ying tung chan and chi man yip · 11/23/2018 · ying tung chan and chi man yip abstract why are...
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SEARCH RELATIVITY
Ying Tung Chan and Chi Man Yip∗
Abstract
Why are the unemployment rates lower at higher educational levels? Why is the aggregate unem-
ployment rate (AUR) uncorrelated with the distribution of educational level? Why is the unemploy-
ment rate of the postgraduates about half the AUR? We propose a theory to answer these questions.
Our theory suggests that a job-finding rate increases with the relative position of a search intensity dis-
tribution (search relativity). We illustrate our theory via a search-theoretic model and derive a novel
formula to disaggregate the AUR into the unemployment rates by educational level simply using one
easily accessible sufficient statistic—the distribution of educational attainment in the labor force. In
most states, the distributions of the predicted and the actual unemployment rates by educational level
are found statistically identical, suggesting that search relativity is one of the key determinants of a
job-finding rate and is the major source of heterogeneity in the unemployment rate.
KEYWORDS: Distribution of Search Intensity, Matching Technology, Unemployment Distribu-
tion.
JEL Classification Numbers: C78, D3, E24, J64.
∗Chan: The Research Institute of Economics and Management, Southwestern University of Finance and Economics,Chengdu, China (email: [email protected]); Yip: Department of Economics, the University of Calgary, 2500 Univer-sity Dr. N.W., Calgary, AB, T2N 1N4 (email: [email protected]) We would like to thank Scott Taylor and Trevor Tombefor their patience, guidance, encouragement support, and our many insightful conversations. We thank David Card, NicoleFortin, Junichi Fujimoto, Fabian Lange, Charles Ka Yui Leung, Lucija Muehlenbachs, Ron Siegel, Pascal St-Amour, StefanStaubli, Atsuko Tanaka, Yikai Wang, Jean-Francois Wen, Alexander Whalley, Russell Wong, and seminar participants at the2018 SOLE@MEA Conference, the 2018 WEAI Conference, the 2018 SEA Conference, the 2017 SOLE Annual Conference,the 2017 ASSA Annual Conference, the 2017 China Meeting of the Econometric Society, the 2016 Canadian Economics As-sociation Annual Conference, the City University of Hong Kong, and the University of Calgary for their useful comments.We owe our classmates Younes Ahmadi, Yutaro Sakai, Yan Song, Meng Sun, and Qian Sun for their suggestions.
1 Introduction
At least since Keynes (1936), economists have been studying the determinants of the average wage
rate and the aggregate unemployment rate (AUR). Over the past few decades, the distributions of
wages, income, wealth, consumption, health, mortality, etc. have attracted much attention from both
empirical and theoretical studies.1 And yet, perhaps surprisingly, works, especially theoretical ones,
on the unemployment distribution and its properties are sparse. This paper investigates why the
unemployment rate differs across demographic groups.
Human capital is one of the key determinants of many labor market variables, such as wages and
their distribution (Mincer, 1958; Ben-Porath, 1967; Griliches, 1977; Becker, 1994). While schooling
is one of the major channels to accumulate human capital, heterogeneity in labor market outcomes
by educational level is huge. Inspired by the human capital theory, this paper pays attention to the
heterogeneity in the educational unemployment rate—the unemployment rate by educational level.
However, the theory proposed by this paper can be applied to explain the heterogeneity in the unem-
ployment rate by any demographic group, not just by educational group.
1.1 Two Seemingly Unrelated Features of Unemployment
We begin with the documentation of two seemingly unrelated features of unemployment, namely the
statistical puzzle of unemployment and the magic number of “one-half”.
The Statistical Puzzle of Unemployment. Figure 1 shows that unemployment rates are lower at
higher educational levels in the United States. Each line represents the unemployment rate of each
educational level during 1994-2015. The higher the educational level, the lower is the unemploy-
ment rate. While these five lines are completely segregated, this negative relationship holds over 20
years. Indeed, the documentation of this relationship has a long history (Ashenfelter and Ham, 1979;
Mincer, 1991), and we extend the documentation period of this relationship to 2015.2
Statistically, the AUR is likely lower at a higher fraction of high-educated workers in any econ-
omy. The AUR is the weighted average of the unemployment rate of each educational category, with
the weight equal to the fraction of each category. While high-educated workers have a lower risk
of unemployment than their low-educated counterparts, the weighted average of the unemployment
rate, the AUR, is expected to decrease with the fraction of high-educated workers.
However, Figure 1 illustrates that the AUR and the fraction of the high-educated are uncorre-
lated during 1994-2015. Moreover, we show in the online Appendix A that these two variables are
1Readers who are interested in the empirical studies on wage distributions and its dynamics are referred to Juhn et al.(1993), DiNardo et al. (1996), Katz et al. (1999), Lemieux (2006), and Autor et al. (2008). Those who interested in thetheoretical works are referred to Galor and Zeira (1993), Burdett and Mortensen (1998), Krusell and Smith (1998), Postel-Vinay and Robin (2002), Shi (2009), Hornstein et al. (2011), Moscarini and Postel-Vinay (2013), Jones and Kim (2014), andGabaix et al. (2016).
2Also, Topel (1993) finds that men of lower wages have a higher risk of unemployment. While men with higher educa-tional levels receive a higher average wage, his finding coheres with our documentation.
1
uncorrelated and their cyclical components are also uncorrelated in each year during 1994-2015. Ob-
viously, the increasing fraction of the high-educated must have adjusted the unemployment rates of
the high- and/or the low-educated to reconcile the two documentations in Figure 1. Interestingly, this
adjustment always ceases at the situation, in which the AUR and the fraction of the high-educated are
uncorrelated, and thus raises a statistical puzzle: what economic mechanism makes the AUR and the
fraction of the high-educated always uncorrelated?
Figure 1: The Statistical Puzzle of Unemployment
06
1218
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oym
ent R
ate
by E
duca
tiona
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inm
ent (
in %
)
1994 2000 2005 2010 2015Year
High-School DropoutsHigh-School GraduatesAssociate's Degree HoldersBachelor's Degree HoldersPostgraduates
ME
NHVT
MA
RI
CTNY NJ
PAOHINIL
MI
WIMN
IA
MO
NDSD
NE
KSDE MD
VA
WV
NC
SCGAFLKY TNAL
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AR LA
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TXMTID
WY
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UT
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ORCA
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HI
02
46
8Ag
greg
ate
Une
mpl
oym
ent R
ate
(in %
)
20 25 30 35 40 45Proportion of High-Educated Workers
Notes: The left figure displays the educational unemployment rate. The right figure displays the correlation between theaggregate unemployment rate and the fraction of high-educated workers in the United States. The high-educated are definedas those who are bachelor’s degree holders and the postgraduates. Each dot illustrates the average aggregate unemploymentrate and the average fraction of high-educated in each state during 1994-2015. DC is excluded because it is an outlier. In thesefigures, data are collected from the Current Population Survey 1994-2015. Samples are restricted to labor force participantsaged 25-60. Samples below 25 years old are excluded because Ph.D holders are likely graduated and enter the labor marketafter 25.
The Magic Number of “One-Half”. Figure 2 displays the correlation between the unemploy-
ment rate of the postgraduates and the AUR in the United States.3 The dash line is the fitted value, and
the solid line represents the mathematical relation in which the unemployment rate of the postgrad-
uates is half the AUR. Clearly, the two lines are close to each other, and the observations lie around
these two lines, suggesting that the unemployment rate of the postgraduates is about half the AUR.
Why is the unemployment rate of the postgraduates about half the AUR? Can it be other numbers? If
not, what economic mechanism creates the number of “one-half”?
The purpose of this paper is twofold: to explain that the heterogeneity in the unemployment rate
emerges mainly via a subtle relation between a job-finding rate and the distribution of search inten-
sity, and to show that the two seemingly unrelated features of unemployment is mainly attributable
to this subtle relation. Despite a voluminous literature on the AUR (Ljungqvist and Sargent, 2008;
3This feature can also be found in Canada and the United Kingdom using the Canadian Labor Force Survey and theUnited Kingdom Labor Force Survey, respectively.
2
Figure 2: The Correlation between the Aggregate Unemployment Rate and the Unemployment Rate ofthe Postgraduates, 1994-2015
MENH
VT
MA RICT NYNJPA
OHIN
IL
MIWI
MN
IAMO
ND SDNE
KS DEMDVA
WV
NCSC
GA
FL
KYTN
AL
MSAR
LAOK
TXMT
ID
WY
CO
NMAZ
UT
NVWA
ORCA
AKHI
01
23
45
Une
mpl
oym
ent R
ate
of th
e Po
stgr
adua
tes
(in %
)
2 3 4 5 6 7Aggregate Unemployment Rate (in %)
Slope: 0.44
Notes: This figure displays the relationship between the aggregate unemployment rate and the unemployment rate of post-graduates. The horizontal axis is the average aggregate unemployment rate of the corresponding state during 1994-2015. Thevertical axis is the average unemployment rate of the postgraduates in the corresponding state during 1994-2015. The dot lineis the fitted value, in which the slope is 0.44. The solid line represents the unemployment rate of the postgraduates is half theaggregate unemployment rate. Data are collected from the Current Population Survey 1994-2015. Samples are restricted tolabor force participants aged 25-60.
Elsby et al., 2009; Davis et al., 2010; Shimer, 2012; Sahin et al., 2014), discussions on unemployment
distributions are rare because the features of an unemployment distribution seem unrelated to each
other. If the properties of the unemployment rate by educational attainment are mastered well, the
properties of the AUR will be well understood, not vice versa. Therefore, to study the two seemingly
unrelated features of unemployment not only opens the “black box” of the unemployment distribu-
tion, but it also complements the literature on the AUR, enhances our understanding of the functioning
of the labor market, and provides insights how labor market policies could affect the unemployment
distribution.
1.2 Search Relativity Theory and Related Literature
Ask anyone about the relationship between a search intensity level and a job-finding rate, and what
first comes to his or her mind is likely that a job-finding rate increases with a search intensity level.
Despite no economy-wide experiment, we may observe that those who search more intensively tend
to get rid of unemployment faster. This may formulate our belief that a higher level of one’s search
intensity increases his or her job-finding rate.
This paper argues that a matching function maps one’s percentile rank in a search intensity dis-
tribution into his or her job-finding rate. Hence, the heterogeneity in job-finding rates arises from
the relative position in an intensity distribution (hereafter, we call it search relativity when appropri-
3
ate). This paper shows that this search relativity accounts in large part for the heterogeneity in the
unemployment rate.
This paper demonstrates our search relativity theory using a search-theoretic model with hetero-
geneous workers. The key innovation of our theory is that an unemployed worker chooses his own
search intensity level to maximize his value function, and his job-finding rate increases with his rela-
tive position in an intensity distribution. In pursuit of a higher job-finding rate, unemployed workers
are required to have a slightly better curriculum vitae, perform slightly better in a job interview, and
devote slightly more time to seek jobs than the candidates ranked slightly higher in the intensity lad-
der. To increase one’s search intensity without climbing up an intensity ladder does not enhance his
job-finding rate. Hence, the determination of the optimal search intensity is challenging because it re-
quires unemployed workers to consider both a marginal search cost and whether an additional search
effort allows them to climb up an intensity ladder. We introduce the notion of the Nash equilibrium to
solve this problem. One of the striking results is that the distributions of search intensity and human
capital in unemployment are identical in the Nash equilibrium.
According to our theory, increasing ones’ search intensity level increases their relative positions
in an intensity ladder and thus their job-finding rates: their unemployment rates decline. Meanwhile,
it creates a negative externality on others: there exist workers whose rank declines. Consequently, the
rise in their search intensity levels has no effect on the AUR; a higher fraction of workers who search
intensively has no effect on the AUR. Analogously, because unemployed workers with a higher edu-
cational level tend to search more intensively, their higher percentile rank in the intensity distribution
returns them higher job-finding rates via a matching function. Consequently, their unemployment
rates are lower, and a higher fraction of high-educated workers and the AUR are independent, ex-
plaining the statistical puzzle of unemployment.
This theory also provides a rationale for the puzzling empirical finding from Kroft and Pope
(2014): the AUR is unaffected by an increasing popularity of job-finding websites like Craigslist.
Theoretically, an increasing popularity of this website may reduce the marginal search cost of Craigslist’s
users and incent them to search more intensively. The increase in their search intensity levels does
enhance their relative positions in an intensity ladder. Meanwhile, it creates a negative externality on
the nonusers of Craigslist: their rank declines. While Craigslist may help their users get rid of unem-
ployment faster, it slows the job search process for the nonusers. Overall, the increasing popularity
of job-finding websites has no effect on the AUR.
The magic number of “one-half” is attributed to a statistical property: percentile ranks of any
distribution are uniformly distributed between zero and one, with its average one-half. With the
highest search benefit, the postgraduates tend to search the most intensively and thus rank top in
an intensity ladder. Their percentile rank of search intensity is always one, twice the average of
any economy. Consequently, the postgraduates’ job-finding rate is twice the average, and thus they
always get a job twice as fast as the average. Therefore, the unemployment rate of the postgraduates
4
is about half the AUR, explaining the puzzling feature of the magic number of “one-half”.4
The success of our theory in explaining the two features suggests that search relativity is one of the
key determinants of a job-finding rate. Since the distributions of search intensity and human capital
in the unemployment are identical in the equilibrium, the distribution of educational level can used
as the proxy for the distribution of human capital (and thus search intensity). This proxy allows us to
derive a novel formula to disaggregate an aggregate unemployment rate into the unemployment rate
by educational attainment without the information on search intensity levels. The formula requires
only two inputs, the AUR and the distribution of educational levels in the labor force, both of which
are easily accessible.
Using the United States Current Population Survey (US CPS) 1994-2015, we disaggregate the
annual AUR into the educational unemployment rates in each of the 50 states. Most of the null
hypotheses, in which the actual and the derived educational unemployment rates are from the same
distribution, cannot be rejected at any conventional level of significance. For example, these null
hypotheses cannot be rejected for high-school graduates and bachelor’s degree holders in 46 and
43 out of 50 states at five percent significance level. Using the least number (two) of the inputs, our
simple formula allows economists, policymakers, and the public to accurately map the AUR to a more
relevant information—the educational unemployment rates. The accuracy of the formula implies that
our model not only explains the two seemingly unrelated features of unemployment qualitatively, but
also predicts the magnitudes of the unemployment rates well over the past two decades.
To the best of our knowledge, there exists no work that documents or explains the two seemingly
unrelated features of unemployment. Cairo and Cajner (2016) is the closest work that studies the
educational unemployment rate. Our paper differs from theirs in the objective. Focusing on the
relationship between the level and the volatility of the unemployment rate, their paper succeeds in
explaining why more-educated workers experience lower unemployment rates and lower employment
volatilities. In contrast, our paper explains the two seemingly unrelated features of unemployment
and how an unemployment distribution emerges, focusing mainly on the relationship between the
AUR and the educational unemployment rate.
Cairo and Cajner (2016) is an extension of the canonical search and matching model with on-
the-job training and endogenous separations. In contrast, our model incorporates search relativity
into the canonical search-theoretic model. Importantly, the derived relation between the AUR and
the educational unemployment rate in our model does not depend on the separation rate. Our work
complements theirs in that the canonical model with search relativity and the endogenous separations,
the key feature in Cairo and Cajner (2016), could explain well both the two seemingly unrelated
features of unemployment and the relation between the level and the volatility of unemployment.
4We realize that the annual salary of the postgraduates, on average, is not the highest in some European countries: theirsearch benefits are not the highest in the unemployment. In this case, their unemployment rates may not be half the AUR.Our theory may still be applicable in the sense that the unemployment rate of any educational group with the highest searchbenefits is half the AUR.
5
This paper is structured as follows. The basic model environment is described in Section 2.
In Section 3, we characterize the steady-state equilibrium and show that the model captures the two
seemingly unrelated features of unemployment qualitatively. Section 4 evaluates the predictive power
of our model. Section 5 concludes the paper.
2 The Basic Model
This section presents our search relativity theory through the workhorse of a search-theoretic model.
We aim to construct a simple, yet intuitive, model to shed light on the mechanism through which
search relativity affects job-seeking behaviors. Despite the abstraction of other labor market fea-
tures, our model is flexible, could be easily extended to incorporate other features, and, of paramount
importance, captures well the features of unemployment rates.
In this model, time is continuous. There is a continuum of profit-maximizing firms who are risk-
neutral and discount future at a rate r > 0. A firm could be filled by at most one worker, and any
worker can take up at most one position. Since one firm is basically equivalent to one vacancy, we
follow the tradition in this literature to address a firm as a vacancy to avoid confusion.
There is a unit measure of utility maximizing workers who are risk-neutral, infinitely lived, and
share an identical real discounted rate r.5 Each is either employed with earnings w, or unemployed
with home production z > 0.6
Workers are ex ante heterogeneous: they differ in a human capital level δ. We denote by H(δ)
and h(δ) the cumulative distribution function and the corresponding density function of δ. For ease
of exposition, we assume that the distribution is continuous over its support, and the lower support
of H(δ) exceeds home production z so that it makes sense that workers are willing to sign a contract
during a job interview.
We denote by JE(δ) and JU (δ) as the value functions of employment and unemployment. An
employed worker of type δ receives a wage w and faces a separation shock at a Poisson rate λ.
When the shock arrives, the employed worker becomes unemployed. Hence, the value function of
5This simplification is common in this literature (Mortensen and Pissarides, 1994; Moen, 1997; Moscarini, 2005; Roger-son et al., 2005; Gonzalez and Shi, 2010; Fujita and Ramey, 2012; Michaillat, 2012). Applications of the search and matchingmodel also assume agents to be risk-averse such as the literature that investigates the optimal unemployment benefits withsearch frictions (Fredriksson and Holmlund, 2006; Guerrieri et al., 2010). Recent literature also investigates job search be-haviors with the preference of ambiguity aversion (Chan and Yip, 2017). Our model can be easily extended to incorporateagents’ ambiguity aversion; we do not do so because the modification of agents’ preference towards ambiguity complicatesour model without providing a richer economic intuition in our context.
6The model assumes that no decision on labor supply, either the number of working hours or labor force participation, ismade. This simplification is standard (Mortensen and Pissarides, 1994; Shimer, 2005; Hall, 2005; Hagedorn and Manovskii,2008; Hall and Milgrom, 2008; Fujita and Ramey, 2012; Michaillat, 2012), and is in line with empirical regularities: cyclicalvariations in total working hours (unemployment) basically arise from changes in the number of employment but not changesin working hours per worker (labor force participation) (Shimer, 2010).
6
employment can be written as follows:
rJE(δ) = w(δ) + λ
(JU (δ)− JE(δ)
). (1)
An unemployed worker with home production z > 0 pays a search cost C : R+ 7→ R+ and selects
the optimal level of search intensity s ≥ 0. We assume that search cost is zero in the absence of
search effort, and the search cost function is a strictly increasing strictly convex function. That is,
C(0) = 0, C ′(s) > 0, and C ′′(s) > 0 for all s ≥ 0.
Each transits from unemployment to employment at a rate of F (s)p, where p ∈ (0, 1) measures
the matching technology of an overall economy.7 We can interpret the p as the measure of the eco-
nomic condition so that p is high in booms and low in slumps. In sharp contrast to the existing
literature, the job-finding rate depends on F (s), the cumulative distribution function of search inten-
sity in unemployment, for several reasons. First, F (s) describes the relative position in the search
intensity distribution. Second, it increases in s so that the job-finding rate increases with the relative
position in a search intensity ladder. Third, the lower bound of the F (s) is zero so that the job-finding
rate is zero if no search effort is made. Fourth, its upper bound is one to guarantee that a job-finding
rate F (s)p less than one.
It is noteworthy that the distribution of search intensity F (s) is endogenous. Increasing one’s
search intensity level without climbing up the search intensity ladder does not enhance the likeli-
hood of getting a job. The selection of the optimal search intensity largely depends on the relative
position in the ladder. 8 In the equilibrium, workers’ strategies map their types δ ≥ z to search
intensities s∗(δ) ∈ R+. Hence, unemployed workers of each type choose s to maximize the value of
unemployment given the strategies of other unemployed workers.
Given JE(δ) and others’ strategies, an unemployed worker of type δ chooses his action s to
maximize his value of unemployment as follows:
rJU (δ) = maxs≥0
{z − C(s) + F (s)p
(JE(δ)− JU (δ)
)}. (2)
One may be concerned that workers of different human capital levels may not compete with
one another in a job search process. In reality, jobs are not perfectly segregated by human capital
level. Therefore, it is reasonable that the high-school dropouts and the high-school graduates may
compete for a similar job type, and the bachelor’s degree holders and the postgraduates also perform
a similar task in their positions. Meanwhile, it is rarely to see that the high-school dropouts and the
postgraduates compete for the same job. We will show that each unemployed worker has to consider
7Since endogenizing market tightness does not provide additional economic insight of this paper but significantly com-plicates the analysis, we assume that the job-finding rate does not depend on market tightness. We will discuss this issue inSection 3 and 4.
8Similar to an auction, each unemployed worker submits his own search intensity to bid for a higher job-finding rateF (s)p, and is rewarded with the rate F (s)p.
7
whether the increase in s improves his relative position in the intensity ladder. In the equilibrium,
workers with a similar human capital level are in a similar position in the intensity ladder; therefore,
workers with a significant difference in a human capital level will not compete with one another in
the equilibrium in deciding ones’ optimal search intensity levels, in line with the reality.
When a vacancy is filled by a worker of type δ, it generates a production value δ and pays a wage
w to the worker of type δ.9 A filled vacancy faces a separation shock at a rate of λ. When the shock
arrives, a filled vacancy becomes unfilled, and receives zero profits. Hence, the asset value of a filled
vacancy JF (δ) can be written as follows:
rJF (δ) = δ − w(δ)− λJF (δ). (3)
Following the existing literature, we assume that wages are determined by maximizing the general-
ized Nash product. Consequently, expected gains from search are split according to the generalized
Nash bargaining solution as follows:
JE(δ)− JU (δ) = β
(JE(δ)− JU (δ) + JF (δ)
), (4)
where β ∈ (0, 1) is a bargaining power of workers. The higher the the β, the higher is workers’
bargaining power. Equating flow in and flow out of unemployment, a steady-state aggregate unem-
ployment rate is given by
λ(1− u) =
∫ ∞z
F (s∗(δ))pdG(δ)︸ ︷︷ ︸Average job-finding Rate
×u, (5)
where G(δ) is the cumulative distribution function of the type of unemployed workers, with g(δ) the
corresponding probability density function. G(δ) is endogenous and, in plain English, captures the
distribution of human capital in the unemployment. The L.H.S. and the R.H.S. of the equation (5) are
the flow in and the flow out of unemployment, where s∗(δ) is the optimal search intensity submitted
by the unemployed worker of type δ, and p∫F (s∗(δ))dG(δ) is the average job-finding rate in the
economy. Similarly, a steady-state unemployment rate of the workers of type δ is given by
λ
(h(δ)− g(δ)u
)= F (s∗(δ))pg(δ)u. (6)
The L.H.S. captures the number of employed workers of type δ flowing into unemployment, where
h(δ)− g(δ)u is the measure of employed workers of type δ. F (s∗(δ))p is the job-finding rate of the
unemployed workers of type δ. So, the R.H.S. captures their flow out of unemployment, where g(δ)u
is the measure of unemployed workers of type δ. In the equilibrium, the flow out and the flow in of
unemployment are equal for workers of each type.
9In fact, the implications of this model do not change if the production value is given by f(δ), where f ′(·) > 0.
8
3 Characterization of the Steady State Equilibrium
This section characterizes the steady-state equilibrium, explores the mathematical properties of the
equilibrium, and provides economic intuitions as to why the search relativity theory could explain the
two seemingly unrelated feature of unemployment. We begin with the definition of the steady-state
equilibrium.
Definition 1. A steady-state equilibrium is defined as {s(δ), JE(δ), JU (δ), JF (δ), u, g(δ)}, for all
δ ≥ z,
1. (Optimal Search Intensity): s(δ) maximizes JU (δ) given the optimal s∗(δ) of other unemployed
workers;
2. (Value Functions): JE(δ), JU (δ), and JF (δ) satisfy equations (1)-(3);
3. (Rent-Sharing): w(δ) maximizes the generalized Nash product, satisfying the sharing rule (4);
4. (Steady-State Accounting): u and g(δ) satisfy equations (5) and (6).
Using equations (1), (3), and (4), the wage equation is given by
w(δ) = rJU (δ) + β(δ − rJU (δ)). (7)
A wage is equal to an outside option value plus a fraction of economic rent. Using equations (2) and
(7), the outside option value is given by
rJU (δ) = maxs≥0
{z − C(s) + F (s)
βp
r + λ(δ − rJU (δ))
}. (8)
Lemma 1. If δ1 > δ2, s∗(δ1) ≥ s∗(δ2) in a steady-state equilibrium.10
Intuitively, the higher the human capital level, the higher is the bargaining wage due to rent-
sharing. Hence, an unemployed worker with more human capital has a higher net search benefit.
Lemma 1 shows that if the net search benefit is large enough to incent workers of a particular type
to search at s, it is also high enough to incent all other workers with more human capital to search
no less than s. In other words, Lemma 1 implies that all the unemployed workers of type x ≤ δ
will search with the search intensity level s∗(x) not higher than s∗(δ), meaning that the distributions
of human capital in the unemployment and search intensity coincide (i.e., G(δ) = F (s∗(δ))) in the
equilibrium.
Lemma 2. G(δ) = F (s∗(δ)) in a steady-state equilibrium.
Using Lemma 2 and equation (5), the steady-state unemployment rate is given by
u =λ
λ+∫∞z F (s∗(x))pdG(x)
=λ
λ+ p∫∞z G(x)dG(x)
=λ
λ+ 12p. (9)
10Proof is provided in the Appendix.
9
Ostensibly, the aggregate unemployment rate (9) is parallel to the one derived from the search and
matching literature: u strictly increases with the job separation rate but decreases with the job-finding
rate.
Using Lemma 2, equation (6), and the aggregate unemployment rate (9), the distribution of human
capital in the unemployment G(δ) is given by
G(δ) =u
2(1− u)
(√1 +
4(1− u)H(δ)
u2− 1
). (10)
Proofs are given in Appendix 6.2. It is noteworthy that G(δ) is endogenous in our model; no further
restriction is imposed on the function G(δ). Being a valid cumulative distribution function, G(δ) has
to satisfy three properties: (i) it increases with δ, (ii) it equals zero at its lower support, and (iii) it
equals one at its upper support. First, differentiating G(δ) with respect to δ, one can show that G(δ)
strictly increases with δ. Second, it is straightforward to verify thatG(δ) approaches zero (one) when
H(δ) approaches zero (one). We can therefore conclude that G(δ) is a valid cumulative distribution
function.
Differentiating both sides of G(δ) = F (s∗(δ)) with respect to δ gives us the corresponding
density function as follows:
g(δ) = f(s∗(δ))ds∗(δ)
dδ. (11)
Differentiating G(δ) with respect to δ and rearranging terms, the unemployment rate associated
with workers of type δ is given by
uδ ≡g(δ)u
h(δ)=
(1 +
4(1− u)H(δ)
u2
)−12
. (12)
Lemma 3. The aggregate unemployment rate u and the unemployment rate of worker of type δ are
given by
u =λ
λ+ 12p
and uδ =
(1 +
4(1− u)H(δ)
u2
)−12
Lemma 3 derives two important formulae of unemployment rates. These two formulae provide
rich intuition on the two seemingly unrelated features of unemployment. The second formula dis-
aggregates the AUR into the unemployment rate of workers of type δ, simply using the distribution
of human capital in the labor force. We will discuss the application of this formula in practice and
evaluate its explanatory power in Section 4.
Using Lemma 3 and equation (10), Lemma 4 follows:
10
Lemma 4. The unemployment distribution G(δ) is given by
G(δ) =1
2
Ψ(u)
Ψ(uδ),
where Ψ(u)Ψ(uδ)
is the unemployment odds ratio;
Ψ(u) ≡ u1−u is the aggregate unemployment odds; and
Ψ(uδ) ≡ uδ1−uδ is the unemployment odds of workers of type δ.
Proofs are given in Appendix 6.3.
Lemma 4 derives another novel formula of the distribution of human capital in unemployment.
In the equilibrium, the cumulative distribution function of the human capital δ in the unemployment
is half of the unemployment odds ratio of the corresponding type of workers. The formula is surpris-
ingly simple and is again a function of two variables, the AUR and the unemployment rate of workers
of type δ. According to Lemma 3, uδ is a function of the aggregate unemployment rate u and the
distribution of human capitalH(δ). This unemployment distribution is thus a function of u andH(δ)
only. We will evaluate its explanatory power in Section 4.
We proceed to show the existence and the uniqueness of the equilibrium. While Lemma 2 states
that G(δ) = F (s∗(δ)), G(δ) is differentiable from the equation (10). Hence, F (s∗(δ)) is differen-
tiable. Differentiating equation (2) with respect to s and using equations (1) and (7) give the first
order condition as follows.
C ′(s∗(δ))︸ ︷︷ ︸Marginal Search Cost
= f(s∗(δ))βp
r + λ(δ − rJU (δ))︸ ︷︷ ︸
Marginal Search Benefit
. (13)
The optimal search intensity equates a marginal search cost to a marginal search benefit. Using
equations (11) and (13), one could verify that search intensity strictly increases with δ.
ds∗(δ)
dδ=βpg(δ)(δ − rJU (δ))
(r + λ)C ′(s∗(δ))> 0. (14)
Using equations (2), (10), (11), and (12), the optimal s∗(δ) in equation (13) can be found by solving
K(δ) ≡ C(s∗(δ)) in the following first-order linear differential equation:11
dK(δ)
dδ= T (δ)(δ − z) + T (δ)K(δ), (15)
where T (δ) ≡ βph(δ)(r+λ)uΦ1(δ)(1+Φ2(δ)) , Φ1(δ) ≡
√1 + 4(1−u)H(δ)
u2, and Φ2(δ) ≡ βλ
r+λ(Φ1(δ) − 1).
Notice that T (δ) and thus T (δ)(δ − z) are continuous functions on a real line; hence, there exists a
unique solution to this initial value problem with the initial condition K(z) = 0. The unique solution
11The derivation is shown in Appendix 6.4.
11
to this initial value problem is given by
C(s∗(δ)) =
∫ δ
zT (x′)(x′ − z)e
∫∞x′ T (y)dydx′ ≥ 0.
Since C(s∗(δ)) strictly increases with s∗, the unique s∗ is given by
s∗(δ) = C−1
(∫ δ
zT (x′)(x′ − z)e
∫∞x′ T (y)dydx′
). (16)
Proposition 1. (Existence of the Equilibrium) There exists a unique steady-state equilibrium defined
in Definition 1. The unique strategy profile is given by equation (16). The steady-state AUR and the
unemployment rate associated with each worker type δ are given by Lemma 3. The equilibrium
distribution of human capital in unemployment is given by Lemma 4.
Theorem 1 summarizes five properties of the derived unemployment rates in the steady-state
equilibrium.
Theorem 1. (Properties of Unemployment Rates) In a steady-state equilibrium,
1. uδ unambiguously decreases with δ;
2. the AUR is independent of the distribution of human capital in the labor force H(δ);
3. the lowest uδ equals u/(2− u);
4. a fall in p increases uδ and u for all δ ≥ z; and
5. if 2H(δ)u ( 1
u − 1) > 1, the unemployment rate of workers with a lower human capital level δ
increases more in slumps.
Proof. See the Appendix 6.5.
The First Property. The first statement indicates that the more human capital the worker has,
the lower is his/her unemployment rate. While more-educated workers, on average, possess more
human capital, the unemployment rates, according to this property, are lower at higher educational
levels. This property is supported by the documentation in Figure 1. Since search benefits increase
with human capital, workers with more human capital tend to search more intensively and are thus
more likely to transit into employment. While this implication is seemingly no difference from the
search-theoretic literature, the mechanisms are different. In contrast to this literature, the level of
search intensity essentially plays no role in determining one’s job-finding rate. Instead, a higher
search intensity level places an unemployed worker at a higher position in the intensity distribution.
It is this higher position in the intensity distribution, not the intensity level, that makes him get rid of
the unemployment faster.
The Second Property. It is noteworthy that the AUR is a function of the job separation rate λ and
the matching technology (the economic condition) p of an overall economy, not the distribution of
12
search intensity or human capital. Mathematically, the AUR is negatively associated with the average
job-finding rate, given by p∫∞z F (s∗(x))dG(x). According to Lemma 2, the distributions of search
intensity and human capital in the unemployment coincide. That is, F (s∗(δ)) = G(δ). The average
job-finding rate reduces to p∫∞z G(x)dG(x). Regardless of the distribution of human capital, the
percentile point follows a uniform distribution with support between zero and one. Since the mean
value of the percentile point is always one half, the average job-finding rate further reduces to p/2,
independent of the distribution of search intensity or human capital.
Consider a worker increases his search intensity level such that his relative position slightly im-
proves. On the one hand, his job-finding rate increases. On the other hand, such the improvement in
the relative position creates a negative externality on others: there exists some unemployed workers
whose relative positions decline. Overall, the average job-finding rate of an economy is unaffected
by the reallocation of the position in the search intensity distribution. This explains why a rise in
one’s search effort could shorten his own average unemployment duration but not the unemployment
spell of the economy as a whole, leaving the AUR independent of the distribution of human capital
or search intensity level. This independence property captures a more general feature of unemploy-
ment than the statistical puzzle of unemployment. While the independence property states that the
AUR is uncorrelated with the distribution of human capital in the labor force, the statistical puzzle
of unemployment documents that the AUR is uncorrelated with the fraction of high-educated work-
ers in the labor force. This property also explains why the natural rate of unemployment remains
steady for quite a long time (at least 50 years in the United States) even though the proportion of
the high-educated increases over several decades (which in turn increases the average human capital
level).
Lastly, one should be aware that it is rather difficult to obtain the information on workers’ search
intensity level (Petrongolo and Pissarides, 2001). We are uncertain how to define and measure search
intensity level in practice. The independence property guarantees that the AUR is free from any
function of search intensity levels, making practitioners easier to uncover the AUR without the infor-
mation on search intensity levels.
The Third Property. This property relates the unemployment rate of workers with the highest
level of human capital to the AUR. Recall from the first property that the unemployment rates are
lower at higher educational levels. Hence, the unemployment rate reaches its minimum when δ tends
to infinity. Using Lemma 3, it is straightforward to show that the lowest unemployment is u/(2− u),
which is about half the AUR.
As indicated in Lemma 1, workers with more human capital tend to search more intensively.
Hence, workers with the highest human capital level will search the most intensively so that they,
once get unemployed, rank top in an intensity ladder (i.e., F (s) = 1). Consequently, their job-
finding rate equals p. As discussed in the first property, the average percentile point of an economy
is always one-half, and thus the average job-finding rate equals p/2. Hence, the unemployed with
the highest human capital get the jobs twice as fast as the average of the economy as to why the
13
unemployment rate of the postgraduates is about half the aggregate unemployment rate, explaining
the magic number of “one-half”. More strikingly, this result holds regardless of the distribution of
human capital.
The Fourth Property. It is well known that p is low and thus the AUR is high in slumps. This
property states that the unemployment rates are countercyclical regardless of human capital level.
The AUR is countercyclical because it is a weighted average of the unemployment rate of all worker
types, and the unemployment rates of all worker types are countercyclical. This property is supported
by Figure 1: the unemployment rates by educational level move up and down in the same way in the
past 20 years. For example, the unemployment rates are high during 2008-2011 because of the 2008
recession.
The Fifth Property. This property illustrates that the increase in the unemployment rate is more
pronounced for workers with less human capital in slumps. Indeed, a fall in p has two forces on
the unemployment rate. First, a lower p mitigates the marginal effect of the heterogeneity in the
search relativity F (s∗(δ)), closing the gaps in the job-finding rates and thus the unemployment rates
between workers of any two types. Second, the heterogeneity in the unemployment rate arises not
from any heterogeneity in the employment but from the heterogeneity in the search relativity in the
unemployment. While the unemployment rates are higher at lower educational levels, the propor-
tion of unemployment is larger among less-educated workers. Hence, the impacts of a lower p are
more pronounced on the less-educated. If 2H(δ)u ( 1
u − 1) > 1, the former effect will be dominated.
Therefore, the increase in the unemployment rate of workers with a lower δ is more pronounced in
slumps.
Clearly, the first three properties of unemployment in Theorem 1 correspond to the two seemingly
unrelated features of unemployment. While the implications of the fourth and the fifth property are
easily seen in Figure 1, we do not discuss it in details. Instead, we relegate to the online Appendix B
the evidence that the unemployment rates are higher in slumps and the effects are more pronounced
for less-educated workers.
However, the fifth property is valid only if 2H(δ)u ( 1
u − 1) holds. The remaining question is how
likely this inequality holds. Notice that 2H(δ)u ( 1
u − 1) strictly increases with H(δ) and decreases
with u. According to the US CPS data, the proportions of high-school graduates or below followed a
decreasing trend and reached its lowest point at 34% in 2015. According to Bureau of Labor Statistics
Data, the highest recorded monthly unemployment rate (since 1948) was 10.8% in November and
December in 1982. With H(δ) = 34% and u = 10.8%, 2H(δ)u ( 1
u − 1) should exceed 52, far above
one. In other words, the smallest value of 2H(δ)u ( 1
u − 1) exceeds 52 during 1948-2015. In fact,
this inequality holds even if the share of high-school graduate or below H(δ) is as low as 5% and
the unemployment rate is as high as 25%. Following the current decreasing trend in the share of
high-school graduate or below (minus 1% each year), we expect that the condition holds not only
during 1948-2015 but also the coming thirty years or even more. In other words, the fifth property
in Theorem 1, together with the other properties, is expected to hold in the United States during
14
1948-2045.
Up to this point, this paper has demonstrated the explanatory power of the search relativity theory.
It is noteworthy that our slight modification of the search-theoretic model is indeed widely applicable.
We purposely abstract other labor market features for ease of exposition; we present a model with
minimal labor market features to shed light on the crucial role of the search relativity in explaining
the heterogeneity in the unemployment rate by educational attainment. Our theory, though simple,
does qualitatively capture the two seemingly unrelated features of unemployment (Property 1-3) and
the dynamics of the unemployment rate over business cycle (Property 4 and 5), suggesting that the
search relativity is a key determinant of the job-finding rate.
We close this section with several remarks. First, Theorem 1 is robust to heterogeneous separation
rates λ. More-educated workers tend to have lower separation rates (Cairo and Cajner, 2016). Hence,
they have higher values of employment and thus higher search benefits. Consequently, Lemma 2 and
thus the following lemmas and theorem hold even though the separation rate is endogenous. Second,
some may interpret z as an unemployment insurance. If z equals a fraction of a wage, search benefits
JE(δ) − JU (δ) are higher for the more-educated even though both their values of employment and
unemployment are higher. Again, Lemma 2 holds and Theorem 1 is robust to heterogeneous home
production values. To verify the credibility of the search relativity theory, we evaluate the predictive
power of the model in the next section.
4 The Evaluation of the Search Relativity Theory
This section purposes to assess the predictive power of the search relativity model beyond the two
seemingly unrelated features of unemployment. In particular, this section evaluates the unemploy-
ment distribution G(δ) implied by Lemma 4 and the unemployment rate by educational attainment
uδ implied by Lemma 3. In the end of this section, we derive a measure of the unemployment sheerly
generated by search friction, in which we call it fundamental frictional unemployment rate. We will
show that search relativity explains most of the unemployment rate (if not considered entirely) that
is generated by search friction, leaving no room that can be explained by the level of search inten-
sity. Throughout this section, the implications are evaluated using the US CPS 1994-2015, and the
samples of examination are restricted to the labor force aged 25-60.
These exercises are important for two reasons. First, one may wonder that the proposed theory of
this paper—Search Relativity Theory—explains the two seemingly unrelated features of unemploy-
ment by chance. The predictive power of the search relativity model does reflect the credibility of the
theory. In other words, this exercise provides additional support on the search relativity theory if the
predictive power of the model is sufficiently high. Second, Lemma 3 and 4 provide two novel for-
mulas of the unemployment rate by educational attainment and the unemployment distribution: the
lemmas provide closed-form expressions linking the two variables to the aggregate unemployment
rate and the distribution of human capital. If the proposed procedures to obtain the unemployment
15
distribution and the unemployment rate by educational attainment are sufficiently easy to implement
without solving systems of equations, the two formulas can then be widely and directly applied in
other contents. Undoubtedly, such the applications hinge on the accuracy of the two formulas. The
evaluations in this section speak directly to the concern of its accuracy.
Analogue to the level of search intensity, it is an empirical challenge in observing and measuring
the relative position of search intensity amongst unemployed workers. Thanks to Lemma 2, we show
that the distributions of search intensity and human capital in the unemployment coincide in the
equilibrium. Thus, Lemma 3 and 4 could make use of the distribution of human capital to measure
the search relativity and derive the formula of the unemployment rate by educational attainment and
the unemployment distribution. In practice, it may also be difficult to observe the exact human capital
level of a worker. We therefore make two assumptions:
Assumption 1. A human capital level strictly increases with educational attainment.
Assumption 2. A human capital distribution is continuous over its support.
We denote by j the educational level: the higher the number of j the higher is the educational
level. Here, an educational level is called higher if the average wage associated to this educational
level is higher during 1994-2015 in the United States so that Assumption 1 is satisfied. We denote
by δ̄j and δj as the highest and the lowest human capital level in the educational group j. Also, we
also denote by Hj as the cumulative distribution function of a worker of type δ̄j in the educational
group j. Assumption 1 implies that δj+1 > δ̄j , and Assumption 2 ensures that given any j, for
every ε > 0 there exists a η > 0 such that |δj+1 − δ̄j | < η implies |H(δj+1) − H(δ̄j)| < ε.
With these assumptions, we now proceed to evaluate the implied unemployment distribution and the
unemployment rate by educational attainment.
4.1 The Evaluation of the Unemployment Distribution
This subsection discusses the unemployment distribution derived from the model. According to
Lemma 4, G(δ) is the cumulative distribution function of human capital in the unemployment, which
is indeed the unemployment distribution across educational groups. For theoretical convenience, the
expression in Lemma 4 is not ready for the implementation. This subsection will derive the formula
that allows economists, policymakers, and the public to generate the unemployment distribution in
practice.
Theorem 2. (Unemployment Distribution) Suppose Assumption 1 and 2 are satisfied. In a steady-
state equilibrium, the cumulative distribution function of the educational group j in the unemploy-
ment Gj is given by
Gj =u
2(1− u)
(√1 +
4(1− u)Hj
u2− 1
).
16
Figure 3: The Predicted and the Actual Unemployment Distribution in the United States
020
4060
8010
0Pr
edic
ted
Une
mpl
oym
ent D
istri
butio
n (in
%)
1994 2000 2005 2010 2015Year
HSD HS AD BA PG 020
4060
8010
0Ac
tual
Une
mpl
oym
ent D
istri
butio
n (i
n %
)
1994 2000 2005 2010 2015Year
HSD HS AD BA PG
Notes: The left (right) figure displays the predicted (actual) unemployment distribution in the United States. Each line indi-cate the cumulative distribution function of the corresponding educational group in the unemployment. HSD, HS, AD, BA,and PG. represent the high-school dropouts, the high-school graduates, individuals with some college education and asso-ciate’s degree holders, bachelor’s degree holders, and the postgraduates., respectively. Each dot illustrates the unemploymentdistribution in the corresponding year. Data are collected from the Current Population Survey 1994-2015 and samples arerestricted to labor force participants aged 25-60.
This theorem derives a novel formula of the unemployment distribution across educational lev-
els. We proceed to verify the predictive power of Gj . According to Theorem 2, the unemployment
distribution is a function of two variables: the aggregate unemployment rate u and the cumulative
distribution function of educational level Hj , both of which are easily obtained from the US CPS.
The predicted and the actual annual Gj are illustrated in Figure 3. As highlighted before, Gjis one for the most productive workers; therefore, it must be identical to the actual Gj for the post-
graduates. The predicted Gj for the associate’s degree holders and the bachelor’s degree holders are
nearly identical to the actual ones over twenty years. However, the predicted Gj’s are about 5-10
percent higher than the actual ones for high-school dropouts and high-school graduates. One of the
possibilities, accounting for the derivation, is that low-educated unemployed workers may decide not
only on their search intensity level, but also on whether to participate to the labor force. Nevertheless,
our model did not explicitly endogenize the decision on the labor force participation and we leave it
for future research avenue.12 This result does provide a solid support that the implied unemployment
distribution from the search relativity model largely coheres with the data. To further illustrate the
performance of this novel formula, we treat each state as a local labor market and conduct the same
exercise in each of the 50 states in the United States. The predicted and the actual values of Gj in
each state are illustrated in the Online Appendix C. Surprisingly, the predictabilities of Theorem 2 in
most of the states are as good as its performance in the country as a whole. We therefore conclude
12For example, incorporating the search relativity in the framework of Alvarez and Shimer (2011) could be a fruitful andinteresting research area.
17
that the prediction of Theorem 2 on the unemployment distribution is fairly good.
4.2 The Unemployment Rate by Educational Attainment
This subsection purposes to assess the predictive power of the formula of the unemployment rate by
educational attainment implied by Lemma 3. For theoretical convenience, the expression in Lemma 3
is not ready for the implementation. This subsection will derive the formula that allows economists,
policymakers, and the public to generate the unemployment rate of various educational groups in
practice. In particular, the expression in Lemma 3 maps an aggregate unemployment rate to the
unemployment rate of the xth percentile of a human capital distribution. But the percentile of each
respondent is unobservable in data. The following theorem presents a disaggregation theory that
allows us to generate the unemployment rates of various educational groups under mild assumptions.
Theorem 3. (Disaggregation Theory) Suppose Assumption 1 and 2 are satisfied. In a steady-state
equilibrium, the unemployment rate of the educational group j is given by
uj =2u
(Bj +Bj−1)(17)
where Bj ≡ (u2 + 4(1− u)Hj)12
Proof. See the Appendix 6.6.
Using the US CPS, we generate the actual and the predicted unemployment rates of high-school
graduates, individuals with some college education and associate’s degree holders, bachelor’s degree
holders, and the postgraduates. Figure 4 displays both the actual (the solid lines) and the predicted
(the dash lines) annual unemployment rates of various educational groups over 1994-2015. According
to Theorem 1, the implied unemployment rates of various educational groups do capture the two
seemingly unrelated features of unemployment and the movement over business cycle. It is therefore
expected that the two sets of values display similar movements over time regardless of educational
attainment. More strikingly, the magnitude of the actual and the predicted unemployment rates are
so close that the null hypothesis that they are drawn from an identical distribution cannot be rejected
by the Wilcoxon rank-sum test for the high-school graduates, the bachelor’s degree holders, and
the postgraduates. Certainly, the disaggregation theory does a superb performance in mapping the
aggregate unemployment rate into the unemployment rate by educational attainment in magnitude in
the United States. Although the null hypothesis is rejected for the sample of individuals with some
college education and associate’s degree holders, the predicted values do display a co-movement with
the actual ones over the entire period of examination.
Again, to provide a further support on Theorem 3, we apply this disaggregation theory to disaggre-
gate the annual aggregate unemployment rates to the unemployment rates of high-school graduates,
associate’s degree holders, bachelor’s degree holders, and postgraduates in each state during 1994-
2015. Both the actual (the solid lines) and the predicted (the dash lines) annual unemployment rates
18
Figure 4: Evaluation of the Disaggregation Theory
02
46
810
12U
nem
ploy
men
t Rat
e (in
%)
1994 2000 2005 2010 2015Year
p-value=0.32
High-School Graduates
02
46
810
12U
nem
ploy
men
t Rat
e (in
%)
1994 2000 2005 2010 2015Year
p-value=0.00
Some College & Associate's Degree Holders
02
46
810
12U
nem
ploy
men
t Rat
e (in
%)
1994 2000 2005 2010 2015Year
p-value=0.42
Bachelor's Degree Holders
02
46
810
12U
nem
ploy
men
t Rat
e (in
%)
1994 2000 2005 2010 2015Year
p-value=0.17
Postgradrates
Notes: This figure displays the predicted unemployment rate (the dotted line) and the actual unemployment rate (the solidline). The predicted unemployment rate is simulated using Theorem 3. Data are from the US CPS. Samples are restricted tothe labor force aged 25-60.
19
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2005
2010
2015
Stat
e: T
X (p
-val
ue=0
.02)
04812 1994
2000
2005
2010
2015
Stat
e: U
T (p
-val
ue=0
.96)
04812 1994
2000
2005
2010
2015
Stat
e: V
A (p
-val
ue=0
.85)
04812 1994
2000
2005
2010
2015
Stat
e: V
T (p
-val
ue=0
.98)
04812 1994
2000
2005
2010
2015
Stat
e: W
A (p
-val
ue=0
.94)
04812 1994
2000
2005
2010
2015
Stat
e: W
I (p-
valu
e=0.
91)
04812 1994
2000
2005
2010
2015
Stat
e: W
V (p
-val
ue=0
.13)
04812 1994
2000
2005
2010
2015
Stat
e: W
Y (p
-val
ue=0
.76)
Not
es:
Thi
sfig
ure
disp
lays
the
pred
icte
dun
empl
oym
ent
rate
(the
dotte
dlin
e)an
dth
eac
tual
unem
ploy
men
tra
te(t
heso
lidlin
e).
The
pred
icte
dun
empl
oym
ent
rate
issi
mul
ated
usin
gT
heor
em3.
Dat
aar
efr
omth
eU
SC
PS.S
ampl
esar
ere
stri
cted
toth
ela
borf
orce
aged
25-6
0.
21
Figu
re6:
Eva
luat
ion
ofth
eD
isag
greg
atio
nT
heor
y(I
ndiv
idua
lsw
ithSo
me
Col
lege
Edu
catio
n&
Ass
ocia
te’s
Deg
ree
Hol
ders
)by
Stat
e
04812 1994
2000
2005
2010
2015
Stat
e: A
K (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: A
L (p
-val
ue=0
.01)
04812 1994
2000
2005
2010
2015
Stat
e: A
R (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: A
Z (p
-val
ue=0
.06)
04812 1994
2000
2005
2010
2015
Stat
e: C
A (p
-val
ue=0
.02)
04812 1994
2000
2005
2010
2015
Stat
e: C
O (p
-val
ue=0
.06)
04812 1994
2000
2005
2010
2015
Stat
e: C
T (p
-val
ue=0
.09)
04812 1994
2000
2005
2010
2015
Stat
e: D
E (p
-val
ue=0
.01)
04812 1994
2000
2005
2010
2015
Stat
e: F
L (p
-val
ue=0
.04)
04812 1994
2000
2005
2010
2015
Stat
e: G
A (p
-val
ue=0
.02)
04812 1994
2000
2005
2010
2015
Stat
e: H
I (p-
valu
e=0.
09)
04812 1994
2000
2005
2010
2015
Stat
e: IA
(p-v
alue
=0.0
3)
04812 1994
2000
2005
2010
2015
Stat
e: ID
(p-v
alue
=0.0
0)
04812 1994
2000
2005
2010
2015
Stat
e: IL
(p-v
alue
=0.0
2)
04812 1994
2000
2005
2010
2015
Stat
e: IN
(p-v
alue
=0.0
1)
04812 1994
2000
2005
2010
2015
Stat
e: K
S (p
-val
ue=0
.01)
04812 1994
2000
2005
2010
2015
Stat
e: K
Y (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: L
A (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: M
A (p
-val
ue=0
.02)
04812 1994
2000
2005
2010
2015
Stat
e: M
D (p
-val
ue=0
.01)
04812 1994
2000
2005
2010
2015
Stat
e: M
E (p
-val
ue=0
.02)
04812 1994
2000
2005
2010
2015
Stat
e: M
I (p-
valu
e=0.
06)
04812 1994
2000
2005
2010
2015
Stat
e: M
N (p
-val
ue=0
.02)
04812 1994
2000
2005
2010
2015
Stat
e: M
O (p
-val
ue=0
.01)
04812 1994
2000
2005
2010
2015
Stat
e: M
S (p
-val
ue=0
.00)
Not
es:
Thi
sfig
ure
disp
lays
the
pred
icte
dun
empl
oym
ent
rate
(the
dotte
dlin
e)an
dth
eac
tual
unem
ploy
men
tra
te(t
heso
lidlin
e).
The
pred
icte
dun
empl
oym
ent
rate
issi
mul
ated
usin
gT
heor
em3.
Dat
aar
efr
omth
eU
SC
PS.S
ampl
esar
ere
stri
cted
toth
ela
borf
orce
aged
25-6
0.
22
Figu
re5:
Eva
luat
ion
ofth
eD
isag
greg
atio
nT
heor
y(I
ndiv
idua
lsw
ithSo
me
Col
lege
Edu
catio
n&
Ass
ocia
te’s
Deg
ree
Hol
ders
)by
Stat
e(c
ont.)
04812 1994
2000
2005
2010
2015
Stat
e: M
T (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: N
C (p
-val
ue=0
.05)
04812 1994
2000
2005
2010
2015
Stat
e: N
D (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: N
E (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: N
H (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: N
J (p
-val
ue=0
.01)
04812 1994
2000
2005
2010
2015
Stat
e: N
M (p
-val
ue=0
.02)
04812 1994
2000
2005
2010
2015
Stat
e: N
V (p
-val
ue=0
.01)
04812 1994
2000
2005
2010
2015
Stat
e: N
Y (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: O
H (p
-val
ue=0
.01)
04812 1994
2000
2005
2010
2015
Stat
e: O
K (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: O
R (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: P
A (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: R
I (p-
valu
e=0.
02)
04812 1994
2000
2005
2010
2015
Stat
e: S
C (p
-val
ue=0
.01)
04812 1994
2000
2005
2010
2015
Stat
e: S
D (p
-val
ue=0
.01)
04812 1994
2000
2005
2010
2015
Stat
e: T
N (p
-val
ue=0
.04)
04812 1994
2000
2005
2010
2015
Stat
e: T
X (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: U
T (p
-val
ue=0
.07)
04812 1994
2000
2005
2010
2015
Stat
e: V
A (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: V
T (p
-val
ue=0
.02)
04812 1994
2000
2005
2010
2015
Stat
e: W
A (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: W
I (p-
valu
e=0.
01)
04812 1994
2000
2005
2010
2015
Stat
e: W
V (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: W
Y (p
-val
ue=0
.00)
Not
es:
Thi
sfig
ure
disp
lays
the
pred
icte
dun
empl
oym
ent
rate
(the
dotte
dlin
e)an
dth
eac
tual
unem
ploy
men
tra
te(t
heso
lidlin
e).
The
pred
icte
dun
empl
oym
ent
rate
issi
mul
ated
usin
gT
heor
em3.
Dat
aar
efr
omth
eU
SC
PS.S
ampl
esar
ere
stri
cted
toth
ela
borf
orce
aged
25-6
0.
23
Figu
re7:
Eva
luat
ion
ofth
eD
isag
greg
atio
nT
heor
y(B
ache
lor’
sD
egre
eH
olde
rs)b
ySt
ate
04812 1994
2000
2005
2010
2015
Stat
e: A
K (p
-val
ue=0
.01)
04812 1994
2000
2005
2010
2015
Stat
e: A
L (p
-val
ue=0
.23)
04812 1994
2000
2005
2010
2015
Stat
e: A
R (p
-val
ue=0
.13)
04812 1994
2000
2005
2010
2015
Stat
e: A
Z (p
-val
ue=0
.32)
04812 1994
2000
2005
2010
2015
Stat
e: C
A (p
-val
ue=0
.21)
04812 1994
2000
2005
2010
2015
Stat
e: C
O (p
-val
ue=0
.09)
04812 1994
2000
2005
2010
2015
Stat
e: C
T (p
-val
ue=0
.26)
04812 1994
2000
2005
2010
2015
Stat
e: D
E (p
-val
ue=0
.85)
04812 1994
2000
2005
2010
2015
Stat
e: F
L (p
-val
ue=0
.26)
04812 1994
2000
2005
2010
2015
Stat
e: G
A (p
-val
ue=0
.66)
04812 1994
2000
2005
2010
2015
Stat
e: H
I (p-
valu
e=0.
98)
04812 1994
2000
2005
2010
2015
Stat
e: IA
(p-v
alue
=0.4
1)
04812 1994
2000
2005
2010
2015
Stat
e: ID
(p-v
alue
=0.8
9)
04812 1994
2000
2005
2010
2015
Stat
e: IL
(p-v
alue
=0.8
3)
04812 1994
2000
2005
2010
2015
Stat
e: IN
(p-v
alue
=0.7
4)
04812 1994
2000
2005
2010
2015
Stat
e: K
S (p
-val
ue=0
.34)
04812 1994
2000
2005
2010
2015
Stat
e: K
Y (p
-val
ue=0
.27)
04812 1994
2000
2005
2010
2015
Stat
e: L
A (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: M
A (p
-val
ue=0
.13)
04812 1994
2000
2005
2010
2015
Stat
e: M
D (p
-val
ue=0
.67)
04812 1994
2000
2005
2010
2015
Stat
e: M
E (p
-val
ue=0
.89)
04812 1994
2000
2005
2010
2015
Stat
e: M
I (p-
valu
e=0.
64)
04812 1994
2000
2005
2010
2015
Stat
e: M
N (p
-val
ue=0
.25)
04812 1994
2000
2005
2010
2015
Stat
e: M
O (p
-val
ue=0
.93)
04812 1994
2000
2005
2010
2015
Stat
e: M
S (p
-val
ue=0
.02)
Not
es:
Thi
sfig
ure
disp
lays
the
pred
icte
dun
empl
oym
ent
rate
(the
dotte
dlin
e)an
dth
eac
tual
unem
ploy
men
tra
te(t
heso
lidlin
e).
The
pred
icte
dun
empl
oym
ent
rate
issi
mul
ated
usin
gT
heor
em3.
Dat
aar
efr
omth
eU
SC
PS.S
ampl
esar
ere
stri
cted
toth
ela
borf
orce
aged
25-6
0.
24
Figu
re6:
Eva
luat
ion
ofth
eD
isag
greg
atio
nT
heor
y(B
ache
lor’
sD
egre
eH
olde
rs)b
ySt
ate
(con
t.)
04812 1994
2000
2005
2010
2015
Stat
e: M
T (p
-val
ue=0
.81)
04812 1994
2000
2005
2010
2015
Stat
e: N
C (p
-val
ue=0
.64)
04812 1994
2000
2005
2010
2015
Stat
e: N
D (p
-val
ue=0
.26)
04812 1994
2000
2005
2010
2015
Stat
e: N
E (p
-val
ue=0
.94)
04812 1994
2000
2005
2010
2015
Stat
e: N
H (p
-val
ue=0
.02)
04812 1994
2000
2005
2010
2015
Stat
e: N
J (p
-val
ue=0
.26)
04812 1994
2000
2005
2010
2015
Stat
e: N
M (p
-val
ue=0
.78)
04812 1994
2000
2005
2010
2015
Stat
e: N
V (p
-val
ue=0
.53)
04812 1994
2000
2005
2010
2015
Stat
e: N
Y (p
-val
ue=0
.03)
04812 1994
2000
2005
2010
2015
Stat
e: O
H (p
-val
ue=0
.98)
04812 1994
2000
2005
2010
2015
Stat
e: O
K (p
-val
ue=0
.91)
04812 1994
2000
2005
2010
2015
Stat
e: O
R (p
-val
ue=0
.61)
04812 1994
2000
2005
2010
2015
Stat
e: P
A (p
-val
ue=0
.72)
04812 1994
2000
2005
2010
2015
Stat
e: R
I (p-
valu
e=0.
71)
04812 1994
2000
2005
2010
2015
Stat
e: S
C (p
-val
ue=0
.22)
04812 1994
2000
2005
2010
2015
Stat
e: S
D (p
-val
ue=0
.04)
04812 1994
2000
2005
2010
2015
Stat
e: T
N (p
-val
ue=0
.50)
04812 1994
2000
2005
2010
2015
Stat
e: T
X (p
-val
ue=0
.81)
04812 1994
2000
2005
2010
2015
Stat
e: U
T (p
-val
ue=0
.27)
04812 1994
2000
2005
2010
2015
Stat
e: V
A (p
-val
ue=0
.34)
04812 1994
2000
2005
2010
2015
Stat
e: V
T (p
-val
ue=0
.11)
04812 1994
2000
2005
2010
2015
Stat
e: W
A (p
-val
ue=0
.23)
04812 1994
2000
2005
2010
2015
Stat
e: W
I (p-
valu
e=0.
78)
04812 1994
2000
2005
2010
2015
Stat
e: W
V (p
-val
ue=0
.04)
04812 1994
2000
2005
2010
2015
Stat
e: W
Y (p
-val
ue=0
.87)
Not
es:
Thi
sfig
ure
disp
lays
the
pred
icte
dun
empl
oym
ent
rate
(the
dotte
dlin
e)an
dth
eac
tual
unem
ploy
men
tra
te(t
heso
lidlin
e).
The
pred
icte
dun
empl
oym
ent
rate
issi
mul
ated
usin
gT
heor
em3.
Dat
aar
efr
omth
eU
SC
PS.S
ampl
esar
ere
stri
cted
toth
ela
borf
orce
aged
25-6
0.
25
Figu
re8:
Eva
luat
ion
ofth
eD
isag
greg
atio
nT
heor
y(P
ostg
radu
ates
)by
Stat
e
04812 1994
2000
2005
2010
2015
Stat
e: A
K (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: A
L (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: A
R (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: A
Z (p
-val
ue=0
.72)
04812 1994
2000
2005
2010
2015
Stat
e: C
A (p
-val
ue=0
.53)
04812 1994
2000
2005
2010
2015
Stat
e: C
O (p
-val
ue=0
.21)
04812 1994
2000
2005
2010
2015
Stat
e: C
T (p
-val
ue=1
.00)
04812 1994
2000
2005
2010
2015
Stat
e: D
E (p
-val
ue=0
.27)
04812 1994
2000
2005
2010
2015
Stat
e: F
L (p
-val
ue=0
.94)
04812 1994
2000
2005
2010
2015
Stat
e: G
A (p
-val
ue=0
.05)
04812 1994
2000
2005
2010
2015
Stat
e: H
I (p-
valu
e=0.
45)
04812 1994
2000
2005
2010
2015
Stat
e: IA
(p-v
alue
=0.2
4)
04812 1994
2000
2005
2010
2015
Stat
e: ID
(p-v
alue
=0.2
5)
04812 1994
2000
2005
2010
2015
Stat
e: IL
(p-v
alue
=0.2
1)
04812 1994
2000
2005
2010
2015
Stat
e: IN
(p-v
alue
=0.0
3)
04812 1994
2000
2005
2010
2015
Stat
e: K
S (p
-val
ue=0
.56)
04812 1994
2000
2005
2010
2015
Stat
e: K
Y (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: L
A (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: M
A (p
-val
ue=0
.93)
04812 1994
2000
2005
2010
2015
Stat
e: M
D (p
-val
ue=0
.32)
04812 1994
2000
2005
2010
2015
Stat
e: M
E (p
-val
ue=0
.22)
04812 1994
2000
2005
2010
2015
Stat
e: M
I (p-
valu
e=0.
01)
04812 1994
2000
2005
2010
2015
Stat
e: M
N (p
-val
ue=0
.61)
04812 1994
2000
2005
2010
2015
Stat
e: M
O (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: M
S (p
-val
ue=0
.00)
Not
es:
Thi
sfig
ure
disp
lays
the
pred
icte
dun
empl
oym
ent
rate
(the
dotte
dlin
e)an
dth
eac
tual
unem
ploy
men
tra
te(t
heso
lidlin
e).
The
pred
icte
dun
empl
oym
ent
rate
issi
mul
ated
usin
gT
heor
em3.
Dat
aar
efr
omth
eU
SC
PS.S
ampl
esar
ere
stri
cted
toth
ela
borf
orce
aged
25-6
0.
26
Figu
re7:
Eva
luat
ion
ofth
eD
isag
greg
atio
nT
heor
y(P
ostg
radu
ates
)by
Stat
e(c
ont.)
04812 1994
2000
2005
2010
2015
Stat
e: M
T (p
-val
ue=0
.76)
04812 1994
2000
2005
2010
2015
Stat
e: N
C (p
-val
ue=0
.27)
04812 1994
2000
2005
2010
2015
Stat
e: N
D (p
-val
ue=0
.94)
04812 1994
2000
2005
2010
2015
Stat
e: N
E (p
-val
ue=0
.72)
04812 1994
2000
2005
2010
2015
Stat
e: N
H (p
-val
ue=0
.20)
04812 1994
2000
2005
2010
2015
Stat
e: N
J (p
-val
ue=0
.32)
04812 1994
2000
2005
2010
2015
Stat
e: N
M (p
-val
ue=0
.02)
04812 1994
2000
2005
2010
2015
Stat
e: N
V (p
-val
ue=0
.80)
04812 1994
2000
2005
2010
2015
Stat
e: N
Y (p
-val
ue=0
.19)
04812 1994
2000
2005
2010
2015
Stat
e: O
H (p
-val
ue=0
.03)
04812 1994
2000
2005
2010
2015
Stat
e: O
K (p
-val
ue=0
.01)
04812 1994
2000
2005
2010
2015
Stat
e: O
R (p
-val
ue=0
.14)
04812 1994
2000
2005
2010
2015
Stat
e: P
A (p
-val
ue=0
.53)
04812 1994
2000
2005
2010
2015
Stat
e: R
I (p-
valu
e=0.
15)
04812 1994
2000
2005
2010
2015
Stat
e: S
C (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: S
D (p
-val
ue=0
.51)
04812 1994
2000
2005
2010
2015
Stat
e: T
N (p
-val
ue=0
.03)
04812 1994
2000
2005
2010
2015
Stat
e: T
X (p
-val
ue=0
.20)
04812 1994
2000
2005
2010
2015
Stat
e: U
T (p
-val
ue=0
.79)
04812 1994
2000
2005
2010
2015
Stat
e: V
A (p
-val
ue=0
.83)
04812 1994
2000
2005
2010
2015
Stat
e: V
T (p
-val
ue=0
.80)
04812 1994
2000
2005
2010
2015
Stat
e: W
A (p
-val
ue=0
.20)
04812 1994
2000
2005
2010
2015
Stat
e: W
I (p-
valu
e=0.
09)
04812 1994
2000
2005
2010
2015
Stat
e: W
V (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: W
Y (p
-val
ue=0
.21)
Not
es:
Thi
sfig
ure
disp
lays
the
pred
icte
dun
empl
oym
ent
rate
(the
dotte
dlin
e)an
dth
eac
tual
unem
ploy
men
tra
te(t
heso
lidlin
e).
The
pred
icte
dun
empl
oym
ent
rate
issi
mul
ated
usin
gT
heor
em3.
Dat
aar
efr
omth
eU
SC
PS.S
ampl
esar
ere
stri
cted
toth
ela
borf
orce
aged
25-6
0.
27
of 1994-2015 are plotted in the Figure 5-8. According to these figures, the observed and the predicted
unemployment rate are close in its magnitude for most of the educational groups and states.
We perform the Wilcoxon rank-sum test for each of the educational groups in each state and
report the associated p-value. Most of the tests cannot reject the null hypothesis that the actual and
the predicted unemployment rates are from an identical distribution. In particular, the null hypothesis
cannot be rejected at one percent level in all of the 50 states for the high-school graduates. Meanwhile,
we cannot reject the null hypothesis in 48 and 39 out of 50 states for bachelor’s degree holders and
the postgraduates. At the conventional significance level (five percent level), we cannot reject the
null hypothesis in over half the United States in these three educational groups. Surprisingly, the
null hypothesis cannot be rejected over half the U.S. for high-school graduates, bachelor’s degree
holders, and postgraduates at 10 percent significance level. More strikingly, in the group of high-
school graduates and bachelor’s degree holders, the null hypothesis cannot be rejected in about half
the United States at 50 percent significance level. These results strongly suggest that the derived
formula (3) performs so well (if not considered as nearly perfectly) in predicting the magnitudes
of the unemployment rate by educational attainment that most of the tests could not reject the null
hypothesis that the derived formula (3) is identical to the underlying data generating process of the
unemployment rate by educational attainment. However, one should be aware that even though the
predicted and the actual values display a close co-movement for the sample of associate’s degree
holders, the accuracy in predicting the magnitude is not as satisfactory as that in other educational
categories. We leave further investigation on this issue for future avenue.
With the superb performance of the search relativity theory, the disaggregation theory in Theorem
2 and Theorem 3 could be widely applied to map the aggregate unemployment rate in the unemploy-
ment distribution and the unemployment rates of the educational groups of interest. But in addition to
the predictability of a formula, there are three other criteria one should be concerned in constructing
a formula, namely the number of input variables, the accessibility of the inputs, and the easiness of
implementation.
First, our theory utilizes the least number of input variables. Conditional on the predictability,
the less the input variables, the better is the theory. To disaggregate the aggregate unemployment
rate, at least two variables are required because the aggregate unemployment rate is identical across
the educational groups. According to Theorem 2 and Theorem 3, the unemployment distribution
and the unemployment rate by educational attainment are functions of two variables: the aggregate
unemployment rate and the cumulative distribution function of educational attainment, free from
other parameters. Hence, our theory requires the least number of input variables.
Second, the accessibility of input variables is also an issue: a theory is useful only if the required
underlying conditions or the input variables are observable. Our theory requires the aggregate unem-
ployment rate and the distribution of educational attainment. Needless to mention, these two variables
are well defined and easily accessible in all developed countries and in most (if not all) developing
ones. On the contrary, it is rather challenging to obtain the information on an individual’s search
28
intensity level, the functional form of a matching function, and a job separation rate. Undoubtedly,
our formulas are superior to others (if others exist) in the accessibility of its inputs.
Third, our formulas are not computer-intensive. Unlike many nowadays fancy estimation or cal-
ibration methods, our formulas involve no recursive structure, systems of equations, or samples of
an enormous number of observations. Indeed, the computation takes almost no time and is easy to
implement for economists, policymakers, and even the public.
Of course, with these three advantages, the predictability of the formulas is our concern. As
demonstrated, not only the predicted and the actual values display a similar movement over time, but
also the Wilcoxon rank-sum test of equality cannot reject most of the null hypotheses that the actual
and the observed values are drawn from the same distribution at any conventional significance level in
almost all the states. We conclude that our theory succeeds not only in explaining the two seemingly
unrelated features of unemployment but also in mapping the aggregate unemployment rate into the
unemployment distribution and the unemployment rate of each educational attainment in at least four
aspects: its predictability, its least number of input variables, its accessibility of the required inputs,
and its easiness of implementation.13
According to the steady-state accounting equation (6), the heterogeneity in the unemployment
rate stems from the differences in the job-finding rate and/or the separation rate. We abstract many
labor market features to shed light on the crucial role of the search relativity in explaining the differ-
ence in the unemployment rate by educational attainment. For example, workers share an identical
job separation rate λ and economic environments, including the economic condition p, the wage
determination method, and workers’ bargaining power β etc. Therefore, the heterogeneity in the un-
employment rates must result from the difference in the job-finding rate. Since we shut down the
channel in which the level of search intensity may increase the job-finding rate, the heterogeneity in
the unemployment rates is solely attributed to search relativity. In other words, the unemployment
distribution is generated by search relativity in our model. The gaps between the actual and the pre-
dicted unemployment rates of various educational groups are therefore attributed to the differences in
labor market features other than search relativity. The superb performances of Theorem 2 and The-
orem 3 suggest that it is the search relativity, not the level of search intensity, the separation rate, or
the market structure, that generates the heterogeneity in the unemployment rates across educational
groups.
4.3 Fundamental Frictional Unemployment Rate
In this subsection, we discuss another disaggregation theory of the search relativity model: the mea-
sure of the fundamental frictional unemployment rate (FFUR). Here, we define the FFUR as the
unemployment rate that associates with the unemployment spell that cannot be further shortened by
13The authors also verified that the formula works well using Canadian Labour Force Survey 1994-2015 and the UnitedKingdom Labour Force Survey 1994-2015. Testing the implications of the search relativity model in other OECD countrieswill definitely be a fruitful research avenue.
29
increasing one’s search intensity or human capital. This subsection asks if workers keep increasing
their search intensity and/or human capital, does their unemployment rate approach zero? In other
words, if search cost is sufficiently low and human capital is sufficiently high, does search frictional
unemployment vanish? If not, what will be the frictional unemployment rate, abstracting the factors
of search effort and human capital?
According to the search relativity theory, those with the highest human capital level should search
the most intensively. Workers with the highest human capital level always ranks top in a search in-
tensity ladder. Further increasing his search intensity and human capital will not increase his relative
position and thus the job-finding rate. With the highest human capital level in the economy, their
unemployment rate in principle sheerly arises from search friction, net of search effort and human
capital. In other words, their unemployment rate remains at the FFUR even though they further
increase their human capital and search intensity.
Using Lemma 3, when H(δ) approaches one, uδ equals u/(2 − u), which is the FFUR. The
FFUR can therefore be interpreted as the efficiency of the matching technology (i.e., the friction) in
the labor market. The lower is the FFUR, the more efficient is the labor market in the job-seeking
process. Truly, to conclude whether a higher natural rate of unemployment is attributable to the
inefficient matching technology or the lack of workers’ search effort is rather hasty. The derived
FFUR plays its role in establishing a positive relationship between the aggregate unemployment rate
and the efficiency of a matching technology. As the FFUR strictly increases with u, we confirm that
a lower natural rate of unemployment does reflect a more efficient matching technology in the labor
market. More importantly, using u/(2− u), the frictional unemployment rate, though unobservable,
can be quantified.
In fact, if the share of the educational group j with the highest human capital level is small
enough, Hj approaches one. If Assumption 1 is satisfied, the postgraduate workers are the ones with
the highest human capital level. With the following assumption, the FFUR is approximately equal to
the unemployment rate of the postgraduates.
Assumption 3. The share of the postgraduates is sufficiently small.
Theorem 4. (Disaggregation Approximation Theory) Suppose Assumption 1 and 3 are satisfied.
In a steady-state equilibrium, the unemployment rate of the postgraduates approximately equals the
FFUR, given by u/(2− u).
This disaggregation approximation theory shows that the unemployment rate of the postgraduates
equals the FFUR, which is a function of the aggregate unemployment rate only. We evaluate this ap-
proximation theory by comparing the FFUR with the actual unemployment rate of the postgraduates.
If the two sets of values are different, we should cast doubt on our arguments in the previous section
that the search relativity, not the level of search intensity, is a key determinant of the job-finding rate
and is the major source of the unemployment distribution.
Figure 9 shows that the FFUR and the actual unemployment rates of the postgraduates exhibit
a similar movement both in recession and expansion. The two sets of values are so close that they
30
overlap for ten consecutive years during 1995-2004. However, the FFURs are slightly higher than the
actual unemployment rate in slumps during 2008-2010. With the exception of the fair performance
in slumps, the approximation method basically performs well in capturing the unemployment rate of
the postgraduates over the past two decades.
Figure 9: Evaluation of Fundamental Frictional Unemployment Rate
02
46
810
12U
nem
ploy
emnt
Rat
e (in
%)
1994 2000 2005 2010 2015Year
Actual Unemployment Rate of the PostgraduatesFFUR
p-value=0.26
Fundamental Frictional Unemployment Rate
Notes: This figure displays the FFUR (the dotted line) and the actual unemployment rate of the postgraduates (the solid line).The FFUR is simulated using Theorem 4. Data are from the US CPS. Samples are restricted to the labor force aged 25-60.
We again generate the FFUR and the unemployment rates of the postgraduates by state and per-
form the the Wilcoxon rank-sum tests in each state. We present the two sets of variables in Figure
10. This exercise is astonishing because the FFUR and the unemployment rates of the postgraduates
are so close that the null hypothesis, that the two sets of values are from an identical distribution,
cannot be rejected in 70 percent of the States at five percent significance level. We cannot reject the
hypothesis in over half the United States at 25 significance level. With only one input variable, the
predictive power of the disaggregation approximation theory is exceptionally high. Moreover, this
approximation theory preserves all the advantages of the disaggregation theory: its predictability, its
least number of input variables, its accessibility of the required input, and its easiness of implemen-
tation. Nevertheless, the drawback of this theory is that it can only be applied to the postgraduates,
not the other educational categories.
Notice that the FFUR is derived by assuming that a job-finding rate is a function of search rel-
ativity. Suppose both ones’ job-finding rate is strictly increasing in both the level and the relative
position of search intensity. The unemployment spell for the postgraduates is expected to be shorter
than the one associated with the FFUR. That is, if the level of search intensity also plays a role in
determining one’s job-finding rate, the unemployment rate of the postgraduates in principle should be
lower than the FFUR. According to Figure 9, the two sets of values are so close that the statistical test
31
Figu
re10
:Eva
luat
ion
ofFu
ndam
enta
lFri
ctio
nalU
nem
ploy
men
tRat
e
04812 1994
2000
2005
2010
2015
Stat
e: A
K (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: A
L (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: A
R (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: A
Z (p
-val
ue=0
.85)
04812 1994
2000
2005
2010
2015
Stat
e: C
A (p
-val
ue=0
.69)
04812 1994
2000
2005
2010
2015
Stat
e: C
O (p
-val
ue=0
.15)
04812 1994
2000
2005
2010
2015
Stat
e: C
T (p
-val
ue=0
.81)
04812 1994
2000
2005
2010
2015
Stat
e: D
E (p
-val
ue=0
.37)
04812 1994
2000
2005
2010
2015
Stat
e: F
L (p
-val
ue=0
.81)
04812 1994
2000
2005
2010
2015
Stat
e: G
A (p
-val
ue=0
.10)
04812 1994
2000
2005
2010
2015
Stat
e: H
I (p-
valu
e=0.
51)
04812 1994
2000
2005
2010
2015
Stat
e: IA
(p-v
alue
=0.2
8)
04812 1994
2000
2005
2010
2015
Stat
e: ID
(p-v
alue
=0.2
9)
04812 1994
2000
2005
2010
2015
Stat
e: IL
(p-v
alue
=0.3
7)
04812 1994
2000
2005
2010
2015
Stat
e: IN
(p-v
alue
=0.0
3)
04812 1994
2000
2005
2010
2015
Stat
e: K
S (p
-val
ue=0
.89)
04812 1994
2000
2005
2010
2015
Stat
e: K
Y (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: L
A (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: M
A (p
-val
ue=0
.64)
04812 1994
2000
2005
2010
2015
Stat
e: M
D (p
-val
ue=0
.53)
04812 1994
2000
2005
2010
2015
Stat
e: M
E (p
-val
ue=0
.34)
04812 1994
2000
2005
2010
2015
Stat
e: M
I (p-
valu
e=0.
01)
04812 1994
2000
2005
2010
2015
Stat
e: M
N (p
-val
ue=0
.40)
04812 1994
2000
2005
2010
2015
Stat
e: M
O (p
-val
ue=0
.01)
04812 1994
2000
2005
2010
2015
Stat
e: M
S (p
-val
ue=0
.00)
Not
es:
Thi
sfig
ure
disp
lays
the
FFU
R(t
hedo
tted
line)
and
the
actu
alun
empl
oym
entr
ate
ofth
epo
stgr
adua
tes
(the
solid
line)
.T
heFF
UR
issi
mul
ated
usin
gT
heor
em4.
Dat
aar
efr
omth
eU
SC
PS.S
ampl
esar
ere
stri
cted
toth
ela
borf
orce
aged
25-6
0.
32
Figu
re9:
Eva
luat
ion
ofFu
ndam
enta
lFri
ctio
nalU
nem
ploy
men
tRat
e
04812 1994
2000
2005
2010
2015
Stat
e: M
T (p
-val
ue=0
.93)
04812 1994
2000
2005
2010
2015
Stat
e: N
C (p
-val
ue=0
.35)
04812 1994
2000
2005
2010
2015
Stat
e: N
D (p
-val
ue=0
.93)
04812 1994
2000
2005
2010
2015
Stat
e: N
E (p
-val
ue=1
.00)
04812 1994
2000
2005
2010
2015
Stat
e: N
H (p
-val
ue=0
.10)
04812 1994
2000
2005
2010
2015
Stat
e: N
J (p
-val
ue=0
.43)
04812 1994
2000
2005
2010
2015
Stat
e: N
M (p
-val
ue=0
.04)
04812 1994
2000
2005
2010
2015
Stat
e: N
V (p
-val
ue=0
.89)
04812 1994
2000
2005
2010
2015
Stat
e: N
Y (p
-val
ue=0
.39)
04812 1994
2000
2005
2010
2015
Stat
e: O
H (p
-val
ue=0
.05)
04812 1994
2000
2005
2010
2015
Stat
e: O
K (p
-val
ue=0
.01)
04812 1994
2000
2005
2010
2015
Stat
e: O
R (p
-val
ue=0
.56)
04812 1994
2000
2005
2010
2015
Stat
e: P
A (p
-val
ue=0
.64)
04812 1994
2000
2005
2010
2015
Stat
e: R
I (p-
valu
e=0.
21)
04812 1994
2000
2005
2010
2015
Stat
e: S
C (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: S
D (p
-val
ue=0
.62)
04812 1994
2000
2005
2010
2015
Stat
e: T
N (p
-val
ue=0
.05)
04812 1994
2000
2005
2010
2015
Stat
e: T
X (p
-val
ue=0
.29)
04812 1994
2000
2005
2010
2015
Stat
e: U
T (p
-val
ue=0
.67)
04812 1994
2000
2005
2010
2015
Stat
e: V
A (p
-val
ue=0
.57)
04812 1994
2000
2005
2010
2015
Stat
e: V
T (p
-val
ue=0
.89)
04812 1994
2000
2005
2010
2015
Stat
e: W
A (p
-val
ue=0
.36)
04812 1994
2000
2005
2010
2015
Stat
e: W
I (p-
valu
e=0.
11)
04812 1994
2000
2005
2010
2015
Stat
e: W
V (p
-val
ue=0
.00)
04812 1994
2000
2005
2010
2015
Stat
e: W
Y (p
-val
ue=0
.24)
Not
es:
Thi
sfig
ure
disp
lays
the
FFU
R(t
hedo
tted
line)
and
the
actu
alun
empl
oym
entr
ate
ofth
epo
stgr
adua
tes
(the
solid
line)
.T
heFF
UR
issi
mul
ated
usin
gT
heor
em4.
Dat
aar
efr
omth
eU
SC
PS.S
ampl
esar
ere
stri
cted
toth
ela
borf
orce
aged
25-6
0.
33
could not reject the null hypothesis that the derived FFUR is identical to the underlying data gener-
ating process of the actual unemployment rates of the postgraduates at any conventional significance
level. It basically leaves no room that can be explained by the level of search intensity. However,
the actual unemployment rates are indeed slightly lower than the FFUR in slumps. One of the possi-
bilities is that the unemployment rate in this period is not only generated by search friction, but also
by rationing unemployment (Michaillat, 2012) and ambiguity unemployment (Chan and Yip, 2017),
both of which are shown to play a crucial role in determining the unemployment rate in economic
downturns.
Therefore, this section not only illustrates the three applications of the search relativity model
but also points to two conclusions. First, while the search relativity is the key determinants of the
job-finding rate, the possibility that the level of search intensity matters in determining the transition
rate is at best low. Second, the search relativity is the major source of unemployment distribution.
5 Conclusion
This paper documents two puzzling phenomena of unemployment, namely the two seemingly unre-
lated features of unemployment. We propose a unified theory to explain the features through the
canonical search and matching model with search relativity. Our model derives the novel formu-
las for the unemployment distribution, the unemployment rate by educational attainment, and the
fundamental frictional unemployment rate, each of which are shown to cohere with the data. The
superb performances in these formulas suggest that the search relativity is the key determinant of the
job-finding rate and is the major source contributing the heterogeneity in the unemployment rate.
This paper brings a new research direction in several aspects. First, this paper succeeds in disag-
gregating the aggregate unemployment rate into the unemployment rate by education attainment. It
will be fruitful and interesting to disaggregate the aggregate unemployment duration into the unem-
ployment duration by educational attainment if the implied duration is derived by a search relativity
model. Second, we show that the formulas fit the US data well. It is a fruitful area to test whether our
search relativity model works well with European labor markets. Third, the proposed theory embeds
the search relativity in a random search model. Incorporating the relativity in a directed search model
with on-the-job search allows the model to capture the on-the-job transition. Such the generalization
completes our understanding in the job-seeking behaviors among both the unemployed and the em-
ployed. Fourth, little is known about the responsiveness of the unemployment rate by educational
attainment over business cycle. To explore the elasticity of the unemployment rate of each educa-
tional attainment with respect to the aggregate unemployment rate will be a fruitful and interesting
area.
34
6 Appendix: Proof
6.1 Proof of Lemma 1
Denote Φ(δ) as βpr+λ(δ − rJU (δ)). If ∂Φ(δ)/∂δ ≤ 0, then ∂rJU (δ)/∂δ ≥ 1. Applying the envelope
theorem to equation (8), ∂Φ(δ)/∂δ ≤ 0 implies that ∂rJU (δ)/∂δ ≤ 0. A contradiction results.
Hence, we can conclude that ∂Φ(δ)/∂δ > 0. We now prove that s∗(δ1) ≥ s∗(δ2) by contradiction.
Suppose it is not the case. Given δ1 > δ2, there exists a steady-state Nash equilibrium such that
s1 < s2, where si is denoted as the optimal search intensity of the worker of type δi. Workers of type
δ2 picks s2 because F (s2)Φ(δ2)− C(s2) ≥ F (s1)Φ(δ2)− C(s1). Therefore, we have
(F (s2)− F (s1))Φ(δ2) ≥ C(s2)− C(s1)
(F (s2)− F (s1))Φ(δ1) > C(s2)− C(s1)
F (s2)Φ(δ1)− C(s2) > F (s1)Φ(δ1)− C(s1)
The second inequality arises because Φ(δ) is strictly increasing in δ and δ1 > δ2. According to the
last inequality, it is strictly better off for workers of type δ1 to choose s2 instead of s1. Therefore, we
can conclude that s∗(δ1) ≥ s∗(δ2) if δ1 > δ2 in a steady-state Nash equilibrium.
6.2 The Derivation of Equation (10)
Notice that G(δ) and F (s∗(δ)) are two endogenous variables. Lemma 2 ensures that these two
variables are equal in the steady-state equilibrium. Nevertheless, the lemma does not specify the
functional form of the two distributions. We conjecture that G(δ) is a differentiable function of δ and
verify this claim later.
Using equation (6), G(δ) is the solution of the following differential equation.
G′(δ)u =λh(δ)
λ+G(δ)p(18)
Notice that u is shown to be independent of δ. Rearranging terms, we have
(λ+G(δ)p)G′(δ)u = λh(δ)
Integrating both side from z to x, we have
u
∫ x
zλ+G(δ)pdG(δ) =
∫ x
zλh(δ)dδ
uλG(δ) + up
∫ x
zG(δ)dG(δ) = λH(x)
upG2(δ)
2= λ(H(δ)−G(δ)u)
35
Solving the quadratic equation gives the solution of G(δ) as in equation (10). Notice that G(δ) is a
differentiable function.
6.3 Proof of Lemma 4
Using Lemma 3 and equation (10), we have
G(δ) =u
2(1− u)
(√1 +
4(1− u)H(δ)
u2− 1
)
=1
2
u
1− u
(1
uδ− 1
)=
1
2Ψ
1− uδuδ
=1
2
Ψ
Ψδ
=1
2Θδ
6.4 The Derivation of Equation (15)
Using F (s∗(δ)) = G(δ) and equation (10),
F (s∗(δ))βp
r + λ=
βλ
r + λ(Φ1(δ)− 1) (19)
where Φ1(δ) =
√1 + 2pH(δ)
λu . Using equations (8) and (19), we have
δ − rJU (δ) =δ + C(s∗(δ))− z
1 + Φ2(δ)
where Φ2(δ) = βλ(Φ1(δ)−1)r+λ . Substituting the above equation, f(s∗(δ))ds∗(δ)/dδ = g(δ) and equa-
tion (12) into equation (13), we have
dC(s∗(δ))
dδ= C ′(s∗(δ))
ds∗(δ)
dδ=
h(δ)
uΦ1(δ)
βp
r + λ
δ + C(s∗(δ))− z1 + Φ2(δ)
= T (δ)(δ − z) + T (δ)C(s∗(δ))
which can be written as equation (15).
6.5 Proof of Theorem 1
Differentiating uδ in Lemma 3 with respect to δ, we have
duδdδ
= −2(1− u)h(δ)
u2
(1 +
4(1− u)H(δ)
u2
)−32
< 0
36
Differentiating the above derivative with respect to u, we have
d2uδdudδ
= −u+ 2(1− u)
u(1− u)
2H(δ)(1− u)− u2
u2 + 4(1− u)H(u)
d2uδ/dudδ > 0 iff 2H(δ)(1− u) < u2.
6.6 Proof of Theorem 3
uj = E(u(δ)
∣∣∣∣δj ≤ δ ≤ δ̄j)
=
∫ δ̄jδju(δ)h(δ)dδ
Hj −Hj−1
=
∫ δ̄jδj
(1 + 4(1−u)H(δ)
u2
)−12h(δ)dδ
Hj −Hj−1
=
∫ HjHj−1
(1 + 4(1−u)x
u2
)−12dx
Hj −Hj−1
=u
Hj −Hj−1
∫ Hj
Hj−1
(u2 + 4(1− u)x
)−12 dx
=u
Hj −Hj−1
1
4(1− u)
∫ Bj
Bj−1
y−12 dy
=u
Hj −Hj−1
1
2(1− u)y
12
∣∣∣∣BjBj−1
=2u
(B12j +B
12j−1)
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