Early-Career Discrimination: Spiraling or Self-Correcting?
Arjada Bardhi

Yingni Guo

Bruno Strulovici∗

July 24, 2024

Abstract
Do workers from social groups with comparable productivity distributions obtain comparable lifetime earnings? We study how a small amount of early-career discrimination propagatesover time when workers’ productivity is revealed through employment. In breakdown learning
environments that primarily track on-the-job failures, such discrimination spirals into a substantial lifetime earnings gap for groups of comparable productivity, whereas in breakthroughlearning environments that track successes, early-career discrimination can be self-corrected,
so comparable groups obtain comparable lifetime earnings. This contrast persists in large
labor markets and with flexible wages, inconclusive learning, investment in productivity, andmisspecified employers’ beliefs.
JEL: C78, D83, J71
Keywords: early-career statistical discrimination, star jobs, guardian jobs, spiraling discrimination, self-correcting discrimination

1

Introduction

Young workers enter the labor market with uncertain productivity levels. To cope with thisuncertainty, employers have been shown to rely on observable characteristics—such as a worker’s
race or gender—as statistical proxies for the worker’s productivity.1 Such early-career statistical
discrimination determines who makes the first cut when job opportunities are scarce. Even if
different social groups have comparable productivity distributions, groups that make the first cut
may be systematically prioritized.
Does the impact of such early-career discrimination on workers’ earnings vanish or intensify overtime? One plausible conjecture is that social groups of comparable productivity obtain comparable∗

Bardhi: Department of Economics, New York University; email: arjada.bardhi@nyu.edu. Guo: Department of
Economics, Northwestern University; email: yingni.guo@northwestern.edu. Strulovici: Department of Economics,
Northwestern University; email: b-strulovici@northwestern.edu. We thank Attila Ambrus, Peter Arcidiacono,
Heski Bar-Isaac, Aislinn Bohren, Yeon-Koo Che, Hanming Fang, Rachel Kranton, Kevin Lang, Erik Madsen,Peter Norman, Wojciech Olszewski, Alessandro Pavan, Giorgio Primicieri, Anja Prummer, Heather Sarsons, Curtis
Taylor, Huseyin Yildirim, and seminar audiences at various seminars and workshops for valuable feedback.
1
Discrimination in hiring practices has been empirically documented by Goldin and Rouse (2000), Pager (2003),
Bertrand and Mullainathan (2004), and other studies surveyed in Bertrand and Duflo (2017).

1

lifetime earnings: over time, employers learn about workers’ productivity from observing theirperformance and reallocate opportunities accordingly. However, an opposite conjecture suggeststhat comparable groups may fare drastically differently: when early opportunities to perform arepivotal to a worker’s career progression, workers favored early on fare substantially better.
This paper shows that how employers learn about workers’ productivity makes a critical difference for which conjecture prevails. In environments that primarily track on-the-job successes,
early-career discrimination has only minor consequences for workers’ later job opportunities andlifetime earnings. In environments that primarily track on-the-job failures, in contrast, earlycareer discrimination significantly affects workers’ lifetime prospects. Moreover, the adverse effecton workers who are discriminated against intensifies with job scarcity: the scarcer jobs are relative
to the size of the workforce, the higher the inequality between groups. Our analysis thus suggests aclassification of learning environments that predicts whether, and if so, under which circumstances
the impact of early-career discrimination vanishes or gets amplified over time.
This contrast between learning environments persists even with flexible wage determination—a possibility that we formalize through a dynamic two-sided matching model. Even between
highly comparable social groups, environments that track failures induce a substantial delay inemployment and a significantly lower wage path for groups that are disfavored at the outset.
Model.

We study labor markets in which (i) workers from different groups compete for tasks, (ii)employers learn about a worker’s productivity only if the worker performs a task, and (iii) groups
have comparable productivity distributions.2 Sarsons (2022) studies one such market in whichmale and female surgeons compete for referrals from physicians. Physicians learn about surgeons’abilities from surgeries they performed in the past. Sarsons (2022) documents comparable ability
distributions for male and female surgeons: the average ability is only slightly lower for female
surgeons in her sample.3 We investigate the consequences of such a small difference.
We begin with a small market that features one employer and two workers identified by theirrespective social groups, a and b. A worker’s productivity is either high or low, and worker a is ex
ante more likely than worker b to have high productivity. In each instant, the employer allocates
the task to one of the two workers—similar to a physician choosing a surgeon for referral—or takes
an outside option if the expected productivity of both workers is too low. The employer’s flow
payoff increases in the productivity of the employed worker, whereas workers benefit from being
allocated the task regardless of their productivity. In section 3, we analyze a large market with acontinuum of workers from each group and a continuum of employers.
The employer learns about a worker’s productivity from their performance. We contrast twolearning environments: breakthrough learning and breakdown learning. In the breakthrough environment, a high-productivity worker generates successes, which we call breakthroughs, at ran2

These stylized features tractably capture a more general setting in which (i’) some tasks are more desirable
than others and desirable tasks are in limited supply, (ii’) workers who perform desirable tasks reveal more about
their productivity than do workers who are either employed in other tasks or unemployed, and (iii’) groups do
not necessarily have comparable productivity distributions. Our focus on groups with comparable productivity
distributions highlights the role played by the learning environment in the dynamics of statistical discrimination.
3
See section 2.2.2 and Figure 1 in Sarsons (2022).

2

domly distributed times, whereas a low-productivity worker generates nothing. In the breakdown
environment, a low-productivity worker generates failures, which we call breakdowns, at randomly distributed times, whereas a high-productivity worker generates nothing. We also analyze
mixed learning environments that combine both breakthroughs from high-productivity workers andbreakdowns from low-productivity workers; the relative frequency of the two determines whether
the environment is breakthrough-salient or breakdown-salient.
The learning environment can be viewed as an intrinsic feature of the job considered. Breakthrough and breakdown environments correspond, respectively, to “star jobs” and “guardian jobs,”
as conceptualized by Jacobs (1981) and Baron and Kreps (1999).

Star jobs—such as high-stakes salespeople, entertainers, and athletes—feature infrequent but large successes. By contrast, guardian jobs—routine surgeons, airline pilots, prison guards, among numerous others—aremarked by infrequent but costly failures.4 (Figure 5 in appendix A.1 compares the performance
distributions of star jobs and guardian jobs.5 )Main results. In both learning environments, the employer first allocates the task to worker a,
who has a higher expected productivity. However, subsequent task allocations differ drastically
across environments. In the breakthrough environment, worker a’s expected productivity declines
gradually in the absence of a breakthrough, until it drops to that of worker b’s. From this point
onward, the employer treats the two workers equally. The length of this grace period over which
the task is allocated exclusively to worker a reflects the difference in the two workers’ expected
productivity at the start. The smaller this initial difference, the shorter the grace period for worker
a, and the smaller the first-hire advantage of worker a. As this difference shrinks to zero, so does
the advantage of worker a. The breakthrough environment is thus self-correcting.
In the breakdown environment, in contrast, the absence of a breakdown from worker a makes
the employer more optimistic about the worker’s productivity. Therefore, the employer allocates
the task exclusively to worker a until a breakdown occurs. Worker b gets a chance to performthe task only if worker a has low productivity and misperforms on the task. As a result, worker
b’s expected lifetime earnings are only a fraction of worker a’s. We show that even if worker a’s
productivity distribution is only slightly superior ex ante, this small initial difference spirals intoa large payoff inequality. This spiraling effect persists even as learning by the employer becomes
arbitrarily fast. It can explain why societies struggle to eliminate inequality in labor markets.
This contrast between breakthrough and breakdown environments extends to mixed learning
environments with both breakthroughs and breakdowns. Any breakthrough-salient environment—
in which breakthroughs arrive faster than breakdowns—continues to be self-correcting, whereas any
4Bose and Lang (2017) argue that most nonmanagerial jobs are guardian jobs and derive the optimal monitoring
policy for such jobs. We instead compare the lifetime impact of early-career discrimination in star jobs to that in
guardian jobs.
5
Similarly to the performance distribution of star jobs in Figure 5, O’Boyle and Aguinis (2012) and Aguinis
and Bradley (2015) show that in occupations centered around star performance, such as entertainers and athletes,
the empirical distribution of performance is indeed right-skewed. This implies that “the majority of individuals are
assumed to perform at an average level, with very few people actually achieving a level of performance that would
place them in the category of being a star performer” (Aguinis and Bradley, 2015, p. 162).

3

breakdown-salient environment gives rise to spiraling. However, the contrast is now more nuanced:
fixing the total arrival rate but varying the arrival rate of breakthroughs relative to breakdowns,
the payoff gap between the two workers changes smoothly as the environment shifts continuously
from pure breakthrough learning to pure breakdown learning. The more comparable the groups
become, the more pronounced is the contrast between breakthrough-salient and breakdown-salient
environments: the payoff gap converges to zero for any breakthrough-salient environment but
converges to the strictly positive gap of a pure breakdown environment for any breakdown-salient
environment.
We further explore this contrast in the large market. We show that the key determinant of
the spiraling effect in the breakdown environment is the scarcity of tasks relative to the size of
the workforce. As tasks become scarcer, the inequality between groups increases. Hence, while
all groups suffer from a decrease in labor demand during economic downturns, groups that are
discriminated against will suffer disproportionately more.
One might conjecture that flexible wages will eliminate inequality between groups of comparable productivity. For instance, Becker (1957) and later Flanagan (1978) have argued that withflexible wages, the wage differential should equalize the employment rates across groups. To evaluate this conjecture, we introduce flexible wages into the large market. From a methodologicalstandpoint, we develop a dynamic two-sided matching model that incorporates both learning andflexible wages, and show that the essentially unique stable stage-game matching is dynamicallystable (Ali and Liu, 2020).
We find that flexible wages do not resolve the severe differential treatment of comparable
groups in the breakdown environment. Intuitively, they do not overcome the tension caused bytask scarcity: when only a subset of workers can be hired, those who generate higher expectedsurplus get hired first. Hence, as in the case of fixed wages, b-workers are not given a chanceto perform unless and until sufficiently many a-workers have experienced breakdowns. Such adelay further implies that employers learn more about a-workers than b-workers. Hence, employed
a-workers (i.e., those who have not generated breakdowns) earn a higher wage than employedb-workers. Breakdown learning thus results in substantial gaps in wage and flow-earnings between
groups of almost identical expected productivity.
Figure 1 illustrates the predicted paths of average wages and those of average flow-earnings.6
Both the gap in average wages and that in average flow-earnings widen early on in the workers’
careers and persist for a substantial amount of time. If tasks are sufficiently scarce relative toworkers, the gap in flow-earnings persists throughout the workers’ careers. While our main characterization assumes a zero minimum wage, we show that the dynamics are similar with a negative
minimum wage. Workers outbid each other to the point that the marginal-productivity worker
among those hired in each instant is paid the negative minimum wage. Since the employment
dynamics are identical to those under a zero minimum wage, b-workers face the same delay in
6
Average flow-earnings are defined as the average payoff across both employed and unemployed workers. The
average wage is taken across employed workers only, so it is higher than average flow-earnings.

4

employment.
Average wage

Average flow-earnings

Group aGroup bGroup aGroup b
0

Time

0

Time

Figure 1: Average wage/flow-earnings under breakdowns for groups of comparable productivity
We extend the contrast between the learning environments to workers investing in their productivity, to inconclusive performance signals, and to prior differences being due to employers’
incorrect beliefs rather than objective differences between groups. In particular, when workers caninvest in their productivity before entering the labor market, we show that the contrast becomeseven sharper. Across all equilibria of the breakdown environment, slightly different groups invest
in significantly different amounts. Inequality across groups is even greater (i.e., spiraling is evenworse) than in the model without investment, since access to investment disproportionately benefits the group that is favored after the investment stage. In the breakthrough environment, incontrast, there generically exists an equilibrium in which comparable groups invest in comparable amounts and obtain comparable lifetime earnings. Hence, the self-correcting property of the
breakthrough environment can persist with investment opportunities.
Empirical implications and evidence.

Our findings are consistent with the persistent gender

pay gap among surgeons documented by Lo Sasso et al. (2011) and Sarsons (2022). In line with
our emphasis on early-career discrimination, a recent statement by the Association of Women
Surgeons finds that “[T]he disparities women face in compensation at entry level positions lead toa persistent trend of unequal pay for equal work throughout the course of their careers.” 7 Our
results are also consistent with empirical evidence of racial wage gaps that are small at early
career stages but widen with labor market experience, as documented by Arcidiacono (2003) and
Arcidiacono, Bayer and Hizmo (2010). We provide a learning-based mechanism that can explain
this growing wage gap across groups.
In contrast to the breakdown environment, the paths of employment rates, average wages, andaverage flow-earnings across the two groups are closer in the breakthrough environment. Lang andLehmann (2012) observe that it is challenging to explain the simultaneous presence of large racial
wage and employment gaps in low-skill occupations and the absence of such gaps in high-skill
occupations. We provide a mechanism that can explain such discrepancies across occupations. To
the extent that low-skill occupations are driven by breakdown learning and high-skill occupations
7
For
more,
see
the
AWS
2017
Statementon
GenderSalaryEquityhttps://womensurg.memberclicks.net/assets/docs/STATEMENT%20ON%20GENDER%20SALARY%20EQUITY.pdf.

5at

by breakthrough learning, we provide an explanation for the more persistent wage gaps and longer
unemployment duration faced by groups discriminated against in low-skill occupations.
Prejudice can be another cause of early-career discrimination: even when different groups havethe same productivity distribution, employers may mistakenly believe that one group’s distribution
is inferior to the other’s. Such prejudice may be caused by inaccurate stereotypes or information
about the workforce that enters a particular occupation. The contrast between breakthrough andbreakdown environments extends to this setting as well, as we show in section 4.4. In a breakdown
environment, prejudice among employers, even if very mild, can have dire consequences for thegroup that is discriminated against.
Lastly, the spiraling of a negligible productivity difference into a substantial payoff gap isreminiscent of the cumulative advantage known as the Matthew effect (Merton, 1968). Scarcity ofopportunities for acknowledgement is at the core of both our argument and that in Merton (1968).
While the Matthew effect has become an umbrella term for cumulative advantage resulting fromvarious mechanisms, our findings uncover a novel learning-based mechanism for this effect andidentify workplace environments that are more prone to it. We revisit this point in section 4.1.

Related literatureFirst and foremost, our paper contributes to the theoretical literature on statistical discrimination
surveyed by Fang and Moro (2011) and Onuchic (2023). Phelps (1972) and the subsequent literature (e.g., Aigner and Cain (1977), Cornell and Welch (1996), and Fershtman and Pavan (2021))
assume a significant, exogenous difference between social groups, which gives rise to inequality
between groups. In contrast, Arrow (1973) and the subsequent literature (e.g., Foster and Vohra(1992), Coate and Loury (1993), Moro and Norman (2004), and Gu and Norman (2020)) assume
no exogenous difference between groups: inequality arises because groups coordinate on different
equilibria or specialize in different roles within an equilibrium.8
Our approach differs from both of these strands of literature. First, we consider groups thatshare arbitrarily similar expected productivity. In the models building on Phelps (1972), inequality across groups disappears as the productivity difference vanishes, whereas our model shows
how a vanishingly small difference can snowball into a large payoff gap. Second, in contrast to
Arrow (1973), the across-group inequality that we uncover is not due to the existence of multiple
equilibria. Third, most papers in both strands do not model group interaction, whereas in ourmodel groups compete for tasks. From this standpoint our paper is related to Cornell and Welch(1996) in the first group and Moro and Norman (2004) in the second. However, these two papers
consider static task allocation, whereas we explore the consequences of dynamic task allocation.
Our analysis also contributes two insights to the literature on cumulative discrimination (e.g.,
Blank, Dabady and Citro (2004), Blank (2005)). First, the nature of the employer learning environment has a critical impact on the magnitude of cumulative discrimination. Second, the prospect
8

Blume (2006) and Kim and Loury (2018) extend the static setup of Coate and Loury (1993) to incorporate
generations of workers. In contrast, we examine a single generation of long-lived workers whose productivity isrevealed gradually while performing tasks.

6

of future cumulative discrimination casts a long shadow on workers’ investment in productivity.
Similarly to Pallais (2014), our findings emphasize the informational value of entry-level jobs. In a
two-stage tournament model, Drugov, Meyer and Möller (2024) show that an organization might
bias the selection process in favor of early strong performers, even when agents are ex ante identicalto the organization.
Our results also speak to the literature on employer learning (e.g., Farber and Gibbons (1996),
Altonji and Pierret (2001)). The learning environment can be interpreted as an intrinsic featureof an occupation. In this respect, our work relates to Altonji (2005), Lange (2007), Antonovics
and Golan (2012), and Mansour (2012). Whereas these models assume that occupations differ
only in the frequency of signals, we allow the direction of these signals to differ across occupations
and demonstrate the importance of such direction. Hurst, Rubinstein and Shimizu (2024) alsofind that the extent of discrimination varies with the nature of the tasks that workers perform;
however, in our framework tasks vary in terms of the underlying learning environment, whereas
in theirs tasks vary in terms of the amount of social interactions.
Our model leverages the tractability of Poisson bandits, which are used widely in strategic
experimentation models (e.g., Keller, Rady and Cripps (2005), Keller and Rady (2010), Strulovici(2010), Keller and Rady (2015)).9 Both Felli and Harris (1996) and this paper use the frameworkof multi-armed bandits to model labor market learning. Felli and Harris (1996) assume one
worker and two employers, whereas our small market has one employer and two workers so job
opportunities are in short supply.10 The contrasting of breakthrough and breakdown learning adds
to a recent literature that compares the implications of good-news learning and bad-news learningin various applications: Board and Meyer-ter-Vehn (2013) on reputational incentives, Halac andPrat (2016) on managerial attention, and Halac and Kremer (2020) on career concerns. Our mixed
learning environment, which features both conclusive breakthroughs and conclusive breakdowns,
is similar to that in Halac and Prat (2016), Che and Hörner (2018), and more recently Lizzeri,
Shmaya and Yariv (2024).
This paper contributes to a growing literature that employs bandit models to study statistical
discrimination. Li, Raymond and Bergman (2024) study an exploration-based screening algorithm
that leads to higher diversity and quality of workers. Learning about minority groups can lag
behind due to early negative signals from minority workers (Lepage, 2023), population imbalances(Komiyama and Noda, 2024), limited attention in search (Che, Kim and Zhong, 2019), and noisierevaluation of minority groups (Fershtman and Pavan, 2021). We show that learning about groups
disfavored at the start can lag behind also due to the nature of employer learning.

9

Other areas of applications include moral hazard (e.g., Bergemann and Hege (2005), Hörner and Samuelson(2013), Halac, Kartik and Liu (2016)), collaboration (e.g., Bonatti and Hörner (2011)), delegation (e.g., Guo (2016)),and contest design (e.g., Halac, Kartik and Liu (2017)).
10
We explore workers’ incentives to invest in productivity in section 4.2. This section is related to Bergemann
and Valimaki (1996), Felli and Harris (1996), and Deb, Mitchell and Pai (2022), since it also models bandit arms
as strategic players. However, all these models assume that the quality of the arms is exogenously given, while itis endogenously determined in our section 4.2.

7

2

Small market

2.1

Framework

Players and types. We consider a small labor market with one employer (“she”) and two workers(each “he”). Time t ∈ [0, ∞) is continuous, and the discount rate is r > 0. Workers belong to one
of two social groups, a or b. We refer to the worker from group i ∈ {a, b} as worker i.
Before time t = 0, workers’ types are drawn independently of each other. Worker i’s type
θi is either high (θi = h) or low (θi = ℓ). The prior probability that worker i has a high typeis pi ∈ (0, 1). The employer knows (pa , pb ), but she does not observe the workers’ types. We
interchangeably refer to pi as the prior belief for worker i or as worker i’s expected productivity
at time 0. We assume that worker a is ex ante more productive: pa > pb . Our focus is on groups
with comparable expected productivity, i.e., when pb is close to pa .
Task allocation.

At each t > 0, the employer allocates a task either to one of the two workers

or to a safe arm. Allocating the task to the safe arm can be interpreted as the employer resorting
to a known outside option, such as a worker with some known productivity or the value fromreallocating resources to other organizational goals. A worker obtains a flow payoff w > 0 whenever
he is assigned the task. Otherwise, his flow payoff is zero. We interpret w as the fixed wage fora worker who performs the task and normalize it without loss to w = 1. The employer obtains aflow payoff v > 0 if the task is allocated to a high-type worker, and a flow payoff normalized tozero if it is allocated to a low-type worker. These payoffs can be interpreted as the employer’s net
payoffs after the wage w is paid. If the employer allocates the task to the safe arm, she earns aflow payoff s ∈ (0, v).11
Learning by allocating tasks.

Learning about a worker’s type proceeds via conclusive Poisson

signals. If worker i is allocated the task over the interval [t, t + dt) and his type is θi = h, apublic breakthrough arrives with probability λh dt and no breakthrough arrives with complementaryprobability. If θi = ℓ instead, a public breakdown arrives with probability λℓ dt and no breakdown
arrives with complementary probability. Thus, a learning environment is characterized by a pairof type-dependent arrival rates (λh , λℓ ). A breakthrough reveals a high type and a breakdown
reveals a low type. We assume for simplicity that the employer’s payoffs are observed only at theend of the horizon, so that the employer learns only from the Poisson signals.12 Based on which
type is revealed more quickly, we distinguish between two classes of environments:(i) breakthrough-salient environments: λh > λℓ > 0;(ii) breakdown-salient environments: λℓ > λh > 0.
11

We multiply players’ lifetime payoffs by r, as in Keller, Rady and Cripps (2005). This normalizes the employer’s
lifetime payoff from always hiring a high type to v and a worker’s lifetime payoff from always being hired to 1.
12This formulation is, however, equivalent to an alternative formulation in which (i) the employer learns throughobservable payoffs, (ii) type h generates a lump-sum payoff bh at arrival rate λh and type ℓ generates a lump-sum
payoff bℓ 6= bh at arrival rate λℓ . Any nonzero values bh , bℓ would be consistent with our analysis as long as
λh bh > s > λℓ bℓ . Our chosen formulation makes it easier to compare payoffs across the learning environments.

8

The relative size of the arrival rates determines whether the employer’s belief drifts down or up
in the absence of any signal. In a breakthrough-salient (resp., breakdown-salient) environment,
the employer becomes more pessimistic (resp., optimistic) that the worker has a high type. In
the symmetric environment with λh = λℓ , such an absence is uninformative of the worker’s type.
We interpret the learning environment as an intrinsic and stylized feature of how performance ismonitored or evaluated at a given job. Breakthroughs correspond to over-performance by high-type
workers, and breakdowns to under-performance by low-type workers. Therefore, breakthroughsalient environments aim at identifying star employees, whereas breakdown-salient ones aim at
identifying misfits.
Central to our analysis are the environments in which only one type of signals is possible.
If λh > 0 and λℓ = 0, we call the environment a pure breakthrough environment or simply abreakthrough environment. If λh = 0 and λℓ > 0, we call it a pure breakdown environment or
simply a breakdown environment. If both λh > 0 and λℓ > 0, we say that the environment ismixed.13
Let p denote the belief threshold below which the employer switches to the safe arm, which isderived in Lemma A.1 and given by:
p :=

rs
.
rv + max{λh , λℓ }(v − s)This threshold depends only on max{λh , λℓ }, the higher of the two arrival rates. As expected, p
is lower than the myopic threshold s/v due to the value of learning for future allocation decisions.
Hereafter, we assume that pi > p for i ∈ {a, b}, so the employer prefers to experiment with both
workers before turning to the safe arm.

2.2

Contrast between the pure learning environments

We first observe a stark contrast between the two pure environments, and then generalize this contrast to mixed environments in section 2.3. We compare the two workers’ payoffs when they sharearbitrarily similar expected productivity, and analyze how this comparison depends on whether
the learning environment is pure breakthrough or pure breakdown.
Pure breakthrough learning. In each instant, the employer allocates the task to the worker
with the higher expected productivity.14 Since pa > pb , the employer first allocates the task toworker a. Because the belief that worker a has a high type drifts down as long as no breakthrough13
In mixed learning environments, signals are still conclusive. Section 4.3 and appendix D.2 show that themain results remain qualitatively unchanged when allowing for inconclusive signals, i.e., when low-type (resp., hightype) workers also generate breakthroughs (resp., breakdowns) but at a lower rate than high-type (resp., low-type)workers. Despite some loss in tractability, we establish the self-correcting property in Proposition 4.3 and thespiraling property for sufficiently impatient players in Proposition 4.4.
14This is true in any learning environment that we consider because (i) workers’ types are binary, and (ii) thearrival rates of signals and the employer’s type-contingent flow payoffs are the same for both workers. For a fixedenvironment, workers’ Gittins indices at each time t—which determine which worker is optimally chosen at time
t—are given by the same increasing function of the workers’ expected productivities pa (t), pb (t) at time t.

9arrives, worker a is effectively given a grace period [0, t∗ ) over which he is employed regardlessof his performance, where t∗ measures how long it takes for the belief about worker a’s type todrift down from pa to pb in the absence of a breakthrough. If worker a generates a breakthroughbefore t∗ , the employer allocates the task to him alone thereafter. Otherwise, starting from t∗ , theemployer splits the task equally between the two workers until either the belief drops down to p
or a breakthrough occurs, so the workers obtain the same continuation payoff starting from t∗ .
The hiring dynamics therefore go through two phases: a first phase during which worker a
is hired exclusively, and a second phase during which the two workers are treated symmetricallystarting from the symmetric belief pb . Importantly, as pb gets close to pa , the first phase [0, t∗ )shrinks to zero. The probability that worker a generates a breakthrough before t∗ converges tozero as well. Hence, as pb ↑ pa , worker a’s advantage vanishes and the two workers obtain similar
expected payoffs. Therefore, we observe a self-correcting property of pure breakthrough learning:a small difference in prior beliefs can result in only a small payoff advantage for worker a. Thisobservation and the next one are generalized in section 2.3 and proved formally in Appendix B.
Observation 2.1 (Self-correcting property of pure breakthrough learning). Let λh > 0 and λℓ = 0.
As pb ↑ pa , the expected payoff of worker b converges to that of worker a.
Pure breakdown learning. Again, the employer first allocates the task to worker a. As longas worker a generates no breakdown, the employer becomes more optimistic that his type is high,so she continues to hire him. Once a breakdown is realized, the employer turns to worker b. If
worker b also generates a breakdown, the employer resorts to the safe arm thereafter.
The lifetime payoff of a high-type worker a is one, since he is never fired, whereas that of a
low-type worker a is r/(λℓ + r). Once hired, the type-by-type continuation payoff of worker b is
the same as that of worker a. However, worker b faces a delay in getting hired. He obtains anopportunity only if worker a is a low type—that is, with probability (1 − pa )—and even then, it
takes time for worker a’s low type to be revealed, which further discounts worker b’s payoff by afactor of λℓ /(λℓ + r). Crucially, the delay faced by worker b is independent of how close pb is to pa :even if pb is just slightly less than pa , worker b obtains a substantially lower payoff than worker a.
In fact, worker a obtains the same payoff as if worker b did not exist: he is the first to be hired
and remains so unless and until he generates a breakdown. This stands in contrast to the purebreakthrough environment, in which worker a loses his preferential status if he fails to generate abreakthrough within the grace period.
Observation 2.2 (Spiraling property of pure breakdown learning). Let λℓ > 0 and λh = 0. As
pb ↑ pa , the ratio of the expected payoff of worker b to that of worker a approaches(1 − pa )λℓ< 1.
λℓ + r(1)

Since groups a and b have comparable productivity distributions, even if the employer wereblind to workers’ group belonging and treated them identically, her payoff would be only slightly
10lower than what she attains when she observes group belonging. In the limit as pb ↑ pa , her payoff
would be the same with and without observing workers’ group belonging. Therefore, making it
more difficult for the employer to observe group belonging would equalize workers’ payoffs without
making the employer worse off.15

2.3

Contrast across mixed learning environments

We now establish that the sharp contrast between the two pure environments observed in section
2.2 extends to mixed learning environments.
Self-correcting property of breakthrough-salient learning. Given that λh > λℓ , the employer’s belief that worker a has a high type drifts down as long as no signal arrives. Similarly tothe pure breakthrough environment, worker a is hired first and he is given a grace period [0, t∗ )over which to perform as long as he generates no breakdown. Here, t∗ is the time it takes for the
belief about worker a’s type to drift down from pa to pb in the absence of a signal:t∗ =1
pa (1 − pb )log
λh − λℓ(1 − pa )pb(2)

If worker a generates a breakthrough before t∗ , the employer allocates the task to him alone
thereafter. If worker a generates a breakdown before t∗ , the employer switches immediately toworker b. Otherwise, starting from t∗ , the employer treats the two workers symmetrically, so theyobtain the same continuation payoff starting from t∗ . As in the pure breakthrough case, as pb ↑ pa ,the grace period [0, t∗ ) shrinks to zero and so does worker a’s advantage.
Proposition 2.1 (Self-correcting property of breakthrough-salient learning). Let λh > λℓ > 0.
As pb ↑ pa , the expected payoff of worker b converges to that of worker a.
Spiraling property of breakdown-salient learning. Given that λℓ > λh , in the absence ofa signal the employer becomes more optimistic that the worker’s type is high. She first allocates
the task to worker a. If a breakthrough is generated, worker a is hired forever. If a breakdown isrealized, the employer switches to worker b. In the absence of either signal, the employer continues
to hire worker a. As in the pure breakdown environment, a high-type worker a is never fired,
whereas a low-type worker a enjoys a substantial period of employment before being eventually
fired. Because the reallocation of tasks is driven entirely by the occurrence of a breakdown, eachplayer’s expected payoff depends on λℓ but not λh . Therefore, the resulting payoff ratio is thesame as in the pure breakdown environment with the same λℓ .
Proposition 2.2 (Spiraling property of breakdown-salient learning). Let λℓ > λh > 0. As pb ↑ pa ,15
In a study of group-blind hiring practices, Goldin and Rouse (2000) show that blind orchestra auditions
substantially increased the likelihood that female musicians advanced to the final round.

11

the ratio of the expected payoff of worker b to that of worker a approaches(1 − pa )λℓ< 1.
λℓ + r(3)

The payoff ratio (3) has two components: (i) the factor (1 − pa ) reflects the fact that worker bobtains a chance only if worker a has a low type, and (ii) the factor λℓ /(λℓ +r) reflects the expected
time it takes for worker a’s low type to be revealed. Even as the revelation of low types becomesinstantaneous—that is, as λℓ → ∞—the payoff ratio approaches (1 − pa ) rather than one due toa strong rank effect. Being the second hire is detrimental to worker b even under instantaneous
employer learning, since worker b never obtains a chance if worker a has a high type.
Spiraling in the symmetric environment. In the environment with λh = λℓ , the employer’s
belief stays constant in the absence of a signal. The employer starts with worker a, and fires him
if and only if a breakdown arrives revealing his low type. Therefore, the task allocation dynamics
and the workers’ payoffs are the same as in any breakdown-salient environment with the same λℓ .
The payoff ratio as pb ↑ pa is that in (3), so spiraling arises.
Remark 1 (Continuum of types). The contrast across learning environments goes beyond binary
types. For example, suppose that worker i’s type θi is uniform on [θ i , θ i + ∆], where θa > θ b > 0
and ∆ > 0 parameterizes the uncertainty about the workers’ types. As θb ↑ θa , the expected
productivity of worker b converges to that of worker a. If worker i is employed, a signal that reveals
his type arrives according to a Poisson process, and its arrival rate is λ(θi ). If λ(θi ) increases inθi —e.g., if λ(θi ) = λθi for some λ > 0,—the learning environment is a breakthrough-salient one
and the employer becomes more pessimistic in the absence of a signal. If λ(θi ) decreases in θi ,the environment is a breakdown-salient one. The self-correcting property of breakthrough-salient
learning and the spiraling property of breakdown-salient learning continue to hold. In particular,
the upward belief drift in the absence of a signal gives worker a a payoff advantage that does notshrink to zero even as θ b ↑ θa .

2.4

Comparative statics with respect to the learning environment

Our analysis so far fixes the learning environment—that is, it fixes the arrival rates (λh , λℓ )—and
examines the limit payoff ratio as pb ↑ pa . This section performs the complementary exercise of
fixing (pa , pb ) with pa > pb and examining how the payoff gap (i.e., the difference in the workers’
expected payoffs) varies with the learning environment. We parametrize a family of learning
environments by the total arrival rate λh + λℓ = λ > 0. This parametrization ranges from thepure breakdown environment (λh , λℓ ) = (0, λ) to the symmetric one (λh , λℓ ) = (λ/2, λ/2) and thepure breakthrough environment (λh , λℓ ) = (λ, 0). We let Ua and Ub denote worker a’s and worker
b’s expected payoffs respectively.
Taken together, Propositions 2.1 and 2.2 imply that the limit of the payoff gap as the groups
become arbitrarily similar is discontinuous at the frontier between the breakdown-salient andbreakthrough-salient environments: for λh < λ/2, the limit payoff gap as pb ↑ pa is strictly12positive, whereas for λh > λ/2 the limit payoff gap as pb ↑ pa is zero. Thus, an environmentthat is slightly breakdown-salient generates much higher inequality between arbitrarily similargroups than one that is slightly breakthrough-salient. We next show that in fact, for any givenpa > pb > p, the payoff gap is continuously differentiable in the arrival rates.
Lemma 2.1. Fix pa > pb > p and λ > 0, and consider the family of environments





λ
λ
λ λ
(λh , λℓ ) =+ δ, − δ : δ ∈ − ,
.
2
2
2 2
The payoff gap (Ua − Ub ) is continuously differentiable in δ.
Ua − Ub
breakthrough-salient

breakdown-salientpa = pb + 0.45
pa = pb + 0.3
pa = pb + 0.03
pa = pb + 0.003
− λ2

0

λ
2

δ

Figure 2: Payoff gap for four different levels of the prior difference. Parameter values are λ = 2,
s = 1/10, v = 1, r = 1, and pb = 1/3.
Figure 2 illustrates the payoff gap as a function of δ for four different levels of the priordifference pa − pb . As the prior difference gets smaller, the contrast between breakthrough-salient
environments and breakdown-salient ones becomes more pronounced as well. The figure also
shows that the payoff gap is much more sensitive to the prior difference in breakthrough-salient
environments, dropping from significant levels down to zero as (pa − pb ) ↓ 0, than in breakdownsalient ones.
Within the class of breakdown-salient environments, the payoff gap increases in δ or, equivalently, decreases in λℓ . We have argued in section 2.3 that the workers’ payoffs depend only on λℓ
in a breakdown-salient environment. The continuation payoff of each worker when this worker ishired decreases as λℓ gets larger, because low types are revealed faster. This shrinks the payoff gap
between the two workers. Moreover, worker b expects to be hired earlier because a low-type worker
a is revealed faster. This further shrinks the payoff gap. Altogether, the payoff gap decreases in
λℓ . Intuitively, faster learning about low types in breakdown-salient environments allows to weed
out low-productivity workers from group a faster and reduce their initial advantage.
Within the class of breakthrough-salient environments, the payoff gap continues to increase inδ or, equivalently, decreases in λℓ for environments sufficiently close to the symmetric one. The
intuition can be understood by considering the environment that is only slightly breakthroughsalient. The employer’s belief about worker a drifts down very slowly in the absence of any signal,13

so pb is almost never reached. Thus, if worker a is fired, this is most likely due to a breakdown
rather than as a result of worker a’s expected productivity drifting down to the firing threshold.
The dynamics are thus similar to those of the breakdown-salient environment and, as in the
breakdown-salient environment, the payoff gap increases in δ for a small enough δ. Proposition
2.3 formalizes these two observations.
Proposition 2.3. Fix pa > pb > p and λ > 0, and consider the family of environments





λ
λ λ
λ
+ δ, − δ : δ ∈ − ,.
(λh , λℓ ) =22
2 2(i) The payoff gap (Ua − Ub ) strictly increases in δ within the breakdown-salient environments.
(ii) There exists a maximal δ̃ ∈ (0, λ/2] such that the payoff gap strictly increases in δ forδ ∈ (0, δ̃) within the breakthrough-salient environments.
Figure 2 shows that for a sufficiently large prior difference, it can be that δ̃ = λ/2, so the
payoff gap strictly increases in δ throughout its domain. The figure moreover suggests that forδ > δ̃ the payoff gap strictly decreases in δ. Extensive numerical computations confirm suchmonotonicity. Two counteracting forces are at play as δ increases. On the one hand, if we consider
the reallocation of tasks that is only driven by breakdowns, a lower λℓ makes the payoff gap larger.
However, increasing (λh − λℓ ) also shortens the grace period for worker a, which in return shrinks
his payoff advantage. Our numerical computations show that if the second force dominates forsome δ, then it dominates for any larger δ′ > δ as well.

3

Large market

The small market of the previous section features a single employer choosing between two workers.
We now shift our focus to a large market with a continuum of workers from each group and acontinuum of employers. Employers and workers are matched dynamically, tasks are scarce, andall learning is public. We establish that the contrast between the breakthrough and the breakdown
environments not only generalizes to this large market but also does so regardless of whether wages
are fixed or flexible. Notably, flexible wages are insufficient to prevent spiraling in the breakdown
environment. Furthermore, in appendix C.2.4 we demonstrate that this holds true even if theminimum wage is strictly negative, allowing the workers to pay employers for early opportunities.
We introduce the framework in section 3.1. Section 3.2 studies the dynamics of task allocation under fixed wages (i.e., constant across time and across all matched employer-worker pairs),
whereas section 3.3 examines the case of flexible wages.

3.1

Framework

There is a unit mass of employers, a mass of size α > 1 of workers from group a, and a mass of
size β > 0 of workers from group b. All employers and workers are long-lived and share the same14

discount rate r > 0. At each instant, each employer has one task to allocate and each worker
can take up at most one task. Crucially, there are more workers than tasks.16 Employers arehomogeneous. At t = 0, each worker’s type is drawn independently from other workers’ types.17
A worker from group i ∈ {a, b} draws a high type with probability pi . Additionally, there is a unit
mass of identical safe arms that employers can take.
Stage-game matchings and stability.

In this large market, a stage-game matching specifies (i)how workers are matched to employers, and (ii) a wage for each matched pair. We use k ∈ [0, α+β]
to index a worker and j ∈ [0, 1] to index an employer. Worker k is from group a if k ∈ [0, α] andfrom group b if k ∈ (α, α + β]. In the stage game, let Dkj ∈ {1, 0} indicate whether worker k andemployer j are matched to each other. If Dkj = 1, let Wkj denote the wage paid by employer j
to worker k. Worker k’s payoff is Wkj and employer j’s expected payoff is pk v − Wkj where pk is
the probability that θk = h.18 All signals are public, so all employers share the same belief about
each worker. If Dkj = 0 for all j, worker k is unmatched and gets zero payoff. If Dkj = 0 for allk, employer j takes a safe arm and gets a payoff of s > 0. Let D be the set of all such stage-gamematchings. In the case of fixed wages, the wage for any matched pair is fixed at w = 1, so we only
need to specify how workers are matched to employers. In the case of flexible wages, by contrast,
any nonnegative wage is allowed.
We now apply the stability concept in Shapley and Shubik (1971) to our setting. Given astage-game matching (D, W ), the pair (k, j) is called a blocking pair if each strictly prefers to bematched to the other at some allowable wage rather than follow (D, W ).
Definition 1. A stage-game matching (D, W ) is stable if (i) there is no matched employer j whostrictly prefers to employ a safe arm instead of its matched worker, and (ii) there exists no blocking
pair.
In Appendices C.1.1 and C.2.1, we show that there is an essentially unique stable stage-gamematching for fixed wages and flexible wages, respectively. In this matching, the workers who are
the most productive and generate more surplus than the safe arms are matched.
Dynamic stability.

In the dynamic game, a time-t history consists of all past matchings andrealized signals until t. Let Ht be the set of all time-t histories and H := ∪t>0 Ht the set of all
histories. A dynamic matching µ = (µt )t>0 specifies a lottery over stage-game matchings for any
history, i.e., µt : Ht → ∆(D) for each t. We define dynamic stability of a matching µ based on thesolution concept of a stable convention in Ali and Liu (2020).19 For a given dynamic matching µ,
16The assumption that α > 1 simplifies exposition since only workers from group a will be matched at t = 0.
However, our results hold qualitatively for α < 1 as well, as long as tasks are scarce, i.e., α + β > 1.
17
We assume that workers do not know their types at time 0: they share the same prior belief as the employers
and all learning is symmetric. This assumption allows us to make a clear parallel with the small-market setting
and is consistent with our focus on workers’ early-career dynamics, where workers do not know their types yet. It
is standard in models of learning in labor markets such as Felli and Harris (1996) and Altonji and Pierret (2001).
18
In the small market, v and 0 are the employer’s payoffs from hiring, respectively, a high type and a low typeafter the fixed wage is paid. In the large market, we use them to represent an employer’s payoffs before any wage
is paid. This notation simplifies exposition within each market.
19
Even though not crucial to our results, we assume that deviation wages are observable to all.

15

let µht denote the continuation matching after some history ht .
Definition 2. A dynamic matching µ is dynamically stable if at every t and every history ht ∈ Ht ,(i) there exists no matched employer j under µht who strictly prefers to take a safe arm over
the time window [t, t + dt), for some dt > 0, and then revert to µht+dt ;(ii) there exists no matched worker k under µht who strictly prefers to be unmatched over[t, t + dt), for some dt > 0, and then revert to µht+dt ;(iii) there exist no worker-employer pair (k, j) who strictly prefer to be matched to each other atsome allowable wage over [t, t + dt), for some dt > 0, and then revert to µht+dt .
In Appendices C.1.2 and C.2.2, we demonstrate that prescribing the essentially unique stablestage-game matching after each history is dynamically stable for both fixed wages and flexible
wages. Let µ∗ denote this dynamic matching. Sections 3.2 and 3.3 delve into the properties of µ∗in the context of fixed wages and flexible wages, respectively.

3.2

Fixed wages

The intuition gained from the small-market analysis of section 2 extends to the large market
with fixed wages: comparable groups continue to have comparable payoffs in the breakthroughenvironment but markedly different payoffs in the breakdown one. In addition to extending theresults from the small-market environment, the large-market analysis sheds new light on the impactof task scarcity on the size of the payoff gap. When tasks are scarcer relative to the workforce,the inequality between groups in the breakdown environment becomes greater.
Broad hiring under breakthrough learning. Mirroring the analysis in section 2.2, the dynamic allocation of tasks goes through two phases. In the first phase, all a-workers take turns
to perform tasks. If an a-worker generates a breakthrough, he “secures his job” with his current
employer: the employer allocates future tasks only to this worker. For those a-workers without abreakthrough, the employers’ belief drops gradually until it reaches pb . At that point, a-workers
without breakthroughs are believed to be as productive as b-workers. The allocation now entersa second phase in which the remaining employers let all remaining a-workers and all b-workers
take turns to perform tasks. Again, those who generate breakthroughs secure their jobs with theircurrent employers.
Breakthrough learning therefore prompts employers to try a broad set of workers. A similar
observation was made in passing by Baron and Kreps (1999) on recruitment for star jobs:
For a star job, the costs of a hiring error are small relative to the upside potential fromfinding an exceptional individual. Therefore, the organization will wish to sample
widely among many employees, looking for the one pearl among the pebbles. (Baronand Kreps (1999), p. 28-29)
Our focus is on the implications of this broad-hiring practice for group inequality. Employers
quickly extend their search to group b, so a b-worker’s payoff converges to an a-worker’s payoff as
pb ↑ pa . Thus, the self-correcting property extends to larger labor markets.
16

Proposition 3.1 (Self-correction in the large market with fixed wages). Let λh > λℓ = 0, α > 1,
and β > 0. In the dynamic matching µ∗ , the expected payoff of an a-worker converges to that of ab-worker as pb ↑ pa .
Narrow hiring under breakdown learning. Breakdown learning, in contrast, leads to sluggishness in experimenting with new workers: if a worker is hired, he remains employed until hegenerates a breakdown. This sluggishness hurts group b disproportionately no matter how close
pb is to pa , thus generalizing the intuition behind Observation 2.2 to larger labor markets.
At the start, a unit mass of a-workers are hired by the unit mass of employers. These workers
remain hired as long as they do not generate breakdowns. When one of these a-workers generates
a breakdown, he is replaced by a new a-worker for as long as one is available. So b-workers must
wait for their turn until all of the a-workers have been tried and sufficiently many of them havegenerated breakdowns. Crucially, this delay does not shrink as pb ↑ pa . Therefore, the expected
payoff of a b-worker remains bounded away from that of an a-worker.
Proposition 3.2 (Spiraling in the large market with fixed wages). Let λℓ > λh = 0, α > 1, and
β > 0. In the dynamic matching µ∗ , as pb ↑ pa , the limiting ratio of the expected payoff of ab-worker to that of an a-worker is strictly less than one.
Task scarcity magnifies inequality under breakdown learning. In this large market, αand β parametrize not only group sizes but also the relative scarcity of the unit mass of tasks.
By varying α and β, we explore how inequality among groups varies with relative task scarcity.
Proposition 3.3 observes that in the breakdown environment, inequality between groups increases
as the size of either group increases while the mass of tasks is kept fixed.
Proposition 3.3 (Inequality increases in task scarcity under breakdown learning). Let λℓ > λh =0, α > 1, and β > 0. In the dynamic matching µ∗ , as pb ↑ pa , the limiting ratio of the expected
payoff of an a-worker to that of a b-worker increases in both α and β.
Increasing β while keeping α fixed intensifies competition within group b but does not affectthe payoff of a-workers. By contrast, increasing α while keeping β fixed hurts both groups: it
intensifies competition within group a while also increasing the delay for group b. We show that
increasing α hurts group b more than it hurts group a, because adding one more a-worker uniformly
delays every b-worker’s employment. Therefore, the scarcer tasks are relative to the labor supply
from either group, the greater is the inequality between groups.
This result suggests that when job opportunities become scarcer, e.g., when labor demand falls
during an economic downturn, inequality deepens. This is consistent with the observation thatwhile all groups suffer during an economic downturn, some suffer disproportionately more.20
20
Estimates from the Pew Research Center (https://www.pewsocialtrends.org/2011/07/26/wealth-gaps-rise-torecord-highs-between-whites-blacks-hispanics/) show that the white-to-black and white-to-Hispanic wealth ratios
were much higher at the peak of the recession in 2009 than they had been since 1984, the first year for which the
U.S. Census Bureau published wealth estimates by race and ethnicity based on the Survey of Income and Program
Participation.

17

3.3

Flexible wages

We now incorporate flexible wages into the large market. First, we delve into how wages are
determined and why prescribing the stable stage-game matching after each history is dynamicallystable. Next, we explain why flexible wages do not fix spiraling in the breakdown environment.
To quantify the extent of this spiraling, we analyze closed-form expressions for key economic
indicators, including the employment rate, the average wage, and the average flow-earnings foreach group. Finally, we discuss why relaxing limited liability does not prevent spiraling.
In the dynamic matching µ∗ , there is a time-dependent marginal productivity pM (t) such that,at each time t, workers whose expected productivity (i.e., whose probability of having a high type)exceeds pM (t) are matched, while workers whose expected productivity lies below pM (t) are not.
Wages take a strikingly simple form: a matched worker with expected productivity pt at time
t is paid a flow wage of (pt − pM (t))v, which is the additional value that he creates relative tothe marginal-productivity worker. Notably, at time t, marginal-productivity workers obtain zeroflow-earnings, the same as all unmatched workers. All employers get the same flow profit of pM (t)v.
Proposition C.2 in the appendix establishes that prescribing the stable stage-game matchingafter each history is dynamically stable. The intuition may be seen from the following three steps.
First, in a stable stage-game matching, an employer’s flow profit from a match is at least as high asthat from the safe arm, so no employer finds it profitable to reject a match and take the safe arm.
Second, no employer-worker pair has a profitable one-shot deviation, since all employers make thesame flow profit. Lastly, one can show that no worker ever finds it profitable to reject a matchin the hope of delaying the arrival of information about his type. This last point follows fromthe fact that a worker’s flow-earnings are convex in his expected productivity pt at time t: flow
earnings take the form of max 0, (pt − pM (t))v , as Figure 3 shows.21 By Jensen’s inequality,

this implies that any signal about the worker’s type at time t—which splits the current belief
about the worker’s type into a lottery over posterior beliefs—increases the worker’s flow-earnings,
in expectation, at all future dates.
Flexible wages do not fix spiraling under breakdown learning. One plausible conjectureis that with flexible wages, workers with similar expected productivity obtain similar earnings. This
would indeed be the case in the one-shot version of the model, because a worker with expected
productivity p obtains flow-earnings max{0, (p − pM )v}, which is indeed a continuous function of
p. In particular, there would be no discontinuity in flow-earnings between an unemployed worker(p < pM ) and a worker who barely makes the cut (p ≈ pM ).
However, in a dynamic setting, employed workers benefit from the information that theygenerate through employment: unlike unemployed workers, they have an opportunity to establish
an increasingly higher reputation and thereby command an increasingly higher wage, which quickly
21
The minimum wage for an employed worker is zero since we normalize workers’ cost of effort on the task to zero.
If this cost were strictly positive, the limited liability constraint would require that the wage be weakly greater than
this cost. In the left panel of Figure 1 the average-wage paths for both groups would shift up by this cost, whereas in
the right panel the average-flow-earnings paths would remain intact once reinterpreted as average-net-flow-earnings
paths. Moreover, the green curve in Figure 3 would be reinterpreted as net flow-earnings.

18

Flow-earnings

Flow-earnings

max{0, (p − pM )v}
w2

pM
b

0

b

pb pa

1

p

b
b

pM
b

b

0

First periodbbpb pa pa

1pSecond period

Figure 3: A two-period example with α = β = 1
sets them apart from unemployed workers. To be sure, employed workers also risk generating a
breakdown, in which case they become unemployed forever. However, such an event can occur
only if a worker has a low type and, even in this case, the event takes time to occur, during which
the low-type worker enjoys the benefits from employment. The accumulated learning for group athus translates into a substantial earnings advantage over group b.
We now expand on this intuition about spiraling in two steps. First, to show how learning
through employment strictly benefits a worker, consider a discretized version of the model with
only two periods and in which α = β = 1, as depicted in Figure 3. In the first period, a-workers
and b-workers, who have comparable expected productivity, have comparable flow-earnings: while
only a-workers are hired, their wage is equal to 0 since pM is equal to pa . The performance of ana-worker in the first period splits the prior belief pa into posterior beliefs 0 and pa . Since earnings
are convex in beliefs, this splitting strictly benefits a-workers, whose expected flow-earnings in thesecond period now equal w2 . Hence, first-period learning causes the gap in flow-earnings to widen
in the second period.
Second, even though the benefit from learning over each short period (i.e., over [t, t + dt)) issmall, this benefit accumulates over time. Because the posterior pa is significantly more likely tooccur than the zero posterior, the delay in employment experienced by b-workers does not vanish
even as pb gets arbitrarily close to pa . By the time employers start hiring b-workers, they havealready gained substantial knowledge about a-workers’ types. Hence, the average flow-earnings of
a-workers are significantly higher than those of b-workers.
For breakthrough learning, by contrast, the delay in employment experienced by b-workers
vanishes as pb ↑ pa . Hence, a-workers do not get a chance to accumulate the benefit from employer
learning. The contrast between the two environments is formally established in Proposition C.3.
Persistent gap in employment, wage, and flow-earnings under breakdown learning.
Besides establishing the fact that spiraling continues to arise with flexible wages, we further
quantify the magnitude of such spiraling. Appendix C.2.3 computes and analyzes closed-form
expressions for the employment rate, the average wage, and the average flow-earnings of eachgroup.

19

We first show that if task scarcity is sufficiently severe—in the sense that there are more hightype workers than tasks—the employment gap persists throughout workers’ careers, even though
it decreases over time (Proposition C.6). Owing to this nonvanishing delay in employment facedby group b, the wage gap is strictly increasing for a substantial amount of time. The wage gap
starts shrinking only after employed b-workers accumulate enough learning, and shrinks to zero
only in the limit t → ∞ (Proposition C.5). See Figure 4 below for an illustration.22
Average-wage gap

0

Average-flow-earnings gap

t

0

t

Figure 4: Average-wage gap and average-flow-earnings gap as pb ↑ pa
The gap in flow-earnings is due to the combination of the wage gap and the employment
gap. Like the wage gap, the gap in flow-earnings expands early in workers’ careers and begins togradually shrink only in the latter part of their careers (Proposition C.4). But unlike the wage
gap, whether the gap in flow-earnings shrinks to zero depends on how scarce the tasks are. If
there are more high-type workers than tasks, this gap remains bounded away from zero even in
the limit t → ∞. This is due to the nonvanishing employment gap in the limit t → ∞.
Spiraling persists even with negative wages.

A natural question is whether spiraling would

disappear if workers were able to accept negative wages in return for early employment opportunities. Appendix C.2.4 shows that the persistence of spiraling under flexible wages is not driven
by the limited liability requirement, which we relax to a strictly negative lower bound on wages.
Intuitively, since both a-workers and b-workers are willing to accept such negative wages, a-workers
also lower their wages and outbid b-workers down to the lower bound. This intensifies competition
among workers and benefits only the employers. The dynamic matching of workers and employers
continues to be the same as that under limited liability, whereas all wages are now reduced by thesame amount as the lower bound. In particular, the marginal-productivity worker at any instant
now pays the employer the maximum amount that he can pay. As long as the lower bound is
not too negative—which corresponds to b-workers having a strictly positive continuation payoff at
t = 0—such a matching continues to be dynamically stable (Proposition C.7). Workers are willing to incur negative flow-earnings so as to generate signals about their productivity. Therefore,b-workers continue to face the exact same delay as under limited liability.

22

Figure 1 and Figure 4 assume the same parameter values: α = 5/4, β = 1, pa = 1/2, λℓ = 1, and r = 1.

20

4Discussion and robustness

4.1

Connection to the Matthew effect

The spiraling of a negligible productivity difference into a substantial payoff gap is reminiscent of
the cumulative advantage known as the Matthew effect (Merton, 1968). In coining the term, Merton (1968) observed that scientists of established reputation tend to receive a disproportionately
larger share of the credit for joint and simultaneous discoveries, which advances their reputation
further. He observed that the effect is intrinsically linked to scarcity of opportunities for acknowledgment. Scarcity of opportunities for workers to prove their productivity is what drives spiraling
in breakdown environments as well.
The Matthew effect has become an umbrella term for cumulative advantage that results fromvarious mechanisms in science and beyond (Rigney (2010)). Merton (1968) observed that in science, reputation buildup could be due to greater visibility in the scientific community, skewed
citation patterns, and institutional prestige and resources. A recent literature in economics explores various such mechanisms.23 We contribute a novel learning-based mechanism through which
the Matthew effect could arise, as well as a classification of workplace environments into more andless prone to this effect. Most closely related to our work is Bar-Isaac and Lévy (2022), in whichtask allocation also provides workers with opportunities to generate signals about productivity.
Whereas that paper focuses on the relationship between signal informativeness and worker’s effort, ours focuses on the importance of the direction of employer learning for whether the Matthew
effect arises.

4.2

Investment in productivity

If the workers had equal access to an opportunity to invest in productivity before entering thelabor market, would their lifetime employment prospects equalize? The answer depends on theequilibrium implications of this investment opportunity. Access to investment could presumably
level the playing field if incentives to invest were slightly stronger for group b, but it could alternatively amplify the expected productivity gap. We study this question in a variation of thesmall-market model with a pre-market investment stage that involves three steps: (i) workers
draw their pre-investment types independently of each other, according to the priors (pa , pb ); (ii)a low-type worker of either group draws his cost of investment c ∈ [0, 1] according to a cumulative
distribution function F and decides whether to invest; (iii) if he invests, he pays cost c and his
type becomes high with probability π ∈ (0, 1). A worker’s investment cost, investment decision,and post-investment type are observed only by this worker. Subsequently, workers enter the labormarket at t = 0.
23
For instance, the Matthew effect could arise due to the development of match-specific skills in labor markets
in Gibbons and Waldman (1999), due to peer effects and differential institutional resources in Oyer (2006), due tothe sensitivity of the production technology to the worker’s ability in Gabaix and Landier (2008), due to heightened
confidence from relative performance in Murphy and Weinhardt (2020), due to the friendliness of the workplaceenvironments and fertility choices in Azmat, Cuñat and Henry (2023) etc.

21

An equilibrium is characterized by a pair of cost thresholds (ca , cb ) and a pair of post-investment
beliefs about each worker’s productivity (qa , qb ). A worker i who has a low type makes theinvestment if his realized cost is below ci , and the employer’s beliefs are consistent with thisinvestment strategy. A key object in the analysis is worker i’s expected benefit Bi (qa , qb ) frominvestment given the employer’s post-investment belief pair (qa , qb ). Lemma 4.1 establishes that,in both the pure breakdown and the pure breakthrough learning environments, if the employer
believes that worker i’s expected productivity post-investment is higher than worker −i’s, then i’sbenefit from investment is strictly higher than −i’s.24 Worker i would be the first to be allocatedthe task: investment is likely to avoid a breakdown or increase the chance of a breakthrough within
the given grace period.
Lemma 4.1. In both pure learning environments, if qi > q−i , then Bi (qa , qb ) > B−i (qa , qb ). For
each i, Bi (qa , qb ) is continuously differentiable in the pure breakthrough environment, but it is
discontinuous at qa = qb in the pure breakdown environment.
The worker who is favored post-investment has a stronger incentive to invest, which in turnrationalizes the employer’s decision to favor this worker in equilibrium. This self-fulfilling force—
also noted by Coate and Loury (1993)—leads to multiple investment equilibria. In fact, investment
can reverse the initial ranking of groups: if (pa − pb ) is sufficiently small, there exist equilibria in
which worker b invests more than worker a and becomes favored post-investment. However, ourfocus is on the inequality across groups rather than on the identity of the favored group per se. We
characterize the lowest payoff inequality attained across all equilibria as pb ↑ pa in each learning
environment.
Proposition 4.1 establishes that the lowest payoff inequality continues to be zero in the purebreakthrough environment. The self-correcting property of breakthroughs is not disturbed by
the presence of investment. The proof builds on two observations. First, when pa = pb , there
always exists a symmetric equilibrium in which the workers use the same cost threshold andtherefore qa = qb . Second, under breakthrough learning the benefit from investment is continuously
differentiable in (qa , qb ). We apply the implicit function theorem to establish that, when pb is
within a small neighborhood of pa , there exists an equilibrium in which cost thresholds (ca , cb ) andpost-investment probabilities (qa , qb ) are within a small neighborhood of those in the symmetricequilibrium. This equilibrium could either preserve or reverse the prior ranking of the workers.
Proposition 4.1. Suppose that F is weakly convex. Generically,25 as pb ↑ pa , there exists anequilibrium in which the two workers’ expected payoffs as well as their post-investment probabilitiesof having a high type converge.
24

In the breakthrough environment, if qa = qb = q, then Ba (q, q) = Bb (q, q) because the employer optimally
splits her task between the workers. The workers’ benefits are equal also in the breakdown environment, assumingthat the employer randomizes equally between workers at t = 0 if qa = qb .
25
The notion of genericity
here is one in which fixing all parameters of the model except for (pa , π), the set ofvalues of (pa , π) ∈ p, 1 × (0, 1) for which the proposition does not hold has measure zero. If F is not weakly
convex, our preliminary analysis suggests that a version of this result continues to hold according to a different,
more involved notion of genericity based on prevalent and shy sets.

22

By contrast, access to investment not only fails to tame the propensity of pure breakdown
learning to magnify small prior differences, but makes it worse. Across all investment equilibria, theexpected payoffs of ex ante comparable workers are even further apart than in the no-investment
benchmark of section 2.1.
Proposition 4.2. As pb ↑ pa , in any pure-strategy equilibrium with qi > q−i , the ratio of the
expected payoff of worker −i to that of worker i is at most (1 − qi )λℓ /(λℓ + r) < 1, which is strictly
lower than the payoff ratio (1 − pa )λℓ /(λℓ + r) in the no-investment benchmark.
Spiraling persists because the benefit from investment is discontinuous in (qa , qb ). We showthat as pb ↑ pa , there exist only two equilibria with a strict ex post ranking of workers and they are
the same modulo the workers’ identities. Inequality between workers increases due to investment.
In the no-investment benchmark, the payoff ratio is pinned down by pa . Here, because the worker
who is favored post-investment—whoever that might be—has a strong enough incentive to invest,
his post-investment probability is strictly higher than pa . Therefore, for any realized investment
cost, the ratio between the payoff of the worker who is discriminated against post-investment tothat of the worker who is favored—after factoring in the investment cost—is lower than the ratio
in the no-investment benchmark. There exists also a mixed equilibrium in which qa = qb —theworkers are post-investment identical—due to worker b investing slightly more than worker a. Thisequilibrium relies on the employer randomizing asymmetrically between workers at t = 0 so as
to provide slightly stronger investment incentives for worker b. Whether such an equilibrium is
empirically plausible depends on the employer’s ability to credibly and precisely randomize in thisway.
Our characterization of equilibria allows us to compare learning environments not only interms of the workers’ payoffs, but also in terms of their investment behaviors. Proposition D.1 inthe appendix shows that with sufficiently fast learning, the worker favored (discriminated) postinvestment invests strictly more (less) under breakdowns than under breakthroughs. This ranking
is robust across all investment equilibria. Therefore, the breakdown environment is marked bygreater polarization in workers’ investment behavior. One key implication is that for sufficiently
fast learning and effective investment (π ≈ 1), the employer strictly prefers the breakdown environment to the breakthrough one because the strong investment incentives provided by the breakdown
environment guarantee that the post-investment favored worker is almost surely (in the limit as
λℓ , λh become arbitrarily large) a high type.

4.3

Inconclusive learning environments

Our baseline model has assumed that signals were type-specific: only the high type can generate abreakthrough and only the low type can generate a breakdown. We now consider environments inwhich both types generate the same signals, albeit at different rates—hence, the arrival of a signal
is inconclusive of the worker’s type. The environment is characterized by a pair of arrival rates(λh , λℓ ) ∈ R2+ such that λθ > 0 is the arrival rate of the signal if the worker’s type is θ ∈ {h, ℓ}. We
define the environment to be an inconclusive breakthrough environment if the signal suggests a high23

type (i.e., λh > λℓ > 0) and an inconclusive breakdown environment otherwise (i.e., 0 < λh < λℓ ).
If λh = λℓ , signals are uninformative.
The self-correcting property extends to inconclusive breakthrough environments. Even though
the employer does not assign the task to worker a indefinitely upon the realization of the firstsignal, there is still a time window [0, t∗ ) over which worker a should generate a signal in order
to continue being allocated the task exclusively. If no signal arrives during this time window, thebelief about worker a’s type drops to pb , at which point both workers receive the same continuation
payoff. It continues to be the case that as pb ↑ pa , the duration t∗ shrinks to zero and hence theprobability that worker a generates a signal within the time window vanishes as well. The twoworkers’ limit payoffs are therefore equal.
Proposition 4.3. For any λh > λℓ , the two workers’ payoffs converge as pb ↑ pa .
The spiraling property generalizes to inconclusive breakdown environments as well, provided
that players are sufficiently impatient. The departure from a conclusive breakdown environment
brings the complication that the employer might hire workers who have generated signals in thepast. But as long as pa > pb , worker a is the first to be hired and stays employed in the absence ofa signal. The expected time until the first signal is significant. If players are sufficiently impatient,
this already leads to a significant payoff advantage for worker a.26 Proposition 4.4 formalizes thisresult.
Proposition 4.4. For any λh < λℓ , the two workers’ payoffs do not converge as pb ↑ pa if:
λh
λℓ
1
pa +
(1 − pa ) < .
λh + rλℓ + r2

4.4

Misspecified prior belief

Spiraling arises in the pure breakdown environment even if the groups are identical but the employer mistakenly misperceives them as different. Suppose the workers have the same probability
ptrue of having a high type, but the employer believes that worker b has a lower probability
pmis < ptrue .27 In this case, even a very slight misspecification grants a large payoff disadvantage
to worker b. Worker a is still hired first based on the employer’s misspecified belief and the workers’
payoffs coincide with those in Proposition 2.2 (with pa and pb replaced by ptrue ).28
The self-correcting property of the breakthrough environment continues to hold as well, inthe sense that a slight misspecification will not have large payoff consequences for the workers.
Duration t∗ , which is analogous to the grace period in (4), corresponds to the time it takes for the
belief about worker a’s type to drift down from ptrue to pmis . As the amount of misspecification
vanishes to zero, so does t∗ . At time t∗ , the true probability that worker a has a high type is pmis ,26
The sufficient condition for spiraling can be also interpreted in terms of arrival rates (λh , λℓ ) rather than the
discount rate r: signals need to be sufficiently infrequent, i.e., λh , λℓ sufficiently small.
27
Bohren et al. (2023) refer to this as “inaccurate statistical discrimination.” Bohren, Imas and Rosenberg (2019)
identify discrimination driven by misspecified beliefs in an experimental setting.
28
See payoff expressions (5) and (6) in appendix B.

24

whereas the true probability that worker b has a high type is ptrue . However, the employer believes
that both probabilities are pmis , so she splits the task equally between workers from t∗ onwards.
We let Ûa (pmis , ptrue ) and Ûb (pmis , ptrue ) be the continuation payoffs of worker a and workerb, respectively, at time t∗ . Because each worker gets half a task but worker a has a lower trueprobability of having a high type, his payoff Ûa (pmis , ptrue ) is lower than Ûb (pmis , ptrue ). Crucially,as pmis converges to ptrue , the two payoffs get arbitrarily close. To extend the proof of Proposition
2.1 to the misspecified-prior case, we let t∗ be the time it takes for the belief to drop from ptrue topmis and replace Ui (pb , pb ) with Ûi (pmis , ptrue ) for the workers’ payoffs.
Belief misspecification is highly relevant to discussions of labor market discrimination. Langand Lehmann (2012) provide evidence that suggests the presence of widespread mild prejudiceamong employers. Our results show that prejudice, even when infinitesimally mild, has very
different implications in different learning environments. The breakthrough environment works
well against prejudice, whereas the breakdown environment propagates it further.

5

Concluding remarks

This paper studies the consequences of different employer learning environments for social groupsof comparable expected productivity. Whether the learning environment is closer to a breakdown
environment or a breakthrough one has important implications for whether discrimination persistsin the long run. Lange (2007) observed that “how economically relevant statistical discrimination
is depends on how fast employers learn about workers’ productive types.” Our analysis provides
an additional perspective: what matters for statistical discrimination is not only the speed of
employer learning, but also the direction of this learning.
Our analysis sheds light on how negative shocks to labor demand during economic downturns
impact inequality across groups. We predict that breakdown-like occupations will be prone tosignificant increases in inequality as jobs become scarcer. To the extent that low-skill occupations tend to be predominantly breakdown environments and high-skill occupations tend to be
breakthrough ones, our result is in line with substantial evidence that the groups who are hit thehardest in recessions are those who are already discriminated against and in low-skill occupations.
Moreover, our results provide a learning-based explanation for the empirical observation that racial
wage gaps are more present in low-skill occupations, which are typically breakdown-like, but arelargely absent in high-skill ones (Lang and Lehmann (2012)). By the same reasoning, we explain
why wage gaps might even widen with labor market experience in low-skill occupations, as documented by Arcidiacono, Bayer and Hizmo (2010). Our theoretical framework and our predictions
regarding the employment gap, the wage gap, and the flow-earnings gap can guide future empiricalinvestigation of discrimination in breakthrough versus breakdown occupations.
Besides these testable predictions, one natural empirical question for which our framework
can be useful is whether temporary affirmative action has long-lasting effects for groups that are
discriminated against (Miller and Segal (2012), Kurtulus (2016), Miller (2017)). The empiricalevidence on this question is mixed. One corollary of our analysis is that in breakdown environ25

ments, a requirement for employers to give a chance to group b first until the two groups’ expected
productivities are equalized would dramatically improve the prospects of b-workers. Once such arequirement is lifted, employers are willing to treat the two groups identically.29 Such recommendation does not apply to breakthrough environments.
Finally, our framework may be used to address questions beyond the scope of the current
paper. First, an employer may have to allocate multiple tasks that entail different employer
learning dynamics. For instance, if an employer has both a breakthrough task and a breakdown
task, how will she allocate the tasks among workers from comparable social groups? Second, incertain contexts the learning environment is an endogenous choice of the employer rather thanexogenously fixed. Our extension with productivity investment by workers identifies circumstances
under which the employer prefers breakdown learning due to stronger investment incentives. Moregenerally, is the endogenous choice of the learning environment more likely to lead to breakdown
or breakthrough learning? If the employer can adjust her choice of the learning environment inresponse to the workers’ expected productivities (as in Che and Mierendorff (2019)), how does
this affect the lifetime payoffs of comparable groups? Third, our framework can prove useful tounderstanding incentives for occupational segregation: workers from groups that are discriminatedagainst have an incentive to sort into breakthrough-like occupations in order to avoid spiraling.
We leave these questions to future research.

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A

Preliminary results

A.1

Distribution of performance signals for star and guardian jobs

Replicating Figure 2-2 in Baron and Kreps (1999), the dashed curves in Figure 5 depict the probability
densities of performance signals for a guardian job and a star job when the support of performance signals
is an interval. The bars depict the probabilities when the performance signals are binary, as in our baselinemodel. “Breakdown” and “no breakdown” correspond to signals in a guardian job, whereas “breakthrough”
and “no breakthrough” to those in a star job. The bars do not condition on a worker’s type, but theywould look similar if the probabilities were conditional on a low type under breakdowns (Figure 5a) andconditional on a high type under breakthroughs (Figure 5b). The figure suggests how to empirically test
whether a job is a star (breakthrough-like) job or a guardian (breakdown-like) one: a right-skewed density
suggests a star job while a left-skewed density suggest a guardian job. See footnote 5 for examples of such
empirical studies.

performance

performance
breakdownno breakthroughno breakdown(a) Guardian jobs

breakthrough(b) Star jobs

Figure 5: Distribution of performance signals (adapted from Baron and Kreps (1999))

A.2

Derivation of the safe-arm threshold p for section 2.1

Lemma A.1. For any environment described by the pair of arrival rates (λh , λℓ ), the belief threshold at
which the employer switches to the safe arm is given by
p :=

rs
.
rv + max{λh , λℓ }(v − s)30

Proof of Lemma A.1. Consider first λh > λℓ . Fixing an arbitrary prior belief p and threshold belief p < p,this corresponds to durationp/(1 − p)1∗.
log
t p, p :=λh − λℓp/(1 − p)Conditional on the worker having a high type, the payoff of the employer is

 

∗
∗
∗
∗
∗
v 1 − e−rt (p,p) + 1 − e−λh t (p,p) e−rt (p,p) v + e−λh t (p,p) e−rt (p,p) s,

whereas conditional on the worker having a low type, the employer’s payoff is∗

λℓ + re−(λℓ +r)t (p,p)
s
λℓ + rbecause the arrival probability of a breakdown at t ≤ t∗ (p, p) is λℓ e−λℓ t and the employer’s payoff is e−rt s.
Hence, the expected payoff of the employer simplifies to∗∗

VBT (p, p) := pv − pe−(λh +r)t (p,p) (v − s) + (1 − p)

λℓ + e−(λℓ +r)t (p,p)
s.
λℓ + r

The smooth pasting condition yields∂VBT (p, p)
=0
∂p

⇒

p=

rs
.
rv + λh (v − s)

Next, consider λℓ > λh . If the worker has a high type, the payoff of the employer is v: the worker
is never fired, despite whether a breakthrough arrives or not. If the worker has a low type, the payoffof the employer equals the continuation payoff from the safe arm once a breakdown is realized, which is
λℓ s/(λℓ + r). Hence, the employer’s expected payoff if she experiments with a worker given prior belief p is
VBD (p) := pv + (1 − p)λℓ s.
λℓ + rAt the threshold p = p, the employer is indifferent between the worker and the safe arm: the value matchingcondition is VBD (p) = s. This implies the thresholdp=

rs.
rv + λℓ (v − s)B

Proofs for section 2

Proof of Proposition 2.1. The employer initially allocates the task exclusively to worker a. If worker agenerates a breakdown, the employer switches to worker b. If worker a generates a breakthrough, worker a
is hired forever. In the absence of either signal, this initial allocation lasts until the employer’s belief thata has a high type decreases to pb , which happens at time t∗ , where t∗ is defined by∗pa e−λh t= pb ,∗−λtpa e h + (1 − pa )e−λℓ t∗i.e., t∗ =pa (1 − pb )1log.
λh − λℓ(1 − pa )pb(4)

If t∗ is reached without a signal, the task is split equally between the two workers until either one of themgenerates a signal or the belief about the workers’ types reaches p. If worker i generates a breakthrough,the task is allocated only to him forever after. If worker i generates a breakdown, the task is allocated only
to worker −i until −i generates a breakdown or p is reached.
We let Ui (pa , pb ) denote worker i’s payoff given belief pair (pa , pb ). Note that Ua (p, p) = Ub (p, p) =: U (p)31∗
for any p ∈ (p, 1). Over [0, t∗ ), worker a generates a breakthrough with probability pa 1 − e−λh t and a∗
breakdown with probability (1 − pa ) 1 − e−λℓ t . If a breakthrough arrives, worker a’s payoff is 1. If a
breakdown arrives at time t, worker a gets (1 − e−rt ). If no signal arrives, worker a’s payoff consists of∗the flow payoff from [0, t∗ ), which is 1 − e−rt , and the continuation payoff from time t∗ onward, which is
U (pb ). Therefore, worker a’s total expected payoff is





∗
∗
∗
λℓ
+
Ua (pa , pb ) = pa 1 − e−λh t + (1 − pa ) 1 − e−λℓ t − (1 − e−(λℓ +r)t )λℓ + r



∗
∗
∗
∗
1 − e−rt + e−rt U (pb )pa e−λh t + (1 − pa )e−λℓ t

If worker a generates a breakdown before t∗ , which occurs with instantaneous probability (1 − pa )e−λℓ t λℓ ,worker b gets discounted payoff e−rt K, where K is the continuation payoff
K := pb (1 − e

−λh t



) + (1 − pb ) 1 − e−λℓ t− (1 − e−(λℓ +r)tλℓ)
λℓ + r+ (pb e−λh t + (1 − pb )e−λℓ t )(1 − e−rt )and t := 1/(λh − λℓ ) log (pb (1 − p))/(p(1 − pb )) is the time it takes for the belief to drop from pb to thesafe-arm threshold p. On the other hand, if a generates a breakthrough over [0, t∗ ), worker b gets zero
payoff. Integrating over all t ∈ [0, t∗ ), this payoff expression becomes(1 − pa )∗λℓ 
1 − e−(λℓ +r)t K.
λℓ + r
∗∗If a does not generate any signals, which happens with probability pa e−λh t + (1 − pa)e−λℓ t , worker b gets∗
e−rt U (pb , pb ). Therefore, worker b’s total expected payoff isUb (pa , pb ) = (1 − pa )


∗
∗
∗
∗
λℓ 
1 − e−(λℓ +r)t K + (pa e−λh t + (1 − pa )e−λℓ t )e−rt U (pb ).
λℓ + r

As pb ↑ pa , t∗ → 0, so the two workers’ payoffs are equal in the limit to U (pb ).

Proof of Proposition 2.2. We first observe that in any breakdown-salient environment, the expected payoffof each player depends on λℓ but not λh . If worker a has a high type, his payoff is 1: because the belief
about θa drifts upwards in the absence of a signal, worker a continues to be hired despite whether hegenerates a breakthrough or not. If he has a low type, his payoff is (1 − e−rt ) if the breakdown arrives att, and this arrival time t follows density λℓ e−λℓ t . Hence, worker a’s expected payoff is
pa + (1 − pa )r,
λℓ + r(5)

which is independent of pb . Similarly, if the employer starts hiring worker b at time t, then worker b’s
payoff isre−rt pb + (1 − pb ).
λℓ + rConditional on worker a having a low type, this time t is distributed according to density λℓ e−λℓ t . Hence,
worker b’s expected payoff isrλℓ
pb + (1 − pb ).
(6)(1 − pa )λℓ + rλℓ + r
Evaluating the limit as pb ↑ pa , worker b’s expected payoff converges toλℓr(1 − pa )pa + (1 − pa ),λℓ + rλℓ + r32

which is equal to the fraction (1 − pa )λℓ /(λℓ + r) of worker a’s payoff.

Proof of Lemma 2.1. For any breakthrough-salient environment λh > λℓ > 0, the proof of Proposition 2.1
derives the payoff gap Ua (pa , pb ) − Ub (pa , pb ), which does not depend on U (pb ), the continuation payoff
defined as in the proof of Proposition 2.1, and it simplifies to




λℓ
−λℓ t∗−(λℓ +r)t∗−λh t∗++ (1 − pa ) 1 − e− (1 − e)Ua (pa , pb ) − Ub (pa , pb ) = pa 1 − e
λℓ + r




∗
∗
∗
∗
λℓ 
1 − e−rt − (1 − pa )pa e−λh t + (1 − pa )e−λℓ t
1 − e−(λℓ +r)t K
λℓ + r



∗
∗
∗
∗
λℓ 
= 1 − pa e−λh t + (1 − pa )e−λℓ t e−rt − (1 − pa )1 − e−(λℓ +r)t (1 + K),
λℓ + rwhere K, t, and t∗ are defined in the proof of Proposition 2.1. Because p, t∗ and K all are continuously
differentiable in (λh , λℓ ), this difference is continuously differentiable in (λh , λℓ ) as well. By a similarargument, the proof of Proposition 2.2 gives the payoff gap in any breakdown-salient environment, which
after substituting in λℓ = λ/2 − δ simplifies to:Ua (pa , pb ) − Ub (pa , pb ) =(−2δ + λ)2 (pa − pb (1 − pa )) + 4(λ − 2δ)pa r + 4r2.
(λ − 2δ + 2r)2

It is immediate that this is continuously differentiable in δ ∈ [−λ/2, 0). So it only remains to check thatthe limits of Ua (pa , pb ) − Ub (pa , pb ) and of its derivative as δ ↑ 0 and δ ↓ 0 coincide.
Substituting (λh , λℓ ) = (λ/2 + δ, λ/2 − δ) into the payoff gap for a breakthrough-salient environment
above and taking the limit δ ↓ 0, we obtainUa (pa , pb ) − Ub (pa , pb ) → 1 −λ(1 − pa )(λ(1 + pb ) + 4r)
λ2 (pa − pb (1 − pa )) + 4λpa r + 4r2=.
(λ + 2r)2(λ + 2r)2Differentiating this payoff gap with respect to δ and taking its limit as δ ↓ 0 gives us
lim

δ→0

8(1 − pa )r(λpb + 2r)∂(Ua (pa , pb ) − Ub (pa , pb ))
> 0.
=
∂δ
(λ + 2r)2

On the other hand, taking the limit of the payoff gap for the breakdown-salient environment above as δ ↑ 0,
we obtain
λ2 (pa − pb (1 − pa )) + 4λpa r + 4r2
Ua (pa , pb ) − Ub (pa , pb ) →(λ + 2r)2
which is exactly the limit obtained from the limit of a breakthrough-salient environment. Moreover,differentiating the breakdown-salient payoff gap with respect to δ and taking its limit as δ ↑ 0 gives
8(1−pa )r(λpb +2r)
. Hence, the payoff gap is continuously differentiable in a neighborhood of δ = 0.

(λ+2r)2
Proof of Proposition 2.3. Consider first the class of breakdown-salient environments parametrized by λ.
From the proof of Lemma 2.1, the payoff gap is
Ua (pa , pb ) − Ub (pa , pb ) =(−2δ + λ)2 (pa − pb (1 − pa )) + 4(λ − 2δ)pa r + 4r2(λ − 2δ + 2r)2

the derivative of which with respect to δ is8(1 − pa )r(2r + λpb − 2δpb )∂(Ua (pa , pb ) − Ub (pa , pb ))>0=∂δ(2r + λ − 2δ)3for δ ∈ [−λ/2, 0]. By the continuity of the first derivative of the payoff gap established in Lemma 2.1, there
exists a maximal δ̃ ∈ (0, λ/2] such that the gap strictly increases in δ for δ ∈ (0, δ̃) as well.


33

C

Proofs for section 3

C.1

Proofs for section 3.2 (Large market with fixed wages)

C.1.1

Stable stage-game matchings

Consider a stage game. Let G denote the CDF of the distribution of the expected productivity pk of workerk ∈ [0, α + β]. Hence, (α + β)G(p) is the mass of workers with pk 6 p. At time 0, pk is either pa or pb , so
βG(p) equals 0 if p < pb , α+β
if pb 6 p < pa , and 1 if p > pa . As workers are matched to employers, more islearned about their types, so G evolves over time. The evolution of G depends, of course, on the learning
environment. Throughout, we let G(p− ) := limx↑p G(x) denote the left-hand limit of G at p.
We first establish that employers are matched to the most productive workers in any stable stage-gamematching, provided that these workers are better than the safe arm. We look at the unit mass of the mostproductive workers, and let p∗ (G) correspond to the least productive worker in this unit mass.
Definition 3. Fix G. Let p∗ (G) be the expected productivity such that:−(1 − G(p∗ (G) ))(α + β) > 1,and(1 − G(p− ))(α + β) < 1,

∀p > p∗ (G).

To simplify notation, we sometimes omit the dependence of p∗ on G when no confusion arises. Let ps
be the belief at which a worker generates the same flow payoff to an employer as a safe arm does, that is
ps v − w = s where w = 1 is the fixed wage. Hence, ps = (s + w)/v. (We assume that pi > ps for i ∈ {a, b}.)Lemma C.1 shows that worker k is matched if pk > max{p∗ , ps } and unmatched if pk < max{p∗ , ps }.
Lemma C.1 (Most productive workers are matched). Fix G and a stable stage-game matching (D, W ).
Let d(p) denote the fraction of workers with expected productivity p who are matched to an employer.
1. Suppose that p∗ > ps . Then d(p) equals 1 if p > p∗ , and 0 if p < p∗ . If G is continuous at p∗ , thend(p∗ ) = 0. If G is discontinuous at p∗ , then d(p∗ ) is given by:(1 − G(p∗ )) (α + β) + d(p∗ ) G(p∗ ) − G(p∗− ) (α + β) = 1.

2. Suppose that p∗ 6 ps . Then d(p) equals 1 if p > ps , and 0 if p < ps . Moreover, d(ps ) can take any
value in [0, 1] subject to:(1 − G(ps )) (α + β) + d(ps ) G(ps ) − G(p−s ) (α + β) 6 1.

Proof. First, an employer never matches with worker k if pk < ps . Second, if a less productive worker ismatched, then a more productive worker must be matched as well. By way of contradiction, suppose thatfor a given p1 < p2 , a p1 worker is matched but a p2 worker is not. The employer who is matched to the
p1 worker can form a blocking pair with the p2 worker since p2 v − w > p1 v − w and w > 0.
We now argue that, if p∗ > ps , no employer takes the safe arm. Suppose otherwise. Then there exists
an unmatched worker k with pk > p∗ > ps . Then an employer who is taking the safe arm can form ablocking pair with this worker k.
We next argue that, if p∗ 6 ps , then all workers whose p is strictly above ps must be matched. Suppose
otherwise that some worker k is unmatched and pk > ps . The mass of workers whose p is weakly above
pk is strictly smaller than 1. Hence, there exists an employer who is either matched to a worker k ′ withpk′ < pk or taking the safe arm. This employer can form a blocking pair with the unmatched worker k.
Lemma C.1 characterizes the set of stable stage-game matchings for a given G. This set need not bea singleton. However, multiplicity arises only when (1 − G(ps )) (α + β) < 1 and G(ps ) − G(p−s ) > 0, as
covered by part 2 of Lemma C.1. In this case, employers are indifferent between a positive mass of workers
whose productivity is ps and the safe arms. Because stable stage-game matchings are specified uniquely
up to this case, we say that the stable stage-game matching is essentially unique.

34

C.1.2

Dynamic stability

In the dynamic setting, G evolves endogenously over time due to learning about workers’ types. To
argue that prescribing the essentially unique stable stage-game matching characterized in Lemma C.1 isdynamically stable, we show that conditions (i)-(iii) in Definition 2 are satisfied. Fix any history ht ∈ Ht .
Let Ght denote the CDF of pk at history ht and µ∗ht the essentially unique stable stage-game matching
given Ght , as in Lemma C.1.
Proposition C.1. In both the pure breakthrough environment and the pure breakdown one, prescribing thestable stage-game matching µ∗ht at every ht is dynamically stable.
Proof.

(i) At ht , a matched employer j’s flow payoff is at least s because max{p∗ (Ght ), ps } > ps . The
distribution G(ht+dt ), and hence j’s continuation payoff from t + dt on, does not depend on j’s
deviation. Hence, she does not strictly prefer to take a safe arm over [t, t + dt) and then revert toµ∗ht+dt .

(ii) Suppose that worker k is matched at history ht according to µ∗ht . Let p(t) be this worker’s expected
productivity at history ht . We focus on the case in which p(t) ∈ (0, 1), since if p(t) = 1 the worker
will be matched forever and if p(t) = 0 the worker will be unmatched forever. We next show thatthe worker does not strictly prefer to stay unmatched for [t, t + dt) and then revert to µ∗ht+dt .
(a) We first consider breakdown learning. Pick any τ > t and suppose the worker is employed over[t, τ ) for as long as no breakdown arrives. Let p(τ ) denote the worker’s expected productivity
at time τ conditional on no breakdown in [t, τ ) and OM (τ ) the expected amount of time thatthe worker is employed in [t, τ ). Then,
Z τ



1 
p(t) + (1 − p(t))e−λℓ (x−t) dx =OM (τ ) =λℓ p(t)(τ − t) + (1 − p(t)) 1 − e−λℓ (τ −t) .
λℓtExpressing p(τ ) in terms of p(t) fromp(τ ) =p(t),p(t) + (1 − p(t))e−λℓ (τ −t)

we obtain that p(τ ) and OM (τ ) satisfy the following condition:(1 − p(t))p(τ )p(τ ) − p(t) p(t).
log+
OM (τ ) =λℓ p(τ )λℓp(t)(1 − p(τ ))It is readily verified that OM (τ ) increases in p(τ ). Staying unmatched over [t, t + dt) and thenreverting to µ∗ht+dt only makes p(τ ), and thus OM (τ ), lower than their respective values onpath for any τ > t. So the worker does not strictly prefer to stay unmatched for [t, t + dt) andthen revert to µ∗ht+dt .
(b) We next consider breakthrough learning. Pick any τ > t. Let Q̃(τ ) denote the probability that
this worker generates a breakthrough in [t, τ ), p(τ ) the worker’s expected productivity at time˜ (τ ) the expected amount of time that theτ conditional on no breakthrough over [t, τ ), and OM
worker is employed over [t, τ ) conditional on no breakthrough. Then, by similar calculations
˜ (τ ) satisfy the following condition:
to those in part (a), we obtain that p(τ ) and OM
˜ (τ ) = p(t) − p(τ ) + p(t) − 1 log (1 − p(t))p(τ ) .
OM
λh (1 − p(τ ))λhp(t)(1 − p(τ ))By Bayes rule,Q̃(τ ) + (1 − Q̃(τ ))p(τ ) = p(t)35⇒Q̃(τ ) =p(t) − p(τ ).
1 − p(τ )˜ (τ ) and Q̃(τ ) decrease in p(τ ). The lower p(τ ) is, the longer
It is readily verified that both OM
the worker has been employed for over [t, τ ) conditional on no breakthrough, and the higher
the probability that this worker has generated a breakthrough over [t, τ ). Staying unmatchedover [t, t + dt) and then reverting to µ∗ht+dt only makes p(τ ) higher than its value on path, so it
˜ (τ ) and Q̃(τ ) lower than their values on path. This is true for every τ > t, so
makes both OM
the worker does not strictly prefer to stay unmatched for [t, t + dt) and then revert to µ∗ht+dt .
(iii) Suppose otherwise that worker k and employer j are not matched to each other under µ∗ht but bothstrictly prefer to be matched over [t, t + dt) and then revert to µ∗ht+dt . Under µ∗ht , any employer’s
flow payoff is at least max{p∗ (Ght ), ps }v − w. If employer j finds it strictly preferable to match withk and then revert to µ∗ht+ dt , it must be that pk > max{p∗ (Ght ), ps }. This means that worker k was
already matched under µ∗ht , so he must not strictly prefer to be matched with j since the wage isfixed.

Next, we use Lemma C.1 to fully characterize the dynamics of task allocation under each pure learning
environment given the evolution of the expected-productivity distribution.

C.1.3

Breakthrough learning

Once a worker generates a breakthrough, he is employed for the rest of time. To track how many workers
have “secured their jobs”, we let m(t) ∈ [0, 1] denote the mass of workers who have generated a breakthroughby t, so (1 − m(t)) is the mass of employers who are still learning about the type of their current match.
At t = 0, all employers are matched to a-workers due to α > 1 and pa > pb . Within the next instant,the belief for those matched a-workers who have not generated a breakthrough drops slightly below pa .
Their employers find it optimal to switch to previously unmatched a-workers, the belief for whom is pa .
This is essentially equivalent to all a-workers being matched and allocated 1/α < 1 of a task at t = 0.
In the next instant, those a-workers who have generated a breakthrough stay matched forever and areallocated one full task thereafter. Those who have not are once again allocated a fraction of a task. This
process goes on until the belief for those a-workers without a breakthrough drops to pb . We let Tb denote
this time, which is deterministic. From Tb onward, employers start allocating tasks to b-workers as well.
This Tb is the delay that is experienced by group b uniformly.
We let q(t) denote the belief for a matched worker who has not generated a breakthrough until time
t. For any t ∈ [0, Tb ), a mass (α − m(t)) of a-workers have not generated a breakthrough. Each has a
1−m(t)∈ (0, 1) of a task. Therefore, the evolution of m(t)high type with probability q(t), and is allocated α−m(t)follows:dm(t) = (α − m(t))q(t)λh

1 − m(t)dt = q(t)λh (1 − m(t))dt
α − m(t)andm(0) = 0.

(7)

By the law of large numbers, for any t ∈ [0, Tb ), q(t) satisfies:q(t)(α − m(t)) + m(t) = pa α=⇒q(t) =

αpa − m(t).
α − m(t)(8)

The value Tb is given by q(Tb ) = pb .
Starting from Tb , employers who did not have a breakthrough over [0, Tb ) start allocating tasks over alarger set of workers: a-workers who have not generated a breakthrough until time Tb and all b-workers.
The method for solving for m(t) and q(t) is similar. The evolution of m(t) is the same as (7). By the lawof large numbers, for any t > Tb , q(t) satisfies:q(t)(α + β − m(t)) + m(t) = pa α + pb β=⇒q(t) =

αpa + βpb − m(t).
α + β − m(t)The process ends when either m(t) reaches 1 or q(t) reaches ps , depending on which event occurs earlier.

36

If m(t) reaches 1 first, then all employers are matched with workers who have generated a breakthrough.
Otherwise, if q(t) drops to ps first, some employers take safe arms.
Proof of Proposition 3.1. We first show that as pb ↑ pa , Tb → 0. By the definition of Tb and the expression
for q(t) in (8), we have thatα(pa − pb ).
m(Tb ) =1 − pbTherefore, as pb ↑ pa , m(Tb ) → 0. Using the fact that (i) m(0) = 0, (ii) m(t) is independent of pb fort < Tb , and (iii) m(t) is strictly increasing in t, we conclude that Tb → 0.
Conditional on reaching Tb without a breakthrough, an a-worker has the same continuation payoff asa b-worker does. As Tb → 0, the probability of a breakthrough over [0, Tb ) goes to zero and so does theflow payoff from being allocated the task over [0, Tb ). Hence, the payoff of an a-worker approaches that ofa b-worker as Tb → 0.


C.1.4

Breakdown learning

Under breakdown learning, a matched worker stays matched as long as no breakdown occurs. At t = 0, a
unit mass of a-workers are matched with employers. When a matched worker generates a breakdown, his
employer replaces him with an a-worker who has never been matched before. This process goes on until all
the a-workers are tried. From that instant onward, an employer who just experienced a breakdown hires ab-worker who has never been tried before. We let Tb denote the first time that a b-worker is hired. Like inthe case of breakthrough learning, this Tb is again the delay that is experienced by group b uniformly.
We let m(t) > 1 be the mass of workers who have been tried before t. Among these workers, one unit
are currently employed, and a mass (m(t) − 1) of workers have generated a breakdown before t. For anyt ∈ [0, Tb ), the mass of employers who are matched to high-type workers are pa m(t), so 1 − pa m(t) arematched to low-type workers. Hence, the evolution of m(t) follows:
dm(t) = (1 − pa m(t))λℓ dt.
This along with the boundary condition m(0) = 1 pins down m(t) for any t ∈ [0, Tb ):m(t) =

1 − (1 − pa )e−λℓ pa t.
pa

If pa α < 1, then Tb is finite and solves m(Tb ) = α. Otherwise Tb is infinity.
Suppose that pa α < 1. For any t > Tb , the mass of employers who are matched to high-type workers
are pa α + pb (m(t) − α). Hence, the evolution of m(t) follows:
dm(t) = (1 − pa α − pb (m(t) − α))λℓ dt.
This along with the boundary condition m(Tb ) = α pins down m(t) for any t > Tb :m(t) =

1 − (1 − αpa )eλℓ pb (Tb −t) − α(pa − pb )pbWe let Ts denote the time at which this process of hiring untried b-workers ends. If pa α + pb β < 1, there
are fewer high-type workers than employers. Therefore, the process of hiring untried b-workers ends whenm(t) reaches α + β. If pa α + pb β > 1, there are weakly more high-type workers than employers, in whichcase the process of hiring untried b-workers never ends (so Ts = ∞). This is because learning becomes
extremely slow when the mass of employers matched with low-type workers approaches zero.
Proof of Proposition 3.2. Suppose first that αpa > 1. A b-worker’s payoff is zero, so the ratio is zero as
well. The statement holds trivially. Next, let 1 < α < 1/pa . This assumption guarantees that 0 < Tb < ∞.
Let V (pi ) denote an i-worker’s continuation payoff from the time he is first allocated the task. From the37

proof of Proposition 2.2, we know that V (pi ) = pi + (1 − pi )r/(λℓ + r). An a-worker’s expected payoff is
!
Z Tb
1
−rt
e V (pa ) dm(t) .
V (pa ) +α0
A b-worker’s expected payoff is
1
β

Z Ts

e−rt V (pb ) dm(t).

Tb

As pb ↑ pa , V (pb ) ↑ V (pa ). But because each b-worker gets a chance strictly later than any a-worker, ab-worker’s expected payoff is strictly lower than that of an a-worker.
Spiraling arises if and only if b-workers are not guaranteed to be allocated the task at time t = 0. Thatis, tasks must be relatively scarce. For simplicity, we assumed that α > 1 so that b-workers never get achance at t = 0. But even if some b-workers get a chance at t = 0, the expected payoffs of the two groups
do not converge as pb ↑ pa for as long as other b-workers are delayed. Proposition 3.3 shows that the larger
the labor force, i.e., the larger the mass of workers relative to the fixed unit mass of tasks, the greater theinequality across groups.
Proof of Proposition 3.3. The rest of this argument supposes that pa (α + β) < 1. The argument forpa (α + β) > 1 is similar, and hence omitted.
Using the expression we have for m(t) and applying the change of variables µℓ = λℓ /r, we compute theexpected payoffs of workers from each group. The ratio of the expected payoff of an a-worker to that of ab-worker is: µ 1p µ 1p
ℓ a
ℓ b
pa −1
αpa −1β(µℓ pb + 1) (µℓ + 1) αp
+
µ
(αp
−
1)
ℓ
a
αpa +βpb −1
a −1


.
−

 µ 1pℓ ba −1
αµℓ (µℓ pa + 1) (αpa − 1) αpaαp
−
αp
−
βp
+
1
a
b
+βpb −1

We take the limit of this ratio as pb ↑ pa and differentiate with respect to α and β. By applying the change
1−pa
1−αpa
of variables z = 1−αp
> 1 to replace α and β and simplify the algebra, it follows> 1 and y = 1−paa (α+β)that these two derivatives are both positive.


C.2

Proofs for section 3.3 (Large market with flexible wages)

C.2.1

Stable stage-game matchings

We first characterize the set of stable stage-game matchings for a given distribution G. Unlike fixed wages,
flexible wages lead to a situation where all employers earn identical profit.
Lemma C.2 (Equal profit across employers and linear wage for matched workers).
stage-game matching,Fix G. In any stable

1. all employers make the same profit. If some employers take safe arms, then this profit is s;
2. if worker k is matched, his wage takes the form of pk v + c1 , where c1 is a constant.
Proof. We first prove that employers make the same profit across all matched worker-employer pairs.
Suppose that workers k1 and k2 are matched to employers j1 and j2 at wages w1 and w2 respectively. Let
p1 and p2 be, respectively, the expected productivity of k1 and k2 . Suppose that employer j1 makes astrictly higher profit than j2 :vp1 − w1 > vp2 − w2 .
Worker k1 and employer j2 can form a blocking pair at wage w1 + ε. Worker k1 ’s payoff improves by ε.
Employer j2 ’s profit improves to vp1 − w1 − ε > vp2 − w2 . Hence, employers must make the same profit
across all matched pairs. This implies that the wage for a matched worker k must take the form of pk v + c1 .
What remains to be shown is that if some employers take safe arms, then all employers make a profit ofs. If an employer makes more than s, he must be matched to a worker. Then an employer who is currently
taking a safe arm can form a blocking pair with this worker.
38Based on Lemma C.2, a stable stage-game matching (D, W ) is without loss characterized by (d(p), w(p)),
where d(p) specifies the fraction of workers with expected productivity p who are matched, and w(p) =vp + c1 is the wage if a worker with expected productivity p is matched.
Lemma C.3 below shows that employers are matched to the most productive workers, provided thatthese workers are better than safe arms. This lemma is similar to Lemma C.1 for the case of fixed wages.
However, since wages are now flexible and can be pushed down to zero, the belief threshold at which theemployers start taking safe arms is s/v instead of ps = (s + w)/v.
Lemma C.3 (Most productive workers are matched). Fix G and a stable stage-game matching (D, W ).
Let d(p) denote the fraction of workers with expected productivity p who are matched to an employer.
1. Suppose that p∗ > s/v. Then d(p) equals 1 if p > p∗ , and 0 if p < p∗ . If G is continuous at p∗ , thend(p∗ ) = 0. If G is discontinuous at p∗ , then d(p∗ ) is given by:(1 − G(p∗ )) (α + β) + d(p∗ ) G(p∗ ) − G(p∗− ) (α + β) = 1.

2. Suppose that p∗ 6 s/v. Then d(p) equals 1 if p > s/v, and 0 if p < s/v. Moreover, d(s/v) can takeany value in [0, 1] subject to:  s 
s ss−(α + β) 6 1.
1−G
(α + β) + d
G
−G
v
v
v
v
Proof. First, an employer never matches with worker k if pk < s/v. Second, if a less productive worker
is matched, then a more productive worker must be matched as well. By way of contradiction, supposethat for a given p1 < p2 , a p1 worker is matched at wage w1 > 0 but a p2 worker is not. The employer
who is matched to the p1 worker can form a blocking pair with the p2 worker at wage w2 = ε > 0 sincep2 v − ε > p1 v − w1 and ε > 0.
We now argue that, if p∗ > s/v, no employer takes the safe arm. Suppose otherwise. Then there exists
an unmatched worker k with pk > p∗ > s/v. Then an employer who is taking the safe arm can form ablocking pair with this worker k.
Lastly, if p∗ 6 s/v, then all workers whose p is strictly above s/v must be matched. Suppose otherwisethat some worker k is unmatched and pk > s/v. The mass of workers whose p is weakly above pk is strictly
smaller than 1. Hence, there exists an employer who is either matched to a worker k ′ with pk′ < pk or
taking the safe arm. This employer can form a blocking pair with the unmatched worker k.
Next, we fully characterize the wage function for matched workers. If p∗ > s/v, we must distinguish twocases depending on whether there exists an unmatched worker whose productivity is arbitrarily close to p∗ .
If such a worker exists, then the wage function is pinned down uniquely. Otherwise, there is a productivity
gap between the least-productive matched worker and the most-productive unmatched worker, so theconstant c1 in the wage function can take a range of values. If p∗ 6 s/v, there always exists a safe armfor employers to take, so the wage function is pinned down uniquely. Whenever unique, the wage for amatched worker k is (pk − max{p∗ , s/v})v.
Lemma C.4 (Wage in stable stage-game matchings). Fix G and a stable stage-game matching (D, W ).
Let d(p) denote the fraction of workers with expected productivity p who are matched to an employer.
1. Suppose that p∗ > s/v.
(1.a) If for any ε > 0,Z p∗

(1 − d(x)) dG(x) > 0,p∗ −εthen c1 = −vp∗ so w(pk ) = (pk − p∗ )v.
(1.b) Otherwise, let p∗∗ be the supremum belief among workers whose belief is strictly smaller thanp∗ . Then the constant c1 in w(pk ) = vpk + c1 can take any value in [−vp∗ , −v max{p∗∗ , s/v}].
2. Suppose that p∗ 6 s/v. Then w(pk ) = (pk − s/v)v.

39

Proof. We begin by showing that the wage function must be w(pk ) = v(pk − p∗ ) in the case of (1.a).
The linearity of w(pk ) follows from Lemma C.2. First, the wage w(p∗ ) cannot be lower than zero becauseof limited liability. Second, if w(p∗ ) > 0, then the employer that is matched to a p∗ worker can form ablocking pair with an unmatched worker whose pk is arbitrarily close to p∗ .
Next we show (1.b). If there exists ε > 0 such thatZ p∗

(1 − d(x))dG(x) = 0,p∗ −εthen it must be that the fraction of workers whose belief is weakly above p∗ is exactly 1. We argue thatthe constant c1 in w(pk ) = vpk + c1 can be anything in:c1 ∈ [−vp∗ , −v max{p∗∗ , s/v}].
Pick any c1 in this range. All the employers get the same profit, which is at least v max{p∗∗ , s/v}. An
employer cannot form a blocking pair with another worker that is hired, since to attract that worker theemployer has to offer a higher wage than vpk + c1 . This will lead to a lower profit for the employer. Also,
the employer cannot form a blocking pair with a worker that is not hired, since the most profit the employer
can make is vp∗∗ , which is smaller than her current profit. Lastly, employers do not strictly prefer to takesafe arms because v max{p∗∗ , s/v} > s.
For the case of p∗ 6 s/v, the proof is similar to that for the case of (1.a), so is omitted.


C.2.2

Dynamic stability

Lemmata C.2 to C.4 characterize the set of stable stage-game matchings. Multiplicity might arise in two
possible forms. First, for certain G’s, employers are indifferent between a positive mass of workers whose
productivity is s/v and the safe arms, as covered by part 2 of Lemma C.3. Second, for certain G’s, theconstant c1 in the wage function w(pk ) = pk v + c1 can take a range of values, as covered by part (1.b)
of Lemma C.4. Whenever such G’s arise in the dynamic setting, we select a stable stage-game matchingthat (i) leaves unmatched the workers whose productivity is s/v, and (iii) assigns the employer-preferredconstant c1 = −vp∗ (G) in the wage function. This selection criterion is for ease of exposition only; thepropositions below hold even with a different selection because such multiplicity arises only at finitely many
instants of the entire time horizon.
Fix any history ht ∈ Ht . Let Ght denote the CDF of pk at history ht , and let µ∗ht be the stablestage-game matching given Ght , as characterized by Lemmata C.2 to C.4 and selected according to theprevious paragraph. For any G, we let pM (G) := max{p∗ (G), s/v} and call it the marginal productivity
given G. We proceed to demonstrate that µ∗ is dynamically stable.
Proposition C.2. In both the pure breakthrough environment and the pure breakdown one, µ∗ is dynamically stable.
Proof. Pick any ht ∈ Ht . We want to show that conditions (i)-(iii) in Definition 2 are satisfied in eachlearning environment.
(i) If employer j is matched to a worker under µ∗ht , her flow payoff on path is at least s. The distribution
G(ht+dt ), and hence j’s continuation payoff from t + dt on, does not depend on j’s deviation. Hence,
she does not strictly prefer to take a safe arm over [t, t + dt) and then revert to µ∗ht+dt .
(ii) Suppose that worker k is matched at history ht according to µ∗ . Let p(t) be this worker’s probabilityof having a high type at history ht . We next show that he does not strictly prefer to stay unmatched
for [t, t + dt) and then revert to µ∗ht+dt .
(a) We first consider breakdown learning. Pick any τ > t + dt. Let Q(τ ) denote the probability
that this worker has generated a breakdown in [t, τ ), and p(τ ) denote the probability that thisworker has a high type at time τ conditional on no breakdown in [t, τ ). By Bayes rule,(1 − Q(τ ))p(τ ) = p(t).

40The worker’s expected flow-earnings at time τ arep(τ ) − pM (Ghτ )(1 − Q(τ )) max 0, p(τ ) − pM (Ghτ ) v = max 0, p(t)
v
p(τ )

(9)

which is weakly increasing in p(τ ). Staying unmatched over [t, t + dt) and then reverting toµ∗ht+dt only makes p(τ ) lower than its value on path, so the worker will not reject the match.
(b) We next consider breakthrough learning. Pick any τ > t + dt. Let Q̃(τ ) denote the probability
that this worker has generated a breakthrough in [t, τ ), and p(τ ) denote the probability that
this worker has a high type at time τ conditional on no breakthrough in [t, τ ). By Bayes rule,Q̃(τ ) + (1 − Q̃(τ ))p(τ ) = p(t).
The worker’s expected flow-earnings at time τ are
Q̃(τ )(1 − pM (Ghτ ))v + (1 − Q̃(τ )) max 0, (p(τ ) − pM (Ghτ ))von= max Q̃(τ )(1 − pM (Ghτ ))v, (p(t) − pM (Ghτ ))v
which is weakly increasing in Q̃(τ ). Staying unmatched over [t, t + dt) and then reverting toµ∗ht+dt only makes Q̃(τ ) lower than its value on path, so the worker will not reject the match.
(iii) Suppose that worker k and employer j are not matched to each other under µ∗ht . We next show that
there is no wage w > 0 such that both k and j strictly prefer to be matched to each other at flow
wage w over [t, t + dt) and then revert to µ∗ht+dt in either learning environment.
If k is matched to another employer under µ∗ht , w needs to be strictly higher than worker k’s current
wage. This implies that employer j’s flow payoff will be strictly lower than his current flow payoff.
Hence, j does not strictly prefer to pair with k over [t, t + dt).
If k is unmatched, this means that pk 6 pM (Ght ). But employer j’s flow payoff on path is at leastpM (Ght )v. So employer j will not find it strictly profitable to be matched to k.

Our next proposition shows that the contrast between breakthrough and breakdown environments interms of group inequality continues to hold. In particular, flexible wages do not close the earnings gap
between group a and b in the breakdown environment.
Proposition C.3. Given matching µ∗ , as pb ↑ pa the average lifetime earnings of a-workers converge tothose of b-workers under breakthroughs but not under breakdowns.
Proof. Consider first the breakthrough environment. Let Tb be as defined in appendix C.1.3. Because
α > 1, for an initial period t ∈ [0, Tb ), only a-workers are matched. If an a-worker has not achieved abreakthrough by Tb , his probability of having a high type is pb . In this case, he has the same continuation
payoff as a b-worker does. As pb ↑ pa , Tb → 0. Hence, an a-worker’s earnings advantage vanishes as well.
We now consider the breakdown environment. Equation (9) in the proof of Proposition C.2 establishedthat a worker who has been matched for longer has higher expected flow-earnings than a worker who has
been matched for a shorter period. Hence, at any t the expected flow-earnings of an a-worker are strictly
higher than those of a b-worker. Moreover, the uniform delay for group b, Tb , does not converge to zero aspb ↑ pa , hence an a-worker’s earnings advantage due to [0, Tb ) does not converge to zero either. Hence, theaverage lifetime earnings of a-workers are strictly higher than those of b-workers.


C.2.3

Wage, earnings, and employment gaps under breakdown learning

In this subsection we normalize v to 1 without loss of generality. We let Ea (τ ) (resp., Eb (τ )) denote theaverage flow-earnings of a-workers (resp., b-workers) at any time τ > 0. To simplify exposition, we assume41

that (i) α > 1, (ii) αpa < 1, and (iii) αpa + βpb > 1. The first two conditions ensure that the uniform
delay for b-workers is positive but finite, i.e., 0 < Tb < ∞. The third condition ensures that the pool of
new workers is not exhausted before all employers are matched to high-type workers. That is, there are
more high-type workers than employers available. In the paragraph after the proof of Proposition C.4, we
discuss what happens when these conditions are not satisfied.
We first solve for the expected flow-earnings at time τ of an i-worker who is first matched at timet 6 τ . From expression (9), this expected flow-earnings are given by
pM (Ghτ ),pi 1 −q(pi , τ − t)where pi is the prior belief of an i-worker, pM (Ghτ ) is the marginal productivity at time τ , and q(pi , τ − t)denotes the employer’s belief at time τ about an i-worker who is first matched at time t 6 τ and has not
generated a breakdown over [t, τ ). The marginal productivity pM (Ghτ ) is given by:(
pa if τ 6 Tb
M
p (Ghτ ) =pb otherwise,
where the delay for group b is Tb = λℓ1pa log



1−pa
1−αpa

q(pi , τ − t) =

. Moreover,pi.
pi + (1 − pi )e−λℓ (τ −t)

In order to calculate the average flow-earnings of i-workers at any τ , we also need the density over the timeat which each i-worker is first matched. From appendix C.1.4, we have the expression for m(t), the massof workers who have been tried until time t:−λℓ pa t 1 − (1 − pa )e
if t 6 Tbpam(t) =
1 − (1 − αpa )eλℓ pb (Tb −t) − α(pa − pb )otherwise.
pb

A unit mass of a-workers are matched at time 0. For any t ∈ (0, Tb ), new a-workers are tried at rate m′ (t).
For any t > Tb , new b-workers are tried at rate m′ (t). Therefore, for any τ > 0, the average flow-earningsof a-workers are!
 Z Tb ∧τ 
pM (Ghτ )pM (Ghτ )1′
pa 1 −+
pa 1 −
m (t) dtαq(pa , τ )q(pa , τ − t)0which simplifies to:(1 − pa ) 1 − e−λℓ pa τ




α
 p1
Ea (τ ) :=
a
pa −1

e−λℓ τ
p
(αp
−
1)
b
a

αp
−1
a


− pa pb + pa
αif τ 6 Tb
otherwise.

The calculation for the average flow-earnings of b-workers is similar. For any τ < Tb , no b-worker is tried,so the average flow-earnings of b-workers are 0. For τ > Tb , the average flow-earnings are:Z
pM (Ghτ )1 τm′ (t) dt.
pb 1 −β Tb
q(pb , τ − t)

42Hence,

0





 ppb
 p1

a
a
p
−1
p
−1
−λ
p
τ
−λ
τ
a
a
ℓ
b
ℓ
Eb (τ ) := (αpa − 1)e− pb αpa −1e+ pb − 1
αpa −1βif τ 6 Tb
otherwise.

At the start of the horizon, there exists a gap in the average flow-earnings between groups because
Ea (τ ) > 0 = Eb (τ ) for any τ ∈ (0, Tb ]. Moreover, this gap persists over the entire horizon and it does notdisappear even in the long run, as the following proposition shows. This is because even as τ → ∞, there
exist a non-zero mass of b-workers who never get tried.
Proposition C.4 (Persistent gap in average flow-earnings under breakdowns). Suppose that α > 1 > pa α
and pa (α + β) > 1. In the limit pb ↑ pa , there exists T̃ ∈ (Tb , ∞) such that the gap in average flowearnings, Ea (τ ) − Eb (τ ), is strictly increasing for τ < T̃ and strictly decreasing for τ > T̃ . The limit
limτ →∞ (Ea (τ ) − Eb (τ )) is strictly positive.
Proof. The assumption that α > 1 > pa α ensures that Tb ∈ (0, ∞). For any τ ∈ [0, Tb ), the gap Ea (τ ) −Eb (τ ) is simply Ea (τ ), which is strictly increasing in τ .
For any τ ∈ [Tb , ∞), the gap Ea (τ ) − Eb (τ ) is increasing in τ if and only if(α + β)



1−pa
1−αpa

 p1 −1aαe−λℓ (1−pa )τ
> 1.

The LHS is decreasing in τ , so this inequality holds when τ is small enough. Since the LHS equals zero
when τ → ∞ and the inequality holds when τ = Tb , the gap in average flow-earnings is first strictlyincreasing and then strictly decreasing. In the limit of τ → ∞, the gap in average flow-earnings is strictly
positive:(1 − pa )(αpa + βpa − 1)
> 0.
lim (Ea (τ ) − Eb (τ )) =τ →∞βIf α < 1, then Tb = 0. If αpa > 1, then Tb = ∞. The results for both cases are similar to those in
Proposition C.4, so we omit them. If pa α + pb β 6 1 instead, all b-workers will obtain a chance in thelong run. Even though for each τ > 0 there exists a non-zero gap in average flow-earnings, as t → ∞ theaverage flow-earnings of the two groups converge.
We next characterize the average wage of a-workers and that of b-workers at each τ . Let Wa (τ ) andWb (τ ) be the average wage for the two groups. Let Q(pi , τ − t) be the probability that no breakdown has
occurred up to time τ if the i-worker is first matched at time t:
Q(pi , τ − t) = (1 − pi )e−λℓ (τ −t) + pi .
The average wage of a-workers at time τ is:
R Tb ∧τ0
q(pa , τ − t) − pM (Ghτ ) m′ (t)Q(pa , τ − t)dt + q(pa , τ ) − pM (Ghτ ) Q(pa , τ ),R Tb ∧τm′ (t)Q(pa , τ − t)dt + Q(pa , τ )0which simplifies to:(p − 1)e−λℓ pa τ − pa + 1 a
αpa eλℓ τ
Wa (τ ) =
 1 − pb
 αpa eλℓ τ + (1 − αpa ) pa −1 pa
αpa −143

if τ 6 Tb
otherwise.

The average wage of b-workers at time τ > Tb is:Rτq(pb , τ − t) − pM (Ghτ ) m′ (t)Q(pb , τ − t)dt
TbRτ,m′ (t)Q(pb , τ − t)dt
Tbwhich simplifies to

Wb (τ ) =



0



  pa −1  ppab
αpa −1if τ 6 Tb


 p1

pa −1
e−λℓ pb τ − pb αpe−λℓ τ + pb − 1
a −1
 p1

a
pa −1
e−λℓ τ − 1
αpa −1aotherwise.

Proposition C.5 (Persistent wage gap under breakdowns). Suppose that α > 1 > pa α and pa (α + β) > 1.
In the limit pb ↑ pa , there exists T̂ ∈ [Tb , ∞) such that the wage gap Wa (τ ) − Wb (τ ) is strictly increasingfor τ < T̂ , and strictly decreasing for τ > T̂ .
Proof. For any τ ∈ [0, Tb ), the wage gap Wa (τ ) − Wb (τ ) is simply Wa (τ ), which is strictly increasing in τ .

− p1
a
pa −1
pa −1
eλℓ τ . We can
For any τ ∈ [Tb , ∞), we apply the change of variables x = αp,y=αpa −1a −1rewrite the wage gap asxy y −pa − y(pa +x−1)−p
a +1
,
(10)
y−1
where x > 1 since 0 < pa < αpa < 1 and y > 1 since τ > Tb . Note also that y is monotone increasing in τ .
This wage gap (10) is increasing in y if and only ifH(y) := xy pa y 2 (pa + x − 1) − pa + 1 + (−(y − 1)pa − 1)(y(pa + x − 1) − pa + 1)2 > 0.

We next argue that H(y) is positive if and only if y is small enough.
′′First, it is readily verified that H(1) = H ′ (1) = 0, H(∞) < 0, and H (4) (y) < 0. This shows that H (y)′′is concave. It is also readily verified that H (∞) < 0. There are three cases to consider regarding theshape of H ′′ (y), with the third case being impossible:′′′′(1) If H (1) > 0, then as y increases, H (y) is first positive and then negative.
′′

′′′

′′

(2) If H (1) 6 0 and H (1) 6 0, then H (y) is negative for all y > 1.
′′′′′

(3) The last case is H (1) 6 0 but H (1) > 0. We show that this is not possible since it requires that2(pa + x) < pa x2 + 2,
pa (x + 6)x + 4(x − 3)x + 6 < 6pa ,which cannot hold simultaneously given that x > 1 and pa ∈ (0, 1).
If case (1) holds, then H(y) is first convex then concave. This, together with H(1) = H ′ (1) = 0 andH(∞) < 0, shows that H(y) is first positive and then negative. If case (2) holds, then H(y) is concave forall y > 1. This, together with H(1) = H ′ (1) = 0, shows that H(y) is negative for y > 1.
Finally, we also characterize the employment gap between groups. Let Pa (τ ) (resp., Pb (τ )) denotethe fraction of a-workers (resp., b-workers) that are allocated a task at time τ . We refer to Pi (τ ) as theemployment rate for group i. The following proposition shows that at any time τ , a-workers have a strictly
higher chance of being employed than b-workers. Moreover, the gap Pa (τ ) − Pb (τ ) does not vanish to zeroeven as τ → ∞.
Proposition C.6 (Persistent employment gap under breakdowns). Suppose that α > 1 > pa α and pa (α +β) > 1. In the limit as pb ↑ pa , Pa (τ ) − Pb (τ ) is weakly decreasing in τ andlim (Pa (τ ) − Pb (τ )) =

τ →∞

44

pa (α + β) − 1> 0.
βEi (τ ). From the equations for Ei (τ ) and Wi (τ ), we calculate
Proof. The employment rate Pi (τ ) equals Wi (τ )Pi (τ ) as pb ↑ pa :
1if τ 6 Tbα1/pa !
Pa (τ ) =
1
−
p
1
a
−λ τ

otherwise,pa + α e ℓ (1 − αpa ) 1 − αp
a



0
1/pa !
Pb (τ ) = 1
1
−
p
a
−λ τ

 β (1 − αpa ) 1 − e ℓ
1 − αpa

if τ 6 Tb

otherwise.

The employment gap Pa (τ ) − Pb (τ ) is given by
Pa (τ ) − Pb (τ ) =

(1

α

pa +

if τ 6 Tb


1
1
α + β



(1 − αpa )e−λℓ (τ −Tb )− β1 (1 − αpa )otherwise.

It can be readily observed that (i) for τ 6 Tb , Pa (τ ) − Pb (τ ) is constant in τ , (ii) for τ > Tb , it strictly. Because pa (α + β) > 1, this limit is strictly
decreases in τ , and (iii) as τ → ∞, Pa (τ ) − Pb (τ ) → pa (α+β)−1
βgreater than 0.
45