successfully hires a worker in
free entry assures that rm owners get nothing if they fail to make a
hire. Hence, the surplus-splitting condition for newly hired workers is:
Hh B h (r | x) , x − U (B u (r)) = η Gh B h (r | x) , x + Hh B h (r | x) , x − U (B u (r)) .
(2.16)
Now, consider a worker in an existing employment relationship who, in the previous period, produced
x−
with résumé
r− .
If productivity transitions to
Because the complete history of
x
x,
then the worker's résumé becomes
r = B e (r− | x− , x).
is assumed to be observable in any consummated match, the worker's
14
résumé is unaected by staying on the job for an additional period. Hence, the surplus-splitting condition
for continuing employees is:
He (r, x) − U (r) = η [Ge (r, x) + He (r, x) − U (r)] .
More succinctly, we can evaluate (2.16) at résumé
ηGs (r, x)
=
rc
≡
bh (r | x) and consolidate it with equation (2.17) to obtain:
(1 − η) [Hs (r, x) − U (rc )]



B u bh (r | x)
if s = h


r
In the above,
rc
(2.17)
if
(2.18)
s = e.
represents the counterfactual résumé that the worker would have if unmatched.
When interpreting this wage-determination process, some caveats are in order. In many models, linear
surplus-splitting conditions can be supported in two dierent ways:
(1) as the axiomatic solution to a
cooperative game, or (2) as the subgame-perfect equilibrium of a non-cooperative game.
associated with Nash [1950]; the latter, with Rubinstein [1982].
directly assert a surplus-splitting rule as a primitive assumption.
The former is
Some authors, such as Pissarides [1994],
16 In the present model, any of the standard
justications are valid for the surplus-splitting condition in continuing matches (2.17).
For new matches,
however, Rubinstein's alternating-oers protocol does not apply: If a worker were contacted by a rm, but
(inexplicably) declined to accept any oer, then she would continue searching with the knowledge that she
was worth hiring. In that case, the worker would have a private signal of her own skill, relative to her next
employer. Non-cooperative bargaining under asymmetric information is an interesting problem, but it's a
17
non-trivial one that falls outside the scope of the current endeavor.
Alternatively, the Nash [1950] solution species a fair bargain (p. 158), dened as an allocation that
satises a collection of axioms. This approach takes as parameters the feasible payos when agents cooperate, along with utility oors that agents obtain without cooperating.
unemployed worker's payo from not forming a match is
U (B u (r)),
Equation (2.16) species that an
the value of continuing unemployment
with public information. This modeling choice sidesteps the complications that would arise with information
asymmetries. As a plausible defense, one can imagine that workers and rms send their applications and
openings to employment agencies. In the event of a compatible match, the agency applies the Nash axioms
16 In
the environment developed by Pissarides [1994], the space of payos is non-convex, which creates complications for either
Nash [1950] or Rubinstein [1982] bargaining. See Shimer [2006] for a discussion.
17 Firms
have no reason to contact workers unless they expect to reach a bargain. Nevertheless, solving the non-cooperative
bargaining game requires computing the payos associated with deviations from equilibrium outcomes, including deviations
that lead to information asymmetries.
15
to determine a fair bargain and connects the trading parties.
contacted with any contract worth more than
U (B u (r)),
An unemployed worker would want to be
and the rm would want to be contacted with any
contract worth more than zero. These conditions will coincide if wages are determined according to equation
(2.18).
2.8 Policies
Agents decide which matches are worth forming and maintaining. In light of the surplus-splitting rules, a
match is worthwhile for a worker if, and only if, the match is worthwhile for the rm. The set of match-specic
productivities that will result in a worker with résumé
r
being hired is:
X h (r) = x ∈ X | Gh (rh0 , x) + Hh (rh0 , x) ≥ U (ru0 ) , rh0 = B h (r | x) , ru0 = B u (r) .
(2.19)
In subsequent periods, agents decide whether to continue or allow the employment relationship to dissolve.
The set of match-specic productivities that will perpetuate a match with a worker who has résumé
current productivity
x
r
and
is:
X e (r | x) = {x0 ∈ X | Ge (re0 , x0 ) + He (re0 , x0 ) ≥ U (re0 ) , re0 = B e (r | x, x0 )} .
(2.20)
2.9 Beliefs
Having dened the meeting rates and the policy functions, it is now possible to compute the belief-updating
rules. Applying Bayes's rule yields:
u
B (r)
=
B h (r | x)
=
B e (r | x, x0 )
=
1 − p (r)
P
1 − p (r)
P
x∈X h (r)
x∈X h (r)
Ω̄ (x | 1)
Ω̄ (x | r)
r
(2.21)
Ω̄ (x | 1)
r
Ω̄ (x | r)
Ω (x, x0 | 1)
r.
Ω (x, x0 | r)
(2.22)
(2.23)
Notice that the amount of information revealed by an unemployment spell is endogenous because
depends on the equilibrium job-nding rate. All workers searching in market
r,
B u (r)
regardless of their actual
p (r) of being seen and screened by a rm. A genuinely high-skill worker
P
rate of p (r)
x∈X h (r) Ω̄ (x | 1), whereas the job-nding rate for the average
skill level, have the same probability
with résumé
r
has a job-nding
worker with résumé
r
is
p (r)
P
x∈X h (r)
Ω̄ (x | r).
Provided that
X h (r)
consists of all values of
x
meeting or
exceeding a reservation productivity, high-skill workers will have a greater probability of being hired than
16
low-skill workers with the same résumé because
two implications. First,
B u (r) < r,
Ω̄ (· | 1)
stochastically dominates
Ω̄ (· | 0).
This feature has
so a worker's résumé will, in fact, deteriorate during unemployment.
Second, the information-updating process will occur more rapidly for workers searching in tight markets:
Holding
X h (r) constant, B u (r) is decreasing in p (r).
Although unemployment will always send a bad signal,
the strength of that signal depends on aggregate conditions. Failing to nd a job in a slack market is more
indicative of rms' tepid appetite for hiring, whereas failing to nd a job in a tight market is more indicative
of a worker's ability to make a match.
3
Equilibrium
3.1 Denition
We are now prepared to dene an equilibrium.
Denition.
A
recursive search equilibrium comprises:
1. Value functions
2. Policies
X h (r)
3. Contact rates
Gs (r, x), V (r), Hs (r, x),
and
p (r)
6. Belief functions
U (r)
X e (r | x)
and
q (r)
4. A market-tightness function
5. A wage function
and
θ (r)
ws (r, x)
B u (r), B h (r | x),
and
B e (r | x, x0 )
subject to:
1. Bellman equations: (2.10), (2.11), (2.14), and (2.15)
2. Optimal match formation and dissolution: (2.19) and (2.20)
3. Matching technology: (2.8) and (2.9)
4. Free entry (2.12)
5. Surplus splitting (2.18)
6. Bayesian updating: (2.21), (2.22), and (2.23).
Several comments are in order.
I deliberately constructed the equilibrium so that it has a tractable
solution. In particular, the assumption of segmented search markets ensures that policies, wages, and worker
ows do not depend on the distribution of workers over states.
18 However, with the equilibrium objects in
hand, the distributional dynamics become a matter of accounting, which I will lay out in Section 3.2. The
model's structure makes it manageable to solve using the method described in Appendix B. That appendix
18 Shi
[2009] refers to this property as block recursivity.
17
reduces the equilibrium conditions to a system of functional equations for the market-tightness function,
the worker's value of unemployment, and the joint value of employment to the worker and the rm. The
equilibrium objects have to be the xed point of a functional operator, so this line of argument suggests an
iterative algorithm for solving the model numerically.
Even though such a computational approach only establishes the existence of an equilibrium, I conjecture
that the equilibrium is unique.
In other models of endogenous information revelation, such as Lockwood
19 In those models, as in mine, the probability
[1991] and Acemo§lu [1995], there can be multiple equilibria.
of nding a job determines the informational content of a worker's joblessness and the skill composition of
the unemployed workforce.
The crucial dierence is, in other models, workers with all dierent expected
skill levels apply to the same vacancies, so prospective employers care about the distribution of information
and skills in the unemployed population. In such environments, self-fullling equilibria are possible: Firms
are reluctant to hire, which makes workers spend more time in unemployment, which makes the pool of
unemployed applicants look worse, which makes rms reluctant to hire. That mechanism for multiplicity
is absent in my model because a rm can choose to seek a worker with a specic expected skill level.
Consequently, rms' hiring decisions alter the distribution of unemployed workers across markets, but the
skill composition of workers within each market remains unchanged. Moreover, there are two markets in this
economy that have full information, corresponding to
r=0
and
r = 1.
Both of these markets are essentially
discrete-time versions of the environment created by Mortensen and Pissarides [1994], which is known to
have a unique solution. It would be somewhat surprising if equilibrium outcomes were uniquely determined
in markets
r ∈ {0, 1},
but not in markets
r ∈ (0, 1).
3.2 Aggregation
As demonstrated above, we can solve for the model's main ingredients independently of the distribution
of workers across states.
However, to compute any kind of aggregate statistics, it's necessary to know
this distribution, which is determined endogenously.
Let
in continuing employment relationships with productivity
F h (r | x)
r.
dF (r)
x
be the measure of employed workers
whose résumé is less than or equal to
be the measure of newly hired workers with productivity
F u (r)
Let
F e (r | x)
x
r,
and a fraction
r
Let
whose résumé is less than or equal to
be the measure of unemployed workers whose résumé is less than or equal to
is the measure of workers born with résumé
r.
of these
dF (r)
r.
Recall that
workers were actually
born with high skills. Because the prior is assumed to be rational and agents engage in Bayesian updating,
r
will be the fraction of résumé-r workers who are indeed highly skilled.
19 Like
me, Gonzalez and Shi [2010], Fernandez-Blanco and Preugschat [2014], and Jarosch and Pilossoph [2014] cannot
provide armative proof of uniqueness, but they cannot nd instances of more than one equilibrium either.
18
I will dene a functional operator that characterizes the law of motion for these measures. We want an
F e, F h, F u
operator that maps
dened in Section 3.1. Let
Dene the mapping
today into
F e, F h,
and
Fu
F e, F h, F u
(1 − µ)
=
Υ2 F , F , F
0
0
(r , x )
Ω (x, x0 | r) d F e (r | x) + F h (r | x)
p (r) Ω̄ (x0 | r) dF u (r)
(1 − µ)
=
(3.1)
Be (r 0 |x,x0 )
ˆ
u
by:
Xˆ
x
h
tomorrow, given the components of the equilibrium
be the respective spaces in which these measures must reside.
Υ : Fe × Fh × Fu → Fe × Fh × Fu
Υ1 F e , F h , F u (r0 , x0 )
e
0
(3.2)
Bh (r 0 |x0 )
Υ3 F e , F h , F u (r0 )
(1 − µ)
=
Xˆ
Ω (x, x0 | r) d F e (r | x) + F h (r | x)
Bd (r 0 |x,x0 )
x,x0
ˆ


+ (1 − µ)
Bu (r 0 )
X
1 − p (r)
Ω̄ (x | r) dF u (r) + µF (r0 ) ,
(3.3)
x∈X h (r)
where I have dened:
B e (r0 | x, x0 ) ≡ [0, be (r0 | x, x0 )] ∩ {r | x0 ∈ X e (r | x)}
B h (r0 | x) ≡ 0, bh (r0 | x) ∩ r | x0 ∈ X h (r)
(3.4)
B d (r0 | x, x0 ) ≡ [0, be (r0 | x, x0 )] ∩ {r | x ∈
/ X e (r | x)}
(3.6)
B u (r0 ) ≡
F e, F h, F u
The measures
{r | B u (r) ≤ r0 } .
To understand the logic of (3.1), notice that
r
and productivity
x.
following period with productivity
if
x0 ∈ X e (r | x),
integrating
over
x
F e (r0 | x0 ),
than or equal to
0
= Υ F e, F h, F u .
(3.8)
dF e (r | x) + dF h (r | x)
is the measure of employed workers
Of these workers, a fraction
x0
and résumé
B e (r | x, x0 ).
(1 − µ) Ω (x, x0 | r)
r0 .
with respect to
F e (r | x) + F h (r | x)
r0
over
if
r ∈ [0, be (r0 | x, x0 )].
r ∈ B e (r0 | x, x0 )
the measure of workers in continuing jobs with productivity
Concisely,
0
(F e ) = Υ1 F e , F h , F
u
will survive to the
These workers will continue in their jobs
and their updated résumés will be weakly less than
(1 − µ) Ω (x, x0 | r)
yields
(3.7)
then evolve according to:
F e, F h, F u
with résumé
(3.5)
x0
Thus,
and summing
and résumé less
. Similar reasoning accounts for the components of
(3.2) and (3.3). Aggregate employment is given by:
e≡
X
F e (1 | x) + F h (1 | x) .
x
19
(3.9)
Likewise, aggregate output is given by:
y≡z
X x F e (1 | x) + F h (1 | x) .
(3.10)
x
Consequently, the evolution of unemployment and output will always depend on the full distribution of
workers across states.
4
Calibration
One time period in the model corresponds to one week.
Some the model's parameters are common in
search models, so I will take values used elsewhere in the literature. The rate of time preference is
ρ=
.04
52 ,
corresponding to a real interest rate of approximately four percent. The probability of an agent dying from
one period to the next is
1
40×52 , implying that the average working life lasts forty years. Workers and rms
have equal surplus shares, with
leisure is
λ =
1
2.
η =
1
2 . Aggregate productivity is normalized as
z = 1.
The ow value of
As will be clear in a moment, the consumption of unemployed workers will equal half
the output of a newly hired worker; this value is in line with other papers in the literature (c.f.
Shimer
[2005] and Hall [2005]). Following much of the literature, I will set the elasticity of the matching function to
=
1
2 . The matching function's scaling factor
ζ
will be chosen to match the steady-state unemployment rate
using a procedure I will describe below. I will follow Hall and Milgrom [2008], who set the per-period cost
of maintaining a vacancy to 43% of one period's productivity. As will become clear, a newly hired worker
produces one unit of output, so I will set
κ = .43.
The details of the productivity processes are as follows. Recall that the support of
I will assume that
x is X = {0, x1 , . . . , xnx }.
{log (x1 ) , . . . , log (xnx )} constitute a 21-point, equally spaced grid from -1 to 1.
ers in continuing employment relationships, I will assume that
log (x)
For work-
follows a discrete approximation to
a random walk with drift, the parameters of which depend on the worker's skill level. There is a constant,
type-specic probability of productivity dropping to zero. Conditional on remaining positive,
x
has a con-
stant, type-specic probability of taking a step up; there is also a constant, type-specic probability of
taking a step down. That is, for
x
1 < k < nx :
Ω (xk , x0 | i) =




(1 − δi ) αi







(1 − δi ) (1 − αi − χi )




(1 − δi ) χi





δi
20
if
x0 = xk+1
if
x0 = xk
(4.1)
0
if
x = xk−1
if
x0 = 0.
At the endpoints of the positive support (i.e.
x = x1
and
x = x nx )
productivity can only move in one
direction, so I will specify:
Ω (x1 , x0 | i) =



(1 − δi ) αi




(1 − δi ) (1 − αi )






δi
0
if
x = x2
if
x0 = x1
if
x0 = 0
Ω (xnx , x0 | i) =
(1 − δi ) χi






δi
if
x0 = xnx
if
x0 = xnx −1
if
x0 = 0.
(4.2)
The stochastic dominance of
Ω (x, x0 | 1)
α0 , χ0 > χ1 ,
In other words, high-skill workers have a higher probability of experiencing
and
δ0 > δ 1 .
over
Ω (x, x0 | 0)



(1 − δi ) (1 − χi )




is captured by the parameter restrictions
α1 >
productivity gains and a lower probability of experiencing productivity losses.
A diculty in choosing specic values for
{αi , χi , δi }i=0,1
is that these parameters quantify the dierence
between types, and we're interested in a situation where a worker's type is not directly observable. In models
with only one type of worker, or with worker types corresponding to observable categories in the data, authors
typically use parameters that are considered reasonable in light of existing microeconomic estimates. My
calibration strategy will be to choose two sets of parameters within the range of reason, but with one on the
high side, and the other on the low side. In Section 6, I will conduct sensitivity analysis to assess the impact
of various parameter choices. Conditional on
x
remaining positive (and not being at an endpoint of
X ),
the
mean and variance of log productivity growth are:
0
x
| x1 < x < xnx , 0 < x0 ,
E log
x
0
x
V log
| x1 < x < xnx , 0 < x0 ,
x
where
dx ≡ log (xnx /x1 ) / (nx − 1)
i
i
=
(αi − χi ) dx .
(4.3)
=
h
i
2
(αi + χi ) − (αi − χi ) d2x ,
(4.4)
is the increment size of the log productivity grid. For each productivity
process i, the above two moments will guide my choice of the two parameters
αi
and
χi .
In the model, wages
and productivity are not identical, yet the quantitative results will show that productivity is highly correlated
with wages for workers who are in continuing employment relationships. Consequently, my calibration for
{αi , χi }i=0,1 will be motivated by existing estimates for wage and earnings processes. I will set the conditional
0
x
.20
.10
mean of log
to
x
52×10 for high-skill workers and 52×10 for low-skill workers. So, over ten years, matchspecic human capital is expected to grow by about 20% for a high-skill worker and about 10% for a low-skill
worker. The rationale for these numbers comes from the empirical literature on the return to tenure, which
labor economists typically interpret as the growth rate of match-specic human capital.
The calibrated
values fall within the range suggested by existing studies: Altonji and Shakotko [1987] nd a cumulative
21
20 For both
ten-year growth rate of 6.6%, whereas Topel [1991] nds a ten-year growth rate in excess of 25%.
skill types, I will set the variance of
log
0
x
x
.0313
52 . The justication for this choice comes from Meghir
to
and Pistaferri [2004], who estimate the variance of permanent innovations to log earnings; they obtain an
unconditional variance of
δ0 =
1
52×2.25 and
δ1 =
.0313
when using annual data pooled across dierent education groups. I will set
1
52×2.75 . These values imply that the average low-skill worker lasts 2.25 years before
21 As a point
suering a terminal productivity shock, whereas the average high-skill worker lasts 2.75 years.
of reference, Shimer [2005] calibrates separation rates so that the average job lasts 2.5 years.
For newly hired workers, I will assume that the initial value of match-specic productivity has the
following distribution:
Ω̄ (x | 1)
to stochastically dominate
is an absorbing state, we can interpret
Ω̄ (x | 0),
ωi
if


1 − ωi
if
i
Ω̄ (x | i) =
For



ω
x=1
(4.5)
x = 0.
it's necessary and sucient that
ω1 > ω0 .
Because
x=0
as the probability of a worker being able to do a job, conditional
on her true type. An advantage of this parsimonious specication is that the dierence in job-nding rates
between high- and low-skill workers can be summarized by the dierence between
skill workers with résumé
jobs with probability
r
nd jobs with probability
ω0 p (r).
The parameters
ω0
and
ω1 p (r),
ω1
ω1
and
ω0 ;
that is, high-
whereas low skill workers with résumé
r
nd
will be important for determining the informational
content of unemployment spells because the belief-updating function is:
B u (r) =
where I have dened
ωr ≡ (1 − r) ω0 + rω1 .
1 − ω1 p (r)
r,
1 − ωr p (r)
(4.6)
For a given meeting rate
p (r),
the dierence between
governs how quickly a worker's résumé is updated while she is searching for a new job:
in
ω1 − ω0 ,
and
B u (r) = r
when
F (r).
and
ω0
is decreasing
ω1 = ω0 .
It remains to to pick numerical values for
for newly born agents
B u (r)
ω1
ω1 , ω0 ,
the matching eciency
ζ,
and the résumé distribution
To accomplish this, I will select parameters to match the primary moments of
interest, namely, the steady-state unemployment rate and the population job-nding rate as a function of
duration. To economize on free parameters, I will assume that
the fraction of high-skill workers in the economy is
F (r) is Beta(αF , 10 − αF ).
αF
10 . I will also adopt the normalization that
This normalization comes with minimal loss of generality: When
20 For a more recent review of the
21 The model features endogenous
desirable. Strictly speaking,
δi
This implies that
p (r)
ω0 + ω1 = 1.
is strictly less than one, doubling
ζ
is
literature and evidence, see Altonji and Williams [2005].
separation because
x
can gradually drift downward to the point that a match is no longer
is not a type-specic separation rate, but a lower bound on the separation rate. In the numerical
exercises I perform, the majority of separations will come from
x dropping to zero immediately, as opposed to declining gradually.
22
observationally equivalent to cutting both
conduct a grid search over values of
ω0
ω1 − ω0
and
and
ω1
αF
in half. What matters is the dierence
ω1 − ω0 .
I will
in order to minimize the quadratic deviation between
the population job-nding hazard implied by the model and the empirical job-nding hazard constructed
using CPS data.
22 I exclude workers who report being on layo as their reason for unemployment because
Fujita and Moscarini [2013] present evidence that many workers on layo are recalled by the employers from
whom they were initially separated. Only 13% of unemployed workers in my sample were on layo; excluding
them slightly diminishes the drop in job-nding rates at low durations.
consist of an equally-spaced grid of 51 points between zero and
1
2 ; the candidate values of
equally-spaced grid of 10 points between 1.5 and 8.5. For each pair
results in a 6% steady-state unemployment rate.
ω1 = .60, αF = 4.0,
and
The candidate values of
(ω1 − ω0 , αF ),
αF
ω1 − ω0
consist of an
I nd a value for
23 This procedure points to parameter values of
ζ
that
ω0 = .40,
ζ = .2612.
Figure A.2 shows the duration-specic job-nding probabilities implied by the model, along with the
duration-specic job-nding probabilities from the CPS data.
Overall, the t is good.
It's worth noting
that this procedure does not place any ex ante restrictions on the shapes of individuals' hazard curves.
Recall from Section 1.2 that most existing theories imply declining hazards, but Gonzalez and Shi [2010] nd
increasing hazards. Negative duration dependence is not hardwired into the present model. Likewise, this
approach allows either true duration dependence or dynamic selection eects to drive the negative correlation
between search durations and job-nding rates. The results in the following section demonstrate that all
of these forces are at work. The large majority of job-seekers have declining hazards, but a minority has
increasing hazards. True duration dependence and dynamic selection eects are both important, but much
of heterogeneity in job-nding rates is a consequence of informational concerns.
5
Results
The market tightness function
θ (r)
summarizes the demand for workers, given their expected skill level,
and determines job-nding rates for the unemployed.
of a worker's résumé.
Figure A.3 shows market tightness as a function
Strikingly, market tightness is not monotonically related to
r.
In the absence of
informational concerns, one would expect the highest-skill workers to be in the highest demand. Indeed, the
22 More
precisely, the criterion being minimized takes the form
(m − m0 )0 W (m − m0 ). Here, m is a 52 × 1 vector, the dth
d weeks in their rst CPS survey and who
element of which is the fraction of workers reporting an unemployment duration of
report being employed in their second CPS survey. When computing these fractions, workers are weighted using the CPS nal
weights. The vector
m0
is the probability of a worker with an unemployment duration of
in the subsequent four weeks. The weighting matrix
W
is diagonal, the
dth
d
weeks in the model nding work
element of which is proportional to the number
of unemployed workers in the CPS who report an unemployment duration of
d
weeks.
Details on the data are available in
Appendix D.
23 Numerically,
it appears that the steady-state unemployment rate is monotonic in
condition was accomplished using a bisection method.
23
ζ,
so nding a value that satised this
markets
r=0
and
r=1
have full information, and
θ (1)
is 17% greater than
θ (0).
However, the dierence
in market tightness between the two markets with full information is dwarfed by the the range of tightnesses
for markets with incomplete information.
Market tightness is highest for workers with résumé .80, and
this market is 5.4 times tighter than the market for workers who are highly skilled with certainty. Because
B u (r) < r,
a worker's résumé unambiguously declines during unemployment, but this does not imply that
a worker's job-nding probability declines as well. If a worker begins her search in a market
r
close to one,
then she will actually move into tighter markets as she spends time in unemployment. Eventually, though,
her résumé will deteriorate until she is to the left of the peak in Figure A.3; subsequently, her job-nding
rate drops with each additional period of joblessness.
In Section 5.1, I will show that the non-monotonicity of the market-tightness function is related to the
value of information: Workers of uncertain quality are willing to take pay cuts to improve their résumés,
whereas workers who are known to have high skills need to be paid high wages. Recruiting workers with
mid-level résumés may result in lower expected productivity, but higher expected prots. Then, in Section
5.2, I will explore the model's quantitative predictions for the job-nding hazard in more detail.
5.1 Wages and the Value of Information
To understand the economic forces behind job creation, it is essential to understand how the value of
information is priced into wages. Holding down a job and staying out of unemployment improve a worker's
résumé. Therefore, the equilibrium wage should reect the amount that a rm pays for a worker's services,
minus the amount that the worker pays for the résumé value of being employed. The following decomposition
makes this point clear.
In a recursive search equilibrium, wages are given by the following function of the worker's
résumé and productivity:
Proposition 1.
ws (r, x)
= ηzx + (1 − η) λ + ηκθ (rc ) − (1 − η) β
X
x0
rc ≡



B u bh (r | x)
if s = h


r
if s = e
re0 ≡ B e (r | x, x0 )
ru0 ≡ B u (rc ) .
Proof.
See Appendix C.
24
Ω (x, x0 | r) [U (re0 ) − U (ru0 )]
(5.1)
We see that wages depend on three terms.
The rst is
ηzx + (1 − η) λ,
a convex combination of the
worker's output while employed and her ow value of leisure while unemployed. This expression is equal
to the wage that would arise in a static model of Nash bargaining. The second term is
ηκθ (rc ),
which I
will call the search wedge. This expression represents the worker's option value of search; when markets are
tight, the worker can leverage the relative ease with which she can nd a new job. Recall from Section 2.7
that
rc
is the counterfactual résumé a worker would have if she were not employed. In a textbook model
of search and bargaining, such as Pissarides [2000], bargained wages are equal to the static Nash outcome,
plus a search wedge.
In this environment, though, the search wedge depends on
θ (rc ),
the tightness of
the market in which the worker would be searching if she were not employed in her current match.
nal term in equation (5.1) is
− (1 − η) βE [U (re0 ) − U (ru0 ) | r, x],
The
where the expectation is taken over
x0 .
I will call this term the information wedge because it represents the information value of one additional
period of employment to the worker. To make this interpretation clear, consider a worker with résumé
r
who has the opportunity to work today, but will enter unemployment tomorrow. If she is employed today,
then her résumé is updated to
re0
tomorrow, and her value of searching will be
today, then she will have counterfactual résumé
ru0 = B u (rc ).
rc ,
If she is unemployed
and tomorrow she will be be unemployed with résumé
In that case, her value of searching will be
U (ru0 ).
By reaching an agreement for one period
of employment, the worker improves her future search prospects by
equals the expected discounted value of
U (re0 ).
U (re0 ) − U (ru0 ),
U (re0 ) − U (ru0 ).
The information wedge
scaled by the rm's surplus share
(1 − η).
Hiring
someone makes that person look good in the eyes of future employers, so a rm extracts some of this value
from the worker.
Figure A.4 shows the wages of newly hired workers, and Figure A.5 shows the decomposition of these
wages.
Even though all workers have the same output in their rst period on the job, wages can vary
substantially, depending on a worker's résumé.
Many people will receive negative wages, with the lowest
pay going to workers who are probably highly skilled, yet who have some uncertainty surrounding their
true types. The reason for this result is that these workers have the information wedges with the largest
magnitudes. For workers whose type is known, i.e. those with résumés
r ∈ {0, 1},
a job is just a source of
consumption; there's no information value from employment. Notice that the information wedges for these
workers is zero: Because the belief-updating functions all map zero to zero and one to one,
for people with
(r
= 0)
r ∈ {0, 1}.
Unsurprisingly, high-skill workers (r
= 1)
U (re0 ) = U (ru0 )
get paid more than low-skill workers
because they have better outside options, reected by larger search wedges. However, the dierence
in wages between the two types with full information is negligible compared to the range of compensations
received by newly hired workers with
Figure A.7 shows
rc = B u bh (r | x)
r ∈ (0, 1).
Figure A.6 shows
U (r),
the value of unemployment, and
, the counterfactual résumé a newly hired worker would have if she
25
were not employed, as a function of the worker's current résumé. The workers with the largest (absolute)
information wedges are those for whom being hired sends the greatest signal (r
U (r)
the signal is of the greatest value (the slope of
− rc
is large), and for whom
is large).
24
Figures A.8 and A.9 show the wages and wedges of workers in continuing employment relationships.
Wages are only slightly increasing in the worker's résumé, but earnings scale up almost linearly with productivity. If it were possible to regress
w
on
x
and
r,
one might conclude productivity is the main determinant
of wages, and informational concerns play only a minor role. Decomposing wages according to equation (5.1)
suggests otherwise. Figure A.9 demonstrates that the search and information wedges are reasonably large,
but for continuing employees, these terms have comparable magnitudes and opposing signs. In other words,
there are two strong eects that come close to canceling out: The workers who get the most information
value from employment are also those who would have the best job-nding prospects if they were to leave
their current employers. The information wedge is smaller for workers in continuing employment relationships, relative to new matches. All employees enjoy the information value of holding down a job, but new
employees also benet from the signal that they were good enough to hire out of unemployment.
The behavior of wages sheds some light on the shape of the market-tightness function, which is the key to
determining the job-nding hazard. For workers in continuing matches, the search wedge is proportional to
market tightness, and it is nearly the mirror image of the information wedge. The tightest markets are the
therefore the ones that provide the greatest résumé value to workers, not the ones that result in the highest
expected output. This fact explains why high-résumé workers have upward-sloping hazard curves: As these
people spend more time in unemployment, they are eectively willing to take a larger pay cut in order to
halt the résumé damage from joblessness. Workers with good, but not pristine, résumés are in the highest
demand because these matches are likely to be fruitful, and rms can hire labor at a relative bargain. In a
full-information environment, the only thing an employer can oer a worker is some fraction of the goods
she is producing. When there is uncertainty about workers' skills, rms can use the value of information to
compensate workers without giving up output.
Some of the model's predictions for wages are quantitatively drastic, but the qualitative predictions are
clear and logical. The most extreme outcome is that many workers literally pay for the opportunity to work
in their rst week on the job, and in subsequent periods, wages come close to tracking productivity. On a
technical level, this lopsided stream of earnings over the life of a job is a symptom of period-by-period surplus
24 Figure
A.9 show only a single line for the information wedge, even though equation (5.1) suggests that the information
wedge is a function of current-period productivity
x.
In fact, under the chosen specication for
wedge is identical for all workers whose productivity is in the interior of
X:
productivity increase and a one-increment productivity decrease, so the distribution over
relatively few workers with the maximal productivity level
xnx ,
the information
re0
does not depend on
x.
There are
and their information wedges are quantitatively very similar.
In equilibrium, there are no workers with the minimal positive productivity
26
Ω (x, x0 | i),
Everyone has the same probability of a one-increment
x1
because there is no match surplus.
25
splitting with linear utility, which implies that workers are not bothered by sudden swings in consumption.
Nevertheless, real-world wage-tenure proles are often increasing and concave. Theorists have proposed a
number of mechanisms to explain this fact, such as workers' ability to acquire skills or rms' attempts to
retain workers in the face of outside oers. The present model provides an alternative explanation, based on
the informational rents that rms extract from new employees. Workers want to show future employers that
they were worth hiring in their current job. The signal value of being hired, for which workers are willing
to take a pay cut, is greater than the signal value of renewing an existing match. Entry-level internships
t comfortably into this theory: They often have short durations and low pay, yet workers accept them to
bolster their résumés. Finally, for the purposes of explaining market tightness, what matters is the present
value of expected prots over the full lifetime of the match, not the timing of prots within the match. When
wages are determined period-by-period, the information value that rms extract from workers is concentrated
in the rst period of the match. Even if payments were smoothed out, workers with uncertain skill levels
would be willing to accept a contract with a lower expected present value of wages; the markets for such
workers would still be tight relative to the markets for workers with known skill levels.
5.2 Unemployment Durations and Job-Finding Rates
Individual Hazards
Now, I will investigate the shapes of individuals' true job-nding hazards. Consider a worker who enters
unemployment with résumé
denotes the function
B u (·)
r0 .
After
t
periods of search, she will have résumé
composed with itself
t
t
ωi p (B u ) (r0 ) .
t
p (B u ) (r0 ) .
ωi
t
(B u ) (·)
Thus, conditional on her
This representation resembles a multiplicative
proportional hazard model: The probability of nding a job equals a function of duration
times a scaling factor
where
times. Consequently, her probability of being seen and
screened by a rm, as a function of unemployment duration, is
true skill level, her job-nding hazard is
t
(B u ) (r0 ),
that depends on an individual worker's underlying type.
t
p (B u ) (r0 ) ,
26 Two things determine
how the worker's job-nding rate evolve over the course of an unemployment spell: (1) how quickly her
résumé deteriorates and (2) how her probability of being seen by a rm changes as a function of her résumé.
To show how information is revealed during unemployment, Figure A.10 displays
initial initial résumés
r0 .
t
(B u ) (r0 )
for several
Everyone's résumé declines in unemployment, but the drop is slower for people
close to zero or one. This pattern comes from two sources. First, workers with résumés close to
1
2
have the most prior uncertainty about their skill levels, so an additional observation carries more weight.
27
with
r0
25 See
Burdett and Coles [2003] and Stevens [2004] for illustrations of how the curvature of the utility function aects wage-
tenure proles.
26 In Section 7, I'll discuss the theory's implications for
27 Conditional on having résumé r , a worker's skill level
the econometric implementation of proportional hazard models.
follows a Bernoulli distribution with parameter
27
r;
the variance of this
Second, the markets for upper-middle-résumé workers are the tightest, and failing to be hired in a tight
market does more damage to a worker's résumé, as seen from equation (2.21). To show how information
aects contact rates, the left panel of Figure A.11 shows
function of résumé; the right panel shows
market
r.
ωr p (r),
p (r),
the probability of being seen by a rm as a
the average job-nding probability of workers searching in
Both functions inherit the hump shape of the market-tightness function
Figure A.12 displays
t
p (B u ) (r0 )
ωi )
is proportional (by a factor of
for various initial résumés
to
t
p (B u ) (r0 )
r0 .
θ (r), shown in Figure A.3.
An individual's job-nding probability
, so this gure illustrates the nature of true duration
dependence.
Evidently, dierent workers can have hazard curves with totally dierent shapes. Microeconomic studies
often try to establish whether true duration dependence is positive or negative, but in the present model, the
worker's initial résumé inuences both the level and the slope of the hazard. Of newly unemployed workers,
13% will exhibit positive duration dependence, in the sense that their job-nding probability will increase if
they spend a second period in unemployment. Nevertheless, because the overwhelming majority of workers
exhibit negative duration dependence, the model is broadly consistent with the experimental studies cited
in the Introduction. Amongst the workers with downward-sloping hazards, those with the higher résumés
will experience a faster drop in their job-nding probabilities. The average worker entering unemployment
has a résumé of .37, and such a worker will see her job-nding probability drop by 29% after 13 weeks, by
40% after 26 weeks, and by 46% after 52 weeks.
Aggregate Hazards
A long-standing problem in the literature is how to account for the negative correlation between job-nding
rates and unemployment durations that appears in aggregate data: Quantitatively, how much of this correlation is due to true duration dependence, and how much to unobserved heterogeneity? In the calibrated
economy, both mechanisms are important. As one measure of true duration dependence, Jarosch and Pilossoph [2014] propose looking at the average change in job-nding rates experienced by workers who have
been unemployed for
t
28 Figure A.13 shows that this metric of true duration dependence exhibits
periods.
a 20% drop o over the course of a year of search. By contrast, Jarosch and Pilossoph [2014] nd that true
duration dependence in their model accounts for virtually no change in the job-nding rate.
We can perform other experiments to isolate the eects heterogeneity and true duration dependence.
distribution is
28 That
(1 − r) r,
which is maximized at
is, for each duration
t,
r=
1
.
2
compute:
ˆ
100 ×
where
bu (·)
p (r)
dFtu (r) ,
p (bu )t (r)
B u (·), (bu )t (·) is bu (·) composed with itself t times, and Ftu (·) is the résumé distribution amongst
t consecutive periods.
Hence, a worker who has a job-nding rate of ωi p (r) in her tth
t
u
job-nding rate of ωi p (b ) (r) at the outset of her unemployment spell.
is the inverse of
workers who have been unemployed for
period of search had a
28
Figure A.14 contrasts three curves. The rst is the average job-nding rate in the unemployed population
as a function of duration. The second is the population job-nding rate that we would observe if workers
experienced duration dependence, but the skill composition of workers remained xed.
29 The drop in the
job-nding rate coming from true duration dependence is appreciable: Holding the composition of workers
xed, the average job-nding rate drops by 20% after 13 weeks, by 34% after 26 weeks, and by 43% after 52
weeks. The third curve in Figure A.14 is the population job-nding rate that we would observe if workers
had heterogeneous but time-invariant job-nding rates; the distribution over job-nding rates is taken to
30 Despite the importance of
be the distribution implied by the model for workers entering unemployment.
true duration dependence, the amount of heterogeneity in the model is also capable of generating a strong
negative correlation between unemployment duration and the probability of nding a job.
More importantly, the heterogeneity amongst job seekers is not purely mechanical; dierences in jobnding rates across workers are endogenously determined economic outcomes. As evidenced by Figure A.11,
the dierence in job nding rates between workers with known skills (i.e.
dierence in job-nding rates between workers with unknown skills (i.e.
r ∈ {0, 1})
r ∈ (0, 1)).
is small relative to the
Incomplete information
provides an economic mechanism for the relationship between search durations and job-nding rates, and
this mechanism operates through the channels of both heterogeneity and true duration dependence. This
point is further illustrated in the next section, which examines an economy with full information.
5.3 A Full-Information Benchmark
To appreciate the role of incomplete information, it's instructive to compare the economy of Section 2 to
a benchmark environment where workers' skill levels are known, but the expected distribution over labor
productivity is unchanged.
distributions
Suppose that a worker born with résumé
r
Ω̄ (x | r) and Ω (x, x0 | r) for her entire lifetime, with certainty.
draws her productivity from the
Instead of representing a worker's
expected skill, the résumé in this context represents the worker's actual, observable skill level: If
then
Ω̄ (x | r1 )
29 This
stochastically dominates
Ω̄ (x | r0 ),
and
Ω (x, x0 | r1 )
stochastically dominates
r1 > r0 ,
Ω (x, x0 | r0 ).
is the average job-nding rate as a function of duration, where the average is taken over skills and initial résumés,
with respect to the distribution of workers entering unemployment:
ˆ
Hazard Due to True Duration Dependence
where
F0u (·)
30 To
=
[rω1 + (1 − r) ω0 ] p (B u )t (r) dF0u (r) ,
is the distribution of résumés amongst workers entering unemployment.
F0u (·) denote the distribution over résumés amongst workers entering unemployment. A measure rdF0u (r)
ω1 p (r), and a measure (1 − r) dF0u (r) has job-nding rate ω0 p (r). If everyone
had a constant probability of nding a job, then the fraction of workers with job-nding rate ωi p (r) remaining in unemployment
t
after t periods is [1 − ωi p (r)] . The average hazard curve, holding all individual job-nding rates constant, is therefore:
´
[1 − ω1 p (r)]t ω1 p (r) r + [1 − ω0 p (r)]t ω1 p (r) (1 − r) dF0u (r)
.
Hazard Due to Heterogeneity =
´
[1 − ω1 p (r)]t r + [1 − ω0 p (r)]t (1 − r) dF0u (r)
be precise, let
of newly unemployed workers has job-nding rate
29
As before, there is a continuum of segmented search markets, indexed by
r ∈ [0, 1],
vacancy chooses to solicit applications from workers with a particular résumé
r.
and a rm posting a
Agents' Bellman equations
are the same as before, except that the belief-updating functions are replaced by identity functions. Such an
environment is like having a continuum of Mortensen-Pissarides economies, each with its own idiosyncratic
productivity process, indexed by
r.
This full-information benchmark diers from the model of Section 2 in two important respects. First, jobnding rates are not subject to duration dependence. A worker born with résumé
consequently, a worker who begins searching in market
of her spell.
r
will continue searching in market
Second, there is no information value from employment.
wedge disappears because
re0 = ru0 = r.
r will have résumé r forever;
r
for the duration
In equation (5.1), the information
Employers can no longer compensate their workers with the résumé
value of holding a job, so the rm's cost of labor goes up. As a result, market tightness declines. Figure A.15
compares the job-nding rate
these functions align at
incomplete.
ωr p (r)
r = 0
and
in the full- and incomplete-information economies. By construction,
r = 1,
but for all
r ∈ (0, 1),
markets are tighter when information is
Furthermore, when types are known, market tightness is monotonically related to skill level;
in contrast, the market-tightness function has a pronounced hump shape when types are unknown. Besides
causing genuine duration dependence, information frictions also widen the range of job-nding rates amongst
the unemployed. Figure A.16 compares the average aggregate job-nding rates, as a function of duration,
between the baseline model and the full-information benchmark.
The full-information economy falls far
short of replicating the negative relationship observed in the data. In part, this is because individual workers
have constant hazards, but this result also comes from the compression of job-nding rates across workers
with dierent expected skill levels. Although the results in Section 5.2 suggest that dynamic selection eects
are important, heterogeneity should not be interpreted as an alternative explanation for information stigma
when accounting for the aggregate relationship between job-nding rates and search durations. Instead, the
model suggests that information stigma contributes to heterogeneity in job-nding rates.
5.4 Skill Decay and Alternative Sources of Duration Dependence
Thus far, information stigma has been the only source of duration dependence that I have considered, but
other explanations are worth exploring. Amongst the competing hypotheses, skill decay during unemployment is the leading alternative. The basic idea is that low-skill workers are in lower demand, so workers will
have declining job-nding rates as they lose human capital during unemployed. Naïvely, one might think
that information stigma and skill decay can both generate negative duration dependence, so a model with
both forces would have a steeper hazard curve than a model with only one.
30
In the environment I have
Table 1: Parameterizing Skill Change
Parametrization
Prob. of Skill Gain On the Job
(γ)
Prob. of Skill Loss O the Job
Baseline
0
0
SC1
0
SC2
1
2×52
1
26
1
26
1
26
1
26
SC3
(τ )
constructed, however, skill loss during unemployment can actually attenuate duration dependence. If skills
evolve over time, then a signal about a worker's current type becomes less valuable. Consequently, the value
of creating a match may deteriorate less over the course of an unemployment spell, even though someone's
résumé may deteriorate more.
The model allows us to explore these forces in more detail. The framework of Section 2 can be extended
to allow workers' skills to change, depending on their employment status.
being unemployed in period
t + 1.
t,
high-skill workers have probability
Likewise, conditional on being employed in period
high-skill workers in period
t + 1.
t,
τ
Suppose that, conditional on
of becoming low-skill workers in period
low-skill workers have probability
γ
of becoming
The worker's actual skill level remains unobservable. Mathematically, the
only things that change are the belief-updating functions, which become:
Ω (x, x0 | 1)
r
Ω (x, x0 | r)
Ω̄ (x | 1)
r
B h (r | x) = (1 − τ )
Ω̄ (x | r)
P
1 − p (r) x∈X h (r) Ω̄ (x | 1)
u
P
B (r) = (1 − τ )
r.
1 − p (r) x∈X h (r) Ω̄ (x | r)
B e (r | x, x0 )
Table 1 shows the parametrizations for
=
γ
γ + (1 − γ)
and
τ
(5.2)
(5.3)
(5.4)
that I will consider. All other parameters are held constant.
As seen in Figure A.1, the average job-nding rate, as a function of duration, appears to level o after about
six months. With this in mind, I set
τ=
1
26 , implying that it takes an average time of six months for someone
to experience a drop in skills. Parametrization SC1 species that skills can be lost during unemployment,
but not cannot be regained on the job.
Parametrizations SC2 and SC3 allow for skill upgrading during
employment; on average, an improvement takes two years under SC2 and six months under SC3.
As a
point of reference, Ljungqvist and Sargent [1998] specify that the average rate of human-capital depreciation
31
during unemployment is twice as large as the rate of human-capital accumulation during employment.
For each parametrization, Figure A.17 shows the average job-nding rate as a function of résumé. The
probability of being hired is less sensitive to changes in a worker's résumé. Also, for most markets, job-nding
31 In
Ljungqvist and Sargent's model, human capital is synonymous with productivity. In my model, skill is a distribution
over productivities, so the parameters of their model cannot be compared directly to mine.
31
rates are lower in the economies with skill change; as seen from equation (5.4), résumés deteriorate more
quickly during unemployment when job-nding rates are high. The drops in most workers' hazard curves are
therefore more acute in the economy that has information stigma as the only source of duration dependence.
On the aggregate level, the average job-nding rate declines more rapidly as a function of duration under
the baseline calibration, relative to SC1-SC3.
The results suggest that, when types are permanent and unobservable, dierences in workers' job-nding
rates have more to do with dierences in how workers value information, as opposed to dierences in expected productivity. In particular, under the baseline specication, rms are most eager to hire the workers
most willing to give up wages in order to improve their résumés. Adding stochastic skill change to the model
diminishes the marginal value of information. Recall that the information wedge in workers' wages is proportional to the expected value of
U (re0 ) − U (ru0 ),
which captures how the résumé improvement from being
employed today increases the value of search tomorrow. The slope of
U (r)
therefore provides an indication
of how much a worker values the marginal résumé improvement she gets from holding a job. Figure A.18
shows
U (r),
the value of being unemployed with résumé
r,
under each parametrization. Notice that
U (r)
is
steeper under the baseline calibration than under SC1-SC3. If workers can lose their skills in unemployment,
there's less benet to being a high-skill worker; if workers can gain skills on the job, there's less harm in
being a low-skill worker. In either case, the possibility of skill change causes
U (r)
to atten out, implying
that information about a worker's quality has less eect on her search prospects.
There are other possible explanations for duration dependence, besides information stigma and skill
decay.
It's possible that workers become discouraged and expend less eort looking for jobs.
One could
extend the environment of Section 2 to allow for variable search intensity, modeled as in Chapter 5 of
Pissarides [2000].
In general, though, the optimal search intensity selected by a worker will be constant
within an unemployment spell unless some other feature of the environment is time-varying. Consequently,
this notion of search intensity can amplify, but not cause, duration dependence. Yet another possibility is
that employment transitions are determined by stock-ow matching. Under this conception of the matching
process, a newly unemployed worker surveys the stock of all existing vacancies to see if there is a suitable
match. If such a job is available, the worker is hired; if not, the worker surveys the ow of new vacancies
as they are posted by rms.
In that case, duration dependence occurs by assumption, as a result of the
matching technology. Also, this explanation only applies to the drop in job-nding rates between the rst
and second periods of search. A promising approach may be to merge stock-ow matching with one of the
mechanisms discussed above. Doing so, however, is not a straightforward extension, so I defer the problem
to future research.
32
6
Sensitivity Analysis
To assess the sensitivity of the results, I will experiment with some alternative parameter values.
In the
model, information about workers is revealed both on and o the job. Although the model's main application
is analyzing stigma eects from unemployment, the quantitative results depend in part on the on-the-job
productivity dierences between employed workers, captured by
dierence between
and
ω1 ,
Ω (x, x0 | 0)
and
Ω (x, x0 | 1)
Ω (x, x0 | 0)
and
Ω (x, x0 | 1).
Although the
plays a non-trivial role, the results are most sensitive to
ω0
which determine the faction of jobs each type of worker is capable of doing and, by extension, the
probability of being hired conditional on being seen by a rm. To illustrate this point, I solve the model with
Ω (x, x0 | 0) = Ω (x, x0 | 1).
δ0 = δ1 =
I set the conditional mean of
log
0
x
x
, given by equation (4.3), to
.15
52×10 , and I set
1
52×2.50 . All other parameters remain the same as before. Under this alternative parametrization,
high- and low-skill workers are equally productive on a job, conditional on being able to do that job at all.
The market-tightness function (not shown) remains hump-shaped, though the maximal value of
and occurs when
r = .69.
θ (r) is lower
Consequently, a larger fraction of newly unemployed workers, 32%, experience
positive duration dependence. A worker entering unemployment with the average résumé will see her jobnding probability drop by 15% after 13 weeks, by 23% after 26 weeks, and by 29% after 52 weeks. This
decline is not as pronounced as the one described in Section 5.2, but it is nevertheless substantial, especially
given that the two skill levels are assumed to be equally productive once a match has formed.
The main parameters that control the degree of information stigma from unemployment are
ω1
which are the respective probabilities of high- and low-skill workers being able to do a given job. If
ω0
and
ω0 ,
ω1
and
were the same, then the probability of making a match, conditional on being seen by a rm, would not
be correlated with on-the-job productivity; in that case, no learning about worker quality would take place
during unemployment.
Conversely, if
ω1 − ω0
increases, the job-nding probability for high-skill workers
goes up relative to low-skill workers with the same résumé, so an additional period of unemployment makes
it seem even more likely that a worker is unskilled. To demonstrate the importance of these parameters, I
re-solve the model with dierent values of
nding rate
ω1 − ω0 ,
keeping
ωr p (r), as a function of a worker's résumé.
As
ω1 +ω0
xed. Figure A.19 plots the average job2
ω1 −ω0
between high- and low-skill workers (i.e. the dierence between
importantly, though, the hump in
ωr p (r) gets more pronounced.
goes up, the dierence in job-nding rates
ω1 p (1)
and
ω0 p (0))
becomes larger. More
For each of these parameter congurations,
Figure A.20 shows the average job-nding rate as a function of duration. The aggregate relationship between
the duration of search and the probability of being hired becomes stronger as
(ω0 , ω1 ) = (.40, .60)
ω1 − ω0
increases. The values
used in the baseline calibration are the ones that come closest to replicating the data.
33
7
Implications for Empirical Research
7.1 The Precision of Beliefs over Time
Because résumés are updated every period, the variance of posterior beliefs about a worker's skill level will
decline stochastically, and with a sucient amount of labor-market experience, a worker's true type will
become known with certainty.
32 If duration dependence is solely a consequence of learning, then a worker
can only experience true duration dependence if there is uncertainty about her skill.
Thus, the model
implies that workers who have been in the market the longest are subject to the least amount of duration
dependence in unemployment.
33 This suggests comparing the relationship between unemployment durations
and job-nding rates for workers with dierent amounts of labor-market experience.
To summarize these correlations, consider the following semiparametric regression, applied to workers
who are surveyed by the CPS in consecutive months and who report being unemployed in the rst month:
job foundi
= h (durationi ) + x0i β + i .
In the above, durationi is the number of weeks that worker
rst survey month;
h (·)
i
(7.1)
claims to have been looking for a job in the
is a smooth function; job foundi is an indicator variable that is equal to one if the
worker reports being employed in the second survey month;
xi
is a vector of covariates; and
i
is a residual.
Data details are in Appendix D. I divide the sample by potential labor-market experience, dened as age
minus years of education, and I t equation (7.1) separately for each quartile of the potential experience
34
distribution.
Figure A.21 shows the estimated values of
distribution.
100 × h (d) /h (0)
for each quartile of the potential-experience
Indeed, it appears that the drop in average job-nding rates, as a function of duration, is
strongest for those in the bottom quartile of the potential experience distribution and weakest for those
in the top quartile.
32 To
see this, let
ri,t
35 The decline is slightly larger for the second quartile than for the third, but taken
denote the résumé of worker
level. Beliefs are a martingale:
Et [ri,t+1 ] = ri,t .
i
at time
t.
Then,
(1 − ri,t ) ri,t
is the posterior variance of a worker's skill
Combining this fact with Jensen's inequality yields:
Et [(1 − ri,t+1 ) ri,t+1 ] < (1 − Et [ri,t+1 ]) Et [ri,t+1 ] = (1 − ri,t ) ri,t ,
where the inequality is strict because
convergence theorem implies
33 Under
(1 − r) r is strictly concave in r, and ri,t+1
a.s.
(1 − ri,t ) ri,t → 0.
In other words,
ri,t
has positive variance. Thus, the supermartingale
converges to zero or one.
the baseline calibration, the model also predicts that experienced workers have less dispersion in their job-nding
rates. However, the prediction that highly experienced workers have at hazards is a general consequence of Bayesian learning,
not a byproduct of a particular choice of parameter values.
34 More
specically, I t a local-polynomial regression, with additional linear covariates, following Robinson [1988].
How-
ever, when performing the local-polynomial regressions, I employ the estimator described by Breidt and Opsomer [2000] for
incorporating survey weights (in this case, the CPS nal weights).
35 As
an aside, younger workers tend to have higher job-nding rates than older workers.
This would just represent a
vertical shift of each of the population hazards depicted in Figure A.21, which are normalized by the job-nding rates of newly
unemployed workers.
34
together, those in the middle two quartiles show a steeper decline than the most-experienced workers, and a
more moderate decline than the least-experienced workers. Figure A.22 shows the results when the sample
excludes workers who are seeking part-time employment. In that case, the relationship between job-nding
rates and durations is the basically same across the bottom three quartiles of the potential-experience
distribution.
However, it remains true that average job-nding rates fall less with duration for the most
experienced workers.
As a point of comparison, Figure A.23 shows the how the average job-nding rate
changes with duration for dierent experience groups, using articial data generated by the theoretical
36 Qualitatively, the model-generated data shows that there correlation between duration and job-
model.
nding rates declines somewhat with experience, although the magnitudes dier from those constructed using
the CPS. Of course, one would ideally want to distinguish between true duration dependence and dynamic
selection eects, and the regression equation (7.1) can only capture correlations. Nevertheless, these results
are at least suggestive of information stigma as a source of duration dependence.
Field experiments, which send fake résumés to real vacancies, provide a possible avenue for measuring
how duration dependence stems from the diusion of information about a worker's skill. When fabricating
résumés, one would want to vary two things independently: the length of the current unemployment spell
and the precision of beliefs at the outset of the unemployment spell. The theory of Bayesian learning suggests
that prospective employers will have more precise beliefs about applicants who are perceived as having spent
more time in the labor market. To vary the amount of information available on an articial résumé, one could
adjust graduation dates and the number of previous jobs held. If applications containing more information
about worker quality exhibited strong negative duration dependence, then that would suggest the importance
of human capital decay, rather than stigma eects.
7.2 Specication and Identication of Proportional Hazards Models
The theoretical results place restrictions on the proper use of the multiplicative proportional hazard model,
which is the most popular tool in the econometric literature.
probability of nding a job after
disturbance and
hT (t),
xi
t
periods of search is
That specication asserts that worker
vi hX (xi ) hT (t),
where
vi
i's
is a mean-one idiosyncratic
is a vector of person-specic predictors. In applications, the main object of interest is
which is called the baseline hazard.
In light of the economic theory, a proportional hazard model
raises two concerns: specication and identication. For a typical cross-sectional dataset, the proportional
hazard model is misspecied. More subtlety, if a prospective employer and an econometrician have the same
information about workers, then the proportional hazard model is correctly specied
36 In
but not identied. To
the theoretical model, age is geometrically distributed, so the quartiles in Figure A.23 correspond to the quartiles of
the potential experience distribution from the CPS. Also, the job-nding rates shown in Figure A.23 are computed directly
from the model-generated data, rather than from estimating the semiparametric regression in equation (7.1).
35
obtain identication of a correctly specied model, it's necessary for an econometrician to have an indicator
of workers' skills that falls outside the information set of rms.
First, consider a random sample of unemployment spells from the model economy. Recall from Section
5.2 that an individual's job-nding hazard is
t
ωi p (B u ) (r0 ) ,
where
r0
denotes the worker's résumé at the
37 There is heterogeneity amongst job seekers in both
r0 ,
so idiosyncratic eects
cannot be multiplicatively separated from the time-varying portion of the hazard.
Graphically, it's clear
outset of the spell.
ωi
and
that a proportional hazard model will not be correctly specied for a sample of workers with dierent initial
résumés: Figure A.12 shows the shapes of some individuals' true hazard curves, and they are obviously not
in constant proportion to one another. In fact, this may be one of the reasons for the diversity of results
obtained using proportional hazard models.
Now, suppose that an econometrician could observe unemployment spells for a collection of workers
who entered unemployment with identical résumés.
vi hX (xi ) hT (t),
where
vi ≡ ωi /ωr0 , hX (xi ) ≡ 1,
In that case, the job-nding rate can be written as
and
t
hT (t) ≡ ωr0 p (B u ) (r0 ) .
r0 ,
So, for a xed
the true
data-generating process is consistent with a multiplicative proportional hazard model. Unfortunately, the
proportional hazard model is not identied with a dataset that has no variation in
r0 .
In the case where the
data contain only one unemployment spell per worker, Elbers and Ridder [1982] prove that identiability
of the proportional hazard model hinges on
variance.
in
xi ,
hX (xi )
being a non-constant function and
xi
having non-zero
38 Even if an econometrician could observe aspects of a worker's publicly observed history to include
they would be relevant only insofar as they determine
r0 .
In other words, Elbers and Ridder [1982]
show that identiability of the proportional hazard model requires observable dierences in workers with the
same baseline hazard, but the theory of information stigma suggests that the only workers who share the
same baseline hazard are observationally identical, in the sense of having the same résumé.
Identication requires the econometrician to have access to some indicator of workers' skills that is
not included in the publicly observable history that is available to prospective employers. For example, a
longitudinal dataset may allow an econometrician to see a worker's wage on her fth job, which would not be
visible during her third unemployment spell; nevertheless, these wages will be correlated with the worker's
skills. Given covariates
xi
that are outside rms' information set, one could specify
hX (xi ) ≡ E [ωi | xi , r0 ];
it would also be necessary to assume that the expectational error in the econometrician's beliefs, given by
vi ≡ ωi /hX (xi ),
37 If
is independent of
xi ,
conditional on
r0 .
there were more than two skill levels, the hazard would still assume this form, but
r
would be vector-valued. The following
arguments will still apply.
38 Observing multiple spells per worker oers little help in this context.
Honoré [1993] proves that variation in
xi
is unnecessary
for identication when we can observe multiple spells per person, but this result assumes a person has the same
vi
and
hT (t)
across spells. Even if a person's skill level remains constant between spells, her résumé almost surely will not. Consequently,
one person's multiple spells are not informative of a single baseline hazard function.
36
8
Conclusion
Summing up, this model generates several novel theoretical results. First, there is a non-monotonic relationship between expected skill and job-nding rates. The total match surplus depends not only on the worker's
productivity when employed, but also on the résumé damage the worker would have experienced when unemployed. Consequently, the tightest markets are generally not those for the most productive workers, but
for those who most value the résumé improvement from being hired.
Second, the model illustrates how
information is priced into wages, which is essential to understanding the patterns in job-nding rates. The
model provides a decomposition of wages that includes a compensating dierential for the value of information. Third, human capital decay, which is an alternative explanation for negative duration dependence, can
actually attenuate the drop in job-nding rates. When skills change over time, information about a worker's
current skill level is less valuable, thus blunting the above mechanism.
Besides the theoretical results, the quantitative results from the calibrated model suggest that informational concerns can play a large role in shaping job-nding rates.
The model can replicate the aggregate
relationship between search durations and job-nding rates reasonably well. For individual workers, hazards
can change substantially over an unemployment spell. Although heterogeneity in job-nding rates can generate a strong aggregate correlation between search durations and job-nding rates, incomplete information
is responsible for much of the heterogeneity in job-nding rates, in addition to causing true duration dependence for individuals. The overwhelming majority of job-seekers experience negative duration dependence,
which is qualitatively consistent with the micro evidence.
In addition to the implications for empirical research discussed in Section 7, the model also suggests
some directions for future theoretical research.
There have been relatively few macroeconomic models of
information stigma. And, as I have discussed, most of those models do not consider how the information
value of employment inuences future unemployment experiences. Whereas I have embellished the structure
of workers' skills and information, other authors have taken a closer look at the characteristics and behaviors
of rms. In particular, Jarosch and Pilossoph [2014] argue that rm heterogeneity is important, because the
rms that do not bother interviewing the long-term unemployed would not have been compatible with those
workers anyway. An environment that incorporates the mechanisms in each of our models would provide a
fruitful avenue for future research. Another extension would be an examination of business cycles. Although
I have focused on the steady state, the environment I have developed can accommodate aggregate shocks.
This aspect makes the model a fairly general framework for analyzing information dynamics in labor markets.
37
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European Economic Review, 50(4):811830,
2006.
Robert Shimer. The probability of nding a job.
American Economic Review, 98(2):26873, 2008.
Margaret Stevens. Wage-tenure contracts in a frictional labour market: Firms' strategies for recruitment
and retention.
The Review of Economic Studies, 71(2):535551, 2004.
40
Robert Topel.
Specic capital, mobility, and wages:
Wages rise with job seniority.
Journal of Political
Economy, 99(1):14576, 1991.
Gerard J. Van den Berg. Duration models: Specication, identication and multiple durations.
of Econometrics, 5:33813460, 2001.
41
Handbook
A
Figures
Figure A.1: Job-Finding Rates as a Function of Duration
0.7
P[ Job Found Next Month ]
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0
10
20
30
Duration (Weeks)
40
50
Each x in the scatter plot represents the fraction of unemployed workers of a given duration who found a job
by the time they were surveyed in the subsequent sample month. The number of workers who found a job
and the number of workers with a given unemployment duration are weighted with CPS nal weights. The
size of each x is proportional to the weighted number of workers who report that unemployment duration.
The solid line is a local-polynomial kernel regression estimate. Data details are in Appendix D.
42
Figure A.2: Fitting the Population Hazard
0.7
One−Month Job−Finding Prob.
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0
10
20
30
Duration
40
50
The scatter plot is the same as the one in Figure A.1, except that the above excludes workers on layo. The
solid line is the relationship implied by the model.
Figure A.3: Market Tightness
8
7
6
θ(r)
5
4
3
2
1
0
0.2
0.4
0.6
r
43
0.8
1
Figure A.4: Wages of New Hires
1.5
1
0.5
wh(r|1)
0
−0.5
−1
−1.5
−2
−2.5
−3
0
0.2
0.4
0.6
0.8
1
r
Figure A.5: Wedges in the Wage Equation for New Hires
2
1
0
Wedge
−1
−2
−3
−4
−5
−6
0
Search Wedge
Information Wedge
0.2
0.4
0.6
r
44
0.8
1
Figure A.6: Value of Unemployment:
U (r)
1.12
1.1
( 1 − β ) U(r)
1.08
1.06
1.04
1.02
1
0
0.2
0.4
0.6
0.8
1
r
Figure A.7: Signal from Hiring: Counterfactual Résumés
rc
1
0.9
Bu(bh(r))
o
45
0.8
0.7
rc
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
r
45
0.8
1
Figure A.8: Wages of Continuing Employees
2
1.8
we(r,x)
1.6
1.4
1.2
1
1
0.8
3
2
0.5
1
0
0
r
x
Figure A.9: Wedges in the Wage Equation for Continuing Employees
2
1.5
Search Wedge
Information Wedge
Wedge
1
0.5
0
−0.5
−1
−1.5
0
0.2
0.4
0.6
r
46
0.8
1
Figure A.10: Résumé Updating During Unemployment
1
Updated Résumé
0.8
0.6
0.4
0.2
0
0
1
20
0.5
40
60
0
Initial Résumé
Unemp. Duration (Wks)
Figure A.11: Meeting and Matching Probabilities as a Function of Résumé
Matching Rate
Meeting Rate
0.75
0.5
0.7
0.45
0.65
0.4
0.6
0.35
ω p(r)
0.5
r
p(r)
0.55
0.45
0.3
0.25
0.4
0.2
0.35
0.15
0.3
0.25
0
0.5
r
0.1
0
1
47
0.5
r
1
Figure A.12: Contact Rates as a Function of Unemployment Duration
1
0.9
Probability: p(r)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0
1
20
0.5
40
60
0
Initial Résumé
Unemp. Duration (Wks)
Figure A.13: Average Drop in Individual Hazards, by Duration
Relative Job-Finding Rate
100
95
90
85
80
75
0
10
20
30
Unemp. Duration (Wks)
48
40
50
Figure A.14: The Role of Heterogeneity in the Population Hazard
0.25
Job−Finding Rate (Wkly)
Average Hazard
Hazard due to Heterogeneity
Hazard due to Stigma
0.2
0.15
0.1
10
20
30
40
Unemp. Duration (Wks)
50
60
Figure A.15: Job-Finding Rates in the Full-Information Economy
0.5
0.45
Incomplete Info. Economy
Full Info. Economy
0.4
ωr p(r)
0.35
0.3
0.25
0.2
0.15
0.1
0
0.2
0.4
0.6
r
49
0.8
1
Figure A.16: Aggregate Average Job-Finding Rates in the Full-Information Economy
0.25
Job−Finding Prob. (Wkly.)
Incomplete Info. Economy
Full Info. Economy
0.2
0.15
0.1
0
10
20
30
40
Duration (Weeks)
50
Figure A.17: Skill Change and the Job-Finding Rate
60
ωr p (r)
0.5
Avg. Job−Finding Prob. ωr p(r)
0.45
0.4
Baseline
SC1
SC2
SC3
0.35
0.3
0.25
0.2
0.15
0.1
0
0.2
0.4
0.6
Résumé
50
0.8
1
Figure A.18: Skill Change and the Value of Unemployment
U (r)
1.12
1.1
( 1 − β ) U(r)
1.08
1.06
1.04
Baseline
SC1
SC2
SC3
1.02
1
0
0.2
0.4
0.6
0.8
1
Résumé
Figure A.19: Job-Finding Rates by Résumé, Alternative
ω1 − ω0
0.7
0.6
ωr p(r)
0.5
0.4
0.3
0.2
0.1
0
0.3
0.2
ω1 − ω0
1
0.1
0.5
0
0
Résumé
51
Figure A.20: Average Aggregate Job-Finding Rates by Duration, Alternative
ω1 − ω0
0.4
0.35
Probability
0.3
0.25
0.2
0.15
0.1
0.05
0
0.3
0.2
20
0.1
40
60
0
Duration
ω1 − ω0
Figure A.21: Job-Finding Rates by Duration and Potential Experience
100
1st Quartile of Potential Experience
2nd Quartile of Potential Experience
3rd Quartile of Potential Experience
4th Quartile of Potential Experience
Normalized Job-Finding Rate
95
90
85
80
75
70
65
60
0
10
20
30
Duration (Weeks)
52
40
50
Figure A.22: Job-Finding Rates by Duration and Potential Experience, Excluding Part Time
Normalized Job-Finding Rate
100
1st Quartile of Potential Experience
2nd Quartile of Potential Experience
3rd Quartile of Potential Experience
4th Quartile of Potential Experience
95
90
85
80
75
70
65
0
10
20
30
40
50
Duration (Weeks)
Figure A.23: Job-Finding Rates by Duration and Potential Experience, Model-Generated Data
100
1st Quartile of Potential Experience
2nd Quartile of Potential Experience
3rd Quartile of Potential Experience
4th Quartile of Potential Experience
Normalized Job-Finding Prob.
95
90
85
80
75
70
65
0
5
10
15
20
25
30
35
Unemp. Duration (Weeks)
53
40
45
50
B
Computational Outline
I will begin by combining expressions to arrive at a concise system of functional equations that must be
satised in equilibrium. These equilibrium functions must be the xed point of an operator, which I will
dene below, so the computational strategy will be to iterate on this operator. I then provide some details
about how this iterative procedure is executed in practice.
Dene:
Cs (r, , x) ≡ Gs (r, x) + Hs (r, x) .
It will be convenient to evaluate the surplus-splitting condition (2.18) at
Doing so, and using the denition of
Cs (r, x),
(B.1)
(B e (r | x, x0 ) , x0 , e) and B h (r | x) , x0 , h .
yields:
Ge (B e (r | x, x0 ) , x0 ) = (1 − η) [Ce (B e (r | x, x0 ) , x0 ) − U (B e (r | x, x0 ))]
Gh B h (r | x) , x0
= (1 − η) Ch B h (r | x) , x0 − U (B u (r)) .
(B.2)
(B.3)
Combining the above with (2.10) and (2.14) gives us:
Cs (r, , x) = zx + β
X
Ω (x, x0 | r) max {Ce (B e (r | x, x0 ) , x0 ) , U (B e (r | x, x0 ))} .
(B.4)
x0
Notice that the right-hand side does not depend on
write
C (r, x).
s, so neither does the left-hand side.
Hence, I will simply
We can write the value of unemployment (2.15) as:
U (r) = λ + β
u
U (B (r)) + ηζθ (r)
1−
X
h
0
u
Ω̄ (x | r) max C B (r | x) , x − U (B (r)) , 0
!
.
(B.5)
x
Using the surplus-splitting condition for new hires allows us to write the free-entry condition as:
θ (r) = β
X
1−η
1−
ζθ (r)
Ω̄ (x | r) max C B h (r | x) , x0 − U (B u (r)) , 0 .
κ
x
(B.6)
The above allows us to reduce further the value of unemployment to:
U (r) = λ +
ηκ
θ (r) + βU (B u (r)) .
1−η
54
(B.7)
Given
C (r, x), U (r),
and
θ (r),
the policy functions can be written as:
X e (r | x)
=
X h (r)
=
Similarly, once we know
θ (r),
{x0 | C (re0 , x0 ) ≥ U (re0 ) , re0 = B e (r | x, x0 )}
x | C (re0 , x) ≥ U (ru0 ) , rh0 = B h (r | x) , ru0 = B u (r) .
(B.8)
(B.9)
we know the belief functions. Thus, solving for an equilibrium amounts to
nding a solution to the following system of functional equations:
C (x, r)
= zx + β
X
Ω (x, x0 | r) max {C (B e (r | x, x0 ) , x0 ) , U (B e (r | x, x0 ))}
(B.10)
x0
U (r)
θ (r)
Dene
T (C, U, θ)
guesses for
ηκ
θ (r) + βU (B u (r))
1−η
X
1−η
1−
Ω̄ (x | r) max C B h (r | x) , x0 − U (B u (r)) , 0 .
ζθ (r)
= β
κ
x
= λ+
as the operator mapping
(C, U, θ),
I iterate on the
T
(C, U, θ)
(B.11)
(B.12)
to the right-hand side of (B.10)-(B.12). Given initial
operator until I attain practical convergence.
Practically, I perform these iterations as follows. I will solve for the functions of interest on a grid of
values
0 = r1 ≤ · · · ≤ rnr = 1.
Dene:
0
r
≡
x
≡
r1
···
rnr
x0
x1
···
(B.13)
0
where
x0
nr
is understood to be zero. Dene the
nr × (nx + 1)
xnx
,
matrix
C

(B.14)
and the vectors
 C (r1 , x0 ) · · · C (r1 , xnx )

.
.
..
.
.
C ≡ 
.
.
.


C (rnr , x0 ) · · · C (rnr , xnx )
0
U ≡
U (r1 ) · · · U (rnr )
and
Θ
by:






(B.15)
(B.16)
0
Θ ≡
U
θ (r1 ) · · ·
55
θ (rnr )
.
(B.17)
nr × (nx + 1)
Dene the

2
matrix:
e
e
 B (r1 | x0 , x0 ) · · ·

.
..
.
B ≡
.
.


B e (rnr | x0 , x0 ) · · ·
B (r1 | x0 , xnx )
e
···
B (r1 | xnx , x0 )
···
.
.
.
..
.
.
.
e
B e (rnr | x0 , xnx ) · · ·
.
B e (rnr | xnx , x0 ) · · ·

e
B (r1 | xnx , xnx ) 

.
,
.
.


B e (rnr | xnx , xnx )
(B.18)
where elements of
interpolation of
Be
are imputed to be zero for infeasible
11×(nx +1) ⊗ C
along the grid given by
Ce =
where each
Ceh
is
nr × (nx + 1),
Ξei
where
Ω (xh , · | i)
Dene
Ue
Ce
be the column-wise
Ce0
Ce1
Cenx
···
element of
Ceh
,
(B.19)
C (B e (rj | xh , xk ) , xk ).
is
1(nx +1)×1 ⊗ Ω0i Inx +1 ⊗ 1(nx +1)×1

0
Ω (x0 , · | i)
0(nx +1)×1
···
0(nx +1)×1


.
..
.
 0(n +1)×1 Ω (x1 , · | i)0
.
.
x

= 
.
..
..

.
.
.

.
0(nx +1)×1

0
0(nx +1)×1
···
0(nx +1)×1 Ω (xnx , · | i)
Dene:
≡
represents the
(h + 1)
st
row of
as the column-wise interpolation of
U =
(j, k + 1) element of Ueh
is
Ωi ,
and





,



(B.20)
represents element-by-element multiplication.
U ⊗ 11×(nx +1)2
along the grid given by
e
where the
transitions. Let
That is:
(j, k + 1)
and the
Be .
(x, x0 )
Be .
That is:
Ue0
Ue1
···
U (B e (rj | xh , xk )).
Uenx
,
(B.21)
Observe that column
(h + 1) of max {Ce , Ue } Ξei
is
0
max {Ceh , Ueh } Ω (xh , · | i) , which gives the expected value (over x0 ) of max {C (B e (r | xh , x0 ) , x0 ) , U (B e (r | xh , x0 ))}
under the distribution
Ωi ,
conditional on starting with
TC (C, U, Θ)
x = xh .
The updated value of
= z1nr ×1 x0 + β (max {Ce , Ue } Ξe1 ) r11×(nx +1)
C
is given by:
+β (max {Ce , Ue } Ξe0 ) 1nr ×(nx +1) − r11×(nx +1) .
Dene the
nr × 1
(B.22)
vector:
u
B ≡
0
u
B (r1 ) · · ·
56
u
B (rnr )
.
(B.23)
Dene
Uu
as the interpolation of
U
along the grid given by
TU (C, U, Θ) = λ1nr ×1 +
Dene the
nr × 1
Bu .
The updated value of
U
is given by:
ηκ
Θ + βUu .
1−η
(B.24)
vector:
Bh ≡
0
h
B (r1 ) · · ·
h
B (rnr )
.
(B.25)
The above incorporates the specication assumed in the calibration exercise; namely, all newly hired workers
have the same productivity
the grid given by
Bh .
x = 1.
Ch
Dene
as the column-wise interpolation of
C ( : , I [x0 = 1] )
Dene:
P ≡ min ζΘ1− , 1
value of
Θ
(B.27)
min {·} operator and the power (1 − ) are understood to apply element-by-element.
The updated
is given by:
TΘ (C, U, Θ) = β
C
(B.26)
≡ rω1 + (1nr ×1 − r) ω0 ,
ω
~r
where the
along
1−η
κ
Pω
~ r max Ch − Uu , 0 .
(B.28)
Proof of Proposition 1
Evaluating equation (B.11) at
Hs (r, x) − U (rc )
=
rc
and subtracting it from
ws (r, x) + β
X
Hs (r, x),
as dened in equation (2.14), yields:
Ω (x, x0 | r) max {He (B e (r | x, x0 ) , x0 ) − U (B e (r | x, x0 )) , 0}
x0
−λ +
X
Ω (x, x0 | r) [U (B e (r | x, x0 )) − U (B u (rc ))] − β
x0
κη
θ (rc )
1−η
X
η
β
Ω (x, x0 | r) max {Ge (B e (r | x, x0 ) , x0 ) , 0}
1−η
0
x
X
κη
0
−λ + β
Ω (x, x | r) [U (B e (r | x, x0 )) − U (B u (rc ))] −
θ (rc ) ,
1−η
0
= ws (r, x) +
(C.1)
x
where the second line uses the surplus-splitting condition (2.18) evaluated at
(r, x, s).
Multiplying the above by
1 − η,
multiplying equation (2.10) by
η,
(B e (r | x, x0 ) , x0 , e)
equating the results using the
surplus-splitting condition (2.18), and rearranging terms completes the proof.
57
instead of
D
Data
I use the basic monthly CPS data from 1994 to 2013, available from the NBER's website, maintained by
39 Although earlier data are available, the CPS underwent a redesign in 1994, and it's known
Jean Roth.
that this change altered the measured distribution of unemployment durations. The CPS is a rotating panel,
where the same household is interviewed for 4 months, not interviewed for 8 months, and then interviewed
for 4 months. Questions include labor force status and, if unemployed, the duration of search. The goal is
to identify unemployed people who have been unemployed for a given duration, and we want to know what
fraction of these people found jobs by the following interview date.
structure of the CPS, it's necessary to match people across months.
However, to exploit the longitudinal
To do this, I use a method similar
to the one used by Shimer [2008]. I merge the monthly les on the basis of household identier (hrhhid),
sample identier (hrsample), serial sux (hrsersuf ), household number (hrhhnum), and person line number
(pulineno). When I nd observations that match across consecutive months, I drop observations in which
the person's race, sex, or indication of Hispanic origin changes.
changes by two or more years between months.
I also drop observations in which age
The unemployment duration data are heavily imputed,
except for workers in their rst or fth months in sample. To minimize the eects of imputation, I conne
my sample to workers who were unemployed in their rst or fth interview month.
Ultimately, I obtain
194,715 observations where a worker was unemployed in one month, and I can observe the same worker in
the following month.
In Section 7.1, I explore the data by tting a semiparametric regression, as specied by equation (7.1).
The non-linear component
h (·) is assumed to be locally cubic.
The covariates in
xi include race, sex, Hispanic
origin, reason for unemployment, an indicator for being a high-school dropout, an indicator for being a highschool graduate, an indicator for whether the worker was seeking part-time employment, the state-level
unemployment rate, and a quadratic function of age.
The sample excludes people on temporary layo
because many of them are recalled by their previous employers (Fujita and Moscarini [2013]). I also conne
the sample to workers ages 18 to 64. With these restrictions, there are 155,638 observations in the sample;
of these, 13% indicate that they are seeking part-time employment.
Potential labor-market experience is
dened as age minus years of education. Because the CPS asks people about their level of schooling, not
their years of schooling, this requires imputing a certain number of years for each level of schooling. I impute
that those who did not reach 9th grade received 13 years of education; those who reached a maximum of
9th grade (10th grade, 11th grade) received 14 years (15 years, 16 years) of education; those who reached
12th grade but did not graduate from high school received 17 years of education; those who attended some
39 See:
http://www.nber.org/data/cps_basic.html.
58
college but did not graduate received 19 years of education; and those whose highest degree is a high-school
diploma (associate's degree, bachelor's degree, master's degree, professional degree, doctorate) received 18
years (20 years, 22 years, 23 years, 24 years, 26 years) of education.
59