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SEARCH RELATIVITY
Ying Tung Chan and Chi Man Yip∗
Abstract
Why is the unemployment rate of the postgraduates about half the aggregate unemploy-ment rate (AUR)? What makes the unemployment rate decrease with educational level and,simultaneously, the AUR and the fraction of high-educated workers in an economy uncorre-lated? We develop a search-theoretic model to answer these questions. In contrast to leadingsearch-theoretic models, we assume that a job-finding rate increases with the relative positionin the search intensity distribution, not the intensity level. We derive a novel formula to disag-gregate the AUR into the unemployment rate by educational attainment. The formula generatesthe unemployment rates that nearly perfectly match the data, suggesting that the relative po-sition of search intensity is a key determinant of a job-finding rate and is the major source ofheterogeneity in the unemployment rate.
KEYWORDS: Distribution of Search Intensity, Matching Technology, Search and Match-ing Theory, Unemployment Distribution.
JEL Classification Numbers: C78, D3, E24, J64.
∗Chan: The Research Institute of Economics and Management, Southwestern University of Finance and Eco-nomics, Chengdu, China (email: [email protected]); Yip: Department of Economics, the University of Calgary,2500 University Dr. N.W., Calgary, AB, T2N 1N4 (email: [email protected]) We would like to thank Scott Taylorand Trevor Tombe for their patience, guidance, encouragement support, and our many insightful conversations. Wethank David Card, Nicole Fortin, Junichi Fujimoto, Charles Ka Yui Leung, Lucija Muehlenbachs, Pascal St-Amour,Stefan Staubli, Atsuko Tanaka, Yikai Wang, Jean-Francois Wen, Alexander Whalley, and Russell Wong and seminarparticipants at the 2017 ASSA Annual Conference, the 2017 China Meeting of the Econometric Society, the 2017SOLE Annual Conference, the 2016 Canadian Economics Association Annual Conference, the City University ofHong Kong, and the University of Calgary for their useful comments. We owe our classmates Younes Ahmadi, YutaroSakai, Yan Song, Meng Sun, Qian Sun, and Qiongda Zhao for their suggestions.
1 IntroductionFor data limitation and theoretical convenience, a voluminous literature studied average wagesand aggregate unemployment rates. Over the past few decades, wage distributions receiveda lot of attention from both empirical and theoretical studies.1 And yet, perhaps surprisingly,works, especially theoretical ones, on unemployment distributions and its properties are sparse.For example, Albrecht and Vroman (2002), Wong (2003), and Dolado et al. (2009) performwell in modeling between and within group wage inequality dynamics. While an unemploy-ment distribution is out of the scope, most of these works unsurprisingly predict that unem-ployment rates are identical for both high- and low-skill workers.
1.1 Two Seemingly Unrelated Features of Unemployment
This paper studies the unemployment distribution by educational attainment. We begin withthe documentation of two seemingly unrelated features of unemployment, namely the statisti-cal puzzle of unemployment and the magic number of “one-half”.
The Statistical Puzzle of Unemployment. Figure 1 shows that unemployment rates arelower at higher educational levels in the U.S. over twenty years.2 Statistically, an economywith a higher fraction of high-educated workers is likely to have a lower aggregate unemploy-ment rate. However, Figure 1 illustrates that the aggregate unemployment rate and the fractionof the high-educated are uncorrelated and thus raises a statistical puzzle: what economic mech-anism makes the unemployment rate decrease with educational level and, simultaneously, theaggregate unemployment rate and the fraction of the high-educated uncorrelated?3
The Magic Number of “One-Half”. Figure 2 displays the correlation between the unem-ployment rate of the postgraduates and the aggregate unemployment rate. The dash line is thefitted value and the solid line represents the mathematical relation in which the unemploymentrate of the postgraduates is half the aggregate unemployment rate. Clearly, the two lines areclose to each other and the observations lie around these two lines, suggesting that the unem-ployment rates of the postgraduates are about half the aggregate unemployment rates in eachof the 50 states. Why is the unemployment rate of the postgraduates about half the aggregateunemployment rate? Can it be other numbers? If not, what economic mechanism creates the
1Readers who are interested in the empirical studies on wage distributions and its dynamics are referred to Juhnet al. (1993), DiNardo et al. (1996), Katz et al. (1999), Lemieux (2006), and Autor et al. (2008). Those who interestedin the theoretical works are referred to Galor and Zeira (1993), Burdett and Mortensen (1998), Krusell and Smith(1998), Postel-Vinay and Robin (2002), Hornstein et al. (2011), Moscarini and Postel-Vinay (2013), Jones and Kim(2014), and Gabaix et al. (2016).
2This relationship is also documented in Ashenfelter and Ham (1979) and Mincer (1991). Also, Topel (1993) findsthat men of lower wages have a higher risk of unemployment. If men with higher educational levels receive higheraverage wages, his finding coheres with our documentation.
3We relegate to the online Appendix A further evidence that these two variables are uncorrelated in each year of1994-2015 and the cyclical components of these two variables are also uncorrelated.
1
Figure 1: The Statistical Puzzle of Unemployment0
612
18U
nem
ploy
men
t Rat
e by
Edu
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nal A
ttain
men
t (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
MS
AR LA
OK
TXMTID
WY
CO
NMAZ
UT
NVWA
ORCA
AK
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 unemployment rate by educational attainment. The right figure displays the correla-tion between the aggregate unemployment rate and the proportion of high-educated workers in the United States. Thehigh-educated are defined as those who are bachelor’s degree holders and/or the postgraduates. Each dot illustrates theaverage aggregate unemployment rate and the average fraction of high-educated in each state during 1994-2015. DCis excluded because it is an outlier. In these figures, data are collected from the Current Population Survey 1994-2015.Samples are restricted to labor force participants aged 25-60.
Figure 2: The Correlation between the Aggregate Unemployment Rate and the UnemploymentRate of the 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 rateof postgraduates. The horizontal axis is the average aggregate unemployment rate of the corresponding state during1994-2015. The vertical axis is the average unemployment rate of the postgraduates in the corresponding state during1994-2015. The dot line is the fitted value, in which the slope is 0.44. The solid line represents the unemployment rateof the postgraduates is half the aggregate unemployment rate. Data are collected from the Current Population Survey1994-2015. Samples are restricted to labor force participants aged 25-60.
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magic number of “one-half”?The purpose of this paper is twofold: to explain that the unemployment distribution emerges
mainly via a subtle relation between a job-finding rate and the distribution of search inten-sity, and to show that it is this subtle relation that generates the two seemingly unrelated fea-tures of unemployment. Despite a voluminous literature on the aggregate unemployment rate(Ljungqvist and Sargent, 2008; Elsby et al., 2009; Davis et al., 2010; Shimer, 2012; Sahinet al., 2014), the discussions on unemployment distributions are rare because the features ofthe unemployment distribution seem unrelated to each other. If the properties of the unem-ployment rate by educational attainment are mastered well, the properties of the aggregateunemployment rate will be well understood, not vice versa. Therefore, to study the two seem-ingly unrelated features of unemployment not only opens the “black box” of unemploymentdistribution but it also complements the literature on the aggregate unemployment rate, en-hances our understanding of the functioning of the labor market, and provides insights howlabor market policies affect the unemployment distribution.
1.2 Search Relativity Theory and Related Literature
Ask anyone about the relationship between a job-finding rate and a search intensity level, andwhat first comes to his or her mind is likely that a job-finding rate increases with a searchintensity level. Despite no economy-wide experiment, we may observe that those who searchmore intensively tend to get rid of unemployment faster. This may formulate our belief thathigher levels of ones’ search intensity increase their job-finding rates.
This paper argues that a matching function maps a search intensity level into its percentilepoint in a search intensity distribution, and a job-finding rate increases with this percentilepoint. Hence, the heterogeneity in job-finding rates arise from the relative position in an in-tensity distribution (hereafter, we call it search relativity when appropriate), not its level. Thispaper shows that search relativity accounts in large part for the heterogeneity in the unemploy-ment rate and is thus the key factor that generates an unemployment distribution.
This paper demonstrates the search relativity theory using a search and matching modelwith heterogeneous workers. The key innovation of our theory is that an unemployed workerchooses his own search intensity level to maximize his value function and one’s job-findingrate increases with his relative position in an intensity distribution. In pursuit of a higherjob-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 thecandidates ranked slightly higher in the intensity ladder. To increase one’s search intensitywithout climbing up the intensity ladder does not enhance one’s job-finding rate. The determi-nation of the optimal search intensity therefore requires unemployed workers to consider boththe marginal search cost and whether an additional search effort allows them to climb up theintensity ladder.
According to our theory, increasing ones’ search intensity level increases their relative
3
positions in the intensity ladder and thus their job-finding rates: their unemployment rates de-cline. Meanwhile, it creates a negative externality on others: there exist workers whose rankdeclines. Consequently, the rise in their search intensity levels has no effect on the aggre-gate unemployment rate; a higher fraction of workers who search intensively has no effect onthe aggregate unemployment rate. Analogously, because unemployed workers with a highereducational level tend to search more intensively, the matching function returns them higherpercentile points and thus higher job-finding rates. Consequently, their unemployment ratesare lower and a higher fraction of high-educated workers and the aggregate unemploymentrates are independent, explaining the statistical puzzle of unemployment.
This theory provides a rationale for the puzzling empirical finding from Kroft and Pope(2014): an aggregate unemployment rate is unaffected by an increasing popularity of job-finding websites like Craigslist. Theoretically, an increasing popularity of such the websitesmay reduce the search cost of the Craigslist’s users and incent them to search more intensively.The increase in their search intensity levels does enhance their relative positions in the intensityladder. Meanwhile, it creates a negative externality on the nonusers of Craigslist: their rankdeclines. Overall, a rise in the job-finding rate for the Craigslist’s users has no effect on theaggregate unemployment rate.
The magic number of “one-half” is attributed to a statistical property. A matching functionmaps an intensity level to its percentile points in the distribution; the percentile points are uni-formly distributed between zero and one with mean one-half. With the highest search benefit,the postgraduates tend to search the most intensively and thus rank top in the intensity ladder.Their percentile point of search intensity is always one, twice the average of the economy.Consequently, the postgraduates’ job-finding rate is twice the average and thus they alwaysget a job twice as fast as the average. Therefore, the unemployment rate of the postgraduatesis about half the aggregate unemployment rate, explaining the puzzling feature of the magicnumber of “one-half”.
The success of our theory in explaining the two features suggests that search relativityis one of the key determinants of a job-finding rate. One of the striking results is that thedistributions of search intensity and human capital in the unemployment are identical in theequilibrium. Using the distribution of educational level as a proxy of the distribution of hu-man capital (and thus search intensity), this paper derives a novel formula to disaggregate anaggregate unemployment rate into the unemployment rate by educational attainment withoutthe need of the information on search intensity levels. Instead, the formula requires only twoinputs, the aggregate unemployment rate and the distribution of educational attainment, bothof which are easily accessible.
Using the United States Current Population Survey (US CPS) 1994-2015, we disaggregatethe annual aggregate unemployment rates into the unemployment rates of various educationalgroups in each of the 50 states. Most of the null hypotheses, in which the actual and the derivedunemployment rates of a particular educational group are from the same distribution, cannotbe rejected at any conventional significance level. For example, the null hypotheses cannot be
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rejected for high-school graduates and bachelor’s degree holders in 46 and 43 out of 50 statesat five percent significance level, respectively. Using the least number (two) of the inputs, oursimple formula allows economists, policymakers, and the public to accurately map the aggre-gate unemployment rate to a more relevant information to the public—the unemployment rateby educational attainment. Furthermore, we derive a formula to generate the unemploymentdistribution simply using the aggregate unemployment rate and the distribution of educationalattainment. We show that the predicted and the actual unemployment distribution are fairlyclose to each other. These results imply that our model not only explains qualitatively thetwo seemingly unrelated features of unemployment, but also predicts the magnitudes of theunemployment rates and the unemployment distribution well over the past two decades.
To the best of our knowledge, there exists no work that documents or explains the twoseemingly unrelated features of unemployment. Cairo and Cajner (2016) is the closest workthat studies the unemployment rate by educational attainment. Our paper differs from theirsin the objective. Focusing on the relationship between the level and the volatility of the un-employment rate, their paper succeeds in explaining why more-educated workers experiencelower unemployment rates and lower employment volatilities. In contrast, our paper explainsthe two seemingly unrelated features of unemployment and how an unemployment distributionemerges, focusing mainly on the relationship between the aggregate unemployment rate andthe unemployment rate by educational attainment.
Cairo and Cajner (2016) is an extension of the canonical search and matching model withon-the-job training and endogenous separations. In contrast, our model incorporates searchrelativity into the canonical search and matching model. Importantly, the derived relation be-tween the aggregate unemployment rate and the unemployment rate by educational attainmentin our model does not depend on the separation rate. Our work complements theirs in that thecanonical model with search relativity and the endogenous separations, the feature in Cairo andCajner (2016), could explain well both the two seemingly unrelated features of unemploymentand the relation between the level and the volatility of the unemployment.
This paper is structured as follows. The basic model environment is described in Section2. In Section 3, we characterize the steady-state equilibrium and show that the model capturesthe two seemingly unrelated features of unemployment qualitatively. Section 4 evaluates thepredictive power of the implications of our model. Section 5 concludes the paper.
2 The Basic ModelThis section presents our search relativity theory through the workhorse of a search equilibriummodel. We aim to construct a simple, yet intuitive, model to shed light on the mechanismthrough which search relativity affects job-seeking behaviors. Despite the abstraction of otherlabor market features, our model is flexible, could be easily extended to incorporate otherfeatures, and, of paramount importance, captures well the features of the unemployment rates.
5
In this model, time is continuous. There is a continuum of profit-maximizing firms who arerisk-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 toone vacancy, we follow the tradition in this literature to address a firm as a vacancy to avoidconfusion.
There is a unit measure of utility maximizing workers who are risk-neutral, infinitely lived,and share an identical real discounted rate r.4 Each is either employed earning some wage w,or unemployed with home production z > 0.5
Workers are ex ante heterogeneous: they differ in a human capital level δ, which is ex-ogenous. We denote by H(δ) the cumulative distribution function of δ, and the correspondingdensity function as h(δ). For ease of exposition, we assume that the distribution is continuousover its support and the lower support of H(δ) exceeds home production z so that it makessense that workers are willing to sign a contract during a job interview.6
We denote by JE(δ) and JU (δ) as value functions of employment and unemployment,respectively. An employed worker of type δ receives a wage w and faces a separation shock ata Poisson rate λ. When the shock arrives, the employed worker becomes unemployed. Hence,the Hamilton-Jacobian-Bellman equation 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+ andselects the optimal level of search intensity s ≥ 0. We assume that the search cost is zero inthe absence of search effort and the search cost function is a strictly increasing strictly convexfunction. 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)
4This simplification is common in this literature (Mortensen and Pissarides, 1994; Moen, 1997; Moscarini, 2005;Rogerson et al., 2005; Gonzalez and Shi, 2010; Fujita and Ramey, 2012; Michaillat, 2012). Applications of the searchand matching model also assume agents to be risk-averse such as the literature that investigates the optimal unem-ployment benefits with search frictions (Fredriksson and Holmlund, 2006; Guerrieri et al., 2010). Recent literaturealso investigates job search behaviors with the preference of ambiguity aversion (Chan and Yip, 2017). Our modelcan be easily extended to incorporate agents’ ambiguity aversion; we do not do so because the modification of agents’preference towards ambiguity complicates our model without providing a richer economic intuition in our context.
5The model assumes that no decision on labor supply, either the number of working hours or labor force partici-pation, is made. This simplification is standard (Mortensen and Pissarides, 1994; Shimer, 2005; Hall, 2005; Hagedornand Manovskii, 2008; Hall and Milgrom, 2008; Fujita and Ramey, 2012; Michaillat, 2012), and is in line with empiri-cal regularities: cyclical variations in total working hours (unemployment) basically arise from changes in the numberof employment but not changes in working hours per worker (labor force participation) (Shimer, 2010).
6This assumption rules out the possibility of not participating in the labor market. As mentioned, the decisionon labor force participation is beyond the scope of this paper. Our model matches well with the data under thissimplification probably because cyclical variations in unemployment basically arise from changes in the number ofemployment, not changes in labor force participation.
6
measures the matching technology of an overall economy.7 We can also interpret the p as themeasure of the economic condition so that p is high in booms and low in slumps. In sharpcontrast to the existing literature, the job-finding rate also depends on F (s), which describesthe relative position in the search intensity distribution.
To capture the properties of a job-finding rate, we assume that F (s) is a cumulative distri-bution function of search intensity in unemployment for several reasons. First, F (s) increasesin s so that the job-finding rate increases with the relative position in a search intensity ladder.Second, the lower bound of the F (s) is zero so that the job-finding rate is zero if no searcheffort is made. Third, its upper bound is one to guarantee that a job-finding rate F (s)p lessthan one.
It is noteworthy that the distribution of search intensity F (s) is endogenous. The selectionof the optimal search intensity largely depends on the relative position in the ladder.8 In theequilibrium, workers’ strategies map their types δ ≥ z to search intensities s∗ ∈ R+. Hence,unemployed workers of each type choose s to maximize the value of unemployment given thestrategies of other unemployed workers.
Given JE(δ) and others’ strategies, an unemployed worker of type δ chooses his action sto 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 competewith one another in a job search process. In reality, jobs are not perfectly segregated by humancapital level. Therefore, it is reasonable that the high-school dropouts and the high-schoolgraduates may compete for a similar job type, and the bachelor’s degree holders and the post-graduates also perform a similar task in their positions. Meanwhile, it is rarely to see that thehigh-school dropouts and the postgraduates compete for the same job. We will show that eachunemployed worker has to consider whether the increase in s improves his relative positionin the intensity ladder. In the equilibrium, workers with a similar human capital level are ina similar position in the intensity ladder; therefore, workers with a significant difference in ahuman 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 paysa 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,
7Since endogenizing market tightness does not provide additional economic insight of this paper but significantlycomplicates the analysis, we assume that the job-finding rate does not depend on market tightness. We will discussthis issue in Section 3 and 4.
8Similar to an auction, each unemployed worker submits his own search intensity to bid for a higher job-findingrate F (s)p, and is rewarded with the rate F (s)p.
9In fact, the implications of this model do not change if the production value is given by f(δ), where f ′(·) > 0.
7
an asset value of a filled vacancy 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 thegeneralized Nash product. Consequently, expected gains from search are split according to thegeneralized 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 aggregateunemployment rate is given by
λ(1− u) =
∫ ∞z
F (s∗(δ))pdG(δ)︸ ︷︷ ︸Average job-finding Rate
×u (5)
whereG(δ) is a cumulative distribution function of the type of unemployed workers, with g(δ)
the corresponding probability density function. G(δ) is endogenous and can be interpreted asthe unemployment distribution by human capital. The L.H.S. and the R.H.S. of the equation(5) is the flow in and flow out of unemployment, where s∗(δ) is the optimal search intensitysubmitted by the unemployed worker of type δ and p
∫F (s∗(δ))dG(δ) is the average job-
finding rate of the overall economy. Similarly, a steady-state unemployment rate of the workersof 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-findingrate of the unemployed workers of type δ. So, the R.H.S. captures their flow out of unemploy-ment, where g(δ)u is the measure of unemployed workers of type δ. In the equilibrium, theflow out and the flow in of unemployment are equal for workers of each type.
3 Characterization of the Steady State EquilibriumThis section characterizes the steady-state equilibrium, explores the mathematical properties ofthe equilibrium, and provides economic intuitions as to why the search relativity theory couldexplain the two seemingly unrelated feature of unemployment. We begin with the definition ofthe steady-state equilibrium.
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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 un-employed 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 sharingrule (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.
Intuitively, the higher the human capital level, the higher is the bargaining wage due torent-sharing. Hence, an unemployed worker with more human capital has a higher net searchbenefit. The Lemma 1 shows that if the net search benefit is sufficiently large to incent workersof a particular type to search at s, it is also high enough to incent all other workers with morehuman capital to search at the same intensity s. In other words, Lemma 1 implies that all theunemployed workers of type x ≤ δ will search with s∗(x) not higher than s∗(δ), meaningthat 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.
Differentiating both sides of G(δ) = F (s∗(δ)) with respect to δ, we have
g(δ) = f(s∗(δ))ds∗(δ)
dδ(9)
Using Lemma 2 and equation (5) , a 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. (10)
Ostensibly, the aggregate unemployment rate (10) is parallel to the one derived from the searchand matching literature: u strictly increases with the job separation rate but decreases with the
9
job-finding rate. Using Lemma 2, equation (6), and the aggregate unemployment rate (10), theunemployment distribution by human capital G(δ) is given by
G(δ) =u
2(1− u)
(√1 +
4(1− u)H(δ)
u2− 1
). (11)
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 threeproperties: (i) it increases with δ, (ii) it equals zero at its lower support, and (iii) it equalsone 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 cumulativedistribution function.
Differentiating G(δ) with respect to δ and rearranging terms, the unemployment rate asso-ciated to 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 typeδ are given by
u =λ
λ+ 12p, uδ =
(1 +
4(1− u)H(δ)
u2
)−12
Lemma 3 derives a novel formula to disaggregate the aggregate unemployment rate intothe unemployment rate of worker of each type. The formula is a function of two variables,the aggregate unemployment rate and the distribution of human capital. We will discuss theapplication of this formula in practice and evaluate the explanatory power of this formula inSection 4. Using Lemma 3 and equation (11), Lemma 4 follows.
Lemma 4. The unemployment distribution G(δ) is given by
G(δ) =u
2(1− u)
(√1 +
4(1− u)H(δ)
u2− 1
)
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.
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Proofs are given in Appendix 6.3.Lemma 4 derives another novel formula of the unemployment distribution. In the equilib-
rium, the cumulative distribution function of the human capital δ in the unemployment is halfof 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 aggregate unemployment rate and theunemployment rate of workers of type δ. According to Lemma 3, uδ is a function of the ag-gregate unemployment rate u and the distribution of human capital H(δ). This unemploymentdistribution is thus a function of u and H(δ).
We proceed to show the existence and the uniqueness of the equilibrium. Differentiatingequation (2) with respect to s and using equations (1) and (7) give the first order condition asfollows.
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. Usingequations (9) and (13), one could verify that search effort strictly increases with δ.
ds∗(δ)
dδ=βpg(δ)(δ − rJU (δ))
(r + λ)C ′(s∗(δ))> 0 (14)
Using equations (2), (9), (11), and (12), the optimal s∗(δ) in equation (13) can be found bysolving K(δ) ≡ C(s∗(δ)) in the following first order linear differential equation.10
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, thereexists a unique solution to this initial value problem with the initial condition K(z) = 0. Theunique solution 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 equilibriumdefined in Definition 1. The unique strategy profile is given by equation (16). The steady-state
10The derivation is shown in Appendix 6.4.
11
aggregate unemployment rate and the unemployment rate associated to each worker type δ aregiven by Lemma 3. The equilibrium unemployment distribution 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. the uδ unambiguously decreases with δ;
2. u is independent of the distribution of human capital H(δ);
3. the lowest uδ equals u2−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 δ increases more inslumps.
Proof. See the Appendix 6.5.
The First Property. The first statement indicates that the more human capital the workerhas, the lower is his unemployment rate. This implication is seemingly no difference from theresult in the literature: an unemployed worker with a higher search intensity level is more likelyto transit into employment. In our model, the level of search intensity essentially plays no rolein determining one’s job-finding rate. Instead, a higher search intensity places an unemployedworker at a higher position in the intensity distribution. It is this higher position in the intensitydistribution, not the intensity level, that makes him get rid of the unemployment faster. Thisalso explains why while workers with more human capital tend to search more intensively,their unemployment rates are lower.
The Second Property. It is noteworthy that the aggregate unemployment rate is a functionof the job separation rate λ and the matching technology (the economic condition) p of anoverall economy, not the distribution of search intensity or human capital. Mathematically,the aggregate unemployment rate 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 averagejob-finding rate reduces to p
∫∞z G(x)dG(x). Since the mean value of percentile points of any
cumulative distribution function is 1/2, 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 effort such that his relative position slightly im-proves. On the one hand, his job-finding rate increases. On the other hand, such the im-provement in the relative position creates a negative externality on others: there exists someunemployed workers whose relative positions decline. Overall, the average job-finding rateof an economy is unaffected by the reallocation of the ranking, remaining p/2. This explainswhy a rise in one’s search effort could shorten his own average unemployment duration but
12
not the unemployment spell of the economy as a whole, leaving the aggregate unemploymentrate independent of the distribution of human capital or search effort. This independence prop-erty explains the statistical puzzle of unemployment: while the unemployment rates are lowerat higher educational levels, the aggregate unemployment rate and the fraction of the high-educated are uncorrelated.
Furthermore, the considerable improvement in job-search technology, such as job-seekingwebsite, does not help an economy reduce the aggregate unemployment rate (Kroft and Pope,2014). It may increase the job-finding rate of the user of these technologies. But it reducesthe relative position of the nonusers and thus the job-finding rate of those who do not accessto these technologies may decline. Even though every unemployed worker gets access tothe technology, the distribution of the relative position (the rank) in the intensity distributionremains unchanged, leaving the aggregate unemployment rate unchanged. Lastly, the naturalrate of unemployment remains steady for quite a long time (at least 50 years in the UnitedStates) even though the proportion of the high-educated increases over several decades (whichin turn increases the average human capital level and search intensity level).
Second, one should be aware that it is rather difficult to obtain the information on workers’search intensity: it is hard to define and measure search intensity in practice. The independenceproperty guarantees that the derived aggregate unemployment rate is free from any functionof search intensity, making practitioners easier to uncover the aggregate unemployment ratewithout the information of distribution of search intensity.
The Third Property. This property relates the unemployment rate of workers with thehighest level of human capital to the aggregate unemployment rate. Recall from the first prop-erty that the unemployment rates are lower at higher educational levels. Hence, the unemploy-ment rate reaches its minimum when δ tends to infinity. Using Lemma 3, it is straightforwardto show that the lowest unemployment is u
2−u , which is about half the aggregate unemploymentrate.
As indicated in Lemma 1, workers with more human capital tend to search more inten-sively. Hence, workers with the highest human capital level will search the most intensively sothat they, once get unemployed, rank top in an intensity ladder (i.e., F (s) = 1). Consequently,their job-finding rate equals p. The average job-finding rate in an economy equals p/2 as statedin the first property. Hence, the unemployed with the highest human capital get the jobs twiceas fast as the average in an economy as to why the unemployment rate of the postgraduates isabout half the aggregate unemployment rate. More strikingly, this result holds regardless ofthe distribution of human capital, explaining the magic number of “one-half”.
The Fourth Property. The fourth statement indicates that a lower p increases the ag-gregate unemployment rate, which is well known. This property helps us reinterpret the phe-nomenon: the aggregate unemployment rate is countercyclical because it is a weighted averageof the unemployment rate by educational group and the unemployment rates of all groups arecountercyclical.
The Fifth Property. This statement illustrates that the increase in the unemployment rate
13
is more pronounced for workers with less human capital in slumps. Indeed, a fall in p has twoforces on the unemployment rate. First, with a lower p, the marginal effect of the heterogeneityin the search relativity F (s∗(δ)) becomes weaker, closing the gaps in the job-finding rates andthus the unemployment rates between workers of various types. Second, the difference inthe unemployment rate by human capital arises from the heterogeneity in the search intensityin the unemployment, not the employment. Since the proportions of the unemployment arehigher for workers of lower type, the impacts of a lower p are more pronounced for them.If 2H(δ)
u ( 1u − 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 thefifth property are easily seen in Figure 1, we do not discuss it in details. Instead, we relegateto the online Appendix B the evidence that the unemployment rates are higher in slumps andthe 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 questionis how likely this inequality holds. Notice that 2H(δ)
u ( 1u − 1) strictly increases with H(δ) and
decreases with u. According to the US CPS data, the proportions of high-school graduates orbelow followed a decreasing trend and reached its lowest point at 34% in 2015. Accordingto Bureau of Labor Statistics Data, the highest recorded monthly unemployment rate (since1948) 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, the inequality still 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 (minus1% each year), we expect that the condition holds not only during 1948-2015 but also thecoming thirty years or even more. In other words, the fifth property in Theorem 1, togetherwith the other properties, is expected to hold in the United States during 1948-2045.
Up to this point, this paper has demonstrated the explanatory power of the search rela-tivity theory. We should emphasize that our slight modification of the canonical search andmatching model is indeed widely applicable. We purposely abstract other labor market fea-tures for ease of exposition; we present a model with minimal labor market features to shedlight on the crucial role of the search relativity in explaining the difference in the unemploy-ment rate by educational attainment. Our theory, though simple, does qualitatively capture thetwo seemingly unrelated features of unemployment, suggesting that the search relativity is akey determinant of the job-finding rate. To further verify the credibility of the search relativitytheory, we evaluate the predictive power of the model in the next section.
We close this section with several remarks. Theorem 1 is robust to heterogeneous sep-aration rates λ and home production values z. More-educated workers tend to have lowerseparation rates (Cairo and Cajner, 2016). Hence, they have higher values of employment andthus higher search benefits. Consequently, Lemma 2 and thus Theorem 1 remain even though
14
separation rates are endogenous. Some may interpret z as an unemployment insurance, whichis usually larger for more-educated unemployed workers. If z equals a fraction of a wage,search benefits JE(δ)−JU (δ) are higher for the more-educated even though both their valuesof employment and unemployment are higher. Again, Lemma 2 holds and Theorem 1 is robustto heterogeneous home production values.
4 The Evaluation of the Search Relativity TheoryThis section purposes to assess the predictive power of the search relativity model beyond thetwo seemingly unrelated features of unemployment. In particular, this section evaluates theunemployment distribution G(δ) implied by Lemma 4 and the unemployment rate by educa-tional attainment uδ implied by Lemma 3. In the end of this section, we derive a measure of theunemployment sheerly generated by search friction, in which we call it fundamental frictionalunemployment rate. We will show that search relativity explains most of the unemploymentrate (if not considered entirely) that is generated by search friction, leaving no room that canbe explained by the level of search intensity. Throughout this section, the implications areevaluated using the US CPS 1994-2015, and the samples of examination are restricted to thelabor force aged 25-60.
These exercises are important for two reasons. First, one may wonder that the proposedtheory of this paper—Search Relativity Theory—explains the two seemingly unrelated fea-tures of unemployment by chance. The predictive power of the search relativity model doesreflect the credibility of the theory. In other words, this exercise provides additional support onthe search relativity theory if the predictive power of the model is sufficiently high. Second,Lemma 3 and 4 provide two novel formulas of the unemployment rate by educational attain-ment and the unemployment distribution: the lemmas provide closed-form expressions linkingthe two variables to the aggregate unemployment rate and the distribution of human capital.If the proposed procedures to obtain the unemployment distribution and the unemploymentrate by educational attainment are sufficiently easy to implement without solving systems ofequations, the two formulas can then be widely and directly applied in other contents. Un-doubtedly, such the applications hinge on the accuracy of the two formulas. The evaluations inthis section speak directly to the concern of its accuracy.
Analogue to the level of search intensity, it is an empirical challenge in observing andmeasuring the relative position of search intensity amongst unemployed workers. Thanks toLemma 2, we show that the distributions of search intensity and human capital in the unem-ployment coincide in the equilibrium. Thus, Lemma 3 and 4 could make use of the distributionof human capital to measure the search relativity and derive the formula of the unemploymentrate by educational attainment and the unemployment distribution. In practice, it may also bedifficult to observe the exact human capital level of a worker. We therefore make two assump-tions:
15
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 edu-cational level. Here, an educational level is called higher if the average wage associated tothis educational level is higher during 1994-2015 in the United States so that Assumption 1 issatisfied. We denote by δ̄j and δj as the highest and the lowest human capital level in the educa-tional group j. Also, we also denote by Hj as the cumulative distribution function of a workerof type δ̄j in the educational group j. Assumption 1 implies that δj+1 > δ̄j , and Assumption 2ensures 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 impliedunemployment 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. Accordingto Lemma 4, G(δ) is the cumulative distribution function of human capital in the unemploy-ment, which is indeed the unemployment distribution across educational groups. For theo-retical convenience, the expression in Lemma 4 is not ready for the implementation. Thissubsection will derive the formula that allows economists, policymakers, and the public togenerate the unemployment distribution in practice.
Theorem 2. (Unemployment Distribution) Suppose Assumption 1 and 2 are satisfied. In asteady-state equilibrium, the cumulative distribution function of the educational group j in theunemployment Gj is given by
Gj =u
2(1− u)
(√1 +
4(1− u)Hj
u2− 1
).
This theorem derives a novel formula of the unemployment distribution across educationallevels. We proceed to verify the predictive power of Gj . According to Theorem 2, the unem-ployment distribution is a function of two variables: the aggregate unemployment rate u andthe cumulative distribution function of educational level Hj , both of which are easily obtainedfrom the US CPS.
The predicted and the actual annual Gj are illustrated in Figure 3. As highlighted before,Gj is one for the most productive workers; therefore, it must be identical to the actual Gjfor the postgraduates. The predicted Gj for the associate’s degree holders and the bachelor’sdegree holders are nearly identical to the actual ones over twenty years. However, the predictedGj’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-educatedunemployed workers may decide not only on their search intensity level, but also on whether
16
Figure 3: The Predicted and the Actual Unemployment Distribution in the United States0
2040
6080
100
Pred
icte
d U
nem
ploy
men
t Dis
tribu
tion
(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. Eachline indicate 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 collegeeducation and associate’s degree holders, bachelor’s degree holders, and the postgraduates., respectively. Each dotillustrates the unemployment distribution in the corresponding year. Data are collected from the Current PopulationSurvey 1994-2015 and samples are restricted to labor force participants aged 25-60.
to participate to the labor force. Nevertheless, our model did not explicitly endogenize thedecision on the labor force participation and we leave it for future research avenue.11 Thisresult does provide a solid support that the implied unemployment distribution from the searchrelativity model largely coheres with the data. To further illustrate the performance of thisnovel formula, we treat each state as a local labor market and conduct the same exercise ineach of the 50 states in the United States. The predicted and the actual values of Gj in eachstate are illustrated in the Online Appendix C. Surprisingly, the predictabilities of Theorem 2in most of the states are as good as its performance in the country as a whole. We thereforeconclude 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 unemploymentrate by educational attainment implied by Lemma 3. For theoretical convenience, the expres-sion in Lemma 3 is not ready for the implementation. This subsection will derive the formulathat allows economists, policymakers, and the public to generate the unemployment rate ofvarious educational groups in practice. In particular, the expression in Lemma 3 maps an ag-gregate unemployment rate to the unemployment rate of the xth percentile of a human capital
11For example, incorporating the search relativity in the framework of Alvarez and Shimer (2011) could be afruitful and interesting research area.
17
distribution. But the percentile of each respondent is unobservable in data. The followingtheorem presents a disaggregation theory that allows us to generate the unemployment rates ofvarious 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 solidlines) and the predicted (the dash lines) annual unemployment rates of various educationalgroups over 1994-2015. According to Theorem 1, the implied unemployment rates of variouseducational groups do capture the two seemingly unrelated features of unemployment and themovement over business cycle. It is therefore expected that the two sets of values display simi-lar movements over time regardless of educational attainment. More strikingly, the magnitudeof the actual and the predicted unemployment rates are so close that the null hypothesis thatthey are drawn from an identical distribution cannot be rejected by the Wilcoxon rank-sum testfor the high-school graduates, the bachelor’s degree holders, and the postgraduates. Certainly,the disaggregation theory does a superb performance in mapping the aggregate unemploymentrate 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 edu-cation and associate’s degree holders, the predicted values do display a co-movement with theactual ones over the entire period of examination.
Again, to provide a further support on Theorem 3, we apply this disaggregation theoryto disaggregate the annual aggregate unemployment rates to the unemployment rates of high-school graduates, associate’s degree holders, bachelor’s degree holders, and postgraduates ineach state during 1994-2015. Both the actual (the solid lines) and the predicted (the dash lines)annual unemployment rates of 1994-2015 are plotted in the Figure 5-8. According to thesefigures, the observed and the predicted unemployment rate are close in its magnitude for mostof the educational groups and states.
We perform the Wilcoxon rank-sum test for each of the educational groups in each stateand report the associated p-value. Most of the tests cannot reject the null hypothesis that theactual 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 50states for bachelor’s degree holders and the postgraduates. At the conventional significance
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 (thesolid line). The predicted unemployment rate is simulated using Theorem 3. Data are from the US CPS. Samples arerestricted to the labor force aged 25-60.
19
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.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
entr
ate
(the
dotte
dlin
e)an
dth
eac
tual
unem
ploy
men
trat
e(t
heso
lidlin
e).T
hepr
edic
ted
unem
ploy
men
trat
eis
sim
ulat
edus
ing
The
orem
3.D
ata
are
from
the
US
CPS
.Sam
ples
are
rest
rict
edto
the
labo
rfor
ceag
ed25
-60.
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
entr
ate
(the
dotte
dlin
e)an
dth
eac
tual
unem
ploy
men
trat
e(t
heso
lidlin
e).T
hepr
edic
ted
unem
ploy
men
trat
eis
sim
ulat
edus
ing
The
orem
3.D
ata
are
from
the
US
CPS
.Sam
ples
are
rest
rict
edto
the
labo
rfor
ceag
ed25
-60.
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
entr
ate
(the
dotte
dlin
e)an
dth
eac
tual
unem
ploy
men
trat
e(t
heso
lidlin
e).T
hepr
edic
ted
unem
ploy
men
trat
eis
sim
ulat
edus
ing
The
orem
3.D
ata
are
from
the
US
CPS
.Sam
ples
are
rest
rict
edto
the
labo
rfor
ceag
ed25
-60.
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
entr
ate
(the
dotte
dlin
e)an
dth
eac
tual
unem
ploy
men
trat
e(t
heso
lidlin
e).T
hepr
edic
ted
unem
ploy
men
trat
eis
sim
ulat
edus
ing
The
orem
3.D
ata
are
from
the
US
CPS
.Sam
ples
are
rest
rict
edto
the
labo
rfor
ceag
ed25
-60.
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
entr
ate
(the
dotte
dlin
e)an
dth
eac
tual
unem
ploy
men
trat
e(t
heso
lidlin
e).T
hepr
edic
ted
unem
ploy
men
trat
eis
sim
ulat
edus
ing
The
orem
3.D
ata
are
from
the
US
CPS
.Sam
ples
are
rest
rict
edto
the
labo
rfor
ceag
ed25
-60.
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
entr
ate
(the
dotte
dlin
e)an
dth
eac
tual
unem
ploy
men
trat
e(t
heso
lidlin
e).T
hepr
edic
ted
unem
ploy
men
trat
eis
sim
ulat
edus
ing
The
orem
3.D
ata
are
from
the
US
CPS
.Sam
ples
are
rest
rict
edto
the
labo
rfor
ceag
ed25
-60.
27
level (five percent level), we cannot reject the null hypothesis in over half the United States inthese three educational groups. Surprisingly, the null hypothesis cannot be rejected over halfthe U.S. for high-school graduates, bachelor’s degree holders, and postgraduates at 10 percentsignificance level. More strikingly, in the group of high-school graduates and bachelor’s degreeholders, the null hypothesis cannot be rejected in about half the United States at 50 percentsignificance 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 rateby educational attainment that most of the tests could not reject the null hypothesis that thederived formula (3) is identical to the underlying data generating process of the unemploymentrate by educational attainment. However, one should be aware that even though the predictedand 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 educationalcategories. We leave further investigation on this issue for future avenue.
With the superb performance of the search relativity theory, the disaggregation theory inTheorem 2 and Theorem 3 could be widely applied to map the aggregate unemployment ratein the unemployment distribution and the unemployment rates of the educational groups ofinterest. But in addition to the predictability of a formula, there are three other criteria oneshould be concerned in constructing a formula, namely the number of input variables, theaccessibility of the inputs, and the easiness of implementation.
First, our theory utilizes the least number of input variables. Conditional on the predictabil-ity, the less the input variables, the better is the theory. To disaggregate the aggregate unem-ployment rate, at least two variables are required because the aggregate unemployment rateis identical across the educational groups. According to Theorem 2 and Theorem 3, the un-employment distribution and the unemployment rate by educational attainment are functionsof two variables: the aggregate unemployment rate and the cumulative distribution functionof educational attainment, free from other parameters. Hence, our theory requires the leastnumber of input variables.
Second, the accessibility of input variables is also an issue: a theory is useful only ifthe required underlying conditions or the input variables are observable. Our theory requiresthe aggregate unemployment rate and the distribution of educational attainment. Needless tomention, these two variables are well defined and easily accessible in all developed countriesand in most (if not all) developing ones. On the contrary, it is rather challenging to obtainthe information on an individual’s search intensity level, the functional form of a matchingfunction, and a job separation rate. Undoubtedly, our formulas are superior to others (if othersexist) in the accessibility of its inputs.
Third, our formulas are not computer-intensive. Unlike many nowadays fancy estimationor calibration methods, our formulas involve no recursive structure, systems of equations, orsamples of an enormous number of observations. Indeed, the computation takes almost notime 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.
28
As demonstrated, not only the predicted and the actual values display a similar movement overtime, but also the Wilcoxon rank-sum test of equality cannot reject most of the null hypothesesthat the actual and the observed values are drawn from the same distribution at any conventionalsignificance level in almost all the states. We conclude that our theory succeeds not only inexplaining the two seemingly unrelated features of unemployment but also in mapping theaggregate unemployment rate into the unemployment distribution and the unemployment rateof each educational attainment in at least four aspects: its predictability, its least number ofinput variables, its accessibility of the required inputs, and its easiness of implementation.12
According to the steady-state accounting equation (6), the heterogeneity in the unemploy-ment rate stems from the differences in the job-finding rate and/or the separation rate. Weabstract many labor market features to shed light on the crucial role of the search relativity inexplaining the difference in the unemployment rate by educational attainment. For example,workers share an identical job separation rate λ and economic environments, including theeconomic condition p, the wage determination method, and workers’ bargaining power β etc.Therefore, the heterogeneity in the unemployment rates must result from the difference in thejob-finding rate. Since we shut down the channel in which the level of search intensity mayincrease the job-finding rate, the heterogeneity in the unemployment rates is solely attributedto search relativity. In other words, the unemployment distribution is generated by search rel-ativity in our model. The gaps between the actual and the predicted unemployment rates ofvarious educational groups are therefore attributed to the differences in labor market featuresother than search relativity. The superb performances of Theorem 2 and Theorem 3 suggestthat it is the search relativity, not the level of search intensity, the separation rate, or the mar-ket structure, that generates the heterogeneity in the unemployment rates across educationalgroups.
4.3 Fundamental Frictional Unemployment Rate
In this subsection, we discuss another disaggregation theory of the search relativity model: themeasure of the fundamental frictional unemployment rate (FFUR). Here, we define the FFURas the unemployment rate that associates with the unemployment spell that cannot be furthershortened by increasing one’s search intensity or human capital. This subsection asks if work-ers keep increasing their search intensity and/or human capital, does their unemployment rateapproach zero? In other words, if search cost is sufficiently low and human capital is suffi-ciently high, does search frictional unemployment vanish? If not, what will be the frictionalunemployment rate, abstracting the factors of search effort and human capital?
According to the search relativity theory, those with the highest human capital level shouldsearch the most intensively. Workers with the highest human capital level always ranks top in
12The authors also verified that the formula works well using Canadian Labour Force Survey 1994-2015 and theUnited Kingdom Labour Force Survey 1994-2015. Testing the implications of the search relativity model in otherOECD countries will definitely be a fruitful research avenue.
29
a search intensity ladder. Further increasing his search intensity and human capital will notincrease his relative position and thus the job-finding rate. With the highest human capitallevel 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 theFFUR 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., thefriction) in the labor market. The lower is the FFUR, the more efficient is the labor marketin the job-seeking process. Truly, to conclude whether a higher natural rate of unemploymentis attributable to the inefficient matching technology or the lack of workers’ search effort israther hasty. The derived FFUR plays its role in establishing a positive relationship betweenthe aggregate unemployment rate and the efficiency of a matching technology. As the FFURstrictly increases with u, we confirm that a lower natural rate of unemployment does reflect amore 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 issmall enough, Hj approaches one. If Assumption 1 is satisfied, the postgraduate workers arethe ones with the highest human capital level. With the following assumption, the FFUR isapproximately 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 satis-fied. In a steady-state equilibrium, the unemployment rate of the postgraduates approximatelyequals the FFUR, given by u/(2− u).
This disaggregation approximation theory shows that the unemployment rate of the post-graduates equals the FFUR, which is a function of the aggregate unemployment rate only. Weevaluate this approximation theory by comparing the FFUR with the actual unemployment rateof the postgraduates. If the two sets of values are different, we should cast doubt on our argu-ments in the previous section that the search relativity, not the level of search intensity, is a keydeterminant 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 postgraduatesexhibit a similar movement both in recession and expansion. The two sets of values are soclose that they overlap for ten consecutive years during 1995-2004. However, the FFURsare slightly higher than the actual unemployment rate in slumps during 2008-2010. With theexception of the fair performance in slumps, the approximation method basically performswell in capturing the unemployment rate of the postgraduates over the past two decades.
We again generate the FFUR and the unemployment rates of the postgraduates by state andperform the the Wilcoxon rank-sum tests in each state. We present the two sets of variablesin Figure 10. This exercise is astonishing because the FFUR and the unemployment rates of
30
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 (thesolid line). The FFUR is simulated using Theorem 4. Data are from the US CPS. Samples are restricted to the laborforce aged 25-60.
the postgraduates are so close that the null hypothesis, that the two sets of values are from anidentical distribution, cannot be rejected in 70 percent of the States at five percent significancelevel. 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 theoryis exceptionally high. Moreover, this approximation theory preserves all the advantages of thedisaggregation theory: its predictability, its least number of input variables, its accessibilityof the required input, and its easiness of implementation. Nevertheless, the drawback of thistheory 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 searchrelativity. Suppose both ones’ job-finding rate is strictly increasing in both the level and therelative position of search intensity. The unemployment spell for the postgraduates is expectedto be shorter than the one associated with the FFUR. That is, if the level of search intensity alsoplays a role in determining one’s job-finding rate, the unemployment rate of the postgraduatesin principle should be lower than the FFUR. According to Figure 9, the two sets of values areso close that the statistical test could not reject the null hypothesis that the derived FFUR isidentical to the underlying data generating process of the actual unemployment rates of thepostgraduates at any conventional significance level. It basically leaves no room that can beexplained by the level of search intensity. However, the actual unemployment rates are indeedslightly lower than the FFUR in slumps. One of the possibilities is that the unemploymentrate in this period is not only generated by search friction, but also by rationing unemployment
31
Figu
re10
:Eva
luat
ion
ofFu
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enta
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ctio
nalU
nem
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men
tRat
e
04812 1994
2000
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-val
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-val
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2000
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2010
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Stat
e: A
Z (p
-val
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.85)
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2000
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A (p
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2000
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.15)
04812 1994
2000
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Stat
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T (p
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04812 1994
2000
2005
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E (p
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ue=0
.37)
04812 1994
2000
2005
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e: F
L (p
-val
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.81)
04812 1994
2000
2005
2010
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e: G
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.10)
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2000
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e: H
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e=0.
51)
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2000
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e: IA
(p-v
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e: ID
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=0.2
9)
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e: IL
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=0.3
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2000
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e: IN
(p-v
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=0.0
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04812 1994
2000
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e: K
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.89)
04812 1994
2000
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e: K
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04812 1994
2000
2005
2010
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Stat
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04812 1994
2000
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2010
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A (p
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04812 1994
2000
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2010
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Stat
e: M
D (p
-val
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.53)
04812 1994
2000
2005
2010
2015
Stat
e: M
E (p
-val
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.34)
04812 1994
2000
2005
2010
2015
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I (p-
valu
e=0.
01)
04812 1994
2000
2005
2010
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Stat
e: M
N (p
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04812 1994
2000
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Stat
e: M
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-val
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2000
2005
2010
2015
Stat
e: M
S (p
-val
ue=0
.00)
Not
es:T
hisfi
gure
disp
lays
the
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R(t
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tted
line)
and
the
actu
alun
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oym
entr
ate
ofth
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stgr
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32
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re9:
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2000
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.93)
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2000
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2015
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C (p
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.35)
04812 1994
2000
2005
2010
2015
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.93)
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2000
2005
2010
2015
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2000
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2010
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2000
2005
2010
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ata
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33
(Michaillat, 2012) and ambiguity unemployment (Chan and Yip, 2017), both of which areshown 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 relativitymodel but also points to two conclusions. First, while the search relativity is the key de-terminants of the job-finding rate, the possibility that the level of search intensity matters indetermining the transition rate is at best low. Second, the search relativity is the major sourceof unemployment distribution.
5 ConclusionThis paper documents two puzzling phenomena of unemployment, namely the two seeminglyunrelated features of unemployment. We propose a unified theory to explain the featuresthrough the canonical search and matching model with search relativity. Our model derivesthe novel formulas for the unemployment distribution, the unemployment rate by educationalattainment, and the fundamental frictional unemployment rate, each of which are shown tocohere with the data. The superb performances in these formulas suggest that the search rel-ativity is the key determinant of the job-finding rate and is the major source contributing theheterogeneity in the unemployment rate.
This paper brings a new research direction in several aspects. First, this paper succeedsin disaggregating the aggregate unemployment rate into the unemployment rate by educationattainment. It will be fruitful and interesting to disaggregate the aggregate unemploymentduration into the unemployment duration by educational attainment if the implied duration isderived 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 labormarkets. 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 modelto capture the on-the-job transition. Such the generalization completes our understanding in thejob-seeking behaviors among both the unemployed and the employed. Fourth, little is knownabout the responsiveness of the unemployment rate by educational attainment over businesscycle. To explore the elasticity of the unemployment rate of each educational attainment withrespect to the aggregate unemployment rate will be a fruitful and interesting area.
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 contradic-tion results. Hence, we can conclude that ∂Φ(δ)/∂δ > 0. We now prove that s∗(δ1) ≥ s∗(δ2)
34
by contradiction. Suppose it is not the case. Given δ1 > δ2, there exists a steady-state Nashequilibrium such that s1 < s2, where si is denoted as the optimal search intensity of the workerof 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. Accordingto 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 (11)
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)
Solving the quadratic equation gives the solution of G(δ) as in equation (11).
35
6.3 Proof of Lemma 4
Using Lemma 3 and equation (11), 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 (11),
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
equation (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
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)
36
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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