The Hazards of Unemployment: A Macroeconomic Model of Job

Search and Résumé Dynamics

Ross Doppelt∗

August 14, 2016

Abstract

I introduce a dynamic general-equilibrium model to investigate the relationship between the duration

of unemployment and the probability of nding a job. Specically, I analyze the hypothesis that a long

unemployment spell sends a negative signal about a worker's quality, thereby aecting her probability

of being hired. In the model, skills are unobservable. I refer to a worker's posterior probability of

being highly skilled, conditional on her labor-market history, as the worker's résumé. Spending time in

unemployment damages a worker's résumé, which alters her job-nding rate: Because high-skill workers

are more likely to form matches, prospective employers infer that unmatched workers are less likely to

be highly skilled. The match surplus incorporates the fact that hiring a worker improves her résumé,

and the theory illustrates how the résumé value of being hired is priced into wages. I calibrate the 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: ross.doppelt@psu.edu. This paper previously circulatedunder the shorter title, The Hazards of Unemployment. Please download the most recent version on my website:https://sites.google.com/site/rossdoppelt/. This work is based on my doctoral thesis at New York University; I am grate-ful to my thesis advisor, Tom Sargent, and committee members, Ricardo Lagos and Gianluca Violante, for their feedback andcriticisms. I also beneted from discussions with many colleagues and numerous seminar participants. Any imperfections arepurely my own.

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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: Firms are less 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 stigma

eects, the notion that the length of an unemployment spell sends a signal about worker quality. A job

seeker may fail to be hired because no rms saw her application, or because she was rejected by all of the

rms that did see her application. Prospective employers cannot observe an applicant's past rejections, but

rms will draw inferences from the duration of unemployment. If good workers are more likely to nd jobs,

then there is reason to think that someone who fails to nd a job 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

quality. 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 labor-

market 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. Search markets are

1Data details are in Appendix D.

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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 résumé damage 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.

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. 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

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

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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 prob-

ability 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.2 The 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 learning during unemployment 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 a minority of 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

2Throughout, when discussing the role of composition eects in shaping the aggregate job-nding hazard, I will use theterms dynamic selection and unobserved heterogeneity interchangeably. I will reserve the term duration dependence torefer to actual changes in an individual's job-nding rate over the course of an unemployment spell.

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unemployment spell, beliefs about a worker's skill 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, informational dynamics are necessary for the calibrated model's ability to ac-

count 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 fea-

tures 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 explanation for negative duration dependence, and the

model can be extended to incorporate skill loss during unemployment and skill gain during employment.

Relative to the baseline model, this modication can actually attenuate the drop in job-nding rates. When

skills change over time, information about an unemployed worker's current skill level is less valuable. Ulti-

mately, 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 résumés, so individual

hazards become atter.

Finally, the theory 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.3 Consequently, older workers have less information revealed about them during an unem-

ployment spell. I document that the correlation between durations and job-nding rates is, in fact, weaker for

more experienced workers. That being said, dierent age groups will have dierent amounts of heterogeneity

in job-nding rates, as well as dierent amounts of true duration dependence. 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 obtain identication of the proportional hazard model, the theory places

3In the model, the résumé distribution will never degenerate because workers die and are replaced by new-born agents, whoseskills are unknown.

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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 learning during unemployment as a source of duration dependence from a theo-

retical macroeconomic perspective. The most recent and closely related examples are Jarosch and Pilossoph

[2016] and Fernandez-Blanco and Preugschat [2015]; earlier contributions include Acemo§lu [1995] and Lock-

wood [1991].4 Those models unanimously conclude that learning during unemployment leads to negative

duration dependence, although the calibration exercises by Jarosch and Pilossoph [2016] and Fernandez-

Blanco and Preugschat [2015] 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 and this modeling choice is crucial for the way agents assess the

value of information. 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

reenters unemployment.5 Such simplications allow authors to make progress on other fronts. However,

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. Consequently, employees accept lower wages in exchange for the résumé value

of being hired. In that respect, the model relates to the literature on career concerns, initiated by Holmström

[1999]. That line of work typically abstracts from unemployment in order to focus on eort provision.6

Instead, I show how preventing the résumé damage from unemployment can contribute signicantly to the

surplus generated by forming a match.

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

4Jarosch and Pilossoph [2016] propose a model with two-sided heterogeneity and positive assortative matching betweenworkers and rms. All workers are assumed to meet rms at a constant and exogenous rate, but as a worker's search dura-tion increases, fewer rms are willing to pay the cost of screening her. Fernandez-Blanco and Preugschat [2015] construct acontract-posting model in which rms oer wages contingent on unemployment duration. Acemo§lu [1995] combines informationasymmetry with human capital depreciation by positing that workers have to exert eort, which is unobservable, to keep theirskills from decaying during unemployment. 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.

5Mathematically, these are almost identical. Lockwood [1991] and Acemo§lu [1995] make the former assumption; Jaroschand Pilossoph [2016] and Fernandez-Blanco and Preugschat [2015] make the latter.

6Gibbons and Murphy [1992], for instance, say that career concerns arise whenever the (internal or external) labor marketuses a worker's current output to update its belief about the worker's ability and then bases future wages on these updatedbeliefs (p. 469). The present model highlights the fact that the labor market updates its belief about the worker's ability onemployment status, as well as output, and these updated beliefs aect job-nding rates, as well as wages.

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upward-sloping hazards. Gonzalez and Shi construct a directed-search model in which workers learn about

their ability while searching for a job.7 Qualitatively, those authors tell a story about discouragement

during unemployment: As spells drag on, workers infer that their 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-sloping individual hazards.8 Nevertheless, on a

theoretical level, their result demonstrates that negative duration dependence is not a trivial outcome when

unemployment duration provides a signal of worker quality. 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 as

information about those skills. One novelty of my model is the analysis of how skill change interacts with

informational dynamics. 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.9 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.10

1.3 Relation to the Empirical Literature

This model speaks to the large empirical literature that tries to measure duration dependence using microe-

conomic 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.11 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

7In Gonzalez and Shi's model, skill corresponds to the probability that a worker can form a productive match, conditionalon meeting a rm. They assume that, conditional on being employed, all workers produce the same amount of output.

8To be fair, much of this evidence came out after Gonzalez and Shi [2010] was published. Also, because of dynamic selectioneects, Gonzalez and Shi's model can generate a negative correlation between job-nding rates and unemployment durationsin the aggregate, even though true duration dependence is positive.

9Seminal papers along these lines include Altonji and Pierret [2001] and Gibbons and Katz [1991].10Morchio [2016], who cites an earlier version of this paper, adopts the rhetorical device of referring to a worker's expected

skill level as her résumé. However, he focuses on learning from on-the-job productivity and separations throughout a worker'scareer. He abstracts from learning during unemployment, which is my main focus.

11See Machin and Manning [1999] and Van den Berg [2001] for reviews of the labor and econometrics literatures.

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theory. Nevertheless, there exists good evidence for negative duration dependence.

Recently, authors have used eld experiments to circumvent the identication issues inherent in using

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. 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. 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.12 Those authors interpret their results as a sign of learning: 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 news media has documented that some rms advertise vacancies

while stating an explicit preference for applicants who are currently or recently employed.13 These stories

conjecture that businesses expect the long-term unemployed to be lower quality workers. 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 help-wanted ads that specically encourage applications from

the unemployed and underemployed.14 By welcoming unemployed applicants, some rms are likely trying

to nd people who can be recruited easily or paid cheaply. This demonstrates that rms target dierent

types of workers, with dierent histories of joblessness. Besides current unemployment status, 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. [2014] provide evidence that workers apply for

lower-skill jobs as their unemployment spells grow longer. Exploiting longitudinal data from a job-posting

website, 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 use unemployment duration to learn about worker quality.15

I will proceed as follows. Section 2 describes the model environment. Section 3 denes an equilibrium

12Using U.S. data, Imbens and Lynch [2006] also nd that hazard curves are steeper in tight labor markets, although Biewenand Stees [2010] nd no such correlation in German data. Both of these studies use observational data, so they are subject tothe same qualications mentioned above.

13See, e.g., Rampell [2011] and Banjo [2012].14Searching for the words unemployed or unemployment on a job-search website will turn up many such postings.15Kudlyak et al. [2014] cite Gonzalez and Shi [2010] as one theory consistent with their results; my model is also consistent

with these ndings.

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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 eects. 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

The following subsections delineate: (1) the agents who inhabit the model, (2) the distinction between a

worker's skill level and her productivity, (3) the information structure of the economy, and (4) the details of

the matching process. Figures 2.1 and 2.2 show how these pieces t together. More specically, Figure 2.1

shows the timing of events for unemployed workers, and Figure 2.2 shows the timing of events for workers

in existing jobs. After describing technology, information, and matching, I present the agents' Bellman

equations and discuss wage determination.

2.1 Preferences and Demographics

There is a unit mass of risk-neutral workers. During periods of unemployment, workers get leisure value λ.

A worker also has a constant probability µ of dying from one period to the next. Each period, a measure µ

of workers die and are replaced by a measure µ of new workers. Agents discount the future at rate ρ; hence,

the eective discount factor is β ≡ 1−µ1+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. A worker's skill level is

not the same as her productivity level; rather, high-skill workers take random productivity draws from a

more favorable distribution. A worker's output is given by zx, where z is aggregate productivity, and x is

match-specic. Match-specic productivity x evolves according to a Markov process that depends on the

worker's type. In particular, x has support X ≡ 0, x1, . . . , xnx, and in all states of the world, the process

governing x for high-skill workers stochastically dominates that for low-skill workers. Upon encountering a

rm, a high-skill worker draws a value of match-specic productivity from a distribution with probability

mass function Ω (x | 1); for low-skill workers, this probability mass function is Ω (x | 0). For the duration of

a job, the probability of a high-skill worker transitioning from x to x′ is given by Ω (x, x′ | 1); for low-skill

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workers, this probability is given by Ω (x, x′ | 0). Conditional on a worker's type, x is independent across

matches: When a worker starts a new job, she takes a fresh draw from Ω (· | 0) or Ω (· | 1), independent

of her productivity on her last job. The distinction between high- and low-skill workers is captured by the

stochastic-dominance assumptions:

∑x≤X

Ω (x | 1) <∑x≤X

Ω (x | 0) . (2.1)

∑x′≤X

Ω (x, x′ | 1) <∑x′≤X

Ω (x, x′ | 0) . (2.2)

I will assume that there is always positive probability of realizing x = 0, which is an absorbing state.16 The

fact that 0 < Ω (0 | 1) < Ω (0 | 0) 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.

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 incom-

plete, it is symmetric: Workers are also learning about their skills through the progression of their careers.17

Notice that public information includes realizations of x in jobs where a worker has been employed not

realizations of x that result in an unemployed worker's application being rejected. Also, if a worker holds a

job with productivity x, and a transition to x′ induces her to enter unemployment, then the realization x′ 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

16To be clear, x dropping to zero does not make a worker permanently unproductive; it just makes her permanently unpro-ductive at her current job.

17It's fair to ask whether an employee's on-the-job productivity is publicly observable. The evidence from empirical microstudies is somewhat mixed: Schönberg [2007] and Kim and Usui [2013] suggest that this assumption is realistic for at least someworkers, though Kahn [2013] suggests that it is not. Nevertheless, the employer-learning literature commonly assumes thatsomeone's idiosyncratic productivity is observable to the entire marketplace, so current and prospective employers are using thesame noisy signals to form inferences about a worker's skill. For example, Altonji and Pierret [2001], Lange [2007], and Kahnand Lange [2014] take this approach.

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is drawn from an exogenous distribution F (·). I will maintain the assumption that, amongst the workers

born with prior probability r of being highly skilled, a fraction r is, in fact, highly skilled.

Denition. A worker's 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

someone's qualications upon entering the labor market, based on features such as academic performance.18

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 x.

3. An employed worker's productivity can transition from x to x′, possibly resulting in separation.

Each of these transitions sends a dierent signal, so there are three belief-updating functions. If an unem-

ployed worker with résumé r fails to nd a job, then her updated résumé is denoted Bu (r). If an unemployed

worker with résumé r is hired with productivity x, then her updated résumé is denoted Bh (r | x). If an

employed worker has résumé r and her productivity transitions from x to x′, then her updated résumé is

denoted Be (r | x, x′).19 Concisely:

Bu (r) ≡ P [highly skilled | unemployed→ unemployed, résumé r] (2.3)

Bh (r | x) ≡ P [highly skilled | unemployed→ x, résumé r] (2.4)

Be (r | x, x′) ≡ P [highly skilled | x→ x′, résumé r] . (2.5)

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 Section 2.9, after I have

described the matching process. When appropriate, I will denote the inverses of these functions by lowercase

letters; e.g., the function bh (r | x) satises r = Bh(bh (r | x) | x

).20

18If employers make inferences about worker quality on the basis of time-invariant characteristics, such as race or sex, thesefeatures would also be subsumed into the prior.

19Recall that a transition from x to x′ may or may not result in the continuation of a match. Regardless, beliefs will beupdated according to Be (r | x, x′) because the transition from x to x′ is assumed to be observable.

20It will become apparent in Section 2.9 that Bh (· | x) and Be (· | x, x′) are guaranteed to be invertible in their rst arguments.In the numerical exercises, Bu (·) is invertible as well.

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Figure 2.1: Timing for Unemployed Workers

It will be convenient to dene for r ∈ [0, 1]:

Ω (x | r) ≡ rΩ (x | 1) + (1− r) Ω (x | 0) . (2.6)

Ω (x, x′ | r) ≡ rΩ (x, x′ | 1) + (1− r) Ω (x, x′ | 0) . (2.7)

Notice that each of the above quantities is a conditional probability. In particular, Ω (x | r) is the posterior

predictive distribution of productivity for a newly hired worker with résumé r. Likewise, Ω (x, x′ | r) is the

probability of productivity transitioning from x to x′ for an employee with résumé r.

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,

12

Figure 2.2: Timing for Employed Workers

and they begin production in the following period. The timing of the matching process, depicted in Figure

2.1, ensures that information between employers and employees will be symmetric: A worker will only be

contacted by a rm if her realization of x makes for a viable match, so a worker who remains unemployed

never nds out if she had a bad x 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 are segmented in order to make the solution tractable.21 Without

segmented search markets, a rm posting a vacancy would have to know the full distribution of résumés in

the unemployed population. The state space is reduced dramatically when all workers within a given market

have the same résumé.

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 θ (r) the market tightness, or the ratio of vacancies to unemployed

workers, in market r. If a worker is searching in market r, then the probability of her application being seen

21Elsewhere, in the directed-search literature, authors focus on the endogenous segmentation of markets. Examples includeGonzalez and Shi [2010] and Menzio and Shi [2010, 2011]. In addition to delivering tractability, the search protocol in thepresent model is clearly a normal-form Nash equilibrium: Workers with résumé r play the strategy of searching in marketr, and rms posting vacancies in market r play a strategy of considering only those applicants who have résumé r. Giventhe recruitment strategies of rms, there's no reason for workers to deviate from their search strategies, and given the searchstrategies of workers, there's no reason for rms to deviate from their recruitment strategies.

13

by a rm is:

p (r) ≡ minζθ (r)

1−ε, 1. (2.8)

The probability of a rm encountering a worker in market r is:

q (r) ≡ minζθ (r)

−ε, 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 s ∈ e, h

denote a worker's tenure status: 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∑x′

Ω (x, x′ | r) max Ge (r′, x′) , 0 (2.10)

s.t.: r′ = Be (r | x, x′) .

Let V (r) be the value associated with posting a vacancy in market r. For a potential rm, there is a constant

cost κ 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

Ω (x | r) max Gh (r′, x) , V (r)

)(2.11)

s.t.: r′ = Bh (r | x) .

Free entry requires that:

V (r) = 0, ∀r. (2.12)

Hence, the value of posting a vacancy in any market must satisfy:

κ = βq (r)∑x

Ω (x | r) maxGh(Bh (r | x) , x

), 0. (2.13)

14

2.6 The Worker's Problem

Let Hs (r, x) be the value associated with being an employed worker in state (r, x, s); let U (r) be the value

associated with being an unemployed worker with résumé r. The employed worker's Bellman equation is:

Hs (r, x) = ws (r, x) + β∑x′

Ω (x, x′ | r) max He (r′, x′) , U (r′) (2.14)

s.t.: r′ = Be (r | x, x′) .

The unemployed worker's Bellman equation is:

U (r) = λ+ β

([1− p (r)]U (r′u) + p (r)

∑x

Ω (x | r) max Hh (r′h, x) , U (r′u)

)(2.15)

s.t.: r′h = Bh (r | x) and r′u = Bu (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) and continuing employees (s = e).

First, consider an unemployed worker searching in market r. If she is not hired, then her résumé is updated

to Bu (r), and she receives value U (Bu (r)). If she is hired with productivity x, then her résumé is updated to

Bh (r | x), and she receives value Hh

(Bh (r | x) , x

). The worker's surplus is therefore Hh

(Bh (r | x) , x

)−

U (Bu (r)). A rm that successfully hires a worker in market r gets value Gh(Bh (r | x) , x

); free entry

ensures that rm owners get nothing if they fail to make a hire. Hence, the surplus-splitting condition for

newly hired workers is:

Hh

(Bh (r | x) , x

)− U (Bu (r)) = η

[Gh(Bh (r | x) , x

)+Hh

(Bh (r | x) , x

)− U (Bu (r))

]. (2.16)

The left-hand side of the above is the worker's surplus, and the term in square brackets on the right-hand

side is the combined surplus of the worker and the rm.

Now, consider a worker in an existing employment relationship who, in the previous period, produced

x− with résumé r−. If productivity transitions to x, then the worker's résumé becomes r = Be (r− | x−, x).

Because the complete history of x is assumed to be observable in any consummated match, the worker's

résumé is unaected by staying on the job for an additional period. Consequently, the worker gets value

15

He (r, x) from continuing employment, and she would get value U (r) if she entered unemployment. The

rm's value from maintaining the match is Ge (r, x), and the rm's value from dissolving the match is zero.

Hence, the surplus-splitting condition for continuing employees is:

He (r, x)− U (r) = η [Ge (r, x) +He (r, x)− U (r)] . (2.17)

Again, the left-hand side of the above is the worker's surplus, and the term in square brackets on the

right-hand side is the combined surplus of the worker and the rm.

More succinctly, we can evaluate (2.16) at résumé bh (r | x) and consolidate it with equation (2.17) to

obtain:

ηGs (r, x) = (1− η) [Hs (r, x)− U (rc)] (2.18)

rc ≡

Bu(bh (r | x)

)if s = h

r if s = e.

In the above, rc 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. The former is

associated with Nash [1950]; the latter, with Rubinstein [1982]. Some authors, such as Pissarides [1994],

directly assert a surplus-splitting rule as a primitive assumption, without invoking any specic game-theoretic

foundations.22 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 non-trivial one that falls outside the scope of

the current endeavor.23

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,

22In the environment developed by Pissarides [1994], the space of payos is non-convex, which creates complications for eitherNash [1950] or Rubinstein [1982] bargaining. See Shimer [2006] for a discussion.

23Firms have no reason to contact workers unless they expect to reach a bargain, so information would continue to be sym-metric in an equilibrium. Nevertheless, solving the non-cooperative bargaining game requires computing the payos associatedwith deviations from equilibrium outcomes, including deviations that lead to information asymmetries.

16

along with utility oors that agents obtain without cooperating. Equation (2.16) species that an unemployed

worker's payo from not forming a match is U (Bu (r)), the value of continuing in 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

to determine a fair bargain and connects the trading parties. An unemployed worker would want to be

contacted with any contract worth more than U (Bu (r)), and the rm would want to be contacted with any

contract worth more than zero. Those conditions will both be satised 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:

Xh (r) =x ∈ X | Gh (r′h, x) +Hh (r′h, x) ≥ U (r′u) , r′h = Bh (r | x) , r′u = Bu (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 r and

current productivity x is:

Xe (r | x) = x′ ∈ X | Ge (r′e, x′) +He (r′e, x

′) ≥ U (r′e) , r′e = Be (r | x, x′) . (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:

Bu (r) =1− p (r)

∑x∈Xh(r) Ω (x | 1)

1− p (r)∑x∈Xh(r) Ω (x | r)

r (2.21)

Bh (r | x) =Ω (x | 1)

Ω (x | r)r (2.22)

Be (r | x, x′) =Ω (x, x′ | 1)

Ω (x, x′ | r)r. (2.23)

17

Notice that the amount of information revealed by an unemployment spell is endogenous, because Bu (r)

depends on the equilibrium job-nding rate. All workers searching in market r, regardless of their actual

skill level, have the same probability p (r) of being seen and screened by a rm. A genuinely high-skill worker

with résumé r has a job-nding rate of p (r)∑x∈Xh(r) Ω (x | 1), whereas the job-nding rate for the average

worker with résumé r is p (r)∑x∈Xh(r) Ω (x | r). Provided that Xh (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

low-skill workers with the same résumé, because Ω (· | 1) stochastically dominates Ω (· | 0). This feature has

two implications. First, Bu (r) < r, 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 Xh (r) constant, Bu (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 Gs (r, x), V (r), Hs (r, x), and U (r)

2. Policies Xh (r) and Xe (r | x)

3. Contact rates p (r) and q (r)

4. A market-tightness function θ (r)

5. A wage function ws (r, x)

6. Belief functions Bu (r), Bh (r | x), and Be (r | x, x′)

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).

18

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.24 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

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

[1991] and Acemo§lu [1995], there can be multiple equilibria.25 In those models, as in mine, the probability

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 those 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 aggregate statistics, it's necessary to know this distribution,

which is determined endogenously. Let F e (r | x) be the measure of employed workers in continuing employ-

ment relationships with productivity x whose résumé is less than or equal to r. Let Fh (r | x) be the measure

24Shi [2009] refers to this property as block recursivity.25Like me, Gonzalez and Shi [2010], Fernandez-Blanco and Preugschat [2015], and Jarosch and Pilossoph [2016] cannot

provide armative proof of uniqueness, but they cannot nd instances of more than one equilibrium, either.

19

of newly hired workers with productivity x whose résumé is less than or equal to r. Let Fu (r) be the mea-

sure of unemployed workers whose résumé is less than or equal to r. Keep in mind that these functions are

measures, though not proper probability distributions; i.e., Fu (1) +∑x F

h (1 | x) + F e (1 | x) = 1. Recall

that dF (r) is the measure of workers born with résumé r, and a fraction r of these dF (r) 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.

I will dene a functional operator that characterizes the law of motion for these measures. We want an

operator that maps(F e, Fh, Fu

)today into

(F e, Fh, Fu

)′tomorrow, given the components of the equilibrium

dened in Section 3.1. Let Fe, Fh, and Fu be the respective spaces in which these measures must reside.

Dene the mapping Υ : Fe ×Fh ×Fu → Fe ×Fh ×Fu by:

Υ1

(F e, Fh, Fu

)(r′, x′) = (1− µ)

∑x

ˆBe(r′|x,x′)

Ω (x, x′ | r) d[F e (r | x) + Fh (r | x)

](3.1)

Υ2

(F e, Fh, Fu

)(r′, x′) = (1− µ)

ˆBh(r′|x′)

p (r) Ω (x′ | r) dFu (r) (3.2)

Υ3

(F e, Fh, Fu

)(r′) = (1− µ)

∑x,x′

ˆBd(r′|x,x′)

Ω (x, x′ | r) d[F e (r | x) + Fh (r | x)

]

+ (1− µ)

ˆBu(r′)

1− p (r)∑

x∈Xh(r)

Ω (x | r)

dFu (r) + µF (r′) , (3.3)

where I have dened:

Be (r′ | x, x′) ≡ [0, be (r′ | x, x′)] ∩ r | x′ ∈ Xe (r | x) (3.4)

Bh (r′ | x) ≡[0, bh (r′ | x)

]∩r | x′ ∈ Xh (r)

(3.5)

Bd (r′ | x, x′) ≡ [0, be (r′ | x, x′)] ∩ r | x /∈ Xe (r | x) (3.6)

Bu (r′) ≡ r | Bu (r) ≤ r′ . (3.7)

The measures(F e, Fh, Fu

)then evolve according to:

(F e, Fh, Fu

)′= Υ

(F e, Fh, Fu

). (3.8)

To understand the logic of (3.1), notice that dF e (r | x) + dFh (r | x) is the measure of employed workers

with résumé r and productivity x. Of these workers, a fraction (1− µ) Ω (x, x′ | r) will survive to the

following period with productivity x′ and résumé Be (r | x, x′). These workers will continue in their jobs

if x′ ∈ Xe (r | x), and their updated résumés will be weakly less than r′ if r ∈ [0, be (r′ | x, x′)]. Thus,

20

integrating (1− µ) Ω (x, x′ | r) with respect to F e (r | x) + Fh (r | x) over r ∈ Be (r′ | x, x′) and summing

over x yields F e (r′ | x′), the measure of workers in continuing jobs with productivity x′ and résumé less

than or equal to r′. Concisely, (F e)′

= Υ1

(F e, Fh, Fu

). Similar reasoning accounts for the components of

(3.2) and (3.3). Aggregate employment is given by:

e ≡∑x

[F e (1 | x) + Fh (1 | x)

]. (3.9)

Likewise, aggregate output is given by:

y ≡ z∑x

x[F e (1 | x) + Fh (1 | x)

]. (3.10)

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 = .0452 ,

corresponding to an annual real interest rate of approximately four percent. The probability of an agent

dying from one period to the next is 140×52 , implying that the average working life lasts forty years. Workers

and rms have equal surplus shares, with η = 12 . Aggregate productivity is normalized as z = 1. The ow

value of leisure is λ = 12 . 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

ε = 12 . The matching function's scaling factor ζ will be chosen to match the steady-state unemployment rate

using a procedure I will describe below. Hall and Milgrom [2008] 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 x is X = 0, x1, . . . , xnx.

I will assume that log (x1) , . . . , log (xnx) constitute a 21-point, equally spaced grid from -1 to 1. There

are two stochastic processes for x, one for each skill type. The parameters governing the evolution of x for

high-skill workers are indexed by i = 1, and the parameters governing the evolution of x for low-skill workers

are indexed by i = 0.

21

For workers in continuing employment relationships, I will assume that log (x) follows a discrete approx-

imation to a random walk with drift, the parameters of which depend on the worker's skill level. There is

a constant, type-specic probability δi of productivity dropping to zero. Conditional on remaining positive,

x has a constant, type-specic probability αi of taking a step up; there is also a constant, type-specic

probability χi of x taking a step down. That is, for 1 < k < nx:

Ω (xk, x′ | i) =

(1− δi)αi if x′ = xk+1

(1− δi) (1− αi − χi) if x′ = xk

(1− δi)χi if x′ = xk−1

δi if x′ = 0.

(4.1)

At the endpoints of the positive support (i.e. x = x1 and x = xnx) productivity can only move in one

direction, so I will specify:

Ω (x1, x′ | i) =

(1− δi)αi if x′ = x2

(1− δi) (1− αi) if x′ = x1

δi if x′ = 0

Ω (xnx , x′ | i) =

(1− δi) (1− χi) if x′ = xnx

(1− δi)χi if x′ = xnx−1

δi if x′ = 0.

(4.2)

The stochastic dominance of Ω (x, x′ | 1) over Ω (x, x′ | 0) is captured by the parameter restrictions α1 >

α0, χ0 > χ1, and δ0 > δ1. In other words, high-skill workers have a higher probability of experiencing

productivity gains and a lower probability of experiencing productivity losses.

A diculty in choosing specic values for αi, χi, δii=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:

E[log

(x′

x

)| x1 < x < xnx

, 0 < x′, i

]= (αi − χi) dx. (4.3)

V[log

(x′

x

)| x1 < x < xnx , 0 < x′, i

]=

[(αi + χi)− (αi − χi)2

]d2x, (4.4)

22

where dx ≡ log (xnx/x1) / (nx − 1) 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, χii=0,1 will be motivated by existing estimates for wage and earnings processes. I will set the conditional

mean of log(x′

x

)to .20

52×10 for high-skill workers and .1052×10 for low-skill workers. So, over ten years, match-

specic productivity is expected to grow by about 20% for a high-skill worker and by 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

ten-year growth rate of 6.6%, whereas Topel [1991] nds a ten-year growth rate in excess of 25%.26 For both

skill types, I will set the variance of log(x′

x

)to .0313

52 . The justication for this choice comes from Meghir

and Pistaferri [2004], who estimate the variance of permanent innovations to log earnings; they obtain an

unconditional variance of .0313 when using annual data pooled across dierent education groups. I will set

δ0 = 152×2.25 and δ1 = 1

52×2.75 . These values imply that the average low-skill worker lasts 2.25 years before

suering a terminal productivity shock, whereas the average high-skill worker lasts 2.75 years.27 As a point

of reference, Shimer [2005] calibrates separation rates so that the average job lasts 2.5 years.

Whereas x follows a random walk with drift over the course of a job, we need to specify how x is initialized

when a match is formed. For newly hired employees, I will assume that there is a type-specic probability

ωi that x = 1; with probability 1 − ωi, x = 0. That is, the initial value of match-specic productivity has

the following distribution:

Ω (x | i) =

ωi if x = 1

1− ωi if x = 0.

(4.5)

Because x = 0 is an absorbing state, we can interpret ωi as the probability of a worker being able to do a job,

conditional on her true type. For Ω (x | 1) to stochastically dominate Ω (x | 0), it's necessary and sucient

that ω1 > ω0. 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 ω1 and ω0; that is, high-

skill workers with résumé r nd jobs with probability ω1p (r), whereas low skill workers with résumé r nd

jobs with probability ω0p (r). The parameters ω0 and ω1 will be important for determining the informational

26For a more recent review of the literature and evidence, see Altonji and Williams [2005].27The model features endogenous separation because x can gradually drift downward to the point that a match is no longer

desirable. Strictly speaking, δi is not a type-specic separation rate, but a lower bound on the separation rate. In the numericalexercises I perform, the majority of separations will come from x dropping to zero immediately, as opposed to declining gradually.

23

content of unemployment spells because the belief-updating function is:

Bu (r) =1− ω1p (r)

1− ωrp (r)r, (4.6)

where I have dened ωr ≡ (1− r)ω0 + rω1. For a given meeting rate p (r), the dierence between ω1 and ω0

governs how quickly a worker's résumé is updated while she is searching for a new job: Bu (r) is decreasing

in ω1 − ω0, and Bu (r) = r when ω1 = ω0.

It remains to to pick numerical values for ω1, ω0, the matching eciency ζ, and the résumé distribution

for newly born agents F (r). 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 F (r) is Beta(αF , 10− αF ). This implies that

the fraction of high-skill workers in the economy is αF

10 . I will also adopt the normalization that ω0 +ω1 = 1.

This normalization comes with minimal loss of generality: When p (r) is strictly less than one, doubling ζ is

observationally equivalent to cutting both ω0 and ω1 in half. What matters is the dierence ω1 − ω0. I will

conduct a grid search over values of ω1 − ω0 and αF 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.28 I exclude workers who report being on layo as their reason for unemployment because

Fujita and Moscarini [2015] 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. The candidate values of ω1 − ω0

consist of an equally-spaced grid of 51 points between zero and 12 ; the candidate values of αF consist of an

equally-spaced grid of 10 points between 1.5 and 8.5. For each pair (ω1 − ω0, αF ), I nd a value for ζ that

results in a 6% steady-state unemployment rate.29 This procedure points to parameter values of ω0 = .40,

ω1 = .60, αF = 4.0, and 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

28More precisely, the criterion being minimized takes the form (m−m0)′W (m−m0). Here, m is a 52× 1 vector, the dth

element of which is the fraction of workers reporting an unemployment duration of d weeks in their rst CPS survey and whoreport being employed in their second CPS survey. When computing these fractions, workers are weighted using the CPS nalweights. The vector m0 is the probability of a worker with an unemployment duration of d weeks in the model nding workin the subsequent four weeks. The weighting matrix W is diagonal, the dth element of which is proportional to the numberof unemployed workers in the CPS who report an unemployment duration of d weeks. Details on the data are available inAppendix D.

29Numerically, it appears that the steady-state unemployment rate is monotonic in ζ, so nding a value that satised thiscondition was accomplished using a bisection method.

24

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. Figure A.3 shows market tightness as a function

of a worker's résumé. 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

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

Bu (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.

Figure A.4 shows the résumé distribution for all workers, employed workers, unemployed workers, and

new entrants into the labor force. The employed tend to have better résumés than the unemployed, reecting

the fact that high-skill workers are more likely to form and maintain matches. Figure A.4 also shows the

eects of learning: Relative to the résumé distribution of new entrants, the population distribution of résumés

has a lot of mass close to r = 0 and r = 1. Experienced workers have less uncertainty about their types,

because their career outcomes send signals to the market about their underlying abilities.

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.

25

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.

Proposition 1. In a recursive search equilibrium, wages are given by the following function of the worker's

résumé and productivity:

ws (r, x) =

Static Nash︷ ︸︸ ︷ηzx+ (1− η)λ+

Search Wedge︷ ︸︸ ︷ηκθ (rc) −

Information Wedge︷ ︸︸ ︷(1− η)βE [U (r′e)− U (r′u) | r, x], (5.1)

where the expectation is taken over x′, and where I have dened:

rc ≡

Bu(bh (r | x)

)if s = h

r if s = e

r′e ≡ Be (r | x, x′)

r′u ≡ Bu (rc) .

Proof. See Appendix C.

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], 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.

The nal term in equation (5.1) is − (1− η)βE [U (r′e)− U (r′u) | r, x]. I will call this term the information

wedge, because it represents the résumé 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 r′e tomorrow,

26

and her value of searching will be U (r′e). If she is unemployed today, then she will have counterfactual

résumé rc, and tomorrow she will be be unemployed with résumé r′u = Bu (rc). In that case, her value of

searching will be U (r′u). By reaching an agreement for one period of employment, the worker improves her

future search prospects by U (r′e)− U (r′u). The information wedge equals the expected discounted value of

U (r′e) − U (r′u), 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 in the form of lower

wages. In other words, wages include a compensating dierential for the résumé value of holding a job and

staying out of unemployment.

Figure A.5 shows the wages of newly hired workers, and Figure A.6 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 types are 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: U (r′e) = U (r′u) for people with r ∈ 0, 1. Unsurprisingly, high-skill workers (r = 1) get

paid more than low-skill workers (r = 0) 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 wages received by newly hired workers with r ∈ (0, 1). Figure A.7 shows U (r), the value of

unemployment, and Figure A.8 shows rc = Bu(bh (r | x)

), the counterfactual résumé a newly hired worker

would have if she were not employed, as a function of the worker's current résumé. The workers with the

largest information wedges are those for whom being hired sends the greatest signal (r− rc is large), and for

whom the signal is of the greatest value (the slope of U (r) is large).

Figures A.9 and A.10 show the wages and wedges of workers in continuing employment relationships.30

Wages are only slightly increasing in the worker's résumé, but earnings scale up almost linearly with produc-

tivity. 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.10 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,

30Figure A.10 show only a single line for the information wedge, even though equation (5.1) suggests that the informationwedge is a function of current-period productivity x. In fact, under the chosen specication for Ω (x, x′ | i), the informationwedge is identical for all workers whose productivity is in the interior of X : Everyone has the same probability of a one-incrementproductivity increase and a one-increment productivity decrease, so the distribution over r′e does not depend on x. There arerelatively few workers with the maximal productivity level xnx , and their information wedges are quantitatively very similar.In equilibrium, there are no workers with the minimal positive productivity x1 because there is no match surplus.

27

there are two strong eects that come close to canceling out: The workers who get the most résumé 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 résumé value of holding down a job, but new employees

also avoid the résumé damage from 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 résumé

improvement 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

splitting with linear utility, which implies that workers are not bothered by sudden swings in consumption.31

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 résumé value of being hired, for which workers are willing

to take a pay cut, is greater than the résumé 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 résumé-improvement value that rms extract from workers is

31See Burdett and Coles [2003] and Stevens [2004] for illustrations of how the curvature of the utility function aects wage-tenure proles.

28

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

Let's investigate the shapes of individuals' true job-nding hazards. Consider a worker who enters unem-

ployment with résumé r0. After t periods of search, she will have résumé (Bu)t(r0), where (Bu)

t(·) denotes

the function Bu (·) composed with itself t times. Consequently, her probability of being seen and screened

by a rm, as a function of unemployment duration, is p(

(Bu)t(r0)

). Thus, conditional on her true skill

level, her job-nding hazard is ωip(

(Bu)t(r0)

). This representation resembles a proportional hazard model:

The probability of nding a job equals a function of duration p(

(Bu)t(r0)

), times a scaling factor ωi that

depends on an individual worker's underlying type.32 Two things determine how the worker's job-nding

rate evolves 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.11 displays (Bu)t(r0) for several

initial initial résumés r0. Everyone's résumé declines in unemployment, but the drop is slower for people

with r0 close to zero or one. This pattern comes from two sources. First, workers with résumés close to 12

have the most prior uncertainty about their skill levels, so an additional observation carries more weight.33

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.12 shows p (r), the probability of being seen by a rm as a

function of résumé; the right panel shows ωrp (r), the average job-nding probability of workers searching in

market r. Both functions inherit the hump shape of the market-tightness function θ (r), shown in Figure A.3.

Figure A.13 displays p(

(Bu)t(r0)

)for various initial résumés r0. An individual's job-nding probability

is proportional (by a factor of ωi) to p(

(Bu)t(r0)

), 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,

32In Section 7, I'll discuss the theory's implications for the econometric implementation of proportional hazard models.33Conditional on having résumé r, a worker's skill level follows a Bernoulli distribution with parameter r; the variance of this

distribution is (1− r) r, which is maximized at r = 12.

29

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 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 cor-

relation 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 Pi-

lossoph [2016] propose looking at the average change in job-nding rates experienced by workers who have

been unemployed for t periods.34 Figure A.14 shows that this metric of true duration dependence exhibits

a 20% drop o over the course of a year of search. By contrast, Jarosch and Pilossoph [2016] 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.

Figure A.15 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.35 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.15 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

be the distribution implied by the model for workers entering unemployment.36 Despite the importance of

34That is, for each duration t, compute:

100׈

p (r)

p((bu)t (r)

)dFut (r) ,

where bu (·) is the inverse of Bu (·), (bu)t (·) is bu (·) composed with itself t times, and Fut (·) is the résumé distribution amongst

workers who have been unemployed for t consecutive periods. Hence, a worker who has a job-nding rate of ωip (r) in her tth

period of search had a job-nding rate of ωip((bu)t (r)

)at the outset of her unemployment spell.

35This 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:

Hazard Due to True Duration Dependence =

ˆ[rω1 + (1− r)ω0] p

((Bu)t (r)

)dFu

0 (r) ,

where Fu0 (·) is the distribution of résumés amongst workers entering unemployment.

36To be precise, let Fu0 (·) denote the distribution over résumés amongst workers entering unemployment. A measure rdFu

0 (r)

30

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 job-

nding rates across workers are endogenously determined economic outcomes. As evidenced by Figure A.12,

the dierence in job-nding rates between workers with known skills (i.e. r ∈ 0, 1) is small relative to the

dierence in job-nding rates between workers with unknown skills (i.e. r ∈ (0, 1)). 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. Suppose that a worker born with résumé r draws her productivity from the

distributions Ω (x | r) and Ω (x, x′ | r) for her entire lifetime, with certainty. Instead of representing a worker's

expected skill, the résumé in this context represents the worker's actual, observable skill level. As before,

there is a continuum of segmented search markets, indexed by r ∈ [0, 1], and a rm posting a vacancy chooses

to solicit applications from workers with a particular résumé r. 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,

job-nding rates are not subject to duration dependence. A worker born with résumé r will have résumé r

forever; consequently, a worker who begins searching in market r will continue searching in market r for the

duration of her spell. Second, there is no résumé value from employment. In equation (5.1), the information

wedge disappears because r′e = r′u = r. 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.16

compares the job-nding rate ωrp (r) in the full- and incomplete-information economies. By construction,

these functions align at r = 0 and r = 1, but for all r ∈ (0, 1), markets are tighter when information is

of newly unemployed workers has job-nding rate ω1p (r), and a measure (1− r) dFu0 (r) has job-nding rate ω0p (r). If everyone

had a constant probability of nding a job, then the fraction of workers with job-nding rate ωip (r) remaining in unemploymentafter t periods is [1− ωip (r)]t. The average hazard curve, holding all individual job-nding rates constant, is therefore:

Hazard Due to Heterogeneity =

´ [[1− ω1p (r)]t ω1p (r) r + [1− ω0p (r)]t ω0p (r) (1− r)

]dFu

0 (r)´ [[1− ω1p (r)]t r + [1− ω0p (r)]t (1− r)

]dFu

0 (r).

31

incomplete. 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.17 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 résumés. Although the results in Section 5.2 suggest that dynamic selection eects are important,

heterogeneity should not be interpreted as an alternative to informational dynamics when accounting for

the aggregate relationship between job-nding rates and search durations. Instead, the model suggests that

informational dynamics contribute to heterogeneity in job-nding rates.

5.4 Skill Decay and Alternative Sources of Duration Dependence

Thus far, informational dynamics have been the only source of duration dependence that I have considered,

but other explanations are worth exploring. Amongst the competing hypotheses, skill decay during unem-

ployment 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 unemployment. Naïvely, one might

think that résumé damage 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. In the environment I have

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. Suppose that, conditional on

being unemployed in period t, high-skill workers have probability τ of becoming low-skill workers in period

t+ 1. Likewise, conditional on being employed in period t, low-skill workers have probability γ of becoming

high-skill workers in period t+ 1. The worker's actual skill level remains unobservable. Mathematically, the

32

Table 1: Parametrizing Skill ChangeParametrization Prob. of Skill Gain On the Job (γ) Prob. of Skill Loss O the Job (τ)

Baseline 0 0SC1 0 1

26

SC2 12×52

126

SC3 126

126

only things that change are the belief-updating functions, which become:

Be (r | x, x′) = γ + (1− γ)Ω (x, x′ | 1)

Ω (x, x′ | r)r (5.2)

Bh (r | x) = (1− τ)Ω (x | 1)

Ω (x | r)r (5.3)

Bu (r) = (1− τ)1− p (r)

∑x∈Xh(r) Ω (x | 1)

1− p (r)∑x∈Xh(r) Ω (x | r)

r. (5.4)

Table 1 shows the parametrizations for γ and τ 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 = 126 , 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

during unemployment is twice as large as the rate of human-capital accumulation during employment.37

For each parametrization, Figure A.18 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

rates are lower in the economies with skill change. The drops in most workers' hazard curves are more acute

in the economy that has résumé damage 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 who

are most willing to give up wages in order to improve their résumés. Adding stochastic skill change to the

model diminishes the marginal value of résumé improvement. Recall that the information wedge in workers'

37In Ljungqvist and Sargent's model, human capital is synonymous with productivity. In my model, skill is a distributionover productivities, so the parameters of their model cannot be compared directly to mine.

33

wages is proportional to the expected value of U (r′e)−U (r′u), 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.19 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 informational dynamics 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.

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 applica-

tion is analyzing learning during unemployment, the quantitative results depend in part on the on-the-job

productivity dierences between employed workers, captured by Ω (x, x′ | 0) and Ω (x, x′ | 1). Although the

dierence between Ω (x, x′ | 0) and Ω (x, x′ | 1) plays a non-trivial role, the results are most sensitive to ω0

and ω1, which determine the fraction 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, x′ | 0) = Ω (x, x′ | 1). I set the conditional mean of log(x′

x

), given by equation (4.3), to .15

52×10 , and I set

34

δ0 = δ1 = 152×2.50 . 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 θ (r) is lower

and occurs when r = .69. Consequently, a larger fraction of newly unemployed workers, 32%, experience

positive duration dependence. A worker entering unemployment with the average résumé will see her job-

nding 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 informational content of unemployment are ω1 and ω0, which

are the respective probabilities of high- and low-skill workers being able to do a given job. If ω1 and ω0

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 ω1 − ω0, keepingω1+ω0

2 xed. Figure A.20 plots the average job-

nding rate ωrp (r), as a function of a worker's résumé. As ω1−ω0 goes up, the dierence in job-nding rates

between high- and low-skill workers (i.e. the dierence between ω1p (1) and ω0p (0)) becomes larger. More

importantly, though, the hump in ωrp (r) gets more pronounced. For each of these parameter congurations,

Figure A.21 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 ω1 − ω0 increases. The values

(ω0, ω1) = (.40, .60) used in the baseline calibration are the ones that come closest to replicating the data.

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. To see this, let ri,t denote the résumé of worker i at time t. Beliefs about a worker's

skill level follow a martingale: Et [ri,t+1] = ri,t. The variance of posterior beliefs is given by (1− ri,t) ri,t,

which is a supermartingale:

Et [(1− ri,t+1) ri,t+1] ≤ (1− Et [ri,t+1])Et [ri,t+1] = (1− ri,t) ri,t,

35

where the inequality comes from Jensen's inequality. Consequently, workers with a lot of labor-market

experience will have little uncertainty about their types. 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. 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) + x′iβ + εi. (7.1)

In the above, durationi is the number of weeks that worker i claims to have been looking for a job in the

rst survey month; h (·) 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 εi 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

distribution.38

Figure A.22 shows the estimated values of 100× h (d) /h (0) for each quartile of the potential-experience

distribution. 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.39 The decline is essentially the same for the second and third quartiles, but taken

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.23 shows the results when the sample

excludes workers who are seeking part-time employment. In that case, it looks clearer that the average

job-nding rates fall less with duration for the most experienced workers. As a point of comparison, Figure

A.24 shows the how the average job-nding rate changes with duration for dierent experience groups,

using articial data generated by the theoretical model.40 Quantitatively, it's unsurprising that the model-

generated lifecycle data do not align perfectly with the lifecycle data in the CPS; that's because the model

38More 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] forincorporating survey weights (in this case, the CPS nal weights).

39As an aside, younger workers tend to have higher job-nding rates than older workers. This would just represent avertical shift of each of the population hazards depicted in Figure A.22, which are normalized by the job-nding rates of newlyunemployed workers.

40In the theoretical model, age is geometrically distributed, so the quartiles in Figure A.24 correspond to the quartilesof the potential experience distribution from the CPS. Also, the model-generated job-nding rates shown in Figure A.24 arecomputed directly using p (r), rather than from estimating the semiparametric regression in equation (7.1).

36

has a very primitive treatment of age, with workers facing a constant probability of dying. Qualitatively,

though, the model-generated data do show that the correlation between duration and job-nding rates

declines somewhat with experience, although the magnitude diers from what we observe in the CPS. More

importantly, the prediction that highly experienced workers have at hazards is a general consequence of

Bayesian learning, not a byproduct of the way the theoretical model is calibrated. Of course, the regression

equation (7.1) can only capture correlations, and ideally, one would want to distinguish between true duration

dependence and dynamic selection eects. Nevertheless, these results are at least suggestive of informational

dynamics as a source of duration dependence.

7.2 Specication and Identication of Proportional Hazards Models

The theoretical results place restrictions on the proper use of the mixed proportional hazard model, which

is the most popular tool in the econometric literature. That specication asserts that worker i's probability

of nding a job after t periods of search is vihX (xi)hT (t), where vi is a mean-one idiosyncratic disturbance

and xi is a vector of person-specic predictors. In applications, the main object of interest is hT (t), 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 but not identied. To 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 ωip(

(Bu)t(r0)

), where r0 denotes the worker's résumé at the

outset of the spell.41 There is heterogeneity amongst job seekers in both ωi and r0, so idiosyncratic eects

cannot be multiplicatively separated from the time-varying portion of the hazard. Graphically, it's clear

that a proportional hazard model will not be correctly specied for a sample of workers with dierent initial

résumés: Figure A.13 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. In that case, the job-nding rate can be written as

vihX (xi)hT (t), where vi ≡ ωi/ωr0 , hX (xi) ≡ 1, and hT (t) ≡ ωr0p(

(Bu)t(r0)

). So, for a xed r0, the

41If there were more than two skill levels, the hazard would still assume this form, but r would be vector-valued. The followingarguments will still apply.

37

true data-generating process is consistent with a mixed 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 hX (xi) being a non-constant function and xi having non-zero

variance.42 Even if an econometrician could observe aspects of a worker's publicly observed history to include

in xi, 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 informational dynamics 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), is independent of xi, conditional on r0. In that case, the proportional hazards model

would be both identied and correctly specied, in the sense that it comports with the theoretical general-

equilibrium framework presented here.

8 Conclusion

Summing up, this model generates several novel theoretical results. First, there is a non-monotonic rela-

tionship 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 work-

ers, 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 résumé value

of employment. Third, human capital decay, which is an alternative explanation for negative duration de-

pendence, 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.

42Observing multiple spells per worker oers little help in this context. Honoré [1993] proves that variation in xi is unnecessaryfor 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 certainly will not. Consequently,one person's multiple spells are not informative of a single baseline hazard function.

38

Besides the theoretical results, the quantitative results from the calibrated model suggest that informa-

tional 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 gen-

erate 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 depen-

dence 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 learning

from the duration of unemployment. And, as I have discussed, most of those models do not consider career

concerns, or how the résumé 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 [2016] 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 mech-

anisms 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 frictional labor markets.

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42

A Figures

Figure A.1: Job-Finding Rates as a Function of Duration

0 10 20 30 40 500.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

Duration (Weeks)

P[ J

ob F

ound

Nex

t Mon

th ]

Each x in the scatter plot represents the fraction of unemployed workers of a given duration who found a jobby the time they were surveyed in the subsequent sample month. The number of workers who found a joband the number of workers with a given unemployment duration are weighted with CPS nal weights. Thesize 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.

43

Figure A.2: Fitting the Population Hazard

0 10 20 30 40 500.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

Duration

One

−M

onth

Job

−F

indi

ng P

rob.

The scatter plot is the same as the one in Figure A.1, except that the above excludes workers on layo. Thesolid line is the relationship implied by the model.

Figure A.3: Market Tightness

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

8

r

θ(r)

44

Figure A.4: The Résumé Distribution

Résumé0 0.2 0.4 0.6 0.8 1

Mea

sure

0

2

4

6

8

10

12

EntrantsPopulationEmployedUnemployed

Figure A.5: Wages of New Hires

0 0.2 0.4 0.6 0.8 1−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

r

wh(r

|1)

45

Figure A.6: Wedges in the Wage Equation for New Hires

0 0.2 0.4 0.6 0.8 1−6

−5

−4

−3

−2

−1

0

1

2

r

Wed

ge

Search WedgeInformation Wedge

Figure A.7: Value of Unemployment: U (r)

0 0.2 0.4 0.6 0.8 11

1.02

1.04

1.06

1.08

1.1

1.12

r

( 1

− β

) U

(r)

46

Figure A.8: Signal from Hiring: Counterfactual Résumés rc

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r

rc

Bu(bh(r))

45o

Figure A.9: Wages of Continuing Employees

00.5

1

01230.8

1

1.2

1.4

1.6

1.8

2

rx

we(r

,x)

47

Figure A.10: Wedges in the Wage Equation for Continuing Employees

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

r

Wed

ge

Search WedgeInformation Wedge

Figure A.11: Résumé Updating During Unemployment

0 20 40 60 00.5

10

0.2

0.4

0.6

0.8

1

Initial RésuméUnemp. Duration (Wks)

Upd

ated

Rés

umé

48

Figure A.12: Meeting and Matching Probabilities as a Function of Résumé

0 0.5 10.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

r

p(r)

Meeting Rate

0 0.5 10.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

r

ωrp(

r)

Matching Rate

Figure A.13: Contact Rates as a Function of Unemployment Duration

020

4060 0

0.510.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Initial RésuméUnemp. Duration (Wks)

Pro

babi

lity:

p(r

)

49

Figure A.14: Average Drop in Individual Hazards, by Duration

0 10 20 30 40 50

Unemp. Duration (Wks)

75

80

85

90

95

100R

elat

ive

Job-

Fin

ding

Rat

e

Figure A.15: The Role of Heterogeneity in the Population Hazard

10 20 30 40 50 600.1

0.15

0.2

0.25

Unemp. Duration (Wks)

Job−

Fin

ding

Rat

e (W

kly)

Average HazardHazard due to HeterogeneityHazard due to Stigma

50

Figure A.16: Job-Finding Rates in the Full-Information Economy

0 0.2 0.4 0.6 0.8 10.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

r

ωr p

(r)

Incomplete Info. EconomyFull Info. Economy

Figure A.17: Aggregate Average Job-Finding Rates in the Full-Information Economy

0 10 20 30 40 50 600.1

0.15

0.2

0.25

Duration (Weeks)

Job−

Fin

ding

Pro

b. (

Wkl

y.)

Incomplete Info. EconomyFull Info. Economy

51

Figure A.18: Skill Change and the Job-Finding Rate ωrp (r)

0 0.2 0.4 0.6 0.8 10.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Résumé

Avg

. Job

−F

indi

ng P

rob.

ωr p

(r)

BaselineSC1SC2SC3

Figure A.19: Skill Change and the Value of Unemployment U (r)

0 0.2 0.4 0.6 0.8 11

1.02

1.04

1.06

1.08

1.1

1.12

Résumé

( 1

− β

) U

(r)

BaselineSC1SC2SC3

52

Figure A.20: Job-Finding Rates by Résumé, Alternative ω1 − ω0

00.5

1

00.1

0.20.3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Résuméω1 − ω

0

ωr p

(r)

Figure A.21: Average Aggregate Job-Finding Rates by Duration, Alternative ω1 − ω0

020

4060 0

0.10.2

0.30.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

ω1 − ω

0Duration

Pro

babi

lity

53

Figure A.22: Job-Finding Rates by Duration and Potential Experience

Duration (Weeks)0 10 20 30 40 50

Nor

mal

ized

Job

-Fin

ding

Rat

e

40

50

60

70

80

90

100

1st Quartile of Potential Experience2nd Quartile of Potential Experience3rd Quartile of Potential Experience4th Quartile of Potential Experience

Figure A.23: Job-Finding Rates by Duration and Potential Experience, Excluding Part Time

Duration (Weeks)0 10 20 30 40 50

Nor

mal

ized

Job

-Fin

ding

Rat

e

50

60

70

80

90

100

1st Quartile of Potential Experience2nd Quartile of Potential Experience3rd Quartile of Potential Experience4th Quartile of Potential Experience

54

Figure A.24: Job-Finding Rates by Duration and Potential Experience, Model-Generated Data

Unemp. Duration (Weeks)0 5 10 15 20 25 30 35 40 45 50

Nor

mal

ized

Job

-Fin

ding

Pro

b.

65

70

75

80

85

90

95

100

1st Quartile of Potential Experience2nd Quartile of Potential Experience3rd Quartile of Potential Experience4th Quartile of Potential Experience

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) . (B.1)

It will be convenient to evaluate the surplus-splitting condition (2.18) at (Be (r | x, x′) , x′, e) and(Bh (r | x) , x′, h

).

Doing so, and using the denition of Cs (r, x), yields:

Ge (Be (r | x, x′) , x′) = (1− η) [Ce (Be (r | x, x′) , x′)− U (Be (r | x, x′))] (B.2)

Gh(Bh (r | x) , x′

)= (1− η)

[Ch(Bh (r | x) , x′

)− U (Bu (r))

]. (B.3)

55

Combining the above with (2.10) and (2.14) gives us:

Cs (r, x) = zx+ β∑x′

Ω (x, x′ | r) max Ce (Be (r | x, x′) , x′) , U (Be (r | x, x′)) . (B.4)

Notice that the right-hand side does not depend on s, so neither does the left-hand side. Hence, I will simply

write C (r, x). We can write the value of unemployment (2.15) as:

U (r) = λ+ β

(U (Bu (r)) + ηζθ (r)

1−ε∑x

Ω (x | r) maxC(Bh (r | x) , x′

)− U (Bu (r)) , 0

). (B.5)

Using the surplus-splitting condition for new hires allows us to write the free-entry condition as:

θ (r) = β1− ηκ

ζθ (r)1−ε∑

x

Ω (x | r) maxC(Bh (r | x) , x′

)− U (Bu (r)) , 0

. (B.6)

The above allows us to reduce further the value of unemployment to:

U (r) = λ+ηκ

1− ηθ (r) + βU (Bu (r)) . (B.7)

Given C (r, x), U (r), and θ (r), the policy functions can be written as:

Xe (r | x) = x′ | C (r′e, x′) ≥ U (r′e) , r

′e = Be (r | x, x′) (B.8)

Xh (r) =x | C (r′e, x) ≥ U (r′u) , r′h = Bh (r | x) , r′u = Bu (r)

. (B.9)

Similarly, once we know θ (r), 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, x′ | r) max C (Be (r | x, x′) , x′) , U (Be (r | x, x′)) (B.10)

U (r) = λ+ηκ

1− ηθ (r) + βU (Bu (r)) (B.11)

θ (r) = β1− ηκ

ζθ (r)1−ε∑

x

Ω (x | r) maxC(Bh (r | x) , x′

)− U (Bu (r)) , 0

. (B.12)

Dene T (C,U, θ) as the operator mapping (C,U, θ) to the right-hand side of (B.10)-(B.12). Given initial

guesses for (C,U, θ), I iterate on the T 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 nr

56

values 0 = r1 ≤ · · · ≤ rnr = 1. Dene:

r ≡[r1 · · · rnr

]′(B.13)

x ≡[x0 x1 · · · xnx

]′, (B.14)

where x0 is understood to be zero. Dene the nr × (nx + 1) matrix C and the vectors U and Θ by:

C ≡

C (r1, x0) · · · C (r1, xnx

)

.... . .

...

C (rnr, x0) · · · C (rnr

, xnx)

(B.15)

U ≡[U (r1) · · · U (rnr )

]′(B.16)

Θ ≡[θ (r1) · · · θ (rnr

)

]′. (B.17)

Dene the nr × (nx + 1)2matrix:

Be ≡

Be (r1 | x0, x0) · · · Be (r1 | x0, xnx

) · · · Be (r1 | xnx, x0) · · · Be (r1 | xnx

, xnx)

.... . .

......

. . ....

Be (rnr| x0, x0) · · · Be (rnr

| x0, xnx) · · · Be (rnr

| xnx, x0) · · · Be (rnr

| xnx, xnx

)

,(B.18)

where elements of Be are imputed to be zero for infeasible (x, x′) transitions. Let Ce be the column-wise

interpolation of 11×(nx+1) ⊗C along the grid given by Be. That is:

Ce =

[Ce

0 Ce1 · · · Ce

nx

], (B.19)

where each Cei is nr × (nx + 1), and the (j, k + 1) element of Ce

i is C (Be (rj | xi, xk) , xk). Dene:

Ξei ≡(1(nx+1)×1 ⊗ Ω′i

)(Inx+1 ⊗ 1(nx+1)×1

)

=

Ω (x0, · | i)′ 0(nx+1)×1 · · · 0(nx+1)×1

0(nx+1)×1 Ω (x1, · | i)′. . .

...

.... . .

. . . 0(nx+1)×1

0(nx+1)×1 · · · 0(nx+1)×1 Ω (xnx, · | i)′

, (B.20)

where Ω (xh, · | i) represents the (h+ 1)st

row of Ωi, and represents element-by-element multiplication.

57

Dene Ue as the column-wise interpolation of U⊗ 11×(nx+1)2 along the grid given by Be. That is:

Ue =

[Ue

0 Ue1 · · · Ue

nx

], (B.21)

where the (j, k + 1) element of Uei is U (Be (rj | xi, xk)). Observe that column (j + 1) of max Ce,UeΞei is

maxCej ,U

ej

Ω (xj , · | i)′, which gives the expected value (over x′) of max C (Be (r | xj , x′) , x′) , U (Be (r | xj , x′))

under the distribution Ωi, conditional on starting with x = xj . The updated value of C is given by:

TC (C,U,Θ) = z1nr×1x′ + β (max Ce,UeΞe1)

(r11×(nx+1)

)+β (max Ce,UeΞe0)

(1nr×(nx+1) − r11×(nx+1)

). (B.22)

Dene the nr × 1 vector:

Bu ≡[Bu (r1) · · · Bu (rnr )

]′. (B.23)

Dene Uu as the interpolation of U along the grid given by Bu. The updated value of U is given by:

TU (C,U,Θ) = λ1nr×1 +ηκ

1− ηΘ + βUu. (B.24)

Dene the nr × 1 vector:

Bh ≡[Bh (r1) · · · Bh (rnr

)

]′. (B.25)

The above incorporates the specication assumed in the calibration exercise; namely, all newly hired workers

have the same productivity x = 1. Dene Ch as the column-wise interpolation of C ( : , I [x′ = 1] ) along

the grid given by Bh. Dene:

P ≡ minζΘ1−ε, 1

(B.26)

~ωr ≡ rω1 + (1nr×1 − r)ω0, (B.27)

where the min · operator and the power (1− ε) are understood to apply element-by-element. The updated

value of Θ is given by:

TΘ (C,U,Θ) = β

(1− ηκ

)P ~ωr

[max

Ch −Uu, 0

]. (B.28)

58

C Proof of Proposition 1

Evaluating equation (B.11) at rc and subtracting it from Hs (r, x), as dened in equation (2.14), yields:

Hs (r, x)− U (rc) = ws (r, x) + β∑x′

Ω (x, x′ | r) max He (Be (r | x, x′) , x′)− U (Be (r | x, x′)) , 0

−λ+∑x′

Ω (x, x′ | r) [U (Be (r | x, x′))− U (Bu (rc))]− βκη

1− ηθ (rc)

= ws (r, x) +η

1− ηβ∑x′

Ω (x, x′ | r) max Ge (Be (r | x, x′) , x′) , 0

−λ+ β∑x′

Ω (x, x′ | r) [U (Be (r | x, x′))− U (Bu (rc))]−κη

1− ηθ (rc) , (C.1)

where the second line uses the surplus-splitting condition (2.18) evaluated at (Be (r | x, x′) , x′, e) instead of

(r, x, s). Multiplying the above by 1− η, we obtain:

(1− η) [Hs (r, x)− U (rc)] = (1− η)ws (r, x)− (1− η)λ− κηθ (rc)

+ (1− η)β∑x′

Ω (x, x′ | r) [U (Be (r | x, x′))− U (Bu (rc))]

+ηβ∑x′

Ω (x, x′ | r) max Ge (Be (r | x, x′) , x′) , 0 . (C.2)

Multiplying equation (2.10) by η, we obtain:

ηGs (r, x) = ηzx− ηws (r, x) + ηβ∑x′

Ω (x, x′ | r) max Ge (Be (r | x, x′) , x′) , 0 . (C.3)

The surplus-splitting condition (2.18) allows us to equate the above two expressions for ηGs (r, x) and

(1− η) [Hs (r, x)− U (rc)]. Rearranging terms completes the proof.

D Data

I use the basic monthly CPS data from 1994 to 2013, available from the NBER's website, maintained by

Jean Roth.43 Although earlier data are available, the CPS underwent a redesign in 1994, and it's known

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. However, to exploit the longitudinal

43See: http://www.nber.org/data/cps_basic.html.

59

structure of the CPS, it's necessary to match people across months. 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. I also drop observations in which age

changes by two or more years between months. 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 high-

school 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 [2015]). 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

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. The sample is divided into quartiles

of the potential experience distribution; the 25th, 50th, and 75th percentiles of the potential experience

distribution are 6 years, 16 years, and 28 years.

60